# Variable Structure Control Essay

2 IEEE TRANSACTIONSON INDUSTRIAL ELECTRONICS,VOL. 40, NO. 1, FEBRUARY 1993 Variable Structure Control: A Survey John Y. Hung, Member, IEEE, Weibing Gao, SeniorMember, IEEE, and James C. Hung, Fellow, IEEE Abstract-A tutorial account of variable structure control with brief discussions about its historical development are pre- sliding mode is presented in this paper.

The purpose is to sented. introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control sys- A. The Basic Notion of VSC tem design approach. This approach is particularly attractive The basic idea of VSC was originally illustrated by a for the control of nonlinear systems.

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Prominent characteristics such as invariance, robustness, order reduction, and control second-order system [431, [ 1541 similar to the following chattering are discussed in detail. Methods for coping with X=y chattering are presented. Both linear and nonlinear systems are considered.

Future research areas are suggested and an exten- +U 2y -x y = (la) sive list of references is included. *x U = where * CONVENTIONS SYMBOLS FOR 4 when s ( x , y ) > 0 =

Capital, italic letters represent matrices, e. g. , A and K . (1b) when s ( x , y ) < 0 = -4 Boldface, roman lower-case letters represent vectors, e. g. , a and k. and Lower-case, italic letters represent scalars, e. g. a and s(x,y) = X U , u=0. 5x+y. (IC) k. A block diagram of the system is shown in Fig. l(a). The dim-x and d i m 4 stand for dimensions of x and B , variable s(x, y) in (1. 14 is the product of two functions respectively. I. INTRODUCTION 0 x and 0. ~y + 0. = = U = V The functions describe lines dividing the phase plane (xy ARIABLE structure control (VSC) with sliding mode plane) into regions where s ( x , y ) has different sign as control was first proposed and elaborated in the shown in Fig. (b). As such, the lines (2) are often called early 1950’s in the Soviet Union by Emelyanov and several switching lines and s(x, y ) is called a switching function. coresearchers [43], [79], [ 1541. In their pioneer works, the The lines also define the set of points in the phase plane plant considered was a linear second-order system mod- where s ( x , y ) = 0.

This set of points is known as the eled in phase variable form. Since then, VSC has devel- switching su\$uce, despite the fact that the set composed of oped into a general design method being examined for a two lines is not a surface in the strict sense. All of these wide spectrum of system types including nonlinear sys- terms are more carefully defined and used later. ems, multi-input/multi-output systems, discrete-time The feedback gain q9 is switched according to (lb), i. e. , models, large-scale and infinite-dimensional systems, and the sign of s ( x , y). Therefore, the system (la, b) is analyti- stochastic systems.

In addition, the objectives of VSC has cally defined in two regions of the phase plane by two been greatly extended from stabilization to other control different mathematical models: functions. The most distinguished feature of VSC is its In region I where s(x, y) = x u > 0, model is ability to result in very robust control systems; in many cases invariant control systems result.

Loosely speaking, x=y the term “invariant” means that the system is completely (3) 2 y – 5x y = 2 y – x – 4x = insensitive to parametric uncertainty and external distur- bances.

Today, research and development continue to u < 0, model is In region I1 where s ( x , y ) =x apply VSC to a wide variety of engineering systems. In x =y this introductory section, the basic notion of VSC and (4) 4x = 2y + 3x. 3 = 2y -x Manuscript received Nov. 26, 1991; revised December 28, 1991, April The phase plane trajectories for (3) and (4) are shown as 30. 1992. and Julv 17. 1992. J. Y. Hung is’with the Electrical Engineering Department, Auburn port&ts in-Fig. 2(a)”and (b). The equilibrium point of (3) University, Auburn, AL 36849. is an unstable focus at the origin.

