Design of an adaptive fuzzy logic controller for nonlinear dynamic system Текст научной статьи по специальности «Медицинские технологии»

УДК 621.3

DESIGN OF AN ADAPTIVE FUZZY LOGIC CONTROLLER FOR NONLINEAR

DYNAMIC SYSTEM

Ho Dac Loc

В этой статье разработан адаптивный нечеткий контроллер для обобщенной высокопорядковой нелинейной континуальной системы. Синтезированный контроллер обеспечивает: 1) глобальную стабильность циклически замкнутой системы; 2) асимптотическую сходимость к нулю ошибки слежения. Нечеткий контроллер синтезируется на основе коллекции правил "Если-то". Параметры членских функций характеризуют лингвистические выражения, измеряющиеся в соответствии с несколькими адаптивными законами с целью управления слежением за установкой по ориентировочной траектории.

In this paper, an adaptive fuzzy controller is designed for a general higher-order nonlinear continuous system. The synthesized controller ensures that 1) the close-loop system is globally stable and 2) the tracking error converges to zero asymptotically. The fuzzy controller is synthesized from a collection of IF-THEN rules. The parameters of the membership functions characterizing the linguistic terms change according to some adaptive law for the purpose of controlling a plant to track a reference trajectory.

1 INTRODUCTION

In recent years the adaptive control of nonlinear systems has attracted a lot of attention. Control methodology called feedback linearization has been proved sound and successful in some problems. The central concept of this approach is to transform the nonlinear system dynamic into an equivalent linear systems, so that conventional linear control techniques can be applied [1, 2]. A key assumption in these studies is that the system nonlinearities are known a priori, or can be linearizable parameterization. This presents a limitation of the theory because the real system, or more precisely the model of a system, may always contain with a nonlinear uncertain elements. Therefore, the design of a robust adaptive controller that deals with a nonlinear system with uncertainties is an important subject. So far, to deal with uncertain nonlinear systems, many adaptive control approaches have been proposed. Adaptive control approaches are applied to the systems with parameterized uncertainties, several results can be found in [3-6]. The above discussion makes apparent that adaptive control research has thus far been directed towards the systems with parametric uncertainties, for the totally unknown nonlinear systems cannot be discussed. The need to deal with uncreasingly complex systems, to accomplish in creasingly demanding design requirements and the need to obtain these requirements with less precise advanced knowledge of the plant and its environment, inspired many works that came mostly from the area of fuzzy logic control.

Recently, the integration of fuzzy logic techniques and conventional control approaches has been an active research

focus. It is claimed that this union will lead to new control algorithms that exploit the advantages of both paradigms. The property of uniformly approximating any nonlinear systems over compact input-space with fuzzy systems [7] lays the foundation for this integration and provides a bridge to convert a set of human heuristic rules into a mathematical description. Wang [8] suggested the utilization of conventional adaptive systems to the fuzzy system framework, and presented an in-depth and thorough analysis of adaptive fuzzy logic control based on the general error dynamics of adaptive system. The most important challenge to this approach is to solve the Lyapunov equation and find the positive definite matrix. Passino et. al. [9] developed a fuzzy model reference learning control by introducing a reference model for defining the desired process characteristics where a fuzzy inverse model has to be specified by the control engineer in advance. Also, many other excellent works on introducing the concept of fuzzy logic into the conventional control techniques and vice versa have been developed in the past years [10, 11-14], to name but a few.

However, most practical applications of fuzzy logic control have been limited to relatively simple problems, due to a lack of formal synthesis techniques which guaranties the very basic requirements of global stability and acceptable performances [15, 16]. The design of the globally stable fuzzy control systems was an open problem until recent efforts presented in [16]. Based on fuzzy logic systems which are capable of approximating, with arbitrary, any real continuous function on a compact set, a globally stable adaptive controller is first synthesized from a collect of IF-THEN rules. The fuzzy systems used to approximate an optimal controller, is adjusted by an adaptive law based on a Lyapu-nov function synthesis approach. However, this adaptive fuzzy control system is limited to the linearizable SISO nonlinear systems with control gain being constant. On the other hand, it cannot eliminate the effect of the modeling errors and of disturbances on the error output of the system. By introducing sliding mode control techniques, [15] designed a fuzzy controller for the same system as [16]. But the uses of the sliding mode, which is a discontinuous control, generally creates various problems, such as the chattering phenomena and possible excitation of high-frequency unmodeled dynamics.

In this paper, we will develop a robust adaptive control scheme for a class of unknown nonlinear systems. The basic architecture of our adaptive fuzzy controller is a standard fuzzy logic controller (FLC) [17] used in most fuzzy control systems, plus an adaptive law used for adjusting the parameters of FLC.

