variable structure control of nonlinear a tutorial structure control of nonlinear multivariable...

Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial

This paper presents, in a tutorial manner, the design of variable structure control WSC) systems for a class of multivariable nonlinear time varying systems. By the use of the Utkin-DraienoviC “method of equivalent control” and generalized Lyapunov stability concepts, VSC design is described in a unified manner. Com- plications that arise due to multiple inputs are then described and several approaches useful in overcoming these complications are then developed.

After this, the paper investigates recent developments and the kinship of VSC and the deterministic approach to the control of uncertain systems. All points are illustrated by numerical examples. In addition, the recent VSC applications literature is surveyed.

I . INTRODUCTION

Variable Structure Control (VSC) i s a viable high-speed switching feedback control (for example, the gains in each feedback path switch between two values according to some rule). This variable structure control law provides an effective and robust means of controlling nonlinear plants. It has i ts roots in relay and bang-bang control theory. The advances in computer technology and high-speed switching circuitry, have made the practical implementation of VSC a reality and of increasing interest to control engineers (see References).

Essentially, VSC utilizes a high-speed switching control law to drive the nonlinear plant’s state trajectory onto a specified and user-chosen surface in the state space (called the sliding or switching surface), and to maintain the plant’s state trajectory on this surface for all subsequent time. This surface is called the switching surface because if the state trajectory of the plant i s “above” the surface a control path has one gain and a different gain if the trajectory drops ”below” the surface. The plant dynamics restricted to this surface represent the controlled system’s behavior. By proper design of the sliding surface, VSC attains the con-

Manuscript received June 3,1987; revised November 16, 1987. R. A. DeCarlo and S. H. Zak arewith the School of Electrical Engi-

neering, Purdue University, West Lafayette, IN 47907, USA. G. P. Matthews is with the Electrical and Electronics Engineering

Department, General Motors Research Laboratories, Warren, MI 48090, USA.

I E E E Log Number 8719082.

ventional goals of control such as stabilization, tracking, regulation, etc.

The purpose of this paper is to furnish quick readable access to key design techniques in VSC (scattered throughout the literature) for a class of nonlinear time-varying systems. Because of the paper’s tutorial nature, the presen- tation includes only several of the basic forms of the many VSC design methods for multivariable, nonlinear, time- varying systems. These basic forms often need tweaking before application.

To minimize confusion and to maintain a unified exposition, our discussion concentrates on systems linear in the control input. Such systems are amenable to Utkin’s methods [I], [2], [4]. Also for simplicity, the paper deals most of the time with ideal VSC-i.e., switching in the control law can occur infinitely fast. The ideal case is much easier to analyzeand provides a baselineagainstwhich onecan mea- sure more realistic designs. Comments on the nonideal case are included for completeness at the end of the paper.

Section I I introduces the reader to the flexibility offered by the variable structure control strategy via the medium of a simple example. Section I l l crafts the setting in which the tutorial development is to proceed. It sets forth the basic definitions such as the system model, the switching surface, the associated notion of a sliding mode, and an overview of the two-phase VSC design process.

Section Vexamines phaseoneof theVSC design process, that of designing a sliding surface so that the plant restricted to the sliding surface has a desired system response. This means that the statevariables of the plant dynamics are con- strained tosatisfyanother set of equationswhich definethe so-called switching surface. An example illustrates the ideas and the relevant literature i s cited.

Section VI discusses the construction of the switched feedback gains necessary to drive the plant’s state trajectory to the sliding surface. These constructions build on the generalized Lyapunov stability theory.

The remainder of the paper deals with applications, the problem of nonideal switching, and the use of the boundary layer concept to alleviate the problem of chattering induced by the high-speed switching. Relationships to the theory of uncertain systems are also pointed out and dis-

212

0018-9219/88/0300-0212$01.00 0 1988 IEEE

PROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988

cussed along with a brief review of the recent VSC applications literature.

U- - 1 I U- I 1 I I . BACKGROUND

The term "variable structure control" arises because the "controller structure" around the plant i s intentionally changed by some external influence to obtain a desired plant behavior or response. For example, consider a plant with two accessible states and one control input as described by the following state equations

A block diagram representation of (2.1) appears in Fig. 1. Let the so-called switching surface be u ( x l , x2) = S I X l + x2 = 0

I 1

Relay

Fig. 1. Block diagram of a second-order system described by (2.1).

and the control law be given by

where

1 u > o

-1 U < 0. sgn (a) =

(2.2)

s1 "small", U = U I

(a)

x 2

U- t 1

Fig. 3. Phase-planediagramsof theclosed-loop system (2.1), (2.2) for different values of s,.

A block diagram of the closed-loop system is depicted in Fig. 2. Let us now investigate the behavior of the system for our system on u = 0 is dependent only on the slope s1 of

the switching line. This means the system i s insensitive to any variation-or perturbation of the plant parameters con- tained in the bottom row of the A matrix of (2.1), i.e., per- turbations in the image of the " B matrix"[O I]'. This i s one dominate motivation for investigating variable structure systems. The motion of Fig. 3(b) is more complex. Here the state trajectory switches to a new parabolic motion every time it intercepts the switching line u = 0. Nevertheless, the parabolic motions "spiral" into the origin.

As a second example consider a plant with two accessible states and one control input of the form U = k l ( x l , x 2 ) x 1 + k 2 ( x 1 , x z ) x 2 where the gains k , ( x l , x,) take on two possible values, say a, or p,. To specifically illustrate the idea, consider the state model

Fig. 2. Block diagram of the closed-loop system with the control strategy (2.2).

different values of the parameter sl, i.e., for different switching surfaces u. The phase-plane plotsof the system (2.1) with control law (2.2) are given in Fig. 3. Fig. 3(a) shows phase- plane trajectories for small s1 > 0 while Fig. 3(b) illustrates those for large s1 > 0. Upward motion in the trajectories is associated with U = + I and downward motion for U =

-1. Suppose the relay element in the block diagram of Fig. 2 has a small delay when switching between the gains "1" and "-1". Consider the resulting system behavior as this delay tends to zero and s1 is small. The behavior of this second-order system on the switching line u = s l x l + x2 = 0 (Fig.3(a)) isdescribed equation x1 + s l x l = 0 . It is important to note, that the behavior of

under the variable structure control law

u(t) = k ( x , ) xAt)

where k ( x l ) can be "-2" or "3". This system illustrated in Fig. 4 has two linear structures,

one each for k ( x l ) = 2 and k ( x l ) = -3. With k ( x l ) = -3, the system has complex eigenvalues and with k ( x l ) = 2, the system has real eigenvalues.

With the switch in the upper position, the feedback pro-

DeCARLO er al.: C O N T R O L OF NONLINEAR MULTIVARIABLE SYSTEMS 213

Fig. 4. Block diagram of a second-order system with variable structure control.

duces an unstable free motion satisfying

p] = [ -2 O 2 '1 p] x,

as shown in Fig. 5(a). With the switch in the lower position, the feedback becomes positive and the system's free motion satisfies

E:] = [" 3 2 '1 p]. x2

Switching of course i s not random. It occurs with respect to a sliding or switching surface, generically denoted as U = 0. To illustrate this notion, consider the surface defined as U = ul(xl, x,) = slxl + x, = 0 with s1 > 1. If the feedback is switched according to

-3, if ul(xl, x,)xl > 0 i 2, if ul(xl, xz)xl < 0 k(x1) =

a behavior illustrated in the phase plane plot of Fig. 6 results.

>1

XP + s1 x 1 = 0

Fig. 6. Path (dotted line) of state trajectory of system with control when perturbed slightly below the asymptote.

The "unstable" equilibrium point (0,O) i s now a saddle point with asymptotes x2 = 3x1 and x, = -xl, as shown in Fig. 5(b). Observe from the dotted line trajectory that if the state vec-

tor i s perturbed below the surface, ul(xl, x,) = slxl + x, =

"'I

1

- x 1

(b)

Fig. 5. Phase plane plots for feedback structures (a) kl(xl) = -3, and (b) k,(xl) = 2.

0, at time to, it circles to the point tl before intercepting the surface again. On the other hand, if the switching surface is u2(xl, x,) = slxl + x, = 0 with s1 < 1, then a perturbation off the surface i s always immediatelyforced back to the surface since the phase-plane velocity vectors always point towards the surface. Fig. 7 illustrates the phenomena.

x 1

2 ) = x 2 + s 1 x 1 =

S I < 1

+ X I = 0

Fig. 7. Phase plane plot of system with control when a sliding mode exists on the switching surface.

0

As suggested by Figs. 3, 6, and 7, different choices of switching surfaces produce radically different system responses.The richnessof variablestructurecontrol comes from this ability to choose various controller structures at different points in time.

The above example also illustrates an important notion

214 PROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988

in VSC. For the switching surface, u2(xl, x,) = slxl + x2 = 0 of Fig. 7, once the state trajectory intercepts the surface it remains on the surface for all subsequent time. This property of remainingon the switching surfaceonce intercepted i s called a sliding mode.

A sliding mode will exist for a system i f in the vicinity o f the switching surface, the state velocity vector (the derivative o f the state vector) is directed towards the surface.

The lack of a sliding mode for the "a = u1 scenario" described in the second example disappears when using a full state feedback control law: u(x) = kl(xl, x2)x1 + k,(xl, x2)x2. With appropriate gain choice, the original system can always be forced to have a sliding mode on any surface U

= slxl + x2 = 0. It was the choice of partial state feedback (U = kl(xl, x2)xl) which prevented the existence of a sliding mode on the surface ul = slxl + x2 = 0 with s, > 1.

Insuring the existence of a sliding mode on the switching surface i s a key necessity in VSC design. Designing the proper surface i s the complementary key problem. Thus VSC design breaks down into two major phases. The first i s theconstruction of the switching surface so that the original system or plant restricted to the surface responds in a desired manner. The second phase entails the development of a switching control law (i.e., appropriate switched feedback gains) which satisfies a set of "sufficient conditions" for the existence and reachability of a sliding mode [I], 121, 181, [ I l l , [W, [151, [191-[211, [24l, [291, [301, 1771.

[14], [82]. Moreover, for a large class of systems, design of linear switching surfaces proves amenable to classical linear controller techniques. Thus for clarity, convenience, and simplicityof exposition, this tutorial will focuson linear switching surfaces of the form

4x1 = Sx(t) = 0 (3.4)

where S i s an m x n matrix.

Sliding Modes

After switching surface design, the next important aspect of VSC is guaranteeing the existence of a sliding mode. A sliding mode exists, i f in the vicinity o f the switching surface, a(~) = 0, the tangent or velocity vectors o f the state trajectory always point toward the switching surface. Con- sequently, if the state trajectory intersects the sliding surface, the value of the state trajectory or "representative point" remains within an E neighborhood of {xla(x) = 0). As a point of information, i f a sliding mode exists on a(x) = 0, then a(x) is termed a sliding surface. As seen in Fig. 8, a

Ill. DEFINITIONS AND PRELIMINARIES

System Model

This paper considers a class of systems having a state model nonlinear in the state vector x(-) and linear in the control vector U ( * ) of the form

X(t) = f(t, XI U ) = f(t, x) + B(t, x) u(t) (3.1)

where the state vector x(t) E R", the control vector u(t) E R m , f (t, x) E R", and B(t, x) E R" '"'; further, each entry in f (t, x) and B(t, x) i s assumed to be continuous with continuous bounded derivative with respect to x.

Each entry ui(t) of the switched control u(t) E R m has the form

u:(t, x) with q(x) > 0

u;(t, x) with uj(x) -= 0 i = 1, , m

(3.2)

where ai(x) = 0 is the ith switching (also called disconti- nuity) surface associated with the (n - m)-dimensional switching surface

(3.3)

u;(t, x) =

a(x) = [aq(x), - * * , um(x)lT = 0.

