Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

Huey-Yang Horng

## Abstract

The proportional-integral-derivative (PID) controllers are widely used in many industrial control applications. However, the lead-lag controllers are more practical. Traditionally, time-domain or frequency-domain methods have been used to design a lead-lag controller in order to meet the design specifications. This chapter will focus on the design of controller by optimizing the time-domain objective function. The proposed objective function includes the first peak time, maximum peak time, rise time, maximum overshoot, maximum undershoot, setting time, and steadystate error. In the study, cuckoo search algorithm is adopted to search the optimal controller parameters. Cuckoo search is a recently developed meta-heuristic optimization method, which is a population-based algorithm inspired by the behavior of some cuckoo species in combination with the Lévy flight behavior. A numerical example is simulated to illustrate the use of the proposed method.

Keywords: PID controller, lead-lag controller, controller design, cuckoo search

## 1. Introduction

In many industrial control applications, proportional-integral-derivative (PID) controllers are probably the most commonly used controllers. Several methods were proposed in the past for tuning the PID controller parameters [1–5]. It is noted the lead-lag controllers provide a more practical alternative. Tan used the Kharitonov and the Hermite-Biehler theorems to compute the parameters of lead-lag controller in [6]. Kuo et al. designed the lead-lag compensator based on vector margin and steady-state error of the step response [7]. Horng used cuckoo search to design leadlag controllers [8]. It is easy to see that the lead-lag controller is just a special case of the general second-order controllers. Therefore, this paper extends the lead-lag controllers to general second-order controllers. The goal of this paper is to propose a simple second-order controller design procedure using the cuckoo search.

The control design specifications may usually be divided into time-domain and frequency-domain specifications. Time-domain specifications (TDS) include the lower and/or upper bounds of the quantities of the time response such as the first peak time, maximum peak time, rise time, maximum overshoot, maximum undershoot, setting time, and steady-state error. Frequency-domain specifications are

usually given in terms of the resonant peak, phase margin, resonant frequency, and bandwidth. In this study, we only consider the time-domain specifications.

A typical unit-step response is shown in Figure 1. In this figure, yss, yM, and ym denote the steady-state value, maximum response value, and the response value where the maximum undershoot occurs, respectively. Moreover, Tr, Tp, and Ts are

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

We now describe seven time-domain specifications (TDS) used in objective

1.Rise time Tr. It is the time required for the step response to rise from 10 to 90%

the rise time, peak time, and settling time, respectively.

DOI: http://dx.doi.org/10.5772/intechopen.89832

4.Maximum overshoot MOS. This is defined as

MOS ¼

5.Maximum undershoot MUS. It is defined as

MUS ¼

stay within a specified �2% of the final value.

Unit-step response for underdamped second-order systems.

2.First peak time T <sup>f</sup> . It is the time to reach the first peak.

8 < :

3.Maximum peak time Tp. It is the time to reach the maximum peak.

yM � yss yss

ym ¼ minð Þ y tð Þ , t ≥T <sup>f</sup> ,

8 ><

>:

yss � ym yss

, if yM > yss,

, if ym < yss,

0, if ym ≥ yss:

6. Settling time Ts. This is the time required for the step response to decrease and

7.Steady-state error Ess. It is the difference between the desired and actual

(2)

(3)

0, if yM ≤ yss:

function in more detail:

responses.

Figure 1.

103

of its final value.

The main contribution of this study is that we provide a simple controller design procedure for simple second-order controllers. For the problem formulation, some of the time-domain specifications (e.g., desired peak time and maximum overshoot) can be used to fully define a desired simple second-order reference model. With this reference model, we may reasonably specify lower and/or upper bounds for other time-domain specifications. Finally, we define the deviation ratios (i.e., percentage errors) and total deviation ratio. The total deviation ratio, which is the weighted sum of the deviation ratios, will be used as the objective function for the problem. Zero objective value means that all specifications are satisfied. However, there is no guarantee of the zero objective value in the most general cases. Some specifications might actually be violated. The design goal is to choose the best controller parameters so that the objective value is as close to zero as possible.

To search for optimal controller parameters, some optimizer must be employed to solve the aforementioned optimization problem. Since there is usually no analytic formula for the objective function, evolutionary computation algorithms are well suited in this situation to solve this optimization problem. In this study, the metaheuristic cuckoo search algorithm is adopted to search the optimal controller parameters.

Cuckoo search algorithm is one of the latest meta-heuristic techniques, developed by Yang and Deb [9, 10]. Nowadays, implementations of cuckoo search has proved to be effective in engineering optimization problems [11–13], for example, optimal power system stabilizers [14], load frequency controller design [15], optimal power system [16], synthesis of analog controller [17], etc. Cuckoo search achieved better results than available methods in most cases which appeared in the literature.

The novelty of this study is that the whole controller design problem can be formulated as an optimization problem by considering most important timedomain performance indices as a whole. Moreover, the meta-heuristic cuckoo search algorithm (or any other powerful evolutionary computation algorithms like genetic algorithm or particle swarm optimization) can be adopted to search for the best controller parameters. A numerical example will be provided to illustrate the design process. To show the wide applicability of the proposed method, four different plants and three different time-domain specifications will be used in the illustrative example.

## 2. Time-domain analysis of control systems

The time response of a control system is typically divided into two parts: the transient response and the steady-state response. The steady-state response is just the part of the total response which remains after the transient has died out. Hence, the steady-state response can still vary in a fixed pattern, such as a ramp function or a parabola function that increases with time.

The underdamped second-order system is a familiar model for physical problems. The detailed understanding of the underdamped response is necessary for both analysis and design. Let us begin by describing the step response for the second-order system. The transfer function of an underdamped second-order system is given by

$$\mathcal{C}(\mathfrak{s}) = \frac{\alpha\_n^2}{s^2 + 2\xi a\_n \mathfrak{s} + a\_n^2}, \quad 0 < \xi < 1. \tag{1}$$

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

A typical unit-step response is shown in Figure 1. In this figure, yss, yM, and ym denote the steady-state value, maximum response value, and the response value where the maximum undershoot occurs, respectively. Moreover, Tr, Tp, and Ts are the rise time, peak time, and settling time, respectively.