The equilibrium point W. Gao is with The Seventh Research Division,Beijing University of of (4) is a saddle at the origin; the saddle point is also Aeronautics and Astronautics, Beijing 100083, China. J. C. Hung is with the Electrical and Computer Engineering Depart- unstable. ent, The University of Tennessee, Knoxville, TN 37996, and is the The phase portrait for the system (la, b) is formed by author to whom correspondence should be addressed. drawing the portrait for (2) in region I of the phase plane IEEE Log Number 9204072. 0278-0046/93\$03. 00 0 1993 IEEE Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply.

HUNG et al. : VARIABLE STRUCTURE CONTROL: A SURVEY 3 U = 0 line, representing motion called a sliding mode. Thus, a phase trajectory of this system generally consists of two parts, representing two modes of the system.

The first part is the reaching mode, also called nonsliding mode, in which the trajectory starting from anywhere on the phase plane moves toward a switching line and reaches the line in finite time.

The second part is the sliding mode in which the trajectory asymptotically tends to the origin of the phase plane, as defined by the differential equation I’ (5).

Four basic notions of this example VSC system should be observed: 1) Since the origin of the phase plane represents the equilibrium state of the system, the sliding mode represents the behavior of the system during the transient period. In other words, the line that de- Fig. 1. A simple VSC example. a) System model. (b) Regions defined by the switching logic. scribes U = 0 defines the transient response of the system during the sliding mode. 2) During the sliding mode, trajectory dynamics ( 5 ) are of a lower order than the original model (1). ) During the sliding mode, system dynamics are solely governed by the parameters that describe the line U = 0. 4) The sliding mode is a trajectory that is not inherent in either of the two structures defined by (2) or (3). During the control process, the structure of the control system (la, b) varies from one structure (2) to another (31, thus earning the name variable structure control.

To em- phasize the important role of the sliding mode, the con- trol is also often called sliding mode control. It should be noted that a variable structure control system can be devised without a sliding mode, but such a system does not possess the associated merits.

In the next section, the general VSC problem is stated. B. Statement of the VSC Problem (C) For a given control system represented by the state equation Fig. 2. Phase portraits. (a) For system (1. 3). (b) For system (1. 4). (c) For system (Lla, b). B(x,t ) u t) = A(x, x (6) n and dim-u m , find: where dim-x and drawing the portrait for (3) in region 11.

The resultant = = portrait is shown in Fig. 2(c). To obtain the complete m switching functions, represented in vector form as phase portrait, the trajectory of the system on the set s(x), and s ( x , y ) = 0 must be described. On the line x = 0, the a variable structure control phase trajectories of regions I and I1 are just joined together without any ambiguity.

On the line u + ( x , t ) when s(x) > 0 u(x, t ) = when s(x) < 0 U-(x,t) = u=OS~+y=OSx+i=0 (5) such that the reaching modes satisfy the reaching which itself is a dynamical equation, the phase portrait is condition, namely, reach the set s(x) = 0 (switching a trajectory along the switching line U = 0 as shown in surface) in finite time. Fig. 2(c). The physical meaning of above statement is as follows: The complete phase portrait of the system shows that there are no unusual motion characteristics on the line 1) Design a switching surface s(x) = 0 to represent a x = 0 other than possible discontinuities on motion direc- desired system dynamics, which is of lower order tion. However, the line U = 0 contains only endpoints of than the given plant. those trajectories coming from both sides of the line. ) Design a variable structure control u(x, t ) such that These points constitute a special trajectory along the any state x outside the switching surface is driven to Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. 4 IEEE TRANSACTIONS O N INDUSTRIAL ELECTRONICS, VOL. 40, NO. I , FEBRUARY 1993 c l x l + c 2 x 2+ … +x, defines a reach the surface in finite time. On the switching Then, a function s(x) = surface, the sliding mode takes place, following the surface desired system dynamics.

In this way, the overall + c 2 x 2+ c3x3 + … +x, = 0 (8) C,Xl VSC system is globally asymptotically stable. in the n-dimensional phase space. If switching is con- C. Brief Theoretical Background strained to occur on this ( n – 1)-dimensional surface, Before the emergence of the early stages of VSC devel- then all points of discontinuity lie on the surface. By opment, its foundation had already been laid.

