**Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company**

Tawanda Mushiri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67762

#### **Abstract**

The processing company under study found out that the boiler was the key machine and needs artificial intelligence monitoring and control. It was simulated under Matlab software and oil level, and pressure and temperature were to be modelled and controlled using the programmable logic controller (PLC) with a fuzzy logic controller as the main brain of control. The company is for processing of fruits to produce juice.

**Keywords:** fuzzy logic, control, critical machinery, monitoring, processing

## **1. Introduction**

Beverage industry denotes the industry accountable for the manufacture of drinks through the usage of highly automated systems, which are responsible for the production of beverages within a short period of time [1–3]. Beverage industry is greatly affected with profitability problems. The ever‐increasing costs of new equipment and spares, perennial foreign currency shortages and the need for improved competitiveness bring about the need for more effective maintenance systems [4–6]. This results in maximum utilization of plant‐installed capacity through improved reliability, uptime, quality and asset life—all achieved at optimal levels of costs versus benefits. Emphasis should be on ensuring that the correct maintenance is being done (doing the right job) rather than merely ensuring that maintenance is being done cor‐ rectly (doing the job right). At the processing company, process losses have traditionally been 1.5% of total losses but increased to 10% in 2011. This has raised concern and hence the need to find new emerging maintenance management philosophies, such as improving maintenance cost effectiveness as one sure way of increasing the overall profitability [7].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **1.1. Background**

Currently, the plant uses run‐to‐failure maintenance as a strategy to take care of its machines. This approach is reported to be creating a high risk to workers, production and property. This is seen by the high rate of unscheduled events associated with breakdowns, emergency equip‐ ment, long working hours and high maintenance costs. This has lowered down the image of the company in terms of competition when compared the benchmarks with other beverage companies in developing nations. The progression towards a global economy has increased the base of competition for almost all companies. By competition, it is suggested that every orga‐ nization out there is always targeting to keep a certain score. The record on a scorecard may be a ratio of more sales, increased revenue or a mounting customer base. Despite the benchmark used for the measurements, for a business to remain competitive, there is a basic company demand to improve at maintaining and taking care of machinery as well as meeting customers' requirements. The competitiveness of production corporations relies on the availability and productivity of their manufacturing equipment, which is possible if all manufacturing losses are recognized and eliminated so that the products can be sold on a marker at a minimum cost.

#### **1.2. Fuzzy logic**

The rules can process simple arithmetic and logical operations with the help of Locator Identifier Separation Protocol (LISP), but a process control algorithm that is based on fuzzy logic is called fuzzy control [8]. The oldest and most common maintenance and repair strat‐ egy are "fix it when it breaks". The appeal of this approach is that no analysis or planning is required [9]. Early detection of faults in a machinery is a key parameter in control and main‐ tenance to avoid failure of equipment [10].

## **2. Critical machinery**

The criticality index is regularly used to decide the extent of the nature of assessment on a given machine by checking the machine's objective [11], excess (that is on the off chance that if the machine falls flat, is there a standby machine which can assume control), repairing expenses, idle sways, well‐being, security and environmental concerns and other considerable features. The criticality record puts all machines into one of three classifications as follows:

**Critical equipment:** This is the fundamental machinery at the processing company. With basic equipment which is the core focus of the procedure, it is understood to require ade‐ quate online state checking to ceaselessly capture much information from the mechanism as could be expected paying little heed to expenditure and is frequently indicated by the plant protection. Estimations, for example, differential development, velocity, spiral relocation and shaft pivotal, dislodging and packaging vibration, temperatures, weights and loads, are taken where conceivable. Such qualities are regularly sustained into a hardware administration pro‐ gramming system, which is fit for slanting the recorded information and giving the admin‐ istrators data, for example, executing information and even anticipating failures and giving conclusions of breakdowns beforehand.

**Vital equipment:** These are units that are essential to the procedure; however, in the event that there is a breakdown, the procedure still proceeds. Excess units (if accessible) would be in this domain. Analysing and regulation of these units are additionally crucial to keep up option strategies given that critical machinery falls flat.

**Balance of industrial equipment or the general purpose:** These are the components that complement the whole plant and are ordinarily assessed, utilizing a handheld information gatherer, as said beforehand, to occasionally build an idea of the strength of the machine.

### **2.1. Types of failure caused by not maintaining machinery**

Breakdowns might be characterized by their seriousness in any of these four classes and are contingent upon the characterized breakdown impacts below:


#### **2.2. Types of condition monitoring techniques**

In line with the research area of interest, condition monitoring will be looked at since it is a branch of Condition Based Maintenance (CBM).

#### *2.2.1. Noise and vibration*

**1.1. Background**

348 Modern Fuzzy Control Systems and Its Applications

**1.2. Fuzzy logic**

tenance to avoid failure of equipment [10].

conclusions of breakdowns beforehand.

**2. Critical machinery**

Currently, the plant uses run‐to‐failure maintenance as a strategy to take care of its machines. This approach is reported to be creating a high risk to workers, production and property. This is seen by the high rate of unscheduled events associated with breakdowns, emergency equip‐ ment, long working hours and high maintenance costs. This has lowered down the image of the company in terms of competition when compared the benchmarks with other beverage companies in developing nations. The progression towards a global economy has increased the base of competition for almost all companies. By competition, it is suggested that every orga‐ nization out there is always targeting to keep a certain score. The record on a scorecard may be a ratio of more sales, increased revenue or a mounting customer base. Despite the benchmark used for the measurements, for a business to remain competitive, there is a basic company demand to improve at maintaining and taking care of machinery as well as meeting customers' requirements. The competitiveness of production corporations relies on the availability and productivity of their manufacturing equipment, which is possible if all manufacturing losses are recognized and eliminated so that the products can be sold on a marker at a minimum cost.

The rules can process simple arithmetic and logical operations with the help of Locator Identifier Separation Protocol (LISP), but a process control algorithm that is based on fuzzy logic is called fuzzy control [8]. The oldest and most common maintenance and repair strat‐ egy are "fix it when it breaks". The appeal of this approach is that no analysis or planning is required [9]. Early detection of faults in a machinery is a key parameter in control and main‐

The criticality index is regularly used to decide the extent of the nature of assessment on a given machine by checking the machine's objective [11], excess (that is on the off chance that if the machine falls flat, is there a standby machine which can assume control), repairing expenses, idle sways, well‐being, security and environmental concerns and other considerable features. The criticality record puts all machines into one of three classifications as follows:

**Critical equipment:** This is the fundamental machinery at the processing company. With basic equipment which is the core focus of the procedure, it is understood to require ade‐ quate online state checking to ceaselessly capture much information from the mechanism as could be expected paying little heed to expenditure and is frequently indicated by the plant protection. Estimations, for example, differential development, velocity, spiral relocation and shaft pivotal, dislodging and packaging vibration, temperatures, weights and loads, are taken where conceivable. Such qualities are regularly sustained into a hardware administration pro‐ gramming system, which is fit for slanting the recorded information and giving the admin‐ istrators data, for example, executing information and even anticipating failures and giving All machines vibrate, and when they are in good condition, their frequency spectrum has a characteristic form; any departure from this form indicates that something is wrong—fatigue, or wear, or ageing of something of some component. Parameters that are useful include amplitude, frequency and phase angle [16].

#### *2.2.2. Amplitude*

This gives an indication of the stress under which a piece of rotating machinery is working, in particular, it can give a measure of the eccentricity (out‐of‐roundness) of a rotor [17].

#### *2.2.3. Frequency*

Through the frequency spectrum, you can detect a fault in rotating machinery. Vibrations fall into two main classes:

Synchronous: Frequencies are in multiples or sub‐multiples of the frequency of rotation, that is, they are harmonics or sub‐harmonics of that frequency.

Asynchronous: These are not related to the rotation frequency; they can be the natural fre‐ quencies of various parts of the system, which can be identified [17].

#### *2.2.4. Phase angle*

It locates the high point in a rotor that is not perfectly circular and thus gauges its out‐of‐bal‐ ance characteristics. Machines must be continuously monitored to denote their state. The fol‐ lowing machines can do these:


#### *2.2.5. Temperature*

Temperature recording is a relatively simple matter at the industrial level. Change of tem‐ perature in rotating machinery is often a sign of deterioration and is therefore something to which close attention should be given. This is currently practised in thermography [18].

#### *2.2.6. Tribology*

An examination of the particles suspended in oil can give very valuable information. The amount of suspended material is an indicator of the state of deterioration of the machine; the composition can identify the source of the wear and thus the component that is failing. The necessary analysis can be done in the laboratory with the electron microscope. Basically, the monitoring techniques that were listed in **Table 1** are almost the major ones currently used.


**Table 1.** Summary of condition monitoring techniques.

## **3. Problem behind the processing company (PC)**

The reactive nature of the corrective maintenance strategy at the company has resulted in loss of trivial production time. Downtime due to breakdown is greatly affecting the production targets and delivery of juice products to customers which had resulted in loss of goodwill and trust from consumers. The processing company (PC) uses a number of raw materials for the production of the fruit juice concentrates: mangoes, guavas, oranges, tomatoes, lemons, grape fruits, granadilla and other continuous inputs such as municipality and borehole water, diesel, electricity and labour. The company has got equipment in place which were mainly supplied by a company in Italy. The mission of the production department is to safely produce quality prod‐ ucts and maintain equipment at a lowest possible cost to meet sales and marketing demands.

#### **3.1. Failure in equipment**

Synchronous: Frequencies are in multiples or sub‐multiples of the frequency of rotation, that

Asynchronous: These are not related to the rotation frequency; they can be the natural fre‐

It locates the high point in a rotor that is not perfectly circular and thus gauges its out‐of‐bal‐ ance characteristics. Machines must be continuously monitored to denote their state. The fol‐

**1.** Clearance recorder: recording the actual movements of the shaft which generate the

**2.** Speed recorder: mounted externally to a machine, and it gives a strong signal at medium

**3.** Accelerometer: also installed to a machine, and it gives a strong signal at high frequencies [17].

Temperature recording is a relatively simple matter at the industrial level. Change of tem‐ perature in rotating machinery is often a sign of deterioration and is therefore something to which close attention should be given. This is currently practised in thermography [18].

An examination of the particles suspended in oil can give very valuable information. The amount of suspended material is an indicator of the state of deterioration of the machine; the composition can identify the source of the wear and thus the component that is failing. The necessary analysis can be done in the laboratory with the electron microscope. Basically, the monitoring techniques that were listed in **Table 1** are almost the major ones currently used.

frequencies, depending on the temperature and the general environment.

is, they are harmonics or sub‐harmonics of that frequency.

**Type Method** Visual Eye

Temperature Thermometer, thermocouple Lubricant monitoring Filtering, spectroscopy Vibration Signal frequency analysis

Corrosion monitoring Eye, corrosometer

**Table 1.** Summary of condition monitoring techniques.

Crack Di‐penetrant analysis, radiography

*2.2.4. Phase angle*

vibrations.

*2.2.5. Temperature*

*2.2.6. Tribology*

lowing machines can do these:

350 Modern Fuzzy Control Systems and Its Applications

quencies of various parts of the system, which can be identified [17].

Using records from previous seasons, equipment with high failure rates and high downtimes is identified. For easy interpretation and identification of this equipment, the information is sorted using Microsoft Excel and graphs are drawn. From the bar graph showing the relative contri‐ butions of equipment to the total downtime, equipment that contributed the most to the total downtime is identified. Breakdown of any of these equipment results in stoppage of the produc‐ tion; therefore, the production loss that is represented by a breakdown of each of these machines is equivalent to the production of the diffuser line, which is 60% of total production (300 TCH of 500 TCH). The following graph in **Figure 1** shows the downtimes caused by the breakdown of each item during the last juice extraction season, and **Figure 1** shows how extraction takes place.

**Figure 1.** Unit and cumulative downtime of equipment.

#### **3.2. Planned stoppages**

The production line is stopped once every 2 weeks. The shutdown is for the whole plant, as some equipment will be due for cleaning and some equipment would be due for servicing. The shutdown starts at 0000 h and servicing starts at 1000 h. The time from 0000 to 1000 h is used to stop the processes, empty some vessels and allow some equipment to cool. This time is also used to plan some of the jobs, perform pre‐task risk assessments, order and transport spares from the stores department and also to service some equipment. Full‐fledged servic‐ ing kicks off at 1000 h and continues up to 1600 h. At 1600 h, the start‐up process begins with steam being released into the vessels to warm them. Production is started around 1700 h when vessels are ready to receive juice, and hot imbibition water is available. The processes are then started, and the levels gradually increased. Full production levels are achieved by 1800 h. The total downtime for this planned stoppage is thus 18 h. A crushing season starts early April and ends in mid‐December. On average, it has 36 weeks. The average number of hours for planned stoppages annually is thus:

$$\text{Total annual planned downtime} = \text{Number of stop pages} \times \text{Downtime per stoppage} \quad \text{(1)}$$

$$\text{Total annual planned downtime} = \frac{36}{2} \times 18 = 324 \,\text{h.} \tag{2}$$

**Figure 2** shows a comparison of planned stoppages and unplanned stoppages. This is a prob‐ lem in this company with more stoppages. This has justified the need for my research, as shown in **Figure 2**.

**Figure 2** is very critical as most of the time, the plant is always shutdown, which results in a great deficiency and loss of goodwill from the customers as the plant is failing to meet the targets.

#### **3.3. Plant performance**

From operating history, the availability performance for each item was evaluated, and the results are shown in **Figure 3**. All availability performance scores were above 90 so for clarity

**Figure 2.** A pie chart for planned and unplanned stoppages.

**Figure 3.** Equipment availability performance including planned downtime.

in comparing, the axis for availability performances was broken such that it starts at 90–100. The main contributor to downtime is planned downtime. This affects the availability of all equipment in the production line, but it is not all equipment that requires servicing every fortnight. Some of the equipment that requires servicing does not need 18 h. In view of this, maintenance schedules currently in use were analysed in order to identify equipment that really requires servicing fortnightly and has the least maintainability. **Figure 3** shows the availability performance of the plant with planned downtime.

#### **3.4. Equipment criticality analysis**

**3.2. Planned stoppages**

352 Modern Fuzzy Control Systems and Its Applications

planned stoppages annually is thus:

shown in **Figure 2**.

**3.3. Plant performance**

targets.

Total annual planned downtime = \_\_36

**Figure 2.** A pie chart for planned and unplanned stoppages.

The production line is stopped once every 2 weeks. The shutdown is for the whole plant, as some equipment will be due for cleaning and some equipment would be due for servicing. The shutdown starts at 0000 h and servicing starts at 1000 h. The time from 0000 to 1000 h is used to stop the processes, empty some vessels and allow some equipment to cool. This time is also used to plan some of the jobs, perform pre‐task risk assessments, order and transport spares from the stores department and also to service some equipment. Full‐fledged servic‐ ing kicks off at 1000 h and continues up to 1600 h. At 1600 h, the start‐up process begins with steam being released into the vessels to warm them. Production is started around 1700 h when vessels are ready to receive juice, and hot imbibition water is available. The processes are then started, and the levels gradually increased. Full production levels are achieved by 1800 h. The total downtime for this planned stoppage is thus 18 h. A crushing season starts early April and ends in mid‐December. On average, it has 36 weeks. The average number of hours for

Total annual planned downtime = Number of stoppages × Downtime per stoppage (1)

**Figure 2** shows a comparison of planned stoppages and unplanned stoppages. This is a prob‐ lem in this company with more stoppages. This has justified the need for my research, as

**Figure 2** is very critical as most of the time, the plant is always shutdown, which results in a great deficiency and loss of goodwill from the customers as the plant is failing to meet the

From operating history, the availability performance for each item was evaluated, and the results are shown in **Figure 3**. All availability performance scores were above 90 so for clarity

<sup>2</sup> × 18 = 324 h. (2)

This section now seeks to identify the equipment that is critical to mainly to production, health, safety, environment and operational costs. This classification of equipment according to risk level is done so as to prioritize equipment in terms of maintenance, work orders, costs and inspection. The criticality of equipment was analysed based on two sides: the effect of failure and its consequence. Equipment history data and interviews were the methodology, which was used to find out which equipment has major effect on production. This was used to give scores to various equipment using the risk analysis matrix balanced score card by Nowlan and Heap, which is shown in Appendix.

#### *3.4.1. Risk analysis matrix balanced score card: Appendix*

As shown in Appendix, there are two categories of scores (Priority and Equipment Score), which are used to calculate the Criticality Score. The Priority Score is obtained from multi‐ plying the results of the three factors (Factor E, Factor F and Factor G). The Equipment Score is obtained from multiplying the results of the four factors (Factor A, Factor B, Factor C and Factor D). This is summarized below.

**Criticality score** (The Criticality Score is obtained by multiplying the Equipment Score by the Priority Score **= ES \* PS**).

**Equipment score** (The equipment score (ES) is obtained by multiplying the results for the four factors **= Factor A \* Factor B \* Factor C \* Factor D**). The factors A–D are shown in Appendices 1–4.

**Priority score** (The priority score is obtained by multiplying the results for the three factors **(= Factor E \* Factor F \* Factor G**) as shown in **Table 2** and **Figure 4**.

**NB: The general details on the factors used are presented in Appendices 1–4. From Table 2, it is clear that criticality equipment ranking for PC is as follows:** boiler, juice transfer pumps, plant conveyor systems, sterilizers and hot break machines, and the boiler is critically affected most as shown in **Figure 5**.

The graph above shows the Key Performance Indicators (KPI) trend in the year 2010–2015. From the graph, the plant availability trend is showing an increment over the years, and the perfor‐ mance rate seems to be showing the same trend; thus, it seems plant availability is proportional to performance rate. Planned downtime trend has shown to be decreasing with the unplanned downtime showing an increase over the years, meaning that the maintenance effort is opposing the world‐class standard as it is showing an increase in reactive domination. OEE is showing a haphazard trend as it seems to be aligning with the performance rate. OEE depends mainly on the performance rate.


**Table 2.** Selection of critical machines.

**Figure 4.** Critical equipment scores.

#### **3.5. Boiler analysis**

**Equipment score** (The equipment score (ES) is obtained by multiplying the results for the four factors **= Factor A \* Factor B \* Factor C \* Factor D**). The factors A–D are shown in Appendices

**Priority score** (The priority score is obtained by multiplying the results for the three factors **(=** 

**NB: The general details on the factors used are presented in Appendices 1–4. From Table 2, it is clear that criticality equipment ranking for PC is as follows:** boiler, juice transfer pumps, plant conveyor systems, sterilizers and hot break machines, and the boiler is critically affected

The graph above shows the Key Performance Indicators (KPI) trend in the year 2010–2015. From the graph, the plant availability trend is showing an increment over the years, and the perfor‐ mance rate seems to be showing the same trend; thus, it seems plant availability is proportional to performance rate. Planned downtime trend has shown to be decreasing with the unplanned downtime showing an increase over the years, meaning that the maintenance effort is opposing the world‐class standard as it is showing an increase in reactive domination. OEE is showing a haphazard trend as it seems to be aligning with the performance rate. OEE depends mainly on

**Equipment Factor A Factor B Factor C Factor D ES Factor E Factor F Factor G PS CS Boiler** 4 4 3 4 **192** 4 4 4 64 **12,288**

**Sterilizers** 4 4 3 2 **96** 2 4 3 24 **2304**

**Evaporators** 2 3 3 2 **36** 3 4 2 24 **864 Heat exchangers** 2 4 3 2 **48** 2 3 2 12 **576**

**Chillers** 3 2 3 1 **18** 2 4 1 8 **144**

**Sorting plant** 4 3 1 2 **24** 1 1 2 2 **48 Compressors** 3 3 2 1 **18** 1 2 1 2 **36 Washing line** 2 2 2 1 **8** 1 2 1 2 **16 Drier** 1 1 1 1 **1** 1 1 1 1 **1 CIP Plant** 1 1 1 1 **1** 1 1 1 1 **1**

4 4 3 3 **144** 4 3 4 48 **6912**

4 4 3 4 **192** 4 2 4 32 **6144**

4 4 3 2 **96** 2 3 3 18 **1728**

4 4 3 2 **96** 2 2 1 4 **384**

4 4 1 1 **16** 1 3 2 6 **96**

**Factor E \* Factor F \* Factor G**) as shown in **Table 2** and **Figure 4**.

1–4.

most as shown in **Figure 5**.

354 Modern Fuzzy Control Systems and Its Applications

the performance rate.

**Pumps and piping**

**Hot break machine**

**Super pulp creamers**

**Aseptic filling machine**

**Table 2.** Selection of critical machines.

**Plant conveyor Systems**

When failure occurs in any manufacturing process, it is critical to identify and analyse the root causes leading to that failure. Hybrid proactive maintenance strategy is condition‐based maintenance complementing preventive maintenance (PM) strategy. PM uses mean time to failure (MTTF) as its pivot in the calculation of PM intervals. This gives calculations for MTTF on unrepairable equipment and mean time before failure (MTBF) on repairable equipment based on the assumption that the failure rate of each component is constant. Failure Modes Effects and Criticality Analysis (FMECA) analysis for the boiler is discussed in detail in a later chapter. The risk priority number chart is drawn with the highest scores corresponding to the most critical boiler component. The root cause failure analysis (RFCA) for the boiler machine is presented in the form of the Ishikawa diagram which was drawn using the Edraw max soft‐ ware as in **Figure 6**. The Ishikawa diagram is also called a fishbone diagram. The gearbox and feed check are reportedly failing almost every week. A Mamdani fuzzy logic controller is fur‐ ther presented in this chapter to monitor parameters such as moisture and dust effects on the feed check valve as well as oil level and torque control on the boiler gearbox. The fuzzy logic framework was determined among other counterfeit shrewd frameworks to be best fitting to comprehend the breakdown difficulties of the heater. A gearbox is continually sticking, and it is difficult to investigate the breakdown of fuzzy logic, and the reason is an instrument that

**Figure 5.** KPI trend analysis.


**Figure 6.** Ishikawa diagram for the root cause failure analysis of a steam boiler failure (Edraw Max).

was utilized for observation. On the risk priority side, it helps me knowing which segments to take and observe the most.

In general, I will do main emphasis on the raw data analysis of the overall plant performance as shown in **Figure 7**. Critical equipment has been selected using the Nowlan and Heap proce‐ dure. The critical components for the plant were found to be the boiler, juice transfer pumps, conveyor system, sterilization unit and hot break machine. This is the equipment that stands to benefit the most from a change in maintenance policy. Because of the time constraint, suit‐ able maintenance strategies will be produced for the boiler of these critical items, which show the greatest opportunity for improvement.

**Figure 7.** Risk priority number for steam boiler.

## **4. Results of the boiler for case study using fuzzy logic**

Analysis of the boiler is carried out in this section, and **Figure 8** shows the risk priority num‐ ber for the steam boiler.

#### **4.1. Fuzzy logic systems (using Matlab software)**

The purpose of the gearbox in a boiler setup is to change the speed ratio between the motor and kicker. The gearbox is seen to be continually sticking, and it is difficult to investigate the underlying reason for the breakdowns along these lines; fuzzy rationale framework is created to screen the torque and oil level when the boiler container is stacked. The gearbox must be halted when terrible conditions exist in the evaporator as an integrated upkeep pro‐ cedure to abstain from sticking, henceforth, canny fuzzy rationale control. From the inter‐ views with the engineers and plant artisans, as well as maintenance of history log books, it was noted down that oil level and torque are the major factors which are contributing to the jamming of the gearbox whenever the boiler is being loaded. In this chapter, the oil level is being controlled and is to be modified with specific levels: high, acceptable and low. Controlled torque is to be modified with specific levels: high, normal and low. This is high‐ lighted beneath.

#### *4.1.1. Oil level and torque control*

was utilized for observation. On the risk priority side, it helps me knowing which segments

**Figure 6.** Ishikawa diagram for the root cause failure analysis of a steam boiler failure (Edraw Max).

In general, I will do main emphasis on the raw data analysis of the overall plant performance as shown in **Figure 7**. Critical equipment has been selected using the Nowlan and Heap proce‐ dure. The critical components for the plant were found to be the boiler, juice transfer pumps, conveyor system, sterilization unit and hot break machine. This is the equipment that stands to benefit the most from a change in maintenance policy. Because of the time constraint, suit‐ able maintenance strategies will be produced for the boiler of these critical items, which show

to take and observe the most.

356 Modern Fuzzy Control Systems and Its Applications

the greatest opportunity for improvement.

**Figure 7.** Risk priority number for steam boiler.

#### *4.1.1.1. Effects of oil level and torque control on gearbox jamming*

Moisture inside the valves may result from the compressed air passing through the valve. The effects of moisture on the valves include:

• Moisture may cause rusting in the moving parts of the valve.

**Figure 8.** Risk priority number for steam boiler.


The following are the range of values of moisture, which are tolerated and some not tolerated inside the valve:

#### *4.1.1.2. Oil level control*

#### *4.1.1.2.1. For a range of 0–100%*

This is the aggregate scope of oil level either acknowledged or unaccepted in the gearbox.

• 0–40%

This alerts the artisans for refiling when oil is about to run out.

• 40–80%

At this level, it is acceptable, and the gearbox can continue operating.

• 80–100%

The level is fine and the gearbox has to keep running.

#### *4.1.1.3. Torque control*

#### *4.1.1.3.1. For a range of 0–12,000 Nm*

This is the total range for torque in the gearbox at any time (unaccepted or accepted).

• 0–2000 Nm

The range is very fine for starting the gearbox and is fine for running the boiler.

• 2000–8000 Nm

The torque is very fine for the gearbox to continue running.

• 8000–12,000 Nm

This range is very high, and it needs close monitoring, otherwise, stop the boiler.

#### *4.1.1.4. Output control*

#### *4.1.1.4.1. For gearbox range of 0–100%*

The gearbox has to run or stop, either of the 2, that is the meaning of 0 for stop and 100% for running that is the 0 or 1 for logic:

• 0 stop

• Increased rate of wear of the valve material may also result due to moisture. Moisture washes the lubrication away, which will result in the eventual failure or malfunctioning of

• The industrial processes, which rely on the full functionality of the pneumatic control valves, may be jeopardized, and this usually results in costly breakdowns of the machine.

• Air‐ or gas‐operated instruments may give inaccurate readings due to corrosion of the ma‐

• The rubber diaphragms inside the pneumatic valves can be stiffened and will eventually

The following are the range of values of moisture, which are tolerated and some not tolerated

This is the aggregate scope of oil level either acknowledged or unaccepted in the gearbox.

This is the total range for torque in the gearbox at any time (unaccepted or accepted).

The range is very fine for starting the gearbox and is fine for running the boiler.

This range is very high, and it needs close monitoring, otherwise, stop the boiler.

terial and hence interrupting the plant processes.

rupture due to the moisture flowing through them.

This alerts the artisans for refiling when oil is about to run out.

The level is fine and the gearbox has to keep running.

The torque is very fine for the gearbox to continue running.

At this level, it is acceptable, and the gearbox can continue operating.

the valve.

358 Modern Fuzzy Control Systems and Its Applications

inside the valve:

• 0–40%

• 40–80%

• 80–100%

*4.1.1.3. Torque control*

• 0–2000 Nm

• 2000–8000 Nm

• 8000–12,000 Nm

*4.1.1.3.1. For a range of 0–12,000 Nm*

*4.1.1.2. Oil level control*

*4.1.1.2.1. For a range of 0–100%*

At this range for the stop signal, the gearbox is to stop being controlled by torque and oil level

• 1 run

This involves the range 0–100%. All the conditions are being met (oil level and torque), and the gearbox can run effectively.

### *4.1.2. Simulation of the effects of oil level and torque on the gearbox using fuzzy logic*

The rule base consists of a collection of expert rules, which are required to meet the control goals. These control rules can be developed from survey results, common sense, general prin‐ ciples and intuitive knowledge. The *IF* − *THEN or IF* − *AND* − *THEN* rules are mainly going to be used in designing the controller as shown in **Figure 9**. The situation for which the rules are projected is given by the IF part. The fuzzy system reaction in this state will be given by the THEN part. **Figure 9** below shows the fuzzy logic (Matlab software) screenshot for the two inputs (oil level and torque) and the output parameter (gearbox).

**Figure 9.** FIS editor for gearbox.

#### **4.2. Control of gearbox jamming using fuzzy logic tool box**

#### *4.2.1. Membership function editor for oil level and torque monitoring*

The membership function editor is the central concept of the fuzzy set, which has values rang‐ ing from 0 to 1 in the y axis. The ranges of control were done in **Tables 3**–**5**. Membership function editor is the stage whereby the ranges, as explained above, are inserted into the fuzzy logic Matlab software and the screenshot of the data is taken as shown in **Figure 10**. It helps to display all membership functions connected with the input and output for the entire Fuzzy Inference System (FIS). In this case, a triangular fuzzy set membership function is used for oil and torque monitoring. This is shown clearly in figures below with ranges for oil level monitor‐ ing varying from low, acceptable and high and the numerical values being added. For torque control, the corresponding numerical values range from low, normal and high. The gearbox is either run or stopped, depending on the oil level and torque conditions as in **Table 5**.

The membership functions editor for the output is shown in **Figure 11**, which is the gearbox that is plotted in the form of a triangle. Fuzzy logic is an artificial intelligence software which can store the output ranges in its memory and can learn the system to give solutions in what can be done. The Matlab software of fuzzy logic consists of the rule editor function which allows for the generation of the rules. This is done after inserting the range of values for oil level and torque monitoring and the gearbox outputs. The IF…..THEN……ELSE rules are being used for retrofitting as shown in **Figure 12**.


**Table 3.** Input 1: oil level.


#### **Table 4.** Input 2: torque control.


**Table 5.** Output 1: gearbox.

Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company http://dx.doi.org/10.5772/67762 361


**Figure 10.** (a) Membership function editor for the output, gearbox and (b) membership function editor for oil level monitoring.

**Figure 11.** Membership function editor for torque control.

**4.2. Control of gearbox jamming using fuzzy logic tool box**

360 Modern Fuzzy Control Systems and Its Applications

being used for retrofitting as shown in **Figure 12**.