We now describe seven time-domain specifications (TDS) used in objective function in more detail:


$$\text{MOS} = \begin{cases} \frac{\mathcal{Y}\_M - \mathcal{Y}\_{\text{sr}}}{\mathcal{Y}\_{\text{sr}}}, & \text{if } \mathcal{Y}\_M > \mathcal{Y}\_{\text{sr}}, \\ & \mathbf{0}, \qquad \text{if } \mathcal{Y}\_M \le \mathcal{Y}\_{\text{sr}}. \end{cases} \tag{2}$$

5.Maximum undershoot MUS. It is defined as

$$\mathcal{Y}\_m = \min(\mathcal{Y}(t)), \quad t \ge T\_f,$$

$$\text{MUS} = \begin{cases} \frac{\mathcal{Y}\_{ss} - \mathcal{Y}\_m}{\mathcal{Y}\_{ss}}, & \text{if } \mathcal{Y}\_m < \mathcal{Y}\_{ss}, \\\\ 0, & \text{if } \mathcal{Y}\_m \ge \mathcal{Y}\_{ss}. \end{cases} \tag{3}$$


Figure 1. Unit-step response for underdamped second-order systems.

usually given in terms of the resonant peak, phase margin, resonant frequency, and

The main contribution of this study is that we provide a simple controller design procedure for simple second-order controllers. For the problem formulation, some of the time-domain specifications (e.g., desired peak time and maximum overshoot) can be used to fully define a desired simple second-order reference model. With this reference model, we may reasonably specify lower and/or upper bounds for other time-domain specifications. Finally, we define the deviation ratios (i.e., percentage errors) and total deviation ratio. The total deviation ratio, which is the weighted sum of the deviation ratios, will be used as the objective function for the problem. Zero objective value means that all specifications are satisfied. However, there is no guarantee of the zero objective value in the most general cases. Some specifications might actually be violated. The design goal is to choose the best controller parameters so that the objective value is as close to zero as possible.

To search for optimal controller parameters, some optimizer must be employed to solve the aforementioned optimization problem. Since there is usually no analytic formula for the objective function, evolutionary computation algorithms are well suited in this situation to solve this optimization problem. In this study, the metaheuristic cuckoo search algorithm is adopted to search the optimal controller

Cuckoo search algorithm is one of the latest meta-heuristic techniques, developed by Yang and Deb [9, 10]. Nowadays, implementations of cuckoo search has proved to be effective in engineering optimization problems [11–13], for example, optimal power system stabilizers [14], load frequency controller design [15], optimal power system [16], synthesis of analog controller [17], etc. Cuckoo search achieved better results than available methods in most cases which appeared in the literature. The novelty of this study is that the whole controller design problem can be formulated as an optimization problem by considering most important timedomain performance indices as a whole. Moreover, the meta-heuristic cuckoo search algorithm (or any other powerful evolutionary computation algorithms like genetic algorithm or particle swarm optimization) can be adopted to search for the best controller parameters. A numerical example will be provided to illustrate the design process. To show the wide applicability of the proposed method, four different plants and three different time-domain specifications will be used in the

The time response of a control system is typically divided into two parts: the transient response and the steady-state response. The steady-state response is just the part of the total response which remains after the transient has died out. Hence, the steady-state response can still vary in a fixed pattern, such as a ramp function or

The underdamped second-order system is a familiar model for physical problems. The detailed understanding of the underdamped response is necessary for both analysis and design. Let us begin by describing the step response for the second-order system. The transfer function of an underdamped second-order sys-

> n s<sup>2</sup> þ 2ξωns þ ω<sup>2</sup>

n

, 0<ξ<1: (1)

C sðÞ¼ <sup>ω</sup><sup>2</sup>

parameters.

illustrative example.

tem is given by

102

2. Time-domain analysis of control systems

a parabola function that increases with time.

bandwidth. In this study, we only consider the time-domain specifications.

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Figure 2. Unit-step response for some underdamped higher-order systems.

Notice that for second-order systems, the first peak time is always the maximum peak time. However, for underdamped higher-order systems, they may not be the same, as illustrated in Figure 2. In general, the response of an underdamped highorder system is similar to that of an underdamped second-order system.

Notice also that the following relationships hold for second-order systems, which will be used in the illustrative example:

$$\xi = \frac{|\ln(\text{MOS})|}{\sqrt{\pi^2 + \ln^2(\text{MOS})}} \tag{4}$$

DR TDS K ð Þ ð Þ , a, b,c, d

Unity feedback system with second-order controller.

DOI: http://dx.doi.org/10.5772/intechopen.89832

TDS Kð Þ� , a, b,c, d TDS\_ub

TDS\_lb � TDS Kð Þ , a, b,c, d

desired interval. That is, the specification is fully satisfied.

tion ratio (TDR), used in this study as follows:

importance of the corresponding terms.

design specifications.

105

4. Cuckoo search algorithm

TDS\_ub , if TDS Kð Þ , <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>&</sup>gt;TDS\_ub,

(7)

�= X 7

i¼1 wi

(8)

TDS\_lb , if TDS Kð Þ , <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>&</sup>lt; TDS\_lb,

0, otherwise,

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

where TDS Kð Þ , a, b,c, d is any one of the time-domain specifications defined in the last section. In practical designs, some tolerances in time-domain specifications are allowed. Therefore, each of the time-domain specifications has a corresponding lower limit (TDS\_lb) and an upper bound (TDS\_ub). DR TDS K ð Þ ð Þ , a, b,c, d is a measure of how the actual quantity is close to the desired interval specified by lower limit and upper limit. For example, if the value of DR Tð Þ <sup>r</sup>ð Þ K, a, b,c, d is zero, where the relevant quantity is the rise time, then this means that the rise time lies in the

Next, we define the objective function TDR Kð Þ , a, b,c, d , called the total devia-

<sup>þ</sup>w<sup>4</sup> � ð Þ DR MOS K ð Þ ð Þ , <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>5</sup> � ð Þ DR MUS K ð Þ ð Þ , <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>2</sup>

<sup>þ</sup>w<sup>6</sup> � ð Þ DR Tð Þ <sup>s</sup>ð Þ <sup>K</sup>, <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>7</sup> � ð Þ DR Eð Þ ssð Þ <sup>K</sup>, <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>2</sup>

TDR Kð Þ¼½ , <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>w</sup><sup>1</sup> � ð Þ DR Tð Þ <sup>r</sup>ð Þ <sup>K</sup>, <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> <sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>2</sup> � DR T <sup>f</sup>ð Þ <sup>K</sup>, <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> � � � � <sup>2</sup>

