**4.1.1 Mamdani's method**

Mamdani's method is the most commonly used in applications, due to its simple structure fuzzy calculations. This method as a simple FIS method is used to solve almost general decision making problems for practical issues.

Let *X* be the universe of discourse and its elements be denoted as *x*. In the fuzzy theory, fuzzy set *A* of universe *X* is defined by function *μA(x)* called the membership function of set.

$$\begin{aligned} \mu\_A(\mathbf{x}): \mathbf{X} \in [0, 1] \text{ / where } \mu\_A(\mathbf{x}) = 1 \quad \text{if } \mathbf{x} \text{ is totally in } A;\\ \mu\_A(\mathbf{x}) = 0 \text{ if } \mathbf{x} \text{ is not in } A;\\ 0 < \mu\_A(\mathbf{x}) < 1 \text{ if } \mathbf{x} \text{ is partly in } A. \end{aligned}$$

Fuzzy Inference System as a Tool for Management of Concrete Bridges 455

The last step in the fuzzy inference process is defuzzification. Fuzziness helps us to evaluate

The input for the defuzzification process is the aggregate output fuzzy set and the output is

Fig. 3 depicts the flowchart for fuzzy logic analysis based on Mamdani's fuzzy inference

1

1

1

Fig. 3. Flowchart for fuzzy logic analysis based on Mamdani's fuzzy inference method

Adaptive Neuro Fuzzy Inference System **(**ANFIS) is a multilayer feed-forward network which uses neural network learning algorithms and fuzzy reasoning to map inputs into an

Input Membership Function

Perform Fuzzy Implication (i.e., map input to output)

Perform Fuzzy Aggregation (i.e., combine output membership functions Onto a single UOD)

> Perform Defuzzification (i.e., transform fuzzy output to non-fuzzy discrete number)

Output Membership Function

the rules, but the final output of a fuzzy system has to be a crisp number.

**Step 4.** Defuzzification

a single number (Esragh & Mamdani, 1981).

Define Universe of Discourse (UOD)


Define Membership Functions for Input and Output

Gaussian

Define Fuzzy Rule Base Relating Input and Output If … Then … If … Then … If … Then …

Perform Fuzzification Process (i.e. map non-fuzzy number onto UOD)

**4.1.2 Adaptive Neuro Fuzzy Inference System (ANFIS) method** 

method (Symans & Kelly, 1999).

1

1

(Symans & Kelly, 1999)

This set allows a continuum of possible choices. For any element *x* of universe *X*, membership function *μA(x)* equals the degree to which *x* is an element of set *A*. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element *x* in set *A*. Any universe of discourse consists of some sets describing some attributes to the output. The main idea of fuzzy set theory is dealing with linguistic variables (Tarighat & Miyamoto, 2009).

A linguistic variable is a fuzzy variable. For example, the statement "a is b" implies that the linguistic variable *a* takes the linguistic value *b*. In fuzzy systems, linguistic variables are used in fuzzy rules. The range of possible values of a linguistic variable represents the universe of discourse of that variable. A fuzzy rule can be defined as a conditional statement in the form:

IF (*x* is a) THEN (*y is* b)

where *x* and *y* are linguistic variables; and a and b are linguistic values determined by fuzzy sets on the universe of discourses *X* and *Y*, respectively. The main and most important characteristic of fuzzy systems is that fuzzy rules relate fuzzy sets to each other. Fuzzy sets provide the basis for output estimation model. The model is based on relationships among some fuzzy input parameters (Baldwin, 1981).

All these definitions and arrangements are used to infer output based on the inputs. The most commonly used fuzzy inference technique is the so-called Mamdani method. Mamdani method is widely accepted for capturing expert knowledge. It allows describing the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference entails a substantial computational burden.

The Mamdani-style fuzzy inference process is performed in four steps:

**Step 1.** Fuzzification of the input variables

The first step is to take the crisp inputs, *x1* and *y1*, and determine the degree to which these inputs belong to each of the appropriate fuzzy sets.

#### **Step 2.** Rule evaluation

The second step is to take the fuzzified inputs and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator *(AND or OR)* is used to obtain a single number that represents the result of the antecedent evaluation. This number (the truth value) is then applied to the consequent membership function. Now the result of the antecedent evaluation can be applied to the membership function of the consequent. The most common method of correlating the rule consequent with the truth value of the rule antecedent is to cut the consequent membership function at the level of the antecedent truth.