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2 DESCRIPTION OF FUZZY LOGIC SYSTEMS

The basic configuration of the fuzzy logic system is shown in Fig. 1. The fuzzy logic system performs a mapping from

UiRn to R . Let U = U1 x ... x Un where Ut eR , i = 1, 2, ..., n . The fuzzy rule base consists of a collection of fuzzy IF-THEN rules

Rl :

IF x, is F[ AND x, is F\ AND...AND x is Fl

11 2 2 n n

THEN yl is Cl (l = 1, ...,M),

f 11 (ft mF{ (x}>)

<( x ) = 1T Z( x ),

n m fi (x)

Z l (x ) = ' =1 i

(ft mFl(4 l = 1 i = 1

(1) Figure 1 - Basic configuration of a fuzzy logic system

where x = (x^, ..., xn)T e U, y e VcR are the input and output of fuzzy logic systems, respectively, Fj, Cj are the fuzzy sets defined on Ui and R , respectively. The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in R , based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point x = (x.[, ..., xn )T into a fuzzy set in U. The deffuzifier maps a fuzzy set in V to a crisp point in V.

The fuzzy logic systems of Fig.1 comprise a very rich class of static systems mapping from U c. Rn to V c. R , because many different choices are available within each block, and in addition, many combinations of these choices can result in a useful subclass of fuzzy logic systems. One subclass of fuzzy logic systems is used here as building blocks of our adaptive fuzzy controller, and is described by the following important result.

Lemma 1. The fuzzy logic systems with center-average defuzzifier, product inference and singleton fuzzifier are in the following form

There are two main reasons for using the fuzzy logic system (3) as basic building blocks of adaptive fuzzy controller. First, it was proved that the fuzzy logic system in the form of (3) are universal approximates, i.e., for any given real continuous function g on the compact set U, there exits a fuzzy logic system in form of (9) such that it can uniformly approximate g over U to arbitrary accuracy. Therefore, the fuzzy logic system (3) are qualified as building blocks of adaptive fuzzy controllers for nonlinear systems. Second, the fuzzy logic system (3) are constructed from the fuzzy IF-THEN rules of (1), therefore, linguistic information from human experts can be directly incorporated into the controllers.

3 BASIC IDEAS OF CONSTRUCTING STABLE ADAPTIVE FUZZY CONTROLLERS

In this section, we first setup the control objectives, and then show how to develop an adaptive fuzzy logic controller to achieve these control objectives. Consider the n th-oder nonlinear systems of the form

x i = f\ (X)

x 2 = f2 (x )

u (x ) = m-'—1-, (2)

(ilm f ( 4)

l = 1 i = 1

where ll is the point at which machieves its maximum

value, and we assume that m ci (l1 ) = 1 . Eq.(2) can be written as

(5)

(3)

where l = (l1, ..., ) is a parameter vector, and

Z(x) = (Zi(x), .••, Zm(x))T is regressive vector, Z(x) is defined fuzzy basic function (FBF) []

(4)

xn = fn( x) + u

y = h (x ),

where x = (xi, ..., xn)T is the state vector, f., i = 1, n ;

h(.) are unknown functions, u e R and y e R are the input and output of system, respectively.

Control objectives: Determine a feedback control u based on fuzzy logic system (3) and an adaptive law for adjusting the parameter vector l such that the following conditions are met:

a) lim Vm -y\ = lim \ym - h(x)| = 0 where ym is the t 1 t

output tracking signal;

b) the cost function J = f[Y( x ) 2 + Y( x )2 ] dt is mini-

0

mized, Y(x) is a differentiable function of state variables and Y( 0) = 0 .

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ISSN 1607-3274 "Pa^ioe^eKTpoHiKa. iH^opMaTHKa. YnpaBëiHHfl" № 1, 2003

We now show the basic ideas of how to construct a direct adaptive fuzzy controller to achieve these control objectives.

To begin, let e = (e, e)T = (e^, e2)T and

Y(x) = e1 + ae2 . To ensure the control objective b), we

have

Y(e) + Y(e) = 0 .

Using (5) we have:

p5Y(e)l -1

L ¡xn \

Y( e) +

9Y( e)

d x J1 i = 1 oxi

f ( X )

i = 1 dxi ' dxn

n -1¡Y ¡Y

i = 13 x/! 9x

fn

¡Y

.9x„.

■ n -1¡Y "

Y+ - if

■ ! = \ Cxi H

xn = fn(x) + u + u- u.