The Switching Surface

The switching surface a(x) = 0 i s a (n - mbdimensional manifold in R" determined by the intersection of m (n - 1)-dimensional switching surfaces aj(x) = 0. The switching surfaces are designed such that the system response restricted to u(x) = 0 has a desired behavior such as stability or tracking. Switching surface design i s taken up in a later section.

Although general nonlinear switching surfaces (3.3) are possible, linear ones are more prevalent in design [I], [6],

Fig. 8. A situation in which sliding exists only on the intersection of the two sliding surfaces.

sliding mode may not exist on u,(x) = 0 separately, but only on the intersection.

An ideal sliding mode exists only when the state trajectory x(t) of the controlled plant satisfies cr[x(t)] = 0 at every t 2 to for some to. This requires infinitely fast switching. In actual systems, all facilities responsible for the switching control function have imperfections such as delay, hyster- esis, etc., which force switching to occur at a finite frequency. The representative point then oscillates within a neighborhood of the switching surface. This oscillation i s called chattering. If the frequency of the switching is very high compared with the dynamic response of the system, the imperfections and the finite switching frequencies are often but not always negligible. Hence our subsequent development considers primarily ideal sliding modes. The

DeCARLO et al.: CONTROL OF NONLINEAR MULTIVARIABLE SYSTEMS 215

problem of chattering and techniques for circumventing the problem are discussed in a later section of the paper.

Conditions for the Existence o f a Sliding Mode

Existence of a sliding mode [I]-[3], [5] requires stability of the state trajectory to the sliding surface a(x) = 0 at least in a neighborhood of {x la(x) = 0}-Le., the representative point must approach the surface at least asymptotically. The largest such neighborhood is called the region ofattraction. Geometrically, the tangent vector or time derivative of the state vector must point toward the sliding surface in the region of attraction [I], [8]. For a rigorous mathematical discussion of the existence of sliding modes for such systems see [I], [2], [8], [Ill, [45], [&I. These types of systems are referred to as discontinuous systems in the literature.

The existence problem resembles a generalized stability problem, hence the second method of Lyapunov provides a natural setting for analysis. Specifically, stability to the switching surface requires selecting a generalized Lyapu- nov function V(t, x) which i s positive definite and has a negative time derivative in the region of attraction. Formally stated:

Definition I [2]: A domain D in the manifold U = 0 i s a sliding mode domain if for each E > 0, there i s 6 > 0, such that any motion starting within a n-dimensional &vicinity of D may leavethe n-dimensional e-vicinity of Donlythrough the n-dimensional e-vicinity of the boundary of D. See Fig. 9.

6 - vicinity of boundary point of D

/ ‘boundary po iEof D

X,

Fig. 9. Two-dimensional illustration of domain of sliding mode.

Since the region D lies on the surface ~ ( x ) = 0, dimension [D] = n - m. Hence:

Theorem 1: For the n - m-dimensional domain D to be the domain of a sliding mode, it i s sufficient that in some n-dimensional domain Q 3 D, there exists a function V(t, x, a) continuously differentiable with respect to all of its arguments, satisfying the following conditions:

1) V(t, x, U) is positive definite with respect to U , i.e., V(t, x, U) > 0 with U # 0 and arbitrary t, x, and V(t, x, 0) = 0; and on the sphere 11u11 = p for all x E Q and any t the relations

i) inf V(t, x, a) = h, h, > 0 (3.5) IIUII = p

ii) sup V(t, x, a) = Hw H, > 0 (3.6) llull = P

hold, where h, and H, depend on p (hp # 0 if p # 0). 2) The total time derivative of V(t, x, a) for the system

(3.1) has a negative supremum for all x ECI except for x on the switching surface where the control inputs are unde- fined, and hence the derivative of V(t, x, U) does not exist.

A sliding mode is globally reachable if the domain of attraction i s the entire state space. Otherwise the domain of attraction is a subset of the state space.

Thestructureof thefunction V(t, x,a)determines theease with which one computes the actual feedback gains imple- menting a VSC design. For poorly chosen Lyapunov functions, the feedback gain computations can be untenable.

For all single input systems a suitable Lyapunov function is V(t, x) = .5a2(x) which clearly is globally positive definite. In VSC, uwil l depend on the control and hence if switched feedback gains can be chosen so that

Proof: The proof i s given in [I].

du2 da dt dt

. s - = u - < o (3.7)

in the domain of attraction, then the state trajectory con- verges to the surface and is restricted to the surface for all subsequent time. The feedback gains which would implement an associated VSC design are straightforward to compute in this case [I], 121, [81, [Ill, [MI, 1291.

Illustrative Design Examples

To illustrate the single input VSC design procedure consider the single pendulum system of [22], [23] having nonlinear state model x(t) = A(x) x(t) + Bu(t) where x = [xl, x2IT

r o i L XI _I

with control u(t) = kl(x)x, + k2(x)x2 where

ai(x), if a(x)xi > 0

pi(x), if o(x)xi < 0 k;(x) =

and a(x) = [sl s2]x. The feedbacks aj(x) and &(x) are chosen so that ~ ( x ) ir(x) < 0. Some simple substitutions show that

i f

al(x) = a1 < min - = -1 x1 Lsinxlx”I

x, [‘inxxl’] pl(x) = p1 > max - = 1

and a2 < -(s1/s2) and p2 > -(s1/s2). Hence, computation of the feedback gains i s straightforward.

For multivariable systems, useful Lyapunov functions prove difficult to find, except in special cases [I], [21. Spe- cifically:

1) Suppose there exists a positive definite, symmetric transformation W(t, x) such that R(t, x) = - W(t, x) SNt, x)


where R = [rij] has the property rii > EY=l,j+i (r i j l for i = 1, . . . , m, i.e., R(t, x) is diagonally dominant. If so, the recommended formof V(t,x, a) isaquadratic in a(t,x)withcoef- ficients depending on t and x.

2) Suppose SB(t, x) i s symmetric. The recommended form here i s V(t, x, U) = a'Ru where R i s symmetric and diagonalizes SB.

3) Suppose SB(r,x) is diagonallydominant. Here the recommended form of V(t, x, a) is a simple quadratic V(t, x, a) = a(x)' W(t, x) a(x) where W(t, x) i s a nonsingular diagonal matrix.

4) Finally suppose SB(t, x) i s diagonal. The recommended form of v(t, x, a) i s V(t, x, a) = u'u.

For these cases, the necessary feedback gains needed to implement a VSC design are straightforward to compute. For other cases, one usually executes some type of transformation to obtain such a form. Section VI provides a detailed discussion of case 4) above.

To illustrate that a poor choice of a Lyapunov function for a particular sliding surface combined with a naive choice of a control law may lead to extreme difficulties in solving for the necessary control gains, consider the hypothetical multivariable system

-

x(t) = AX@)

where

0 1 0

A = [ 0 0 1

-1 -2 -3-

with the sliding surface

t Bu(t) (3.8a)

and the control law u(t) = *x(t) where \k = [$ i j ] and

f o r i = 1 , 2 and j = 1 , 2 , 3. Observe that

arb = (1 - *lq + 2*',,)x,a, + (-1 + 9 1 1 - *2l)XlUZ

+ (3 - 4 1 2 + 2 4 2 2 ) X 2 f f l + (*12 - 422)XZU2

+ (5 - 9 1 3 + 2423)xg~l + ( - 4 + * I3 - Q23)xj~p

(3.11)

A sliding mode exists if (3.11) i s negative in the domain of attraction. In the single-input case this i s usually accomplished by making each term in the sum negative. By con- sidering just the first two terms in (3.11), it i s necessary to simultaneously satisfy

(1 - 911 + 2*21)XlI71 < 0

(-1 + 4 1 1 - *'21)Xlff2 < 0. (3.12)

There are four cases to consider.

Case 1: xlal > 0 and xlq > 0

Case 2: xlal > 0 and xla2 < 0

Case 3: xlal < 0 and xlq > 0

Case 4: xlul < 0 and xlq < 0.

After some calculations, one concludes that

i) 9.2: C O

ii) *i > 0

iii) 1 + > 4; > 1 + 24;.

Given i) and ii), it i s not possible to satisfy iii). Because of this and the obvious difficulty of solving (3.12)

directly, the use of the Lyapunov function V = .5aro and the control law (3.10) make this problem more difficult than necessary. By using a different Lyapunov function or another control law it is relatively straightforward to compute the control gains. For example, let

v = .5aJu

and

u(x) = -(Sf?)-' SAX - (SB)-' 2?- II #'

This control forces o r b = -1 < 0; thus a sliding mode exists on the sliding surface (3.9) and is reachable for all x E R3. Also, a different sliding surface will work, for example if

2E -1

s-[: -1 €I where E i s a small positive constant, it is possible to solve for the gains *$ since SB i s diagonally dominant. Lastly, we point out that using a multi-input diagonalization method, described in a later part of the paper, controller design for this problem is easily accomplished.

Design Procedure Overview

From the above discussion it becomes clear that VSC design breaks down into two phases. Phase 1 entails constructing switching surfaces so that the system restricted to the switching surfaces produces a desired behavior. Phase2 entails constructing switched feedback gainswhich drive the plant state trajectory to the sliding surface and maintain it there.

The actual details of this procedure are developed in Sec- tions IV through VII.

IV. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO VSC SYSTEMS

VSC produces system dynamicswith discontinuous right- hand sides due to the switching action of the controller. Thus they fail to satisfy conventional existence and uniqueness results of differential equation theory. Nevertheless an important aspect of VSC design i s the presumption that the plant behaves in a unique way when restricted to a(x) = 0. Therefore the problem of existence and uniqueness of differential equations with discontinuous right-hand sides is of fundamental importance.

Various types of existence and uniqueness theorems can be found in [I], [8], [35]. Supplementary material also exists in [52], [53]. However, one of the earliest and conceptually straightforward approaches i s the method of Filippov [46]. We will briefly review this method as background to the


above referenced results and as an aid in understanding variable structure system behavior on the switching surface.

Consider the following nth order, single input system

X(t) = f (t , x, U) (4.1)

with the following general control strategy

U +( t , x),

u - ( t , x ) ,

if u(x) > 0

if u(x) c 0. (4.2)

It can be shown from Filippov's work in [46] that the state trajectories of (4.1) with control (4.2) on ~ ( x ) = Oare the solutions of the equation (see Fig. IO)

xu = af + + (1 - a) f - = f O , 0 5 a 5 1 (4.3)

/

\ r r - 0

Fig. 10. illustration of the Filippov method of determining thedesired velocityvector f'for motion in the sliding mode.

where f + = f (t , x , U +), f - = f (t , x, U -), and f o is the resulting velocityvector of the state trajectory in a sliding mode. Solving the equation (du, f ' ) = 0 for a yields

(do, f - ) a = (da, ( f - - f +))

provided 1) (du, ( f - - f ')) > 0, and 2) (du, f + ) 5 0 and (du, f - ) 0 where the notation (a, b ) denotes the inner product of a and b also written as "a * b", and du = grad 44.

Therefore one may conclude that on the average, the solution to (4.1) with control (4.2) exists and is uniquely defined on a(x) = 0. Notice also that this technique can be used to determine the behavior of the plant in a sliding mode.

V. SLIDING SURFACE DESIGN

Filippov's method is one possible technique for determining the system motion in a sliding mode as outlined in the previous section. In particular, computation of f o rep- resented the "average" velocity (x ) of the state trajectory restricted to the switching surface. A more straightforward technique easily applicable to multi-input systems is the method of equivalent control, as proposed by Utkin in [ I ] , [4], and Draienovit in [77.