#### **Step 3.** Aggregation of the rule outputs

Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents previously found and combine them into a single fuzzy set.

The input of the aggregation process is the list of found consequent membership functions, and the output is one fuzzy set for each output variable.

### **Step 4.** Defuzzification

454 Fuzzy Inference System – Theory and Applications

This set allows a continuum of possible choices. For any element *x* of universe *X*, membership function *μA(x)* equals the degree to which *x* is an element of set *A*. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element *x* in set *A*. Any universe of discourse consists of some sets describing some attributes to the output. The main idea of fuzzy set theory is dealing with linguistic variables

A linguistic variable is a fuzzy variable. For example, the statement "a is b" implies that the linguistic variable *a* takes the linguistic value *b*. In fuzzy systems, linguistic variables are used in fuzzy rules. The range of possible values of a linguistic variable represents the universe of discourse of that variable. A fuzzy rule can be defined as a conditional statement

where *x* and *y* are linguistic variables; and a and b are linguistic values determined by fuzzy sets on the universe of discourses *X* and *Y*, respectively. The main and most important characteristic of fuzzy systems is that fuzzy rules relate fuzzy sets to each other. Fuzzy sets provide the basis for output estimation model. The model is based on relationships among

All these definitions and arrangements are used to infer output based on the inputs. The most commonly used fuzzy inference technique is the so-called Mamdani method. Mamdani method is widely accepted for capturing expert knowledge. It allows describing the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy

The first step is to take the crisp inputs, *x1* and *y1*, and determine the degree to which these

The second step is to take the fuzzified inputs and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator *(AND or OR)* is used to obtain a single number that represents the result of the antecedent evaluation. This number (the truth value) is then applied to the consequent membership function. Now the result of the antecedent evaluation can be applied to the membership function of the consequent. The most common method of correlating the rule consequent with the truth value of the rule antecedent is to cut the consequent membership function at the level of the

Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents previously found and combine them into a single fuzzy set. The input of the aggregation process is the list of found consequent membership functions,

(Tarighat & Miyamoto, 2009).

in the form:

IF (*x* is a) THEN (*y is* b)

**Step 2.** Rule evaluation

antecedent truth.

some fuzzy input parameters (Baldwin, 1981).

**Step 1.** Fuzzification of the input variables

**Step 3.** Aggregation of the rule outputs

and the output is one fuzzy set for each output variable.

inference entails a substantial computational burden.

inputs belong to each of the appropriate fuzzy sets.

The Mamdani-style fuzzy inference process is performed in four steps:

The last step in the fuzzy inference process is defuzzification. Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number.

The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number (Esragh & Mamdani, 1981).

Fig. 3 depicts the flowchart for fuzzy logic analysis based on Mamdani's fuzzy inference method (Symans & Kelly, 1999).

Fig. 3. Flowchart for fuzzy logic analysis based on Mamdani's fuzzy inference method (Symans & Kelly, 1999)

#### **4.1.2 Adaptive Neuro Fuzzy Inference System (ANFIS) method**

Adaptive Neuro Fuzzy Inference System **(**ANFIS) is a multilayer feed-forward network which uses neural network learning algorithms and fuzzy reasoning to map inputs into an

Fuzzy Inference System as a Tool for Management of Concrete Bridges 457

 *Ol,i=μBi-2(y), for i=3, 4*  where *x (or y)* is the input to node *i* and *Ai (or Bi-2)* is an attribute associated with this node. In other words, *Ol,i* is the membership grade of a fuzzy set *A (= A1, A2, B1 or B2)* and it specifies the degree to which the given input *x (or y)* satisfies the quantifier *A*. Here the membership function for *A* can be any appropriate parameterized membership function such as the

> A 2 <sup>1</sup> ( )

*x c a*

*i b i i*

1

where *{ai, bi, ci}* is the premise parameters set. Changing the values of these parameters leads to change of the bell-shaped function. Therefore various forms of membership

Layer 2: Every node in this layer is a fixed node labeled Prod, whose output is the product

Each node output represents the firing strength of a rule. In general, any other T-norm

Layer 3: Every node in this layer is a fixed node labeled Norm. The *ith* node calculates the

 3, 1 2 1,2 *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>w</sup> O w <sup>i</sup>*

where *wi* is a normalized firing strength from layer 3 and *{pi, qi, ri}* is the parameter set of

Layer 5: The only node of this layer is a fixed node labeled Sum, which computes the overall

5,1

*i i <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup>*

operators that perform fuzzy *AND* can be used as the node function in this layer.

ratio of the *ith* rule's firing strength to the sum of all rules' firing strengths:

Layer 4: Every node *i* in this layer is an adaptive node with a node function

this node. Parameters in this layer are referred to as consequent parameters.

