A fuzzy variable structure approach to feedback regulation of uncertain dynamical systems, with application to robotics

NOR'I~ -HOLLAND

A Fuzzy Variable Structure Approach to Feedback Regulation of Uncertain Dynamical Systems, with Application to Robotics MOUNIR BEN GHALIA

Center for Manufacturing Research and Technology Utilization, Tennessee Technological University, Cookeville, Tennessee 38505 and ALI T. ALOUANI

Department of Electrical Engineering, Tennessee Technological University, Cooke~:ille, Tennessee 38505

ABSTRACT This paper develops a fuzzy logic-based control design methodology for a large class of nonlinear uncertain dynamical systems. The proposed control design approach combines the powerful tools of fuzzy logic and approximate reasoning with the advanced mathematical synthesis techniques used in variable structure control systems theory. The rationale for the proposed control design approach is motivated by the results of a recent study that has rigorously established some connections between fuzzy logic and variable structure control systems. The results of this paper represent a step in the right direction for systematic design of a less expert-dependent fuzzy logic control. To illustrate the merits of the new control approach, the latter is applied to a 2 degree of freedom robot manipulator. Preliminary simulation results suggest that the proposed control design deals effectively with the chattering problem encountered when the classical variable structure control is used alone.

1.

INTRODUCTION

F u z z y logic c o n t r o l ( F L C ) [14] is a r u l e - b a s e d type o f c o n t r o l t h a t uses c o n c e p t s f r o m fuzzy set t h e o r y [38] a n d fuzzy logic ( F L ) [39]. It has e m e r g e d as a p a r a d i g m o f intelligent c o n t r o l c a p a b l e o f d e a l i n g with c o m p l e x a n d i l l - d e f i n e d systems f o r which t h e a p p l i c a t i o n o f c o n v e n t i o n a l c o n t r o l t e c h n i q u e s is n o t s t r a i g h t f o r w a r d o r feasible. T h e r e a r e d i f f e r e n t

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factors that may categorize a dynamical system as being complex. These factors are mostly related to the degree of knowledge one has about the system and the complex physical or chemical relations governing the system's behavior. As the complexity of dynamical systems increases, it becomes more difficult to derive control laws for them and, thereby, it becomes more difficult to apply conventional control theory in this context, at least not in an immediate fashion. In this situation, FLC seems to be a promising control strategy, which uses human experts' knowledge in an approximate way in order to generate control rules in the form of situation ~ action rules. The human experts' knowledge consists of accumulated experience with the operation of the dynamical systems under consideration and the way to control them in order to achieve a desired objective. The control rules are often defined in linguistic terms and, accordingly, fuzzy logic controllers are sometimes called fuzzy linguistic controllers. There has been a variety of successful FLC system applications; see, for instance, [9] and the references in [4, 14]. However, even though these successful applications have helped in assessing and demonstrating the merits of FLC, there are still problems concerning the design and analysis of FLC systems, which, although partially treated in the literature, remain unresolved. In fact, to date even though the main procedure of designing FL controllers is known, it is still not well understood why they are successful and how they can deal with complex and ill-defined industrial processes to which they are applied. The above raised questions about the reasons behind the strength of FLC may be answered using intuitive statements very much like what has been done in the literature so far. However, in the current state of FLC theory, it is still more difficult to give rational answers to these questions within an analytical framework. A major reason behind the difficulties encountered whenever it is endeavored to analyze the properties of FLC systems in a similar fashion, as is done in conventional control systems theory, is the problem of the analytical formulation of the FLC law. In fact, to date the eccentricity of the structure of FL controllers is not well understood. However, research in the analytical formulation of the FLC is one possible avenue toward systematic synthesis and tuning of FL controllers that would be able to respond to the peculiarities of a given situation. Some attempts to identify the connection between fuzzy logic controllers and proportional-integral (PI) or proportional-integral-derivative (PID) controllers have appeared in [32, 35, 36]. While in [32], an empirical procedure is used to establish the relation between FLC and PID control, some rigorous proofs have been presented in [35, 36] to establish the same

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type of connection. Some analytical studies on the algebraic formulation of fuzzy logic controllers can also be found in [13], where the control law of a certain class of fuzzy logic controllers has been formulated as an algebraic nonlinear function. However, this result requires the fuzzy controller to satisfy certain strict assumptions, which in a practical situation may limit the performance of the fuzzy controller and lead to a conservative design. The: type of connection of prime interest to this paper is the one betwee.n FLC and variable structure control (VSC) [29]. This type of connection has been discussed by several researchers [5, 7, 12, 20, 21, 28]. In [7, 28] the connection identified between FLC and VSC was shallow and in [5, ]2] some empirical methods and intuitive statements have been used to conclude that a certain class of fuzzy logic controllers work like variable structure controllers. A deeper examination of the nature of this connection has been presented by Palm [20, 21], where some recent results in the develepment of VSC [24] have been used in the comparison between FLC and VSC. However, the work in [20, 21] also does not present a rigorous proof of the connection between FLC and VSC and is limited to identifying this type of connection using "intuitive" statements and general examination of the structure of fuzzy logic controllers. In fact, Palm concluded the existence of a switching function in a class of fuzzy logic controllers by a mere examination of the control rule base that is of the MacVicar-Whelan-type [18, 34]. However, what distinguishes Palm's work from lhe previous attempts is the exploration of this connection to establish certain relations between the scaling factors [4] of a class of fuzzy logic controllers and the system unmodeled frequencies [24] that, unfortunately, may not be known in general. A rigorous theoretical proof that shows the existence as well as the algebraic form of the inherent switching function in the structure of this class of fuzzy logic controllers has been recently developed by the authors [1]. The identification of the connection between FLC and VSC has suggesteci the integration of the two control approaches in control system desigr~ applications [2, 6-8, 15, 16, 20, 21, 27]. However, there has not been a unified integration approach of these two control strategies. To better understand the different attempts made to combine FLC and VSC, it is important to notice that a VSC is composed of two parts [24]: (a) a compensation part that is computed using the information on the system model and (b) a discontinuous (switching) part that depends on a switching function defined in the state space (or the error space for that matter). In [20, 21], a control law is designed by replacing the discontinuous part by a FLC law and keeping the compensation part if the plant system model is available. In [7, 15], a fuzzy logic controller is designed using a MacVicarWhehm-type rule base [18], where the control rule antecedents, namely,