The Method of Equivalent Control

The method of equivalent control i s a means for determining the system motion restricted to the switching surface a ( ~ ) = 0. Suppose at to, the state trajectory of the plant intercepts the switching surface and a sliding mode exists for t 2 to. The existence of a sliding mode implies 1) u(x(t)) = 0, and 2) a(x(t)) = 0 for all t 2 to. From the chain rule [au/ax]x = 0. Substituting for x yields

[E]x = [E] [ f ( t , x) + B(t, X)Ueq] = 0

where ueq is the so-called equivalent control which solves this equation. After substituting this ueq into (3.1), the motion of (3.1) describes the behavior of the system restricted to the switching surface provided the initial condition x(t,J satisfies u(x(to)) = 0.

To compute ueq, let us assume that the matrix product [aalax] B(t, x ) i s nonsingular for all t and x. Then

Therefore, given u(x(to)) = 0, the dynamics of the system on the switching surface for t 2 to i s given by

-1

x = [ I - B(t, x ) [E B(t, x)] E] f ( t , x). (5.2)

In the special case of a linear switching surface u(x) = Sx = 0, a d a x = S, (5.2) reduces to

x = [l - B(t, x ) [SB(t, x)l-'Sl f ( t , x). (5.3)

This structure can be advantageously exploited in switching surface design.

Observe that (5.2) in conjunction with the constraint a(x) = 0 determines the system motion on the switching surface. As such, the motion on the switching surface will be governed by a reduced order set of equations. This order reduction comes about because of the set of state variable constraints, u(x) = 0.

The remaining parts of the section will describe 1) how one determines a reduced order set of dynamical equations governing the system motion on the switching surface, and 2) howtochoose surface parameterssfor a linear switching surface u(x) = Sx = 0, so that the system in a sliding mode exhibits the desired behavior.

Before closing this subsection, the reader should note that some control applications require a time-varying switching surface u(t, x ) = 0. In this case, U(t, x) = (adat) + (au/ax)x and the equivalent control takes the form

-1

ueq = -[E B(t, x ) ] [E f ( t , x ) + - ""1 (5.4) at

For simplicity of exposition, we have avoided the added complexity of the time-varying surface throughout most of the paper. Generalizations incorporating a time-varying component of a(t, x) are straightforward to construct. The time-varying surface structure will appear briefly when discussing diagonalization methods in Section VI and more commonly when discussing uncertain systems and VSC in Section VII.


Reduction of Order

For sake of clarity, we concentrate on the case of linear switching surface, u(x) = Sx = 0. As mentioned above, in a sliding mode, the equivalent system must satisfy not only the n-dimensional state dynamics (5.2), but also the "m" algebraic equations, a(x) = 0. The use of both constraints reduces the system dynamics from an nth-order model to an (n - m)th-order model.

Specifically, suppose the nonlinear system of (3.1) i s restricted to the switching surface of (3.4), i.e., a(x) = Sx = 0, with the system dynamics given by (5.3). Then it i s possible to solve for m of the state variables, in terms of the remaining n - m state variables, if the rank [SI = m.

The condition that rank [SI = m holds under the earlier assumption that [aalax] B(t, x ) is nonsingular for all t and x. To obtain the solution, solve form of the state variables (e.g., x,-,+~, , x,,) in terms of the n - m remaining state variables. Substitute these relations into the remaining n - m equations of (5.3) and the equations corresponding to the m state variables. The resultant (n - m)th-order system fully describes the equivalent system given an initial condition satisfying dx) = 0.

Example 5.4: To clarify the above procedure and to pave the way for later examples consider the system x( t ) = A(t, x ) x(t) + BuW, where

7 -.

(5.5)

Assume the third and fifth rows of A(t, x ) have nonlinear time-varying entries which are bounded: ayin 5 aij(t, x ) 5 a y for all x E R" and t E [tol a).

The method of equivalent control produces the following equivalent system (as per (5.3))

X(t) = [I - S(SS)-'S] A(t, X ) x(t) (5.6)

provided a(x(ro)) = 0 for some to. If the linear switching surface parameters are

s11 s12 513 s14 '151

s21 s22 s23 s24 s25

(5.7)

then r -7

SB = 1::: l::]. (5.8)

To simplify the example let us choose ~ 1 3 . ~ 2 5 - s15s23 = 1. Specifically, choose ~ 1 3 = 2, s15 = ~ 2 3 = sZ5 = 1. Then

(SB)-' = . (5.9)

This produces

X(t) = [I - S(SS)-lS] A([, X ) X ( t )

1 0 0 0

(5.10)

subject to a(x) = 0, in which case r - l

[' 1 '1 1 1";1 xg = -[I:: s12 s22 '14] s24 111. (5.11)

x4

Solving for x3 and x5 yields - -

The reduced order equivalent linear time-invariant system is

where i1 = xl , i2 = xZl 2, = xq. To see how control design might be accomplished, sup-

pose a design constraint requires the spectrum of the equivalent system be { -1, -2, -3}; the desired characteristic polynomial is

aA(h) = h3 + 6X2 + 11X + 6.

The characteristic polynomial of the equivalent system given in (5.13) is

a,4(X) = x3 + (SI2 - s22 + 2s24 - s14)X2

+ (s12s24 - s14s22 + s11 - s2l)X + (s11s24 - s14s21).

Equating coefficients of like powers of X produces the set of equations

s12

1 ~ 2 4 -sZ -1

S Z


One solution accomplishing the control design objective i s

1- 1 1.8333 2 -6 1 S = [

1 1.8333 1 0 1

In conclusion, the reduced order equivalent system with the desired eigenvalues i s = An, where

A = io 0 0

L-1 -1.83333 - 6 1

This example worked out so cleanly because the original system dynamics were given in the Luenberger canonical form. Systems not in this form often require a transformation to a more general form called the regular form [28].

Regular Form and the Reduced Order Dynamics

The regular form of the plant dynamics (3.1), is

x, = f l ( t , x)

Xi(z = f&t, X ) + B2(tr X ) U (5.14)

where x1 E R"-m and x, E Rm. A system in this form has sim- ply computed reduced order equivalent dynamics, also referred to as the system equations of slow motion. The computation of this form assumes BZ(t, x) i s an m x m nonsingular mapping. This assumption i s necessaryforthe existence of the equivalent control.

To compute the reduced order dynamics, assume a linear switching surface (this will be generalized later) of the form

a(x) = [S, S,] p] = 0. (5.15)

Without loss of generality assume S2 i s nonsingular. Thus in a sliding mode

x, = -s; 's1x, (5.16)

and

x, = f l ( t , x) = f,(t, XI, -S;'S,x1) (5.17)

which i s the reduced order dynamics. Observe that if f, has the very desirable linear structure

XI = fi(t, X ) = Ai Ix , + A12~2

then the reduced order dynamics becomes

XI = [A,, - A12S;1S1]~1 (5.18)

which has the feedback structure "All + A12F" with F = -S;'S1 and A,, playing the role of the input matrix. If the pair (A,,, A,,) is controllable, then it is possible to effectively use classical feedback control design techniques to compute an F such that Al l + AI2F has desired characteristics. Having found F, one can compute [S, Sd such that F = -S;'S1, thus completing the switching surfacedesign. Note that one can use pole placement techniques, linear optimal control techniques, etc., to design F. For more details of the linear case see Young et a/. [25] and El-Ghezawi et a/. [14].

For the more general case of a nonlinear switching sur-

face consider

a(x) = a,(x,) + S2x, = 0 (5.19)

which i s linear in x2 and possibly nonlinear in x,. For this case, the reduced order dynamics in a sliding mode will have the form

x, = f,(t, XI, -S;lu,(x,)). (5.20)

An example of designing a nonlinear switching surface will be given later.

The next important question i s how one transforms the given system dynamics (3.1) to the regular form of (5.14). We first consider the case of a linear switching surface of (5.15) and a nonsingular linear time invariant transformation z = Tx. Taking the time derivative of z yields

z = Tx = T f ( t , X ) + TB(t, X ) U . (5.21)

If it i s true that

(5.22)

then in the new coordinates the dynamics of the plant (3.1) become

Z, = f,(t, z)

2, = f2 ( t , z) + B,ct, z)u. (5.23)

Hence in a sliding mode the equivalent reduced order dynamics are given by (5.17) modulo the coordinate change, i.e.,

z, = il(t, Z,, -S;'S,Z1) (5.24)

where [s, s2] = [SI &IT- ' . If there i s no linear transformation such that (5.22) i s sat-

isfied, then one must resort to nonlinear transformations of the form

[;::: ::I z = T(t, x) = (5.25)

where 1) T ( . , e ) i s a diffeomorphic transformation, 2) T,(. , e ) : R x R" + R"-"', and T 2 ( * , e ) : R X R" + Rm. Diffeo- morphic [66], [67l means that there exists a continuous differentiable inverse mapping T(r, z) = x satisfying T(t, 0) = 0 for all t .

Differentiating z in (5.25) with respect to time produces

z = - aT (t , x)X + - aT (t , X I .

ax at (5.26)

Substituting (3.1) into (5.26) yields

aT aT aT Z = - f ( t , X ) + - B(t, X ) ~ ( t ) + -. (5.27)

ax ax at

If the transformation has the property that

B(t, x) = B(t, X ) = (5.28) ax

L a x J

then in the new coordinates, the equations describing our system or plant are


i2 ?,U, z) + s^,(t, z)u. (5.29)

The problem of converting a nonlinear system to a canonical form, in particular the regular form, was explored in 1281, 1661, 1671 among others.

VI. CONTROLLER DESIGN

Controller design is the second phase of the VSC design procedure mentioned earlier. Here the goal i s to determine switched feedbackgains which will drive the plant state trajectory to the switching surface and maintain a sliding mode condition. The presumption is that the sliding surface has been designed. In general, the control is an m-vector u(t) each of whose entries have the structure of the form

where 4x1 = [ul(x), * . , um(x)lJ = 0.

Diagonalization Methods

Our purpose hereistodescribetwodifferentapproaches to controller design labeled in the literature as diagonalization methods. The essential feature of these methods i s conversion of a multi-input design problem into m single- input design problems.

Method 1 entails constructing a new control vector U* via a nonsingular transformation, Q-'(t, x) [aulax] B(t, x) of the original control U defined as

where Q(t, x) i s an arbitrary m x m diagonal matrix with elements qi(t, x) (i = 1, . . , m) such that inf Iqi(t, x)) > 0 for all t 2 0 and all x. The actual conversion of the m-input design problem to "m" single input design problems is accomplished by the [adax] B(t, x)-term with the diagonal entries of Q-'(t, x) merely allowing flexibility in the design, for example by weighting the various control channels of U * .

Often Q(t, x) i s chosen as the identity. In terms of U * the state dynamics become

-1

x(t) = f ( t , x) + B(f, x) [E (x) B(t, XI] Q(f, x) u*(t) . (6.3)

Although this new control structure looks more complicated, the structure of U(x) = 0 permits one to independently choose the m-entries of U* to satisfy the sufficient conditions for the existence and reachability of a sliding mode. Once U* i s known it can be unraveled by inverting thetransformation to yield the required u.To see this, recall that for existence and reachability of a sliding mode it i s enough to satisfy the condition uT(x) U(x) < 0. In terms of U*

aa ax

U(X) = - (x) f ( t , X) + Q(t, X) u*(t) . (6.4)

Thus i f the entries U * + and U*- are chosen to satisfy

qi(t, X) U ; + < -vu;(x) f ( t , X)

n

= -igl s j j f i ( t , x) when ui(x) > 0 (6.5a)

9;(t, x) U ; - > -Vu;(t) f ( t , x)

n

= -jFl s i i f i ( t , x) when ui(x) < 0 (6.5b)

then sufficient conditions for the existence and reachability are satisfied where sji equals thej-entry of Vu,(x) which is the ith row of (adax). In particular, the conditions of (6.5) force each term in the summation of arbto be negative definite. As mentioned, the control actually implemented i s

u(t) = - B(t, X) Q(t, X) u*(t) . c: )-I (6.6)

Other sufficient conditions for the existence of a sliding mode can also be used.