In Eq. (8), ωi, i ¼ 1, 2, ⋯, 7 represents weights that reflect the relative

Eqs. (6)–(8) are improved versions of those in Ref. [8]. Once we defined the deviation ratio and total deviation ratio, the problem of the design controller becomes the minimization of TDR Kð Þ , a, b,c, d for all possible parameters. Now, we can use various optimization methods to implement the controller design. In the paper, the cuckoo search algorithm is used to minimize the objective function. Further, if the TDR Kð Þ , a, b,c, d is zero, all specifications are within the range of the

In this section, the cuckoo search algorithm is briefly introduced. This algorithm was proposed by Yang and Deb in [9, 10]. Cuckoo search represents an optimized

<sup>þ</sup>w<sup>3</sup> � DR Tpð Þ <sup>K</sup>, <sup>a</sup>, <sup>b</sup>,c, <sup>d</sup> � � � � <sup>2</sup>

¼

Figure 3.

8 >>>>><

>>>>>:

$$\rho\_n \quad = \frac{\pi}{T\_p \sqrt{1 - \xi^2}} \tag{5}$$

In the design process, a second-order system with satisfactory performances is designated as the reference standard. All the desired specifications with lower and upper bounds are tabulated. Note that the lower bounds of maximum overshot, maximum undershot, setting time, and steady error are set to zero.

## 3. Proposed controller

The transfer function of general second-order controller is written as

$$G\_c(s) = K \left(\frac{cs^2 + ds + 1}{as^2 + bs + 1}\right),\tag{6}$$

where K >0, a, b, c, d∈ ℜ: It is easy to see that the phase lead-lag controller is just a special case of general second-order controller. The overall control system in this study is shown in Figure 3, where Gpð Þs is the transfer function of the plant. Besides, r tð Þ, y tð Þ, and e tð Þ denote the reference input, output, and error signal.

For the feedback control system shown in Figure 3, the overall response is determined by the parameters of the controller. To establish the proposed timedomain objective function, we first define a function called the deviation ratio (DR), where TDS stands for the time-domain specifications described in Section 2: Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

#### Figure 3.

Notice that for second-order systems, the first peak time is always the maximum peak time. However, for underdamped higher-order systems, they may not be the same, as illustrated in Figure 2. In general, the response of an underdamped high-

order system is similar to that of an underdamped second-order system.

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which will be used in the illustrative example:

Unit-step response for some underdamped higher-order systems.

3. Proposed controller

104

Figure 2.

Notice also that the following relationships hold for second-order systems,

<sup>ξ</sup> <sup>¼</sup> j j lnð Þ MOS ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup><sup>2</sup> <sup>þ</sup> ln<sup>2</sup>

In the design process, a second-order system with satisfactory performances is designated as the reference standard. All the desired specifications with lower and upper bounds are tabulated. Note that the lower bounds of maximum overshot,

<sup>ω</sup><sup>n</sup> <sup>¼</sup> <sup>π</sup> Tp

maximum undershot, setting time, and steady error are set to zero.

The transfer function of general second-order controller is written as

GcðÞ¼ <sup>s</sup> <sup>K</sup> cs<sup>2</sup> <sup>þ</sup> ds <sup>þ</sup> <sup>1</sup>

as<sup>2</sup> þ bs þ 1 � �

where K >0, a, b, c, d∈ ℜ: It is easy to see that the phase lead-lag controller is just a special case of general second-order controller. The overall control system in this study is shown in Figure 3, where Gpð Þs is the transfer function of the plant. Besides, r tð Þ, y tð Þ, and e tð Þ denote the reference input, output, and error signal. For the feedback control system shown in Figure 3, the overall response is determined by the parameters of the controller. To establish the proposed timedomain objective function, we first define a function called the deviation ratio (DR), where TDS stands for the time-domain specifications described in Section 2:

ð Þ MOS

ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>q</sup> (4)

<sup>1</sup> � <sup>ξ</sup><sup>2</sup> <sup>p</sup> (5)

, (6)

Unity feedback system with second-order controller.

$$= \begin{cases} \begin{array}{l} \text{DR}(\text{TDS}(K,a,b,b,c,d)) \\ \text{TDS}(K,a,b,c,d) - \text{TDS\\_ub} \end{array} , & \text{if } \text{TDS}(K,a,b,c,d) > \text{TDS\\_ub}, \\\ b = \begin{cases} \begin{array}{l} \text{TDS\\_lb} - \text{TDS}(K,a,b,c,d) \\ \text{TDS\\_lb} \end{array} , & \text{if } \text{TDS}(K,a,b,c,d) < \text{TDS\\_lb}, \\\ 0, & \text{otherwise}, \end{cases} \end{cases} \end{cases} \tag{7}$$

where TDS Kð Þ , a, b,c, d is any one of the time-domain specifications defined in the last section. In practical designs, some tolerances in time-domain specifications are allowed. Therefore, each of the time-domain specifications has a corresponding lower limit (TDS\_lb) and an upper bound (TDS\_ub). DR TDS K ð Þ ð Þ , a, b,c, d is a measure of how the actual quantity is close to the desired interval specified by lower limit and upper limit. For example, if the value of DR Tð Þ <sup>r</sup>ð Þ K, a, b,c, d is zero, where the relevant quantity is the rise time, then this means that the rise time lies in the desired interval. That is, the specification is fully satisfied.

Next, we define the objective function TDR Kð Þ , a, b,c, d , called the total deviation ratio (TDR), used in this study as follows:

$$\begin{aligned} \text{TDR}(K, a, b, c, d) &= \left[ w\_1 \cdot \left( \text{DR}(T\_r(K, a, b, c, d)) \right)^2 + w\_2 \cdot \left( \text{DR}(T\_f(K, a, b, c, d)) \right)^2 \right] \\ &+ w\_3 \cdot \left( \text{DR}(T\_p(K, a, b, c, d)) \right)^2 \\ &+ w\_4 \cdot \left( \text{DR}(\text{MOS}(K, a, b, c, d)) \right)^2 + w\_5 \cdot \left( \text{DR}(\text{MUS}(K, a, b, c, d)) \right)^2 \\ &+ w\_6 \cdot \left( \text{DR}(T\_l(K, a, b, c, d)) \right)^2 + w\_7 \cdot \left( \text{DR}(E\_{\text{Z}}(K, a, b, c, d)) \right)^2 \right) / \sum\_{i=1}^7 w\_i \end{aligned} \tag{8}$$

In Eq. (8), ωi, i ¼ 1, 2, ⋯, 7 represents weights that reflect the relative importance of the corresponding terms.

