Many techniques have been reported [1-3] to solve the problem of unknown function identification. Intelligent techniques have also been applied to achieve this goal [4]. The identificantion method is presented [1-3] as an optimization problem where the objective function is defined as the total square error between the output and the reference signal. Since neural networks and fuzzy logic systems are universal approximators [5,6] nonlinear functions approximated by these approximators have widely been developed for many practical applications [7,8]. Moreover, many researches [8,9] combining fuzzy logic with neural networks have been developed to improve the efficiency of function approximation. Therefore, a fuzzy approximator was developed to obtain the input-output transfer characteristic of the nonlinear function.
Traditionally, fuzzy logic systems and/or neural networks are trained by using gradient-based methods, which may fall into a local minimum during the learning process. Unfortunately, such techniques suffer from many difficulties such as the choice of starting guess, convergence, etc. Moreover, since the cost function generally has multiple local minima, the attainment of the global optimum by these nonlinear optimization techniques is difficult [10].
In fuzzy set theory, the selection of appropriate membership functions has been an important issue for engineering problems [8]. It is important that the fuzzy membership functions are updated iteratively and automatically, because a change in fuzzy membership functions may alter the performance of the fuzzy logic system significantly.
Several researchers have proposed many methods to adjust the parameters of triangular or Gaussian membership function [11-14]. The fuzzy B-spline membership functions (BMFs) constructed in [10] possess the property of local control and have been successfully applied to fuzzy-neural control [15]. This is mainly due to the local control property of B-spline curve, i.e., the BMF has the elegant property of being locally tuned in a learning process. Several learning algorithms have been proposed in [7,15,16] to deal with the tuning of the BMFs.
To search for global optimal solutions, genetic algorithms [17-33] have drawn significant attentions in various fields due to their capability of directed random search.
Thanks to a probabilistic search procedure based on the mechanics of natural selection and natural genetics, the genetic algorithms are highly effective and robust over a broad spectrum of problems [34-36]. This motivates the use of the genetic algorithms [37-42]
to overcome the problems encountered by the conventional learning methods for fuzzy-neural networks. In the traditional GAs, the natural parameter set of the optimization problem needs to be coded as a finite length string. The coding operation maps a real number to a fixed length binary string. However, the coding process will lose some information due to truncation. Furthermore, one must face a trade-off problem between the length of the coding string and the resolution of the parameter value. To increase the resolution, a longer binary string must be chosen and hence, will slow down the convergence rate. It is well known that searching speed of the conventional genetic algorithms is not desirable. Such conventional genetic algorithms are inherently disadvantaged in dealing with a vast number (over 100) of adjustable parameters in the fuzzy-neural networks. Thus, a framework to automatically tune the
of the fuzzy-neural networks to approximate nonlinear functions using a simplified genetic algorithm (SGA) is proposed in this paper.
To start with, chromosomes consisting of adjustable parameters of the fuzzy-neural networks are coded as a vector with real number components and searched by the SGA.
The fitness value of each chromosome is obtained via a mapping from the error function, which is the difference between the outputs of the fuzzy-neural network and the desired outputs. Thus, an optimal set of adjustable parameters of the fuzzy-neural network can be obtained by repeating genetic operations, i.e., crossover and mutation, so that an optimal fuzzy neural network satisfying an error bound condition can be evolutionarily obtained. Because of the use of the SGA, faster convergence of the evolution process to search for an optimal fuzzy neural network can be achieved.
In the last years, the adaptive control of nonlinear systems has been an exciting research area. The control scheme via feedback linearization for nonlinear systems has been proposed in [43-46]. The fundamental idea of feedback linearization is to transform a nonlinear system dynamic into a linear one. Therefore, linear control or fuzzy control techniques can be used to acquire the desired performance. Recently, an adaptive fuzzy neural control system [47,48] has been proposed to incorporate with the expert information systematically, and the stability can be guaranteed by universal approximation theorem [49]. For systems with a high degree of nonlinear uncertainty, such as chemical process, aircraft, etc., they are very difficult to control using the conventional control theory. However, human operators can often successfully control them. Based on the fact that fuzzy logic and/or neural networks systems are capable of uniformly approximating a nonlinear function. Adaptive control is a popular technique of system identification and controller design to obtain a model of a system from input-output data and to design a controller. Using the conventional adaptive control, the adaptive fuzzy neural control has direct and indirect adaptive control categories [48].