The second method of diagonalization requires a nonsingular transformation of U rather than the control U . In particular, consider the new switching surface

U*& x) = nct, x) u(x) = 0 (6.7)

for an appropriate transformation Q(t, x). This method is based on the fact that the equivalent system is invariant to a npnsingular switching surface transformation as verified in the following theorem.

Theorem 2 [?I: Suppose that the original system is given by (3.1), (3.2) with switching surfaces uj(t, x) = 0 (i = 1, a . . , m), then the sliding motion (trajectory of the equivalent system) i s invariant to the transformation of the switching surface u*(t, x) = h(t, x) u(t, x) = 0 E R m , if llhll and l l f l - ' l / are bounded for all t, x E S E R x R".

Proof: First, by the method of equivalent control

au au ax a t

U* = n(t, X) - (x) ( f (t , X) + Bu,,) + Cl - 00 = 0. (6.8)

Since O(t, x) i s a nonsingular m x m matrix

(6.9)

which differs from U,, only by the term ((adax) B ) - ' Q - ' hu. However, on the switching surface, U = 0, thus

(6.10)

Hence the equivalent systems are identical and the motions in the sliding mode coincide.

Loosely stated, Theorem 2 says that the motion in the sliding mode is independent of a nonsingular possibly time- varying transformation of the switching surfaces. Observe that any nonsingular transformation fi with bounded deriv- atives will produce the same "equivalent" system.

In this second diagonalization procedure, we select

DeCARLO er al.: CONTROL OF NONLINEAR MULTIVARIABLE SYSTEMS 221

Wt, x ) so that Wt, x ) @a/ax) (x) B(t, x ) i s a diagonal matrix, say Q(t ,x ) = diag[q;(t,x)]whoseentriesare bounded awayfrom zero. Specifically select Q(t, x ) as

-1

Nt, x) = Q(t, x)[$ (x) B(t, x,] (6.11)

for appropriate Q(t, x). Again Q(t, x ) i s often chosen as the identity matrix.

In order to determine the existence and reachability conditions it i s necessary to compute U* as

* f ( t , x ) + Q(f , X ) U + o(t, X ) n-'(t, x)a*

A = ao(t, X ) + Q(t, X ) U + a& x) (6.12)

where again the control term U "enters" U* via the diagonal matrix Q(t , x).

Sufficient conditions for reachability/existence of sliding mode are met if for any point in the state space and for all t a to, $( t , x ) and $( t , x ) are of opposite sign. Specifically, this requires that

q,(t, x ) U +

qi(t, x ) U; > -af(t, x ) - a&, x), for U: 0 (6.13)

-uf( t , x ) - ajn(t, x), for U: > o

where a,& x) i s the ith entry of

A U& X ) = B(t, X) n - l ( t , X ) a*(t, x) .

Example 6.74: To clarify the diagonalization methods,

(6.15a)

consider the system of Example 5.4 where

x(t) = ~ ( t , X ) x(t) + Bu(t)

and

B = 1 0 . (6.15b)

The surface a(x) = Sx = 0 was designed to have

The objective of this example is to illustrate phase 2 of the VSC controller design process using the first and second diagonalization methods described above. The first

design employs method 1 which transforms the control U as per (6.2)

u*(t) = Q-'(t, x ) SB(t, X ) u(t) (6.17)

where Q(t, x) i s a nonsingular diagonal matrix such that inf Iqi(t, x)l > 0. For simplicity choose

Q = [: Q-' = [i :d. (6.18)

The choice was random. However, the diagonal entries of Q can be chosen to weight different control channels or to compensate somewhat for the "distortion" introduced by [SB(t, x ) ] - ' . As per (6.3), the state dynamics driven by U* are

x(t) = ~ ( t , X ) ~ ( t ) + ~ ( t , x ) [ s B ( ~ , x1i-l Q(t, x ) u*(t). (6.19)

In computing the feedback gains to meet the existence conditions, (6.4) becomes

ir(t) = Sx(t) = SA(t, x ) x(t) + Q(t, x ) u*(t, x).

Since Q(t, x ) i s diagonal, using (6.5), sufficient conditions for the existence of a sliding mode are

q;(t, X ) U: < -[s;l, * * . t si51 A(t, X) x(t),

if U; > 0, i = 1, 2

9;(C X ) U; > - [s i l l * * , si51 A(t, X ) x(t),

if a; < 0, i = 1, 2. (6.20)

It follows tha for the first switching surface al(x) = [sll, . . . , s151 x(t) = Slx(t) one has

SIA(r, x ) x(f) = {(2all + aZ1) x1 + (1 + 2a12 + a22x2)x2

h

+ (1.8333 + 2a13 + az3) x3 + (2a14 + a24)x4

+ ( .al5 + aZ5 - 6 ) x 5 } . (6.21)

Recall the assumption: arin 5 aji(t, x ) 5 a y , i = 1,2 , j = 1, - , 5 . Under this assumption and control law U* = Kx where K = [ k p I, to satisfy the existence condition of (6.20), kli must satisfy the following:

-(2a,mlax + a y ) ,

> -(2a?;'" + a$n),

< -(I + 2al;"" + a r ) ,

> -(I + 2aEin + ag"),

if ulxl > 0

if alxl < 0

if ulx2 > 0

ifa1x2 < o

i i

kll =

4 . 8 3 3 3 + 2a;3"" + a F ) ,

'13 = t> -(1.8333 + 2 a p + agn),

- ( 2 a r + a,","),

k15 = [

if (I1x3 > 0

if < 0

if 61x4 > 0

> + a;"), if 01x4 < O

k12 =

- ( 2 a y + a r - 6),

> -(2aZn + agn - 6),

if alxs > 0

if ~ 1 x 5 < 0. (6.22)

With regard to the second switching surface, u ~ ( x ) = [ s~I ,

S2A(t, x ) x(t) = {(all + a24 x1 + (1 + a12 + aZ2x2) xz

k14 =

A . . . , sZ5] x(t) = Szx(f), one has

+ (1.83333 + a13 + a23)x3

+ (al4 + az4) x4 + (al5 + aZ5)x5} . (6.23)


Let U: = K2x; to satisfy the existence conditions (6.201, k , must satisfy the following:

< -1/2(ar,a" + a y ) ,

> -1/2(a;;'" + agin),

< -1/2(1 + a y + a y ) ,

> -1/2(1 + a;;'" + ag"),

if q x l > 0

if q x l < o

if u2x2 > 0

if u2x2 < 0

I I

I

k21 =

-1/2(1.8333 + a y + a y ) ,

> -1/2(1.8333 + aT," + ag"),

< -1/2(a;",ax + a?),

> -1/2(a$" + a$"',

- 1 / 2 ( a y + a y ) ,

if ~ 2 x 3 > 0

if g2x3 < 0

k22 =

k23 = [ if ~ 2 x 4 > 0

if ~ 2 x 4 < o

if ~ 2 x 5 > 0

k24 =

I > -1/2(a?'5'" + ag"), if u2x5 < 0. (6.24)

Summarizing, u*(t) = Kx(t) where the entries of K are specified by (6.22) and (6.24). Since u(t) = (SB)-'Qu*(t, x), the actual control i s u(t) = [U#), u2(t)] ', so that

k25 =

r l

I x1 I

L x 5 J This completes the illustration of the first method of diag-

onalization. Attention now turns to the second method. Again assume that the switching surface design is complete. In the second diagonalization method, the objective is to decouple the controls by making a nonsingular transformation of the switching surface. The control components, the entries of U , now switch on u:(x) = 0, i = 1, 2, with U: (x) given by (6.7). To perform the diagonalization choose Q(t, x) according to (6.11) where (adax) = S. If Q i s chosen as per (6.18), then

nu, X) = Q(t, x)[SB(t, XI]-'

1 -1 1 -1

= [: 2] = [-2 41- (6*26)

To construct a controller meeting the existence conditions of a sliding mode consider the derivative of U* given by (6.12), noting that h = 0

u*(t, X) = Q(t, x) SA@, x) x(t) + Q(t, x) u(t) (6.27)

where

Q(t, x) SA(t, x) x(t)

1 + a12x2 + a13x3 + a14x4 + (al5 - 6)x5

+ 2(1 + a22) x2 + 2(a23 + 1.8333)x3

+ 2a2,x4 + 2(a25 + 6 ) ~ s (6.28)

and Qu = [ul 2 ~ 2 1 ' .

The conditions for the existence of a sliding mode (6.13), for this example are

qlu: < -4'(t, XI, i = 1 , 2 (6.29)

q,u; > - U % x),

where up(x, t) i s the ith component of uo(t, x) = Q(t, x) SA(t, x) x(t). - . , 5. Since ay'" s a,(t, x) 5 a y , i = 1 , 2 , / = 1, . , 5 , then to satisfy (6.29), it is required that

< - a y ,

> -a;",

< - a y ,

> -ag",

< - a y ,

> -a;",

c -(I + a y ) ,

> -(I + a;"),

< - a y ,

> -aT;",

< -(1.8333 + a y ) ,

> -(1.8333 + a z " ) ,

< - a y , > - p i n

14 , < - a y ,

> -a;",

< -(a;;"" - 6),

Let U = Kx where K = [k,], i = 1, 2, j = 1,

if ulxl > 0

if ulxl < 0

if u2xl > 0

if u2xl < 0

if u1x2 > 0

i f u1x2 < 0

i i i i i i i i

kll =

k21 =

if u2x2 > 0

if uzx2 < 0

k12 =

if ~ 1 x 3 > 0

if u1x3 < 0

k22 =

if ~ 2 x 3 > 0

if ~ 2 x 3 < 0

k13 =

if u1x4 > 0

if ~ 1 x 4 < 0

if ~ 2 x 4 > 0

i f ~ 2 x 4 < 0

k23 =

k14 =

if ulxs > 0 i > -(a;" - 6), if u1x5 < 0

k24 =

k15 =

-(6 + a y ) ,

> -(6 + ag") ,

if ~ 2 x 5 > 0

if ~ 2 x 5 < 0. (6.30) i k25 =

Since Q(t, x) in (6.26) is constant, h = 0 and the above are also sufficient conditions for reaching the sliding surface.

As a second rather important illustration of the first diagonalization method, recall the example developed in (3.8) through (3.11). This multi-input example demonstrated how a poor choice of Lyapunov function in conjunction with a naive choice of controller led to an inconsistent solution of the equations defining the switched feedback gains needed to drive the state trajectory of the plant (3.8) to the switching surface of (3.9). The use of a diagonalization method circumvents this difficulty by converting the problem to one where the intuitive choice of Lyapunovfunction actually will work.

Using the first diagonalization method with Q = I and with B defined in (3.8b) and the switching surface S defined in (3.9), then as per (6.2), the new control has the form


where

k l , if uixi > 0

k i , if ujxi c 0. (6.32)

To compute the values of these gains consider

X 1 + 3x2 + 5x3 a = SAX + U* =

= [ - x , - 4x, ] + [:I. (6.33)

This leads to

U101 = (1 + k l l ) ~ 1 x 1 + (3 + k12) ~ 1 x 2 + (5 + k 1 3 ) ~ 1 ~ 3

6 2 0 2 = (k21 - 1) ~ 2 x 2 + k 2 2 ~ 2 ~ 2 + (k23 - 4 ) U 2 X 3 .