**Table 3.** Input 1: oil level.

**Table 4.** Input 2: torque control.

**Table 5.** Output 1: gearbox.

0–40 Low 40–80 Acceptable 80–100 High

**Input range (%) Fuzzy variable name**

**Crisp input range (%) Fuzzy variable name**

0–2000 Low 2000–8000 Normal 8000–12,000 High

Indicator Symbol Stop 0 Run 1

*4.2.1. Membership function editor for oil level and torque monitoring*

The membership function editor is the central concept of the fuzzy set, which has values rang‐ ing from 0 to 1 in the y axis. The ranges of control were done in **Tables 3**–**5**. Membership function editor is the stage whereby the ranges, as explained above, are inserted into the fuzzy logic Matlab software and the screenshot of the data is taken as shown in **Figure 10**. It helps to display all membership functions connected with the input and output for the entire Fuzzy Inference System (FIS). In this case, a triangular fuzzy set membership function is used for oil and torque monitoring. This is shown clearly in figures below with ranges for oil level monitor‐ ing varying from low, acceptable and high and the numerical values being added. For torque control, the corresponding numerical values range from low, normal and high. The gearbox is

either run or stopped, depending on the oil level and torque conditions as in **Table 5**.

The membership functions editor for the output is shown in **Figure 11**, which is the gearbox that is plotted in the form of a triangle. Fuzzy logic is an artificial intelligence software which can store the output ranges in its memory and can learn the system to give solutions in what can be done. The Matlab software of fuzzy logic consists of the rule editor function which allows for the generation of the rules. This is done after inserting the range of values for oil level and torque monitoring and the gearbox outputs. The IF…..THEN……ELSE rules are

**Figure 12.** Rule editor for the fuzzy logic control.

After the rules are inserted, the overall results of combined effects of oil level and torque monitoring on gearbox monitoring are as discussed below. The rule viewer is an intelligent software, which is used to view the rules created using the rule editor. The rule viewer for the range values for oil level, torque and gearbox operation is discussed clearly in the figure below. However, the values obtained in the rule viewer are just optimum ranges and cannot be concluded as the best control values. Conclusions are only taken after the defuzzification process as described later in the chapter. The surface viewer shows a three‐dimensional struc‐ ture, which gives a conclusion to the range of values that are required to keep the gearbox functioning. This can be put in two‐dimensional structure to highlight, in a simplified man‐ ner, the range of values recommended to keep the gearbox functioning as in **Figure 13**.

#### *4.2.2. Analysis of the effects of oil level and torque on gearbox*

#### *4.2.2.1. Control of gearbox oil level*

From the surface viewer of the gearbox against the oil level shown in **Figures 14** and **15**, it is seen that the gearbox is maintained and remains constant, at a 50% capacity if the oil level is 40%. If the oil level is at 40%, smooth running of the boiler is experienced. If the oil level by any means goes below 40%, stop the gearbox to avoid breakdown of the gearbox. It is seen that the fuzzy logic engine will instruct the gearbox to stop running without any human inter‐ vention so as to reduce the frequency of breakdown of the boiler machine through an effective gearbox operation. **Figure 14** shows a three‐dimensional, and **Figure 15** is a two‐dimensional image.

#### *4.2.2.2. Control of gearbox torque*

From the graph, **Figure 14** of gearbox operation against torque control, it is shown that the gearbox should never be operated at torque greater than 8000 Nm as shown in **Figure 16**. The gearbox is seen to run smoothly at a 50% capacity if torque ranges from 0 to 8000 Nm. From

**Figure 13.** Rule viewer of the gearbox monitoring.

Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company http://dx.doi.org/10.5772/67762 363

**Figure 14.** Surface viewer for gearbox monitoring.

After the rules are inserted, the overall results of combined effects of oil level and torque monitoring on gearbox monitoring are as discussed below. The rule viewer is an intelligent software, which is used to view the rules created using the rule editor. The rule viewer for the range values for oil level, torque and gearbox operation is discussed clearly in the figure below. However, the values obtained in the rule viewer are just optimum ranges and cannot be concluded as the best control values. Conclusions are only taken after the defuzzification process as described later in the chapter. The surface viewer shows a three‐dimensional struc‐ ture, which gives a conclusion to the range of values that are required to keep the gearbox functioning. This can be put in two‐dimensional structure to highlight, in a simplified man‐ ner, the range of values recommended to keep the gearbox functioning as in **Figure 13**.

From the surface viewer of the gearbox against the oil level shown in **Figures 14** and **15**, it is seen that the gearbox is maintained and remains constant, at a 50% capacity if the oil level is 40%. If the oil level is at 40%, smooth running of the boiler is experienced. If the oil level by any means goes below 40%, stop the gearbox to avoid breakdown of the gearbox. It is seen that the fuzzy logic engine will instruct the gearbox to stop running without any human inter‐ vention so as to reduce the frequency of breakdown of the boiler machine through an effective gearbox operation. **Figure 14** shows a three‐dimensional, and **Figure 15** is a two‐dimensional

From the graph, **Figure 14** of gearbox operation against torque control, it is shown that the gearbox should never be operated at torque greater than 8000 Nm as shown in **Figure 16**. The gearbox is seen to run smoothly at a 50% capacity if torque ranges from 0 to 8000 Nm. From

*4.2.2. Analysis of the effects of oil level and torque on gearbox*

*4.2.2.1. Control of gearbox oil level*

362 Modern Fuzzy Control Systems and Its Applications

*4.2.2.2. Control of gearbox torque*

**Figure 13.** Rule viewer of the gearbox monitoring.

image.

**Figure 15.** Surface viewer of oil level versus gearbox operation.

**Figure 16.** Surface viewer of gearbox control using torque.

this figure, it can be concluded that the required range for the gearbox to run smoothly is 0–8000 Nm of torque; otherwise, stop the gearbox and hence the boiler. Matlab software has stored this data in fuzzy logic and will automatically stop the gearbox operation and hence the boiler if the above conditions are not met.

#### *4.2.3. Moisture and dust control on boiler feed check valves*

#### *4.2.3.1. Effects of moisture and dust levels on feed check valves*

Moisture inside the valves may result from the compressed air passing through the valve. The effects of moisture on the valves include:


The following are the range of values of moisture which are tolerated and not tolerated inside the valve:

#### *4.2.3.2. Moisture level control*

#### *4.2.3.2.1. For a range of 1–5%*

This is the total range of moisture in the feed check valve at any time (either accepted or unaccepted).

• 0–1%

This defines that the amount of the moisture in the feed check valve is very minimum in such a way that it causes little or no damage to the valve.

• 1–3%

At this range, the valve works effectively but further exposure to moisture may result in a breakdown.

• 3–5%

This is the range when a maximum amount of moisture is experienced inside the valves. This range is unacceptable at all times as it leads to breakdown of the valves (rust, corrosion, wear and eventually breakdown of the boiler).

### *4.2.3.3. Dust level control*

### *4.2.3.3.1. For a range of 0–1%*

This is the total range of values of dust inside the feed check valve (accepted or unaccepted).

• 0–0.02%

this figure, it can be concluded that the required range for the gearbox to run smoothly is 0–8000 Nm of torque; otherwise, stop the gearbox and hence the boiler. Matlab software has stored this data in fuzzy logic and will automatically stop the gearbox operation and hence

Moisture inside the valves may result from the compressed air passing through the valve. The

• Increased rate of wear of the valve material may also result due to the moisture. The mois‐ ture washes the lubrication away which will result in the eventual failure or malfunction‐

• The industrial processes which rely on the full functionality of the pneumatic control valves may be jeopardized, and this usually results in costly breakdowns of the machine.

• Air‐ or gas‐operated instruments may give inaccurate readings due to corrosion of the ma‐

• The rubber diaphragms inside the pneumatic valves can be stiffened and will eventually

The following are the range of values of moisture which are tolerated and not tolerated inside

This is the total range of moisture in the feed check valve at any time (either accepted or

This defines that the amount of the moisture in the feed check valve is very minimum in such

At this range, the valve works effectively but further exposure to moisture may result in a

This is the range when a maximum amount of moisture is experienced inside the valves. This range is unacceptable at all times as it leads to breakdown of the valves (rust, corrosion, wear

the boiler if the above conditions are not met.

364 Modern Fuzzy Control Systems and Its Applications

effects of moisture on the valves include:

ing of the valve.

the valve:

unaccepted).

• 0–1%

• 1–3%

• 3–5%

breakdown.

*4.2.3.2. Moisture level control*

*4.2.3.2.1. For a range of 1–5%*

*4.2.3. Moisture and dust control on boiler feed check valves*

*4.2.3.1. Effects of moisture and dust levels on feed check valves*

terial and hence, interrupting the plant processes.

rupture due to the moisture flowing through them.

a way that it causes little or no damage to the valve.

and eventually breakdown of the boiler).

• Moisture may cause rusting in the moving parts of the valve.

At this range, the valves can run effectively without any damage.

• 0.02–0.5%

This amount of dust entering the valve may cause the valve to malfunction with further expo‐ sure. However, at this range, the feed check valves are seen to be operating effectively.

• 0.5–1%

These ranges of dust are not acceptable at all. The excessive exposure of dust inside the valve may result in scoring, wear and eventually total failure of the boiler machine.

### *4.2.3.4. Output control*

The feed check valve has to either open or close, either of the two, and that is the meaning of 0 for close and 1 for open.

• 0 close

At this range, for close signal, the valve will close if the if the dust and moisture levels inside the valves are unacceptable.

• 1 open

This involves the acceptable range. All the conditions are being met (dust and moisture level), and the valve can effectively open at any angle between 0° and 90° depending on the amount of feed water to be pumped into the boiler.

#### *4.2.4. Simulation of the effects of dust and moisture levels on feed check valves using fuzzy logic*

The moisture and dust level variations in the boiler machine can be monitored using an artifi‐ cial intelligence software. The inference engine of fuzzy logic can store two input parameters (moisture and dust level) and one output parameter (feed check valve operation) as shown by the print screen below from the laptop in **Figure 17**.

Control of feed check valve operation using fuzzy logic tool box is shown in **Tables 6**–**8**.

#### *4.2.5. Membership function editor for moisture and dust monitoring*

This is shown clearly in **Figures 18**–**20** with ranges for moisture level varying from low, accept‐ able and high and the numerical values being added. For dust control, the corresponding numerical values range from low, normal and high. The valve either closes or opens depend‐ ing on moisture and dust levels. This is done to control the feed check valves so that they won#x2019;t operate in unfavourable conditions which might result in the failure of the boiler.

**Figure 17.** FIS Editor for feed check valve.

The following rules were developed in **Figure 20** and will instruct the controller what to do.

After the rules are inserted, the overall results of combined effects of moisture and dust levels on feed check valve are as discussed below. The rule viewer is an intelligent software which is used to view the rules created using the rule editor and is shown in **Figure 21**. The rule viewer for the range values for moisture level, dust and valve operation is discussed clearly


**Table 6.** Input 1: moisture level.


**Table 7.** Input 2: dust level.


**Table 8.** Output 1: feed check valve operation.

Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company http://dx.doi.org/10.5772/67762 367

**Figure 18.** Membership function editor for moisture.

The following rules were developed in **Figure 20** and will instruct the controller what to do. After the rules are inserted, the overall results of combined effects of moisture and dust levels on feed check valve are as discussed below. The rule viewer is an intelligent software which is used to view the rules created using the rule editor and is shown in **Figure 21**. The rule viewer for the range values for moisture level, dust and valve operation is discussed clearly

**Input range (%) Fuzzy variable name**

**Input range (%) Fuzzy variable name**

**Output range (%) Fuzzy variable name**

0 Closed 1 Open

**Table 8.** Output 1: feed check valve operation.

0–0.02 Acceptable 0.02–0.5 Average 0.5–1 Not acceptable

0–1 Low 1–3 Medium 3–5 High

**Figure 17.** FIS Editor for feed check valve.

366 Modern Fuzzy Control Systems and Its Applications

**Table 6.** Input 1: moisture level.

**Table 7.** Input 2: dust level.

in **Figure 21**. The surface viewer shows a three‐dimensional layout in **Figure 22**, and the blue lines indicate the values of moisture level and dust level, which may result in the failure of the valve. The required values which should be maintained are shown by the yellow surfaces. However, the values obtained from the rule viewer are just ranges and not concluded as the best control values. Conclusions are only taken after the defuzzification process.

**Figure 19.** Membership function editor for the output and feed check valve.

**Figure 20.** Rule editor for feed check valve.

#### *4.2.6. Analysis of the effects dust and moisture on feed check valve*

#### *4.2.6.1. Control of feed check dust level*

From the surface viewer of the feed check valve control against the amount of dust, it is seen that the valve remains constant at 45% corresponding to a dust range of 0–0.55%. The dust level should be maintained at this range. If the dust level is increased from 0.55 to 0.58%, there is a linear decrease in the feed check valve's opening angle from 45 to 0°. As the level of dust is further increased from 0.58 to 0.95%, the valve is kept constant at 0°. It is therefore concluded that dust level should remain between 0 and 0.55% so that the valves will run smoothly; oth‐ erwise, boiler breakdown will result. Fuzzy logic engine is an artificial intelligence system

**Figure 21.** Rule viewer for the feed check valve monitoring.

Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company http://dx.doi.org/10.5772/67762 369

**Figure 22.** Surface viewer for the feed check valve monitoring.

which will instruct the valves to immediately close as soon as the dust level exceeds 0.55%, as in **Figure 23**, without any human intervention.

#### *4.2.6.2. Control of moisture level*

*4.2.6. Analysis of the effects dust and moisture on feed check valve*

From the surface viewer of the feed check valve control against the amount of dust, it is seen that the valve remains constant at 45% corresponding to a dust range of 0–0.55%. The dust level should be maintained at this range. If the dust level is increased from 0.55 to 0.58%, there is a linear decrease in the feed check valve's opening angle from 45 to 0°. As the level of dust is further increased from 0.58 to 0.95%, the valve is kept constant at 0°. It is therefore concluded that dust level should remain between 0 and 0.55% so that the valves will run smoothly; oth‐ erwise, boiler breakdown will result. Fuzzy logic engine is an artificial intelligence system

*4.2.6.1. Control of feed check dust level*

**Figure 21.** Rule viewer for the feed check valve monitoring.

**Figure 20.** Rule editor for feed check valve.

368 Modern Fuzzy Control Systems and Its Applications

**Figure 24** indicates that moisture level does not contribute much to the valve opening. From 0 to 85%, the feed check valves operate at a constant capacity of 45%. For smooth operation of the valves, moisture level should not exceed 85%; otherwise, fuzzy logic will set the valves to close automatically. This helps to prevent the breakdown of feed check valves.

**Figure 23.** Surface viewer of gearbox control using dust level.

**Figure 24.** Surface viewer of feed check valve control using moisture.

## **5. Recommendations and conclusions**

**Improve management commitment:** The success of any system depends on the commitment of the management. Design and implementation of a hybrid proactive approach require a greater initial investment for it to be successful. Since systems are people driven, it is certain to require worker participation. Management needs to look up at providing resources for in‐house training for the fuzzy logic system.

### **Appendices for the processing company**

#### **Appendix 1: Factor A**


#### **Appendix 2: Factor B**


### **Appendix 3: Factor C**


### **Appendix 4: Factor D**


#### **Appendix 5: Priority score**

The priority score is obtained by multiplying the results for the three factors:

#### **PS = Factor E \* Factor F \* Factor G** (3)

#### **Factor E**

**5. Recommendations and conclusions**

370 Modern Fuzzy Control Systems and Its Applications

**Figure 24.** Surface viewer of feed check valve control using moisture.

in‐house training for the fuzzy logic system.

**Appendix 1: Factor A**

production to stop

**Appendices for the processing company**

**Improve management commitment:** The success of any system depends on the commitment of the management. Design and implementation of a hybrid proactive approach require a greater initial investment for it to be successful. Since systems are people driven, it is certain to require worker participation. Management needs to look up at providing resources for

**Effect on production output (Factor A) Factor score**

4

No significant impact/standby equipment is available 1 Minor impact on production. Unlikely to affect other areas of the plant 2 Failure would have significant impact on output and may affect other sections 3

Major impact on the plant#x2019;s operations; failure would cause over 40% of plant


#### **Appendix 6: Factor F**


#### **Appendix 7: Factor G**


#### **Appendix 8: Fire Tube boiler's detailed FMECA analysis**


Fuzzy Logic Application, Control and Monitoring of Critical Machine Parameters in a Processing Company http://dx.doi.org/10.5772/67762 373

**Appendix 6: Factor F**

372 Modern Fuzzy Control Systems and Its Applications

**Appendix 7: Factor G**

**Component name**

Feed check valve

Combustion chamber

Servo motor To generate

**Component function**

torque to turn fan rotor

To regulate the supply of water which is pumped into the boiler by a feed pump.

To burn fuel Leakage

**Appendix 8: Fire Tube boiler's detailed FMECA analysis**

**Effect(s) of failure**

Excessive vibration, noise. motor overheat and sparks of fires are produced

Overpressure in system and leakages resulting in fire

Combustion gases entering fire room

**Failure mode(s)**

Bearing failure, coil shorts, overheating

Fails to open Remains open Crack valve

Too much fuel being fired, and excess air

**Occurrence index (O)**

**Severity index (S)**

6 7 3 126

6 8 2 96

7 7 5 245

**Detection index (D)**

**Risk priority number (O)\*(S)\*(D)**

**Cause(s) of failure**

Worn bearing, lubrication failure, burnt brushes and broken fan bolts

Internal valve malfunction Operator error Calibration error

through the soot blower casing seal

**Downtime/repair time (Factor F) Downtime Factor score**

**Waste (Factor G) Quantity F** No waste is generated under normal operating conditions 0% 1 Small amounts of waste are produced by failure 2% 2 Waste is produced during production that is significant 5% 3 Quantities of waste are significant and warrant immediate attention 10% 4

Minor 0–30 min 1 Significant 30–120 min 2 Major 2–8 h 3 Severe >8 h 4




## **Author details**

**Component name**

rotor bearing

Pressure relief valve

Blow off cock

Oil Control Unit

Water indicator gauge glass

Super heater

**Component function**

374 Modern Fuzzy Control Systems and Its Applications

To absorb the axial thrust and support the rotor

the speed ratio between motor and kicker

To measure the pressure of the steam boiler

the flow of steam from the boiler to the main steam pipe

To empty the boiler when discharging mud, scale or sediments

To circulate lubricant at specified flow rate and pressure

To indicate the water level inside the boiler

To increase temperature of saturated steam without raising its pressure

Gearbox To change

Stop valve To control

**Cause(s) of failure**

Excessive vibrations

Starting under load

Internal valve malfunction Operator error and calibration error

Internal valve malfunction Operator error and calibration error

Internal valve malfunction Operator error and calibration error

Blocked filters, worn impellor, high lubricant temperature

Salt deposition due to high water level in drum

Blocked tube Starvation of tube Erosion of tube due to high excessive air **Effect(s) of failure**

Rapture of the boiler container or pipes

Overpressure: protection compromised

Overpressure in system resulting in fire, leakages

Delivery pressure lower than specified, flow rate lower than specified and cavitation

Fails to react to water rise/ drop above/ below the preset value

Cracking of carbon steel

**Failure mode(s)**

Jammed closed

Fails to open Fails to close

Blockage, wear and cavitation

Fails to react to water rise/ drop above/ below the preset value

Dirty on surface Noisy operation Impact wear Corrosion on surface

**Occurrence index (O)**

Bearing wear Wear 3 6 2 36

Wear of gears Impact wear 3 8 4 96

**Severity index (S)**

5 9 4 180

7 6 8 336

6 6 8 288

6 9 7 378

6 3 8 144

Leakages 7 9 9 567

**Detection index (D)**

**Risk priority number (O)\*(S)\*(D)**

#### Tawanda Mushiri

Address all correspondence to: tawanda.mushiri@gmail.com

Faculty of Engineering and the Built Environment, University of Johannesburg, Johannesburg, South Africa

## **References**


## **Use of Fuzzy Logic for Design and Control of Nonlinear MIMO Systems**

[8] Liang, S.Y., Hecker, R.L., & Landers, R.G. (2002). Machining process monitoring and con‐ trol: the state‐of‐the‐art. In ASME 2002 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, pp. 599‐610. University of

[9] Nagilla, S.R. (2015). Non‐thermal reliability considerations for. Arlington: University of

[10] Nwaobi, G.C. (2008). Energy power, digital infrastructure and e‐learning platforms:

[11] Okah‐Avae. (1981). Condition monitoring – a maintenance management strategy for

[12] Papadopoulos, C., & Dimarogonas, A. (1992). Coupled vibration of cracked shafts.

[13] Pine, B.J. (1993). Mass customization: the new frontier in business competition. Harvard

[14] Beverage Industry Environmental Roundtable. (2014). Beverage industry sector guidance for greenhouse gas emissions reporting. Beverage Industry Environmental Roundtable.

[15] Scheffer, C., & Girdhar, P. (2004). Practical machinery vibration analysis and predictive

[16] Seemann, K.W. (2000). Technacy education: towards holistic pedagogy and epistemol‐ ogy in general and indigenous/cross‐cultural technology education. School of Tourism

[17] Tavner, P.J., & Penman, J. (1987). Condition monitoring of electrical machines*.* Research

[18] Zhang, Z. (2011). Energy and environmental policy in China: towards a low‐carbon

Missouri, United States of America.

376 Modern Fuzzy Control Systems and Its Applications

African experience. Journal of Energy.

Business Press. Harvard.

Studies Press.

maintenance*.* Elsevier. Amsterdam.

industrial machinery. Benin: University of Benin‐City.

and Hospitality Management Papers, 9. United Kingdom.

economy*.* Edward Elgar Publishing. Massachusetts, USA.

Journal of Vibration and Acoustics, 114, 461‐467.

Texas.

Pavol Fedor and Daniela Perduková

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68050

#### **Abstract**

Standard analytical methods are often ineffective or even useless for design of nonlinear control systems with imprecisely known parameters. The use of fuzzy logic principles presents one possible way to control such systems which can be used both for modeling and design of the control. The advantage of using this method consists in its simplicity and easy way of developing the algorithm, which in the phase of designing the controllers and also for modeling the features of the designed structures, allows the use of computer technology. Simplicity of the proposed structure (usually with the PI controllers) and determination of their parameters without any need for complex mathematical description present another considerable advantage of the used method. This chapter presents two typical examples of designing the control of nonlinear multi‐input multi‐output (MIMO) systems from the field of mechatronic systems based on fuzzy logic principles.

**Keywords:** fuzzy PI torque controller, asynchronous motor, continuous line, black‐box modeling, fuzzy model‐based control

#### **1. Introduction**

There are many processes in technological practice the analytical description of which is rather complicated. This can be due to their complexity, nonlinearity, transfer lags, complicated measurement of important parameters, etc. However, information on the performance of these processes can often be obtained experimentally (by suitably chosen measurements or by monitoring their responses to the control activities of the operator). In these situations, fuzzy systems can always be considered as an alternative for system modeling and control.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Applying fuzzy logic in fuzzy controller design is very often a suitable possibility of solving problem issues in control in various fields of industry because these controllers are an effective tool for achieving high‐quality properties of the controlled systems [1–9]. The disadvantage in this case is the unsystematic approach to their synthesis and a relatively demanding analysis of their stability. A fuzzy controller design is primarily based on the fuzzification of its range of inputs and the setting up of rules of its behavior within this range. The behavior of classic fuzzy controllers was designed on basis of linguistic rules obtained from experts. However, this knowledge is not always easily obtained, especially in cases of higher‐order nonlinear systems [10, 11]. For this reason, special attention has been focused in recent years on the design of fuzzy control systems that are not based on the search for expert linguistic rules [12–17]. Control methods based on the controlled system fuzzy model have many modifications that depend on the particular application [6, 13–17], while the quality of the fuzzy model of the controlled system is also of significant importance.

It has been proved that fuzzy modeling can be recognized as one of the nonlinear black‐box modeling techniques [11, 12, 18–20]. When designing a black‐box fuzzy system, it is necessary to identify its qualitative properties only on the basis of experimentally measured data, while neither its structure nor its parameters are known. That often results in problems with inconsistency of the database, problems with covering the entire space of possible inputs, etc. [20–22], which makes the fuzzy model unusable in practical applications. In the design of a black‐box fuzzy model of a dynamic system, a suitable method for the selection of qualitative properties from the collected database always needs to be applied. The functional dependencies between inputs and outputs can then be used for developing a suitable nonparametric fuzzy model of the process that can be applied in the design of their control [23–27].

Two typical examples for designing the control of nonlinear MIMO systems using fuzzy approach are presented in this chapter:


### **2. Design of fuzzy torque controller for asynchronous motor drive**

An asynchronous motor represents a strongly nonlinear fifth‐order system, whose good quality vector torque control is solved by relatively complex mathematical transformations and leads to a complicated control structure [28, 29]. Therefore, a fuzzy system for design of torque controller for asynchronous motor drive has been used. The fuzzy controller design is based on the concept of identifying the time sequence of the input signal into the controlled system that will provide the control target in terms of the selected optimality criterion. Fuzzy controller design method is characterized by simplicity, and quality of control is appropriate to the considered drive.

API‐type discrete controller is generally described by the equation:

Applying fuzzy logic in fuzzy controller design is very often a suitable possibility of solving problem issues in control in various fields of industry because these controllers are an effective tool for achieving high‐quality properties of the controlled systems [1–9]. The disadvantage in this case is the unsystematic approach to their synthesis and a relatively demanding analysis of their stability. A fuzzy controller design is primarily based on the fuzzification of its range of inputs and the setting up of rules of its behavior within this range. The behavior of classic fuzzy controllers was designed on basis of linguistic rules obtained from experts. However, this knowledge is not always easily obtained, especially in cases of higher‐order nonlinear systems [10, 11]. For this reason, special attention has been focused in recent years on the design of fuzzy control systems that are not based on the search for expert linguistic rules [12–17]. Control methods based on the controlled system fuzzy model have many modifications that depend on the particular application [6, 13–17], while the quality of the fuzzy

It has been proved that fuzzy modeling can be recognized as one of the nonlinear black‐box modeling techniques [11, 12, 18–20]. When designing a black‐box fuzzy system, it is necessary to identify its qualitative properties only on the basis of experimentally measured data, while neither its structure nor its parameters are known. That often results in problems with inconsistency of the database, problems with covering the entire space of possible inputs, etc. [20–22], which makes the fuzzy model unusable in practical applications. In the design of a black‐box fuzzy model of a dynamic system, a suitable method for the selection of qualitative properties from the collected database always needs to be applied. The functional dependencies between inputs and outputs can then be used for developing a suitable nonparametric

fuzzy model of the process that can be applied in the design of their control [23–27].

Two typical examples for designing the control of nonlinear MIMO systems using fuzzy

• The design of fuzzy PI torque controller of the PI type for a drive with an induction motor, whose parameters and rules are obtained by searching control input of such a vector which

• The design of control for middle part of a continuous line for material processing by tension, where the continuous line presents a nonlinear MIMO system. Its control requires to ensure decoupled control of individual subsystems, because the output quality of the processed material depends directly on quality of the control. The controllers of the subsystems ensuring such decoupling usually are of complex structures and, when designing them by analytic methods, they are often unrealizable. When the continuous line is presented by a fuzzy model, it is possible to design simple controllers of the PI type ensuring

model of the controlled system is also of significant importance.

is optimal in terms of the selected criterion of optimality.

high‐quality dynamical properties of the controlled system.

**2. Design of fuzzy torque controller for asynchronous motor drive**

An asynchronous motor represents a strongly nonlinear fifth‐order system, whose good quality vector torque control is solved by relatively complex mathematical transformations

approach are presented in this chapter:

378 Modern Fuzzy Control Systems and Its Applications

$$u\_k = \left. u\_{k-1} + q\_0 \, e\_k + q\_1 \, e\_{k-1} \right. \tag{1}$$

where *u*<sup>k</sup> and *u*k−1 are values of controller output in the relevant sampling steps, *e*<sup>k</sup> and *e*k−1 are values of the control error, and *q*<sup>0</sup> and *q*<sup>1</sup> are parameters of controller [28]. From these follows, it is possible structure, shown in **Figure 1**.

A discrete fuzzy PI controller can be described, for example, by the following rules:

$$IF\ e\_{\boldsymbol{\upiota}}\ \boldsymbol{\upiota}\ldots\boldsymbol{\upiota}\ e\_{\boldsymbol{\upiota}-1}\ \boldsymbol{\upiota}\ldots\boldsymbol{\upiota}\ \boldsymbol{\upPi}\boldsymbol{\upPi}\ \boldsymbol{\upiota}\boldsymbol{\upiota}\ldots\boldsymbol{\upiota}\tag{2}$$

where quantities *e*<sup>k</sup> , *e*k−1, and *du*<sup>k</sup> are fuzzy variables that describe the relevant workspace of the fuzzy PI controller. The fuzzy controller design procedure consists of the following three steps:

**Step 1.** Finding the optimal sequence of input values

Fuzzy rules and fuzzification of the fuzzy PI controller workspace can be identified by means of relations expressed by triplets [*e*<sup>k</sup> *e*k−1 *du*<sup>k</sup> ] that have been obtained for its optimal behavior. This behavior is represented by the time sequence of input vector *d***u**opt, at which the value of the optimality criterion is minimal. It is suitable to choose this criterion in the next form of the integral of the quadratic deviation of the system output *y* from the desired value *w*:

$$\mathbf{J}(e) = \int \left( w(t) - \mathbf{y}(t) \right)^2 dt \tag{3}$$

When obtaining the sequence of values of vector *d***u**opt, we apply to the drive input various sequences of input vector and evaluate the criterion value Eq. (3). This is a standard optimization task that can be solved, for example, by suitable geometric division of the drive workspace (which leads to rather high computing demands in the design process), or by applying the genetic algorithm method, which significantly speeds up the whole process of identification.

**Figure 1.** Block diagram of fuzzy controller.

#### **Step 2.** Finding the database of optimal data

Having found the optimal input sequence *d***u**opt for the controlled drive, we then set up the database of triplets [*e*<sup>k</sup> *e*k−1 *du*<sup>k</sup> ], which describes relations between the inputs and the outputs of the optimal fuzzy PI controller.