Elements of solving (8) for x , and then substituting the result in (71, the foundation consist of the theory of oscillation and the the differential equation of the sliding mode is found: qualitative theory of differential equations. Brief discus- x1 = x 2 sions of these elements are given below. (a) Phase Plane Method: As a powerful graphical tool (9) for studying second-order dynamic systems, the phase -clxl c 2 x 2- * – . C n p I X , Xnpi = – 1 plane method was well established in the realm encom- passing the qualitative (geometric) theory of differential In summary, the coefficients in the switching function (8) equations and oscillation theory. The classical literature define the characteristic equation of the sliding mode if of Andronov et al. 161 and Flugge-Lotz [52] cited many the system model is described in controllable canonic early works in these areas. In their outstanding works, two form. contributions provided the foundation for he emergence Method 2: This method obtains the differential equa- of vsc: tion of sliding mode by means of two coordinate transfor- mations. Consider a linear, time-invariant plant and a 1) Regionwise linearization of nonlinear dynamic sys- switching function described by tems in which linearization of nonlinear systems was applied in partitioned regions of the phase plane. X=Ax+Bu This gave the initial prototype VSC systems. (x) = c x 2) The sliding mode motion, a term first used by Nikol- where dim-x = n and dim-u = dim-s = m. The dynamics ski [ l l l l . This was the first concept of sliding mode of the sliding mode could be more easily described if the control. state vector was composed with s as m of the state (b) Theory of DiSferential Equations with a Nonanalytic variables.

Therefore, the objective is to transform the Right-Hand Side: Two kinds of nonanalyticity are of im- model (10) into a form that contains s as m of the state portance with respect to VSC: variables. The plant is first transformed by y = Tlx to the form [1581 1) Finite discontinuous right-hand side, which is the relay type discontinuity, and Y , =A,lYl +&Y2 2) Double-valued right-hand side, which is the relay – type discontinuity with hysteresis. Bu =Az,y, +&y2 Y2 The problem is that a differential equation is not defined A # 0. where dim-y, = n – m , dim-y2 = m, and det at the point where the right-hand side of the equation is second transformation not analytic because the existence and uniqueness of the [ ]; solutions at these points are not guaranteed.

Hence, the 7: \$] = phase plane method cannot give a complete solution without defining an auxiliary equation at these points. The brings (11) to the form I601 auxiliary equation is the model of switching that occurs in VSC systems with discontinuous control. Five methods Y, = & y , +&S have been suggested to define the differential equation + A22s+ B u for the system at points of discontinuous dynamics.

They s =&yl are summarized as follows. When the system is in the sliding mode, the dynamics Method 1: First, transform the system model to control- satisfy s = 0 and S = 0. From (121, the differential equa- lable canonic form. For example, the controllable canonic tion for the sliding mode is easily solved by setting s = 0: form for a time-invariant, single-input linear system is x, = x 2 =Ally,. 13) Yl Method 3: Fillipov established a systematic mathemati- cal theory for differential equations with discontinuities [%I, [51], [110]. Consider a general plant (7) x,-1 =x,, x f(x,u) (14) = n , dim-u m. and switching function s(x) where dim-x = = Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. HUNG et al. VARIABLE STRUCTURE CONTROL: A SURVEY 5 Given that the system is of variable structure, the system dynamics can be described by two structures >0 f + ( x , u ) when s(x) f(x) = (15) < 0. when s(x) f-(x,u) = The system dynamics are not directly defined on s(x) = 0 Fig. 3. Fillipov’sconstruction of the equivalent control. by (1. 15). Instead, Fillipov describes the dynamics on s(x) = 0 as a type of “average” of the two structures in differential equations in the original state space, but they (15) are seldom used in practice.