Therefore, sufficient conditions for stability to the switching surface are

k $ c -1,

k , > -1,

if ulxl > 0

if ulxl < 0 kll =

k:2 < -3, if u1x2 > 0 i kG > -3, if ulx2 c 0 k12 =

k2: < 4, if ~ 2 x 3 > 0 i k s > 4, if ~ 2 x 3 < 0. k23 =

Thus in terms of the original plant controller

An implementation of the control can be accomplished using the structure of Fig. 11.

P x(O’

(sei’ Q

Fig. 11. Implementation of the controller of (6.34).

Method of Control Hierarchy

As an alternative to the diagonalization methods described earlier it i s often possible to define a hierarchy of controls. One then employs this control hierarchy in designing a controller.

With this approach a hierarchy of control channels is established so that, for example, the first control u1 drives the system from an initial condition onto the surface u1 = 0. The second control then drives the system onto the inter-

mode on u1 = 0. The third control u,drives the system along the intersection of the surfaces uI = 0 and u2 = 0 to the intersection of the first three switching surfaces. This hierarchy of controls i s continued until the last control U, drives the system to a sliding mode on the intersection of all the m switching surfaces.

Design of the control entry uk presupposes i) existence of a sliding mode on ai = 0, j = 1, . 9 , k - 1 for any possible value of the controls uk through U,, and ii) knowledge of the system structure in these sliding modes. Since all controls Uk, k c m, depend on the values taken on by the control U,,,, U, must precede the design of u,, , -~, u , - ~ , * * * , U‘. In addition, design of the control u2 presupposes the system structure obtained assuming a sliding mode exists on u1 = 0. This system structure results by replacing u1 with the Utkin-Draienovik equivalent control uleq. Call the resulting system structure E’.

To determine E’, consider (aul/ax) x = 0, which, using (3.1), implies that

(6.35)

where bi i s the j th column of B(t, x) . Note that this rela- tionship requires (dullax) bl # 0. (In fact, it i s necessary to assume that (auj/ax) bj # 0 for all i.) Substituting (6.35) into (3.1) produces the equivalent system model

x = f’(t, x ) + B’(t, x) [I] u m

for an appropriately formed f ’(f, x ) and B’(t, X I . Design of u3 presupposes the system structure obtained

by supposing a sliding mode exists on U, = 0 for the system structure E’. This implies a sliding mode exists on u1 = u2 = 0. Call the resulting structure E’. Of course E 2 is specified by replacing u2 in E’ by uleq. In general, uk+l is designed supposing a sliding mode exists on uk = 0 for the system structure C k - ’ and hence on ai = 0, j = 1, . * . , k. The new system structure is Ek and is denoted by the dynamics

rUk+’i

Before designing the first control U,,, it isclearly necessary to sequentiallydeterminethesetof equivalent systems {E’, ~ 2 , ” . , ~ m - 1 }. Given Em-’, U, has gains chosen to satisfy

the reachabilityand existenceconditionsforasliding mode on U, = 0. After this, one presumes the system structure

and proceeds to find gains for U,,,-’ so that a sliding mode exists on U,-’ = 0 given the gains computed for U,.

To see how the existence and reachability conditions are determined at the (k + 1)-step, realize that a sliding mode exists and is reachable on uk + = 0 provided uk + is chosen so that

E m - 2

section of u1 = 0 and 0, = 0, while u1 maintains a sliding ‘ J k + l u k + l (6.37)


for ail values of uk+2, * - 9 , U,. Observe that uk+' has the form

m - k

, = 1 u k + i = V U k + l f k + ,z V u k + l b f U k + ;

where b : i s the ith column of Bk(t, x ) and Vuk+' = (auk +,,/ax.

To insure the existence of a sliding mode on uk+' = 0, (6.37) must hold for all U ; , i = k + 2, * * , m. Specifically, this condition has the form

V u k + l b $ u ; + l < min U k + 2 , " ' r U m

m I - c Vuk+lb ,k -ku j , if uk+l > 0 i = k + 2

(6.38a)

m

if uk+' < 0.

(6.38 b)

The maximal and minimal values in (6.38) indicate that uk+luk+l < 0 no matter which of the two values U : or U,: i s taken on by the components of the control U; (i = k + 2, , m ) .

Summarizing, we introduce a hierarchy of controls whereby u1 guarantees motion in a sliding mode along u1 = 0 for any value of u2, * . , U,. The second component u2 guarantees motion along the intersection of u1 = 0 and u2 = 0 for any values of uj, * . * , U,,,, and so on. The sig- nificance of this method is that a sufficient condition for a sliding mode is obtained from the sliding mode existence condition for the scalar case.

Example 6.39: To clarify the method of control hierarchy, consider the state model

x = Ax + Bu

where

A =

Let u = [Ul, u2, u J r = Sx = 0 where

1 2 1 1 0 1

S = O 1 0 1 1 1 = s 2

0 1 0 0 d A [::I Following the preceding algorithm, first find U, = S'Ax + S'BU = 0 where S' i s the first row of S. Hence

S'A = [2, 5 , 2, 4, -1, -21, S'B = [2, 1, I].

Solving for U', in terms of x, u2 and ug yields

Uleq = -3[2, 5, 2, 4, -1, -21 x - i [ U Z , Ugl.

Inserting ul, into x = Ax + Bu, yields x = A'x + B'u', where

0 -112 -1 -2 112 1

0 1 0 0 0

'1 L o 0 0 - 1 - 2

0 0 0 0

Next, solve u2 = S2A'x + S2B1u' = 0 for u2,, in terms of x and u3, where S2A1 = [0, -1/2, 1, 1, -1/2, 01, S2B1 = [1/2, 1/21, and u2, is given by

u2, = [O, 1, -2, -2, 1, O I X - u3.

Inserting u2, into x = A'x + B'u' yields x = A2x + B2u2 where

0 1 0

0 - 1 0 -

0 0

1 0

A 2 = i 0 0 0 0

0 0

-1 0

1 0

1 1

0 0

0 -1 -

2

1

-2 :j 0

B 2 = ,1 Now that A', A', B', B2 are known, the second half of the

algorithm is used to determine the controls. Starting with k = 2, (6.38) gives

S 3 b : u : < -S3A2x

S 3 b : u ; > -S3A2x.


Since S3b: = 1, and S3A2 = [0,1,0,1, -1, -21, to satisfy the above inequalities it i s sufficient that

U ; < -S3A2x,

U ; > -S3A2x,

if U, > 0

if U, < 0.

TO determine the second control we again apply (6.38) obtaining

S2b:u; < min [-S2A’x - S2b:u3] U 3

and

S2b:u; > max [-S2A’x - S2b:u3].

From our previous calculations we have S2b: = 1/2, S2bi = 112, and S2A’ = [0, -112, 1, 1, -112, 01. To satisfy (6.38) it is sufficient that

-(2S2A1x + U ? ) ,

U ; > -(2S2A’x + U ; ) ,

U 3

if u2 > 0

if u2 < 0. u2 = y <

Finally the first control U , must satisfy

S’blu: < min [-S’Ax - S’b2u2 - S’b3u31 U2,U3

and

S’blu; > max [-S’Ax - S’b2u2 - S’b3u3]

where B(t, x) = [b,, b2, b3]. Since S’bl = 2, S’b2 = 1, and S’A = [2, 5, 2, 4, -1, -21 to satisfy the above inequalities it i s sufficient that

U?., U 3

U : < -0.5(S1Ax + U ; + U ; ) ,

ul- > -0.5(S1Ax + U: + U ; ) ,

if u1 > 0

if u1 < 0. U1 =[

In this example the control hierarchy i s u1 then u2 and then u3, but i s it the optimum order? Is there another order in which thecontrol gains are smaller?Theanswer depends on the initial condition.

Other Approaches

In addition to the diagonalization methods and hierar- chial control method mentioned above, other approaches are possible. In theory an infinite variety of control strategies of the form (6.1) are possible. An alternative structure for the control of (6.1) i s

uj = uj4 + UjN (6.40)

where uj4 i s the ith component of the equivalent control (which i s continuous) and where ujN is the discontinuous or switched part of (6.1). For controllers having the structure of (6.40), the following i s true:

Let us assume that (adax) B(t, x) = I, the identity. Then ~ ( x ) = U,.+ This condition allows an easy verification of the sufficiency conditions for the existence and reachability of a

sliding mode, i.e., the condition that ujuj < 0 when uj(x) # 0. Below are five possible discontinuous control structures for UN.

I ) Relays with constant gains:

Observe that th is controller will meet the sufficiency condition for the existence of a sliding mode since

uiuj = ajuj(x) sgn (ui(x)) < 0, if u,(x) # 0.

2) Relays with state dependent gains:

CY; sgn (uj(x)), uj(x) # 0, m i ( . ) < 0 I U;(X) = 0, UiN(X) =

Again it i s straightforward to check that

u;u; = al(x) ul(x) sgn (u,(x)) < 0,

For an example of cq(x) consider q(x) = &(U;~(X) + 7 ; ) with Pi < 0, y i > 0, where k i s a natural number.

if u,(x) z 0.

3) Linear feedback with switched gains:

“jj, ujxj > 0 i Pij, uixl < 0

ujuj = Uj($j’X’ + $;2x2 + * * . + $,nXn) < 0.

uiN(x) = $x; $ = [$;j], $,j =

with ai, < 0 and Pi, > 0. Thus again

4) Linear continuous feedback:

u;N(x) = CU;U;(X) and a; < 0. The condition for the existence of a sliding mode is

ujuj = aju;(x) < 0 or more generally

where L E R m X m is positive definite constant matrix. The condition for the existence of a sliding mode is easily checked

uN(X) = -I!.&)

uT(x) u(x) = -uT(x) Lu(x) < 0, if u(x) + 0. 5) Univector nonlinearity with scale factor:

The existence conditions are

uT(x) u(x) = Ilu(x)II p < 0, if u(x) + 0. Given a nonlinear system, if a linear behavior is required

in a sliding mode then one may need to use a nonlinear switching surface. A good control for this purpose i s the controller of (6.41). To illustrate the above point in the context of this control structure consider the following example.

Example 6.42: Consider a simple robotic manipulator driven by a dc armature control dc motor [72] modeled by the following equations

[ = [l:n(xl) + x3] + [!] U f(x) + Bu. (6.43)

x2 + x3


For any switching surface u(xl, x,, x3) = 0, development of the control requires [(adax) B] be nonsingular. The structure of B then implies that adax, # 0. Without loss of generality, we set adax, = 1. Hence a general structure for a nonlinear switching surface in this example (similar to (5.19)) is

a(x) = x3 + fJI(X,, x,) = 0.

Using a control structure of (6.40) and (6.41) yields

U = ueq + 01 sgn [a(x)], CY < 0.

Note that the above control somewhat resembles a control strategy resulting from the quadratic form-based design of Utkin [I].

It i s easy to check that

U2

14 air = O1 - < 0

since sgn (U) = u/(a(. Thus as expected we are assured the existence of a sliding mode.

The next step i s to determine the structure of the switching surface and the equivalent system. Suppose we desire the system to exhibit a linear behavior in a sliding mode described by the equation

for appropriate a, and a2, both positive. Computing ueq according to (5.1) results in

x2

Substituting the above into the plant dynamics implies that on a(x) = 0

which follows according to (5.20). Comparing (6.44) and (6.45) requires that

al(x,, x,) = sin(x,) + alxl + a2x2

a nonlinear switching surface, i.e., thedesired linear behavior with the controller structure of (6.40), (6.41) requires the use of a nonlinear switching surface. For other examples of using a nonlinear switching surface see for example [73].