Direct adaptive fuzzy neural control has been discussed in [47] and [48], in which the adaptive FNN controller uses a fuzzy logic system as a controller.
In this paper, we propose a method for designing both SGA-based indirect and direct adaptive fuzzy-neural controller for unknown nonlinear dynamical systems, in which the system state can be measured or not. The free parameters of the adaptive fuzzy-neural controller can be tuned on-line via the SGA approach. Also a supervisory controller is incorporated into the both adaptive control categories. If the closed-loop system controlled by the adaptive controller tends to unstable, especially in the transient period, the supervisory controller will be activated to work with the adaptive controller to stabilize the closed-loop system. On the other hand, if the adaptive controller works well, the supervisory controller will be deactivated.
Paragraphs are so arranged that Chapter 2 describes the fuzzy theory and the construction of the fuzzy-neural network. Chapter 3 describes B-spline membership functions. Chapter 4 gives details of the proposed simplified genetic algorithm (SGA).
Designing indirect adaptive FNN controllers is shown in Chapter 5. An example is illustrated in Chapter 5 to show the effectiveness of this approach. In chapter6 GA-based output-feedback direct adaptive fuzzy-neural controller (GODAF) is proposed to show the effectiveness of the stability in accordance with the output-feedback systems. The conclusion shows the last chapter.
CHAPER 2 FUZZY CONTROL SYSTEM
Fuzzy control was first introduced in early 1970’s [50] in an attempt to design controllers for systems that are structurally difficult to model due to naturally existing non-linearity and model complexities.
Since Mandani and his co-workers have successfully applied the fuzzy logic controller (FLC) to steam engine control, the fuzzy control theory has been widely applied to many fields [51][52][53]. The characteristic of FLC is that it adopts the linguistic control strategy to control plants without realizing their mathematic models.
The linguistic control strategy of FLC is constructed according to the operator experience and/or expert knowledge. Therefore, the FLC can control the complex and ill-defined industrial processes as well as the skilled operators do. Experiences show that the FLC yields results superior to those obtained by traditional control algorithm in the complex situation where the system model or parameters are difficult to obtain.
Typically, a fuzzy control system consists of four components: a fuzzification interface, a rule base, an inference engine, and a defuzzification interface as shown in Figure 2.1. More detail descriptions for each component are stated as follows:
2.1 Fuzzifiers
On account of the most applications, the inputs and the outputs of the fuzzy system are real valued numbers. So we must make the fuzzifier defined as a mapping from a real valued point x*∈U ⊂Rnto the fuzzy set A in U. we have three criteria to design the fuzzifier. In this paper, we present three fuzzifiers as follows:
1. Singleton fuzzifier: the singleton fuzzifier maps a real valued point x*∈Uinto the fuzzy singleton A in U, in which the membership value is 1 at x* and 0 at other points in U, i.e.,
=
= 0 otherwise ) 1
(
x*
x x
µA (2-1) 2. Triangular fuzzifier: the triangular fuzzifier maps x*∈Uinto the fuzzy set A in U, in which the membership function is written as:
− − − − = < =
=
otherwise 0
, 2 , 1 ,
|
| if
|) 1 |
(
*
*
|) 1 |
) ( (
*
*
1
* 1
1 x x b i n
b x x b
x x
x n i
n n
A
µ (2-2)
Fig. 2-1: The fuzzy system architecture
X in U
Fuzzy Rule Base
Fuzzifier Fuzzy Inference Defuzzifier
Fuzzy sets
in U Fuzzy sets
in V Y in V
X in U
Fuzzy Rule Base
Fuzzifier Fuzzy Inference Defuzzifier
Fuzzy sets
in U Fuzzy sets
in V Y in V
Fuzzy Rule Base
Fuzzifier Fuzzy Inference Defuzzifier
Fuzzy sets
in U Fuzzy sets
in V Y in V
where biare positive parameters and symbol * is often chosen as algebraic product or min.