Method 5 is devoted to 0 establishing a more general theoretical foundation. pfi+(l 1 (16) f,(x) p)f,, p I I = X – where D. Brief History of VSC The history of VSC development is marked by three limf+(x,u) and f , = limf-(x,u) f:= tages of development-the early stage of VSC, the stage s-0 s-0 of VSC for multi-input linear systems, and the stage of are functions of x. The term p is also a function of x and complete VSC development. Each of them is briefly de- can be specified in such a way that the “average” dynamic scribed below. f,(x) is tangent to the surface s(x) = 0. The geometric (a) The Early Stage of VSC (1957-1970): VSC systems concept is illustrated in Fig. 3. studied during this period possessed three characteristics.

Method 4: A fourth method for describing the dynamics The first characteristic is that each system was modeled by of the sliding mode is called the equivalent control ap- either a high-order, linear differential equation with a proach [153]. Consider the system single input: + B(x)u x = A(x) (17) + a , x ( ” – ’ ) ( t ) + . -. + a , . i ( t ) + a, = b u ( t ) (20) x(“)(t) with a switching function 4×1.

The first step of the equiva- lent control approach is to find the input u,(x) such that or by its equivalent state space model in controllable the state trajectory stays on the switching surface s(x) = 0. canonic form (7). The second characteristic is that the Once the equivalent control input is known, the sliding switching surface was defined in a special quadratic form: mode dynamics can be described by substituting u,(x> in + c 2 x 2+ s(x) = x , ( c l x I +c,x,). (21) *. * (17).

The equivalent control is found by recognizing that The third characteristic is that the structure of the control S(x> = 0 is a necessary condition for the state trajectory to was described by stay on the switching surface s(x) = 0. Differentiating 4×1 >0 where I+ a when s(x) ! U = \$x, = with respect to time along the trajectory of (17) gives (22) < 0. p when s(x) = dS dS + –B(x)u 0.

S(x) –A(x) (18) = = Equations (20)-(22) describe a single-input, linear system dX dX with switched feedback gain. Numerous VSC papers were yields the equivalent control Solving (18) for U published in this area and their results were well docu- mented [61, [43], [79], [154]. The issues studied in detail included: 1) Existence of the sliding mode where the existence of the matrix inverse is a necessary 2) Stability of the sliding mode condition.

Systems with time-varying coefficients 3) Method 5: The theory of general dynamical system con- 4) Effects of system parameter perturbations and out- cerns a differential equation with its right-hand side being side disturbances a set-mapping function; the theory is naturally applicable 5 ) Systems having unmeasurable state variables for describing the dynamics of a VSC system on its The type of systems with the aforementioned three switching surface [121]. It has been applied to min-max control, which can be considered a special case of VSC characteristics is quite restrictive. The quadratic nature of the switching function (21) and the structure of the con- 1641, 1661.

In all five methods just described, the objective is to trol (22) are sufficient to guarantee a sliding mode re- sponse, but other structures exist that are easier to imple- find the differential equation of the sliding mode, which is ment and are more flexible in design. In addition, the defined only on the switching surface. Methods 1 and 2 controllable canonic representation cannot always reveal are often more simple to use than the latter three, even the general nature the VSC systems. hough their results do not describe the sliding motion in (b) The Stage of VSC for Multi-Input Linear Systems the original state space because some state transforma- (2970-1980): During this period, the theory of VSC for tion is used. Methods 3 and 4 establish the sliding mode Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. 6 IEEE TRANSACTIONS ON INDUSTRIAL El. ECTRONICS, VOL. 40, NO. I , FEBRUARY 1993 general linear systems was more firmly established; the able structure system response. The extensions of these general system is of the form concepts to nonlinear systems are then presented.

Further extensions of variable structure concepts to other control + Bu (231 x =Ax objectives besides stabilization and regulation are briefly described. Known studies of variable structure control for where dim-x = n and dim-u = m. an m-dimensional lin- some special systems are noted.