*overall output O w f*

Outputs of this layer are called normalized firing strengths.

output as the summation of all incoming signals:

*Ol,i=μAi(x), for i=1, 2, or* (1)

*O2,i =wi = μAi (x) μBi (y), i = 1, 2* (3)

4, *i i* ( ) *O w f w px qy r i i i ii* (5)

*w f*

*w*

(6)

*w w* (4)

(2)

Following few paragraphs contain brief description of the different layers: Layer 1: Every node i in this layer is an adaptive node with a node function:

*x*

generalized bell function:

functions for fuzzy set *A* are possible.

of all the incoming signals:

output. It is a fuzzy inference system implemented in the framework of adaptive neural networks.

In order to explain ANFIS a fuzzy inference system with two inputs x and y and one output z is considered (Jang et al., 1997). In a first-order Sugeno fuzzy model with two fuzzy if-then rules we have:

Rule 1: If x is A1 and y is B1, then f1=p1x+q1y+r1

Rule 2: If x is A2 and y is B2, then f2=p2x+q2y+r2

Fig. 4 shows the reasoning procedure for the considered Sugeno model. Fig. 5 depicts the ANFIS architecture. As it is shown nodes of the same layer have similar functions. The output of the *ith* node in layer l is as *Ol,i*.

Fig. 4. A two-input first-order Sugeno fuzzy model with two rules

Fig. 5. Equivalent ANFIS architecture

output. It is a fuzzy inference system implemented in the framework of adaptive neural

In order to explain ANFIS a fuzzy inference system with two inputs x and y and one output z is considered (Jang et al., 1997). In a first-order Sugeno fuzzy model with two fuzzy if-then

Fig. 4 shows the reasoning procedure for the considered Sugeno model. Fig. 5 depicts the ANFIS architecture. As it is shown nodes of the same layer have similar functions. The

networks.

rules we have:

Rule 1: If x is A1 and y is B1, then f1=p1x+q1y+r1 Rule 2: If x is A2 and y is B2, then f2=p2x+q2y+r2

Fig. 4. A two-input first-order Sugeno fuzzy model with two rules

output of the *ith* node in layer l is as *Ol,i*.

Fig. 5. Equivalent ANFIS architecture

Following few paragraphs contain brief description of the different layers:

Layer 1: Every node i in this layer is an adaptive node with a node function:

$$\begin{aligned} \text{O}\_{l,i} &= \mu\_{\text{Ai}}(\text{x}), \qquad \text{for } i = 1, 2, or \\ \text{O}\_{l,i} &= \mu\_{\text{Bi}:2}(y), \qquad \text{for } i = 3, 4 \end{aligned} \tag{1}$$

where *x (or y)* is the input to node *i* and *Ai (or Bi-2)* is an attribute associated with this node. In other words, *Ol,i* is the membership grade of a fuzzy set *A (= A1, A2, B1 or B2)* and it specifies the degree to which the given input *x (or y)* satisfies the quantifier *A*. Here the membership function for *A* can be any appropriate parameterized membership function such as the generalized bell function:

$$\mu\_{\rm A}(\mathbf{x}) = \frac{1}{1 + \left| \frac{\mathbf{x} - c\_i}{a\_i} \right|^{2h\_i}} \tag{2}$$

where *{ai, bi, ci}* is the premise parameters set. Changing the values of these parameters leads to change of the bell-shaped function. Therefore various forms of membership functions for fuzzy set *A* are possible.

Layer 2: Every node in this layer is a fixed node labeled Prod, whose output is the product of all the incoming signals:

$$\mathbf{U}\_{2,i} = \mathbf{w}\_{\mathbf{i}} = \mu\_{Ai} \text{ (x) } \mu\_{Bi} \text{ (y), } i = \mathbf{1}, \text{ 2} \tag{3}$$

Each node output represents the firing strength of a rule. In general, any other T-norm operators that perform fuzzy *AND* can be used as the node function in this layer.