Eqs. (6)–(8) are improved versions of those in Ref. [8]. Once we defined the deviation ratio and total deviation ratio, the problem of the design controller becomes the minimization of TDR Kð Þ , a, b,c, d for all possible parameters. Now, we can use various optimization methods to implement the controller design. In the paper, the cuckoo search algorithm is used to minimize the objective function. Further, if the TDR Kð Þ , a, b,c, d is zero, all specifications are within the range of the design specifications.

## 4. Cuckoo search algorithm

In this section, the cuckoo search algorithm is briefly introduced. This algorithm was proposed by Yang and Deb in [9, 10]. Cuckoo search represents an optimized

meta-heuristic algorithm that is biologically inspired by the way cuckoo looks for a nest where they could lay eggs in combination with the Lévy flight behavior of some birds and fruit flies.

ð Þ K, a, b,c, d and use cuckoo search algorithm to find the minimum value of TDR Kð Þ , a, b,c, d in Eq. (8). To save computation time, the initial populations of 25 host nests are selected using Routh-Hurwitz criterion (for linear time-invariant

pa ¼ 0:25. The optimization process is hierarchical. First, run cuckoo search 25 times to get minimum values. Then the 25 minimum values become elite group. Next, run cuckoo search one more time, and use the group as initial host nests. The

In the following design procedure, we set maximum generation to be 1000 and

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

In the section, a numerical example is provided to illustrate the design procedure. In the following simulations, four different plants are used for comparison,

As an instance, the step response of the uncompensated system of Plant 1 is shown in Figure 4. The peak time is estimated to be 2.4 s, the maximum overshoot is 26.5398%, and the steady-state error Ess is 0.5. Assume we are not satisfied with these time-domain performance indices. Consequently, compensation must be

Experiment 1. Keep in mind that our reference model is a simple second-order system defined in Eq. (1). Assume that the desired peak time Tp is set to be 1 s and the maximum tolerable overshoot is 0.03. Now we can use Eqs. (4) and (5) to calculate the corresponding damping ratio ξ and natural frequency ωn, respectively. This determines the desired reference model in Eq. (1). Based on the resulting reference model, we can calculate its seven performance indices depicted in Section 2 in order to establish reasonable bounds for these performance indices. We may put 2% tolerance on rise time Tr, first peak time T <sup>f</sup> , and maximum peak time Tp. Only upper bounds are specified for the remaining four performance indices. The full specifications for Experiment 1 are listed in the second column of Table 2. We set all the weights to be 1, i.e., ω<sup>i</sup> ¼ 1, i ¼ 1, 2, … , 7. Suppose the maximum steady-state error is limited to [0.001, 0.022], and then we can set the range of K to [7.4091, 166.5000]; the ranges of a, b, c, and d are all set to ½ � �100, 100 . The

The time-domain performances of the resulting closed-loop systems are shown in Table 5. Notice that the final objective values for four plants shown in the final row of Table 5 are all zeros, which means that all seven design specifications are all

Transfer function System type Closed-loop stability

<sup>s</sup>2þ12sþ<sup>20</sup> <sup>0</sup> Stable

ð Þ <sup>s</sup>þ<sup>1</sup> ð Þ <sup>s</sup>þ<sup>2</sup> ð Þ <sup>s</sup>þ<sup>4</sup> <sup>0</sup> Unstable

s sð Þ <sup>þ</sup><sup>1</sup> <sup>1</sup> Stable

The time responses due to unit-step input of the resulting four closed-loop

Plant 2 GpðÞ¼ <sup>s</sup> <sup>100</sup>sþ<sup>1</sup> <sup>s</sup>ð Þ <sup>10</sup>sþ<sup>1</sup> ð Þ <sup>s</sup>þ<sup>1</sup> <sup>1</sup> Stable

plants) so that the closed-loop systems are stable.

DOI: http://dx.doi.org/10.5772/intechopen.89832

6. Illustrative example

which are described in Table 1.

designed for better performances.

final result is the parameters for the second-order controller.

specified ranges of controller parameters are listed in Table 3. The resulting control parameters are shown in Table 4.

systems are shown in Figure 5. These responses look quite nice.

met using our controller design methods.

Plant 1 GpðÞ¼ <sup>s</sup> <sup>120</sup>

Plant 3 GpðÞ¼ <sup>s</sup> <sup>100</sup>

Plant 4 GpðÞ¼ <sup>s</sup> <sup>1</sup>

Description of the four plants.

Table 1.

107

Now we briefly describe some breeding behaviors of cuckoos. As pointed out in [9], some cuckoo species often lay the eggs in the nests of other host birds, especially those that just spawned eggs. Some host birds can engage direct conflict with the intruding cuckoos. If a host bird discovers the eggs are not their own, they may either throw these alien eggs away or simply abandon its nest and build a new nest elsewhere [9]. Some cuckoo species are often very specialized in the mimicry in color and pattern of the eggs of a few chosen host species, which reduces the probability of their eggs being abandoned and thus increases their reproductively.

In the optimization algorithm, each nest represents a potential solution. The process of cuckoo search algorithm is simplified by three rules [10]:


There are many variants of the cuckoo search algorithms. In the following, we describe a commonly used version. This algorithm uses a combination of a local random walk and the global random walk, controlled by a switching parameter pa. This allows for proper balance between exploration and exploitation of the solution space. The local and global random walks for generating the new solution of for cuckoo i can be written as, respectively,

$$\mathbf{x}\_{i}^{t+1} = \mathbf{x}\_{i}^{t} + a\mathbf{s} \otimes H(p\_{a} - \boldsymbol{\varepsilon}) \otimes \left(\mathbf{x}\_{j}^{t} - \mathbf{x}\_{k}^{t}\right) \tag{9}$$

$$\mathbf{x}\_{i}^{t+1} = \mathbf{x}\_{i}^{t} + aL(\mathbf{s}, \lambda) \tag{10}$$

$$L(\mathfrak{s}, \lambda) = \frac{\lambda \Gamma(\lambda) \sin(\pi \lambda/2)}{\pi} \cdot \frac{1}{s^{1+\lambda}}, \quad \mathfrak{s} \ge s\_0 \ge 0 \tag{11}$$

In Eqs. (9)–(11), the notations are explained as follows:


## 5. Design procedure

In the design procedure, a second-order system with satisfactory performances is designated as the reference standard. All of the seven desired specifications with lower bounds and upper bounds are tabulated. After that, we set solution vector as Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

ð Þ K, a, b,c, d and use cuckoo search algorithm to find the minimum value of TDR Kð Þ , a, b,c, d in Eq. (8). To save computation time, the initial populations of 25 host nests are selected using Routh-Hurwitz criterion (for linear time-invariant plants) so that the closed-loop systems are stable.