Since application of VSC ear vector switching function 4×1 was postulated for the has been widespread in recent years, some notable contri- VSC. The control structure for each of the m inputs was butions are listed. Finally, a very extensive list of refer- described as ences is included. Other survey papers by Utkin [1551- @,+(x) when s,(x) > 0 u,(x) 11571, DeCarlo er al. [35], and a monograph by Gao [601 = (24) are also recommended. @,-(x) when s,(x) < 0 i 1,2;.. m = = 11. VSC FOR LINEAR SYSTEMS and every scalar switching function s,(x) was linear in the state variables rather than quadratic. However, the estab- The discussion in this section concerns the general lished VSC theory did not attract much attention for linear time-invariant system represented by the state many practical applications. The reason seems to be equation twofold.

First, VSC theory was overshadowed by the popu- (25) X=Ax+Bu lar linear control system design techniques. Second, the The state vector x is n-dimensional and the input U is important robustness properties of the VSC system were m-dimensional. The m column vectors of the B matrix not yet fully recognized or appreciated. are designated as b,, for i = 1 to m. (b) The Present Stage of Adilancing Der>elopment (1980-Present): Since 1980, two developments have greatly A. Basic Definitions enhanced the attention given to VSC systems.

The first is Basic terminology of variable structure systems with the existence of a genral VSC design method for complex sliding mode are more carefully defined in this section. systems. The second is a full recognition of the property Definition 1: The structure in a VSC system is governed of perfect robustness of a VSC system with respect to by the sign of a vector-valued function s(x), which is system perturbation and disturbances.

As a result, re- defined to be the switching function. A switching function search and development of VSC methods have been is generally assumed to be m dimensional and linear, i. e. , greatly accelerated, both in theory and in applications. The R & D work may be classified into five categories: cx (26) s(x) = 1) Development for different system models.

This in- where cludes the development of VSC theory for nonlinear … S(X) = [s,(x) s? (x) (27) systems, discrete-time systems, systems with time S,,,(X)]” . ; delay, stochastic systems, large-scale systems and and infinite-dimensional systems. 4I c ;. [ 2) Extension of the objectives of control. The functions … 28) = 7- of VSC have been extended beyond system stabiliza- Thus tion to include motion following or tracking, model following, model reaching, adaptive and optimal con- s,(x) (29) c7’x. = trol, and state observation. Each scalar switching function s,(x) describes a linear 3 ) Exploration of additional properties of VSC.

Such surface s,(x) = 0, which is defined to be ii switching sur- properties include invariance of the sliding mode to face. The term switching manifold is often used. In addi- system perturbations, robustness of the reaching or tion, the surface can be called a switching hyperplane nonsliding mode, and the elimination or reduction of because the switching function is linear. Notice in the control chatter. ntroductory exampled, however, that the set s ( x , y) = 0 4) Establishment of VSC laws that possess certain consisted of two intersecting lines. Such a set is not a characteristics. manifold in the mathematical sense. Hence, the terms 5 ) Applications in various engineering problems. manifold and hyperplane are avoided in the remainder of The purpose of this tutorial paper is to present the this paper. fundamental theory and main results for the design of Let x,) be the initial state of the system at the initial VSC systems.

The basic notions and a brief history of time to, x(t) be the state at any time t, and S be a VSC development have already been presented. More switching surface that includes the origin x = 0. precise definitions and deeper concepts are presented Definition 2: If, for any xo in S, we have x(t) in S for all t ; to, then x(t) is a sliding motion or sliding mode of the next by considering VSC for linear plants. Concepts dis- cussed are the design of switching surfaces, characteriza- system.

Definition 3: If every point in S is an end point, that is, tion of the sliding mode and the reaching or nonsliding for every point in S there are trajectories reaching it from mode, control law design, and basic properties of a vari- Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. HUNG et al. :VARIABLE STRUCTURE CONTROL: A SURVEY both sides of S, then switching surface S is called a sliding suflace. Definition 4: The condition under which the state will move toward and reach a sliding surface is called a reaching condition.

