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Sliding mode control of distributed parameter systems
Pergamon
0005-1098(94) E0037-I
Automatica, Vol.30, No. 12, pp. 1961-1966, 1994 ElsevierScienceLtd Printedin Great Britain 0005-1098/94 $7.00+ 0.00
Brief Paper
Sliding Mode Control of Distributed Parameter Systems*t KAM-CHI LI,; TIN-PUI LEUNG§ and YUE-MING HUll Key
Words--Distributed parameter systems; sliding states; variable structure control; synthesis methods; robustness. Al~straet--The sliding mode control problem for distributed parameter systems (DPS) is studied. Two mathematical description theorems of sliding modes are obtained using a novel analysis approach. The existence conditions of sliding modes are proposed for general DPS, and the sliding mode (SM) controllers are also designed straightforwardly for a class of DPS. An important feature of the theory is that the system is made insensitive to system uncertainties once the sliding manifold is reached.
few results exist for DPS yet, since the assumption on Lipschitzian right sides of state equations, which is quite natural for lumped parameter systems, is no longer valid for DPS. The fundamental SM control problem for DPS was treated first in Orlov and Utkin (1987) and a mathematical description theorem of the sliding mode was proved under some harsh conditions; some existence conditions of the sliding mode were also proposed for a class of DPS via Lyapunov functions (Utkin, 1992; Orlov and Utkin, 1987; Utkin and Orlov, 1990; Breger, 1981; Franke, 1981). However, the existence condition of the sliding mode, which is one crucial problem in the SM control theory, remains unknown for general DPS. On the other hand, a set of examples proved the SM properties obtained for lumped parameter systems to be applicable to DPS as well, but did not satisfy the harsh conditions (e.g. the strong positiveness of operators) in Orlov and Utkin (1987). Therefore, the SM control theory for DPS needs to be developed urgently in order to provide an efficient, robust and adaptive control approach for a wide class of distributed parameter industrial processes. The paper is organized as follows: in Section 2, the mathematical description theorems of the sliding modes for DPS are given by a regularization principle and operator properties. In Section 3, the existence conditions of the sliding mode are proposed for general DPS, and the design methods of SM controllers are also given straightforwardly for a class of DPS with strictly dissipative operators.
1. Introduction THE DPS THEORY has been established to an extent within the last 30 years because of its wide applications in various industrial processes (Curtain and Pritchard, 1978). But a gap can still be observed between theory and application in the field of automatic control. One of the reasons for this gap may be the fact that a great deal of the theory requires exact knowledge of the plant to be controlled. Another one is that finite-dimensional approximate techniques are often used in the control of actual distributed parameter plants. Therefore, in the case of system uncertainties or approximations, the finest control law when applied to the actual plant will yield a poor result and may even produce an unstable system. Hence, it is worthwhile to establish a robust control approach which avoids time-consuming parameter or system identifications and nevertheless provides a control system with the desired dynamics in the presence of plant uncertainties. Research in recent years has shown that the theory of variable structure systems plays an important role in the robust and adaptive control design of industrial processes. This approach is based on the switching functions of the state variables which are used to create a sliding manifold, when this manifold is attained the switching functions keep the trajectory on the manifold, thus yielding desired system dynamics, which become insensitive to system uncertainties or disturbances. Although the sliding mode (SM) control theory for lumped parameter systems has been developed satisfactorily and applied to robotic manipulators during the last 30 years (Hung et al., 1993; Utkin, 1992; Matthews and DeCarlo, 1988; Hu, 1991; Elmali and Olgac, 1992), relatively
2. Mathematical description o f sliding mode 2.1. Problem formulation. Consider the following nonlinear distributed parameter control systems Yc = A x +f(t, x) + Bu,
x(O) = Xo,
(1)
where x(t) is the state variable with values in the reflexive real Banach space V; u(t,x) is the control variable with values in the linear space V,; A is an unbounded linear operator and D(A) = V; f(t, x) is a continuous function from R × V to V; B • L ( V t , V ) (L(V~,V) is a linear space of continuous linear operators which transform V1 to V). Hence, equation (1) may be used to describe a wide class of actual control problems, such as heat processes and flexible manipulators (Orlov and Utkin, 1987; Chen and Yeung, 1991; Pazy, 1983). Let V2~- V~ be a Banach space and C • L(V, V2). We wish to control DPS (1) by using feedback control laws which are discontinuous along the manifolds S(x) = Cx = 0. Obviously, the traditional existence and uniqueness conditions of the solution for the initial problem are no longer valid since the right-hand sides are discontinuous on the manifolds. Therefore, when the control laws are applied to equation (1), a suitable mathematical description of equation (1) on the manifolds needs to be found in order to analyze the dynamic behavior of motion in the sliding mode. A standard technique to consider such a problem is to use the equivalent control method (Utkin, 1992; Orlov and Utkin, 1987). In this section, we will prove that this method is also available for a class of DPS by using a novel analysis approach.