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the error and the rate-of-change of error are replaced by linguistic labels describing the value of a switching function and its rate-of-change, respectively. In [10], a Takagi-Sugeno-Kang fuzzy model (TSK model) [26, 31] composed a weighted average of linear systems is constructed to approximate a nonlinear system and a linear controller is constructed for each linear model. The overall controller is then computed as the sum of the weighted average of the different linear controllers and a switching control term using constant gain. In [2], the switching part of the control law is approximated by a TSK-type FLC [31]. However, the algorithm requires the knowledge of the desired switching control input in order to adjust the parameters of the TSK-type FLC. A similar approach is suggested in [16]; however, there the TSK-type FLC is used to approximate the compensation term and not the switching term, which is computed as in [24]. In [6], a fuzzy logic controller of the type used in [7, 15] is designed to compute the switching part of the overall control law. In [8], a fuzzy logic system is combined with a variable structure controller in order to attenuate the chattering that occurs when the latter is used alone. Two approaches have been proposed. The first uses a low-pass filter whose bandwidth is tuned by the fuzzy logic system. The second approach considers a fuzzy partition of the state space into several regions and a crisp control function is computed for each region. The overall control input is computed as a weighted average of the different control functions. This paper presents a more rigorous integration between FLC and VSC in that the mathematical tools used in VSC theory are rigorously redefined within the paradigm of fuzzy set theory. The study presented in this paper represents part of an ongoing research aimed at advancing the theory of FLC by combining the latter and the analytical methods for design used in classical and modern control theory. The objective of this paper is to introduce the rule-based control techniques and FL as important ingredients for conventional control theory, which currently cannot process high-level information in an imprecise environment. More specifically, the present study combines the powerful tools of approximate reasoning and fuzzy logic used in FLC design with the advanced mathematical synthesis techniques used in VSC theory for the purpose of developing a new control approach that integrates FLC and VSC techniques by considering the advantages that each one of them can offer. The new control approach is called fuzzy variable structure control

(FVSC). The layout of this paper is as follows. In Section 2, the basic concepts of fuzzy sets and fuzzy rule-based controllers are discussed. The connection between FLC and VSC is discussed in Section 3. In Section 4, the fuzzy variable structure controller is developed and the rationale for its structure

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is discussed. Simulation results of the application of the proposed control approach to a 2 degree of freedom robot manipulator are presented in Section 5. Finally, Section 6 presents some conclusions and some recommendations for further research. 2.

F'UZZY SETS AND F U Z Z Y L O G I C C O N T R O L

In this section we review some concepts that will be referred to in subsequent sections. In particular, we review the concept of fuzzy sets and FLC. However, the discussion presented here is not exhaustive and is only meant to motivate the presentation of the material covered in this paper. For a more detailed discussion on the material presented in this section, see [4, 14, 42].

2.1. gSSENTIALS OF FUZZY SET THEORY A fuzzy set [38] is a generalization of the classical notion of a set. Whilst the characteristic function of a given classical set [23] can take values from the pair set {0,1}, i.e., an object either belongs to or does not belong to the set, the characteristic function of a fuzzy set can take on values in the whole interval [0,1]. For example, suppose that it is desired to interpret the term comfortable regarding a room temperature T°F in the interval of temperatures J = [20°F, 120°F] and let C denote the subset of comfortable temperatures (C c~D. In the classical set theory [23], the subset C can be characterized by a function /Zc, called characteristic function of the subset C, defined as follows:

Itc(T) =

1, if T~C,

0,

if Tq~C,

(1)

i.e., if txc(T)= 1, then the temperature T is comfortable; otherwise it is not. Alternatively, in the fuzzy set theory the subset C, called fuzzy subset of J " in this context, has a more realistic characterization in that its characteristic function is defined as

{~(0,1], / x c ( T ) --

=0,

ifT~C, if Tq~C,

(2)

i.e., a temperature T belongs to the subset C with a certain degree of membership tzc(T)~ [0,1]. The characteristic function /zc is called the

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M. BEN GHALIA AND A. T. ALOUANI

membership function in the fuzzy set theory. From the above discussion, a classical set C can be regarded as a special type of fuzzy set in which the range o f / z c is {0, 1} c [0,1]. z2. THE BASIS OF FUZZY LOGIC CONTROL Before giving the main idea of FLC, a brief review of approximate reasoning using fuzzy logic is in order. In the classical two-valued logic [23], it is assumed that every proposition is either true (1) or false (0). However, in fuzzy logic [39] the degree of truth of a proposition can assume any value in the interval [0,1]. Consider the following fuzzy if-then rule:

if(x is Lx) then (y is

ty),

(3)

where x and y are linguistic variables defined on the universes of discourses ~-~and y , respectively; L x and Ly are linguistic values of x and y, respectively. The meanings of L x and Ly are defined by the fuzzy subsets L x and Ly, respectively. The meaning representation of the fuzzy if-then rule (3) is given by a fuzzy relation R defined as /~ =/x~ : ,~ X y---* [0, 1],

(4)

( x , y ) --*/zn(x, y ) , where

~R( x, y ) = ixL~( x ) A tzL,( y ) .

(5)

Note that other types of fuzzy relations can be defined, for example, using the product instead of min [14]. Suppose we have the fuzzy proposition "x is L'x," whose meaning is given by the membership function /zL,(x). The problem in approximate reasoning [41] is to infer the meaning of the fuzzy proposition conclusion "y is L'y." The fuzzy implication inference for approximate reasoning is based on the compositional rule of inference introduced by Zadeh [40], i.e., £'y =L'~o/~,

(6)

F U Z Z Y VARIABLE STRUCTURE A P P R O A C H

247

-t where Ly represents the fuzzy subset corresponding to the linguistic value Ey, and " o " denotes the compositional rule of inference operation defined as

P'L',( Y ) = /x/,'x o ~ (Y) = sup ( /XL,(X) A P'R ( X, y ) ).

(7)

x~,~

In this case, the composition " o " is called the sup-min composition. Other types of compositions can be defined as well, such as the sup-product composition [14]. An important part of a FL controller is its fuzzy rule base, which consi,;ts of control rules in the form of Rule¢': If (x is Lt~) then (y is

Lty),

/ = 1 .... ,L,

(8)

where L is the total number of control rules in the fuzzy rule base, Ltx and Lyt are linguistic variables representing the process input variable x and the control output variable y, respectively. Once the fuzzy control action inferred by each rule has been determined as explained previously [refer to Equation (6)], the final crisp control action is computed using defuzzification. The latter consists of a mapping from the space of fuzzy control actions into the space of crisp control actions. Several defuzzification methods have been proposed in the literature, such as the center of area, the mean of maximum [14], and the weigbLted average method [37]. Because of its simplicity, the weighted average method is widely used by researchers. Using the weighted average defuzzification approach, the crisp control action is computed as

y* =

E ~= L 1 ~L'o ~,',(Yt)

,

(9)

where Yt (I = 1.... , L) can be chosen as the mean of all the outputs y ~ y at which /XL; reaches its maximum. Using (5) and (7), Equation (9) becomes

y* =

(10)

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M. BEN G H A L I A AND A. T. A L O U A N I

Note that it is common in the literature to consider /~'x(x) as a fuzzy singleton, i.e.,

1, /zc'(x) =

0,

if x=x*, if x 4 x * .