In the following design procedure, we set maximum generation to be 1000 and pa ¼ 0:25. The optimization process is hierarchical. First, run cuckoo search 25 times to get minimum values. Then the 25 minimum values become elite group. Next, run cuckoo search one more time, and use the group as initial host nests. The final result is the parameters for the second-order controller.

## 6. Illustrative example

meta-heuristic algorithm that is biologically inspired by the way cuckoo looks for a nest where they could lay eggs in combination with the Lévy flight behavior of some

process of cuckoo search algorithm is simplified by three rules [10]:

that the egg laid by a cuckoo is discovered by the host bird.

xtþ<sup>1</sup> <sup>i</sup> <sup>¼</sup> xt

L sð Þ¼ , <sup>λ</sup> <sup>λ</sup>Γð Þ<sup>λ</sup> sin πλ

In Eqs. (9)–(11), the notations are explained as follows:

⊗ entry-wise multiplication of two vectors

pa switching parameter used to switch local and global walks ε a random number drawn from a uniform distribution

L sð Þ , λ Lévy distribution used to define the step size of random walk

In the design procedure, a second-order system with satisfactory performances is designated as the reference standard. All of the seven desired specifications with lower bounds and upper bounds are tabulated. After that, we set solution vector as

<sup>k</sup> current positions of cuckoos j and k selected by random permutation

1.Each cuckoo lays an egg in a randomly selected nest.

Swarm Intelligence - Recent Advances, New Perspectives and Applications

2.The best nests will carry over to the next generation.

cuckoo i can be written as, respectively,

xtþ<sup>1</sup>

s step size

5. Design procedure

H Heaviside function

xt

xt j , xt

106

x<sup>t</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>x</sup><sup>t</sup>

<sup>i</sup> next position of cuckoo i

<sup>i</sup> current position of cuckoo i α step size scaling factor

Now we briefly describe some breeding behaviors of cuckoos. As pointed out in [9], some cuckoo species often lay the eggs in the nests of other host birds, especially those that just spawned eggs. Some host birds can engage direct conflict with the intruding cuckoos. If a host bird discovers the eggs are not their own, they may either throw these alien eggs away or simply abandon its nest and build a new nest elsewhere [9]. Some cuckoo species are often very specialized in the mimicry in color and pattern of the eggs of a few chosen host species, which reduces the probability of their eggs being abandoned and thus increases their reproductively. In the optimization algorithm, each nest represents a potential solution. The

3.The number of available host nest is fixed, and there is a positive probability

There are many variants of the cuckoo search algorithms. In the following, we describe a commonly used version. This algorithm uses a combination of a local random walk and the global random walk, controlled by a switching parameter pa. This allows for proper balance between exploration and exploitation of the solution space. The local and global random walks for generating the new solution of for

<sup>i</sup> <sup>þ</sup> <sup>α</sup><sup>s</sup> <sup>⊗</sup> H pa � <sup>ε</sup> <sup>⊗</sup> xt

=<sup>2</sup> ð Þ <sup>π</sup> � <sup>1</sup>

<sup>j</sup> � xt k 

<sup>i</sup> þ αL sð Þ , λ (10)

<sup>s</sup><sup>1</sup>þ<sup>λ</sup> , <sup>s</sup>≥s<sup>0</sup> <sup>≥</sup> <sup>0</sup> (11)

(9)

birds and fruit flies.

In the section, a numerical example is provided to illustrate the design procedure. In the following simulations, four different plants are used for comparison, which are described in Table 1.

As an instance, the step response of the uncompensated system of Plant 1 is shown in Figure 4. The peak time is estimated to be 2.4 s, the maximum overshoot is 26.5398%, and the steady-state error Ess is 0.5. Assume we are not satisfied with these time-domain performance indices. Consequently, compensation must be designed for better performances.

Experiment 1. Keep in mind that our reference model is a simple second-order system defined in Eq. (1). Assume that the desired peak time Tp is set to be 1 s and the maximum tolerable overshoot is 0.03. Now we can use Eqs. (4) and (5) to calculate the corresponding damping ratio ξ and natural frequency ωn, respectively. This determines the desired reference model in Eq. (1). Based on the resulting reference model, we can calculate its seven performance indices depicted in Section 2 in order to establish reasonable bounds for these performance indices. We may put 2% tolerance on rise time Tr, first peak time T <sup>f</sup> , and maximum peak time Tp. Only upper bounds are specified for the remaining four performance indices. The full specifications for Experiment 1 are listed in the second column of Table 2.

We set all the weights to be 1, i.e., ω<sup>i</sup> ¼ 1, i ¼ 1, 2, … , 7. Suppose the maximum steady-state error is limited to [0.001, 0.022], and then we can set the range of K to [7.4091, 166.5000]; the ranges of a, b, c, and d are all set to ½ � �100, 100 . The specified ranges of controller parameters are listed in Table 3.

The resulting control parameters are shown in Table 4.

The time-domain performances of the resulting closed-loop systems are shown in Table 5. Notice that the final objective values for four plants shown in the final row of Table 5 are all zeros, which means that all seven design specifications are all met using our controller design methods.


The time responses due to unit-step input of the resulting four closed-loop systems are shown in Figure 5. These responses look quite nice.

Table 1. Description of the four plants.

#### Figure 4.

Step response of the Plant 1 without controller.