3. Gaussian fuzzifier: the Gaussian fuzifier maps x*∈U into the fuzzy set A in U, in which the membership function is written as:
* 2 2
1 1*
1 ) ( )
(
*
* )
( n
n n x x x
x
A x e δ e δ
µ = − − − − (2-3) where δiare positive parameters and symbol * is often chosen as algebraic product or min.
Finally, we summarize the above fuzzifiers. The singleton fuzzifier greatly simplifies the computation involved in the fuzzy inference engine for all membership functions.
And the Gaussian and triangular fuzzifiers do, too. The Gaussian and triangular fuzzifiers can restrain noise in the input, but the singleton fuzzifier cannot.
2.2 Deffuzzifiers
The defuzzifier is defined as a mapping from a fuzzy set D in V ⊂R to a crisp point
* . V
y ∈ Hence, the task of the defuzzifier is to specify a point in V that represents the fuzzy set D. The three criteria should be considered as follows:
1. Plausibility: The point y*can represent D from intuitive point of view.
2. Computational simplicity: The criterion is important for fuzzy control.
3. Continuity: A small change in D should not result in a large change in y*. Hence, there are three types of defuzifiers.
1. Center of gravity Defuzzifier
The center of gravity defuzzifier specifies y*as the center of the area covered by the membership function of D.
∫
= ∫
V D
V D
dy y
dy y y y
) (
)
* ( µ
µ (2-4)
where ∫V is the conventional integral and V be universe of discourse and y be a mapping from V.
2. Center Average Defuzzifier
Let ylbe the center of the lth fuzzy set and wlbe its height. The center average defuzzifier presents y*as
∑
∑
=
= =M
l l
M
l l
l
w w y y
1 1
* (2-5) 3. Average Maximum Defuzzifier
The maximum defuzzifier chooses y*as the point in V, at which µD( y) achieves its maximum value. Define
)}
( sup ) (
| { )
(D y V y y
hgt D
V y
D µ
µ = ∈
∈
= . (2-6) hgt(D) is a set of all point in V, at which µD( y) achieves its maximum value. The maximum defuzzifier y* is defined as an arbitrary element in hgt(D), i.e.,
y*=any point in hgt(D)
The mean of maximum defuzzifier is defined as:
∫
= ∫
hgt D
hgt D
dy y
dy y y
y ( )
)
* (
µ
µ (2-7)
where ∫hgt(D)is an integration for the continuous part of hgt(D) and it is a summation for the discrete part of hgt(D).
2.3 Fuzzy Rule Base
The fuzzy rule base consists of fuzzy IF-THEN rules. It is a heart of the fuzzy system in the sense. And all other components are used to implement these rules in a reasonable and efficient manner. Hence, the fuzzy rule base comprises the following fuzzy IF-THEN rules:
Rule i: IF xl is A1i and …and xn is Ani THEN y is Di (2-8) The canonical fuzzy IF-THEN rules in the form of (2-13) includes the following ones:
(1) Partial rules:
IF x1 is A1i and …and xm is Ami THEN y is Di (2-9) (2) Or rules
IF x1 is A1i and …and xm is Ami or xm+1 is Ami+1 and …xn isAniTHEN y is Di (2-10) (3) Fuzzy statements
y is Di (2-11) 2.4 Fuzzy Inference
The fuzzy inference is a reasoning method using the fuzzy theory, and whereby the expert knowledge is presented using linguistic rules. For example: IF premise THEN conclusions, where premise is a statement in the fuzzy logic.
The fuzzy inference introduced as follows:
Product Inference : (y)=max=1 [sup∈ ( A(x)⋅
∏
A (xi)⋅ D(y))]U x M
D l l
i µ
µ µ
µ (2-12)
Minimum Inference: ( ) max[supmin( ( ), ( 1)..., ( ), ( ))]
1 x 1 x x y
y l l
l An n D
A A U
x M
D l µ µ µ µ
µ = = ∈ (2-13)
The product inference and minimum inference are the most commonly used fuzzy inference in the fuzzy system and other fuzzy applications.