*Received 8 December 1992; revised 9 June 1993; received in final form 27 February 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Utkin under the direction of Editor Huibert Kwakernaak. Corresponding author Dr Yue-Ming Hu. Tel. +862711140; Fax +862 0551 6862. t The project was supported in part by the National Natural Science Foundation of China and the Croucher Foundation of Hong Kong. :~Department of Electronic Engineering, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong. § Department of Mechanical and Marine Engineering, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong. II Department of Automation, South China University of Technology, Guangzhou 510641, P.R.C. 1961
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Brief Papers
2.2. Definitions and lemmas. For the sake of brevity, the norm of each space in this paper will be designated by tt.11. Let V* be the dual space of Banach space V and V*(x) be the value of x * • V* at x • V. Defining the semi-inner product [., .] as follows [x, y] = inf {x*(x):x* • F(y)}
(2)
where F(y) is the duality set for y e V defined by
F(y) = {x*:x* • X and x * ( y ) = Ilxll2= IIY IF'}.
(3)
indicates that Cx = 0 if and only if Px = O, which will play an important role in the design of the sliding mode controls. 2.3. Mathematical description theorems o f sliding mode. To obtain the mathematical description equation of the sliding mode for equation (1), we will use the equivalent control method and the regularization principle (Orlov and Utkin, 1987). Suppose that the sliding motion does occur on the sliding manifold S(x) = 0. Then, using the equivalent control method, we can easily obtain the sliding mode equation as follows .f = n o x + (1 - P)f(t, x), (9)
From the H a h n - B a n a c h theorem it follows that F(y) is nonempty for every y • V. In fact, for any z • V, Ilzll = 1. By the Hahn-Banach theorem, f * • V* exists such that f*(z)=l and I I f * l l = l . Let x * = l l Y t l f * , then x * • V * , IIx*ll = ItYlI, x*(y) = IlYlIf*(Y) = IlYlIZf(Y/IlYlI) = ttYll2, that is, F ( y ) is nonempty. Evidently, for the defined x* • V*, we have lx*(x)l-< IIx*tl llxll = Ilxll IlYlI. From equations (2)-(3) we deduce that the semi-inner product [., .] has the following properties: (1) I[x,y]l -< tlxll IlYlI; (2) [x,x] = Ilxl12; and (3) [ x + a y , y ] = [ x , y ] + a IlYlle. Particularly, if V is a Hilbert space, then it is easy to prove that the semi-inner product defined as above is identical with the one given in V.
where Ao and P are defined by equation (6). However, as imperfect factors (e.g. switching delays, errors in the control laws, approximate description of the plant based on incomplete knowledge) exist for the practical systems, the real sliding motion does not occur on the manifold S ( x ) = 0 exactly but in a small a-vicinity of the manifold. To prove that the real sliding mode can be described approximately by equation (9), we will use the regularization principle. Let, in a a-vicinity of the sliding manifold S ( x ) = 0 , control u in equation (1) be replaced by the control ~ such that the solution of equation (1) corresponding to this control does exist for the time interval [to, q] and belongs to the boundary layer IIS(x)ll -< 8. Then, we have:
Definition 1. A linear operator A is called upper-bounded if there is a constant d such that
Theorem 1. Let the following conditions be satisfied. (i) Operator A generates a strongly continuous semigroup T(t) and has a nonempty resolvent set p(A). (ii) Operators A and P are commutable:
[Ax, x] <- d Ilxll2.
(4)
A linear operator A is called strictly dissipative if there is a positive number a such that
[Ax, x] <- - a Ilxll2.
(5)
Evidently, a strictly dissipative operator is upper-bounded. For a wide class of DPS, the linear operator A in state equation (1) does exist upper-bounded although it is generally unbounded. Well-known strictly dissipative operators are second-order partial derivative operators in a heat process (Orlov and Utkin, 1987) and fourth-order partial derivative operators in a flexible manipulator system (Hu, 1991; Chen and Yeung, 1991). Using the properties of semi-inner product [.,.] and operator A (Klaus, 1977; Pazy, 1983), we have:
Lemma 1. If A is an upper-bounded linear operator, then a strongly continuous semigroup T(t) (t->0) must exist such that (i) A is the infinitesimal generator of T(t); and (ii) liT(011 -0. For linear operators A, B and C in the above, operators P and Ao are defined as follows
P=B(CB)
'C;
A,,=(1-P)A,
(6)
where operator CB is supposed to be continuously invertible.