The time-domain performances of the resulting closed-loop systems are shown in Table 7. Notice again that the final objective values for four plants are all zeros, which means that all seven design specifications are all met using our controller

Plant 1 Plant 2 Plant 3 Plant 4

Plant 1 Plant 2 Plant 3 Plant 4

K 8.4963 995.7351 14.6760 934.2808 a 100.0000 1.6668 1.7609 0.1364 b 30.4229 8.8714 60.2882 0.8138 c 3.1336 98.4428 0.6624 86.8253 d 7.3230 48.7921 1.2683 85.5579

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

Tr 0.4804 0.4734 0.4923 0.4735 T <sup>f</sup> 1.0131 1.0131 0.9804 1.0131 Tp 1.0131 1.0131 0.9804 1.0131 MOS 0.0000 0.0294 0.0297 0.0281 MUS 0.0179 0.0004 0.0093 0.0004 Ts 0.7717 1.0295 0.9968 1.0295 Ess 0.0192 0.0010 0.0054 0.0011 TDR 0000

The time responses due to unit-step input of the resulting four closed-loop

Time-domain performance indices of the resulting systems in Experiment 1.

Experiment 3. In this experiment, assume that the first peak time and the maximum peak time are both set to 2.0 s. Following the procedure in Experiment 1, the full design specifications for Experiment 3 are also shown in Table 2. The specified ranges of controller parameters are listed in Table 3. The resulting control

The time-domain performances of the resulting closed-loop systems are shown in Table 9. Notice that the final objective values for four plants are zeros, which means that all seven design specifications are met using our controller design

The time responses due to unit-step input of the resulting four closed-loop

In the illustrative example above, four different plants and three different timedomain specifications were used. As illustrated in the preceding example, before searching the best controller parameters, we first use some of the time-domain specifications (e.g., desired peak time and maximum overshoot) to fully specify the desired simple second-order reference model. Then we may use this reference model to define reasonable bounds for other time-domain specifications. Finally, we define the deviation ratios (i.e., percentage errors) and total deviation ratio as the objective function. If the final value of the objective function is zero, then all seven specifications are satisfied. But, in the most general cases, there is no

design methods.

Table 5.

Table 4.

Resulting controller parameters in Experiment 1.

DOI: http://dx.doi.org/10.5772/intechopen.89832

methods.

109

systems are shown in Figure 6.

parameters are shown in Table 8.

systems are shown in Figure 7.

#### Table 2.

Design specifications.


#### Table 3.

Search ranges of controller parameters.

Experiment 2. In this experiment, assume that the first peak time and the maximum peak time are both set to 1.5 s. Following the procedure in Experiment 1, the full design specifications for Experiment 2 are also listed in Table 2. The specified ranges of controller parameters are listed in Table 3. The resulting control parameters are shown in Table 6.


Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

Table 4.

Resulting controller parameters in Experiment 1.


#### Table 5.

Time-domain performance indices of the resulting systems in Experiment 1.

The time-domain performances of the resulting closed-loop systems are shown in Table 7. Notice again that the final objective values for four plants are all zeros, which means that all seven design specifications are all met using our controller design methods.

The time responses due to unit-step input of the resulting four closed-loop systems are shown in Figure 6.

Experiment 3. In this experiment, assume that the first peak time and the maximum peak time are both set to 2.0 s. Following the procedure in Experiment 1, the full design specifications for Experiment 3 are also shown in Table 2. The specified ranges of controller parameters are listed in Table 3. The resulting control parameters are shown in Table 8.

The time-domain performances of the resulting closed-loop systems are shown in Table 9. Notice that the final objective values for four plants are zeros, which means that all seven design specifications are met using our controller design methods.

The time responses due to unit-step input of the resulting four closed-loop systems are shown in Figure 7.

In the illustrative example above, four different plants and three different timedomain specifications were used. As illustrated in the preceding example, before searching the best controller parameters, we first use some of the time-domain specifications (e.g., desired peak time and maximum overshoot) to fully specify the desired simple second-order reference model. Then we may use this reference model to define reasonable bounds for other time-domain specifications. Finally, we define the deviation ratios (i.e., percentage errors) and total deviation ratio as the objective function. If the final value of the objective function is zero, then all seven specifications are satisfied. But, in the most general cases, there is no

Experiment 2. In this experiment, assume that the first peak time and the maximum peak time are both set to 1.5 s. Following the procedure in Experiment 1, the full design specifications for Experiment 2 are also listed in Table 2. The specified ranges of controller parameters are listed in Table 3. The resulting control

Experiment 1 Experiment 2 Experiment 3

Tr [0.4730, 0.4923] [0.7095, 0.7385] [0.9461, 0.9847] T <sup>f</sup> [0.9788, 1.0188] [1.4682, 1.5281] [1.9576, 2.0375] Tp [0.9788, 1.0188] [1.4682, 1.5281] [1.9576, 2.0375] MOS ≤0:03 ≤0:03 ≤ 0:03 MUS ≤0:02 ≤0:02 ≤0:02 Ts ≤1:0724 ≤1:6086 ≤2:1448 Ess ≤0:02 ≤0:02 ≤0:02

Swarm Intelligence - Recent Advances, New Perspectives and Applications

Plant 1 Plant 2 Plant 3 Plant 4

K [7.4091,166.5000] [45.4545, 1000] [7.4091, 166.5000] [45.4545, 1000] a [100,100] [100,100] [100,100] [100,100] b [100,100] [100,100] [100,100] [100,100] c [100,100] [100,100] [100,100] [100,100] d [100,100] [100,100] [100,100] [100,100]

parameters are shown in Table 6.

Search ranges of controller parameters.

Figure 4.

Table 2.

Table 3.

108

Design specifications.

Step response of the Plant 1 without controller.

Figure 5. Step responses of four resulting systems in Experiment 1.


#### Table 6.

Resulting controller parameters in Experiment 2.


guarantee that the final value of the objective function will always be zero. In that case, some specifications might be violated. The design goal is to choose the best controller parameters so that the objective value is as close to zero as possible. In the

Time-domain performance indices of the resulting systems in Experiment 3.

Plant 1 Plant 2 Plant 3 Plant 4

Plant 1 Plant 2 Plant 3 Plant 4

K 8.4963 995.7351 14.6760 934.2808 a 100.0000 1.6668 1.7609 0.1364 b 30.4229 8.8714 60.2882 0.8138 c 3.1336 98.4428 0.6624 86.8253 d 7.3230 48.7921 1.2683 85.5579

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

Tr 0.9656 0.9468 0.9794 0.9467 T <sup>f</sup> 1.9935 2.0262 1.9609 2.0262 Tp 1.9935 2.0262 1.9609 2.0262 MOS 0.0024 0.0292 0.0220 0.0284 MUS 0.0154 0.0004 0.0138 0.0004 Ts 1.4927 2.0589 1.9935 2.0589 Ess 0.0198 0.0010 0.0199 0.0011 TDR 0000

Figure 6.