CHATPER 3 B-spline Fuzzy-Neural Network (B-splineFNN)
The B-spline membership function (BMFs) introduced in [7,16] is adopted in this plan as the fuzzy membership function. The fuzzy B-spline membership functions (BMFs) constructed in possess the property of local control and have been successfully applied to fuzzy-neural control. This is mainly due to the local control property of B-spline curve, i.e., the BMF has the elegant property of being locally tuned in a learning process. In this chapter, the property of B-spline curves will be discussed.
3.1 Knot vector and B-spline Curves
A spline is a function, usually constructed using low–order polynomial pieces, and joined at breakpoints with certain smoothness conditions. The breakpoints are called knots. For α order, r+1 control points, the B-spline basis functions have the knot vector T ={ti,i =0,1,l,r+α} with t0 <t1 <t2 <m<tr+α. The following mixed types of knot vectors are adopted in this theis.
1. The knot vector is set to open uniform. The knot vector is defined as
To define B-spline basis functions, we need a parameter, the degree of these basis functions, α . This i-th B-spline basis function of degreeα , written as Ni,α(t), is defined recursively as following:
This above is usually referred to as the B-spline blend function. This definition looks complicated. But, it is not difficult to understand. If the degree is one (i.e., α=1), these basis functions are step functions. That is, basis functionNi,1(t)is 1 if t lies on the i-th span [ti, ti+1]. To understand the way of computingNi,α(t)forα greater than 1, let us use the triangular computation scheme. All knot spans are on the left (first) column and all degree zero basis functions on the second, shown in Fig. 3-1.
For r+1control points, {p0,p1,,pr}, the ith B-spline blending function of order α
3.2 B-spline Membership Function (BMF)
As a result, the B-spline membership function (BMF)µA(xq) introduced in [7,16]
is expressed as
∑
= membership functions and use the SGA (to be introduced in chapter 4) to obtain a set of optimal control points of BMFs. To avoid the increased number of control points, we use the BMFs as the version of fixed number of control points in [15], which is shown in Fig. 3-2.fuzzy variable x0
(x )0
Fig. 3-2. Illustration of fixed number of control points of BMF's of order 2.
Fig. 3-1 All knot spans are on the left (first) column and all degree one basis functions on the second
3.4 The Configuration of A B-spline FNN
Fig. 3-3 shows the configuration of a typical fuzzy-neural network. The system has a total of four layers. Nodes at layer I are input nodes (linguistic nodes) that represent input linguistic variables. Nodes at layer II are term nodes which act as BMFs to represent the terms of the respective linguistic variables. Each node at layer III is a fuzzy rule. Layer IV is the output layer.
3.5 A B-spline FNN Inference Method
Given the training input data xq,q=1,2,m,n, and the output data yp, p=1,2,m,m, the ith fuzzy rule has the following form:
i output ypof the fuzzy inference can be derived from the following equations:
∑
weighting vector. We assume that each input has the same number of BMFs and the ith BMF of the qth input has r+1 control points denoted asFig. 3-3. The configuration of a fuzzy neural network.
1
i
cq ={ciqj |cqji = pj, j=0,1,2,r}=[cqi0 cqi1cqri ]T.
Each input has z fuzzy sets (BMFs). If there are h rules in fuzzy rule base, then the adjustable set of all the control points is defined as
T T z n zT T T zT T
Tc c c c c c
c
c=[ 11 12 h 1 12 22 h 2 h ]
={cqji |i=1,2,h,z,q=1,2,h,n,j=0,1,h,r} (3-7) Hence, the objective of the learning algorithm is to minimize the error function:
2
*) (
) ,
( p p p
p w c y y
e = − (3-8) and
2
* ||
||
) ,
(w c Y Y
E = − (3-9) where w=[w1Tw2TwTm]T is a weighting vector of the fuzzy neural network,
T zT T
Tc c c
c
c=[ 11 12 1 12 c22Tc2zTcnzT]T is a control point vector of the BMFs, ]
[y1 y2 ym
Y = is an m-dimensional vector of the current outputs, and ]
[ 1∗ 2∗ ∗
∗ = y y ym
Y is an m-dimensional vector of the desired outputs acquired from specialists.