From the definitions above, it is shown next that an nth-order system with m inputs will have 2″ – 1 switch- ing surfaces. Fig. 4. Geometric interpretation of various switching surfaces. Let : be the ith row vector of the matrix C. Then c CTX = 0 defines a surface Si of dimension ( n – 1). These are called switching schemes and are described in There are m such surfaces: more detail in the next section. Si = {xls,(x) = cTx = 0) i = 1 , 2 ; – . , m . (30) B. Switching Schemes Consider the intersection of two surfaces S, and S,, As discussed earlier, a system having m inputs can have i Z j .

Their intersection is an ( n – 2)-dimensional m switching functions and up to 2″ – 1 sliding surfaces. switching surface. The total number of such intersec- Although some authors consider dynamics constrained to tions equals the number of combinations of m sur- the eventual sliding surface SE to be the sliding mode, the faces Si taken two at a time sliding mode can actually begin in a number of different ways, hereafter referred to as a switching scheme. The m! (m 1) – number of switching schemes that exist depends on the order of entering different sliding modes. (a) Fixed-Order Switching Scheme: In this scheme, slid- These switching surfaces are mathematically de- ing modes take place in a preassigned order as the system scribed by state traverses the state space.

For example, the state can s,(x) = cTx = 0} first move from the initial state x o onto the switching and S,, = {xls,(x) = cTx = 0 surface S,, which has dimension n – 1. The sliding mode (31) can then move to the surface SI, = ( S , n S,), which has i,j=1,2;.. ,m,i l T sgn(s) = C n, n. = r=l ‘(‘1 … fm(sm;lT. = [ f l ( ~ , ) The scheme is intended for large-scale systems and gives The scalar functions f , satisfy the condition good results [88], [143].

In general, the free-order switching scheme is the best s,f,(s,) ; 0 s, # 0 , i 1 to m . when = among all schemes. But, for large scale systems, the decentralized switching scheme is prefered. Equation (38) is called the reaching law. Various choices of Q and K specify different rates for s and yield differ- C. Reaching Conditions and the Reaching Mode ent structures in the reaching law.

Three examples are The condition under which the state will move toward 1) The constant rate reaching law and reach a sliding surface is called a reaching condition. The system trajectory under the reaching condition is Q sgn(s) (39) S = – called the reaching mode or reaching phase. Three ap- 2) The constant plus proportional rate reaching law proaches for specifying the reaching condition have been proposed.

Ks S -Qsgn(s) (40) = – (a) The Direct Switching Function Approach: The earli- est reaching condition proposed [431, [1541 was 3) The power rate reaching law ; 0, when to (54). The nonlinear systems considered in this paper will have the same numbers of outputs and inputs. Time- Finally, the overall system dynamics are given by invariant systems are modeled by omitting the variable t + a ( x ) + P(X)U x =Ax in (541455). where Three points are stressed at the outset for nonlinear systems: A diag[A,, A 2 , . . . A m ] = 1) Basic concepts and the fundamental theory of VSC are similar to those for linear systems. = a(. ) = 2) The derivation of a control u(x) is similarly simple and staightfonvard. 3) But the analysis of the sliding mode and the search Note that there can be coupling between the m subsys- for the corresponding switching function become a tems (57b). more difficult problem. ) The Input-Output Decoupled Form: Consider a single To study the stability of sliding modes occurring in nonlin- input/single output (SISO) nonlinear system with out put ear systems, various state transformations are used to put y = c(x, U ) . Let y”) denote the rth differentiation of y the differential equations of the system in one of several with respect to time. The number of differentiations of possible canonical forms. Reaching laws are generally the output function y = c(x, U ) required for the input U tailored to take advantage of canonical form characteris- to appear explicitly is called the relative degree r of the system. Note that the relative degree of a system cannot tics. be greater than the system order n. This concept of A.