As a final point, the type of controller discussed in this subsection i s further developed in the next section to account for the problem of uncertain parameters in the plant model.

VII. OVERVIEW OF UNCERTAIN SYSTEM THEORY, VSC, AND

CHATTERING

Introduction

The purposes of this section are the exposition of VSC for uncertain systems, unification of the theories of VSC and deterministic methods of controlling uncertain systems, and a discussion of chattering. The motivation for exploring uncertain systems is the fact that model identi-

fication of real-world systems introduces parameter errors. Hence models contain uncertain parameters which are often known to lie within upper and lower bounds. Awhole body of literature has arisen in recent years concerned with the deterministic stabilization of systems having uncertain parameters lying within known bounds. Such control strategies are based on the second method of Lyapunov.

On the other hand, VSC controllers are based on the Gen- eralized Lyapunov Second Method. Hence, one expects some fundamental links in the two theories. Using the control structures described in Section VI under the heading "Other Approaches" we will establish these links.

First, a description of the uncertain plant and a brief review of the basic definitions of deterministic control of uncertain systems will be given. Following this we will out- IinetheVSCapproach to uncertain system control.Thiswill then lead us to the expected fundamental connections. Lastly, we will focus on improving controller performance by reducing or eliminating chattering through the introduction of the so-called boundary layer controllers. This, in fact, i s a natural outgrowth of the theory of uncertain systems although it was developed independently in the VSC context [75], [76], [30], [7l.

Deterministic Control o f Uncertain Sys tems

uncertainties consider the following state dynamics To represent uncertainties in the plant from parameter

x(t) = [ f (t , x(t)) + A f (t , x(t)),

+ [B(t, x(tN + A M , x(t), r(t))l u(t) (7.1)

where r(t) is a vector function (Lebesgue measurable) of uncertain parameters whose values belong to some closed and bounded set. The formulation presumes no statistical information on the uncertainties. The plant uncertainties A fand A B (arising from r(t)) are required to lie in the image of B(t, x) for all values of t and x. This requirement is the so- called matching condition [47l, [50], [51] [61]. Assuming the satisfaction of the matchingconditions, it i s possibleto lump the total plant uncertainty into a single vector e(t, x(t), r(t), u(t)) and represent the uncertain plant as

x = f(t, x) + B(t, x) U + B(t, x) e(t, x, r, U )

X(t0) = xg. (7.2)

With regard to a stabilization analysis of the above model (7.2) the following definitions are pertinent:

Definition 2: Let x(-): [to, w] -, R" be a solution of (7.2). x(*) i s uniformly bounded if for each xo there i s a positive finite constant, d(xo), (0 < d(xo) < 00) such that IIx(S112 < d(xo) for all t E [ t o , a ] where 11 - 11, is the usual Euclidean vector norm [47l, [a].

Definition 3: Solutions to (7.2) are uniformly ultimately boundedwith respect to some closed bounded set S C R" if for each xo there is a non-negative constant T(xo, S ) < 03 such that x(t) E S for all t > to + T(xo, SI.

The problem is to find a state feedback u(t, x): R x R" -P

Rm such that for any initial condition xo and for all uncer- taintiesr(t)asolutionx(.):[t,,, OD) -+ R"of (7.2)existsandevery such solution is uniformly bounded.

The literaturecontains two main approachesforthe solution of the above stabilization problem, the so-called min- max controller discussed by Gutman and Palmor [50] and


the Corless-Leitmann approach [47J. These approaches begin with a nominal system defined by

x = f (t , x) X(t0) = xg (7.3)

assuming that x = 0 is an equilibrium point, i.e., f ( t , 0) =

0 for all t. Both approaches require this nominal system to be uniformly asymptotically stable, i.e.,

1) for any E > 0, there is a 6(e) > 0 such that a trajectory starting within a &)-neighborhood of x = 0 remains for all subsequent time within the €-neighborhood of "O",

2) there i s a 6, such that a trajectory originating within a &-neighborhood of x = 0 tends to zero as t -+ W.

It turns out that if there exists a (Lyapunov) function U . 1 : R x R" -+ R + with a continuous derivative, and there exist continuous strictly increasing functions n(-): R , + R,, i = 1, 2, 3, with the properties

y;(O) = 0, i = 1, 2, 3

such that for all (t, x) E R x R"

and

where V,Visacolumn vector, then the nominal system (7.3, i s uniformly asymptotically stable. The objective i s to use this nominal Lyapunov function V ( * ) and bounds on the uncertainty e(t, x, r, U ) to develop conditions on the state feedback control U = u(t, x) guaranteeing uniform boundedness of the closed loop state trajectory of the plant of (7.2). This background sets up a discussion of the two meth- odsof stabilization of uncertain systems.The min-maxcon- trol method comes first.

In the min-max approach one assumes a stable nominal system (7.3) with Lyapunov function V(t, x). A Lyapunov function candidate for the closed loop plant, (7.2), with U

= u(t, x), i s again V(t, XI. The objective i s to choose u(t, x) to make the derivative of V(t, x) negative on the trajectories of the closed loop system, i.e., choose U = u(t, x) such that

[a,': 1 av at V(t, x) = - + ( V L V ) x = - + (V,TV) f

+ ( v , T v ) ~ ( u + e) c 0. (7.5)

Since (7.4b) holds, (7.5) holds if U = u(t, x) is chosen such that

min max ( v , T v ) B(u + e) I o (7.6)

for all ( t , x) E R x R" and all admissible controls and admissible uncertainties. Assuming BT(t , x) V,V(t , x) i s nonzero, the control

u e

where p(t, x) i s a scalar function satisfying p(t, x) 2 IIe(t, x, r, u)l12 can be shown by direct substitution to satisfy (7.6).

If B r ( t , x) V,V(t , x) i s zero then take

U E { U I U E R m and I( U 11 5 pft, x)} . (7.8)

The reader should note that the set

can be thought of as a switching surface. One of the main goals of this section i s to show that the controller of (7.7) can be made to behave as a VSC controller with the switching surface (7.9).

A close inspection of (7.7) reveals that this control i s discontinuous in the state since, for example, in the single input case it reduces to U = -sgn (BT( t , x) V,V(t , x))

Since the above control is discontinuous it may excite unmodeled high-frequency dynamics of the plant. To avoid this problem, it i s necessary to modify this controller by introducing the so-called boundary layer controller which continuously approximates the discontinuous action of (7.7) in a neighborhood of the switching surface, (7.9). Let p(t, x) be any continuous function such that p(t, x) = -[ulllu)llp when llall = E. Then the structure of the boundary layer controller is

p(t, x).

i f llull E

Ibll (7.10) I-' p(t, x), if llull < E.

U = u(t, x) =

Unfortunately, this controller does not guarantee asymptotic stability but rather uniform ultimate boundedness as per Definition 3. For a proof of this fact in a slightly modified context see Corless and Leitmann [47l. This completes our brief review of the deterministic control of uncertain systems.

The VSC Approach to the Control of Uncertain Systems

Here again consider the uncertain plant as described in (7.2). In theVSCapproach it i s not necessaryforthe nominal system (7.3) to be stable. However, the equivalent system, i.e., the restriction of (7.3) to the switching surface u(t, x) = 0, must be asymptotically stable.

The VSC control structure for plant (7.3) will be

U = u q + UN (7.11)

where ueq i s the equivalent control for (7.3) assuming al l uncertainties e(t, x, r, U ) are zero and uN i s to be designed to account for nonzero uncertainties. Recall from Section VI under the heading "Other Approaches" that the structure of (7.11) makes determination of the reachability and existence conditions for a sliding mode more straightforward to compute.

Proceeding in the usual fashion, with the switching surface u(t, x) = 0, one may compute

-1

Ueq = -[E B] [; + 5 f ] (7.12)

assuming as usual that [(adax) B] is nonsingular and that e(t, x, r, U ) = 0. It i s now necessary to account for uncertainties and develop an expression for U,.,.

To develop this thread assume as in the previous subsection that

(7.13) IIe(t, x, r, u)l12 s d t , x)


where p(t, x) i s a non-negative scalar valued function. Also introduce the scalar valued function

at , x) = a + dt, x) (7.14)

where a > 0. This particular structure simplifies some of the derivations.

Before specifying the control structure, we choose the most simple generalized Lyapunov function

V(t, x) = .5aJ(t, x) a(t, x). (7.15)

As usual, in order to insure the existence of a sliding mode and attractiveness to the surface, it i s sufficient [ I ] tochoose a variable structure controller so that

d V A dt - (t, x) = V = arb c 0

whenever a(t, x) # 0 where

aa aa at ax

b(t, x) = - + - x.

(7.16)

(7.17)

Thecontroller form given in (6.41) in conjunction with the controller of (7.7) suggests the VSC form

(7.18)

when a(t, x) # 0 and where

(7.19)

where V, V(t, x) i s the gradient of the generalized Lyapunov function (7.15). This is different from the V(t, x) used to develop (7.7). If a(t, x) = 0, then set u(t, x) = ueq(t, x).

In ordertoverifythevalidityof this controller notice that (suppressing t and x arguments)

aa aa at ax

V = a J - + a J - ( f + B u + Be). (7.20)

Substituting (7.18) into (7.20) and manipulating produces

i / = a J - + + J - f f g J - f - a J - aa aa aa aa at ax ax at

(7.21)

verifying the negative definiteness of V. This establishes attractiveness to the switching surface.

A Comparison of the VSC and Deterministic Controllers

The similarity of the structure of (7.7) and (7.18) i s clear in view of the uN term. The controllers, however, generate different responses with respect to different switching surfaces. This is best seen by viewing Fig. 12. Fig. 12(a) illustrates a typical trajectory of the closed loop system (7.2) driven by the controller (7.7). Notice that the control switches with respect to the surface BJV,V = 0 where V i s the Lyapunov function of the nominally stable system. In

(b) Fig. 12. Typical closed-loop system trajectories for the control (a) (73, and (b) (7.18).

some cases for sufficiently large p(t, x), the response will behave as a VSC response in a vicinity of the origin. In fact, if p(t, x) i s sufficiently large the controller exhibits the stan- dard VSC response once the trajectory intercepts the surface B 'V,V = 0 [50].

In Fig. 12(b)we have the usual VSC responsewith respect to the user-chosen switching surface a(t, x) = 0. In addition to stabilization of the closed loop system one may obtain tracking properties as implicitly built into the user-designed switching surface.

Chattering

The VSC controllers developed in Section VI and the uncertain system controllers of this section assure the desired behavior of the closed loop system. These controllers, however, requirean infinitely(in the ideal case) fast switching mechanism. The phenomenon of nonideal but fast switching was labeled as chattering (actually the word stems from the noise generated by the switching element). The high frequency components of the chattering are unde- sirable because they may excite unmodeled high-frequency plant dynamics which could result in unforseen instabilities. The boundary layer controller of (7.10) helps to eliminate the effects of chattering. Let us now refine this notion of boundary layer and boundary layer controller.

Define the set

{XI Ila(x)II 5 E, E > 0)

as the so-called boundary layer of thickness 2 ~ . Consider the control law (suppressing t and x arguments)

C U e q + Pf if )1u11 < E

DeCARLO er al.: CONTROL OF NONLINEAR MULTIVARIABLE SYSTEMS 229

where ueq i s given by (7.12) and wherep = p(t, x) is any continuous function such that

whenever ) )U ) ) = E and \)p)l 6 . This control guarantees attractiveness to the boundary layer and inside the boundary layer, (7.22) offers a continuous approximation to the discontinuous control action of (7.18). As shown in Corless and Leitmann [47], one i s not guaranteed asymptotic stability but ultimate boundedness of trajectories to within a neighborhood of the origin depending on E.