**Step 3.** Designing the fuzzy controller from the optimal data database

From the obtained triplets [*e*<sup>k</sup> *e*k−1 *du*<sup>k</sup> ] of optimal data, it is possible to design a concrete fuzzy controller of various types using standard procedures of clustering the data into significant clusters and describing them by means of rules.

In the concrete application of the said procedure in a drive with asynchronous motor, we will use its analytical model (see Refs. [28, 29]). If we consider a rotating system which rotates with the frequency of the motor's stator field (usually marked by coordinates *x*, *y*), the mathematical model of the device is described by the following equations:

$$\begin{aligned} \mu\_{ss} &= \, \, \mathcal{R}\_s \dot{i}\_{sz} + \frac{d\,\psi\_{ss}}{dt} - \omega\_s \, \psi\_{sy} \\ \mu\_{sy} &= \, \, \mathcal{R}\_s \dot{i}\_{sy} + \frac{d\,\psi\_{sy}}{dt} - \omega\_s \, \psi\_{sz} \\ \quad \text{or} \quad \, \, \mathcal{R}\_r \dot{i}\_{rz} + \frac{d\,\psi\_{rs}}{dt} - \omega\_2 \, \psi\_{ry} \\ \quad \, \, \mathcal{R} &= \, \, \mathcal{R}\_r \dot{i}\_{ry} + \frac{d\,\psi\_{ry}}{dt} + \omega\_2 \, \psi\_{rz} \\ \, \, \mathcal{M}\_e &= \, \frac{3}{2} p \left( \psi\_{ry} \, \dot{i}\_{rx} - \psi\_{xx} \, \dot{i}\_{ry} \right) \end{aligned} \tag{4}$$

Used symbols:


As a standard, asynchronous motors are supplied from static voltage frequency converters in which the stator frequency and voltage rate are *U*<sup>1</sup> /*ω*<sup>1</sup> .

When connected directly to the power supply network, the motor shows a large increase in torque and also in current (**Figure 2**).

Let the aim of the torque controller design be to adjust the slip *ω*<sup>2</sup> (i.e., the difference of *ω*<sup>1</sup> – *ω*m) according to the desired torque value. The structure of the controlled system will be as shown in **Figure 3**.

For finding the optimal input sequence *d***u**opt, we shall use the diagram shown in **Figure 4**.

**Figure 2.** Direct connection of drive with AM.

**Step 2.** Finding the database of optimal data

database of triplets [*e*<sup>k</sup> *e*k−1 *du*<sup>k</sup>

*i* sx, *i*

*i* rx, *i*

*ω*1

*ω*2

*R*s , *R*<sup>r</sup>

*Ψ*s , *Ψ*<sup>r</sup>

*M*e

Used symbols:

of the optimal fuzzy PI controller.

380 Modern Fuzzy Control Systems and Its Applications

From the obtained triplets [*e*<sup>k</sup> *e*k−1 *du*<sup>k</sup>

clusters and describing them by means of rules.

sy components of stator current space vector *i*

ry components of rotor current space vector *i*

*u*sx, *u*sy components of stator voltage space vector *u*<sup>s</sup>

 = *ω*<sup>1</sup> – *ω*<sup>m</sup>

*ω*m the motor mechanical angular speed

*p* number of pole pairs (*p* = 2)

electrical motor moment

slip angular speed *ω*<sup>2</sup>

angular frequency of the stator voltage

stator and rotor phase resistance

AC drive parameters are given in the Appendix.

which the stator frequency and voltage rate are *U*<sup>1</sup>

stator and rotor magnetic flux

Having found the optimal input sequence *d***u**opt for the controlled drive, we then set up the

controller of various types using standard procedures of clustering the data into significant

In the concrete application of the said procedure in a drive with asynchronous motor, we will use its analytical model (see Refs. [28, 29]). If we consider a rotating system which rotates with the frequency of the motor's stator field (usually marked by coordinates *x*, *y*), the mathemati-

> *sx* <sup>+</sup> \_\_\_\_ *d ψsx dt* <sup>−</sup> *<sup>ω</sup><sup>s</sup> <sup>ψ</sup>sy*

*sy* <sup>+</sup> \_\_\_\_ *d ψsy*

*rx* <sup>+</sup> \_\_\_\_ *d ψrx dt* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>ψ</sup>ry* <sup>0</sup> <sup>=</sup> *<sup>R</sup> <sup>r</sup> <sup>i</sup> ry* <sup>+</sup> \_\_\_\_ *d ψry*

<sup>2</sup> *p*(*ψry i*

*dt* <sup>−</sup> *<sup>ω</sup><sup>s</sup> <sup>ψ</sup>sx*

*dt* <sup>+</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>ψ</sup>rx*

*rx* − *ψrx i ry*)

s

r

As a standard, asynchronous motors are supplied from static voltage frequency converters in

/*ω*<sup>1</sup> .

**Step 3.** Designing the fuzzy controller from the optimal data database

cal model of the device is described by the following equations:

*usx* = *Rs i*

*usy* = *Rs i*

0 = *Rr i*

*Me* <sup>=</sup> \_\_3

], which describes relations between the inputs and the outputs

(4)

] of optimal data, it is possible to design a concrete fuzzy

**Figure 3.** Structure of the controlled drive.

**Figure 4.** Block diagram for finding vector *d***u**opt.

We search for the input signal sequence *d***u**opt through such changes of drive input voltage and frequency that will lead to minimal value of the criterion according to Eq. (3). This goal can be achieved, for example, by application of the genetic algorithm method, which efficiently enables finding the extreme of the selected function in a given space of mutations. Having selected input signal sampling time 50 ms and desired value of motor torque Mz = 30, we obtained the optimal input vector [0.165 0.145 0.155 0.12 0.105 0.095 0 0 0 0 0 0], as illustrated in **Figure 5**.

**Figure 5.** Optimal start‐up of drive with AM.

The said optimal input signal sequence was used for generating the database of triplets [*ω*mk*ω*mk−1 (*ω*<sup>1</sup> –*ω*m) *k* ] for the design of the P‐type fuzzy controller. Using the Anfisedit tool in the Matlab program, a static Sugeno‐type fuzzy system was designed, based on the acquired database of measured data ([*ω*mk*ω*mk−1 (*ω*<sup>1</sup> –*ω*m)*<sup>k</sup>* ]). This system has three zones for input data fuzzification, and its internal structure can be seen in **Figure 6**.

The resulting torque control structure of AM with the designed fuzzy controller is shown in **Figure 7**.

The start‐up of the drive with desired torque Mz = 30 Nm using the designed fuzzy controller is shown in **Figure 8**.

The comparison of **Figures 5** and **8** shows that the application of the fuzzy controller resulted in the achievement of the desired torque responses of the AM at start‐up.

#### **2.1. Discussion**

We search for the input signal sequence *d***u**opt through such changes of drive input voltage and frequency that will lead to minimal value of the criterion according to Eq. (3). This goal can be achieved, for example, by application of the genetic algorithm method, which efficiently enables finding the extreme of the selected function in a given space of mutations. Having selected input signal sampling time 50 ms and desired value of motor torque Mz = 30, we obtained the optimal input vector [0.165 0.145 0.155 0.12 0.105 0.095 0 0 0 0 0 0], as illustrated in **Figure 5**.

The said optimal input signal sequence was used for generating the database of triplets

the Matlab program, a static Sugeno‐type fuzzy system was designed, based on the acquired

The resulting torque control structure of AM with the designed fuzzy controller is shown in

The start‐up of the drive with desired torque Mz = 30 Nm using the designed fuzzy controller

The comparison of **Figures 5** and **8** shows that the application of the fuzzy controller resulted

–*ω*m) *k*

] for the design of the P‐type fuzzy controller. Using the Anfisedit tool in

]). This system has three zones for input data

[*ω*mk*ω*mk−1 (*ω*<sup>1</sup>

**Figure 7**.

is shown in **Figure 8**.

–*ω*m) *k*

**Figure 5.** Optimal start‐up of drive with AM.

382 Modern Fuzzy Control Systems and Its Applications

database of measured data ([*ω*mk*ω*mk−1 (*ω*<sup>1</sup>

fuzzification, and its internal structure can be seen in **Figure 6**.

in the achievement of the desired torque responses of the AM at start‐up.

The fuzzy controller designed in this chapter is a PI controller, and it provides optimal dynamics in terms of the selected criterion Eq. (3). The design procedure consists of three steps:

In the first step, we search for such input sequence of input vector **u** of the asynchronous motor that would provide minimal value of function Eq. (3). This can be achieved by modeling

**Figure 6.** Structure of AM fuzzy controller.

**Figure 7.** Structure of torque control in AM with fuzzy controller.

**Figure 8.** Optimal start‐up of drive with AM with fuzzy controller.

various vector **u** input sequences in the system with asynchronous motor, and by a simulation of its behavior for a given sequence. The selection of sampling time for a change of the input vector value and by that also of the number of its samples has to be defined according to the Shannon‐Kotelnikov theorem. Obviously, with an increase in the number of samples also the time required for simulation and for optimal vector **u**opt identification grows. As the whole process of optimal vector identification can be fully automated and the performance of computing means nowadays is sufficient, there is no essential problem to identify the optimal control input signal for a concrete drive with asynchronous motor.

In the second step, using the database obtained by application of the input vector **u**opt to the asynchronous motor, the input/output relations of the optimal PI controller for the given controlled system are defined. Various input and output signals (e.g., depending on their actual measurability …) can be chosen for setting up the suitable database. This makes it possible to adjust the controller design to a concrete drive with concrete technical capabilities.

The relations obtained in the second step of the design procedure, written down in the form of a suitable input and output signals database (mostly in a table), are described in the third step by means of fuzzy logic principles. This way a fuzzy PI controller similar to an optimal continuous PI controller is constructed. Standard computing means for working with fuzzy systems are employed in this step, such as the fuzzy toolbox in Matlab.

The whole procedure of fuzzy PI controller design was verified by simulation of its properties in the concrete control of a drive with asynchronous motor. The results of simulation experiments show that the controller, in spite of its simplicity and the uncomplicated computer oriented design procedure applied, enables considerable improvement in the control circuit dynamic properties also in case of strongly nonlinear higher order controlled systems.

## **3. Multi‐motor drive optimal control using a fuzzy model‐based approach**

Typical representative of multi‐motor drive is the middle part of the continuous line, where the individual working machines are coupled with each other through the material. It can be lines for processing continuous flows of material (e.g., sheet metal strips, tubes, processing lines in paper mills, and printing works) by material traction in the field of elastic or plastic deformation, which influences the material's mechanical properties. It means that the multiple motor drives are complex and coupled MIMO nonlinear systems. Therefore, due to the complexity of their mathematical models, which parameters are difficult to identify, the development of effective control systems is quite complicated task. This chapter presents the design of optimal control of continuous production line using a fuzzy model‐based approach.

The structure of the middle part of the continuous line (further referred to as CL) is shown in **Figure 9**. The structure includes DC motors powered through static transistor converters TC. The working machines of the line are driven by the motors through gearbox *j*; *v*<sup>1</sup> and *v*<sup>2</sup> are machine rolls circumferential velocities, and *F*12 is the tension in the web of material between the two machines. The main line disturbances are tensions before and after the middle part of the considered line which are affecting the first and second drive (*F*01 and *F*23). *K*<sup>v</sup> is circumferential velocity sensors, *K*<sup>F</sup> is tension sensor, *u*v1 and *u*v2 are outputs from velocity sensors, and *u*F12 is output of tension sensor. The controlling voltages *u*<sup>1</sup> , *u*<sup>2</sup> of converters present the input variables of the system. The tension in the web of material *F*12 and the web of material velocity v<sup>2</sup> is the output variables (let us consider *y*<sup>1</sup>  = *u*F12 and *y*<sup>2</sup>  = *u*v2).

The described system with the mechanical coupling of two machines presents a third‐order nonlinear MIMO system with two inputs and two outputs (**Figure 10**), the parameters of the

**Figure 9.** Structure of middle section of continuous line.

various vector **u** input sequences in the system with asynchronous motor, and by a simulation of its behavior for a given sequence. The selection of sampling time for a change of the input vector value and by that also of the number of its samples has to be defined according to the Shannon‐Kotelnikov theorem. Obviously, with an increase in the number of samples also the time required for simulation and for optimal vector **u**opt identification grows. As the whole process of optimal vector identification can be fully automated and the performance of computing means nowadays is sufficient, there is no essential problem to identify the optimal

In the second step, using the database obtained by application of the input vector **u**opt to the asynchronous motor, the input/output relations of the optimal PI controller for the given controlled system are defined. Various input and output signals (e.g., depending on their actual measurability …) can be chosen for setting up the suitable database. This makes it possible to

The relations obtained in the second step of the design procedure, written down in the form of a suitable input and output signals database (mostly in a table), are described in the third step by means of fuzzy logic principles. This way a fuzzy PI controller similar to an optimal continuous PI controller is constructed. Standard computing means for working with fuzzy

The whole procedure of fuzzy PI controller design was verified by simulation of its properties in the concrete control of a drive with asynchronous motor. The results of simulation experiments show that the controller, in spite of its simplicity and the uncomplicated computer oriented design procedure applied, enables considerable improvement in the control circuit dynamic properties also in case of strongly nonlinear higher order controlled systems.

**3. Multi‐motor drive optimal control using a fuzzy model‐based approach**

Typical representative of multi‐motor drive is the middle part of the continuous line, where the individual working machines are coupled with each other through the material. It can be lines for processing continuous flows of material (e.g., sheet metal strips, tubes, processing lines in paper mills, and printing works) by material traction in the field of elastic or plastic deformation, which influences the material's mechanical properties. It means that the multiple motor drives are complex and coupled MIMO nonlinear systems. Therefore, due to the complexity of their mathematical models, which parameters are difficult to identify, the development of effective control systems is quite complicated task. This chapter presents the design of optimal control of continuous production line using a fuzzy model‐based approach. The structure of the middle part of the continuous line (further referred to as CL) is shown in **Figure 9**. The structure includes DC motors powered through static transistor converters TC.

The working machines of the line are driven by the motors through gearbox *j*; *v*<sup>1</sup>

considered line which are affecting the first and second drive (*F*01 and *F*23). *K*<sup>v</sup>

machine rolls circumferential velocities, and *F*12 is the tension in the web of material between the two machines. The main line disturbances are tensions before and after the middle part of the

and *v*<sup>2</sup>

is circumferential

are

adjust the controller design to a concrete drive with concrete technical capabilities.

control input signal for a concrete drive with asynchronous motor.

384 Modern Fuzzy Control Systems and Its Applications

systems are employed in this step, such as the fuzzy toolbox in Matlab.

**Figure 10.** Middle section of continuous line as MIMO system.

system change depending on the mechanical properties of the material and on the speed of its motion. Defining precise parameters of this nonlinear system analytically presents a rather demanding task, and therefore, it is suitable to use for its description a fuzzy system (model) built only on basis of its measured input/output data.

Various fuzzy system structures consisting of static fuzzy subsystems and their dynamic parts can be found in the literature. In setting up the structure of the fuzzy model of a continuous line, we used its state description, where the given state of the system and the given input allow us to define the subsequent state, which can be expressed mathematically by the following equation:

$$\begin{aligned} \mathbf{x}\_{k+1} &= \mathbf{x}\_k + \boldsymbol{\Delta} \, \mathbf{x}\_k \\ \boldsymbol{\Delta} \, \mathbf{x}\_k &= \boldsymbol{f}(\boldsymbol{u}\_{k-1'}, \boldsymbol{y}\_{k-1}) \end{aligned} \tag{5}$$

where **u** is the model's input quantities vector, **x** is the state quantities vector, *f* is the searched for static vector function of the controlled system, and *k* is representing the sampling step.

Construction of the CL fuzzy model consists in determining the fuzzy approximation of this function on basis of the obtained CL inputs and outputs database. Considering the choice of CL input, state and output quantities presented in **Figure 10**, the structure of the proposed CL fuzzy model is shown in **Figure 11**.

The whole design of CL optimal control consists of two steps:

Step 1. The design of the fuzzy model for the middle section of the continuous line.

**Figure 11.** Structure of the discrete CL fuzzy model.

The first step in the design of the fuzzy model for the middle section of the continuous line is the establishment of a consistent database from measured inputs and their corresponding outputs, which covers its entire assumed work space and describes the behavior of the modeled system. For establishing a consistent database, we can use, for example, the method of dividing the input range into *n*‐levels and generating *n*(*n* − 1) transient trajectories between them [12, 31, 32], or the method of exciting the system by a random input signal [10, 30] in case it is not possible (e.g., for operational reasons) to apply a pre‐defined input signal at the system's input. Knowledge of the structure or of the parameters of the modeled system is not required in either of these methods. To define suitable sampling time *T* (according to the Shannon‐Kotelnikov theorem) and approximate times for transitions for database measurement, we performed identification measurements on the physical model of the CL with input signals *u*<sup>1</sup> and *u*<sup>2</sup> . Their value and performance are shown in **Figure 12**. Responses of the CL physical model output quantities to the input signals are illustrated in **Figure 13**.

**Figure 12.** Input signals *u*<sup>1</sup> and *u*<sup>2</sup> for identification measurements.

system change depending on the mechanical properties of the material and on the speed of its motion. Defining precise parameters of this nonlinear system analytically presents a rather demanding task, and therefore, it is suitable to use for its description a fuzzy system (model)

Various fuzzy system structures consisting of static fuzzy subsystems and their dynamic parts can be found in the literature. In setting up the structure of the fuzzy model of a continuous line, we used its state description, where the given state of the system and the given input allow us to define the subsequent state, which can be expressed mathematically by the fol-

where **u** is the model's input quantities vector, **x** is the state quantities vector, *f* is the searched for static vector function of the controlled system, and *k* is representing the sampling step.

Construction of the CL fuzzy model consists in determining the fuzzy approximation of this function on basis of the obtained CL inputs and outputs database. Considering the choice of CL input, state and output quantities presented in **Figure 10**, the structure of the proposed CL

The first step in the design of the fuzzy model for the middle section of the continuous line is the establishment of a consistent database from measured inputs and their corresponding outputs, which covers its entire assumed work space and describes the behavior of the modeled system. For establishing a consistent database, we can use, for example, the method of dividing the input range into *n*‐levels and generating *n*(*n* − 1) transient trajectories between them [12, 31, 32], or the method of exciting the system by a random input signal [10, 30] in case it is not possible (e.g., for operational reasons) to apply a pre‐defined input signal at the system's input. Knowledge of the structure or of the parameters of the modeled system is not required in either of these methods. To define suitable sampling time *T* (according to the Shannon‐Kotelnikov theorem) and approximate times for transitions for database

Step 1. The design of the fuzzy model for the middle section of the continuous line.

 + *Δ x<sup>k</sup> <sup>Δ</sup> <sup>x</sup><sup>k</sup>* <sup>=</sup> *<sup>f</sup>*(*u<sup>k</sup>*−<sup>1</sup>

, *yk*−1) (5)

*x<sup>k</sup>*+1 = *x<sup>k</sup>*

built only on basis of its measured input/output data.

386 Modern Fuzzy Control Systems and Its Applications

lowing equation:

fuzzy model is shown in **Figure 11**.

**Figure 11.** Structure of the discrete CL fuzzy model.

The whole design of CL optimal control consists of two steps:

**Figure 13.** Identification response of CL middle section outputs.

The responses of the CL to inputs *u*<sup>1</sup> and *u*<sup>2</sup> (**Figure 13**) show that this system includes a fast tension subsystem and a slow speed subsystem. The range of input values for input *u*<sup>1</sup> (web tension) is assumed within the interval [−1, 1] and the range of input values for input *u*<sup>2</sup> (line speed) within the interval [−4, 4]. The database for CL fuzzy model set up will be generated so that line speed (input *u*<sup>2</sup> ) will increase in steps each 12 s, and each one second the faster (oscillating) part of the system will be excited by input *u*<sup>1</sup> . The plot of input signals for generating the CL fuzzy model database is shown in **Figure 14**; **Figure 15** shows the output quantities corresponding to these inputs.

The database for CL fuzzy model was generated as demonstrated in **Figure 16**. With sampling time *T* = 0.1 s, we obtained a database with 1000 samples.

**Figure 14.** Identification of transitions for CL fuzzy model database generation.

**Figure 15.** CL output variables corresponding identification transitions of inputs.

**Figure 16.** Generating database for CL fuzzy model.

This measured database can be used to search for two FIS structures of the given nonlinear system which best describe the measured relations between [*u*1*<sup>k</sup>*−1, *u*2*<sup>k</sup>*−1, *x*1*<sup>k</sup>*−1, *x*2*<sup>k</sup>*−1, *x*3*k*−1] → d*y*1*<sup>k</sup>* , and [*u*1*<sup>k</sup>*−1, *u*2*<sup>k</sup>*−1, *x*1*<sup>k</sup>*−1, *x*2*<sup>k</sup>*−1, *x*3*k*−1] → d*y*2*<sup>k</sup>* .

Using the measured database, the particular fuzzy model can be designed by standardly known procedures of cluster analysis and adaptive approaches to improve the quality of modeling and reduce development time. The fundamental features of cluster analysis are reduction of the number of fuzzy rules and provision of good initial rule parameters. For our purpose from the large number of methods for adaptive fuzzy networks development [33–36], we chose the adaptive neuro‐fuzzy inference system (ANFIS) with subtractive clustering [14], which is a fast and robust data analysis method, having the following parameters: range of influence = 0.4, squash factor = 1.25, accept ratio = 0.4, reject ratio = 0.01. Subtractive clustering determines the optimal clusters [34] in a multi‐dimensional input/output space that accurately represent the data [34, 37] and CL behavior. The ANFIS approach uses Gaussian functions for fuzzy sets, linear functions for the rule outputs, and Sugeno's inference mechanism [15]. The results were two static Sugeno‐type fuzzy systems with two rules for each output quantity as is shown in **Figure 17**.

**Figure 17.** CL fuzzy model—SUGENO type with 2 rules.

**Figure 15.** CL output variables corresponding identification transitions of inputs.

**Figure 14.** Identification of transitions for CL fuzzy model database generation.

388 Modern Fuzzy Control Systems and Its Applications

**Figure 16.** Generating database for CL fuzzy model.

**Figure 18.** Performance of randomly generated signals *u*<sup>1</sup> and *u*<sup>2</sup> .

The thus obtained fuzzy systems were implemented into the final continuous line fuzzy model structure, as illustrated in **Figure 11**.

To verify the correctness of the CL fuzzy model, randomly generated signals *u*<sup>1</sup> and *u*<sup>2</sup> were applied to its input, as demonstrated in **Figure 18**.

The comparison of the fuzzy model outputs and CL physical model outputs for these inputs is shown in **Figure 19**.

**Figure 19.** Performance of CL fuzzy model outputs *y*<sup>1</sup> and *y*<sup>2</sup> for randomly generated inputs *u*<sup>1</sup> and *u*<sup>2</sup> .

The obtained results confirm that the designed fuzzy model very well approximates the performance of the continuous line also for randomly generated inputs and can be further used for the design of CL control.

Step 2. Design of optimal controller for middle section of continuous line.

The principal aim of CL control consists in achieving good dynamic control of tension in the material, with the speed of material movement being in accord with the pre‐set CL speed. As it has been said above, this is in fact a nonlinear MIMO system an important feature of which is mutual influencing of the individual input and state quantities that can result in bad quality or even in the destruction of the material being processed. This fact makes the controller design methods and their subsequent resulting structures often very complex and presents an obstacle to their wider practical application in industry. Therefore, our aim was to design a simple CL controller that would ensure the desired dynamics in terms of the selected criterion for systems that are only described by input/output relationships, that is, on basis of their fuzzy model.

For control of middle section of CL (for which fuzzy model was designed), we chose the simplest control structure consisting of two standard PI controllers (one for tension control *F*<sup>12</sup> and one for output velocity control *v*<sup>2</sup> ), as illustrated in **Figure 20**.

Processing of material in a CL is usually carried out in operation cycles during which a required amount of prepared material is processed (e.g., a roll of paper, a sheet metal coil.) An operation cycle includes three stages—line start‐up, line running at constant processing speed, and line delayed shut‐off.

**Figure 20.** Continuous line PI controller structure.

**Figure 19.** Performance of CL fuzzy model outputs *y*<sup>1</sup>

for the design of CL control.

their fuzzy model.

and one for output velocity control *v*<sup>2</sup>

model structure, as illustrated in **Figure 11**.

390 Modern Fuzzy Control Systems and Its Applications

is shown in **Figure 19**.

applied to its input, as demonstrated in **Figure 18**.

and *y*<sup>2</sup>

Step 2. Design of optimal controller for middle section of continuous line.

The obtained results confirm that the designed fuzzy model very well approximates the performance of the continuous line also for randomly generated inputs and can be further used

The principal aim of CL control consists in achieving good dynamic control of tension in the material, with the speed of material movement being in accord with the pre‐set CL speed. As it has been said above, this is in fact a nonlinear MIMO system an important feature of which is mutual influencing of the individual input and state quantities that can result in bad quality or even in the destruction of the material being processed. This fact makes the controller design methods and their subsequent resulting structures often very complex and presents an obstacle to their wider practical application in industry. Therefore, our aim was to design a simple CL controller that would ensure the desired dynamics in terms of the selected criterion for systems that are only described by input/output relationships, that is, on basis of

For control of middle section of CL (for which fuzzy model was designed), we chose the simplest control structure consisting of two standard PI controllers (one for tension control *F*<sup>12</sup>

Processing of material in a CL is usually carried out in operation cycles during which a required amount of prepared material is processed (e.g., a roll of paper, a sheet metal coil.)

), as illustrated in **Figure 20**.

The thus obtained fuzzy systems were implemented into the final continuous line fuzzy

The comparison of the fuzzy model outputs and CL physical model outputs for these inputs

To verify the correctness of the CL fuzzy model, randomly generated signals *u*<sup>1</sup>

for randomly generated inputs *u*<sup>1</sup>

 and *u*<sup>2</sup> . and *u*<sup>2</sup>

were

The objective of the optimization is to find such vector **K**[KPF, KIF, KPV, KIV] of the CL controller parameters for which the selected optimization criterion for a given CL operation cycle would be minimum. Most often this criterion is selected in quadratic form according to following equation

$$\text{J(KO)} = \int \left( \mathbb{C}\_1 \, ^\ast e\_1 \, ^\ast + \mathbb{C}\_2 \, ^\ast e\_2 \, ^\ast \right) dt \tag{6}$$

where *e*<sup>1</sup> is the deviation between the desired and real tensile force of CL, *e*<sup>2</sup> is the deviation between the desired and real output velocity of the CL. Coefficients *C*<sup>1</sup> and *C*<sup>2</sup> determine the importance placed on the control errors of the particular outputs. In this case, we chose *C*<sup>1</sup>  = 5 and *C*<sup>2</sup>  = 1, which in terms of physics can be interpreted as larger emphasis put on the quality of regulation of error *e*<sup>1</sup> (tension in the strip of material which primarily determines its final quality). The optimal value of parameters of vector **K** is identified in the space of real values of gain of proportional and integrating elements of the particular controllers.

Let us note that what we are looking for is the extreme of the function of various variables, where the value of the criterial function for the individual vector **K** is determined by simulation on basis of the experimentally constructed CL fuzzy model according to **Figure 21**.

Several procedures can be applied for the purpose of optimization (e.g., genetic algorithm methods, and network charts). Thanks to today's availability of high‐performance computing means, we chose the method of even geometrical division of the parameter space into equal intervals and of systematic searching within the whole range of the space. The advantage of this approach is that we can always identify the global minimum of the function Eq. (6), the disadvantage may be the time and computing demands in case vector **K** has several parameters, and the division of the space is more dense.

**Figure 21.** Simulation diagram for computing the value of criterial function *J*(**K**).

At the start of the optimization process, we determined the initial values of controller parameters *K*PF0 = 1, *K*IF0 = 1, *K*PV0 = 1, *K*IV0 = 10, and we divided the parameter space of individual gains of controllers by increments Δ*K*PF = 1, Δ*K*IF = 5, Δ*K*PV = 1, Δ*K*IV = 5. The maximum values of individual parameters were defined as *K*PFmax = 10, *K*IFmax = 100, *K*PVmax = 10, *K*IVmax = 100—this was based on physically standard values of proportional and integrating elements of PI controllers.

The value of the criterial function for initial values of vector **K** = **K**<sup>0</sup> and for the chosen CL operation cycle is equal to *J*(**K**<sup>0</sup> ) = 37.19. The time responses of CL physical model output quantities corresponding to this initial PI controller parameter setting are illustrated in **Figure 22**.

For finding optimal controller parameters, an m‐file was created in Matlab program environment. At the end of the search process for optimal vector of CL controller parameters, the value of the optimization criterion was *J*(**K**opt) = 0.3906, which corresponds with optimal CL controller parameter values **K**opt [9, 20, 7, 80]. Time responses of CL physical model output quantities for optimally set parameters of its controller are shown in **Figure 23**.

#### **3.1. Discussion**

The proposed controller has been verified by experimental measurements on a real system which presented the physical model of the continuous line (the parameters of the CL physical model are specified in the Appendix). **Figures 22** and **23** illustrate experimental results of the control of the continuous line middle section for selected operational cycle. In industrial practice, the required tension in the strip of material is set the first and then the line starts up to the desired operational speed.

**Figure 22** shows the selected CL operation cycle in which first the desired value of tension in the strip of material is set to 0.8 N and at time 4 s the line starts up to reach the operational

**Figure 22.** Time responses of CL outputs during operation cycle for **K**<sup>0</sup>**.**

this approach is that we can always identify the global minimum of the function Eq. (6), the disadvantage may be the time and computing demands in case vector **K** has several param-

At the start of the optimization process, we determined the initial values of controller parameters *K*PF0 = 1, *K*IF0 = 1, *K*PV0 = 1, *K*IV0 = 10, and we divided the parameter space of individual gains of controllers by increments Δ*K*PF = 1, Δ*K*IF = 5, Δ*K*PV = 1, Δ*K*IV = 5. The maximum values of individual parameters were defined as *K*PFmax = 10, *K*IFmax = 100, *K*PVmax = 10, *K*IVmax = 100—this was based on physically standard values of proportional and integrating elements of PI controllers.

corresponding to this initial PI controller parameter setting are illustrated in **Figure 22**.

quantities for optimally set parameters of its controller are shown in **Figure 23**.