(11)

3. C O N N E C T I O N B E T W E E N F U Z Z Y L O G I C AND V A R I A B L E S T R U C T U R E C O N T R O L SYSTEMS In this section we review the results of [1], which has analytically established some connections between FL and variable structure control systems. We start first by presenting a brief review of the basic concepts of variable structure control as well as its shortcomings.

3.1. OVERVIEW OF VARIABLE STRUCTURE CONTROL 3.1.1.

Basic Concepts

Variable structure control systems [3] constitute a class of nonlinear feedback control systems whose structure changes depending upon the state of the system. Although neither structure is necessarily stable, their combination results in a sliding mode, that is, the system trajectory slides along a switching surface (also called sliding surface). A VSC with sliding modes is often called sliding mode control (SMC). To describe the basic concepts of VSC, consider a nonlinear system represented by the state equation

2=f(x,t)

+g(x,t)u,

(12)

where the state vector of the system x E ~ , an open set of ~ n , f and g are smooth vector fields, f, g: .9~+ × ~ , and u: ~ - ~ ~ ' c ~ is the control input function. Let s denote a smooth function s: ~-~---,~, with nonzero gradient on ~ . The VSC design consists in achieving the following steps: 1. Design a switching manifold S : in the state space, also called sliding manifold, to represent desired system dynamics. S : is defined by

=01, where s ( x ) ~ ~ is called the switching function.

(13)

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249

2. Design a variable structure control law u+(x),

u(t)=

u-(x),

when s( x) > 0, whens(x)<0,

(14)

such that the state trajectories of (12) are forced to reach the sliding manifold S ~' and, from then on, their motion is constrained to stay on S ~, where the sliding mode takes place (see Figure 1). Once the system state is on the sliding surface, the system remains insensitive to parameter perturbation and external disturbances because the system is forced to constrain its evolution in the predetermined sliding surtace. Therefore, it is important to determine the conditions under which the state will move toward and reach the sliding surface. This condition is called the reaching condition. The phase during which the state trajectories are driven toward the sliding manifold is called the reaching phase or the reaching mode. To specify the reaching condition, a Lyapunov function candidate may be defined as [25]

V(x,t) = ½s2(x).

(15)

U +

SU_ o s(x)=O Fig. 1. Slidingmode on a switchingmanifold.

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M. BEN GHALIA AND A. T. ALOUANI

The reaching condition for the existence of the sliding mode motion of system under consideration is given by [25]

I2(x,t) =s(x)~(x) <0,

for x ~ " - 5

~.

(16)

3.1.2. Shortcomingsof VSC Despite the benefits of VSC, the latter suffers from two major shortcomings. First, the insensitivity property of a VSC system is present only when the system is in the sliding mode. In other words, state trajectories starting off the sliding manifold remain sensitive to parameter perturbations and external disturbances [30]. Moreover, the convergence to the sliding manifold may only be asymptotic, resulting in a long reaching time (time needed for the state trajectory to hit the switching manifold) during which the advantages of VSC are not achieved [11]. The second shortcoming is control chattering. In the design and analysis of FSC systems, it is assumed that the control can be switched from one structure to another infinitely fast. However, in practice it is impossible to achieve a high switching control, which is a requirement for most VSC designs. This is due to several reasons. One of them is the existence of switching time delays resulting from delays in control computation. Another reason is the limitations of physical actuators, which cannot handle the switching of control signal at infinite rate. As a result of these imperfections in switching between control structures, the system trajectory chatters instead of sliding along the switching surface. This chattering is undesirable in practice because it may serve as a source to excite the unmodeled high-frequency dynamics of the system [3, 30]. To overcome the first shortcoming, that is, to reduce the reaching time, the use of high-gain control signal was suggested [33]. However, this may result in the saturation of the actuator and also in higher chattering phenomenon, which is undesirable in a physical system. Alternatively, a time-varying switching surface was suggested [25] in order to eliminate the reaching phase, where the initial tracking control errors were assumed to be zero. However, this assumption rules out many practical situations in which the system initial conditions are located arbitrarily. The problem of chattering has been addressed by many researchers. In [25], the discontinuous control is approximated inside a boundary layer located around the switching surface. However, although chattering can be reduced, robustness and tracking accuracy are compromised.

FUZZY VARIABLE STRUCTURE APPROACH

251

3.2. F U Z Z Y LOGIC CONTROL IS A GENERAL TYPE OF VARIABLE STRUCTURE CONTROL

This section reviews the results of [1], which has theoretically shown that FLC, whose rule base is of the MacVicar-Whelan type, is a general type of VSC. Suppose that it is desired to regulate the states of a nonlinear process P to some predetermined fixed values using FLC. Let the nonlinear process P be described by a vector differential equation of the form

e(t)

(17)

where z ( t ) ¢ ~ " represents the state vector of the process, u ( t ) e 2 p is the control input vector, and f is an unknown nonlinear function. To simplify the analysis, we consider (a) single-input single-output process, i.e., m = 1 and p = 1, and (b) the reference signal r = 0. The input vector to the FL controller is x = [xl, x2], where x 1 is the deviation of the process state from the reference set point r, i.e., xl = z - r = z (because in this :regulation problein the reference signal r is assumed to be zero), and x2 =:~1 = 2 is the rate of change of the state variable z. The output of the FL controller is the control variable u. Let Kx,, Kx~, and K~ denote the scaling factors of the input variables x 1 and x2, and the output variable u, respectively. The variables x p x z, and u are normalized to a common universe of discourse represented by the interval [ - a , +a], where a is a positive number that is usually chosen in accordance with the values of the scaling factors. Note that this can be easily extended to the case where x~, x 2, and u are normalized to different uniw~rses of discourse. The normalized variables are denoted by X~N, X2N and u N. We define XN:=[XIN,X2N] T. Each input variable X~N is characterized by two fuzzy subsets ?Xiu and ]Vx,N given by

) =

(XiN+a)/2a,

if - a
o,

if XiN ~ --a, if Xiu >~ -I-a,

1,

(a --XiN)/2a ,

=

o,

if - a "~XiN < + a , if XiN>~ + a , if XiN~ - a ,

(18)

(19)

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M. BEN GHALIA AND A. T. ALOUANI

for i = 1, 2. The fuzzy subsets defined on the universe of discourse of u N are the following fuzzy singletons: A?,x=6(--a ),

2,N=6(0),

and

/5, = 6 ( a ) .