Lemma 2. Operators P and A0 defined by equation (6) have the following properties: (4) p2 = p; P(I - P) = 0; (1 - p)2 = 1 - P; (5) PB = B; CP = C; PAo=O; and (6) let x = x ( t ) be any solution of equation (1), then S(x) - Cx ~ 0 (as t ~ ~) if and only if Px ~ 0 (as t ~ ~). Proof. Properties (4) and (5) can be obtained immediately via equation (6). Property (6) is obtained by the following inequalities IIPxll = IIB(CB) 'Cxll <-ItB(CB) 111 Ilfxll
(7)
IlCxll = IlfPxll <-ttCII Ilexll
(8)
and
PAx=APx,
(iii) f ( t , x ) is Lipschitzian with respect to x and measurable with respect to t. (iv) Control ff is bounded in any bounded region, and for any -to • D ( A ) , equation (1) exists as a unique bounded solution Y(t) under u = ff and this solution belongs to the boundary layer ilS(Y)II-< 8. Then, for the class of DPS (1) considered, we have lim ]]£(t) - x(t)t[ = 0.
uniformly for all t e [to, q],
g~O
where x(t) is the solution of the ideal sliding mode equation (9) under S(x(to)) = O, x(to) • D(Ao), and Pl£(to) -x(to)It-<
/3a (/3 > 0). The proof of the theorem is given in the Appendix. By Lemma 1 we obtain the following result which coincides with theorem 5.1 given in Utkin and Orlov (1990).
Corollary 1. If A is an upper-bounded operator and conditions (ii)-(iv) of Theorem 1 are satisfied, then the conclusion of Theorem 1 is true. Condition (i) in Theorem 1 is obviously essential for infinite-dimensional systems. Therefore, not only the previously known conditions (1) and (2) given in Orlov and Utkin (1987) are weakened considerably, but also the analysis approach is improved. Furthermore, the conclusion of the above theorem can be easily proved to be true on the infinite time interval [to, ,e) for the class of DPS with stable sliding modes. Theorem 2. Under conditions (ii)-(iv) of Theorem l, the conclusion of Theorem 1 is also true on the time interval [to, oc) if: (v) operator A has a nonempty resolvent set and generates a strongly continuous semigroup T(t) such that IIT(t)ll<-Mexp{-at},
t>-O;and
(l(I)
(vi) a positive measurable function g(t) exists such that (a)
Ilf(t,x)-f(t,y)ll<-g(t)llx-yll;
(11)
(b)
dz <- L(t - 10),
(12)
(c) Property (4) in l_emma 2 shows that operator P defined by equation (6) is a projective operator in V, while property (3)
x ED(A).
MLIll-PII
The proof of the theorem is given in the Appendix.
(13)
Brief Papers Conditions (v) and (vi) in Theorem 2 are, in fact, sufficient conditions to guarantee the asymptotic stability of the sliding mode, which will be proved in the next section. Particularly, if A is a strictly dissipative operator, f ( t , x ) is globally Lipschitzian with respect to x with Lipschitz constant L such that L I l I - P l l < a , then the conclusion of Theorem 2 is obviously true. It should be noted that, by the above theorems, we can uniformly approximate any ideal sliding state by real states realizing only approximately the sliding conditions, regardless of the disturbances causing the sliding error. That is, all types of regularizations for system (1) lead to the same sliding mode equation (9) which may be obtained with the use of the equivalent control method. Therefore, equation (9) is the well-posed mathematical description of the sliding mode for DPS (1), 3. Design o f sliding mode controllers The design of the sliding mode control consists of two steps: firstly, a sliding mode is designed to have the prescribed dynamic properties by a proper choice of operator C; secondly, the discontinuous control law u(t, x) is designed to guarantee the existence of the sliding mode on manifold S(x) = Cx =0. A n invariance control system is then designed, as indicated by Orlov and Utkin (1987) and Hu (1991), if system uncertainties or parameter variations belong to the span of input operator B. 3.1. Existence conditions o f sliding mode. Although some existence conditions of the sliding mode have been proposed for a set of concrete DPS (Utkin, 1992; Orlov and Utkin, 1987; Hu, 1991), general existence conditions of sliding mode for DPS have not yet been obtained. In this section, we will give some existence conditions for the sliding mode in order to design the discontinuous control law u(t,x) straightforwardly. Definition 2. Let y(t) be any continuous function in V or I/2 defining the left derivative of IIy(t)ll as follows
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proper sliding manifold S(x) = Cx = 0 so that the motion in the sliding mode has prescribed properties. By Theorem 1, the sliding mode equation is given by equation (9). Theorem 4. The sliding motion is asymptotically stable if the conditions in Theorem 2 are satisfied. Proof Since A and P are commutable by the assumption, from equation (9), we deduce = A ( I - P)x + (1 - P)f(t, x).