For finding optimal controller parameters, an m‐file was created in Matlab program environment. At the end of the search process for optimal vector of CL controller parameters, the value of the optimization criterion was *J*(**K**opt) = 0.3906, which corresponds with optimal CL controller parameter values **K**opt [9, 20, 7, 80]. Time responses of CL physical model output

The proposed controller has been verified by experimental measurements on a real system which presented the physical model of the continuous line (the parameters of the CL physical model are specified in the Appendix). **Figures 22** and **23** illustrate experimental results of the control of the continuous line middle section for selected operational cycle. In industrial practice, the required tension in the strip of material is set the first and then the line starts up

**Figure 22** shows the selected CL operation cycle in which first the desired value of tension in the strip of material is set to 0.8 N and at time 4 s the line starts up to reach the operational

) = 37.19. The time responses of CL physical model output quantities

and for the chosen CL oper-

The value of the criterial function for initial values of vector **K** = **K**<sup>0</sup>

**Figure 21.** Simulation diagram for computing the value of criterial function *J*(**K**).

ation cycle is equal to *J*(**K**<sup>0</sup>

**3.1. Discussion**

to the desired operational speed.

eters, and the division of the space is more dense.

392 Modern Fuzzy Control Systems and Its Applications

speed. At time 20 s, failure *F*01 = 0.7\*FN occurs at entry of the line middle section (caused e.g., by a change in material thickness—material weld). The Figure shows that at initial values of parameters of CL speed and tension PI controllers, the deviation in tension from the desired value is up to 10%, the speed deviation is up to 25%, and at certain moments, the line also runs in the opposite direction. Autonomy and invariance in terms of failures are poor in dynamic states. On the contrary, when we set optimal values of parameters of CL speed and tension PI controllers, we can see—as illustrated in **Figure 23**—that tension in the line is maintained also in dynamic states within the range of 2% (which ensures high material processing quality during the whole operation cycle), and line speed only briefly falls outside the desired value by approx. 8% at a moment of influence by an external step disturbance. Optimal setting of the CL PI controller parameters therefore ensures good quality dynamics, autonomy and invariance of the controlled system against failures.

For the control of continuous line tension and velocity, a very simple control structure with two PI controllers was designed. We looked for four optimal parameters in the structure, such that would best satisfy the chosen quadratic optimality criterion for the given operation cycle of the line.

The quality of the designed controllers depends on the quality of the constructed fuzzy model which very well approximates the performance of the modeled system and can be further employed in the design of various CL control structures and also in the identification of non‐measurable additive disturbances influencing the system, principally in real time.

The quality of the proposed controller depends on a large extent on a good quality of the nonlinear system fuzzy model which is constructed in the first step of the design procedure The model is constructed only on basis of suitably measured relations between the system's inputs and outputs, without the necessity of preliminary knowledge of its internal structure and parameters. The fuzzy model design is based on the basic idea of dynamic system description in state space.

**Figure 23.** Time responses of CL outputs during operation cycle for **K**opt**.**

The quality of the proposed controller depends on a large extent on a good quality of the nonlinear system fuzzy model which is constructed in the first step of the design procedure (see Section 4). The model is constructed only on basis of suitably measured relations between the system's inputs and outputs, without the necessity of preliminary knowledge of its internal structure and parameters. The fuzzy model design is based on the basic idea of dynamic system description in state space.

With this method, no principal limitations for the investigated system's nonlinearities are defined, and therefore, there is good reason to assume that the presented method will find wide use in multi‐motor drives in steel industry, paper‐making, printing and textile industries, in the production of synthetic fibers and foils in the chemical industry and in other industries.

### **Acknowledgements**

The authors wish to thank the project VEGA 1/0464/15 for its support.

## **Appendix**

#### AC drive parameters:


Stator phase resistance: *r*<sup>1</sup>  = 0.267 Ω, rotor phase resistance: *r*<sup>2</sup>  = 0.54 Ω

Main inductance: *L*h = 96 mH, leakage inductance: *L*S1 = *L*S2 = 2.75 mH

Slip angular speed: *ω*<sup>2</sup>  = *ω*<sup>1</sup> –*ω*<sup>g</sup> , *ω*<sup>0</sup>  = *K*11 (*R*<sup>1</sup>  + (*M*<sup>2</sup> /*L*2 )·*ω*<sup>g</sup> ) = 143.33 s−1, *ω*<sup>g</sup>  = 5.46 s−1

Mechanical angular speed of the motor:*ω*m, angular frequency of the stator voltage: *ω*<sup>1</sup>

Number of pole pairs: *n*p = 2

Parameters of the CL physical model:

DC motors:


Converters: *T*TM = 0.1 ms

Current sensor: *K*<sup>I</sup>  = 2V/A, velocity sensor: *K*<sup>v</sup>  = 6.6 V/m s−1, tension sensor: *K*<sup>F</sup>  = 0.022 V/N Working rolls: *r* = 0.04 m, *v*max = 1.5 m s−1.

## **Author details**

Pavol Fedor and Daniela Perduková\*

\*Address all correspondence to: daniela.perdukova@tuke.sk

Department of Electrical Engineering and Mechatronics, Technical University of Kosice, Kosice, Slovak Republic

## **References**

The quality of the proposed controller depends on a large extent on a good quality of the nonlinear system fuzzy model which is constructed in the first step of the design procedure (see Section 4). The model is constructed only on basis of suitably measured relations between the system's inputs and outputs, without the necessity of preliminary knowledge of its internal structure and parameters. The fuzzy model design is based on the basic idea of dynamic sys-

With this method, no principal limitations for the investigated system's nonlinearities are defined, and therefore, there is good reason to assume that the presented method will find wide use in multi‐motor drives in steel industry, paper‐making, printing and textile industries, in the production of synthetic fibers and foils in the chemical industry and in other industries.

The authors wish to thank the project VEGA 1/0464/15 for its support.

*M*N = 98.8 Nm *J* = 0.11 kgm<sup>2</sup> *M* = 0*,*064 H *R*<sup>1</sup>

 = 0.267 Ω, rotor phase resistance: *r*<sup>2</sup>

Mechanical angular speed of the motor:*ω*m, angular frequency of the stator voltage: *ω*<sup>1</sup>

 + (*M*<sup>2</sup> /*L*2 )·*ω*<sup>g</sup>

Main inductance: *L*h = 96 mH, leakage inductance: *L*S1 = *L*S2 = 2.75 mH

 = *ω*<sup>1</sup> –*ω*<sup>g</sup> , *ω*<sup>0</sup>  = *K*11 (*R*<sup>1</sup>

*P*N = 15 kW *U*1N = 220 V *I*1N = 29.5 A *n*N = 1450 rev/min

 = 0.36 Ω *K*11 = 277.08 H–1 *K*12 = –269 H–1 *K*12 = –269 H–1

 = 0.54 Ω

 = 5.46 s−1

) = 143.33 s−1, *ω*<sup>g</sup>

 = 0.178 Ω

**Figure 23.** Time responses of CL outputs during operation cycle for **K**opt**.**

tem description in state space.

394 Modern Fuzzy Control Systems and Its Applications

**Acknowledgements**

**Appendix**

*R*2

AC drive parameters:

Stator phase resistance: *r*<sup>1</sup>

Number of pole pairs: *n*p = 2 Parameters of the CL physical model:

Slip angular speed: *ω*<sup>2</sup>

DC motors:


[24] Tang K.S., Man K.F., Chen G., Kwong S. An optimal fuzzy PID controller. IEEE Transactions Industrial Electronics. 2001;**48**(4):757‐765.

[9] Fedor P., Perduková D. A simple fuzzy controller structure. Acta Electrotechnica et

[11] Fedor P., Perduková D. Fuzzy model of a nonlinear mechatronic system. Procedia

[12] Feng G. A survey on analysis and design of model‐based fuzzy control systems. IEEE

[13] Qi P., Liu Ch., Ataka A., Lam H.K., Althoefer K. Kinematic control of continuum manipulators using a fuzzy‐model‐based approach. IEEE Transactions on Industrial Electronics.

[14] Kiriakidis K. Fuzzy model‐based control of complex systems. IEEE Transactions on

[15] Wu Ch., Wang J., Li H., Liang H. Fuzzy‐model‐based control for nonlinear networked systems with random packet losses. In: 5th International Conference on Information Science and Technology (ICIST); 2015. pp. 455‐460. doi:10.1109/ICIST.2015.7289015 [16] Fedor P., Perduková D. Model based fuzzy control applied to a real nonlinear mechanical system. Iranian Journal of Science and Technology, Transactions of Mechanical

[17] Li B., Fan X., Zhang D., Jiang G. Modeling and fuzzy sliding decoupling control of looper multivariable system. In: Chinese Control and Decision Conference (CCDC); 2016. pp.

[18] Sjoberg J., Zhang Q., Ljung L., Benveniste A., Delyon B., Glorennec P., Hjalmarsson H., Juditsky A. Nonlinear black‐box modeling in system identification: a unified overview.

[19] Tečec Z., Petrović I., Matuško J. A Takagi‐Sugeno fuzzy model of synchronous generator unit for power system stability application. AUTOMATIKA: Journal for Control,

[20] Fedor P., Perduková D. Fuzzy model for middle section of continuous line. International

[21] Liu X., Xioung Z., Chen L., Zhu Z. A new Takagi‐Sugeno fuzzy approach of process modeling and fault detection. In: 35th Chinese Control Conference – CCC; 2016. pp.

[22] Johansen T.A. Fuzzy model based control: Stability robustness and performance issues.

[23] Girovský P. Fuzzy control of synchronous motor with permanent magnet. Acta

Measurement, Electronics. Computing and Communications. 2010;**51**(2):12‐137.

[10] Babuška R. Fuzzy modeling for control. Boston: MA: Kluwer; 1998.

Transactions on Fuzzy Systems. 2006;**14**(5):676‐697.

Informatica. 2005;**5**(4):53‐56.

396 Modern Fuzzy Control Systems and Its Applications

Engineering. 2014;**96**:91-100.

2016;**63**(9):5022‐5035.

Fuzzy Systems. 1998;**6**(4):517‐529.

Engineering. 2016;**40**(2):113‐124.

Automatica. 1995;**31**:1691‐1724.

2467‐2472. doi:10.1109/CCDC.2016.7531400

7126‐7130. doi:10.1109/ChiCC.2016.7554483

Electrotechnica et Informatica. 2016;**16**(4):17‐20.

Journal of Engineering Research in Africa. 2015;**18**:75‐84.

IEEE Transactions on Fuzzy Systems. 1994;**2**(1):221‐233.


## **Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller**

Hidehiro Ikeda

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68319

#### Abstract

Vibration suppression control of the mechanical system is a very important technology for realizing high precision, high speed response and energy saving. In general, the mechanical system is modeled with a multi-mass resonance system, and vibration suppression control is applied. This chapter presents a novel controller design method for the speed control system to suppress the resonance vibration of two-mass resonance system and three-mass resonance system. The target systems are constructed by a motor, finite rigid shafts, and loads. The control system consists of a speed fuzzy controller and a proportional-integral (PI) current controller to realize precise speed and torque response. In order to implement the experimental system, the system is treated as the digital control. This chapter also utilizes a differential evolution (DE) to determine five optimal controller parameters (three scaling factors of the fuzzy controller and two controller gains of PI current controller. Finally, this chapter verified the effectiveness to suppress the resonance vibrations and the robustness of the proposed method by the computer simulations and the experiments by using the test experimental setup.

Keywords: multi-mass resonance system, vibration suppression control, fuzzy controller, differential evolution

## 1. Introduction

Recently, motor drive system, which consists of several motors, shafts, gears, and loads, is widely utilized in industrial fields. These mechanical systems are made a request the highspeed response, weight reduction, miniaturization, and high precision requirements for various industrial applications.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Hence, in industrial field, the system is treated as a multi-mass resonance system, which consists of several inertial moments, torsional shafts, and gear coupling. The first-order approximation model of multi-mass resonances model is two-mass resonance model. For instance, several control methods, which are PID control (Proportional plus Integral plus Derivative Control) with a resonance ratio control using the disturbance observer, coefficient diagram method (CDM), full state feedback control with the state observer, the pole placement method, fractional order PIDk control, and H<sup>∞</sup> control method, are effective to control for twomass resonance system [1–3]. Ikeda et al. [4] have explained the effectiveness of the controller design technique using the pole placement method for the two-mass position control system.

However, the resonance system is required more high precision and high response speed control in recent years. Therefore, it is necessary to deal with a higher order model of the resonance system. For instance, the drive train of the electric vehicle is constructed the fourmass system. Likewise, the ball screw drive stage is typically four-mass system. The thermal power generation system composed of multiple turbines and generators is modeled as twelvemass resonance system. Thus, several vibration suppression control methods on three-mass resonance system or more have been proposed [5, 6]. Here, modified-IPD speed controller using Taguchi Method has been proposed in Refs. [7, 8].

Meanwhile, the state equations of the controlled object and its parameters are required to design the control systems. Refs. [9, 10] previously proposed a controller gain tuning method for a vibration suppression-type speed controller using fictitious reference iterative tuning (FRIT) for single-input multi-variable control objects without knowledge of the system state equations and the parameters.

In contrast, a fuzzy control system can be assumed as one method for solving these problems. A fuzzy control system using a fuzzy inference is the embodiment of non-mathematical control algorithm, which is constructed by experience and intuition. Several applications brought in the fuzzy control system to motor drive system [11–14].

This chapter proposes a vibration suppression controller by using a fuzzy inference. The control system consists of a speed fuzzy controller and a proportional-integral (PI) current controller to realize precise speed and torque response on two or three inertial resonance system. In the control system, only motor side state variables are utilized for controlling the resonance system. Additionally, this chapter treats with the proposed control system as the digital control system. Here, the proposed control system is new system that I improved to apply the control system which I already proposed for simulation model in Refs. [13, 14] to experimental actual equipment.

The fuzzy controller has three scaling factors, and the PI current controller has two controller gains. In this chapter, a differential evolution algorithm (DE) is utilized the determination of these five controller parameters [13–18]. DE, which was proposed by Price and Storn, is one of the evolutionary optimization strategies. By using DE, it is easy and fast to determine the proper controller parameters.

Lastly, the validity of the controller design, the robustness, and the control effectiveness of the proposed method was verified using the simulations and the experiments by using the test experimental set up.

### 2. Multi-mass vibration suppression control system

#### 2.1. 2-mass model

Hence, in industrial field, the system is treated as a multi-mass resonance system, which consists of several inertial moments, torsional shafts, and gear coupling. The first-order approximation model of multi-mass resonances model is two-mass resonance model. For instance, several control methods, which are PID control (Proportional plus Integral plus Derivative Control) with a resonance ratio control using the disturbance observer, coefficient diagram method (CDM), full state feedback control with the state observer, the pole placement method, fractional order PIDk control, and H<sup>∞</sup> control method, are effective to control for twomass resonance system [1–3]. Ikeda et al. [4] have explained the effectiveness of the controller design technique using the pole placement method for the two-mass position control system. However, the resonance system is required more high precision and high response speed control in recent years. Therefore, it is necessary to deal with a higher order model of the resonance system. For instance, the drive train of the electric vehicle is constructed the fourmass system. Likewise, the ball screw drive stage is typically four-mass system. The thermal power generation system composed of multiple turbines and generators is modeled as twelvemass resonance system. Thus, several vibration suppression control methods on three-mass resonance system or more have been proposed [5, 6]. Here, modified-IPD speed controller

Meanwhile, the state equations of the controlled object and its parameters are required to design the control systems. Refs. [9, 10] previously proposed a controller gain tuning method for a vibration suppression-type speed controller using fictitious reference iterative tuning (FRIT) for single-input multi-variable control objects without knowledge of the system state

In contrast, a fuzzy control system can be assumed as one method for solving these problems. A fuzzy control system using a fuzzy inference is the embodiment of non-mathematical control algorithm, which is constructed by experience and intuition. Several applications

This chapter proposes a vibration suppression controller by using a fuzzy inference. The control system consists of a speed fuzzy controller and a proportional-integral (PI) current controller to realize precise speed and torque response on two or three inertial resonance system. In the control system, only motor side state variables are utilized for controlling the resonance system. Additionally, this chapter treats with the proposed control system as the digital control system. Here, the proposed control system is new system that I improved to apply the control system which I already proposed for simulation model in Refs. [13, 14] to

The fuzzy controller has three scaling factors, and the PI current controller has two controller gains. In this chapter, a differential evolution algorithm (DE) is utilized the determination of these five controller parameters [13–18]. DE, which was proposed by Price and Storn, is one of the evolutionary optimization strategies. By using DE, it is easy and fast to determine the

Lastly, the validity of the controller design, the robustness, and the control effectiveness of the proposed method was verified using the simulations and the experiments by using the test

using Taguchi Method has been proposed in Refs. [7, 8].

brought in the fuzzy control system to motor drive system [11–14].

equations and the parameters.

400 Modern Fuzzy Control Systems and Its Applications

experimental actual equipment.

proper controller parameters.

experimental set up.

Figure 1 shows the two-mass resonance model. The model is configured of two rigid inertial masses with a torsional shaft, where ωM, Tdis, ωL, Tin, JM, JL, Ks, and TL denote the motor angular speed, the torsional torque, the load angular speed, the input torque, the inertia of motor, the inertia of load, the shaft torsional stiffness, and the load torque, respectively.

If all the state variables can be observed by several sensors and all the system parameters are known or identified, it is easy to construct the optimal control system. However, in general, it is difficult to measure the state variables of the load side due to constraints on scarce measurement environment and sensor installation location. Therefore, in this chapter, we use only the motor side variables. Furthermore, we contemplate for the current minor control in order to compensate torque response. Eq. (1) shows the continuous state equation of two-mass resonance model, where the viscous friction is not considered.

$$
\frac{d}{dt} \begin{pmatrix} \omega\_M \\ \omega\_L \\ T\_{\text{dis}} \end{pmatrix} = \begin{pmatrix} 0 & 0 & -\frac{1}{f\_M} \\ 0 & 0 & \frac{1}{f\_L} \\ 0 & 0 & \frac{1}{f\_L} \\ K\_s & -K\_s & 0 \end{pmatrix} \begin{pmatrix} \omega\_M \\ \omega\_L \\ T\_{\text{dis}} \end{pmatrix} + \begin{pmatrix} \frac{1}{f\_M} \\ 0 \\ 0 \\ 0 \end{pmatrix} T\_{in} + \begin{pmatrix} 0 \\ -\frac{1}{f\_L} \\ 0 \end{pmatrix} T\_L \tag{1}
$$

Eq. (2) shows the transfer function of two-mass model, which input signal is Tin and output signal is ωM.

$$\frac{\omega\_M}{T\_{in}} = \frac{\mathbf{s}^2 + \omega\_a^2}{f\_M \mathbf{s} (\mathbf{s}^2 + \omega\_r^2)}\tag{2}$$

where ω<sup>r</sup> is a resonance frequency and ω<sup>a</sup> is an anti-resonance frequency. Here, we use the DC servo motor as the driving motor. Eq. (3) is the voltage equation of dc servo motor, where Ra is the armature resistance, La is the armature inductance, Ke is the back-emf constant, and K<sup>0</sup> is the converter gains of the DC power supply. Input torque is calculated by Tin = Ktia, where Kt is the torque constant.

$$L\_a \frac{d\dot{\mathbf{i}}\_a}{dt} + R\_a \dot{\mathbf{i}}\_a = K\_0 \mu\_c - \omega\_M \tag{3}$$

Figure 2 is indicative of the block diagram of the two-mass resonance system.

Figure 1. 2-mass model.

Figure 2. Block diagram of two-mass resonance model.

The inertia ratio R of two-mass model is given by Eq. (4), where JMn and JLn represent the nominal values of the motor and load inertias, respectively.

$$R = \frac{I\_{Ln}}{I\_{Mn}} \tag{4}$$

#### 2.2. Three-mass model

Similarly to two-mass resonance model, Figure 3 reveals the three-mass model. The model consists of three rigid inertias and two shafts. Here, Jc and JL are the load 1 inertia moment and the load 2 inertia moment, respectively. Furthermore, ωc, ωL, Tdis1, Tdis2, Ks1, and Ks<sup>2</sup> denote load 1 angular speed, load 2 angular speed, shaft 1 torsional torque, shaft 2 torsional torque, the shaft 1 stiffness, and the shaft 2 stiffness, respectively.

$$
\frac{d}{dt} \begin{pmatrix} \omega\_{\mathcal{M}} \\ \omega\_{\mathcal{M}} \\ \omega\_{\mathcal{M}} \\ T\_{\mathrm{dis}} \\ T\_{\mathrm{dis}} \end{pmatrix} \begin{pmatrix} \omega\_{\mathcal{M}} \\ \omega\_{\mathcal{M}} \\ T\_{\mathrm{dis}} \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & -\frac{1}{f\_{\mathcal{M}}} & 0 \\ 0 & 0 & 0 & \frac{1}{f\_{\mathcal{L}}} & -\frac{1}{f\_{\mathcal{L}}} \\ 0 & 0 & 0 & 0 & \frac{1}{f\_{\mathcal{L}}} \\ 0 & 0 & 0 & 0 & \frac{1}{f\_{\mathcal{L}}} \\ K\_{\mathrm{s1}} & -K\_{\mathrm{s1}} & 0 & 0 & 0 \\ 0 & K\_{\mathrm{s2}} & -K\_{\mathrm{s2}} & 0 & 0 \end{pmatrix} \begin{pmatrix} \omega\_{\mathcal{M}} \\ \omega\_{\mathcal{M}} \\ 0 \\ T\_{\mathrm{dis}} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \\ 0 \\ -\frac{1}{f\_{\mathcal{L}}} \\ 0 \\ 0 \end{pmatrix} T\_{\mathcal{L}} \tag{5}
$$

Figure 3. Three-mass model.

The state equation of three-mass resonance model is shown in Eq. (5). Then, Eq. (6) shows the continuous transfer function of three-mass resonance model, which input signal is Tin and output signal is ωM.

$$\frac{\omega\_M}{T\_{in}} = \frac{(\mathbf{s}^2 + \omega\_{d1}^2)(\mathbf{s}^2 + \omega\_{d2}^2)}{f\_M s (\mathbf{s}^2 + \omega\_{r1}^2)(\mathbf{s}^2 + \omega\_{r2}^2)}\tag{6}$$

In this equation, ω indicates the angular frequency, where ωr1, ωr2, ωa1, and ωa<sup>2</sup> are the resonance frequencies, and anti-resonance frequency, respectively. Then, the block diagram realized by using above equations is shown in Figure 4.

#### 2.3. Experimental set up

ð4Þ

The inertia ratio R of two-mass model is given by Eq. (4), where JMn and JLn represent the

<sup>R</sup> <sup>¼</sup> JLn JMn

Similarly to two-mass resonance model, Figure 3 reveals the three-mass model. The model consists of three rigid inertias and two shafts. Here, Jc and JL are the load 1 inertia moment and the load 2 inertia moment, respectively. Furthermore, ωc, ωL, Tdis1, Tdis2, Ks1, and Ks<sup>2</sup> denote load 1 angular speed, load 2 angular speed, shaft 1 torsional torque, shaft 2 torsional torque, the shaft 1

JM

Jc

0

1

0

BBBBB@

ω<sup>M</sup> ω<sup>M</sup> ω<sup>M</sup> Tdis Tdis 1

1

1

CCCCCCA TL

ð5Þ

0

BBBBBB@

CCCCCCCA Tin þ

0

BBBBBBB@

CCCCCA þ

CCCCCCCCCCCA

� 1 Jc

JL

00 0 � <sup>1</sup>

00 0 <sup>1</sup>

00 0 0 <sup>1</sup>

Ks<sup>1</sup> �Ks<sup>1</sup> 0 00 0 Ks<sup>2</sup> �Ks<sup>2</sup> 0 0

nominal values of the motor and load inertias, respectively.

stiffness, and the shaft 2 stiffness, respectively.

Figure 2. Block diagram of two-mass resonance model.

402 Modern Fuzzy Control Systems and Its Applications

0

BBBBBBBBBBB@

2.2. Three-mass model

d dt ω<sup>M</sup> ω<sup>M</sup> ω<sup>M</sup> Tdis Tdis 1

0 @

CCCCCA

Figure 3. Three-mass model.

ω<sup>M</sup> ω<sup>L</sup> Tdis 1 A ¼

0

BBBBB@

This chapter confirms the effectiveness and performance of the proposed method by experiments using the experimental equipment.

Figure 5 is the appearance of the experimental system constructed in this research. The twomass resonance system is simulated by utilizing the dc servo motor and the dc generator with a finite rigid coupling. The controller is realized on a digital signal processor, which calculates the PWM signal to a four-quadrant dc chopper.

The DSP board (PE-PRO/F28335 Starter Kit, Myway Plus Corp.) consists of the DSP (TMS320F28335PGFA), a digital input/output (I/O), ABZ counters for encoder signals, analogto-digital (AD) converters and digital-to-analog (DA) converters [19]. The motor and load angles and angular speeds are detected using 5000 pulses-per-revolution encoders. The current of dc servo motor is measured by the current sensor and AD converter.

Figure 4. Block diagram of three-mass model.

Figure 5. Experimental apparatus.

The control frequency and the detection frequency of the encoder are both 1 ms, and the detection period for the current is 10 μsec. The design language used was C. Then, while considering the application of the system to specific apparatus, we constructed a digital control system that contains a discrete controller. In addition, we used MATLAB/Simulink software for the proposed off-line tuning process based on simulation and constructed the fuzzy control system as a continuous system [20]. The disturbance is added to the dc generator as the torque by using the electric load device on constant current mode. Figures 6 and 7 show the apparatus of the two-mass model and three-mass model used in the experimental set up, respectively. Figure 8 shows the experimental system configuration. For reference, the nominal parameters

Figure 6. Photograph of two-mass resonance model.

Figure 7. Photograph of three-mass resonance model.

Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller http://dx.doi.org/10.5772/intechopen.68319 405

Figure 8. Configuration of experimental system (two-mass resonance model).

The control frequency and the detection frequency of the encoder are both 1 ms, and the detection period for the current is 10 μsec. The design language used was C. Then, while considering the application of the system to specific apparatus, we constructed a digital control system that contains a discrete controller. In addition, we used MATLAB/Simulink software for the proposed off-line tuning process based on simulation and constructed the fuzzy control system as a continuous system [20]. The disturbance is added to the dc generator as the torque by using the electric load device on constant current mode. Figures 6 and 7 show the apparatus of the two-mass model and three-mass model used in the experimental set up, respectively. Figure 8 shows the experimental system configuration. For reference, the nominal parameters

Figure 6. Photograph of two-mass resonance model.

Figure 7. Photograph of three-mass resonance model.

Figure 5. Experimental apparatus.

404 Modern Fuzzy Control Systems and Its Applications

of the experimental two-mass model and three-mass model are given in Tables 1 and 2, respectively.

Figure 9 shows an example of experimental result using two-mass model. These step waves are the motor and load angular speeds with direct current voltage input. Similarly, Figure 10 shows an example of experimental result using three-mass model, which are the motor and


Table 1. Nominal parameters of two-mass experimental model.


Table 2. Nominal parameters of three-mass experimental model.

Figure 9. Angular speeds (ω<sup>M</sup> and ωL) of the step responses to a DC voltage input (two-mass model).

Figure 10. Angular speeds (ω<sup>M</sup> and ωL) of the step responses to a DC voltage input (three-mass model).

load angular speeds with same above condition. In these figures, the resonance vibrations can be observed. The purpose of this research is to suppress these resonance vibrations.

#### 3. Proposed fuzzy control system

#### 3.1. Fuzzy speed controller

Fuzzy controller, which is executed by the fuzzy set and the fuzzy inference, can control for nonlinear systems or uncertain model. Figure 11 indicates the proposed fuzzy speed controller in this chapter. The speed controller is based on fuzzy control. The current controller is typical PI controller. Furthermore, the load side state variables are not utilized for control, where S1, S2, and S<sup>3</sup> are the parameters to determine the scale of the membership function, which are called scaling factors or scaling coefficient.Kpc and Kic are the current PI controller gains. Eq. (7) shows the transfer function of current PI controller. Additionally, this chapter uses the discrete control system.

$$
\mu\_c(k) = \left(K\_{\text{pc}} + \frac{1}{s} K\_{\text{ic}}\right) e(k) \tag{7}
$$

Figure 12 is indicative of the membership function for the premise variables. This membership function is a shape of triangle with a dense center. Figure 13 indicates the membership Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller http://dx.doi.org/10.5772/intechopen.68319 407

Figure 11. Block diagram of the proposed control system.

Figure 12. Membership functions of the antecedence.

load angular speeds with same above condition. In these figures, the resonance vibrations can

Fuzzy controller, which is executed by the fuzzy set and the fuzzy inference, can control for nonlinear systems or uncertain model. Figure 11 indicates the proposed fuzzy speed controller in this chapter. The speed controller is based on fuzzy control. The current controller is typical PI controller. Furthermore, the load side state variables are not utilized for control, where S1, S2, and S<sup>3</sup> are the parameters to determine the scale of the membership function, which are called scaling factors or scaling coefficient.Kpc and Kic are the current PI controller gains. Eq. (7) shows the transfer

> 1 s Kic

Figure 12 is indicative of the membership function for the premise variables. This membership function is a shape of triangle with a dense center. Figure 13 indicates the membership

eðkÞ ð7Þ

function of current PI controller. Additionally, this chapter uses the discrete control system.

ucðkÞ ¼ Kpc þ

be observed. The purpose of this research is to suppress these resonance vibrations.

Figure 10. Angular speeds (ω<sup>M</sup> and ωL) of the step responses to a DC voltage input (three-mass model).

Figure 9. Angular speeds (ω<sup>M</sup> and ωL) of the step responses to a DC voltage input (two-mass model).