Table 8.

Table 9.

111

Step responses of four resulting systems in Experiment 2.

DOI: http://dx.doi.org/10.5772/intechopen.89832

Resulting controller parameters in Experiment 3.

#### Table 7.

Time-domain performance indices of the resulting systems in Experiment 2.

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

Figure 6. Step responses of four resulting systems in Experiment 2.


#### Table 8.

Figure 5.

Table 6.

Table 7.

110

Step responses of four resulting systems in Experiment 1.

Swarm Intelligence - Recent Advances, New Perspectives and Applications

Resulting controller parameters in Experiment 2.

Plant 1 Plant 2 Plant 3 Plant 4

Plant 1 Plant 2 Plant 3 Plant 4

K 8.4963 995.7351 14.6760 934.2808 a 100.0000 1.6668 1.7609 0.1364 b 30.4229 8.8714 60.2882 0.8138 c 3.1336 98.4428 0.6624 86.8253 d 7.3230 48.7921 1.2683 85.5579

Tr 0.7133 0.7105 0.7384 0.7102 T <sup>f</sup> 1.5197 1.5197 1.4952 1.5197 Tp 1.5197 1.5197 1.4952 1.5197 MOS 0.0103 0.0282 0.0299 0.0281 MUS 0.0128 0.0004 0.0139 0.0004 Ts 1.0929 1.5442 1.5197 1.5442 Ess 0.0198 0.0010 0.0129 0.0010 TDR 0000

Time-domain performance indices of the resulting systems in Experiment 2.

Resulting controller parameters in Experiment 3.


#### Table 9.

Time-domain performance indices of the resulting systems in Experiment 3.

guarantee that the final value of the objective function will always be zero. In that case, some specifications might be violated. The design goal is to choose the best controller parameters so that the objective value is as close to zero as possible. In the

Figure 7. Step responses of four resulting systems in Experiment 3.

simulations above, it is fortunate that the final objective values of all simulations are zeros and all seven time-domain specifications are met. This also shows that the cuckoo search is an excellent optimizer for searching best controller parameters.

## 7. Conclusion

In this study, a design procedure of second-order controller for various plants has been proposed. The final controller was obtained by minimizing a time-domain cost function which is weighted sum of important time-domain performance indices including the rise time, first peak time, maximum peak time, maximum overshoot, maximum undershoot, setting time, and steady-state error. In our approach, the desired design specifications were built via a good second-order reference model. Cuckoo search algorithm was adopted to search the optimal controller parameters. Detailed simulations with different design specifications for four plants were provided to illustrate the use of the proposed design method. From the simulation results, the resulting performances of the closed-loop systems are quite good by using the proposed method, which justifies the usefulness of our method. We wish to point out that the methodology proposed in this study can easily be modified to handle the time delay or nonlinear plants. This constitutes an interesting future research topic.

Author details

Huey-Yang Horng

R.O.C

113

Department of Electronic Engineering, I-Shou University, Kaohsiung City, Taiwan,

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

DOI: http://dx.doi.org/10.5772/intechopen.89832

© 2019 The Author(s). Licensee IntechOpen. 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,

\*Address all correspondence to: hyhorng@isu.edu.tw

provided the original work is properly cited.

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

## Author details

simulations above, it is fortunate that the final objective values of all simulations are zeros and all seven time-domain specifications are met. This also shows that the cuckoo search is an excellent optimizer for searching best controller parameters.

In this study, a design procedure of second-order controller for various plants has been proposed. The final controller was obtained by minimizing a time-domain cost function which is weighted sum of important time-domain performance indices including the rise time, first peak time, maximum peak time, maximum overshoot, maximum undershoot, setting time, and steady-state error. In our approach, the desired design specifications were built via a good second-order reference model. Cuckoo search algorithm was adopted to search the optimal controller parameters. Detailed simulations with different design specifications for four plants were provided to illustrate the use of the proposed design method. From the simulation results, the resulting performances of the closed-loop systems are quite good by using the proposed method, which justifies the usefulness of our method. We wish to point out that the methodology proposed in this study can easily be modified to handle the time delay or nonlinear plants. This constitutes an interesting

7. Conclusion

Step responses of four resulting systems in Experiment 3.

Swarm Intelligence - Recent Advances, New Perspectives and Applications

Figure 7.

future research topic.

112

Huey-Yang Horng Department of Electronic Engineering, I-Shou University, Kaohsiung City, Taiwan, R.O.C

\*Address all correspondence to: hyhorng@isu.edu.tw

© 2019 The Author(s). Licensee IntechOpen. 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.

## References

[1] Panda S, Sahu BK, Mohanty PK. Design and performance analysis of PID controller for an automatic voltage regulator system using simplified particle swarm optimization. Journal of the Franklin Institute. 2012;349(8): 2609-2625

[2] Perng JW, Chen GY, Hsieh SC. Optimal PID controller design based on PSO-RBFNN for wind turbine systems. Energies. 2014;7(1):191-209

[3] Kesarkar AA, Selvaganesan N. Tuning of optimal fractional-order PID controller using an artificial bee colony algorithm. Systems Science & Control Engineering. 2015;3:99-105

[4] Ghosal S, Darbar R, Neogi B, Das A, Tibarewala DN. Application of swarm intelligence computation techniques in PID controller tuning: A review. In: Proceedings of the InConINDIA 2012, AISC 132; 2012. pp. 195-208

[5] Saroja K, Stefka Sharon R, Meena S, Chitra K. Multi-loop PID controller design for distillation column using firefly algorithm. International Journal of Engineering and Technology (IJET). 2017;9(2):1404-1410

[6] Tan N. Computation of stabilizing lag/lead controller parameters. Computers and Electrical Engineering. 2003;29:835-849

[7] Kuo YS, Lin JY, Tang JC, Hsieh JG. Lead-lag compensator design based on vector margin and steady-state error of the step response via particle swarm optimization. In: International Conference on Fuzzy Theory and Its Applications (iFuzzy); 2016. pp. 96-101

[8] Horng HY. Design of lead-lag controller via time-domain objective function by using cuckoo search. In: Lecture Notes in Electrical Engineering (ICITES2013); Vol. 293; 2014. pp. 1083-1091

[9] Yang XS, Deb S. Cuckoo search via Lévy flights. In: World Congress on Nature & Biologically Inspired Computing, December 2009; India. USA: IEEE Publications; 2009. pp. 210-214

of Engineering Research & Technology

DOI: http://dx.doi.org/10.5772/intechopen.89832

Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search

[17] Wongkaew B, Puangdownreong D.

nternational journal of circuits Systems and Signal Processing. 2019;13:79-84

Application of cuckoo search to synthesize analog controllersi.