x = (1 - P)x + Px = (! - P)x.
Ily(t)ll = lim sup Ily(t)ll - I l y ( t - h)ll r~o h E(0.r] h
( I - p ) e = I - P.
(21)
(1 - P ) ~ = A ( I - P ) x + (1 - P ) f ( t , (1 - P ) x )
(22)
It follows that
which is equivalent to the equation (I = A q + (1 - P)f(t, q),
Lemma 3. If y(t) is differentiable, then (i) D-[[y(t)[[ exists; (ii) Ily(t)ll D - I[y(t)ll --- [y(t), y(t)]; (15) (iii) Ily(t)ll is strictly decreasing if D Ily(t)ll < 0 ; and (iv) IIy(t)l[ is strictly decreasing if [~(t), y(t)] < 0.
The proof of the lemma is given in the Appendix. Let [., .] be a semi-inner product in V2 defined in the same way as the one defined in V. Then, we have: Theorem 3. Let x = x(t) be any solution of equation (1) if
(16)
or
[P£, ex] <- -/3 Ilexll,
then the solution x = x ( t ) S(x) = Cx = 0 in finite time.
must
reach
(17) the
manifold
Proof In fact, equation (16) [or (17)] together with equation (15) implies that
D - IIS(x)ll <- -/3,
(or D
Ilexll ~ -/3).
(23)
f0
r ( t - r)(1 - P)f(r, q ( r ) ) dr,
q(t) = T ( t - to)q(to) +
(24)
since A generates a strongly continuous semigroup T(t) and f(t, q) satisfies condition (iii) in the Theorem 1. Using the given conditions (v) and (vi), we obtain IIq(t)ll --- Me - ' ( ' - ' ° ) IIq(to)ll + M I l l - ell
f,
e -~(' ~)g(r) IIg(r)ll dt.
(25)
~t 0
Using equations (20) and (25) and Gronwall's inequality, we obtain
By equations (2) and (3), we have the following important conclusions.
(/3 > 0)
q(to) = (1 - P)x(to),
where q = ( 1 - P)x. Therefore, by a similar proof to the global existence theorem given in Alain (1981), we know that the solution of equation (23) does exist and satisfies the integral equation
(14)
if the limit exists.
IS(x), S(x)]-< -/3 IIS(x)ll,
(20)
On the other hand, by Lemma 2, we have
× D
(19)
Evidently, Px = 0 if S ( x ) = Cx =0. Therefore, for the sliding motion x = x(t), we have
(18)
The conclusion is, therefore, true by equation (18) and Lemma 3. Theorem 3 shows that either equation (16) or (17) can guarantee the existence of a sliding mode on S ( x ) = 0 for DPS (1). It will be seen that condition (4) is more convenient than condition (3) for the design of SM controls. Evidently, we can obtain similar existence conditions by using the right derivative of IIS(x)ll. 3.2. Design o f sliding manifold. The first step is to design a
IIq(t)ll -< M e -'~t'-'°) IIq(t0)ll + M II1 - ell x
f,
e - ° t ' - ~ ) g ( r ) IIg(r)ll dt.
(25)
~t 0
Using equations (20) and (25) and Gronwall's inequality, we obtain IIx(t)ll = IIq(t)ll <- M IIq(to)ll exp
{f0
[ - ~ + M I1! - PII g(r)] d r
<- M II1 - PII IIX(to)tl exp { - ( ~ - M L II1 - PII)(t - to)}.