3. Proposed fuzzy control system

406 Modern Fuzzy Control Systems and Its Applications

3.1. Fuzzy speed controller

Figure 13. Membership functions of the consequence.

function, which is formed uniformly triangle for the consequent variable. Here, the s denotes the scaling factor. PB, PM, PS, ZE, NS, NM, and NB are the linguistic variables of the fuzzy control where, PB indicates positive big, PM indicates positive medium, PS indicates positive small, ZE indicates zero, NS indicates negative small, NM indicates negative medium, and NB indicates negative big, respectively. The premise variables are eωM(k) and ΔeωM(k).

$$
\sigma\_{\omega \mathbf{M}}(\mathbf{k}) = \omega\_{\rm ref} - \omega\_{\rm M}(\mathbf{k}) \tag{8}
$$

$$
\Delta \varepsilon\_{aM}(k) = \varepsilon\_{aM}(k) - \varepsilon\_{aM}(k-1) \tag{9}
$$

Then, the consequence variable is the variation width of the current input Δiref(k). Therefore, the proposed fuzzy controller is nearly same as the proportional-derivative (PD) type controller.

Figure 14 is indicative of the fuzzy rule table. The rule is included the rising correction of the angular speed response.


Figure 14. Control rule table.

#### 3.2. Design method of controller parameters by differential evolution

In this chapter, five parameters (S1, S2, S3, Kpc and Kic) of the proposed controller have to be designed. However, it is difficult to determine them by trial and error or some. Therefore, this chapter proposes the differential evolution (DE) to search the optimal controller parameters. Here, DE is one of evolutionary optimized solution search methods. DE is the optimization method-based multi-point search method. In particular, basic GA expresses parameter by binary coding, whereas DE uses the parameters by real variable vector. The DE design is conducted by the initial population, the mutation, the crossover, and the selection. The design flow of DE is shown in Figure 15. In this chapter, DE/rand/1/bin design strategy is used for the determination of five controller parameters.

where D is the number of design parameter vectors, NP is the number of members in each population. Each parameter vector is represented by the parameter vector (target vector) xi,G, where G denotes one generation. The mutation vector vi,G is calculated by Eq. (10). From this equation, F indicates the step width (scaling factor) of DE design, and CR indicates of the crossover rate, where r1, r2, and r<sup>3</sup> are different values.

$$\mathbf{v}\_{i,\mathbf{G}+1} = \mathbf{x}\_{r1,\mathbf{G}} + F(\mathbf{x}\_{r1,\mathbf{G}} - \mathbf{x}\_{r3,\mathbf{G}}), \quad r\_1 \neq r\_2 \neq r\_3 \neq i \tag{10}$$

$$\mathfrak{u}\_{\mathfrak{j},\mathbb{G}+1} = \begin{cases} \mathfrak{v}\_{\mathfrak{j},\mathbb{G}+1} & \text{rand} \le \text{CR or } j = \text{ST} \\ \mathfrak{x}\_{\mathfrak{j},\mathbb{G}} & \text{rand} > \text{CR or } j \ne \text{ST} \end{cases} \tag{11}$$

Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller http://dx.doi.org/10.5772/intechopen.68319 409

Figure 15. Flow of DE algorithm.

Then, the consequence variable is the variation width of the current input Δiref(k). Therefore, the proposed fuzzy controller is nearly same as the proportional-derivative (PD) type controller. Figure 14 is indicative of the fuzzy rule table. The rule is included the rising correction of the

3.2. Design method of controller parameters by differential evolution

determination of five controller parameters.

crossover rate, where r1, r2, and r<sup>3</sup> are different values.

In this chapter, five parameters (S1, S2, S3, Kpc and Kic) of the proposed controller have to be designed. However, it is difficult to determine them by trial and error or some. Therefore, this chapter proposes the differential evolution (DE) to search the optimal controller parameters. Here, DE is one of evolutionary optimized solution search methods. DE is the optimization method-based multi-point search method. In particular, basic GA expresses parameter by binary coding, whereas DE uses the parameters by real variable vector. The DE design is conducted by the initial population, the mutation, the crossover, and the selection. The design flow of DE is shown in Figure 15. In this chapter, DE/rand/1/bin design strategy is used for the

where D is the number of design parameter vectors, NP is the number of members in each population. Each parameter vector is represented by the parameter vector (target vector) xi,G, where G denotes one generation. The mutation vector vi,G is calculated by Eq. (10). From this equation, F indicates the step width (scaling factor) of DE design, and CR indicates of the

<sup>u</sup>j,Gþ<sup>1</sup> <sup>¼</sup> <sup>v</sup>j,Gþ<sup>1</sup> rand <sup>≤</sup>CR or <sup>j</sup> <sup>¼</sup> ST

xj,G rand > CR or j 6¼ ST

vi,Gþ<sup>1</sup> ¼ xr1,G þ Fðxr1,G � xr3,GÞ, r<sup>1</sup> 6¼ r<sup>2</sup> 6¼ r<sup>3</sup> 6¼ i ð10Þ

ð11Þ

angular speed response.

408 Modern Fuzzy Control Systems and Its Applications

Figure 14. Control rule table.

In Eq. (11), uj,G+1 is the vector of trial parameter, the rand is random value, and ST indicates the start point. The selection is utilized next algorithm,

$$\mathbf{x}\_{i,\mathcal{G}+1} = \begin{cases} \mathbf{u}\_{i,\mathcal{G}+1} & \text{if } y(\mathbf{u}\_{i,\mathcal{G}+1}) > y(\mathbf{x}\_{i,\mathcal{G}}) \text{ for maximization problems} \\ \mathbf{x}\_{i,\mathcal{G}} & \text{otherwise} \end{cases} \tag{12}$$

As previously described, the proposed method uses five control parameters (S1, S2, S3, Kpc, and Kic). The population size is 2000, the order of each vector is 20, and the coefficient of membership function F is 0.5. Moreover, the rate of crossover CR is 0.9. Then, the performance index function is shown in Eq. (13). Meanwhile, this chapter utilizes the inverse of y as a fitness function.

$$y = \int\_0^\infty t \sqrt{\left(\omega\_{ref} - \omega\_{\mathcal{L}}\right)^2} dt\tag{13}$$

## 4. Simulation and experimental results

#### 4.1. Verification results of computer simulation

Next, the simulation results of the proposed method are demonstrated by computer simulation.

Table 3 shows the results of design parameter using the proposed method for two-mass model. Figure 16 is indicative of the transition of the maximum fitness function. In this simulation design, the step response and the disturbance response have been evaluated. Furthermore, the inertia ratio R is 1.07, and the stiffness of shaft Ksn has been set to 18.5 Nm/rad in the simulation design.


Table 3. Results of design parameter calculated by DE.

Figure 16. Convergence of index function y.

Figures 17 and 18 show the step responses that were obtained for the motor and load angular speeds, and armature current when using the proposed method. In this chapter, ωref is 30 rad/s, the DC voltage input is 25 V, and the disturbance input TL is changed from 0 to 20% at t = 0.3 s. As shown by these figures, good waves are observed for the reference-following, vibration

Figure 17. Simulation results ω<sup>M</sup> and ω<sup>L</sup> (two-mass, R = 1.07, Ksn = 18.5 Nm/rad).

Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller http://dx.doi.org/10.5772/intechopen.68319 411

Figure 18. Simulation results ia (two-mass, R = 1.07, Ksn = 18.5 Nm/rad).

suppression, and the disturbance performance. Figure 19 is indicative of the search process of the S<sup>1</sup> vector. Similarly, Figures 20–23 show the transition of the S<sup>2</sup> vector, S<sup>3</sup> vector, Kpc vector and Kic vector, respectively. In particular, from Figure 23 and Table 3, Kic is 1.0 · 10<sup>6</sup> of the design limitation value. Therefore, integral gain of the current PI controller can be omitted for this control object.

Figure 19. Transition of scaling factor S1.

4. Simulation and experimental results

410 Modern Fuzzy Control Systems and Its Applications

4.1. Verification results of computer simulation

Table 3. Results of design parameter calculated by DE.

0

Figure 17. Simulation results ω<sup>M</sup> and ω<sup>L</sup> (two-mass, R = 1.07, Ksn = 18.5 Nm/rad).

0.01

Evaluation

Figure 16. Convergence of index function y.

0.02

Next, the simulation results of the proposed method are demonstrated by computer simulation. Table 3 shows the results of design parameter using the proposed method for two-mass model. Figure 16 is indicative of the transition of the maximum fitness function. In this simulation design, the step response and the disturbance response have been evaluated. Furthermore, the inertia ratio R is 1.07, and the stiffness of shaft Ksn has been set to 18.5 Nm/rad in the simulation design.

S<sup>1</sup> S<sup>2</sup> S<sup>3</sup> Kpc Kic 8.486 0.4802 0.4001 4.678 1.0 · 10<sup>6</sup>

Figures 17 and 18 show the step responses that were obtained for the motor and load angular speeds, and armature current when using the proposed method. In this chapter, ωref is 30 rad/s, the DC voltage input is 25 V, and the disturbance input TL is changed from 0 to 20% at t = 0.3 s. As shown by these figures, good waves are observed for the reference-following, vibration

0 100 200 300 400 500

iteration

Figure 20. Transition of scaling factor S2.

Figure 21. Transition of scaling factor S3.

Figure 22. Transition of current proportional gain Kpc.

Figure 23. Transition of current integral gain Kic.

#### 4.2. Experimental results

#### 4.2.1. 2-mass model

Next, the experimental results by using the proposed method are illustrated in this section. Figures 24 and 25 show the experimental results of two-mass model using the proposed method, where the condition (R = 1.07, Ksn = 18.5 Nm/rad) is same as the above simulation results shown in Figures 17 and 18. From these figures, it is observed that the resonance vibrations between the motor and the load angular speed (ω<sup>M</sup> and ωL) have been suppressed very well. Furthermore, after inputting disturbance, it can be seen that the angular speeds immediately have followed the reference speed ωref without resonance vibrations. Hence, the validity of the control system, which consists of the proposed method, can be confirmed.

Figure 24. Experimental results for ω<sup>M</sup> and ω<sup>L</sup> obtained using the proposed method (two-mass, R = 1.07, Ksn = 18.5 Nm/rad).

Figure 25. Experimental results for ia obtained using the proposed method (two-mass, R = 1.07, Ks = 18.5 Nm/rad).

### 5. Effects of parameter variation

4.2. Experimental results

Figure 23. Transition of current integral gain Kic.

0

150

*Kic*

Figure 22. Transition of current proportional gain Kpc.

412 Modern Fuzzy Control Systems and Its Applications

300

0

3

*Kpc*

6

Next, the experimental results by using the proposed method are illustrated in this section. Figures 24 and 25 show the experimental results of two-mass model using the proposed method, where the condition (R = 1.07, Ksn = 18.5 Nm/rad) is same as the above simulation results shown in Figures 17 and 18. From these figures, it is observed that the resonance vibrations between the motor and the load angular speed (ω<sup>M</sup> and ωL) have been suppressed very well. Furthermore, after inputting disturbance, it can be seen that the angular speeds immediately have followed the reference speed ωref without resonance vibrations. Hence, the validity of the control system, which consists of the proposed method, can be confirmed.

Figure 24. Experimental results for ω<sup>M</sup> and ω<sup>L</sup> obtained using the proposed method (two-mass, R = 1.07, Ksn = 18.5 Nm/rad).

0 100 200 300 400 500

0 100 200 300 400 500

iteration

iteration

4.2.1. 2-mass model

Next, it is described the effectiveness of robustness by the proposed design method. This section evaluates the robustness to variations in the ratio of inertia and the stiffness of the rigid shaft based on a nominal value.

Figures 26 and 27 show the experimental results of the motor and load angular speeds obtained for the inertia ratio variation when using the same controller gains that were designed using the

Figure 26. Robustness verification results (two-mass, R = 0.42, Ksn = 18.5 Nm/rad).

Figure 27. Robustness verification results (two-mass, R = 2.67, Ksn = 18.5 Nm/rad).

proposed method, when R = [0.42, 2.65], where the disturbance torque input was skipped. From these figure, although it can be observed some overshoot and resonance vibration, the good results can be confirmed that were obtained for the design condition.

Figure 28 shows the experimental results of the motor and load angular speeds obtained for the stiffness of shaft variation using the same controller gains, when Ksn = 70.7. From this figure, it can be seen some resonance vibrations. However, the vibrations rapidly have been suppressed well.

Similarly, Figure 29 shows the experimental results when Ksn = 3.1. As can be seen, the motor and load angular speeds oscillated and overshot. Therefore, if the stiffness of shaft of the experimental model is less than the design value, the settling time to suppress the resonance vibration becomes longer, although the proposed control system is not unstable. In addition, Figure 30 shows the experimental result when the control parameter redesigned with the stiffness of shaft Ksn as the nominal value of experimental model. Good responses can be observed in this figure.

Furthermore, the proposed fuzzy control system is applied to a three-mass resonance model. Figure 31 shows the experimental results of the motor and load angular speeds when using the same controller gains designed for two-mass model (R = 1.07, Ksn = 18.5 Nm/rad, where the nominal parameters of the three-mass experimental setup are JMn = 2.774 · 10<sup>4</sup> kgm<sup>2</sup> , JLn = 2.940 · 10<sup>4</sup> kgm<sup>2</sup> , Ks1<sup>n</sup> = 18.5 Nm/rad, Ks2<sup>n</sup> = 18.5 Nm/rad. From this figure, the effectiveness of the proposed method can be confirmed in a similar manner to the two-mass model case.

Figure 28. Robustness verification results (two-mass, R = 1.07, Ksn = 70.7 Nm/rad).

Figure 29. Robustness verification results (two-mass, R = 1.07, Ksn = 3.1 Nm/rad).

Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller http://dx.doi.org/10.5772/intechopen.68319 415

Figure 30. Experimental results for ω<sup>M</sup> and ω<sup>L</sup> redesigned using the proposed method (two-mass, R = 1.07, Ksn = 3.1 Nm/rad).

Figure 31. Robustness verification results (three-mass,JMn = 2.774 · 104kgm2 ,Jcn = 1.112 · 10<sup>4</sup> kgm2 ,JLn = 2.940 · 104kgm2 , Ks1<sup>n</sup> = 18.5 Nm/rad, Ks2<sup>n</sup> = 18.5 Nm/rad).

#### 6. Conclusions

, JLn =

proposed method, when R = [0.42, 2.65], where the disturbance torque input was skipped. From these figure, although it can be observed some overshoot and resonance vibration, the good

Figure 28 shows the experimental results of the motor and load angular speeds obtained for the stiffness of shaft variation using the same controller gains, when Ksn = 70.7. From this figure, it can be seen some resonance vibrations. However, the vibrations rapidly have been suppressed well. Similarly, Figure 29 shows the experimental results when Ksn = 3.1. As can be seen, the motor and load angular speeds oscillated and overshot. Therefore, if the stiffness of shaft of the experimental model is less than the design value, the settling time to suppress the resonance vibration becomes longer, although the proposed control system is not unstable. In addition, Figure 30 shows the experimental result when the control parameter redesigned with the stiffness of shaft Ksn as the nominal value of experimental model. Good responses can be observed in this figure. Furthermore, the proposed fuzzy control system is applied to a three-mass resonance model. Figure 31 shows the experimental results of the motor and load angular speeds when using the same controller gains designed for two-mass model (R = 1.07, Ksn = 18.5 Nm/rad, where the nominal parameters of the three-mass experimental setup are JMn = 2.774 · 10<sup>4</sup> kgm<sup>2</sup>

of the proposed method can be confirmed in a similar manner to the two-mass model case.

, Ks1<sup>n</sup> = 18.5 Nm/rad, Ks2<sup>n</sup> = 18.5 Nm/rad. From this figure, the effectiveness

results can be confirmed that were obtained for the design condition.

414 Modern Fuzzy Control Systems and Its Applications

Figure 29. Robustness verification results (two-mass, R = 1.07, Ksn = 3.1 Nm/rad).

Figure 28. Robustness verification results (two-mass, R = 1.07, Ksn = 70.7 Nm/rad).

2.940 · 10<sup>4</sup> kgm<sup>2</sup>

This chapter proposed the speed control system to suppress the resonance vibration of multiinertial model, especially two-mass system and three-mass system. The controller has been constructed with the digital fuzzy controller for speed control and the digital PI controller for current control. In the control system, only motor side state variables have been used for controlling the resonance system. Additionally, this chapter utilized the DE to determine these five controller parameters. Finally, the validity of the controller design, the robustness, and the control effectiveness of the proposed method has been verified using the simulations and the experiments by using the test experimental set up.

### Author details

Hidehiro Ikeda

Address all correspondence to: ikeda@nishitech.ac.jp

Nishinippon Institute of Technology, Kitakyushu, Japan

## References


References

1999;46(1):162–168

416 Modern Fuzzy Control Systems and Its Applications

ICCAS 2008, pp. 1367–1372

tions. 1999;119-D(4):544–545

IEE Japan. 1999;119-D(11):1386–1392

Proceedings of 2003 IEEE Conference. 2003;1:60–65

Mechatronic Systems, S10-19 to S10-25; 2006.

IECON 2015, TS-48, YF-008451; 2015. pp. 1825–1830

International Journal of ETAE. 2013;3(4):64–66

Journal of AEST. 2011;8(2):291–296

Systems. 2014;3(2);184–189

Industrial Electronics, 2007;54(2):1193–1206

[1] Hori Y, Sawada H, Chun Y. Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system. IEEE Transactions on Industrial Electronics.

[2] Kwang-Ho Yoon, Jong-Kwang Lee, Ki-Ho Kim, Byung-Suk Park, Ji-Sup Yoon. Hybrid robust controller design for a two mass system with disturbance compensation. In: Proc.

[3] Szabat K, Kowalska TO. Vibration suppression in a two-mass drives system using PI speed controller and additional feedbacks – comparative study. IEEE Transactions on

[4] Ikeda H, Hanamoto T, Tsuji T, Tanaka Y. Position control of 2-mass systems with speed minor loop designed by pole placement method. IEEJ Transactions on Industry Applica-

[5] Zhang G, Furusho J. Control of three-inertia system by PI/PID control. Transactions of

[6] Eker I, Vural M. Experimental online identification of a three-mass mechanical system.

[7] Ikeda H, Hanamoto T, Tsuji T. Design of multi-inertia digital speed control system using Taguchi method. In: Proceedings of ICEM 2008, Paper ID 1167, PB.3.9; 2008. pp. 1–6 [8] Ikeda H, Hanamoto T, Tsuji T, Tomizuka M. Design of vibration suppression controller for 3-inertia systems using Taguchi method. In: Proceedings of SPEEDAM 2006,

[9] Ikeda H, Hanamoto T. Design of vibration suppression controller for 2-inertia system by fictitious reference iterative tuning. In: Proceedings of ICEE 2015, ICEE15A-123; 2015. p. 6

[10] Ikeda H, Ajishi H, Hanamoto T. Application of fictitious reference iterative tuning to vibration suppression controller for 2-inertia resonance system. In: Proceedings of

[11] Malhotra R, Kaur T, Deol GS. DC motor control using fuzzy logic controller. International

[12] Chakravorty J, Sharma R. Fuzzy logic based method of speed control of DC motor.

[13] Ikeda H, Hanamoto T. Fuzzy controller of multi-inertia resonance system designed by differential evolution. In: Proceedings of ICEMS 2013, MC-1883; 2013. pp. 2291–2295 [14] Ikeda H, Hanamoto T. Fuzzy controller of three-inertia resonance system designed by differential evolution. Journal of International Conference on Electrical Machines and

## **Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant**

Mao Li

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68411

#### Abstract

In this chapter, design and stability analyses of direct model reference control system based on wastewater treatment plant are addressed. The purpose of controller design includes input saturation control and two-level control system with fuzzy supervisor control. The wastewater treatment plant is a highly uncertain non-linear system and the plant parameter are unknown, therefore controller design are under those condition.

Keywords: fuzzy control, fuzzy supervisor, wastewater treatment plant, adaptive control, model reference adaptive control, Lyapunov function

1. Introduction

The problem to be solved for this chapter is the dissolved oxygen reference trajectory tracking in an aerobic reactor for nutrient removal using direct model reference adaptive controller at the activated sludge wastewater treatment plant (WWTP). The reference trajectory is provided on-line by upper control layer of the overall control system. The controller design utilizes a different time scale in the internal dissolved oxygen dynamic and in disturbance inputs. In this chapter, we introduce two kinds of adaptive control, one is Direct Model Reference Adaptive Control (DMRAC) and another one is fuzzy logic based on DMRAC with two-level control.

## 2. Adaptive control

The basic concept of adaptive control is that it comprises of two main types. The first is called model reference adaptive control (MRAC) mode whereas the second is called self-tuning mode. An adaptive control characteristic is that the control parameter are variable, and those parameters are updated online with the signal in the system.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2.1. Model reference adaptive control

A model reference adaptive control can be divided into four parts, such as plant, reference model, control law and controller. A plant includes unknown parameters. The reference model is described as control system output. The closed-loop control law is adjusting mechanism for adjustable control parameters. The controller updates the adjustable parameter with time-varying control system.

The plant is supposed to exist with known system structure, but plant parameters are unknown in the real. The structure of the dynamic equation is known with some unknown parameters in the nonlinear plants. The number of poles/zero are supposed to be known with unknown location poles/zero.

The reference model is used in order to obtain assignation ideal response of adaptive control system to control output. For the adaptive control system, that mean it is supply the ideal response by adjusting ideal plant parameters in the adaptation mechanism. To design the adaptive control system, first step is choice of the reference model. It is needed to meet two following clauses.


The controllers are composition of several adjustable parameters. This implies that the controllers are distribution signal to each adjustable parameter with online update. The controller needs to have good tracking performance which means it can achieve tracking convergence behaviour. To design controller, two conditions need to be considered.


The linearly parameterized that mean is the control law with linear term of the adjustable parameters. To guarantee stability and tracking performance, adaptive control design is used by linear parameterization of the controller.

Adjusting parameters in adaptive control law is call adaptation mechanism. In the MRAC systems, the adaptation law is used in order to search the plant parameters, therefore the plant out can track set-point (model reference) with good performance by adaptive controller. The difference between ideal adaptive control parameter and real plant parameter is call tracking error. The tracking error converge to zero that implies that adaptive control system is stable.

#### 2.2. Self-tuning model reference adaptive control

The control parameters estimate plant unknown parameters in control system. If a plant parameter is unknown, then a parameter estimator provides estimation values to those plant unknown parameters. If a plant parameter is known, then control parameters would transmit plant parameters by on-line update on model reference. The estimator provides estimation control parameters with on-line update from model reference, it is called self-tuning controller. The self-tuning controller is estimation unknown parameter in the plant at the same time.

The self-turning MRAC manipulate processes:

2.1. Model reference adaptive control

420 Modern Fuzzy Control Systems and Its Applications

rise time, overshoot and settling time.

trajectory by relevant controller parameters.

by linear parameterization of the controller.

2.2. Self-tuning model reference adaptive control

trajectory by adjusting the controller parameters.

ing control system.

location poles/zero.

A model reference adaptive control can be divided into four parts, such as plant, reference model, control law and controller. A plant includes unknown parameters. The reference model is described as control system output. The closed-loop control law is adjusting mechanism for adjustable control parameters. The controller updates the adjustable parameter with time-vary-

The plant is supposed to exist with known system structure, but plant parameters are unknown in the real. The structure of the dynamic equation is known with some unknown parameters in the nonlinear plants. The number of poles/zero are supposed to be known with unknown

The reference model is used in order to obtain assignation ideal response of adaptive control system to control output. For the adaptive control system, that mean it is supply the ideal response by adjusting ideal plant parameters in the adaptation mechanism. To design the adaptive control system, first step is choice of the reference model. It is needed to meet two following clauses.

• The reference model should satisfy the performance of adaptive control system such as

The controllers are composition of several adjustable parameters. This implies that the controllers are distribution signal to each adjustable parameter with online update. The controller needs to have good tracking performance which means it can achieve tracking convergence

• If the plant parameters are known, then the plant out should track model reference

• If the plant parameters are unknown, then the plant out should track model reference

The linearly parameterized that mean is the control law with linear term of the adjustable parameters. To guarantee stability and tracking performance, adaptive control design is used

Adjusting parameters in adaptive control law is call adaptation mechanism. In the MRAC systems, the adaptation law is used in order to search the plant parameters, therefore the plant out can track set-point (model reference) with good performance by adaptive controller. The difference between ideal adaptive control parameter and real plant parameter is call tracking error. The tracking error converge to zero that implies that adaptive control system is stable.

The control parameters estimate plant unknown parameters in control system. If a plant parameter is unknown, then a parameter estimator provides estimation values to those plant unknown parameters. If a plant parameter is known, then control parameters would transmit plant parameters by on-line update on model reference. The estimator provides estimation control parameters

• The ideal plant parameter should be implemented by the adaptive control system.

behaviour. To design controller, two conditions need to be considered.


The estimation parameter can be taken from an ideal parameters and real parameters by plant input/plant output data that are updated on-line with time-varying adaptive control system. The error dynamic is described as the difference between ideal plant parameters and real plant parameters; this implies that if tracking errors converge to zero by adjusting parameters adaptation then plant output complete tracking reference model. It is purpose of self-turning adaptive control design.

The self-turning control includes two types of adaptive controllers, one is called Indirect Model Reference Adaptive Control (IMRAC) and the another one is called Direct Model Reference Adaptive Control. The plant unknown parameters are provided by adaptive controller estimation of those plant parameters. If the estimation plant parameters need transfer into controller parameters,furthermore control law parameters can influent plant unknown parameters. This implies that the control parameters can adjust plant unknown parameters with standard estimation approach. It is called IMRAC. On the other hand, if it does not need transfer process, this method is called DMRAC.

#### 2.3. Direct model reference control design

The property of adaptive control is used for plant with unknown parameters; therefore, choosing the adaptive control law is more implicated in controller design. Since we mention before, adaptive control law produce controller parameters. Also the stability analysis for control system need to be considered in controller design. In this chapter, we used Lyapunov theory to analyse control system whether stable or unstable. The process of adaptive control design includes three steps. The first step is choosing control law (include plant variable parameters). The second step is choosing adaptation law. The final step is stability analyses to guarantee convergence of control system.

## 3. DMRAC with input saturation apply on WWTP

#### 3.1. Introduction

An activated sludge wastewater treatment plant (WWTP) is a complex nonlinear system due to multiple time scale and unmeasurable state variables. In addition, it has time-varying input disturbances and saturation during the WWTP operation; hence, the hierarchical structures which were considered in Refs. [1, 2]. The two-level controller of tracking prescribed a concentration of the dissolved oxygen (DO) trajectory, while the reference of concentration dissolved oxygen (DOref) was developed in Refs. [3, 4]. The activated sludge plant contained two main components, such as bioreactor and settler as illustrated in Figure 1.

Figure 1. Structure of WWTP for nutrient removal.

The microorganism produced the biomass to nutrient removal in the bioreactor. The concentration of dissolved oxygen control is an important state parameter that feeds the microorganisms. The concentration of DO control was considered in Ref. [5]. The upper level controller produced airflow Qref airðtÞ into the aerobic biological reactor zone. The lower level controller produced the concentration of DO to track the Qref airðtÞ set-point trajectory. The airflow is the control input, and the concentration of DO is the control output.

The dissolved oxygen reference trajectory DOrefðt<sup>Þ</sup> set-point was optimized by the upper control layer which was the medium control layer in overall WWTP [1]. The clean water came out from the settler after being separated from the biomass and sludge. The concentration of substrate and biomass were unmeasurable state variables; hence, they were not able to be online updated. The upper layer control with input saturation was presented in Ref. [6]. The saturation function was assumed considering that aeration system controller was ideal and thus the airflow was equal to airflow reference. However, the physical modelling of the wastewater treatment plant was used to design the controller. In this chapter, we consider the upper level controller with input saturation by designing the new direct model reference adaptive control (DMRAC).

#### 3.2. Problem statement

A mathematical model of the WWTP is based on the mass balance equations, which are illustrated in Figure 1. Hence, they represent the plant variables to produce the model in the state-space format [8]:

$$\frac{dX}{dt} = \mu(t)X(t) - D(t)(1+r)X + rD(t)X\_r(t) \tag{1}$$

$$\frac{d\mathbf{S}}{dt} = -\frac{\mu(t)\mathbf{X}(t)}{Y} - D(t)(1+r)\mathbf{S}(t) + D(t)\mathbf{S}\_{\text{in}}(t) \tag{2}$$

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 423

$$\frac{dDO}{dt} = -\frac{K\_0 \mu(t)X(t)}{Y} - D(t)(1+r)DO(t) + k\_{La}(Q\_{air}(t))(DO\_{max} - DO(t)) + D(t)DO\_{in}(t) \tag{3}$$

$$\frac{dX\_\mathbf{r}}{dt} = D(t)(1+r)X(t) - D(t)(\beta+r)X\_\mathbf{r}(t) \tag{4}$$

$$
\mu(t) = \mu\_{\text{max}} \,\, \frac{S(t)}{K\_s + S(t)} \times \frac{DO(t)}{K\_{DO} + DO(t)}\tag{5}
$$

where

disturbances and saturation during the WWTP operation; hence, the hierarchical structures which were considered in Refs. [1, 2]. The two-level controller of tracking prescribed a concentration of the dissolved oxygen (DO) trajectory, while the reference of concentration dissolved oxygen (DOref) was developed in Refs. [3, 4]. The activated sludge plant contained two main

The microorganism produced the biomass to nutrient removal in the bioreactor. The concentration of dissolved oxygen control is an important state parameter that feeds the microorganisms. The concentration of DO control was considered in Ref. [5]. The upper level controller

The dissolved oxygen reference trajectory DOrefðt<sup>Þ</sup> set-point was optimized by the upper control layer which was the medium control layer in overall WWTP [1]. The clean water came out from the settler after being separated from the biomass and sludge. The concentration of substrate and biomass were unmeasurable state variables; hence, they were not able to be online updated. The upper layer control with input saturation was presented in Ref. [6]. The saturation function was assumed considering that aeration system controller was ideal and thus the airflow was equal to airflow reference. However, the physical modelling of the wastewater treatment plant was used to design the controller. In this chapter, we consider the upper level controller with input saturation by designing the new direct model reference

A mathematical model of the WWTP is based on the mass balance equations, which are illustrated in Figure 1. Hence, they represent the plant variables to produce the model in the

dt <sup>¼</sup> <sup>μ</sup>ðtÞXðtÞ � <sup>D</sup>ðtÞð<sup>1</sup> <sup>þ</sup> <sup>r</sup>Þ<sup>X</sup> <sup>þ</sup> rDðtÞXrðtÞ ð1<sup>Þ</sup>

<sup>Y</sup> � <sup>D</sup>ðtÞð<sup>1</sup> <sup>þ</sup> <sup>r</sup>ÞSðtÞ þ <sup>D</sup>ðtÞSinðtÞ ð2<sup>Þ</sup>

airðtÞ into the aerobic biological reactor zone. The lower level controller

airðtÞ set-point trajectory. The airflow is the

components, such as bioreactor and settler as illustrated in Figure 1.

produced airflow Qref

adaptive control (DMRAC).