Layer 3: Every node in this layer is a fixed node labeled Norm. The *ith* node calculates the ratio of the *ith* rule's firing strength to the sum of all rules' firing strengths:

$$CO\_{3,i} = \overline{w\_i} = \frac{w\_i}{w\_1 + w\_2} \qquad \qquad i = 1,2 \tag{4}$$

Outputs of this layer are called normalized firing strengths.

Layer 4: Every node *i* in this layer is an adaptive node with a node function

$$
\overline{O}\_{4,i} = \overline{w}\_i f\_i = \overline{w}\_i (p\_i \mathbf{x} + q\_i y + r\_i) \tag{5}
$$

where *wi* is a normalized firing strength from layer 3 and *{pi, qi, ri}* is the parameter set of this node. Parameters in this layer are referred to as consequent parameters.

Layer 5: The only node of this layer is a fixed node labeled Sum, which computes the overall output as the summation of all incoming signals:

$$lowerall\ output = O\_{5,1} = \sum\_{i} \overline{w\_i} f\_i = \frac{\sum\_{i} w\_i f\_i}{\sum\_{i} w\_i} \tag{6}$$

Fuzzy Inference System as a Tool for Management of Concrete Bridges 459

There are two passes in the hybrid-learning procedure for ANFIS. In the forward pass of the hybrid-learning algorithm, functional signals go forward till layer 4 and the consequent parameters are identified by the least-squares estimate. In the backward pass, the error rates propagate backward and the premise parameters are updated by the gradient descent. When the values of the premise parameters are fixed, the overall output can be expressed as

learning procedure for ANFIS is shown schematically in Fig. 6 (Aydin et al., 2006).

**5.1 A fuzzy system for concrete bridge damage diagnosis (DIASYN system)** 

first of all, appropriate preliminary deterioration diagnosis and modeling.

fuzzy inference system can be used in bridge management issues.

about the impacts of design factors on bridge deterioration.

preliminary but important knowledge on individual bridge defects.

 

*w w ff f ww ww w px qy r w px qy r wxp wyq w r wxp wyq w r*

1 2

12 12 1 2 1 11 2 22 1 1 12 2 2 1 11 2 22 ( )( ) ( ) ( ) () ( ) ( ) ()

which is linear in the consequent parameters *p1, q1, r1, p2, q2, and r2* (Jang et al., 1997; Kandel & Langholz, 1993; Li et al., 2001; Sivanandam et al., 2007). A flowchart of hybrid

**5. Case studies and some typical applications of fuzzy inference system for** 

As it is mentioned in earlier parts of this chapter and is shown in Fig. 1 the main concerns of bridge management systems are diagnosis of the encountered problems (deteriorations and/or damage detection) and finding the current condition of the bridge structure. It was also discussed that diagnosis and current condition determination accompany with ambiguity. In this section some case studies and applications are presented to show how

Bridge management systems (BMSs) are being developed in recent years to assist various authorities on the decision making in various stages of bridge maintenance, which requires,

Diagnosis Synthesis (DIASYN) is a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims to provide bridge designers with valuable information

DIASYN is supposed to be a concept demonstration system for providing the bridge maintenance engineers and the bridge design engineers with assistance to obtain

The DIASYN system incorporates a fuzzy reasoning process containing a rule base with its acquisition and update facility and a fuzzy inference engine with an explanation facility, and a user interface with option selecting capacity. Fuzzy logic is utilized to handle uncertainties and imprecision involved. The rules are if-then statements that describe associations between fuzzy parameters. Given the required input data, the inference engine evaluates the rules and generates an appropriate conclusion. Users can choose to make diagnoses of new cases or to update the rule base with new training data through the user

1 2

(7)

a linear combination of the consequent parameters

**managing bridges** 

interface.