The reader might peruse [30], [51] which offer an alternate class of controllers than those above. Specifically [50], [60], [61] offer a discussion of such controllers for linear plants.

Finally, the reader might note that in the control of dc motors chattering i s of minimal concern since switching can occur in the high kilohertz range if not megahertz range due to advances in power electronics. This, of course, i s well beyond the structural frequencies of mechanical systems involved.

VIII. APPLICATIONS

In [26] Young developed an adaptive VSC for an aircraft control. Calise and Krammer [I31 also investigated VSC for aircraft control. An alternative approach to aircraft control using uncertain system controllers was proposed by Peter- sen [12]. Other applications to spacecraft control can be found in Sira-Ramirez and Dwyer in [63], [64].

In the area of robotic control, Young [27] developed an algorithm based on hierarchical VSC, later refined by Mor- gan and Ozguner [36]. Also Slotine and Sastry [7l used VSC for tracking control of robot manipulators. A model following VSC scheme was developed by Ambrosin0 eta/. [20] and applied to a simple model of a robot manipulator. A similar application can be found in Bailey and Arapostathis [70]. The use of the Filippov method applied to robotic control can be found in Bartolini and Zolezzi [MI, Paden and Sastry [74], and [70]. A combination of VSC and deterministic approach to the control of uncertain systems was proposed by Spong and Sira-Ramirez [54].

There are various applications of VSC to power systems. Among these are Young and Kwatny [6] (who developed an overspeed protection control for an electric power gen- erating plant), Hawley and DeCarlo [22], Lefebvre et a/. [23] and Richter et al. [24]. This was later extended by Matthews and DeCarlo in [78], [81]. Also Sivaramakrishnan et a/. [34] applied the VSC to the design of a variable structure load- frequencycontrollerfor a singlearea power system. In Ben- giamin and Kauffmann [32] an application of VSC to the dc motor position control i s described. While Utkin and Orlov [I91 treat the problem of distributed control using VSC.

In contrast to the above VSC applications other methods applicable to real-world systems can be found in Hunt et a/. [66], [67l and Sain and Peczkowski [65]. A nice review paper on recent trends in nonlinear system feedback control is Kokotovic [58].

IX. CONCLUDING REMARKS

This paper has developed and surveyed the essential concepts of VSC. Recall that the design of a VSC has two steps: 1) design of the switching surface to assure the desired behavior of the plant in a sliding mode, and 2) development

of the control law which forces the system’s trajectory to and maintains it on the sliding surface.

In discussingthe multi-input case,wesawthatthedesign process was complicated by the coupling of the controls through the switching surface. Several different methods were then developed (the diagonalization and hierarchical methods) toeffectivelydecouple thecontrolsand thus simplify the design process. Essentially, these techniques reduced the overall control problem to a series of single input problems.

After this, important connections to the theory and application ofthedeterministiccontrol of uncertain systemwere introduced and developed. The two theories were seen to have a close alignment.

To illustrate that VSC theory i s sufficiently advanced to allow the design of sophisticated control systems, a brief survey of VSC applications as found in the literature was given. Thewarnings of the previous sections combined with the application survey indicate that when VSC can be applied to a particular problem a very high-quality control system results. However, significant research i s needed to successfullyapply VSC to even more general classes of nonlinear systems, for example, systems nonlinear in the control U = u(t, x) and systems in which the matching conditions are not satisfied.

Another problem with VSC is the need for complete state information. Development of switching surfaces and controllers based on measurable output signals represents an open problem and an areaof important research.Thedeve1- opment of nonlinearobservers usingVSCconcepts isastep in this direction. See, for example, Walcott and Zak [69].

Another area filled with fertile research soil i s the rela- tionship of VSC with the recently fabricated Lie Algebraic approach to the control of nonlinear systems. Preliminary resu Its in this direction have been published by Marino [57].

Other exciting open problems fall in the categories of tracking and output regulation [59], [80], discrete variable structure control [62] and large scale systems. In the large scale systems area a number of promising results (espe- cially in the decentralized control framework) have been obtained by DeCarlo and co-workers [22]-[24], 1311, [781-[821.

A very interesting application of the VSC theory is in the design of Pulse-Width-Modulated (PWM) control strategies in nonlinear systems. In particular, Sira-Ramirez [83] established an equivalence between the sliding modes, resulting fromaVSCstrategy,and the responseresuItingfromaPWM control law in nonlinear analytic systems. Specifically, under certain conditions the PWM controlled response represents an ideal sliding motion on an invariant manifold associated with an ideal “average” system. Details of these results and an application to the control design of switch- mode dc-to-dc power converter circuits can be found in

Finally let us close with the point that this paper attempts to present to the reader possible solutions to some ques- tions raised in the report [71].

ACKNOWLEDGMENT

[831*

The authors gratefully acknowledge constructive remarks of the reviewers.

REFERENCES

[I] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems. Moscow, Soviet Union: MIR Publishers, 1978.


[2] -, ”Variable structure systems with sliding modes,” /E€€ Trans. Automat. Contr., vol. AC-22, no. 2, pp. 212-222, 1977.

[3] -,“Variable structure systems-present and future,”Auto- mat. Remote Contr., vol. 44, no. 9, pp. 1105-1120,1983.

[4] -, “Equations of the sliding regime in the discontinuous systems 1,”Automat. Remote Contr., vol. 32, no. 12, pp. 1897- 1907,1971.

[SI V. I. Utkin and K. D. Yang, “Methods for construction of dis- continuity planes in multidimensional variable structure systems,” Automat. Remote Contr., vol. 39, no. IO, pp. 1466-1470, 1978.

[6] K.-K. D. Young and H. G. Kwatny, “Variable structure ser- vomechanism design and applications to overspeed protection control,” Automatica, vol. 18, no. 4, pp. 385-400, 1982.

[;7 J . J. Slotine and S. S. Sastry, “Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators,” lnt. 1. Contr., vol. 38, no. 2, pp. 465-492,1983.

[8] U. Itkis, Control Systems of Variable Structure. New York, NY Wiley, 1976.

[9] -, “Dynamic switching of type-l/type-ll structures in tracking servosystems,” /€€€ Trans. Automat. Contr., vol. AC-28, no. 4, pp. 531-534, 1983.

[IO] B. A. White, “Reduced order switching functions in variable structure control theory,” I € € froc. D. Control Theory & Appl.,

[Ill B. A. White and P. M. Silson, ”Reachability in variable structure control systems,“ I€€ Proc., vol. 131, pt. D., no. 3, May 1984.

[I21 I . R. Petersen, “A procedure for simultaneously stabilizing a collection of single input linear systems using non-linear state feedback control,”Automatica, vol. 23, no. 1, pp. 33-40,1987.

[I31 A. J. Cake and F. Kramer, “A variable structure approach to robust control of VTOL aircraft,” in froc. of the American Control Conf., pp. 1046-1052, June 1982.

[I41 0. M. E. El-Chezawi, A. S.Zinober, and S. A. Billings, ”Analysis and design of variable structure systems using a geometric approach,” lnt. /. Contr., vol. 38, no. 3, pp. 657-671, 1983.

[I51 E. P. Ryan, “A variable structure approach to feedback regulation of uncertain dynamical systems,” Int. /. Contr., vol. 38, no. 6, pp. 1121-1134,1983.

[I61 I. Kadushin and A. M. Steinberg, “Stabilization of nonlinear systems with a dither control,” 1. Math. Anal. Appl., vol. 43, no. 1, pp. 273-284, July 1971.

[17] C. Zames and N. A. Shneydor, “Dither in nonlinear systems,” I € € € Trans. Automat. Contr., vol. AC-21, no. 5, Oct. 1976.

[I81 W. M. Wonham and C. D. Johnson,”Optimal bang-bangcon- trol with quadratic performance index,” in 4th joint Auto- maticControlConf., Minneapolis, MN, pp. 101-112, June 1963.

[I91 Y. V. Orlov and V. I. Utkin, “Use of sliding modes in distributed system control problems,” Automat. Remote Contr., vol. 43, no. 9, pp. 1143-1148,1982.

[20] G. Ambrosino, C. Celentano, and F. Garofalo, “Variable structure model reference adaptive control system,” lnt. 1. Contr., vol. 39, no. 6, pp. 1339-1349, 1984.

[21] W. Hejmo, “On the sensitivity of a time-optimal position control,” /€€€ Trans. Automat. Contr., vol. AC-28, no. 5, pp. 618- 621,1983.

[22] P. Hawley and R. DeCarlo, “Variable structure control of interconnected systems with applications to power systems,” Purdue University, School of Electrical Engineering Tech. Rep. TR-EE-83-9, 1983.

[23] S. Lefebvre, S. Richter, and R. DeCarlo, “Decentralized variable structure control design for a two-pendulum system,” I € € € Trans. Automat. Contr., vol. AC-28, no. 12, pp. 112- 114, Dec. 1983.

[24] S. Richter, S. Lefebvre, and R. DeCarlo, “Control of a class of nonlinear systems by decentralized control,” I€€€ Trans. Automat. Contr., vol. AC-27, no. 2, Apr. 1982.

[25] K.-K. D. Young, P. V. Kokotovic, and V. I. Utkin, “A singular perturbation analysis of high-gain feedback systems,” I€€€ Trans. Automat. Contr., vol. AC-22, no. 6, pp. 931-938, 1977.

[26] K.-K. D.Young,”Designofvariablestructure model following systems,” /€€E Trans. Automat. Contr., vol. AC-23, no. 6, pp.

[27] -, “Controller design for a manipulator using the theory of variable structure systems,” / E € € Trans. Syst. Man Cybern., vol. 8, no. 2, pp. 101-109, 1978.

[28] A. C. Luk‘yanov and V. I . Utkin, “Methods of reducing equa-

VOI. 130, pp. 33-40,1983.

1079-1085,1978.

tions of dynamic systems to regular form,” Automat. Remote Contr., vol. 42, no. 4, pp. 413-422, 1981.

[29] A. S. Zinober, “Controller design using the theory of variable structure systems,” in Self-Tuning and Adaptive Control. London, England: Peter Peregrinus Ltd., 1981, ch. 9, pp. 206- 229.

[30] J. J. Slotine, ”Sliding controller design for non-linear systems,” lnt /. Contr., vol. 40, no. 2, pp. 421-434, 1984.

[31] C. P. Matthews, S. H. Zak, and R. A. DeCarlo, “Decentralized control for aclass of nonlinear systems using avariable structure approach,” in Proc. ofthe22Allerton Conf., pp. 192-200, Oct. 1984.

[32] N. N. Bengiamin and B. Kauffmann, “Variable structure position control,” Control Syst. Mag., pp. 3-8, Aug. 1984.

[33] A. R. Bazelow and J. Raamot, “On the microprocessor solution of ordinary differential equations using integer arith- metic,” /E€€ Trans. Comput., vol. C-32, no. 2, pp. 204-207,1983.

[34] A. Y. Sivaramakrishnan, M. V. Hariharan, and M. C. Srisailam, “Design of a variable-structure load-frequency controller using pole assignment technique,” lnt. 1. Contr., vol. 40, no.

1351 0. Hajek, “Discontinuous differential equations I, 11,‘’). Dif- ferential Equations, vol. 32, no. 2, 1979.