(20)

To regulate the process state, the four following fuzzy control rules are used:

fix,Nand X2N is /~x2"' then

Rule('):

If XlN is

u is N~N"

(21)

Rule(2):

If X1N is /Sx~N and X2N is Nx2N, then u is Z , . .

(22)

Rule(B):

If XlN is A~x,N and X2N is fiX2N' then u is '~UU"

(23)

Rule(4):

If

X1N is Nx,u and X2N is /Qx2N' then u is /;Uu"

(24)

In the normalized space, FLC can be viewed as a mapping, denoted by 3-lc, defined on the normalized input domain ~'~U and takes its values in the normalized output domain ~'U, i.e., YlC:~N~/N

.

(25)

The mapping J l c is the result of the composition of all the operations performed in the FL controller, i.e.,

~'-lC(XN) =D "I'P'F( xN) =UN,

(26)

where "-" denotes the composition operator, ff is the fuzzification mapping, fi is the operator that computes the possibility measure [42] of fuzzy set input with respect to the fuzzy sets antecedents in the control rules, I is the function of the inference engine that determines the output fuzzy set, and D is the defuzzification operator. Now, the objective is to determine an algebraic expression of the FLC mapping, Y l c [1].

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253

LEMMA 1. The algebraic expression of J l c is given by

--a(X1N +X2N )

u u =3-lc(x N) = 2 min(Za -IXlu[, 2a -IXzu I)" Proof. See [1].

(27)

[]

THEOREM 1. The fuzzy logic controller defined by its mapping J l c given in (26) and by its control rules (21)-(24) is a variable structure controller with

the swi!tching function defined as S(X1,X2) = KxlX1 + Kx2X2.

Proof. See [1].

(28) []

REMARK 1. The algebraic form of the switching function inherent in the MacVicar-Whelan-type FLC has been derived for the case of a fuzzy controller with two input and one output variables and whose MacVicar-Whelan-type rule base is composed of four rules [1]. Equation (28) gives the algebraic form of the FLC switching function and shows how it is dependent on the input variables scaling factors. In [20, 21], the same class of FLC was considered; however, the switching function used is not the one inherent in the structure of the fuzzy controller, but it is arbitrarily designed. 4.

FUZZY VARIABLE STRUCTURE CONTROLLER

4.1. STATEMENT OF THE PROBLEM In this paper, we consider a class of uncertain nonlinear systems described by the following vector differential equation:

:~( t) = f ( x,t) +g( x , t ) u +w( x , p , t ) ,

(29)

where x: ~ + --*,~, an open set of .~n, is the state vector of the system, f: ~'+×S~ n is a smooth vector field, g: ~ ' + × ~ - o ~ 'nxm is a matrix function whose columns are smooth vector fields, u = [ u I ..... u m ] r ~ ~'a × "'" x ~'m = ~', an open set of ~,m, is the control input vector, p is an uncertain parameter vector, and w contains system model uncertainties and external disturbances.

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The problem to be undertaken is that of designing a globally stabilizing feedback regulator, such that, for an arbitrary initial condition X(to), the state trajectory x(t) of the feedback system satisfies the property x(t) ~ 0 as t ~ .

4.2. FEEDBACK REGULATOR DESIGN The key point of the approach proposed in this paper is the integration of FLC and VSC into a novel robust nonlinear control called fuzzy variable structure control (FVSC). This integration of the two control strategies is motivated by the results presented in [1]. Since we have an m-dimensional control input vector, the desired sliding mode dynamics are specified by designing m switching functions [30], which are denoted by si: S ~ , for i = 1..... m. The desired system dynamics are therefore represented by the sliding manifolds given by

~i:=(x~lsi(x)

=0},

i = 1 ..... m.

(30)

Let s(x) denote the m-dimensional switching function defined by S(X) := [ S I ( X ) , . . . , S m ( X ) ] T

(31)

Since the reaching mode represents an important part of the system transient dynamics, special attention has to be given to the specification of the control law that allows desired system dynamics to be achieved in the reaching mode. In addition, the control law has to satisfy the reaching condition (16), which guarantees the existence of sliding modes on the switching manifold. The control of the system dynamics during the reaching phase may be made possible by specifying the dynamics of the switching function s(x) [19]. More specifically, the dynamics of the switching function s(x) are described by a differential equation of the form [19] ~(x) = - K s g n ( s ) ,

(32)

where K = diag[K 1, K 2..... Kin], K i > 0, i = 1.... , m, sgn stands for the sign function, and sgn(s) = [sgn(s l) ..... sgn(Sm)] T. By specifying the dynamics of the function s(x), we can predetermine the speed with which the system state approaches the switching manifold. For each switching function si(x), this speed is given by K i.

FUZZY VARIABLE STRUCTURE APPROACH

255

After having chosen the dynamics of the reaching mode, we now determine the associated control law, called the reaching control law and denoted by UR. Differentiating s(x) with respect to time along the trajectour of (29) gives

~(x) = Os Os Os - ~ f ( x ) + -$-~g(x)u + - ~ w ( x , p , t ) = - K sgn(s).

(33)

Solving (33) for the control law gives c~S

1

[.s--$-~f(x) + -.s~ w ( x , p , t )

with the assumption that the inverse of the matrix [(c)s/Ox)g(x)] -1 exists. Since w is unknown, the control Un as given in (34) cannot be computed. The computation of (34) is application dependent and will be discussed in Section 5, where the feedback regulation of robot manipulators is treated.

,t.2.1. Equir, alent Control Law ]in the traditional sliding mode control design, the system motion that takes place strictly on the constrained manifold Sp is called the ideal sliding mode [11]. If the control input can be switched at an infinitely fast rate, then an ideal switching mode may be obtained. The condition ~(x)=0 is necessary for the state trajectory to stay on the switching manifold ~ . The control input under the ideal sliding mode is called the equivalent control and is denoted by UE. The latter is the computed control input such that the state trajectory slides on the switching manifold Sp. To compute the equivalent control liE, we differentiate s(x) with respect to time along the trajectory (29) and we let ~(x)= 0,

~( x) = os Os os -~f(x)+-g--~g(x)u+-g-~w(x,p,t)=O.