3.2. Problem statement

state-space format [8]:

produced the concentration of DO to track the Qref

Figure 1. Structure of WWTP for nutrient removal.

422 Modern Fuzzy Control Systems and Its Applications

dX

dt ¼ � <sup>μ</sup>ðtÞXðt<sup>Þ</sup>

dS

control input, and the concentration of DO is the control output.

$$D(t) = \frac{Q\_{\rm in}}{V\_{\rm a}}; \quad r = \frac{Q\_{\rm r}}{Q\_{\rm in}}; \quad \beta = \frac{Q\_{\rm w}}{Q\_{\rm in}} \tag{6}$$

XðtÞ, SðtÞ, DOmax, XrðtÞ, DðtÞ, SinðtÞ, DOinðtÞ, Y, μðtÞ, μmax, KS, KDO, QairðtÞ, K0, r, β, QinðtÞ, QrðtÞ, QwðtÞ, V<sup>a</sup> are biomass concentration, substrate concentration, maximum dissolved concentration, recycled biomass concentration, dilution rate, substrate concentration in the influent, dissolved oxygen concentration in the influent, biomass yield factor, biomass growth rate, maximum specific growth rate, affinity constant, saturation constant, aeration rate, model constant, recycled sludge rate, removed sludge rate, influent flow rate, effluent flow rate, recycled flow rate, waste flow rate and aerator volume, respectively.

The function kLaðQairðtÞÞ is the oxygen transfer, which depends on the aeration actuating system and sludge conditions [4]. In this chapter, it is assumed that

$$k\_{\rm La}(t) = \alpha Q\_{\rm air}(t) + \delta \tag{7}$$

where α and δ are two known constant values relating to oxygen transfer.

As only the DO output is considered in this chapter, the model is sufficiently accurate. Otherwise, more detailed model, for example, the ASM3, than it should be utilized as it has been done in Ref. [7]. In Figure 2, the detailed structure of the activated sludge WWTP for nutrient removal with the airflow actuator is illustrated. The actuator dynamics are described by a complex hybrid model. The output of aeration control system airflow output QairðtÞ needs to

Figure 2. Structure of wastewater treatment plant.

Figure 3. Structure of DO control system with input saturation.

follow the reference airflow input Qref airðtÞ, which was described in Ref. [8]. The plant input with time-varying disturbances are dissolved oxygen concentration in the fluent DOinðtÞ, the substrate concentration in the influent SinðtÞ and influent flow rate QinðtÞ. The controller needs to have high performance to enables the airflow output to track the reference airflow input.

The structure of DO control system for nutrient removal with input saturation at activated sludge WWTP is illustrated in Figure 3.

The aeration controller designed in this chapter considers the aeration control system as the input <sup>Q</sup>airðt<sup>Þ</sup> with input saturation to achieve <sup>Q</sup>ref airðtÞ ¼ QairðtÞ. This is the main difference in comparison with Ref. [8]. If the plant dynamics have several serially coupled reactors, the decentralized controller needs to consider the input saturation [6]. Previous papers [1–3] considered the two-level controller to remove the nutrient in the activated sludge WWTP. The upper layer controller generated the expected Qref airðtÞ into each of the aerobic biological reactor zones. The input of the lower layer controller was QairðtÞ, which needs to track prescribed upper layer output Qref airðtÞ for each of the reactor zones. If the upper layer controller had an input saturation condition, it influenced global control system stability and performance. As the plant dynamics have very high order and nonlinear dynamics as in Eqs. (1)–(5). The fixed parameter linear controller could not continue to keep the expected performance under full range of operating conditions. This was verified in Ref. [4] by using fixed parameter PI controller in low layer control. The upper layer controller used a fuzzy supervised controller. It obtained the expected performance. In practice, if the disturbance of the input becomes large, fast varying and with saturation input, the PI controller becomes very complex. The DMRAC with input saturation in upper layer control is considered in this chapter, which is not based on previous papers [7]. The DOðtÞ of the DMRAC input-output model rearranges the state-space model from Eqs. (1)–(4). As the state variable are not measureable, the unknown quantities in this input-output model will integrate into one term known as respiration. The parameter adaptation laws of the adaptive controller enable the respiration to be estimated indirectly and automatically.

#### 3.3. DMRAC design

The direct state-space model of WWTP is represented in Eqs. (1)–(5). The dynamics are uncertain and nonlinear. The state variables XrðtÞ, XðtÞ, SðtÞ are unmeasurable. The state variables QairðtÞ, QinðtÞ, QwðtÞ, DOðtÞ are measured by on-line updates. To design the direct model reference adaptive controller, we shall derive dissolved oxygen dynamics model in input-output format of first-order and with input disturbance and input saturation. We shall rewrite Eq. (3) by substituting Eqs. (5) and (7). The term DðtÞDOinðtÞ can be neglected in statespace model, since it is very small in comparison with other state variables. The dissolved oxygen input-output model (DOIOM) is derived as follows:

$$\frac{dDO}{dt} = -a\_p(t)DO(t) - c\_p(t)f(DO(t)) + b\_p(t)Q\_{\text{air}}(t) + d\_p \tag{8}$$

where apðtÞ, cpðtÞ, bpðtÞ, dp are DOIOM parameters and

follow the reference airflow input Qref

424 Modern Fuzzy Control Systems and Its Applications

sludge WWTP is illustrated in Figure 3.

layer controller generated the expected Qref

output Qref

3.3. DMRAC design

input <sup>Q</sup>airðt<sup>Þ</sup> with input saturation to achieve <sup>Q</sup>ref

Figure 3. Structure of DO control system with input saturation.

airðtÞ, which was described in Ref. [8]. The plant input with

airðtÞ ¼ QairðtÞ. This is the main difference in

airðtÞ into each of the aerobic biological reactor zones.

time-varying disturbances are dissolved oxygen concentration in the fluent DOinðtÞ, the substrate concentration in the influent SinðtÞ and influent flow rate QinðtÞ. The controller needs to have high performance to enables the airflow output to track the reference airflow input.

The structure of DO control system for nutrient removal with input saturation at activated

The aeration controller designed in this chapter considers the aeration control system as the

comparison with Ref. [8]. If the plant dynamics have several serially coupled reactors, the decentralized controller needs to consider the input saturation [6]. Previous papers [1–3] considered the two-level controller to remove the nutrient in the activated sludge WWTP. The upper

The input of the lower layer controller was QairðtÞ, which needs to track prescribed upper layer

condition, it influenced global control system stability and performance. As the plant dynamics have very high order and nonlinear dynamics as in Eqs. (1)–(5). The fixed parameter linear controller could not continue to keep the expected performance under full range of operating conditions. This was verified in Ref. [4] by using fixed parameter PI controller in low layer control. The upper layer controller used a fuzzy supervised controller. It obtained the expected performance. In practice, if the disturbance of the input becomes large, fast varying and with saturation input, the PI controller becomes very complex. The DMRAC with input saturation in upper layer control is considered in this chapter, which is not based on previous papers [7]. The DOðtÞ of the DMRAC input-output model rearranges the state-space model from Eqs. (1)–(4). As the state variable are not measureable, the unknown quantities in this input-output model will integrate into one term known as respiration. The parameter adaptation laws of the adaptive

The direct state-space model of WWTP is represented in Eqs. (1)–(5). The dynamics are uncertain and nonlinear. The state variables XrðtÞ, XðtÞ, SðtÞ are unmeasurable. The state

controller enable the respiration to be estimated indirectly and automatically.

airðtÞ for each of the reactor zones. If the upper layer controller had an input saturation

$$\begin{aligned} a\_p(t) &= \frac{Q\_{\text{in}}(1+r)}{V\_a} + \delta \\ c\_p(t) &= \frac{K\_0 X(t)}{Y} \frac{\mu\_{\text{max}} S(t)}{(K\_S + S(t))} \\ f(\text{DO}(t)) &= \frac{\text{DO}(t)}{(K\_{\text{DO}} + \text{DO}(t))} \\ b\_p(t) &= \alpha (\text{DO}\_{\text{max}} - \text{DO}(t)) \\ d\_p &= \delta \text{DO}\_{\text{max}} \end{aligned} \tag{9}$$

The parameters apðtÞ and cpðtÞ are slowly varying and unknown. The parameters apðtÞ is dependent on upper control layer which operates in the time scale and is slower in comparison with DOðtÞ control time scale (6) and (9). The parameter cpðtÞ is dependent on X and S, and is slower in comparison with DOðtÞ (1) and (15). The parameter bpðtÞ is dependent on the fast internal dynamics of DOðtÞ time scale (9). Hence, DOðtÞ is fast varying and known. The parameter dp is slowly varying and known. The model reference dynamics (MRD) generate achieved DOðtÞ dynamics.

The DOðt<sup>Þ</sup> tracks the prescribed dissolved oxygen trajectory DOrefðt<sup>Þ</sup> by the controller. The MRD equation is as follows:

$$\frac{dDO\_{m, \text{ref}}}{dt} = -a\_m DO\_{m, \text{ref}}(t) + b\_m DO^{\text{ref}}(t) \tag{10}$$

where DOm:refðtÞ is the reference dynamics output and the parameters am and bm are constant. The DOðtÞ dynamics SISO input-output model with input saturation yields is as follows:

$$\frac{dDO}{dt} = -a\_p(t)DO(t) - c\_p(t)f(DO(t)) + b\_p(t)W(t) + d\_p \tag{11}$$

where WðtÞ is assumed saturation control input with constraint (SCIC) WðtÞ ¼ saturationðQairðtÞÞ,

$$\mathcal{W}(t) = \begin{cases} Q\_{\rm air}^{\mathcal{L}}(t), & \text{if} \quad Q\_{\rm air}(t) > Q\_{\rm air}^{\mathcal{L}}(t) \\ Q\_{\rm air}(t), & \text{if} \quad Q\_{\rm air}^{\mathcal{L}}(t) \le Q\_{\rm air}(t) \le Q\_{\rm air}^{\mathcal{U}}(t) \\ Q\_{\rm air}^{\mathcal{U}}(t), & \text{if} \quad Q\_{\rm air}(t) < Q\_{\rm air}^{\mathcal{U}}(t) \end{cases} \tag{12}$$

where Q<sup>L</sup> airðt<sup>Þ</sup> and <sup>Q</sup><sup>U</sup> airðtÞ are actuator lower and upper constant bounds. If under the input saturation condition, the filter tracking error nðtÞ is increasing, then global stability will be unstable for the control system. The model reference adaptive control law without input saturation was proposed in Ref. [5]. This motivates us to develop a new control law in comparison with Ref. [8] by explicitly considering the influence of the actuator input saturation nonlinearity.

The filter tracking error is applied as follows:

$$m(t) = e(t) - \lambda(t)\tag{13}$$

where eðtÞ represents the difference between DOðtÞ and DO<sup>m</sup>; refðtÞ with on-line update.

$$e(t) = DO(t) - DO\_{\text{m, ref}}(t)\tag{14}$$

We define the auxiliary signal as follows:

$$\frac{d\lambda}{dt} = b\_p \Lambda Q\_{\text{air}}(t) - \Phi \lambda(t) \tag{15}$$

where Φ is small position constant parameter. The parameter ΔQairðtÞ is the difference between SCIC WðtÞ and control input QairðtÞ.

The affine MRAC law is applied as follows:

$$\begin{split} W(t) &= \frac{1}{b\_p} a\_{DO}(t)DO(t) + \frac{1}{b\_p} a\_f(t)f(DO(t)) \\ &+ \frac{1}{b\_p} a\_{DO^{\text{ref}}}(t)DO^{\text{ref}}(t) - \frac{1}{b\_p} d\_p \\ &- \frac{1}{b\_p} \Phi(t)(DO(t) - DO\_{\text{m,ref}}(t)) \end{split} \tag{16}$$

The MRD in Eq. (10) has linear dynamics. The terms <sup>1</sup> bp afðtÞfðDOðtÞÞ and �<sup>1</sup> bp dp in Eq. (16) can be cancelled by closed-loop with an impact of the nonlinear and additive terms in Eq. (8). The control input saturation is described by the last term in Eq. (16) which is retained in DOIOM. The fifth term in Eq. (16) is updated on-line. The parameters aDOðtÞ, afðtÞ and aDOrefðtÞ are updated by adaptive control law. The MRD is achieved in the closed-loop for ideal parameter. Closing the loop by Eq. (16) yields:

$$\begin{split} \frac{dDO}{dt} &= -(a\_p(t) - a\_{DO}(t))DO(t) \\ &- (c\_p(t) - a\_{\not\!f}(t))f(DO(t)) \\ &+ a\_{\text{DO}^{\text{ref}}}(t)DO^{\text{ref}}(t) \\ &- \Phi(DO(t) - DO\_\mathfrak{m}(t)) \end{split} \tag{17}$$

and

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 427

$$-(a\_\mathbb{P}(t) - \hat{a}\_{\text{DO}}(t)) = -a\_\text{m} \tag{18}$$

$$-\left(c\_{\mathbb{P}}(t) - \hat{a}\_{t}(t)\right) = 0\tag{19}$$

$$
\hat{a}\_{\rm DO}(t) = b\_{\rm m} \tag{20}
$$

$$-\Phi(DO(t) - DO\_{\mathfrak{m}}(t)) = -\Phi e(t) \tag{21}$$

where ^aDOðtÞ, ^afðtÞ and ^aDOrefðtÞ are the ideal parameters, which can now be obtained as follows:

$$
\hat{a}\_{\rm DO}(t) = -a\_{\rm m} + a\_{\rm p}(t) \tag{22}
$$

$$
\hat{a}\_{\mathbf{f}}(t) = c\_{\mathbf{p}}(t) \tag{23}
$$

$$
\hat{a}\_{\rm DO}{}^{\rm ref}(t) = b\_{\rm m} \tag{24}
$$

The parameter adaption laws which can achieve stability for a DMRAC system with SISO-controlled plant were derived in Ref. [6]. It was a first-order dynamic system composed of the mixed linear uncertainty in constant but not time-varying parameters and additive structured nonlinear. Applying these laws to Eq. (8) yields:

$$\frac{da\_{\rm DO}}{dt} = -\gamma\_1 e(t) D O(t) \tag{25}$$

$$\frac{da\_f}{dt} = -\gamma\_2 e(t) f(DO(t))\tag{26}$$

$$\frac{da\_{\rm DO}}{dt} = -\gamma\_3 e(t) D\mathcal{O}^{\text{ref}}(t) \tag{27}$$

γ1, γ<sup>2</sup> and γ<sup>3</sup> are small enough positive constants representing the parameter adaptation gains which are used to control the parameter adaptation rates. In order to guarantee the stability of the closed-loop system, these rates shall be harmonized with the process variable rates. The DMRAC structure is presented in Figure 4.

#### 3.4. Stability analysis

where Q<sup>L</sup>

airðt<sup>Þ</sup> and <sup>Q</sup><sup>U</sup>

426 Modern Fuzzy Control Systems and Its Applications

The filter tracking error is applied as follows:

We define the auxiliary signal as follows:

SCIC WðtÞ and control input QairðtÞ.

Closing the loop by Eq. (16) yields:

and

The affine MRAC law is applied as follows:

airðtÞ are actuator lower and upper constant bounds. If under the input

nðtÞ ¼ eðtÞ � λðtÞ ð13Þ

eðtÞ ¼ DOðtÞ � DO<sup>m</sup>; refðtÞ ð14Þ

dt <sup>¼</sup> bpΔQairðtÞ � ΦλðtÞ ð15<sup>Þ</sup>

afðtÞfðDOðtÞÞ

afðtÞfðDOðtÞÞ and �<sup>1</sup>

bp

dp in Eq. (16) can be

ð16Þ

ð17Þ

bp dp

saturation condition, the filter tracking error nðtÞ is increasing, then global stability will be unstable for the control system. The model reference adaptive control law without input saturation was proposed in Ref. [5]. This motivates us to develop a new control law in comparison with Ref. [8] by explicitly considering the influence of the actuator input saturation nonlinearity.

where eðtÞ represents the difference between DOðtÞ and DO<sup>m</sup>; refðtÞ with on-line update.

where Φ is small position constant parameter. The parameter ΔQairðtÞ is the difference between

aDOðtÞDOðtÞ þ <sup>1</sup>

aDOrefðtÞDOrefðtÞ � <sup>1</sup>

cancelled by closed-loop with an impact of the nonlinear and additive terms in Eq. (8). The control input saturation is described by the last term in Eq. (16) which is retained in DOIOM. The fifth term in Eq. (16) is updated on-line. The parameters aDOðtÞ, afðtÞ and aDOrefðtÞ are updated by adaptive control law. The MRD is achieved in the closed-loop for ideal parameter.

dt ¼ �ðapðtÞ � aDOðtÞÞDOðt<sup>Þ</sup>

<sup>þ</sup> <sup>a</sup>DOrefðtÞDOrefðt<sup>Þ</sup>

� ðcpðtÞ � afðtÞÞfðDOðtÞÞ

� ΦðDOðtÞ � DOmðtÞÞ

ΦðtÞðDOðtÞ � DO<sup>m</sup>; refðtÞÞ

bp

bp

dλ

<sup>W</sup>ðtÞ ¼ <sup>1</sup> bp

The MRD in Eq. (10) has linear dynamics. The terms <sup>1</sup>

þ 1 bp

� 1 bp

dDO

The estimated parameters aDOðtÞ, afðtÞ and aDOrefðtÞ are updated on-line by the adaptation laws (25)–(27). The error between estimated parameter and ideal parameters are denoted as ΔaDOðtÞ, ΔafðtÞ and ΔaDOrefðtÞ:

$$
\Delta \mathfrak{a}\_{\rm DO}(t) = \mathfrak{a}\_{\rm DO}(t) - \hat{\mathfrak{a}}\_{\rm DO}(t) \tag{28}
$$

$$
\Delta a\_{\mathbf{f}}(t) = a\_{\mathbf{f}}(t) - \hat{a}\_{\mathbf{f}}(t) \tag{29}
$$

$$
\Delta a\_{\rm DO}{}^{\rm nt}(t) = a\_{\rm DO}{}^{\rm nt}(t) - \hat{a}\_{\rm DO}{}^{\rm nt}(t) \tag{30}
$$

Figure 4. DMRAC structure.

Considering the following Lyapunov function:

$$V(t) = \frac{1}{2}n^2(t) + \frac{1}{2}\Delta a\_{\rm DO}{}^2(t) + \frac{1}{2}\Delta a\_{\rm I}{}^2(t) + \frac{1}{2}\Delta a\_{\rm DO}{}^2(t)\tag{31}$$

Hence,

$$\begin{split} \frac{dV(t)}{dt} &= n(t)n(t) + (a\_{\rm DO}(t) - \hat{a}\_{\rm DO}(t))(a\_{\rmDO}(t)) - a\_{\rm DO}(t)) = \frac{1}{\mathcal{Y}1} \\ &+ (a\_{\rm t}(t) - a\_{\rm t}(t))(\hat{a}\_{\rm t}(t) - a\_{\rm t}(t))\frac{1}{\mathcal{Y}\_2} \\ &+ (a\_{\rm DO^{\rm ref}}(t)) - \hat{a}\_{\rm DO^{\rm ref}}(t))(a\_{\rm DO^{\rm ref}}(t) - a\_{\rm DO^{\rm ref}}(t))\frac{1}{\mathcal{Y}\_3} \end{split} \tag{32}$$

It follows from Eqs. (13), (15), (10) and (17) that

$$\begin{aligned} n(t) &= (a\_{\rm DO}(t) - \hat{a}\_{\rm DO}(t)DO(t)) \\ &+ (a\_{\rm t}(t) - a\_{\rm t}(t))f(DO(t)) \\ &+ (a\_{\rm DO}(t)) - \hat{a}\_{\rm DO}(t))DO\_{\rm m,ref}(t) \\ &- \Phi n(t) \end{aligned} \tag{33}$$

Applying Eqs. (33), (25) (26) and (27) into (32), yields:

$$\frac{dV(t)}{dt} = -\Phi n^2(t)\tag{34}$$

Summarizing the result of the Lyapunov function (RLF) with input saturation closed-loop DOðtÞ dynamic system, it can be seen that RLF progressively approaches zero. If the RLF approaches zero, then filter tracking error approaches zero, when time approaches infinity, Φ is small position constant and <sup>n</sup><sup>2</sup>ðt<sup>Þ</sup> is positive variable. To find the bounded saturation control input by limiting error between control output and dissolved oxygen trajectory reference with auxiliary signal, yields:

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 429

$$\text{limit}\{e(t) - \lambda(t)\} \le 0\tag{35}$$

If time approaches infinity, the eðtÞ approaches zero. To confirm whether the auxiliary signal is negative or positive when time goes to infinity by considering the following Lyapunov function, yields:

$$V\_{\lambda}(t) = \frac{1}{2}\lambda(t)\tag{36}$$

Hence

Considering the following Lyapunov function:

dVðtÞ

It follows from Eqs. (13), (15), (10) and (17) that

Applying Eqs. (33), (25) (26) and (27) into (32), yields:

auxiliary signal, yields:

Hence,

Figure 4. DMRAC structure.

428 Modern Fuzzy Control Systems and Its Applications

<sup>V</sup>ðtÞ ¼ <sup>1</sup> 2 n2 <sup>ð</sup>tÞ þ <sup>1</sup> 2 ΔaDO 2 <sup>ð</sup>tÞ þ <sup>1</sup> 2 Δa<sup>f</sup> 2 <sup>ð</sup>tÞ þ <sup>1</sup> 2

<sup>Δ</sup>aDOref <sup>2</sup>

γ3

dt <sup>¼</sup> <sup>n</sup>ðtÞnðtÞþðaDOðtÞ � ^aDOðtÞÞðaDOðtÞÞ � <sup>a</sup>DOðtÞÞ ¼ <sup>1</sup>

þ ðaDOrefðtÞÞ � ^aDOrefðtÞÞðaDOrefðtÞ � <sup>a</sup>DOrefðtÞÞ <sup>1</sup>

nðtÞ¼ðaDOðtÞ � ^aDOðtÞDOðtÞÞ

� ΦnðtÞ

dVðtÞ

dt ¼ �Φn<sup>2</sup>

Summarizing the result of the Lyapunov function (RLF) with input saturation closed-loop DOðtÞ dynamic system, it can be seen that RLF progressively approaches zero. If the RLF approaches zero, then filter tracking error approaches zero, when time approaches infinity, Φ is small position constant and <sup>n</sup><sup>2</sup>ðt<sup>Þ</sup> is positive variable. To find the bounded saturation control input by limiting error between control output and dissolved oxygen trajectory reference with

þ ðafðtÞ � afðtÞÞfðDOðtÞÞ

þ ðaDOrefðtÞÞ � ^aDOrefðtÞÞDO<sup>m</sup>;refðtÞ

γ2

þ ðafðtÞ � <sup>a</sup>fðtÞÞð^afðtÞ � <sup>a</sup>fðtÞÞ <sup>1</sup>

ðtÞ ð31Þ

ð32Þ

ð33Þ

γ1

ðtÞ ð34Þ

$$\frac{dV\_{\lambda}(t)}{dt} = \lambda(t)\lambda(t)\tag{37}$$

Applying Eq. (12) into Eq. (37), yields:

$$\frac{dV\_{\lambda}(t)}{dt} = \lambda(t)b\_{\text{p}}(t)\Lambda Q\_{\text{air}}(t) - \lambda(t)\lambda(t)\Phi \tag{38}$$

$$\frac{dV\_{\lambda}(t)}{dt} = \lambda(t)$$

Assume term

$$h\_{\rm p}(t)\Delta Q\_{\rm air}(t) = \Delta Q\_{\rm air}^{\rm plant}(t)\tag{39}$$

It follows Eqs. (38) and (39) so that

$$\frac{dV\_{\lambda}(t)}{dt} = -\lambda^2(t)\Phi + \frac{1}{2}\lambda^2(t) + \frac{1}{2}\Delta Q\_{\text{air}}^{\text{plant}^2}(t) \tag{40}$$

Now we assume

$$a\Phi = \frac{1}{2}a\_0 \quad \text{} \quad a\_0 < 0,\tag{41}$$

where a<sup>0</sup> is small positive constant value. Applying Eq. (41) into Eq. (40) yields:

$$\frac{dV\_{\lambda}(t)}{dt} = -2V\_{\lambda}(t)a\_0 + \frac{1}{2}\Delta Q\_{\text{air}}^{\text{plant}^2}(t)\tag{42}$$

The second term <sup>1</sup> 2ΔQplant<sup>2</sup> air ðtÞ is bounded. To find bound of first term by integral, yields:

$$V\_{\lambda}(t) = \frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{4a\_0} + (V\_{\lambda,0}(t) + \frac{\Delta Q\_{\text{air}}^{\text{plant}^2}}{4a\_0})e(t)^{-2a\_0} \tag{43}$$

where the V<sup>λ</sup>:<sup>0</sup>ðtÞ is the initial value of the VλðtÞ. As time approaches infinity and a<sup>0</sup> is large enough for a positive value, then second term is equal to zero in (37).

$$V\_{\lambda}(t) = \frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{4a\_0} \tag{44}$$

It follows from Eqs. (36) and (44) and the limitation VλðtÞ as negative or zero by squared.

$$V\_{\lambda}^{2}(t) = \frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{4a\_0}$$

$$\frac{1}{2}\lambda^2(t) = \frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{4a\_0} \tag{45}$$

$$\lambda(t) = \sqrt{\frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{2a\_0}}$$

It follows from Eqs. (35) and (41) that

$$e(t) \le \sqrt{\frac{\Delta Q\_{\text{air}}^{\text{plant}^2}(t)}{2a\_0}}\tag{46}$$

If the value of a<sup>0</sup> is large enough, then the tracking error eðtÞ is closer to zero. The control system will be more stable. Finally, the standard application of the Barbalat's lemma allows concluding the adaptive control system that achieves the asymptotic tracking of DOrefðt<sup>Þ</sup> under-bounded aDOðtÞ, ^aD^ <sup>O</sup>^ ðtÞ, afðtÞ, afðtÞ, aDOrefðtÞ, aDOrefðtÞ if (a) the parameter in adaptive control law are close enough to the set-point in initial condition; (b) the parameter adaptation rates are positive small enough and (c) the saturation input is small enough, the control parameters bounded are stabilized

Figure 5. DO and DOref with input saturation.

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 431

Figure 6. DO and DOref with input saturation and disturbances.

#### 3.5. Simulation results

<sup>V</sup>λðtÞ ¼ <sup>Δ</sup>Qplant2

It follows from Eqs. (36) and (44) and the limitation VλðtÞ as negative or zero by squared.

<sup>2</sup>ðtÞ ¼ <sup>Δ</sup>Qplant2

<sup>ð</sup>tÞ ¼ <sup>Δ</sup>Qplant2

s

s

If the value of a<sup>0</sup> is large enough, then the tracking error eðtÞ is closer to zero. The control system will be more stable. Finally, the standard application of the Barbalat's lemma allows concluding the adaptive control system that achieves the asymptotic tracking of DOrefðt<sup>Þ</sup> under-bounded aDOðtÞ, ^aD^ <sup>O</sup>^ ðtÞ, afðtÞ, afðtÞ, aDOrefðtÞ, aDOrefðtÞ if (a) the parameter in adaptive control law are close enough to the set-point in initial condition; (b) the parameter adaptation rates are positive small enough and (c) the saturation input is small enough, the control

V<sup>λ</sup>

1 2 λ2

It follows from Eqs. (35) and (41) that

430 Modern Fuzzy Control Systems and Its Applications

parameters bounded are stabilized

Figure 5. DO and DOref with input saturation.

λðtÞ ¼

eðtÞ ≤

air ðtÞ 4a<sup>0</sup>

air ðtÞ 4a<sup>0</sup>

air ðtÞ 4a<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔQplant<sup>2</sup> air ðtÞ 2a<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔQplant<sup>2</sup> air ðtÞ 2a<sup>0</sup>

ð44Þ

ð45Þ

ð46Þ

The simulation data are based on the real record. We assumed WWTP without disturbance, the very good DOðt<sup>Þ</sup> tracking model reference DOrefðt<sup>Þ</sup> performance has been shown in Figure 5. The real plant contained some disturbances such as effluent flow rate, recycled flow rate and waste flow rate. The controller that we have designed still indicated perfect tracking performance in Figure 6.

### 4. Fuzzy supervisor based on multiple DMRAC

#### 4.1. Introduction

In this section, we consider that fuzzy control are based on multiple DMRAC. The fuzzy control represents upper level control and DMRAC represents lower level control. More detail information are described in the next section.

#### 4.2. Problem statement

In Ref. [4] is descried two-level controller tracking previously set-point of DO trajectory in several serially coupled reactors for the nutrient removal served by one actuator system with several air blower at WWTP. The upper level control delivers airflow into each bioreactors to be bioreactor set-point trajectory close to ideal trajectory. The lower level control is used for the concentration of DO trajectory flowing the set-point. The structure of WWTP with coupled reactors is illustrated in Figure 7. The structure of WWTP is different from Section 5 that

Figure 7. The structure of wastewater treatment plant for nutrient removal with coupled reactors.

contains two bioreactors. The capacity limit is that total airflow Qmax air ðtÞ should be small or equal to sum of all the adaptive control signals Qair:<sup>k</sup>ðtÞ.

$$\sum\_{k=1}^{q} Q\_{\text{air},k}(t) \le Q\_{\text{air}}^{\text{max}}(t) \tag{47}$$

where k is the number of bioreactor.