[36] R. C. Morgan and U. Ozgunner, “A decentralized variable structure control algorithm for robotic manipulators,” /€E€ 1. Robotics Automat., vol. RA-I, no. 1, pp. 57-65, Mar. 1985.

[371 T. A. Bezvodinskaya and E. F. Sabaev, “Stability conditions in the large for variable structure system,” Automat. Remote Contr., vol. 35, no. IO, pp. 1596-1599,1974.

[38] G. I . Lozgachev, “The construction of a Lyapunov function for variable structure systems,” Automat. Remote Contr., vol. 33, no. 8, pp. 1391-1393,1972.

[39] A. J. Calise and K. V. Raman, ”A servo compensator design approach for variable structure systems,” presented at the 19th Allerton Conf., University of Illinois, Sept. 30-Oct. 2,1981.

[40] Y. F. ltkis and A. V. Leibovitch, “Application of systems variable structure for centralized control of objects with variable parameters,” Automat. Remote Contr., no. 8, pp. 67- 70, 1970.

[41] 0. M. E. El-Ghezawi, A. S. I. Zinober, D. H. Owens, and S. A. Billings, “Computation of the zeros and zero directions of linear multivariable systems,”lnt.I. Contr., p p 833-843,1972.

[42] M. A. Aizerman and E. S. Pyatniskii, ”Foundations of a theory of discontinuous systems 1,”Automat. RemoteContr., vol. 35, no. 7, pp. 1066-1080,1974.

[43] -, “Foundations of a theory of discontinuous systems 11,” Automat. Remote Contr., vol. 35, no. 8, pp. 1242-1263, 1974.

[44] C. Bartolini and T. Zolezzi, “Variable structure nonlinear in the control law,” I€€€ Trans. Automat. Contr., vol. AC-30, pp.

[45] V. I . Utkin, “Equations of the sliding regime in discontinuous systems 11,” Automat. Remote Contr., vol. 33, pp. 211-218, 1972.

[46] A. F. Filippov, “Differential equations with discontinuous right hand sides,” Am. Math Soc. Transl., vol. 42, pp. 199-231, 1964.

[47] M. J. Corless and G. Leitmann, “Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems,” I € € € Trans. Automat. Contr., vol. AC-26, no. 5, pp. 1139-1144,1981.

[48] C. Ambrosino, G. Celentano, and F. Carofalo, “Robust model tracking control for a class of non-linear plants,” /€E€ Trans. Automat. Contr., vol. AC-30, no. 3, pp. 275-279,1985.

[49] Y.-Y. Hsu and W.-C. Chan, “Optimal variable-structure controller for DC motor speed control,” froc. Inst. €lec. €ng., vol. 131, no. 6, part D, pp. 233-237,1984.

[SO] S. Cutman and Z. Palmor, “Properties of min-max controllers in uncertain dynamical systems,” SIAM 1. Contr. Optimiza- tion, vol. 20, no. 6, pp. 850-861,- 1982.

[SI] B. R. Barmish, M. Corless, and G. Leitmann, “A new class of stabilizing controllers for uncertain dynamical systems,” SAM/. Contr. Optimization, vol. 21, no. 2, pp. 246-255,1983.

[52] F. H. Clarke, Optimization and Nonsmooth Analysis, Cana- dian Math. Soc. Ser. in Math. New York, NY: Wiley-lnter- science, 1983.

[53] R. T. Rockafellar, Convex Analysis, Princeton Mathematics Ser., vol. 28. Princeton, NJ: Princeton Press, 1970.

[541 M. W. Spong and H. Sira-Ramirez, ”Robust control design

3, pp. 487-498,1984.

681-684, Jul. 1985.


techniques for a class of nonlinear systems,” in Proc. 7986 American Contr. Conf. (Seattle, WA), pp. 1515-1522, June 1986. M. W. Spong, K. Khorasani, and P. V. Kokotovic, ”An integral manifold approach to the feedback control of flexible joint robots,” / € € E 1. Robotics Automat., vol. RA-3, no. 4, pp. 291- 300,1987. M. W. Spong, “Robust stabilization for a class of nonlinear systems,” presented at the 7th Int. Symp. on the MTNS, Stockholm, June 1985. R. Marino, “High-gain feedback in nonlinear control systems,” Int. J. Contr., vol. 42, no. 6, pp. 1369-1385, 1985. P. V. Kokotovic, “Recent trends in feedback design: An overview,” Automatica, vol. 21, no. 3, pp. 123-132, 1968. A. S. Vostrikov, V. I. Utkin, and C. A. Frantsuzova, “System with state vector derivative in the contro1,”Automat. Remote Contr., vol. 43, no. 3, pp. 283-286, 1982. L. C. Chouinard, J. P. Dauer, and C. Leitmann, ”Properties of matrices used in uncertain linear control systems,” SIAM I. Conk Optimization, vol. 23, no. 3, pp. 381-389, 1985. I. R. Petersen, “Structural stabilization of uncertain systems: Necessity of the matching condition,” SIAM I. Contr. Opti- mization, vol. 23, no. 2, pp. 286-296, 1985. C. Milosavljevic, “General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable structure systems,”Automat. Remote Contr., vol. 46, no. 3, pt. 1, pp. 307-314,1985. H. Sira-Ramirez and T. A. Dwyer, Ill, ”Variable structure controller design for spacecraft nutation damping,” /€€€ Trans. Automat. Contr., vol. AC-32, no. 5, pp. 435-438, May 1987. T. A. W. Dwyer, Ill, and H. Sira-Ramirez, “Variable structure control of spacecraft attitude manuevers,” to be published J. Guidance, Dynamics and Control, 1987. M. K. Sain and J. L. Peczkowski, “Nonlinear control by coor- dinated feedback synthesis with gas turbine applications,” in Proc. American Contr. Conf. (Boston, MA), pp. 1121-1128, June 1985. L. R. Hunt, R. Su, and C. Meyer, “Global transformation of nonlinear systems,’’ /€E€ Trans. Automat. Contr., vol. AC-28, no. 1, pp. 24-31, 1983. - , “Design for multi-input nonlinear systems,” in Differ- ential Geometric Control Theory, R. W. Brockett et al., Eds. Cambridge, MA: Birkhauser Boston, 1983, pp. 268-298. S. M. Madani-Esfahani, R. A. DeCarlo, M. J. Corless, and S. H. Zak, “On deterministic control of uncertain nonlinear systems,” in Proc. 7986American Contr. Conf. (Seattle, WA), pp. 1523-1528, June 1986. B. L. Walcott and S . H. Zak, “State observation of nonlinear uncertain dynamical systems,‘’ I€€€ Trans. Automat. Contr., vol. AC-32, no. 2, pp. 166-170,1987. E. Bailey and A. Arapostathis, “Simple sliding mode control scheme applied to robot manipulators,” lnt. I . Contr., vol. 45, no. 4, pp. 1197-1209,1987. “Challenges to control: A collective view,” Report of the workshop held at the University of Santa Clara on September 18-19,1986, /€€€Trans. Automat, Contr., vol. AC-32, no. 4, pp. 275-285, Apr. 1987. S. H. Zak and C. A. MacCarley, “State-feedback control of non-linear systems,“lnt. J. Contr., vol. 43, no. 5, pp. 1497-1514, 1986. H. Sira-Ramirez, “Harmonic response of variable-structure- controlled Van der Pol oscillators,” /E€€ Trans. Circuits Syst., vol. CAS-34, no. 1, pp. 103-106, Jan. 1987. B. E. Paden and S. S. Sastry, “A calculus for computing Filip- pov’s differential inclusion with application to the variable structure control of robot manipulators,” /€€E Trans. Circuits Syst., vol. CAS-34, no. 1, pp. 73-82, Jan. 1987. A. S. Vostrikov, Control of Dynamical Systems (in Russian). Novosibirsk, Soviet Union: Novosibirsk Institute of Electrical Eng., 1979. - , Optimal and Adaptive Systems (in Russian). Novosi- birsk, Soviet Union: Novosibirsk Institute of Electrical Eng., 1977. B. Draienovit, “The invariance conditions in variable structure systems,” Automatica, vol. 5, no. 3, pp. 287-295, 1969. C. Matthews, R. DeCarlo, P. Hawlev, and S. Lefebvre, “Toward

machine connected to an infinite bus,” /E€€ Trans. Automat. Contr., vol. AC-31, no. 11, Dec. 1986.

[79] C. Matthews and R. DeCarlo, ”Decentralized variable structure control of interconnected multi-input multi-output nonlinear systems,” Circuits, Systems, and Signal Processing, vol. 6, no. 3, 1987.

[80] -, “A variable structure control approach to the nonlinear decentralized tracking problem,” presented at the Proc. of the 24th Annual Allerton Conf. on Communication, Control, and Computers, Monticello, IL, Oct. 1986.

[SI] C. Matthews, C. DeMarco, and R. DeCarlo, “A decentralized VSC design for two interconnected synchronous power machines,” presented at the Proc. of the 1987American Con- trol Conf., Minneapolis, MN, June 1987.

[82] C. Matthews and R. DeCarlo, “Decentralized tracking for a class of interconnected nonlinear systems using variable structure control,” Automatica, Mar. 1988.

[83] H. Sira-Ramirez, “A geometric approach to pulse-width-modulated control design,” presented at the Proc. 26th IEEE Conf. Decision and Control, Los Angeles, CA, Dec. 1987.

Raymond A. DeCarlo (Senior Member, IEEE) was born in Philadelphia in 1950. He received the B.S. and M.S. degrees in electrical engineering from the University of Notre Dame in 1972 and 1974, respectively. In 1976, he received the Ph.D. degree under the direction of Dr. Richard Saeks from Texas Tech University. His doctoral research centered on Nyquist stability theory with applications to multidimensional digital filters.

After graduating, he became a Lecturer at Texas Tech for one year, and then became an Assistant Professor of Electrical Engi- neering at Purdue University in the Fall of 1977, and an Associate Professor in 1982. He has worked at the General Motors Research Laboratories during the summers of 1985 and 1986. His research interests include interconnected dynamical systems, analog and analog-digital fault diagnosis, decentralized control of large-scale systems, decentralized identification and estimation of interconnected systems, and computer-aided circuit design.

Dr. DeCarlo is past Associate Editor for Technical Notes and Cor- respondence and past Associate Editor for Survey and Tutorial papers, both for the IEEETRANSACTIONSON AUTOMATIC CONTROL. Pres- ently, he is the Secretary-Administrator of the Control Systems Society and also a member of the Board of Governors of the Soci- ety.

Stanislaw H. Zak (Member, IEEE) received the Ph.D. degree from the Technical Uni- versity of Warsaw, Poland, in 1977.

From 1977 to 1980, he was a Faculty Mem- ber in the Institute of Control and Indus- trial Electronics, Technical University of Warsaw. From 1980 until 1983, he was a Vis- iting Assistant Professor in the Department of Electrical Engineering, University of Min- nesota. In August 1983, he joined theschool of Electrical Engineering at Purdue Univer-

sity. His primary areas of research aresystemstheory, time-delay systems, nonlinear systems control, and computer algebra.

Gregory P. Matthews (Member, IEEE) was born in Pontiac, MI, on December 3,1956. He received the B.S. degree in electrical engineering from the University of Michi- gan in 1979, and the M.S. and Ph.D. degrees in electrical engineering from Purdue Uni- versity, West Lafayette, IN, in 1982 and 1985, respectively.

Since 1986 he has been a member of the General Motors Research Laboratories Electrical and Electronics Engineering

Department, Warren, MI. His interests include the control of large afeasiblevar’iablestructurecontroldesign for asynchronous scale and nonlinear systems.


variable structure control of nonlinear a tutorial structure control of nonlinear multivariable...

Documents