(26) The conclusion follows by equations (13) and (26). By Theorem 4, the sliding motion will tend to the equilibrium point x = 0 as t--* ~ if we select an operator C e L(V, V2) such that equation (11) is satisfied. Note that for the sliding mode, the order of system (9) is naturally reduced since S(x) = Cx = 0. Therefore, we can use the pole assignment technique or the optimal control method under the quadratic cost functional to design a proper operator C so that the sliding mode has prescribed properties. This problem has been treated in Hu (1991). 3.3. Design o f sliding mode controllers. The second step is to design a discontinuous control law so that the sliding motion occurs on the manifold S(x) = 0. Theorem 5. If .4 satisfies the conditions in Corollary 1, then the following SM control law u = - a t ( C B ) - t C x - (CB)-~Cf(t, x) - az(CB)-1Cx/IIPx II,
(27) guarantees the existence of a sliding mode on manifold S(x) = Cx = 0 for DPS, where al -> a and a2 > 0.
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Brief Papers
Proof In fact, by equations (1), (4) and (27), we have [P.f, Px] = [PAx + Pf(t, x) + PBu. Px] = [APx - a , P x -a2Px/llPxll, Px] -<(a - a , ) I I P x l l 2 - a 2 IIPxll-< - a 2 IIPxll. The result follows from Theorem 3. For the actual system, parameter variations and disturbances exist on the system. In order to ensure that the system possesses a robustness property, we only need to select a~ and a2 in equation (27) to be suitably large. In fact, let h(t, x) be the function of system uncertainties or parameter variations which are not accessible for measurement. Then, using the equivalent control method, we know that the sliding mode does not depend on h(t, x) if it belongs to the span of operator B, and the sliding motion can also occur on manifold S(x) = 0 in such a case.
Theorem 6. If the bounds of system uncertainties or parameter variations are known: IIh(t, x)ll -< H(t, x), then the SM control law (27) guarantees the existence of a sliding mode on manifold S(x) = Cx = 0 for the disturbed system .i = Ax + f(t, x) + Bu + h(t, x),
(28)
if A, f(t,x) satisfies the conditions in Corollary 1, and constants a~ and a2 in equation (27) are chosen such that al>a;
a2>llPllH(t,x)+ao;
a0=const>0.
(29)
Proof In fact, using the above assumptions, we have [PSc,Px] = [PAx + Pf(t, x) + PBu + Ph(t, x), Px] <-(a-al)
liP.rile + IIPll Ilh(t,x)ll IIPxll-a2 IIPxll
<- - a 0 Ilexll.
(30)
The conclusion follows from equation (30) and Theorem 3. Obviously, by using existence condition (17), we can use the upper boundedness of operator A directly and thus design the sliding mode controller straightforwardly. Hence, the controller in the form of equation (27) has bounded gain in any bounded region, which is important for the actual implementation. If we use the existence condition (16), which is naturally a generalization of the one provided for lumped parameter systems (Utkin, 1992), then, formally, we have
[S(x),S(x)]=[CAx+Cf(t,x)+CBu, Cx]
(31)
the term [CAx, Cx] will be difficult to estimate since C and A are not commutable generally. The example given in Orlov and Utkin (1987) showed that the condition (16) was inapplicable to the general multi-dimensional heat process. However, the problem proposed in Orlov and Utkin (1987) can easily be solved by using condition (17). In fact, let A = aoe/oy 2, D(A) = {Q E H l (0, 1); Q'(0) - Q'(1) = 0}, then operator A satisfies
[AQ, Q] <- --An,in IIQII 2, where h.mi, = min {h.,: 1 -< i -< n}, II. II is the norm of Sobolev space Hi(0, 1), operator C is chosen as a m × n matrix. Therefore, the conditions of Theorems 1 and 5 are satisfied, and hence the sliding mode controller can be designed directly using the proposed method. For details please refer to Hu and Zhou (1991). Morever, it is not difficult to prove that the real sliding mode obtained from the above-designed approach does belong to the boundary layer IIS(x)II <- 6 after some T0.
4. Conclusions Mathematical descriptions and the existence conditions of sliding modes are crucial problems in the design of sliding mode control systems. However, for general DPS, there is, as yet, no efficient approach to solve them since unbounded operators are often involved in such systems. Based on the operator properties presented in this paper, some essential problems in the SM control theory of DPS are solved perfectly. The major contribution of the present work is fourfold. Firstly, the equivalent control method is proved to
be applicable to a wide class of DPS under general assumptions; secondly, sufficient conditions which guarantee the asymptotic stability of sliding mode are obtained and the equivalent control method is proved to be available in the finite time interval for the class of DPS which satisfy the proposed stability conditions of the sliding mode: thirdly, general existence conditions of the sliding mode are also presented: and fourthly, SM controllers are designed straightforwardly for the class of DPS with upper-bounded operators. It should be noted that the sliding mode control strategy, in this paper, is discussed only in state space. Obviously, discontinuous control (27) requires that the state should be directly accessible for measurement, which is far too strong for DPS. However, this problem can be solved in practical implementations by two standard methods. One of them is to discuss the SM control problem in output space using the same idea as above, if the output system is available for control. Another one is to design the control laws using various approximation techniques. For results related to this problem see Hu (1991) and Chen and Yeung (1991).