(35)

Solving (35) for UE yields

Os

-1 c)s

--~w(x,p,t)].

(36)

Here again the control UE as given in (36) cannot be computed since w is un~mown. The solution to this problem is discussed in Section 5.

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4. 2. 2.

M. BEN GHALIA AND A. T. ALOUANI

Design Considerations

The above formulation gives our main framework for the synthesis of FVSC. In what follows we discuss some design considerations that represent specific guidelines for the selection of the final structure of the controller and for the solution of chattering alleviation. These considerations are as follows: (a) The invariance property of VSC systems is present on the sliding manifold (switching manifold) and not during the reaching phase in which the system is affected by the system perturbations and the external disturbances. Therefore, it is important to minimize the time required to reach the sliding manifold. One way of reducing the reaching time is to apply a very large gain all along the reaching phase. However, this may not be practically possible due to saturation problems and also due to the chattering effects that may result. (b) Selecting smaller values of the control gain K results in less chattering while on the sliding mode. Therefore, once the system trajectory is about to reach the sliding manifold, K should be kept as small as possible in order to reduce the chattering effects. (c) The assumption of saying that the control input can be switched from one value to another infinitely fast cannot be achieved in practice. Therefore, an ideal switching surface [11] is not realistic. This is due to the presence of finite time delays for control computation and also to the limitations of physical actuators. The fact that the control input cannot be switched at an infinitely fast rate is responsible for the occurrence of the chattering phenomenon during the sliding and the steady state modes. Consequently, a "more realistic" switching surface has to be considered.

4.2.3.

Fuzzy Rule Base for the Selection of the Control Gain K

In the light of the design considerations given above, the control gain K has to be selected according to the following rules. At each time instant t, si(x(t)) is the algebraic value of the switching function s i (i = 1..... m). These rules are:

Rule 1:

If I s i ( x ( t ) ) l is SL, then Ki(t ) =KL.

(37)

Rule 2:

If I s i ( x ( t ) ) l is SM, then Ki(t ) =KM.

(38)

257

FUZZY VARIABLE STRUCTURE APPROACH

Rule 3:

If [si( x( t ) ) [is SS, then Ki( t ) =KS.

(39)

Rule 4:

If [si(x(t))[ is SZ, then Ki(t ) = K Z .

(40)

where SL (large Is/I), SM (medium Isgl),SS (small [si[), and SZ (zero Isil) are fuzzy sets of the variables Isi(x(t))l (i = 1,..., m). KL, KM, KS, and KZ are different values of the control gain Ki corresponding to a large, medium, small, and zero gain value, respectively. Given the value of ]si(x(t))l at the time instant t, the value of the control gain Ki (i = 1 , . . . , m ) at time t is infer:red using the above four fuzzy if-then rules and the weighted average defuzzification scheme as follows:

Ki(t )

=

~SL(Isil)gL + ~SM(Isil)gM+ ~ss(Isil)gS + ~sz(Isil)gZ ~SL(Is~l) + ~SM(IS~I)+ ~Ss(IS~I) + ~Sz(ISgl)

(41)

This strategy of selecting the variable control gain K(t) has the following advantages over choosing a fixed control gain K: (a) A large control gain is applied only when the system state is far away from the sliding manifold. (b) When the system state is close to the sliding manifold, a small control gain is used. Consequently, this presents a potential solution for chattering reduction.

4. 2.4. Concept of Fuzzy Sliding and Reaching Modes I_x:t _Es denote the set of all possible values that the switching function s can take, i.e.,

"Zs = { S( x ) l x ~ y }

.

(42)

Using the notion of fuzzy sets, we introduce the following definitions: DEFINmON 1. Fuzzy Sliding Mode. The meaning of the linguistic expression "the system motion takes place on the switching manifold" is represented by a fuzzy subset ~ of _E~ and is called the fuzzy sliding mode. The

258

M. BEN G H A L I A AND A. T. A L O U A N I

latter is totally defined by its corresponding membership function /Xs, defined as =1,

if s ( x ) = 0 ,

tZs,(S(X)) ~ [ 0 , 1 ) , 0,

(43)

i f s ( x ) ~ B ~ ( 0 , e), elsewhere,

where B,(0, e) is a closed ball of _E, centered at 0 and of radius e. REMARK 2. In the case where the switching manifold is a line, the set B,(0, e) reduces to the interval [ - e , e]. DEFINITION 2. Fuzzy reaching mode. A fuzzy reaching mode is a fuzzy subset Sr of _Es. It is the complementary mode of the fuzzy sliding mode. This means that fuzzy reaching mode is characterized by its membership function given by

(44)

/Zs,( s ( x ) ) = 1 -/Zs,.( s ( x ) ) .

Figure 2 shows the membership functions of the fuzzy sliding mode and the fuzzy reaching mode in the case where the switching manifold is a line.

Hs~

Its

~s~

s

/ o.o C B

I A

Fig. 2. Membership functions of fuzzy sliding and reaching modes.

,

S

FUZZY VARIABLE STRUCTURE APPROACH

259

Note that an ideal sliding mode [11] Si~ may be characterized by its membership function given by

tZsid(s(x)) =

1, O,

if s(x) = 0 , if s ( x ) - ~ O .

(45)

Characterizing the sliding mode as in (45) is ideal and cannot be achieved in practice. Instead, a realistic characterization of the sliding mode, as given in (43), should be considered. REMARK 3. The proposed fuzzy sliding mode is different from the real sliding mode defined in the literature [11]. The real sliding mode characterize, s the motion in the neighborhood of the sliding manifold, for example, defined by the interval [ - E , E] in the case where the switching manifold is a line. Following our above definitions, a real sliding mode will be characterized by a membership function that has the value 1 everywhere on [ - e, e ]. Therefore, ideal and real sliding modes are special cases of the fuzzy sliding mode as previously defined.

4.2.5.