(IJERT). 2017;6(11):24-28

115

[10] Yang XS, Deb S. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation. 2010;1(4):330-343

[11] Fister I Jr, Fister D, Fister I. Comprehensive review of cuckoo search variants and hybrids. International Journal of Mathematical Modelling and Numerical Optimisation. 2013;4(4): 387-409

[12] Yang XS, Deb S. Cuckoo search: Recent advances and applications. Neural Computing and Applications. 2014;24(1):169-174

[13] Shehab M, Khader AT, Al-Betar MA. A survey on applications and variants of the cuckoo search algorithm. Applied Soft Computing. 2017;61: 1041-1059

[14] Abd Elazim SM, Ali ES. Optimal power system stabilizers design via cuckoo search algorithm. International Journal of Electrical Power & Energy Systems. 2016;75:99-107

[15] Abdelaziz AY, Ali ES. Load frequency controller design via artificial cuckoo search algorithm. Electric Power Components & Systems. 2016;44(1): 90-98

[16] Anh L, Tam NM, Nghia LT, Anh QH. Optimal power system stabilizer parameters tuned by cuckoo search algorithm. International Journal Design of the Second-Order Controller by Time-Domain Objective Functions Using Cuckoo Search DOI: http://dx.doi.org/10.5772/intechopen.89832

of Engineering Research & Technology (IJERT). 2017;6(11):24-28

References

2609-2625

[1] Panda S, Sahu BK, Mohanty PK. Design and performance analysis of PID controller for an automatic voltage regulator system using simplified particle swarm optimization. Journal of the Franklin Institute. 2012;349(8):

Swarm Intelligence - Recent Advances, New Perspectives and Applications

(ICITES2013); Vol. 293; 2014.

[9] Yang XS, Deb S. Cuckoo search via Lévy flights. In: World Congress on Nature & Biologically Inspired Computing, December 2009; India. USA: IEEE Publications; 2009.

[10] Yang XS, Deb S. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation.

[11] Fister I Jr, Fister D, Fister I.

Comprehensive review of cuckoo search variants and hybrids. International Journal of Mathematical Modelling and Numerical Optimisation. 2013;4(4):

[12] Yang XS, Deb S. Cuckoo search: Recent advances and applications. Neural Computing and Applications.

[13] Shehab M, Khader AT, Al-Betar MA. A survey on applications and variants of the cuckoo search algorithm. Applied Soft Computing. 2017;61:

[14] Abd Elazim SM, Ali ES. Optimal power system stabilizers design via cuckoo search algorithm. International Journal of Electrical Power & Energy

Systems. 2016;75:99-107

[15] Abdelaziz AY, Ali ES. Load

[16] Anh L, Tam NM, Nghia LT, Anh QH. Optimal power system stabilizer parameters tuned by cuckoo search algorithm. International Journal

frequency controller design via artificial cuckoo search algorithm. Electric Power Components & Systems. 2016;44(1):

pp. 1083-1091

pp. 210-214

387-409

1041-1059

90-98

2010;1(4):330-343

2014;24(1):169-174

[2] Perng JW, Chen GY, Hsieh SC. Optimal PID controller design based on PSO-RBFNN for wind turbine systems.

[3] Kesarkar AA, Selvaganesan N. Tuning of optimal fractional-order PID controller using an artificial bee colony algorithm. Systems Science & Control

[4] Ghosal S, Darbar R, Neogi B, Das A, Tibarewala DN. Application of swarm intelligence computation techniques in PID controller tuning: A review. In: Proceedings of the InConINDIA 2012,

[5] Saroja K, Stefka Sharon R, Meena S, Chitra K. Multi-loop PID controller design for distillation column using firefly algorithm. International Journal of Engineering and Technology (IJET).

[6] Tan N. Computation of stabilizing lag/lead controller parameters.

Computers and Electrical Engineering.

[7] Kuo YS, Lin JY, Tang JC, Hsieh JG. Lead-lag compensator design based on vector margin and steady-state error of the step response via particle swarm optimization. In: International Conference on Fuzzy Theory and Its Applications (iFuzzy); 2016. pp. 96-101

[8] Horng HY. Design of lead-lag controller via time-domain objective function by using cuckoo search. In: Lecture Notes in Electrical Engineering

Energies. 2014;7(1):191-209

Engineering. 2015;3:99-105

AISC 132; 2012. pp. 195-208

2017;9(2):1404-1410

2003;29:835-849

114

[17] Wongkaew B, Puangdownreong D. Application of cuckoo search to synthesize analog controllersi. nternational journal of circuits Systems and Signal Processing. 2019;13:79-84

## *Edited by Javier Del Ser, Esther Villar and Eneko Osaba*

Swarm Intelligence has emerged as one of the most studied artificial intelligence branches during the last decade, constituting the fastest growing stream in the bioinspired computation community. A clear trend can be deduced analyzing some of the most renowned scientific databases available, showing that the interest aroused by this branch has increased at a notable pace in the last years. This book describes the prominent theories and recent developments of Swarm Intelligence methods, and their application in all fields covered by engineering. This book unleashes a great opportunity for researchers, lecturers, and practitioners interested in Swarm Intelligence, optimization problems, and artificial intelligence.

Published in London, UK © 2019 IntechOpen © gonin / iStock

Swarm Intelligence - Recent Advances, New Perspectives and Applications

IntechOpen Book Series

Artificial Intelligence, Volume 2

Swarm Intelligence

Recent Advances,

New Perspectives and Applications

*Edited by Javier Del Ser,* 

*Esther Villar and Eneko Osaba*