References Alain, H. (1981). Nonlinear Evolution Equations--Global Behavior of Solutions. Springer, New York. Breger, A. M., et al. (1980). Sliding modes in control of distributed plants subjected to a mobile multicycle signal. Aut. Remote Control, 41, 72-83. Chen, Y. P. and K. S. Yeung (1991). Sliding mode control of multi-link manipulators. Int. J. Control, 54, 257-278. Curtain, R. F. and A. J. Pritchard (1978). Infinite Dimensional Linear Systems Theory. Springer, New York. Elmali, H. and N. Olgac (1992). Robust output tracking control of nonliner MIMO systems via sliding mode technique. Automatica, 28, 145-151. Franke, D. (1981). Control of distributed parameter systems with independent linear and bilinear modes. In Proc. 3rd IMA Conf. on Control Theory, 827-841. Academic Press, London. Hale, J. K. (1980). Ordinary Differential Equations. Krieger, Huntington. Hu, Y. M. (1991). Variable Structure Control of Distributed Parameter Systems: Theory and Applications, Ph.D thesis, South China University of Technology, Guangzhou, P.R.C. and also published by the National Defence Industry Press, Beijing, P.R.C., 1994 (in Chinese). Hu, Y. M. and Q. J. Zhou (1991). Variable structure control of distributed parameter systems. Control Theory and Applications, 8, 38-43. Hung, J. Y., W. Gao and J. C. Hung (t993). Variable structure control: a survey. IEEE Trans. Industr. Electron. 40, 2-22. Ktaus, D. (1977). Ordinary Differential Equations in Banach Space. Springer, New York. Matthews, G. P. and R. A. DeCarlo (1988). Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica, 24, 187-193. Orlov, Yu. V. and V. I. Utkin (1987). Sliding mode control in indefinite-dimensional systems. Automatica, 23, 753-757. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York. Tzafestas, S. G. (1982). Distributed Parameter Control Systems. Pergamon, Oxford. Utkin, V. I. (1992). Sliding Modes in Control and Optimizations. Springer, New York. Utkin, V. I. and Yu, V. Orlov (1990). Sliding Mode Control in Infinite-dimensional Systems. Nauka, Moscow.
Appendix Proof of Theorem 1. In the boundary layer IlS(x)ll <-& for any real sliding mode x(t), we have S(2) = C A 2 + Cf(t, £) + CB5
(A.I)
Brief which implies
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Consequently, since
if= -(CB)-IC[AZ+f(t,i)]+(CB)-1CZ
(A.2)
The real sliding mode equation is therefore obtained by substituting equation (A.2) into equation (1) as follows (A.3)
= (1 - P ) A E + (1 - P ) f ( t , £) + P~
E(t) - x(t) = P(£(t) - x(t)) + e(t) = P£(t) + e(t),
and IlP£(t)ll <-IIB(CB)-lll IIC.fll <-IIB(CB)-lll 8 it follows that
which is equivalent to
liE(t) -x(t)I1 - (/33 + IIB(CB) -111)8,
~. = A ~ + (I - P ) f ( t , E),
(A.4)
~(to) = (I - P)E(to),
since A and P are commutable, here ~ = (I-P)JT. h(t) = (1 - P ) f ( t , E), then
Let
and the conclusion is obtained. P r o o f o f Theorem 2. In fact, by equations (A.6) and (A.9)
and the given conditions, we have Ile(t)ll -< M/3 Ill - PII 8 exp { - o t ( t - to)} (A.5)
where L is the local Lipschitz constant. Equation (A.5) implies h e L([to, tl], V) since E is assumed to be bounded and f ( t , 0) is measurable. Therefore, the solution of equation (A.4) can be represented in the integral form
f(t)
= T(t -
to)z(to) +
f,i
T(t - t)(1 - P)f(t, ~ ( t ) ) d t
(A.6)
+
L
M Ill-PII g(r)liE(t) -x(t)ll
× exp { - t r ( t - z)} dr.