#### 4.3. Controller design

#### 4.3.1. Lower DMRAC design

As mentioned in Section 3, the process of DMRAC design is explained in detail; therefore, in this section we provide essential equations. The state-space format is same with last section for single reactor as in Eqs. (1)–(6).

• The dissolved oxygen input-output model (DOIOM) with coupled bioreactors is derived as follows:

$$\frac{dDO\_i}{dt} = -a\_{p.i}(t)DO\_i(t) - c\_{p.i}(t)f(DO\_i(t)) + b\_{p.i}(t)Q\_{\text{air.}i}(t) + d\_{p.i} \tag{48}$$

where ap:<sup>i</sup>ðtÞ, cp:<sup>i</sup>ðtÞ, bp:<sup>i</sup>ðtÞ, dp:<sup>i</sup> are DOIOM parameters and

$$\begin{aligned} a\_{p,i} &= \frac{Q\_{in}(1+r\_1)}{V\_{a,i}} + \delta\_i\\ c\_{p,i} &= \frac{K\_{0,i}X\_i(t)}{Y\_i} \frac{\mu\_{\max,i}S\_i(t)}{K\_{s,i} + S\_i(t)}\\ b\_{p,i} &= \alpha\_i(DO\_{\max,i} - DO\_i(t))\\ d\_{p,i} &= \delta\_i DO\_{\max,i} \end{aligned} \tag{49}$$

where i ¼ 1, 2

• The plant parameters status are exactly same with single reactor. The model reference dynamics equation is set as:

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 433

$$\frac{dDO\_{m.ref,j}}{dt} = -a\_{m.j}DO\_{m.ref,j}(t) + b\_{m.j}DO\_{j}^{ref}(t) \tag{50}$$

where j ¼ 1, 2

• The affine model reference adaptive control law is applied as follows:

$$Q\_{\rm air.k}(t) = a\_{\rm DO.k}(t)DO\_k(t) + a\_{f.k}(t)f(DO\_k(t)) + a\_{\rm DO\_k^{\rm ref}}(t)DO\_k^{\rm ref}(t) - \frac{\delta\_k DO\_{\rm max.k}}{b\_{p.k}(t)} \tag{51}$$

where k ¼ 1, 2

contains two bioreactors. The capacity limit is that total airflow Qmax

where ap:<sup>i</sup>ðtÞ, cp:<sup>i</sup>ðtÞ, bp:<sup>i</sup>ðtÞ, dp:<sup>i</sup> are DOIOM parameters and

X q

Figure 7. The structure of wastewater treatment plant for nutrient removal with coupled reactors.

<sup>Q</sup>air:<sup>k</sup>ðt<sup>Þ</sup> <sup>≤</sup> <sup>Q</sup>max

As mentioned in Section 3, the process of DMRAC design is explained in detail; therefore, in this section we provide essential equations. The state-space format is same with last section for

• The dissolved oxygen input-output model (DOIOM) with coupled bioreactors is derived

ap:<sup>i</sup> <sup>¼</sup> Qinð<sup>1</sup> <sup>þ</sup> <sup>r</sup>1<sup>Þ</sup> Va:<sup>i</sup>

cp:<sup>i</sup> <sup>¼</sup> <sup>K</sup><sup>0</sup>:iXiðt<sup>Þ</sup> Yi

dp:<sup>i</sup> ¼ δiDOmax:<sup>i</sup>

dt ¼ �ap:<sup>i</sup>ðtÞDOiðtÞ � cp:<sup>i</sup>ðtÞfðDOiðtÞÞ þ bp:<sup>i</sup>ðtÞQair:<sup>i</sup>ðtÞ þ dp:<sup>i</sup> <sup>ð</sup>48<sup>Þ</sup>

þ δ<sup>i</sup>

SiðtÞ Ks:<sup>i</sup> þ SiðtÞ

μmax:<sup>i</sup>

bp:<sup>i</sup> ¼ αiðDOmax:<sup>i</sup> � DOiðtÞÞ

• The plant parameters status are exactly same with single reactor. The model reference

k ¼ 1

equal to sum of all the adaptive control signals Qair:<sup>k</sup>ðtÞ.

where k is the number of bioreactor.

432 Modern Fuzzy Control Systems and Its Applications

4.3. Controller design

as follows:

where i ¼ 1, 2

dynamics equation is set as:

4.3.1. Lower DMRAC design

single reactor as in Eqs. (1)–(6).

dDOi

air ðtÞ should be small or

ð49Þ

air ðtÞ ð47Þ

• Model reference adaptive control law is used as:

$$\frac{da\_{DO\_n}}{dt} = -\gamma\_{z\text{one}.z.1}e\_l(t)DO\_l(t)\tag{52}$$

$$\frac{da\_{fn}}{dt} = -\gamma\_{\text{zone}.z.2}e\_l(t)f(\text{DO}\_l(t))\tag{53}$$

where n ¼ 1, 2; z ¼ 1, 2; l ¼ 1, 2; i ¼ 1, 2.

#### 4.3.2. Fuzzy supervisor design

The purpose of fuzzy supervisor is to divide total airflow Qmax air ðtÞ into two lower control signal, but those should satisfy capacity limit (47). Each of the bioreactors airflow restrict lower control output by MRAC. This implies that if fuzzy supervisor delivers airflow big enough then bioreactor output is more close to set-point trajectories (model reference). The error dynamic described each bioreactors output approaching uniform level. The fuzzy supervisor is designed as following:

#### 4.3.2.1. Step 1: Fuzzification

Linguistic variable is at lower level for each DMRAC error dynamics. Those error dynamics are divided into three types such as small, medium and big by percentage of lower level error dynamics (54). Membership function used in this chapter are Sigmoidal condition (55) and Gauss condition (56).

$$V(t) = e\_i(t)^{-1} \mathbf{g} \sum\_{i=2}^{q} e\_i(t) \tag{54}$$

where VðtÞ is percentage of lower level error dynamics for each airflow.

eiðtÞ is DMRAC error dynamics for each bioreactors.

$$\mu(v(t)) = \frac{1}{1 + \exp\left(-a(v(t)) - c\right)}\tag{55}$$

where a, c are membership function shape parameters.

$$\mu(v(t)) = \begin{cases} \exp\left(-\left(v(t)\right) - c\_1\right)^2 / \sigma\_1^2) \\ 1; \\ \exp\left(-\left(v(t)\right) - c\_2\right)^2 / \sigma\_2^2) \end{cases} \tag{56}$$

#### where a, σ are membership function shape parameters.

4.3.2.2. Step 2: Fuzzy rule

4.3.2.2.1. First rule If error dynamics is small and sum of level airflow is greater than total airflow,

Then bioreactor receives corresponding percentage of total airflow.

$$\text{If } V(t) \text{ is small and } \sum\_{i=1}^{q} \mathbb{Q}\_{\text{air},i}(t) \ge \mathbb{Q}\_{\text{air}}^{\text{max}}(t).$$

Then

$$Q\_{\text{air},i.2}^{\text{supervisor}}(t) = \frac{Q\_{\text{air}}^{\text{max}}(t)}{\sum\_{i=1}^{q} Q\_{\text{air},i}(t)} \times Q\_{\text{air},i} \,\text{g}\,\text{10\%}\tag{57}$$

4.3.2.2.2. Second rule

If error dynamic medium and sum of level airflow is greater than total airflow, Then bioreactor receives corresponding percentage of total airflow.

$$\text{If } V(t) \text{ is medium and } \sum\_{i=1}^{q} Q\_{\text{air},i}(t) \ge Q\_{\text{air}}^{\text{max}}(t)$$

Then

$$Q\_{\text{air}.1.2}^{\text{supervisor}}(t) = \frac{Q\_{\text{air}}^{\text{max}}(t)}{\sum\_{i=1}^{q} Q\_{\text{air}.i}(t)} \times Q\_{\text{air}.i} \,\text{g}\,\text{30\%}\tag{58}$$

4.3.2.2.3. Third rule

If error dynamic is big and sum of level airflow is greater than total airflow,

Then bioreactor receive corresponding percentage of total airflow.

$$\text{If } V(t) \text{ is big and } \sum\_{i=1}^{q} \mathbb{Q}\_{\text{air},i}(t) \ge \mathbb{Q}\_{\text{air}}^{\text{max}}(t)$$

Then

Design and Stability Analysis of Fuzzy‐Based Adaptive Controller for Wastewater Treatment Plant http://dx.doi.org/10.5772/intechopen.68411 435

$$Q\_{\rm air}^{\rm supervisc}(t) = \frac{Q\_{\rm air}^{\rm max}(t)}{\displaystyle} \times Q\_{\rm air,i} \,\text{g}\,\text{60\%}\tag{59}$$

#### 4.3.2.3. Step 3: Defuzzification

Each of the bioreactors obtain airflow by a fuzzy value.

#### 5. Summary

where a, c are membership function shape parameters.

434 Modern Fuzzy Control Systems and Its Applications

where a, σ are membership function shape parameters.

Then bioreactor receives corresponding percentage of total airflow.

<sup>Q</sup>air:<sup>i</sup>ðt<sup>Þ</sup> <sup>≥</sup> <sup>Q</sup>max

Qsupervisor

Then bioreactor receives corresponding percentage of total airflow.

q

i ¼ 1

q

i ¼ 1

4.3.2.2. Step 2: Fuzzy rule

If <sup>V</sup>ðt<sup>Þ</sup> is small and <sup>X</sup>

4.3.2.2.2. Second rule

4.3.2.2.3. Third rule

If <sup>V</sup>ðt<sup>Þ</sup> is big and <sup>X</sup>

q

i ¼ 1

If <sup>V</sup>ðt<sup>Þ</sup> is medium and <sup>X</sup>

Then

Then

Then

4.3.2.2.1. First rule

μðvðtÞÞ ¼

If error dynamics is small and sum of level airflow is greater than total airflow,

air ðtÞ

air:i:<sup>2</sup> <sup>ð</sup>tÞ ¼ <sup>Q</sup>max

If error dynamic medium and sum of level airflow is greater than total airflow,

<sup>Q</sup>air:<sup>i</sup>ðt<sup>Þ</sup> <sup>≥</sup> <sup>Q</sup>max

If error dynamic is big and sum of level airflow is greater than total airflow,

air ðtÞ

Then bioreactor receive corresponding percentage of total airflow.

<sup>Q</sup>air:<sup>i</sup>ðt<sup>Þ</sup> <sup>≥</sup> <sup>Q</sup>max

Qsupervisor

air ðtÞ

air:1:<sup>2</sup> <sup>ð</sup>tÞ ¼ <sup>Q</sup>max

1;

8 < :

exp ð�ðvðtÞÞ � c1Þ

exp ð�ðvðtÞÞ � c2Þ

air ðtÞ

Qair:<sup>i</sup>ðtÞ

air ðtÞ

Qair:<sup>i</sup>ðtÞ

X q

i ¼ 1

X q

i ¼ 1

2 =σ<sup>2</sup> 1Þ

2 =σ<sup>2</sup> 2Þ

� Qair:<sup>i</sup> g10% ð57Þ

� Qair:<sup>i</sup> g30% ð58Þ

ð56Þ

In this chapter, we considered two different adaptive control. The first adaptive control is applied on WWTP with control input saturation. The second adaptive control descried that how upper level fuzzy control working is based on lower level DMRC applied on the coupling bioreactors of WWTP.

#### Author details

Mao Li

Address all correspondence to: limaomxl554@gmail.com

The University of Birmingham, Birmingham, UK

#### References


## **A Model for Evaluating Soil Vulnerability to Erosion Using Remote Sensing Data and A Fuzzy Logic System**

Ignacio Meléndez-Pastor, Jose Navarro Pedreño, Ignacio Gómez Lucas and Antonis A. Zorpas

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67989

#### **Abstract**

[5] Olsson G, Newell R. Wastewater Treatment System. Modelling, Diagnosis and Control.

[6] Zubowicz T, Brdys MA, Piotrowski R. Intelligent PI controller and its application to dissolved oxygen tracking problem. Journal of Automation Mobile Robotics & Intelligent

[7] Li M, Brdys MA. Direct model reference adaptive control of nutrient removal at activated sludge wastewater treatment plant. In: The 20th International Conference on Methods and

[8] Duzinkiewicz K, Brdys MA, Kurek W, Piotrowski R. Genetic hybrid predictive controller for optimized dissolved-oxygen tracking at lower control level. IEEE Transaction on Con-

Models in Automation and Robotics; 24–27 August 2015; Miedzyzdroje, Poland.

London, UK: IWA Publishing; 1999.

trol System Technology. 2009;17:1183–1192.

Systems. 2010;4(3):16–24.

436 Modern Fuzzy Control Systems and Its Applications

Soil vulnerability is the capacity of one or more of the ecological functions of the soil system to be harmed. It is a complex concept which requires the identification of multiple environmental factors and land management at different temporal and space scales. The employment of geospatial information with good update capabilities could be a satisfac‐ tory tool to assess potential soil vulnerability changes in large areas. This chapter pres‐ ents the application of two land degradation case studies which is simple, synoptic, and suitable for continuous monitoring model based on the fuzzy logic. The model combines topography and vegetation status information to assess soil vulnerability to land deg‐ radation. Topographic parameters were obtained from digital elevation models (DEM), and vegetation status information was derived from the computation of the normalized difference vegetation index (NDVI) satellite images. This spectral index provides rele‐ vance and is updated for each scene, evidences about the biomass and soil productivity, and vegetation density cover or vegetation stress (e.g., forest fires, droughts). Modeled output maps are suitable for temporal change analysis, which allows the identification of the effect of land management practices, soil and vegetation regeneration, or climate effects.

**Keywords:** soil vulnerability, fuzzy logic, remote sensing, soil erosion, soil degradation

## **1. Introduction**

Soil is considered a nonrenewable resource, and it is essential for food security and for our sus‐ tainable future and is defined as the top layer of the earth's crust formed by mineral particles, organic matter, water, air, and living organism [1]. Desertification associated to soil erosion

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

processes is the main topic to protect and maintain soils. Moreover, desertification and land degradation are terms that are used in order to indicate ecosystem productivity and are associ‐ ated with loses of vegetation covered [2]. The United Nations program against desertification is crucial as it focuses on fighting hunger and poverty, foresting stability, and building resilience to climate change in some of the world's most vulnerable areas [3]. Soil erosion, water‐holding capacity, salinity, sodicity, losses of nutrients, etc. are common indicators used to categorize land degradation [1]. The genesis of soil is a long process, and the formation of a layer of 30 cm depth takes from 1000 to 10,000 years [4]. This scenery created between soil degradation and the long process of formation gives us few alternatives about soil conservation, which is essen‐ tial for humans.

Soil vulnerability is the capacity of one or more of the ecological functions of the soil system to be harmed, i.e., biomass production, filtering, buffering and transformation medium, gene reserve, and protection of flora and fauna [5]. In other words, it is the sensitivity of soil against degrading processes.

The most important degrading process worldwide related to desertification, among others, is soil erosion as a result of climate change. Climate change is one of the major drivers of an ecosystem shift in a decertified state [6]. The vegetation plays a key role protecting soil. Moreover, soil erosion is associated to biomass production which has an important role in the future energy situation [7]. There are several factors affecting soil erosion. However, vegeta‐ tion and topography play a crucial role in the vulnerability of soil. The first one is the most important source of organic matter to soil, protecting it against rain and wind (water and wind erosion) [8]. The second one determines and facilitates the losses of soil due to erosive processes and transport.

Soils with low vegetation cover and high slopes are more vulnerable against degrading pro‐ cesses. For this reason, it is very important to know these factors to establish the vulnerability of a soil in order to protect it, prevent erosion, and keep resources for a sustainable future use.

Many countries like Cyprus, Greek Islands, Spain, and Italy are affected by soil erosion, and there are many problems to access or obtain soil data because of the orography, accessibility, or other environmental factors. It is estimated that more than 115 million hectares which rep‐ resents 12% of European total land are subject to erosion [9]. Moreover, it is estimated that, in the Mediterranean area (consists of a vulnerable area due to climate change effects), water ero‐ sion could affect the loss of 20/40 ton/ha of soil after a single cloudburst, and in extreme cases, the erosion could be even more than 100 ton/ha [10, 11]. For this reason, it is important to have techniques that facilitate the creation and population of spatial information. These techniques are englobed in digital soil mapping (DSM) which are systems composed by numerical models inferring the spatial and temporal variations of soil types and soil properties from soil observa‐ tion and knowledge and from related environmental variables [12]. It is important to have an overview of the state and properties of soils for decision‐makers and to facilitate the population and access to soil data. We cannot forget that land and water resources are central to agriculture and rural development and are intrinsically linked to global challenges of food insecurity and poverty, climate change adaptation and mitigation, as well as degradation and depletion of natural resources that affect the livelihoods of millions of rural people across the world. [13]. The use of easy models that can be applied by administration for decision‐making plays a key role for resource conservation and people. In this sense, the models based on remote sensing data and mathematical models as those based on fuzzy logic systems can be helpful.

processes is the main topic to protect and maintain soils. Moreover, desertification and land degradation are terms that are used in order to indicate ecosystem productivity and are associ‐ ated with loses of vegetation covered [2]. The United Nations program against desertification is crucial as it focuses on fighting hunger and poverty, foresting stability, and building resilience to climate change in some of the world's most vulnerable areas [3]. Soil erosion, water‐holding capacity, salinity, sodicity, losses of nutrients, etc. are common indicators used to categorize land degradation [1]. The genesis of soil is a long process, and the formation of a layer of 30 cm depth takes from 1000 to 10,000 years [4]. This scenery created between soil degradation and the long process of formation gives us few alternatives about soil conservation, which is essen‐

Soil vulnerability is the capacity of one or more of the ecological functions of the soil system to be harmed, i.e., biomass production, filtering, buffering and transformation medium, gene reserve, and protection of flora and fauna [5]. In other words, it is the sensitivity of soil against

The most important degrading process worldwide related to desertification, among others, is soil erosion as a result of climate change. Climate change is one of the major drivers of an ecosystem shift in a decertified state [6]. The vegetation plays a key role protecting soil. Moreover, soil erosion is associated to biomass production which has an important role in the future energy situation [7]. There are several factors affecting soil erosion. However, vegeta‐ tion and topography play a crucial role in the vulnerability of soil. The first one is the most important source of organic matter to soil, protecting it against rain and wind (water and wind erosion) [8]. The second one determines and facilitates the losses of soil due to erosive

Soils with low vegetation cover and high slopes are more vulnerable against degrading pro‐ cesses. For this reason, it is very important to know these factors to establish the vulnerability of a soil in order to protect it, prevent erosion, and keep resources for a sustainable future use. Many countries like Cyprus, Greek Islands, Spain, and Italy are affected by soil erosion, and there are many problems to access or obtain soil data because of the orography, accessibility, or other environmental factors. It is estimated that more than 115 million hectares which rep‐ resents 12% of European total land are subject to erosion [9]. Moreover, it is estimated that, in the Mediterranean area (consists of a vulnerable area due to climate change effects), water ero‐ sion could affect the loss of 20/40 ton/ha of soil after a single cloudburst, and in extreme cases, the erosion could be even more than 100 ton/ha [10, 11]. For this reason, it is important to have techniques that facilitate the creation and population of spatial information. These techniques are englobed in digital soil mapping (DSM) which are systems composed by numerical models inferring the spatial and temporal variations of soil types and soil properties from soil observa‐ tion and knowledge and from related environmental variables [12]. It is important to have an overview of the state and properties of soils for decision‐makers and to facilitate the population and access to soil data. We cannot forget that land and water resources are central to agriculture and rural development and are intrinsically linked to global challenges of food insecurity and poverty, climate change adaptation and mitigation, as well as degradation and depletion of natural resources that affect the livelihoods of millions of rural people across the world. [13].

tial for humans.

438 Modern Fuzzy Control Systems and Its Applications

degrading processes.

processes and transport.

The models based on fuzzy logic systems are related to fuzzy set theory which plays a pri‐ mary role in fuzzy logic. Zadeh developed the theoretical basis of this theory and defined the concept of fuzzy sets as "A fuzzy set A in X is characterized by a membership function fA(x) which associates with each point in X a real number in the interval [0, 1], with the value of fA(x) at x representing the 'grade of membership' of x in A. The nearer the value of fA(x) to unity, the higher the grade of membership of x in A" [14]. The more interesting property is that fuzzy sets work well with data uncertainties. A strict categorization (with discrete classes) of environmental/soil factors falls within subjectivity and carelessness of uncertain‐ ties (e.g., error measurements, selection of decimals, missed data, etc.). Fuzzy sets are used for classifications where the classes do not have sharply defined boundaries [15]. This approach is easily applied for environmental phenomena study, such as soil nutrient losses, atmospheric pollutant dispersion, forest productivity, etc., due to the great spatial variability and possible discontinuity of those phenomena.

Fuzzy theory provides a rich mathematical basis for understanding decision problems and for constructing decision rules in multi‐criteria evaluation and combination [16]. A wide variety of fuzzy logic approaches have been developed in order to expand the concept of fuzzy sets. Fuzzy measure refers to any set of function which is monotonic with respect to set member‐ ship [17]. The variety of functions that can be applied is great. In fact, a fuzzy function is a generalization of the concept of a classical function. A classical function f is a mapping (corre‐ spondence) from the domain D of definition of the function into a space S; f(D) ⊆ S is called the range of f. Different features of the classical concept of a function can be considered fuzzy [18]. Fuzzy measures include Bayesian probabilities, beliefs and plausibilities of Dempster‐Shafer theory, and membership grades of fuzzy sets, providing a framework for the methodologies of uncertainty studies [19].

In this process, fuzzy sets and soil erosion are combined in order to facilitate decision‐making based on easy criteria that can be applied by administration. Desertification, climate change, and soil productivity have to be considered. The need of global understanding of local pro‐ cesses and the possibilities of the computers and remote sensing techniques are the bases of this methodology. For these environmental, social, and economic reasons, it is necessary to have a tool for taking decisions. On the one hand, decision theory is concerned with the logic by which one arrives at a choice between alternatives [20]. On the other hand, the procedure by which one selects among different alternatives is enabled by a decision support system (DSS). The study of decision support systems is an applied discipline that uses knowledge and especially theory from other disciplines [21]. In this case, the knowledge of soil science and mathematical modeling helps us build an effective DSS.

A type of decision support system widely used is the multi‐criteria evaluation (MCE) based on multi‐criteria analysis (MCA) of analytical hierarchy process (AHP) [22, 23]. The AHP analy‐ sis is based on three basic principles as indicated by Zorpas and Saranti [22], and those include the preferences configuration, the interruption of the problem into subproblems, and finally the pair‐wise comparison of criteria/subcriteria with the proposed alternative scenarios. The MCA [22, 23] focuses on four main stages starting with recognition of the problem (if possible to split in subproblems) and the formation of a hierarchical structure, followed by the pair‐ wise comparison of decision elements used to derive normalized absolute scales of numbers whose elements are then used as priorities, and continued with the control of the priorities set (to solve the problem) and finally the assessment and the evaluation of the alternative scenarios to solve the specific problem. This methodology can be used from decision‐makers, engineers, consultants, researchers, and governmental and local authorities in order to evalu‐ ate or propose specific solution in a specific problem [21].

The MCE explores how to combine the information from several criteria to form a single index of evaluation using discrete or continuous factors [21–23]. For this purpose in envi‐ ronmental sciences, thematic information aggregation procedures are used in the process of criteria combination. Two traditions of aggregation procedures have been extensively used for MEC [24]: (1) Boolean overlay where all criteria are assessed by thresholds of suitability to produce Boolean maps using logical operators such union, intersection, or negation and (2) weighted linear combination (WLC) where continuous criteria are standardized (gener‐ ally by a simple linear transformation) to a common numeric range and then combined by weighted averaging. Both aggregation criteria are rather inflexible as consequence of their inherent logic aggregation (type of operators and properties).

Aggregation operations on fuzzy sets are operations by which several fuzzy sets are com‐ bined in a desirable way (assuming some rules and operators) to produce a single fuzzy set [25]. Fuzzy measures provide a theoretical base to explore an expand understanding of MCE processes and the design of new aggregation operators [24]. Two traditions of logic operators have been extensively used since decades [26]: (1) MIN and MAX operators for Boolean overlay and fuzzy membership aggregation and (2) averaging operator for weighted linear combination. Jiang and Eastman [24] suggested the use of weighted linear combina‐ tions as a fuzzy set membership operator together with the MIN and MAX operators, in the framework of fuzzy measures. This is a very flexible approach for fuzzy set aggregations. In this chapter, the proposed soil vulnerability model related to soil erosion is based on this approach for fuzzy set aggregation and easy functions for the analysis.

## **2. A fuzzy logic model to evaluate soil vulnerability**

Soil erosion may be considered the most important degradation processes in this approach, and the vulnerability of soils is based on the most important criteria affecting this process. Two types of information used by this model are (1) topographic parameters and (2) veg‐ etation. Remote sensing techniques allow us to have data from any part of the Earth. For instance, digital elevation models (DEMs) which can describe the topography of any region can be derived from data obtained by the Shuttle Radar Topography Mission of NASA [18] and many other platforms. Remote sensing techniques can also provide vegetation status data for the analysis. For instance, data obtained from Landsat missions [27] or other programs facilitate the calculus or vegetation index like normalized difference vegetation index (NDVI).

The analytical capabilities derived from the use of DEM are enormous, ranging from basic topographic feature estimation, to flood simulations, and others. Moreover, remote sensing has a great potential to obtain imagery from optical, thermal, and microwave spectral regions across wide regions of the planet with a great temporal frequency, enough accuracy, and open source. Image processing methods of remotely‐sensed data could extract valuable and spe‐ cialized information of selected targets (e.g., soils, vegetation, waters, etc.). An added value of image processing techniques is the capability to analyze temporal series of data which can be useful to study the vulnerability and changes occurred in soils along time.

the pair‐wise comparison of criteria/subcriteria with the proposed alternative scenarios. The MCA [22, 23] focuses on four main stages starting with recognition of the problem (if possible to split in subproblems) and the formation of a hierarchical structure, followed by the pair‐ wise comparison of decision elements used to derive normalized absolute scales of numbers whose elements are then used as priorities, and continued with the control of the priorities set (to solve the problem) and finally the assessment and the evaluation of the alternative scenarios to solve the specific problem. This methodology can be used from decision‐makers, engineers, consultants, researchers, and governmental and local authorities in order to evalu‐

The MCE explores how to combine the information from several criteria to form a single index of evaluation using discrete or continuous factors [21–23]. For this purpose in envi‐ ronmental sciences, thematic information aggregation procedures are used in the process of criteria combination. Two traditions of aggregation procedures have been extensively used for MEC [24]: (1) Boolean overlay where all criteria are assessed by thresholds of suitability to produce Boolean maps using logical operators such union, intersection, or negation and (2) weighted linear combination (WLC) where continuous criteria are standardized (gener‐ ally by a simple linear transformation) to a common numeric range and then combined by weighted averaging. Both aggregation criteria are rather inflexible as consequence of their

Aggregation operations on fuzzy sets are operations by which several fuzzy sets are com‐ bined in a desirable way (assuming some rules and operators) to produce a single fuzzy set [25]. Fuzzy measures provide a theoretical base to explore an expand understanding of MCE processes and the design of new aggregation operators [24]. Two traditions of logic operators have been extensively used since decades [26]: (1) MIN and MAX operators for Boolean overlay and fuzzy membership aggregation and (2) averaging operator for weighted linear combination. Jiang and Eastman [24] suggested the use of weighted linear combina‐ tions as a fuzzy set membership operator together with the MIN and MAX operators, in the framework of fuzzy measures. This is a very flexible approach for fuzzy set aggregations. In this chapter, the proposed soil vulnerability model related to soil erosion is based on this

Soil erosion may be considered the most important degradation processes in this approach, and the vulnerability of soils is based on the most important criteria affecting this process. Two types of information used by this model are (1) topographic parameters and (2) veg‐ etation. Remote sensing techniques allow us to have data from any part of the Earth. For instance, digital elevation models (DEMs) which can describe the topography of any region can be derived from data obtained by the Shuttle Radar Topography Mission of NASA [18] and many other platforms. Remote sensing techniques can also provide vegetation status data for the analysis. For instance, data obtained from Landsat missions [27] or other programs facilitate the calculus or vegetation index like normalized difference vegetation index (NDVI).

ate or propose specific solution in a specific problem [21].

440 Modern Fuzzy Control Systems and Its Applications

inherent logic aggregation (type of operators and properties).

approach for fuzzy set aggregation and easy functions for the analysis.

**2. A fuzzy logic model to evaluate soil vulnerability**

The proposed model is based on a set of three initial inputs, selected considering the previous indications (**Figure 1**):


**Figure 1.** Flowchart of the fuzzy logic model to soil vulnerability evaluation.

(lower incident radiation levels). This parameter could suggest information about soil moisture, water availability by plants, etc. This general rule could be dismissed in special local situations (e.g., for places where wet maritime winds impact against slopes closed to a south aspect). This factor is of great importance in plant distribution and growth and soil development and properties [30].

• Normalized difference vegetation index (NDVI) is computed as a normalized ratio between near infrared (NIR) and red spectral bands of remotely‐sensed imagery. NDVI values range between −1 and +1. This vegetation index is strongly related with several vegetation pa‐ rameters such as changes in biomass and chlorophyll content [31, 32]. NDVI is related with other vegetation parameters too [33] such as leaf water content, CO2 net flux, absorbed pho‐ tosynthetically active radiation (APAR), and leaf area index (LAI), among others. A high NDVI value (usually over 0.3–0.5) indicates a vigorous and dense vegetal coverage.