CHAPTER 4 Design of the FNN Identifiers by the Simplified Genetic Algorithms
In this chapter, a novel approach to adjust both control points of B-spline membership functions (BMFs) and weightings of fuzzy-neural networks using a simplified genetic algorithm (SGA) is proposed. Fuzzy-neural networks are traditionally trained by using gradient-based methods, and may fall into local minimum during the learning process.
Genetic algorithms have drawn significant attentions in various fields due to their capabilities of directed random search for global optimization. This motivates the use of the genetic algorithms to overcome the problem encountered by the conventional learning methods. However, it is well known that searching speed of the conventional genetic algorithms is not desirable. Thus far, such conventional genetic algorithms are inherently disadvantaged in dealing with a vast amount (over 100) of adjustable parameters in the fuzzy-neural networks. In this chapter, the SGA is proposed by using a sequential-search-based crossover point (SSCP) method in which a better crossover point is determined and only the gene at the specified crossover point is crossed as a single point crossover operation. Chromosomes consisting of both the control points of BMF’s and the weightings of fuzzy-neural networks are coded as an adjustable vector with real number components and searched by the SGA. Because of the use of the SGA, faster convergence of the evolution process to search for an optimal fuzzy-neural network can be achieved. Nonlinear functions approximated by using the fuzzy-neural networks via the SGA are demonstrated in this chapter to illustrate the effectiveness and applicability of the proposed method.
4.1 The Simplified Genetic Algorithm
To overcome the problems encountered by conventional genetic algorithms, we propose a simplified genetic algorithm (SGA) with a novel structure different from the conventional GAs to deal with a complicated situation where a vast number (over 100) of adjustable parameters are searched in the fuzzy-neural network.
4.2 Basic Concept of GAs
GAs are powerful search optimization algorithms based on the mechanics of natural selection and natural genetics. GAs can be characterized by the following features [24]:
• A scheme for encoding solutions to the problem, referred to as chromosomes;
• An evaluation function (referred to as a fitness function) that rates each chromosome relative to the others in the current set of chromosomes (referred to as a population);
• An initialization procedure for a population of chromosomes;
• A set of operators which are used to manipulate the genetic composition of the population (such as recombination, mutation, crossover, etc);
Basically, GAs are probabilistic algorithms, which maintain a population of individuals (chromosomes, vectors) for iteration. Each chromosome represents a potential solution to the problem at hand, and is evaluated to give some measure of its fitness. Then, selecting the more fit individuals forms a new population. Some members of the new population undergo transformations by means of genetic operators to form new solutions. After some number of generations, it is hoped that the system converges
with a near-optimal solution.
There are two primary groups of genetic operators, crossover and mutation, used by most researchers. Crossover combines the features of two parent chromosomes to form two similar offspring by swapping corresponding segments of the parents. The intuition behind the applicability of the crossover operator is information exchange between potential solutions. Mutation, on the other hand, arbitrarily alters one or more genes of a selected chromosome, by a random change with a probability equal to the mutation rate.
The intuition behind the mutation operator is the introduction of some extra variability into the population.
The GA described above, however, is a conventional one. In this chapter, we propose a simplified genetic algorithm (SGA), which are characterized by three simplified processes. Firstly, the population size is fixed and can be reduced to a minimum size of 4. Secondly, the crossover operator is simplified to be a single point crossover. Thirdly, only one chromosome in a population is selected for mutation.
Details will be discussed in the following section.
4.3 Evolutionary Processes of the Simplified Genetic Algorithm (SGA)
The adjusted parameters of FNN, which is both the control-points and weights or alternative, can be defined. In this section, we define the evolutionary processes of SGA by using both parameters w and c . For learning the adjustable parameters of the fuzzy-neural network shown in Chapter 3.4, we define the chromosome as
The adjusted parameters of FNN, which is both the control-points and weights or alternative, can be defined. In this section, we define the evolutionary processes of SGA by using both parameters w and c . For learning the adjustable parameters of the fuzzy-neural network shown in Chapter 3.4, we define the chromosome as