It can be observed that the ANFIS architecture has two adaptive layers: Layers 1 and 4. Layer 1 has modifiable parameters *{ai,bi, ci}* and *{aj,bj, cj}* related to the input MFs. Layer 4 has modifiable parameters *{pij,qij, rij}* pertaining to the first-order polynomial. The task of the learning algorithm for this ANFIS architecture is to tune all the modifiable parameters to make the ANFIS output match the training data. Learning or adjusting these modifiable parameters is a two-step process, which is known as the hybrid learning algorithm. In the forward pass of the hybrid learning algorithm, the premise parameters are hold fixed, node outputs go forward until layer 4 and the consequent parameters are identified by the least squares method. In the backward pass, the consequent parameters are held fixed, the error signals propagate backward and the premise parameters are updated by the gradient descent method. The detailed algorithm and mathematical background of the hybrid learning algorithm can be found in (Jang et al., 1997; Wang & Elhag, 2008).

The basic learning rule of ANFIS is the back propagation gradient descent, which calculates error signals (defined as the derivative of the squared error with respect to each node's output) recursively from the output layer backward to the input nodes. This learning rule is exactly the same as the back-propagation learning rule used in the common feed-forward neural networks. From the ANFIS architecture in Fig. 5, it is observed that given the values of premise parameters, the overall output *f* can be expressed as a linear combination of the consequent parameters. On the basis of this observation, a hybrid-learning rule is employed here, which combines the gradient descent and the least-squares method to find a feasible set of antecedent and consequent parameters. The details of the hybrid rule are given in (Jang et al., 1997), where it is also claimed to be significantly faster than the classical backpropagation method.

Fig. 6. Hybrid learning procedure of ANFIS

It can be observed that the ANFIS architecture has two adaptive layers: Layers 1 and 4. Layer 1 has modifiable parameters *{ai,bi, ci}* and *{aj,bj, cj}* related to the input MFs. Layer 4 has modifiable parameters *{pij,qij, rij}* pertaining to the first-order polynomial. The task of the learning algorithm for this ANFIS architecture is to tune all the modifiable parameters to make the ANFIS output match the training data. Learning or adjusting these modifiable parameters is a two-step process, which is known as the hybrid learning algorithm. In the forward pass of the hybrid learning algorithm, the premise parameters are hold fixed, node outputs go forward until layer 4 and the consequent parameters are identified by the least squares method. In the backward pass, the consequent parameters are held fixed, the error signals propagate backward and the premise parameters are updated by the gradient descent method. The detailed algorithm and mathematical background of the hybrid

The basic learning rule of ANFIS is the back propagation gradient descent, which calculates error signals (defined as the derivative of the squared error with respect to each node's output) recursively from the output layer backward to the input nodes. This learning rule is exactly the same as the back-propagation learning rule used in the common feed-forward neural networks. From the ANFIS architecture in Fig. 5, it is observed that given the values of premise parameters, the overall output *f* can be expressed as a linear combination of the consequent parameters. On the basis of this observation, a hybrid-learning rule is employed here, which combines the gradient descent and the least-squares method to find a feasible set of antecedent and consequent parameters. The details of the hybrid rule are given in (Jang et al., 1997), where it is also claimed to be significantly faster than the classical back-

learning algorithm can be found in (Jang et al., 1997; Wang & Elhag, 2008).

propagation method.

Fig. 6. Hybrid learning procedure of ANFIS

There are two passes in the hybrid-learning procedure for ANFIS. In the forward pass of the hybrid-learning algorithm, functional signals go forward till layer 4 and the consequent parameters are identified by the least-squares estimate. In the backward pass, the error rates propagate backward and the premise parameters are updated by the gradient descent. When the values of the premise parameters are fixed, the overall output can be expressed as a linear combination of the consequent parameters

$$\begin{aligned} f &= \frac{w\_1}{w\_1 + w\_2} f\_1 + \frac{w\_2}{w\_1 + w\_2} f\_2 \\ &= \overline{w}\_1 (p\_1 x + q\_1 y + r\_1) + \overline{w}\_2 (p\_2 x + q\_2 y + r\_2) \\ &= (\overline{w}\_1 x) p\_1 + (\overline{w}\_1 y) q\_1 + (\overline{w}\_1) r\_1 + (\overline{w}\_2 x) p\_2 + (\overline{w}\_2 y) q\_2 + (\overline{w}\_2) r\_2 \end{aligned} \tag{7}$$

which is linear in the consequent parameters *p1, q1, r1, p2, q2, and r2* (Jang et al., 1997; Kandel & Langholz, 1993; Li et al., 2001; Sivanandam et al., 2007). A flowchart of hybrid learning procedure for ANFIS is shown schematically in Fig. 6 (Aydin et al., 2006).