■<--Y"

■ n - 1

Y + - gf

i = 1

Next, we develop an adaptive law to adjust the parameters of vector 1, which provides that the close-loop system is generally stable. Define the Lyapunov function in the form

v = 2 Y( e )2 + 2?geer,

(12)

(6)

where 9 = 1 -1*, g is positive constant. Using (11) we have a derivative of Lyapunov function as:

Vp _ ? 3Y. _ " - 1 3Y, ( ) + ^Y ff() + " 1 (7\

Y = Ix x' = I dXf'(x) + aT(x) +u], (7)

i = j i i = 1 ax i uxn

where u is the optimal control. Solving Eq.(6), using (7), we have

v= yy + !eT e = - y 2 + ¡-¡YYeTZ( e) +1 e T1 = g- - dx„ - - y~ -

YZ( e) + 1 1 .¡x- " g-J

= - Y2 + eT

If we choose the adaptive law

(8)

1 = -g ¡YyZ( e),

(13)

(14)

If the functions f (.) , i = 1, n and h(.) are known, the control (8) ensures the control objective a). We can show it using the follow equation:

Y+Y =Y + i jYfr+!- (fn+u) =

= 0. (9)

From (9) we can see that: lim |Y| = 0 or

t ® ~

lim \ym - h(x)| = 0 - the main control objective. t ® ¥ 1

Since fi(x) , i = 1, n and h(x) are unknown, the optimal control u of (8) cannot be implemented. Our purpose is to design a fuzzy logic system to approximate this optimal control.

Eq. (5) can be written as:

x 1 = f1 (x) x 2 = f2 (x )

then (13) becomes V = -Y2 < 0 , which guaranties that the close-loop adaptive system is generally stable.

The overall scheme of the developed adaptive fuzzy control system is shown in Fig. 2. Now, we make some few remarks.

Remark 1. The developed, in this paper, adaptive fuzzy control system using the error vector as input signal for the controller is suitable in situations, where the state vector is not measured variables.

Remark 2. The simple adaptive law makes easy to implement the adaptive fuzzy control. The quality of the control system depends on form and parameters of function Y .

Remark 3. In using the adaptive law (14), the choice of the constant g is important. At present, there has been no theoretical guidance about the choice of g; trial-and-error seems to be only practical option.

ym .ISA c FLC u Plant

(10)

Substituting (8) and approximated control signal by

fuzzy logic system u = 1 * TZ( e) into (10), we have:

x 1 = f (X ) X 2 = f2 (- )

Adaptiv e

1(0)

+ (1T -1 *T )Z( e). (11)

Figure 2 - The overall scheme of adaptive fuzzy control system

4 SIMULATION

In this section, we apply the direct adaptive fuzzy controller in the last section to control an unstable nonlinear dynamic systems.

Example1. In this example, we use our direct adaptive fuzzy controller to regulate the plant:

y

xn = -

IHOOPMATHKA

X2 = -0,1 x2 -x^ + 12cos(t) + u + d(t).

(15)

If without control, the system is chaotic. Here the objective is to use the developed adaptive fuzzy logic controller to let system output track the desired sin input and square input, respectively. Moreover, a random disturbance with 0 mean and 1 variance is added to verify the performance of the control system. Fig.3 and Fig.4 are the simulation results, which indicate that the tracking performance is very good even in the presence of random disturbance.

The step respond of close-loop control system is shown in Fig. 5, in which we can see the adaptive fuzzy controller could regulate the plant and the close-loop system is stable.

the adaptive fuzzy controller could regulate the plant and the close-loop system is stable.

Figure 5 - The step respond of the control system

The step respond of the control system in situation, where g has different values, is shown in Fig.6. From this we can see that the quality of the transient process depends on g.

Figure 3 - Control results with AFLC (a = 1; g = 500 )

I

r

E !

E OS

I

f / ■i \ / r :

■5

»

g=3

J g=2

-

CT

III 1 g=JU

Figure 4 - Control results with AFLC (a = 1; g = 500 )

Example 2. In this example, we apply the adaptive fuzzy controller to control the following nonlinear dynamic system:

Figure 6 - The step respond

5 CONCLUSION

In this paper, we developed an adaptive fuzzy logic controller which: 1) does not require an accurate mathematical model of plant under control, 2) uses the error vector as controller's input, therefore it does not require all that components of state vector to be measurable, and 3) guarantees the global stability of close-loop system. The simulation results show that the adaptive fuzzy controller could successful control the unknown nonlinear dynamic system.

fx 1 x2'

x 2 = - X2 + sat( x3), X3 X3 + u,

i> =x 1.

REFERENCES

(16) 1. A. Isodori. Nonlinear control systems, Springer, New York, 1989.

2. S. Sastry, A. Isodori. Adaptive control of linearizable systems, IEEE Trans. Automat. Control 34 (1989) 1123-1131.

3. I. Kanellakopoulos, P.V. Kokotovic, R. Maritio. An extended direct scheme for robust adaptive nonlinear control, Automat-

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Jamil Ahmad: HYBRID LEARNING ALGORITHM FOR NEURAL NETWORKS

ica 27 (1991) 247-255.