Canonical Forms relative degree for a SISO nonlinear system is compatible 1) Reduced Form: The state vector x is partitioned into with the definition of relative degree for linear systems, x1 and x2, where dim-x, = dim-u = m and dim-x, = n – i. e. , the excess of poles over zeros. m. In addition, the input matrix takes the special form Relative degree in a MIMO system is defined in a Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. HUNG et al. : VARIABLE STRUCTURE CONTROL: A SURVEY 11 similar fashion.

For a given output y,, define the relative mensional switching surface equation degree r, to be the smallest integer such that at least one S(XI,X2) = 0 s(x) (62) = of the inputs appears explicitly in a differentiation. Since there are m outputs in the system, there are m such Theoretically, this equation can be solved for x 2 in terms integers r,. The relative degree is denoted by the set of x , in the form of x2 = d x , ) . This is equivalent to ( r , ; . . , rnl)and the total relatilie degree is defined by r = r1 expressing the switching function as + … +r,. s(x) x2 w(xl). 63) If the total relative degree r equals the system order n , = – then there are conditions such that the system Thus the problem of determining the switch function 4×1 + B(x)u is to find d x , ) such that the sliding mode is asymptoti- =A(x) X cally stable. That is, the system (56a), with x2 = d w , ) is (59) c(x) Y = asymptotically stable.

In general, it is difficult to find a desired w(x,; unless matrix A , in (56) is of a certain can be represented as a set of m decoupled higher-order special structure. At any rate, the dimension of x, is differential equations of the form n – m, less than that of the original system. Once d x , ) is found, so is s(x). Then the VSC u(x) may be designed using the reaching law method in a way similar to that for )… y‘,‘l-”. . . . ; y n l)… )y \$ , – ’ ) ) u , linear systems. 1,… , m.

Using the reaching law (38) in the system (56) yields the (60) i = VSC signal Necessary and sufficient conditions for this transforma- tion have been established in [541, [781. When the total ( + kf(s) relative degree r ; n, then the system can be represented -B*(x) Q sgn(s) = ) . U – dY2 ]-I( in the normal form, which is presented next. 4) Normal Form: Under certain mild restrictions, a dS dS + -A*(x) system model (54) that has total relative degree r ; n can +—A,(x) dx2 dx, be transformed into the normal form using differential geometry methods [24], [78].

The state transformation is 2) VSC for the Controllability Form: In this form, the divided into two groups of scalar functions. The first entire system is decomposed into m subsystems, so it is group of functions z f , ,are simply the m output functions convenient to use a decentralized sliding mode scheme y, = c,(x,u) and their derivatives up through order J = r, with decoupled switching functions, such as – 1, where r, is the relative degree of the ith output. The i l;.. ,m. , = cTx, (65) total number of state transformation functions in this first = group equals r = r1 + … +r,,,. The new state vector is The stability of the sliding mode of each subsystem is completed by finding n – r additional state transforma- guaranteed if c; is appropriately chosen. From (57), one tions rlk, k = l;.. , n – r , which are independent of each sees that x, is a substate vector in the phase variable form other and the functions zl,,.