Slope, aspect, and NDVI are the parameters computed in our model and could be extracted for wide regions from remote sensing data. Those parameters are computed following the flowchart as **Figure 1** presents, with an easy function that can be implemented. The simplic‐ ity of the model may be the limiting factor because only three inputs are taken into account.

However, this tool is good enough to have an overview of the soil erosion and could be a good tool for regional soil vulnerability assessment and long‐term monitoring with a low economic cost. A near real‐time monitoring of soil vulnerability, depending on the availability of data from satellite, could be performed and used by land managers and scientist. The main advan‐ tage in this sense is that field campaign to check soil, vegetation status, slope, and other filed parameters is initially not necessary. Moreover, the centered field work in determined areas reduces the cost of large field campaigns of wide territories.

The proposed model considers input parameters as fuzzy sets (factors) to be standardized by the definition of individual membership functions. A great variety of fuzzy set membership functions have been developed.

Based on Eastman [16], fuzzy set membership functions can be defined by three parameters:


The definition of these parameters provides a great number of possibilities to define the effect of individual factors in the observed phenomena. The shape and type of membership func‐ tions should be defined according to the knowledge of environmental phenomena. Weighted linear combination is used for the standardized aggregation factors. This method allows the possibility to assign equal or differential weights to factors in function of experts' knowledge about their relative importance for the studied phenomena. Finally, it is important to mention that this simple model approach could be completed with other factors to be considered in the aggregation process or in other stages of the model. Equilibrium must be found between model simplicity and valuable obtained data to soil conservation.

## **3. Soil vulnerability under high slope changes: a case study**

(lower incident radiation levels). This parameter could suggest information about soil moisture, water availability by plants, etc. This general rule could be dismissed in special local situations (e.g., for places where wet maritime winds impact against slopes closed to a south aspect). This factor is of great importance in plant distribution and growth and

• Normalized difference vegetation index (NDVI) is computed as a normalized ratio between near infrared (NIR) and red spectral bands of remotely‐sensed imagery. NDVI values range between −1 and +1. This vegetation index is strongly related with several vegetation pa‐ rameters such as changes in biomass and chlorophyll content [31, 32]. NDVI is related with

tosynthetically active radiation (APAR), and leaf area index (LAI), among others. A high

NDVI value (usually over 0.3–0.5) indicates a vigorous and dense vegetal coverage.

Slope, aspect, and NDVI are the parameters computed in our model and could be extracted for wide regions from remote sensing data. Those parameters are computed following the flowchart as **Figure 1** presents, with an easy function that can be implemented. The simplic‐ ity of the model may be the limiting factor because only three inputs are taken into account.

However, this tool is good enough to have an overview of the soil erosion and could be a good tool for regional soil vulnerability assessment and long‐term monitoring with a low economic cost. A near real‐time monitoring of soil vulnerability, depending on the availability of data from satellite, could be performed and used by land managers and scientist. The main advan‐ tage in this sense is that field campaign to check soil, vegetation status, slope, and other filed parameters is initially not necessary. Moreover, the centered field work in determined areas

The proposed model considers input parameters as fuzzy sets (factors) to be standardized by the definition of individual membership functions. A great variety of fuzzy set membership

Based on Eastman [16], fuzzy set membership functions can be defined by three parameters:

• Shape of the function that provides three possible options: (1) monotonically increasing which begins at 0 and then rise and stay at 1, (2) monotonically decreasing which is op‐ posite to the previous one, and (3) symmetric which firstly rise and then fall again

• Type of function with three possible options: (1) sigmoidal produced by a cosine function, (2) J‐shaped produced by a sine function, and (3) linear produced by a linear function

The definition of these parameters provides a great number of possibilities to define the effect of individual factors in the observed phenomena. The shape and type of membership func‐ tions should be defined according to the knowledge of environmental phenomena. Weighted linear combination is used for the standardized aggregation factors. This method allows the possibility to assign equal or differential weights to factors in function of experts' knowledge about their relative importance for the studied phenomena. Finally, it is important to mention that this simple model approach could be completed with other factors to be considered in

net flux, absorbed pho‐

other vegetation parameters too [33] such as leaf water content, CO2

reduces the cost of large field campaigns of wide territories.

• Control points that govern the shape of the curve

functions have been developed.

soil development and properties [30].

442 Modern Fuzzy Control Systems and Its Applications

A test of soil vulnerability evaluation of this model after a fire event was done, considering that the area to test the model has many changes in slope and is close to the sea, affected from marine breeze and moisture. The selected area is located on the south‐east coast of Spain (Alicante province) in the area of "La Granadella" (38.73°N, 0.19°E). This test site supported a fire event from 26 to 30 on August 2000. The topography is characterized by a highly rough relief (cliffs above the sea of more than 150 m) and a wet Mediterranean climate (700–1000 mm/year of rainfall). This is an interesting test area which combines a wet climate and a complex and pre‐ cipitous relief and exhibits a great recurrence of fires.

The hypothesis to verify with the fuzzy logic model is that vegetation regeneration has a primary role in soil vulnerability reduction. The potential advantage for the use of fuzzy sets with respect to other methods of factor combination must be owed to their great flexibility derived from the defined membership functions.

Vegetation status information was obtained from two satellite scenes acquired by the multi‐ spectral advanced spaceborne thermal emission and reflection radiometer (ASTER): (http:// asterweb.jpl.nasa.gov/) sensor. The first scene was acquired 4th October in 2000, some days after the forest fire, while the second scene was acquired 7th June in 2003, being a reasonable period for a substantial evolution in the landscape.

ASTER sensor is composed of three subsystems in function of their spectral and spatial reso‐ lution characteristics: visible near‐infrared radiometer (VNIR) with 15 m of spatial resolution and stereoscopic capability, short‐wave infrared radiometer (SWIR) with 30 m of spatial reso‐ lution, and thermal infrared radiometer (TIR) with 90 m of spatial resolution [34]. Only VNIR system bands were employed for our analyses. Both scenes were in origin preprocessed to an ASTER high‐level product denominated AST\_07 which contain data of surface reflectance [35].

Topographic information was obtained from a high‐resolution digital elevation model (**Figure 2**) that was computed using the previous vector cartography (scale 1:10,000). Triangulated irregu‐ lar network (TIN) polygons were computed with the nodes of the vector cartography and used to develop the raster DEM. The spatial resolution of the DEM was adjusted to ASTER‐VNIR imagery (15 m). Satellite images were geometrically corrected using the high‐resolution DEM and additional cartography in order to minimize the positional errors among the different sources of information. The root‐mean‐square error (RMSE) of the geometric correction was less than half a pixel for both ASTER scenes.

Topographic parameters (slope and aspect) were computed using the DEM. Both parameters were quantized as degrees. Normalized difference vegetation index is derived from a spectral band transformation between near‐infrared and red bands. The original NDVI formulation has been attributed to Rouse et al. [36] and their research with early Landsat images. The original NDVI formulation has been adapted to subsequent sensors whose spectral characteristics are

**Figure 2.** Digital elevation model (DEM) of "Sierra de Escalona" test site.

different among them. The following equation is the adaptation of NDVI for ASTER spectral bands (Eq. (1)):

$$\text{NDVI} = \frac{\rho\_{\text{vNu3}} - \rho\_{\text{vNu2}}}{\rho\_{\text{vNu3}} + \rho\_{\text{vNu2}}} \tag{1}$$

where *ρ*VNIR2 is the spectral reflectance for the second band of the ASTER‐VNIR subsystem and *ρ*VNIR3 is the spectral reflectance for the third band of the same subsystem. The computation of NDVI is more suitable with surface reflectance data (like AST\_07 high‐level ASTER product) in order to minimize differential wavelength‐dependent atmospheric disturbances. NDVI was computed in the same way for the 2000 and 2003 scenes (**Figure 3 (a)** and **(b)**).

Soil vulnerability estimations were computed for both 2000 and 2003 scenarios. Individual fuzzy set membership functions are characterized by their shape and type and were defined for each of the considered soil vulnerability factors (i.e., slope, aspect, and NDVI). In this sense, **Table 1** provides a synthesis of model parameters used with the fuzzy logic model. The proposed model employed the following membership function for input variables: (1) NDVI, a monotonically decreasing linear function; (2) slope, a monotonically increasing J‐shaped function; and (3) aspect, a symmetric linear function. Membership functions required the defi‐ nition of several control points (CP) for a potential model generalization.

Fuzzy membership functions were combined weighting the relative importance of each one using a linear system. Jiang and Eastman [24] suggested the use of weighted linear combina‐ tions as a fuzzy set membership operator together with the MIN and MAX operators, as a very flexible approach for fuzzy set aggregations.

This weighting process considers the relative importance of each variable in the modeling process. Vegetation and slope were considered the most important factors for the soil vulnerability model. Further details of the definition of membership functions and model calibration are available in Melendez‐Pastor et al. [37]. Finally, a temporal change analysis of soil vulnerability was com‐ puted using the percentage of change procedure employing the following formulation (Eq. (2)):

$$\text{Temporal change (\% )} = \frac{\left(t\_2 - t\_i\right)}{t\_i} \cdot 100\tag{2}$$

where *t* 1 and *t* 2 are the soil vulnerability estimations for 2000 and 2003, respectively.

The application of the fuzzy logic model for this study area covers two environmental facts, an (almost) invariant one associated to the slope and orientation (which are mainly deter‐ mined by the geomorphology of the landscape) and a highly changing parameter as the veg‐ etation through NDVI. Landscape topography greatly affects soil profile formation [30, 38], while vegetation status and dynamics are highly related with the quality of the soils [39]. The DEM analysis revealed that the study area has a mean slope value of 16 degree, with a maxi‐ mum slope value of 78 degrees. Relief is mainly configured on a NE‐SW direction with pref‐ erential slope orientation to the SE direction. NDVI mean values varied from 0.31 (standard deviation of 0.12) on 2000 to 0.38 (standard deviation of 0.09) on 2003.

Soil vulnerability simulations (**Figure 3 (c)** and **(d)**) revealed the severe impact of the forest fire on vegetation. The model marks the importance of vegetation dynamics, the relation with the presence of soil although both are limited to the position as the model indicates (slope and

different among them. The following equation is the adaptation of NDVI for ASTER spectral

where *ρ*VNIR2 is the spectral reflectance for the second band of the ASTER‐VNIR subsystem and *ρ*VNIR3 is the spectral reflectance for the third band of the same subsystem. The computation of NDVI is more suitable with surface reflectance data (like AST\_07 high‐level ASTER product) in order to minimize differential wavelength‐dependent atmospheric disturbances. NDVI

Soil vulnerability estimations were computed for both 2000 and 2003 scenarios. Individual fuzzy set membership functions are characterized by their shape and type and were defined for each of the considered soil vulnerability factors (i.e., slope, aspect, and NDVI). In this sense, **Table 1** provides a synthesis of model parameters used with the fuzzy logic model. The proposed model employed the following membership function for input variables: (1) NDVI, a monotonically decreasing linear function; (2) slope, a monotonically increasing J‐shaped function; and (3) aspect, a symmetric linear function. Membership functions required the defi‐

Fuzzy membership functions were combined weighting the relative importance of each one using a linear system. Jiang and Eastman [24] suggested the use of weighted linear combina‐ tions as a fuzzy set membership operator together with the MIN and MAX operators, as a

This weighting process considers the relative importance of each variable in the modeling process. Vegetation and slope were considered the most important factors for the soil vulnerability model.

was computed in the same way for the 2000 and 2003 scenes (**Figure 3 (a)** and **(b)**).

nition of several control points (CP) for a potential model generalization.

very flexible approach for fuzzy set aggregations.

*ρ*VNIR<sup>3</sup> + *ρ*VNIR<sup>2</sup>

(1)

NDVI <sup>=</sup> *<sup>ρ</sup>*VNIR<sup>3</sup> <sup>−</sup> *<sup>ρ</sup>* \_\_\_\_\_\_\_\_\_\_ VNIR<sup>2</sup>

**Figure 2.** Digital elevation model (DEM) of "Sierra de Escalona" test site.

444 Modern Fuzzy Control Systems and Its Applications

bands (Eq. (1)):

**Figure 3.** NDVI estimations for 2000 (a) and 2003 (b). Soil vulnerability estimations for 2000 (c) and 2003 (d). Soil vulnerability results area in 8 bits of quantization.


**Table 1.** Model parameters for the soil vulnerability factors slope, aspect, and NDVI.

aspect). A characteristic pattern of soil vulnerability mitigation by intense vegetation regen‐ eration could be advertised along the valleys where NDVI has increased faster since 2000. The temporal change estimation of soil vulnerability simulations (**Figure 4**) remarks the local high postfire ecosystem regeneration. Change rates are up to a 30% of less soil vulnerability within burned area. Soil vulnerability change map also highlighted areas where soil vulnerability has increased. Those changes correspond to land use conversions to urban (change rates up to a 30% of more soil vulnerability) and vegetation status variations.

**Figure 4.** Temporal change (%) of soil vulnerability estimation. Green pixels correspond to less vulnerable areas on 2003.

## **4. Soil vulnerability and vegetation as a limiting factor**

aspect). A characteristic pattern of soil vulnerability mitigation by intense vegetation regen‐ eration could be advertised along the valleys where NDVI has increased faster since 2000. The temporal change estimation of soil vulnerability simulations (**Figure 4**) remarks the local high postfire ecosystem regeneration. Change rates are up to a 30% of less soil vulnerability within burned area. Soil vulnerability change map also highlighted areas where soil vulnerability has increased. Those changes correspond to land use conversions to urban (change rates up

Shape Monotonically increasing Symmetric Monotonically decreasing

**Figure 4.** Temporal change (%) of soil vulnerability estimation. Green pixels correspond to less vulnerable areas on 2003.

to a 30% of more soil vulnerability) and vegetation status variations.

**Factors Slope Aspect NDVI**

Weights 0.35 0.05 0.6

**Table 1.** Model parameters for the soil vulnerability factors slope, aspect, and NDVI.

CP 1 0 0 0 CP 2 90 180 1 CP 3 No 180 No CP 4 No 360 No Type J‐shaped Linear Linear

Membership functions

446 Modern Fuzzy Control Systems and Its Applications

The proposed model was also engaged in a research of soil vulnerability evaluation in a semiarid area in the south of Alicante province (Spain). This study area is located in a por‐ tion of the "Sierra de Escalona" (37.97°N, 0.86°W), an area with altitudes ranging from 80 to 340 m.a.s.l., a semiarid Mediterranean climate with hot summers and scarce precipitations (less than 250–300 mm), and fragile soils severely affected by erosion and land degradation processes. Dominant land cover classes are intensive agriculture (citrus and almond trees) at low to medium slopes, xerophytic shrubs in abandoned fields and marginal areas, and sparse *Pinus halepensis* stands at the highest slopes. A large reservoir fed by a transbasin diversion is located in the north of the study area. This is an interesting test area which combines a semi‐ arid climate, intensive exploitation of soil resources by agriculture, and high erosion rates resulting transport of sediments and nutrients to the reservoir.

The hypothesis to verify with this fuzzy logic model application is that soil vulnerability is enhanced during drought periods when vegetation status is less protective against land degradation drivers. The engagement of satellite images allowed the estimation of soil vul‐ nerability changes within a drought period and between hydrologic years with different pre‐ cipitation patterns (drought vs. normal year).

A comparison of the changes regarding soil vulnerability between late spring/early sum‐ mer and late summer for different hydrologic years was included in this research. The selec‐ tions of the dates were based on meteorological information and satellite image availability. Meteorological information for the nearby Pilar de la Horadada town meteorological stations was obtained from the Spanish Agroclimatic information System for Irrigation (Ministry of Agriculture and Fisheries, Food and Environment). Precipitation data indicated that the 2000 water year was characterized by a severe drought (169.8 mm), while 2003 was a more regular hydrologic year (253.0 mm). This information was the starting point for the compila‐ tion of vegetation and topography datasets.

Four satellite scenes acquired by the multispectral ASTER sensor were employed to obtain vegetation information. Two scenes were acquired for the 2000 hydrologic year (June 30 and August 1) and two other images for the 2003 water year (May 22 and August 10). The first images correspond to the end of spring and early summer, while the second scenes corre‐ spond with the end of summer. Only VNIR system bands (15 m of spatial resolution) were employed for our analyses. Both scenes were in origin preprocessed to the ASTER high‐level product of surface reflectance (AST\_07).

Topographic information was obtained from a high‐resolution digital elevation model (**Figure 5**) that was computed using the previous vector cartography (scale 1:10,000). Triangulated irregu‐ lar network (TIN) polygons were computed with the nodes of the vector cartography and used to develop the raster DEM in the same way as the previous case study. The spatial resolution of the DEM was adjusted to ASTER‐VNIR imagery (15 m). Satellite images were geometrically corrected using the high‐resolution DEM and additional cartography in order to minimize the positional errors among the different sources of information. The root‐mean‐square error (RMSE) of the geometric correction was less than half a pixel for both ASTER scenes.

**Figure 5.** Digital elevation model (DEM) of "La Granadella" test site.

Slope and aspect were computed using the DEM and quantized as degrees. Normalized differ‐ ence vegetation indices were computed with the surface reflectance data spectral band transfor‐ mation between near‐infrared and red bands according to Eq. (1). Soil vulnerability estimations were computed for the four dates of the ASTER scenes (**Table 2**). The proposed model employed the following membership function and weights for input variables: (1) NDVI, a monotoni‐ cally decreasing linear function with a weight value of 0.4, (2) slope, a monotonically increasing J‐shaped function with a weight value of 0.4, and (3) aspect, a symmetric linear function with a weight value of 0.2. Membership function control points were the same of the previous case study. Further details of the definition of membership functions and model calibration are available in Melendez‐Pastor et al. [40]. Finally, a temporal change analysis of soil vulnerability was com‐ puted using the percentage of change procedure employing the formulation of Eq. (2). Temporal change analyses were done between different seasons for the same year estimations (E01 vs. E02 and E31 vs. E32) and between the same seasons for different years (E01 vs. E31 and E02 vs. E32).

NDVI estimations highlighted the critical importance of hydrologic year accumulated precipi‐ tation for the maintenance of vegetation status for nonirrigated areas. In **Figure 6 (a)** and **(b)**,


**Table 2.** Soil vulnerability estimations and employed ASTER scenes. Hydrologic year accumulated precipitation for Pilar de la Horadada meteorological station is shown.

the largest portion of the study area corresponds with nonirrigated crops, sclerophyllous vege‐ tation, and some coniferous forest areas. They had very low NDVI values, while irrigated areas and some wet ravines exhibited quite high NDVI values (0.4–0.6). These NDVI images corre‐ spond with an intense drought water year with less than 168 mm of accumulated precipitation.

Climatic conditions for the hydrologic year 2003 were very different with almost 50% more precipitation. Greener areas in **Figure 6 (c)** and **(d)** correspond with the irrigated crops and also with some coniferous forest and dense sclerophyllous vegetation areas. More intense NDVI changes for irrigated crops were associated with the harvest of seasonal crops.

High values of soil vulnerability estimations were obtained for almost the whole study area in all the scenes (**Figure 7**), due to the abrupt orography and the lack of vigorous vegetation cover. Southern slopes with high slope values are shown as the most vulner‐ able areas since its vegetation cover, and edaphic development is very scarce. The lowest soil vulnerability estimations were obtained for some crop fields with permanent crops in flat areas.

Temporal change analyses between different seasons for the same year estimations were done. **Figure 8 (a)** corresponds with the change in 2000 (E01 vs. E02), and **Figure 8 (b)** is for the change in 2003 (E31 vs. E32). The most remarkable changes were associated with the irrigated crops. Temporal crops had been collected within the period of time of the change analysis, and the soil vulnerability increased by the elimination of the vegetation cover. On the other hand, temporal change between the same seasons but for different years (2000 vs. 2003) was ana‐ lyzed. **Figure 8 (c)** corresponds with the change analysis for late spring‐early summer (E01 vs. E31), while **Figure 8 (d)** corresponds with the change analysis for late summer (E02 vs. E32).

**Figure 6.** NDVI images for the estimations E01 (a), E02 (b), E31 (c), and E32 (d).

Slope and aspect were computed using the DEM and quantized as degrees. Normalized differ‐ ence vegetation indices were computed with the surface reflectance data spectral band transfor‐ mation between near‐infrared and red bands according to Eq. (1). Soil vulnerability estimations were computed for the four dates of the ASTER scenes (**Table 2**). The proposed model employed the following membership function and weights for input variables: (1) NDVI, a monotoni‐ cally decreasing linear function with a weight value of 0.4, (2) slope, a monotonically increasing J‐shaped function with a weight value of 0.4, and (3) aspect, a symmetric linear function with a weight value of 0.2. Membership function control points were the same of the previous case study. Further details of the definition of membership functions and model calibration are available in Melendez‐Pastor et al. [40]. Finally, a temporal change analysis of soil vulnerability was com‐ puted using the percentage of change procedure employing the formulation of Eq. (2). Temporal change analyses were done between different seasons for the same year estimations (E01 vs. E02 and E31 vs. E32) and between the same seasons for different years (E01 vs. E31 and E02 vs. E32). NDVI estimations highlighted the critical importance of hydrologic year accumulated precipi‐ tation for the maintenance of vegetation status for nonirrigated areas. In **Figure 6 (a)** and **(b)**,

**Figure 5.** Digital elevation model (DEM) of "La Granadella" test site.

448 Modern Fuzzy Control Systems and Its Applications

**Estimation Year Season ASTER scene Accumulated** 

**Table 2.** Soil vulnerability estimations and employed ASTER scenes. Hydrologic year accumulated precipitation for Pilar

E01 2000 Spring summer 06‐30‐2000 167.2 E02 2000 Late summer 08‐01‐2000 167.8 E31 2003 Spring summer 05‐22‐2003 246.4 E32 2003 Late summer 08‐10‐2003 250.0

de la Horadada meteorological station is shown.

**precipitation**

**Figure 7.** Soil vulnerability estimations E01 (a), E02 (b), E31 (c), and E32 (d).

The intense effect of drought mitigation in soil vulnerability values was estimated. This effect was more intense for the change analysis of late spring‐early summer when very few areas exhibited an increase of soil vulnerability estimations. **Figure 8 (c)** indicated a large increase of soil vulnerability in areas of the center and east of the image, which are produced by crop

**Figure 8.** Temporal change (%) of soil vulnerability estimation E01 vs. E02 (a), E31 vs. E32 (b), E01 vs. E31 (c), and E02 vs. E32 (d).

replacements. Two higher soil vulnerability areas in the north of the image are part of the reservoir basin without a soil profile. Late summer temporal changes were less evident for the nonirrigated areas by the absence of precipitations in 2000 and 2003. Soil vulnerability reductions were associated to irrigated areas less affected by the typical summer drought of the study area.

## **5. Conclusions**

The main advantage of the proposed model is the fact that it is associated to the easy data collection and computation, which can be used to evaluate soil vulnerability and erosion. The tests applied as an example remarks the great potential of the proposed approach and its great sensitivity to evaluate the actual state and detect temporal changes of soil vulnerability as a dynamic key parameter which plays significant role for soil conservation.

Our studies verified the utility of this simple and easy tool to update with satellite images and fuzzy logic model approach. The applicability of this approach for large and sparsely populated areas with limited field information could be useful in order to promote better land management strategies and more in‐depth analysis of soil degradation processes. Decision support systems for local and regional authorities and land management can be tools that help decision‐makers arrive to an adequate response for environment, society, and sustain‐ able future.

## **Author details**

The intense effect of drought mitigation in soil vulnerability values was estimated. This effect was more intense for the change analysis of late spring‐early summer when very few areas exhibited an increase of soil vulnerability estimations. **Figure 8 (c)** indicated a large increase of soil vulnerability in areas of the center and east of the image, which are produced by crop

**Figure 7.** Soil vulnerability estimations E01 (a), E02 (b), E31 (c), and E32 (d).

450 Modern Fuzzy Control Systems and Its Applications

**Figure 8.** Temporal change (%) of soil vulnerability estimation E01 vs. E02 (a), E31 vs. E32 (b), E01 vs. E31 (c), and E02

vs. E32 (d).

Ignacio Meléndez‐Pastor<sup>1</sup> , Jose Navarro Pedreño<sup>1</sup> \*, Ignacio Gómez Lucas<sup>1</sup> and Antonis A. Zorpas2

\*Address all correspondence to: jonavar@umh.es

1 Department of Agrochemistry and the Environment, Miguel Hernández University of Elche, Elche, Spain

2 Faculty of Pure and Applied Sciences, Environmental Conservation and Management, Laboratory of Chemical Engineering and Engineering Sustainability, Cyprus Open University, Nicosia, Cyprus

## **References**

[1] Doula MK, Sarris A. Soil Environment. In: Poulopoulos SG, Inglezakis JV editors. Environment and Development: Basic Principles, Human Activities and Environmental Implications. Netherlands: Elsevier; 2016. pp. 213‐216.


[2] D'Odorico P, Ravi S. Land degradation and environmental change. In: Sivanpillai R, Shroder JF Jr editors. Biological and Environmental Hazards, Risks, and Disasters.

[3] FAO. Soil Is A Non‐Renewable Resource. FAO fact sheet. Job Number I4373E/1/2015; 2015. 4 p. Available from: http://www.fao.org/documents/card/en/c/ec28fc04‐3d38‐4e35‐

[4] Häberli R, Luscher C, Praplan Chastonay B, Wyss C. L'affaire sol. Pour une politique raisonnée de l'utilisation du sol. Rapport final du programme national de recherche "Utilisation du sol en Suisse" (PNR 22), Switzerland: Georg Editeur SA; 1991. 192 p. [5] Blum WEH. The challenge of soil protection in Europe. Environmental Conservation.

[6] D'Odorico P, Bhattachan A, Davis KF, Ravi S, Runyan CW. Global desertification:

[7] Kort J, Collins M, Ditsch D. A review of soil erosion potential associated with biomass

[8] Department of Agriculture RSA. Soil erosion. Resource Centre, Directorate Agricultural

[9] COM/2006/0231. Communication from the Commission to the Council, the European Parlia‐ ment, the European Economic and Social Committee and the Committee of the Regions— Thematic Strategy for Soil Protection [SEC(2006)620] [SEC(2006)1165]. Available from: http:// eur‐lex.europa.eu/legal‐content/EN/TXT/?uri=CELEX:52006DC0231 [Accessed: 2017‐04‐03]

[10] Franchis L, Ibañez F, Fox T. Plan Blue Papers‐2, Threats to Soil in Mediterranean Countries‐

[11] COM 2006/0232. Proposal for a Directive of the European Parliament and of the Council establishing a framework for the protection of soil and amending Directive 2004/35/EC. Available from: http://eur‐lex.europa.eu/legal‐content/EN/TXT/?uri=CELEX%3A52006

[12] Lagachierie P. Digital soil mapping: A state of the art. In: Hartemink AE, McBratney A, Mendonça‐Santos ML editors. Digital Soil Mapping with Limited Data. Netherlands:

[13] FAO. The State of the World's Land and Water Resources for Food and Agriculture (SOLAW)—Managing Systems at Risk. Italy and United Kingdom: Food and Agriculture

[15] Urbanski JA. The use of fuzzy sets in the evaluation of the environment of coastal waters. International Journal of Geographical Information Sciences. 1999;**13**(7):723‐730. [16] Eastman JR. IDRISI Kilimanjaro. Guide to GIS and Image Processing. USA: Clark Uni‐

Organization of the United Nations and Earthscan; 2011. 308 p.

[14] Zadeh LA. Fuzzy sets. Information and Control. 1965;**8**:338‐353.

Drivers and feedbacks. Advances in Water Resources. 2013;**51**:326‐344.

Information Services. South Africa: Department of Agriculture; 2008. 7 p.

Netherlands: Elsevier. 2016. pp. 219‐227.

452 Modern Fuzzy Control Systems and Its Applications

8d9b‐e4427e20a4f7/ [Accessed: 2017‐04‐03]

crops. Biomass and Bioenergy. 1998;**14**(4):351‐359.

Document Review. France: UNEP; 2003. 78 p.

PC0232 [Accessed: 2017‐04‐03]

Springer; 2008. pp. 3‐14.

versity; 2003. 328 p.

1990;**17**:72‐74.


[35] Abrams M, Hook S, Ramachandran B. ASTER User Handbook. Version 2; USA: Jet Propulsion Laboratory‐NASA/California Institute of Technology; 2002. 135 p.

[36] Rouse JW, Hass RH, Schell JA, Deering DW. Monitoring vegetation systems in the great plains with ERTS. In Proceedings, Third Earth Resources Technology Satellite‐1

[37] Melendez‐Pastor I, Navarro‐Pedreño J, Koch M, Gómez I, Hernández EI. Evaluation of land degradation after forest fire with a fuzzy logic model. Environmental Engineering

[39] Morgan RPC. Erosión y conservación del suelo. Spain: Mundi‐Prensa; 1997. 348 p.

[40] Melendez‐Pastor I, Córdoba Sola P, Navarro‐Pedreño J, Gómez I. Evaluación de la vul‐ nerabilidad a la degradación por erosión en suelos mediante un modelo de lógica bor‐

Symposium. NASA SP‐351. USA: NASA; 1974; pp. 3010‐3017.

[38] Jenny H. Factors of Soil Formation. USA: McGraw‐Hill; 1941.

rosa. Revista de Ciências Agrarias. 2010;**33**:171‐181.

and Management Journal. 2013;**12**:2087‐2096.

454 Modern Fuzzy Control Systems and Its Applications

## *Edited by S. Ramakrishnan*

Control systems play an important role in engineering. Fuzzy logic is the natural choice for designing control applications and is the most popular and appropriate for the control of home and industrial appliances. Academic and industrial experts are constantly researching and proposing innovative and effective fuzzy control systems. This book is an edited volume and has 21 innovative chapters arranged into five sections covering applications of fuzzy control systems in energy and power systems, navigation systems, imaging, and industrial engineering. Overall, this book provides a rich set of modern fuzzy control systems and their applications and will be a useful resource for the graduate students, researchers, and practicing engineers in the field of electrical engineering.

Modern Fuzzy Control Systems and Its Applications

Modern Fuzzy Control

Systems and Its Applications

*Edited by S. Ramakrishnan*

Photo by solarseven / iStock