4. I. Kanellakopoulos, P.V. Kokotovic, A.S. Morse. Systematic design of adaptive controllers for feedback linearization systems, IEEE Trans. Automat. Control 36 (1991) 1241-1253.

5. R. Marino, P. Tomci. Global adaptive output feedback control of nonlinera systems, part 1: nonlinear parameterization, IEEE Trans. Automat. Control 38 (1993) 17-32.

6. M.M. Polycarpou, P.A. Loannou. A robust adaptive nonlinear control design, Automatica 32 (1996) 423-427.

7. L.X. Wang. Adaptive fuzzy systems and control, Prentice-Hall, Englewood Cliffs, NJ, 1994.

8. L. X. Wang. Fuzzy systems are universal approximators, Proceedings of IEEE Conference on Fuzzy systems, San Diego, 1992, pp. 1163-1170.

9. R. Palm. Sliding mode fuzzy control, Proceedings of IEEE Conference on fuzzy systems, San Diego, 1982, pp. 519-526.

10. D. Driancov, R. Palm, (Eds.). Advances in fuzzy control, Heidelberg, 1998.

11. G.C. Hwang, S.L. Lin. A stability approach to fuzzy control

design for nonlinear systems, Fuzzy Sets and Systems 48

(1992) 279-287.

12. T.A. Johansen. Fuzzy model based control: stability, robustness, and performance issues, IEEE Trans. Fuzzy Systems 2 (3) (1994) 221-234.

13. W.A. Kwong, K.M. Passino. Dynamically focused fuzzy learning control, IEEE Trans. SMS 26 (1996) 53-74.

14. J.R. Layne, K.M. Passino. Fuzzy model reference learning control for cargo ship steering, IEEE Control Systems Mag. 13 (6)

(1993) 23-34.

15. C.Y. Sue, Y. Stepanenko. Adaptive control for a class of nonlinear systems with fuzzy logic, Trans. Fuzzy Systems 29

(1994) 285-294.

16. Wang Li-Xin. Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Systems 1 (1993) 146-155.

17. C.C. Lee. Fuzzy logic in control systems: Fuzzy logic controllers, parts I and II, IEEE Trans. Syst., Man, Cybern., 20 (2) (1990) 404-435.

YAK 004.93

HYBRID LEARNING ALGORITHM FOR NEURAL NETWORKS

Jamil Ahmad

Рассмотрен гибридный алгоритм для обучения нейросети, который адаптирует веса дважды на одной итерации. Этот новый алгоритм представляет собой комбинацию алгоритма обучения машины, предложенный Габором в 1960-х годах и метода наименьших квадратов. Этот гибридный алгоритм использует два разных уравнения, основанных на минимизации среднеквадратической ошибки для оптимизации весов. Алгоритм показал лучшие способности, чем метод наименьших квадратов и метод обратного распространения ошибки относительно задачи распространения образов.

This paper presents a hybrid algorithm for the learning mechanism of the neural network model which adjusts weights twice in any single iteration. This new algorithm is a combination of a machine learning algorithm proposed by Gabor in 1960s and LMS algorithm. This hybrid algorithm uses two different equations based on mean square errors to optimize weights. The algorithm showed better performance when compared with Least Mean Square (LMS) and Back Propagation (BP) learning algorithms using pattern recognition problem.

1 INTRODUCTION

In recent years, ANN models have made great leaps in solving complex problems such as prediction, classification, speech analysis, image analysis, and pattern recognition [15]. A number of sophisticated learning models have been developed to solve variety of problems. In spite the remarkable achievement by ANN model in some application areas, there is still space for improvement. Mostly, these models are suffered from problems of slow convergence and its structure definition. This paper presents a hybrid approach by combining two algorithms into a single learning model. This hybrid algorithm is mainly derived from the Gabor theory of Communication and Machine Learning, [6-7] which is modified by merging it with LMS based learning algorithm. Further information about the algorithm can be found in [8]. The algorithm is compared with standard BP [9] and LMS [10], with the pattern recognition problem. Various parameters such as initialization of weights, learning rate,

and learning curve are also investigated with the help of experimental study.

2 THE LEARNING PARADIGM

The general structure of the proposed hybrid learning algorithm is shown in Figure no. 1, which shows working mechanism of the algorithm. It can be noted that the algorithm adjusts weight in two stages. Both stages are carried out in each training run simultaneously, i.e., the adjustment of the weights takes place twice in single iteration. In the first stage, the algorithm calculates three errors associated with each weight and uses them to modify the associated weight (only one weight). Subsequently, the algorithm uses mean square errors to adjust all the weights which is considered as a second stage of the model. The flowchart for the proposed system is also shown in Figure no. 2.

Input

Figure 1 - General structure of the proposed Hybrid learning algorithm