Fuzzy Rule Base for the Computation of the Overall Control

To guarantee a smooth transition of the control law when the system mode', changes from the reaching mode to the sliding mode, two fuzzy rules are used:

Sr, then u i= Uni;

Rule~¿):

If s i ( x ( t ) ) is

Rule(2):

If s i ( x ( t ) ) is S s, then u i = UEi,

for i = 1..... m, where URi (UEi) is the ith component of UR (UE). The abow~ fuzzy rules mean that if the system motion corresponds to the fuzzy reaching mode, then the control applied to the system is the reaching control Un. On the other hand, if the system motion corresponds to the fuzzy sliding mode, then the control applied to the system is the equivalent control Ue. The degree to which the actual system motion corresponds to one of these modes is given by the degree of membership obtained by matching

260

M. BEN G H A L I A AND A. T. A L O U A N I

the computed switching function, S i ( X ( t ) ) , with the membership functions ]&s,, and ]&ss~ depicted in Figure 2. The proposed overall control law is given by

u,(s) =

m.,(s,) vRi + m . ( si ) uEi m,,(s,) + m,,(s,) '

i = 1 ..... m.

(46)

Note that from Equation (46), the control input gradually switches from URi to UEi when the system trajectory is moving toward the sliding manifold. In fact, by referring to Figure 2, the closer the system trajectory gets to the sliding manifold, the smaller ]&s,~(si) and the larger ]&s~(si) become. When the system trajectory is far from the sliding manifold ~see, for example, the reference point A in Figure 2), then ]&s,~(si)= 1 and ]&s,(Si) = 0, consequently only the reaching control Uni is active. At the reference point B in Figure 2, both control laws, URi and UEi, are active. However, the more the state trajectory gets closer to the sliding manifold, the smaller is the contribution of the reaching control URi to the overall control, until the system reaches the sliding manifold (point C in Figure 2), where the reaching control is turned off completely and the system is totally under the equivalent control Uei. The overall control can be put in a vector form as LI = Id,rUR q- ]&sUE,

(47)

/Xs,1 ]&s,~ ) ]&r= diag ]&s, + ]&s,~ " ' " ]&s,,, + ]&s~,,

(48)

where

and ['~Ssl

]&Ssm

]&, = diag ]&s. + ]&s,, ' " " ]&s.~+ ]&Ssm

)

(49)

"

In summary, the proposed FVSC algorithm consists of the following steps: (i) Design a switching function s(x) such that the sliding manifold S p, defined in the state space by (13), represents desired system dynamics as in traditional VSC. (ii) Design a fuzzy logic system to compute the control gain K as in (41).

FUZZY VARIABLE STRUCTURE APPROACH

261

(iii) Compute the reaching control law UR. (iv) Compute the equivalent control law UE. (iv) Construct the sliding mode fuzzy set S s and the reaching mode fuzzy set S r (see Figure 2). (vi) Design a fuzzy logic system to compute the final control input as giwm in (47). A block diagram of the proposed FVSC is shown in Figure 3. 5. APPLICATION TO ROBOT MANIPULATORS ~1.

CONTROLLER DEIGN

To evaluate the performance of the proposed controller, a simulation study was conducted on a 2 degree-of-freedom revolute joint robot manipulator shown in Figure 4. The physical properties of each link i are specified by the following four parameters: mass m i, moment of inertia about center of gravity/~, length li, and the distance between the center of gravity and the previous joint lci. The dynamic equation is given by [17]

(50)

M ( O ) O + h ( O , O) = T + d ( O , O , p , t ) ,

X

Plant

UE

1

Switching Function

Equivalent Control

(i)

(iv)

Fuzzy Logic S y s t e m II

(v)-(vi) I.I R

Reaching Control

(iii) Fig. 3.

l '

K

Block diagram of FVSC system.

Fuzzy Logic System I

(ii)

262

M. BEN GHALIA AND A. T. ALOUANI

12 Link

2

[ 2~ m 2

I

f

Theta

1 \ I

cl

"-

Link ~

.

.

q .

.

.

~

/

,

,_1o

leta ii ~m d ,.Jo

I

~

'Tau 2 i nt

2

1

nt. Ta u

Fig. 4. The 2 degree-of-freedomrobot manipulator.

where 0=[01,02] T is the vector of joint angles, M(O) is a 2 × 2 positive definite inertia matrix, h(O, O) is a two-dimensional vector containing the centrifugal, coriolis, and gravitational forces, 7=[~-1,~-2] T is the control torque vector, and d(O, O,p, t) is a two-dimensional vector that represents the combined effects of model uncertainties and external disturbances. We have

M(0)=

[ ~1 + ~2 + 2t~3 cos(02)

c~2+ c% cos(02) ]

ae+~3cos(02)

°~2

,

h(O,O) = [ -2a3 sin(Oz)OlO2-a3sin(O2)O~+a4 cos(Ol)g+as cos(Ol+ Oz)g] a3 sin(O2) 012+ % cos(01 + 02)g

FUZZY VARIABLE STRUCTURE APPROACH

263

with

oeI = mllZl + m 2 +11,

O~2 =

a 4 =m1121 +m21 l,

m2122

+

12 ,

Ol 3

=m2lll(. 2 ,

ee5 =m21c2.

Define

~-I:'21-(:1

(51)

It is easy to show that (50) can be put in the form ~ =f(x) +g(x)u+w(x,p,t),

(52)

where u = 7,

w(x,p,t)=

[

o

M_l(x)d(x,p,t)

1 ,

The following set point regulation problem is considered. For any initial state O(t o) and 600) and desired position 0 d, find a feedback regulator u(O, 0), such that O(t) ~ 0 d and O(t) ~ O. Define an error vector

e:=

[] [ 0.]~I el = e2

O+ O+ 0 a

-0+0 -0

][x,+0.~

d =

-x2

(53)

1"

Define a two-dimensional switching function vector

s(e)=[C

1][el]=Cel+e2,e2

(54)

264

M. BEN GHALIA AND A. T. ALOUANI

where C = diag(c 1, c2). Differentiating (54)with respect to time yields

=c l+ 2=c01+M l(x)(h(x) - d ( x , p , 0 - . ) .

(55)

The reaching law is defined as in (32) and is rewritten here for convenience: = - K sgn(s).

(56)

Differentiating (54) with respect to time along the trajectory of (52) gives

s=Cel +e2=Cel + M - l ( x ) ( h ( x ) - d ( x , p , t ) - u ) .

(57)

Solving for the reaching control law UR using (55) and (57) yields

UR=M(x)(Ksgn(s)q-Cel)+h(x) - d ( x , p , t ) .

(58)

The equivalent control law Ue is computed by letting ~ = 0 in (57):

UE =M(x)Cb 1+h(x) - d ( x , p , t ) .