(A.14)
Since liE(t)- x(t)ll = II~(t) - z ( t ) l l <-IIe2?(t)ll + Ile(t)H, follows that
it
liE(t) - x(t)11 -< IIB(CB)-III 8 + lie(t)II <(M/3 II1-PII + IIB(CB)-IlI)8
by the corollary 4.2.2 given in Pazy (1983), since the solution of equation (A.4) exists by assumption (4) and h ~ L([to, tll, V). Where r ( t ) is the strongly continuous semigroup generated by operator .4 such that IIT(t)II -< M exp {jot}.
(A.7)
On the other hand, for the ideal sliding mode x(t), we have P x = 0. From equation (9) it follows that (A.8)
(1 - P):~ = A ( I - P ) x + (1 - P ) f ( t , (1 - P)x),
+
f,i
M II1-PII g ( t ) l i E ( r ) - x ( r ) l l
x exp { - a ( t - z)} dr.
(A.15)
Using Gronwall's inequality, we obtain liE(t) - x(t)II <- dl 8 + d2B[1 - exp { - ( a - M L Ill - Pll)(t - to)}], where
(A.16)
d~ = IIB(CB) -~ II + Mfl I l l - ell;
since A and P are commutable and x = ( l - P ) x . Let z = (1 - P)x, then equation (A.8) is equivalent to the form
d2 = c l M L III - P l l / ( a - M L I11 - Pll).
(A.9)
The results now follow directly from equations (13) and (A.16).
By a similar proof to the global existence theorem given in Alain, (1981), we know that the solution of equation (A.9) does exist and satisfies the integral equation
P r o o f o f L e m m a 3. (1) For any h > 0 and 0 < 0 -< 1, we have
= A z + (1 - P ) f ( t , z),
Z(t) = T ( t - to)Z(to) +
Z(to) = (1 - P)x(to).
f,0
IlY - OhY~ll = I1(1 - O)y + Oy - Oh fll -< 0 IlY - h)ll + (1 - O) IlYlI.
T ( t - "¢)(1 - P ) f ( t , z(z)) dz.
This implies (A.10)
Let e ( t ) = ~ . ( t ) - z(t). Then by equations (A.6), (A.7) and
IIY II - l[Y - Ohp II ~ I1YII - IIY - h))II hO
(A.10), we have
Therefore Ill Y II - IIY - hy II]/h is increasing with respect to h and upper-bounded, since
lie(011 -< M exp {p(t - to)} Ile(to)ll +Mlll-PII
h
L
IIf(z,£(r))-f(z,z(r))ll
Ilyll - Ily - h3~ll < [[y[[- (llyll - IIh~ll) h h
II.~ll.
x exp {p(t - z)} d r This implies that
<- M/3 Ill - P]I 6 exp {p(t - to)} + ML Ill-PII
lim sup IIy II - IIy - h3~ll r ~ O O--
[lle(r)l[ + IIB(CB)-'II 8]
× exp {p(t - z)} dr.
(A.11)
Using Gronwall's inequality, we obtain
exists. On the other hand, we have I(lly(t)ll- [ly(t - h )[I) - (lly(t)[I - Ily(t) - hp(t)lDI
lie(t) II < 131B exp {(p + ~2)(t - to)},
(A.12)
= I Ily(t) - h.9(t)ll - Ily(t - h)lll - IIy(t) - y ( t - h) - h.O(t)l I
where
=o(h).
131 = M/3 Ill - PII + M L Ill - PII IIB(CB)-lll x sup
f,
exp{-p(z-to)}dr;
~2=MLII1-PII.
to~t~tl ~10
Therefore,/33 > 0 exists such that Ile(t)ll -
Therefore, D - Ily(t)ll does exist. (2) For any h > 0 , using the properties (1)-(3) of the semi-inner product [., .], we obtain [y(t) - y ( t - h), y(t)] = [y(t), y(t)] - [y(t - h), y(t)]
t ~ [to, td.
(A.13)
> I[y(t)l] 2 - Ily(t)ll [ly(t - h)l[
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Brief Papers which is concluded by the following inequality:
which implies
- h )"v(t) l >-[[y(t)H Ily(t)ll - Ily(t- h)H
[y(t)-Y(th
[y(t)- y(t-h), y(t)] _[5,(t), y(t)] = [y(t)-y(t-h)-~(t)'y(t)]h
Let h ~ 0 It follows that
<- Y(t)-h--(t-h) 9(t)
(ly(t)ll
[)(t), y(t)] -> Ily(t)ll D - Ity(t)ll since
ft~/y't'-y(t-h) y(t)l]=[.9(O,y(t)],
lim h--o [
h
'
J
and the assumption that ~(t) exists. The second conclusion is therefore true, Conclusions (3) and (4) are obviously true by the properties of left derivative (Hale, 1980) and the second conclusion.