Since t h e variables z,,, are and the equations of the subsystem are associated with the outputs, they can be considered the “external” dynamic variable. The functions rll, are inter- = XI2 i l l preted as being “internal” variables. The system dynamics Xf2 = x,3 are represented in terms of z f , ,and 77 = 7,-,IT as follows. + plul i l;.. ,m. i f n , a,(x) = m 0 becomes Therefore the switch plane C ~ X= , E i 1,”. , m + 77) ~i(z, V)U, =z ii,r, = x p )+ ~ i , k ( ~ , … ~ , ( n z – l p+ C f , , X , + k=l x(rl,-I) 0 . (67) = l rl I1 6= n dim q (61) r. (z,q) = – The VSC can be obtained from (57) and (65) using the In (61), a, and P I , , are scalar-valued functions, and y is a reaching law method. vector-valued nonlinear function. The reduced form, con- Since each subsystem is in controllable canonic form, trollability canonic form, and the input-output decoupled the dynamics of the sliding modes are easily determined. On the sliding surface, the n,th Recall that dim-x, form are all special cases of the normal form. component of the state vector x, can be represented in B. VSC for Canonical Forms terms of the remaining components by solving (63), s, = 0. ) VSC for the Reduced Form: Let the state vector be Substituting the result in (57b) give the dynamics of the partitioned according to (56). Consider a general m-di- sliding mode s,. The model is in controllable canonic form Authorized licensed use limited to: College of Engineering – Trivandrum. Downloaded on October 27, 2009 at 01:50 from IEEE Xplore. Restrictions apply. 12 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40,NO. 1 , FEBRUARY 1YY3 with order n , – 1, and the characteristic polynomial is the sliding mode, and 3) the steady-state mode.

The specified by the coefficients of the vector c , . Therefore, “steady state” of a system is here defined to be the final stability of each sliding mode is guaranteed by choosing periodic state of the system, which includes the constant the elements of c , to match a desired characteristic equa- state as a special case. Different forms of steady state tion. include zero-error state, constant offset state, and limit- 3) VSC for the Input-Output Decoupled Form: This case cycle state.

Since finite delays are inherent in all physical actuators, is very similar to the controllability form case, with the only difference in the meaning of the transformed state the sliding and steady-state modes are always accompa- variables. Variable structure control issues remain un- nied by objectionable chattering. The performance of a changed. The case was studied in [143], [148], where the VSC system is therefore measured by its response charac- teristics, robustness, and severity of chattering. switching function used took a special form A.

Characteristics of the Reaching Mode To better understand the reaching mode dynamics in a VSC system, a reaching space representation has been It should be pointed out that it is very advisable to proposed [611. Consider an nth-order system with m in- apply the centralized VSC where a set of subsystem puts and a switching function s, where dim-s = m. Define switching function is given by an m-dimensional reaching space R , whose coordinates 1 to m . i s,(xI)= C = are the m scalar functions s,. Thus, a reaching law dif- ~ X , erential equation such as 4 VSC for the Normal Form: The known results for ) variable structure control of a system in normal form (61) Kf(s) -Qsgn(s) S (68) = – assume stability of the zero dynamics. Zero dynamics are dynamics of the system under the condition that outputs represents the dynamics of the reaching mode in the and their derivatives equal zero.

Therefore, the zero dy- reaching space rather than in the original state space namics are described by the internal dynamic variables coordinates. Given ffs), (68) can be integrated to yield a with z = 0: solution s(t;,which describes a unique trajectory in R . i = Y(0;77). This trajectory is completely determined by the reaching law’s design and it yields certain important information Then, the switching function for a system in the normal about the reaching mode. For example, the reaching time form can be chosen as

T from the initial state s ~ to the sliding surface s = 0, is , i I;.. , m. s, = cTzl more easily determined in reaching space coordinates. = Further discussion on reaching space can be found in Gao Among the literature treating the normal form can be and Hung in this issue [63]. found the case of r = n (total relative degree = system order) 1491, the single-input case [138], and the multiple- B. characteristics of the Sliding Mode input case 1551, [901, [98].

The main issue in this approach The behavior of the sliding mode for linear plants has is how to guarantee the stability of the zero dynamics been discussed in Section 11. One of the main results is ;7 = 7(0,77) and how to relate to the formulation of the that a desired sliding mode dynamics can be achieved by output vector function y = c(x). an appropriate design of the switching function. In this 5 ) VSC for More General Models: As stated earlier, the ection, another characteristic of the sliding mode is con- most general nonlinear model takes the form x = ffx,U, t). sidered, namely, robustness or insensitivity to certain Works in this area exist [131, [141, [135] and deal with

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