(59)

Since d(x, p, t) is an unknown quantity, the control laws UR and liE, given by (58) and (59), respectively, cannot be computed. As a solution to this problem, the term d(x,p,t) is replaced by a known quantity d(x,t). The choice of d(x, t) is discussed next. Replacing d(x, p, t) by d(x, t) in (58) yields

Un=M(x)(Ksgn(s ) + C ~ 1 ) + h ( x ) -cl(x,p,t).

(60)

In (57), the control u is substituted by the reaching control law UR given by (60), and after appropriate computation we obtain = - K s g n ( s ) +M-'(x)(d(x,t)

-a(x,p,t)).

(61)

Note that the reaching law given in (61) has an extra term M-~(x) × (d(x, t ) - d(x, p, t)) by comparison to the reaching law given in (56). For the sliding mode to exist, the reaching condition sr~ < 0 must be satisfied and, consequently, the term d(x, t) has to be chosen appropriately.

FUZZY VARIABLE STRUCTURE APPROACH

265

sT~ using (61) yields

Computing

sT~ = -Klsl+sTM-l( x)( d( x,t) - d ( x,p,t) ).

(62)

Define

'tm( X,t) :=M-l( x)d( x,t),

dm(x,p,t):=M-l(x)d(x,p,t).

(63)

Using (63), Equation (62) can be written as 2

sT~=--K]s]+

(64) i=1

where the subscript i denotes the ith component of the corresponding vector. For the quantity sr~ in (64) to be negative, the following conditions mu,;t be satisfied:

si(dmi(X,t)-d,,i(x,p,t))
i=1,2.

(65)

d(x, p, t) is bounded and so is din(x, p, t), i.e.,

dmi(X,t) 4dmi(X,p,t)<<.dmi(X,t),

i=1,2,

(66)

where d_mi and dmi (i = 1, 2) are assumed to be known. To aid in the selection of d(x, t) such that the conditions in (65) are satisfied, the following two cases are examined: (a) If si>O, then we need to have to have Clmi= d_mi. (b) If si < 0, then we need to have dmi is to have d,, i = drag.

dmi-dmi <0. A possible choice for

dmi is

clmi dmi > 0. A possible choice for -

Now define

d'~=~(dm-dm), 1

--

d"=~(dm+d_m). 1

--

(67)

266

M. B E N G H A L I A

It is easy to s h o w t h a t t h e c h o i c e s o f c a n be c o m b i n e d as follows:

c[,,,(x,t)

c[,,,(x, t)

AND

A. T. A L O U A N I

s u g g e s t e d in c a s e s (a) a n d (b)

d"n(x,t)-diag(d;~l(x,t),d;,z(X,t))sgn(s

).

(68)

U s i n g (68) a n d (63), t h e t e r m d ( x , t) is t h e r e f o r e c h o s e n as

a~(x,t)= M (

X

)[d,,(x,t)-dlag(dm](X,t),d'm2(X,t))sgn(s)]. I!

/

•

(69)

O n c e a~ has b e e n d e t e r m i n e d , t h e c o n t r o l laws UE a n d UR c a n b e c o m p u t e d a n d t h e n t h e F V S C is d e s i g n e d using t h e a l g o r i t h m d e v e l o p e d in S e c t i o n 4.

600 Applied Torques at Joints 1 and 2

600 Applied Torques at Joints 1 and 2

"~ 400 ......................

400 . . . . . . . . . . . . . .

U'I ..................

= 200 E o b"

i

0

...................

:. . . . . . . . . . . . . . .

E

u2

200

¢

. . . . . . . . . . . . . . . . . . .

0

: ................

1

2

.........

u2

O

. . . . . . . . . . .

: ...............

i 1

I

0

. . . . . . . . . . . . . .

Z

z

3

!

.........

i 2

Time (seconds)

Time (seconds)

Switching Functions sl and s2 / 0.5 .................................................

Switching Functions sl and s2 / 0.5 ................... ::.................................

sl

s1

-0.5

-1

I

0

1 2 Time (seconds)

3

-1

0

2 Time (seconds)

Fig. 5. Case I: Comparison of the performance of feedback regulation of a two-link robot manipulator using (a) VSC (left side) and (b) the proposed FVSC (right side) in the case where there is no disturbance (i.e., d I = d e = 0).

FUZZY

5.2.

VARIABLE

APPROACH

STRUCTURE

267

NUMERICAL SOLUTIONS

In o u r n u m e r i c a l s i m u l a t i o n we have c h o s e n 1 1 = / 2 = 1 m,

m~ = 2 0 kg,

lcl=lc2=0.5m,

11 = 0.8 kgm 2 ,

12 = 0.2 kgm 2 ,

m 2 = 1 0 kg,

g = 9.81 m / s 2.

a d l r = [0,0]r. T h e initial T h e d e s i r e d set p o i n t has b e e n c h o s e n as 0 d = [ ~ad, ~2 state o f t h e m a n i p u l a t o r has b e e n c h o s e n as

6',(0) = 0.36 r a d ,

02(0 ) = 0.4 rad,

0](0) = 02(0 ) = 0 r a d s 1.

T h e p a r a m e t e r s o f the switching functions are c h o s e n as c I = C 2 = 2 s 1 T h e s a m p l i n g interval A T = 0.01 s is u s e d in t h e simulation. T h e p e r f o r m a n c e o f the p r o p o s e d F V S C is c o m p a r e d with that of the t r a d i t i o n a l VSC. T h e results o f the s i m u l a t i o n a r e shown in F i g u r e s 5 - 8 .

Angle Positions of Joints 1 and 2

Angle Positions of Joints 1 and 2 L.,

o~ e-,

0.5 ~.5 ~

-

-

i

.....................

o

:-2,

O

01..........

<

0

O

1 2 Time (seconds)

3

<

2

Time (seconds)

Velocities of Joints and 2 0.5 . . . . . . . . ................ !.............

Velocities of Joints 0.5

...................

and 2

i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Joint I i

-0.5

-0.5

.............. ........................ :...................... 8

>

-1

0

2 Time (seconds)

3

-1 0

Fig. 6. Case I (continued).

2 Time (seconds)

268

M. B E N G H A L I A

600 Applied Torques at Joints 1 and 2

600 Applied Torques at Joints 1 and 2 400 . . . . . . . . . . . . . ....... U I [IIIllulilIlUll

Imll

................. lil

:

A N D A. T. A L O U A N I

"~ 400 z

I IIIIII

:

~

o

200

°

0 '7

....

0

.................... i. . . . . . . . . . . . . . . .

:................

i 0

2 Time (seconds)

3

0

Switchin Functions sl and s2 0.5 ...................................................

2

Switching Functions s 1 and s2 0.5 ............................................... /

sl

sl

-0.5 -I

1

Time (seconds)

-0.5 -1

0

2

i

0

Time (seconds)

J

1 2 Time (seconds)

3

Fig. 7. Case II: Comparison of the performance of feedback regulation of a two-link robot manipulator using (a) VSC (left side) and (b) the proposed FVSC (right side) in the presence of disturbance [ d l = d2 = 5 sin(0.4t)].

F o r t h e t r a d i t i o n a l V S C , t h e c o n t r o l g a i n K = [ K 1, K 2]T has b e e n k e p t c o n s t a n t a n d e q u a l to [2.5,2.0] 7. F o r t h e p r o p o s e d F V S C , t h e fuzzy s u b s e t s S Z , SS, S M , S L , S r, a n d S s, d e f i n e d o n t h e u n i v e r s e o f d i s c o u r s e o f t h e switching f u n c t i o n s s i (i = 1,2), a r e listed in T a b l e 1. T h e levels o f c o n t r o l g a i n s for t h e F V S C a r e

K L 1 = 2.5,

KM

KL 2 = 2.0,

K M 2 = 0.5,

1 =

0.5,

K S 1 = 0.01,

K Z 1 = 0.0;

KS 2

KZ 2 = 0.0.

=

0.01,

N o t e t h a t for c o m p a r i s o n r e a s o n s , t h e m a x i m u m c o n t r o l g a i n s K L i (i = 1, 2) o f F V S C a r e c h o s e n to b e e q u a l to t h e c o n t r o l g a i n s K i (i = 1,2) u s e d in VSC.

F U Z Z Y VARIABLE S T R U C T U R E A P P R O A C H

269

Angle Positions of Joints 1 and 2

Angle Positions of Joints 1 and 2 0

"

i

~

.............................

0 . 5 0 ..................................................... ~ ~ . Q

O

O

<

0

3

<

0

2 Time (seconds)

Time (seconds) Velocities of Joints 1 and 2 --.~ 0.5 ~......................T.......................!...................... Joint 1

i

0

2 Time (seconds)

Velocities of Joints 1 and 2 0.5

0I

:.= . ..............i........................ ~> -o 51 ~ ...................... !...................... g 3

3

...................... i........................ }

Joint 1!

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

{

71f" .............!1........................ ' 0 2

. . . . . . . . . . . . . . . . . . . . . .

3

Time (seconds)

Fig. 8. Case II (continued).

Figures 5 and 6 show the performance of VSC (shown on the left side of the figures) and the proposed FVSC (shown on the right side of the figures) when there is no disturbance (i.e., d 1 = d 2 = 0). Figures 5 and 6 show the chattering phenomenon that occurs when using the traditional VSC and the good performance of the closed loop system using the proposed FVSC. It can be seen that not only the chatterings are eliminated, but also the amplitudes of the control input signals are greatly reduced. Figures 7 and 8 compare the performances of VSC (shown on the left side of the figures) and the proposed FVSC (shown on the right side of the figures) in the presence of system disturbance which is chosen to be periodic [22]: d I = d 2 = 5 sin(0At)(therefore, _d= - 5 and d = 5). Comparing the performance of VSC with that of FVSC, it can be seen that FVSC results in less chattering and better set point regulation performance than VSC. To quantitatively compare the performance of the proposed FVSC with that of a traditional VSC, the integral of the square of the switching

270

M. B E N G H A L I A

A N D A. T. A L O U A N I

TABLE 1 Fuzzy Sets Definitions for the Switching Functions s~ and s 2 (a) Fuzzy sets characterizing the switching functions used in the rule-based fuzzy logic system that computes the control gains K~ and K, ]si], i - 1,2

SZ SS SM SL

0

0.00001

0.05

0.1

3.0

1.0 0 0 0

0 1.0 0 0

0 0 1.0 0

0 0 0 1.0

0 0 0 1.0

(b) Fuzzy sets characterizing the reaching mode and the sliding mode si, i = l , 2 3.0 S,. S,

- 0.002

0

0.002

3.0

1.0 0

0 1.0

1.0 0

1.0 0

1.0 0

functions s~ a n d s 2 a r e c o m p u t e d o v e r a t i m e p e r i o d o f 3 s. T h e s e p e r f o r m a n c e indices a r e d e n o t e d by I S E 1 and I S E 2 for s I a n d s 2, r e s p e c tively, a n d a r e r e p o r t e d in T a b l e 2. 6.

CONCLUSIONS

In this p a p e r , a n e w c o n t r o l design t e c h n i q u e for u n c e r t a i n n o n l i n e a r systems was p r o p o s e d . This t e c h n i q u e c o m b i n e s the c o n c e p t o f a p p r o x i m a t e r e a s o n i n g and fuzzy logic with the a d v a n c e d m a t h e m a t i c a l synthesis t e c h n i q u e s used in v a r i a b l e s t r u c t u r e control. A m o n g the objectives o f this p a p e r is to i n t r o d u c e the r u l e - b a s e d c o n t r o l t e c h n i q u e s a n d fuzzy logic as i m p o r t a n t i n g r e d i e n t s for v a r i a b l e s t r u c t u r e c o n t r o l systems theory. Certain concepts, namely, sliding a n d r e a c h i n g m o d e s , u s e d in v a r i a b l e struc-

TABLE 2 Comparison of the Performance of the Proposed FVSC with that of a Traditional VSC using the Performance Indices ISE 1 and ISE 2

Traditional VSC Proposed FVSC

ISE 1 No dist. With dist.

ISE 2 No dist. With dist.

0.0681 0.0550

0.0034 0.0028

0.0728 0.0573

0.0043 0.0041

FUZZY VARIABLE STRUCTURE APPROACH

271

ture control theory have been redefined within the framework of fuzzy sets. The integration of fuzzy logic and variable structure control provided a powerful solution to the feedback regulation of uncertain systems and also to the chattering problem, which is the main reason behind the hindrance to the wide spread of variable structure control in many industrial control applications. A simulation example of a 2 degree-of-freedom robot manipulator is given in support of the proposed fuzzy variable structure approach. In dealing with the model uncertainties and external disturbances for the application at hand, it was assumed that an upper and lower bound was known about their effects. Using the theory of fuzzy sets, future work will deal more effectively with the model uncertainties and external disturbar~ces for which only a vague knowledge is available. This study is part of ongoing research aimed at bringing the rule-based structure of fuzzy logic control to an algebraic formulation that will help bridge the gap between this type of control and the analytical methods for design used in classical and m o d e m control theory.

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