0005-1098(94) E0037-I
Automatica, Vol.30, No. 12, pp. 1961-1966, 1994 ElsevierScienceLtd Printedin Great Britain 0005-1098/94 $7.00+ 0.00
Brief Paper
Sliding Mode Control of Distributed Parameter Systems*t KAM-CHI LI,; TIN-PUI LEUNG§ and YUE-MING HUll Key
Words--Distributed parameter systems; sliding states; variable structure control; synthesis methods; robustness. Al~straet--The sliding mode control problem for distributed parameter systems (DPS) is studied. Two mathematical description theorems of sliding modes are obtained using a novel analysis approach. The existence conditions of sliding modes are proposed for general DPS, and the sliding mode (SM) controllers are also designed straightforwardly for a class of DPS. An important feature of the theory is that the system is made insensitive to system uncertainties once the sliding manifold is reached.
few results exist for DPS yet, since the assumption on Lipschitzian right sides of state equations, which is quite natural for lumped parameter systems, is no longer valid for DPS. The fundamental SM control problem for DPS was treated first in Orlov and Utkin (1987) and a mathematical description theorem of the sliding mode was proved under some harsh conditions; some existence conditions of the sliding mode were also proposed for a class of DPS via Lyapunov functions (Utkin, 1992; Orlov and Utkin, 1987; Utkin and Orlov, 1990; Breger, 1981; Franke, 1981). However, the existence condition of the sliding mode, which is one crucial problem in the SM control theory, remains unknown for general DPS. On the other hand, a set of examples proved the SM properties obtained for lumped parameter systems to be applicable to DPS as well, but did not satisfy the harsh conditions (e.g. the strong positiveness of operators) in Orlov and Utkin (1987). Therefore, the SM control theory for DPS needs to be developed urgently in order to provide an efficient, robust and adaptive control approach for a wide class of distributed parameter industrial processes. The paper is organized as follows: in Section 2, the mathematical description theorems of the sliding modes for DPS are given by a regularization principle and operator properties. In Section 3, the existence conditions of the sliding mode are proposed for general DPS, and the design methods of SM controllers are also given straightforwardly for a class of DPS with strictly dissipative operators.
1. Introduction THE DPS THEORY has been established to an extent within the last 30 years because of its wide applications in various industrial processes (Curtain and Pritchard, 1978). But a gap can still be observed between theory and application in the field of automatic control. One of the reasons for this gap may be the fact that a great deal of the theory requires exact knowledge of the plant to be controlled. Another one is that finite-dimensional approximate techniques are often used in the control of actual distributed parameter plants. Therefore, in the case of system uncertainties or approximations, the finest control law when applied to the actual plant will yield a poor result and may even produce an unstable system. Hence, it is worthwhile to establish a robust control approach which avoids time-consuming parameter or system identifications and nevertheless provides a control system with the desired dynamics in the presence of plant uncertainties. Research in recent years has shown that the theory of variable structure systems plays an important role in the robust and adaptive control design of industrial processes. This approach is based on the switching functions of the state variables which are used to create a sliding manifold, when this manifold is attained the switching functions keep the trajectory on the manifold, thus yielding desired system dynamics, which become insensitive to system uncertainties or disturbances. Although the sliding mode (SM) control theory for lumped parameter systems has been developed satisfactorily and applied to robotic manipulators during the last 30 years (Hung et al., 1993; Utkin, 1992; Matthews and DeCarlo, 1988; Hu, 1991; Elmali and Olgac, 1992), relatively
2. Mathematical description o f sliding mode 2.1. Problem formulation. Consider the following nonlinear distributed parameter control systems Yc = A x +f(t, x) + Bu,
x(O) = Xo,
(1)
where x(t) is the state variable with values in the reflexive real Banach space V; u(t,x) is the control variable with values in the linear space V,; A is an unbounded linear operator and D(A) = V; f(t, x) is a continuous function from R × V to V; B • L ( V t , V ) (L(V~,V) is a linear space of continuous linear operators which transform V1 to V). Hence, equation (1) may be used to describe a wide class of actual control problems, such as heat processes and flexible manipulators (Orlov and Utkin, 1987; Chen and Yeung, 1991; Pazy, 1983). Let V2~- V~ be a Banach space and C • L(V, V2). We wish to control DPS (1) by using feedback control laws which are discontinuous along the manifolds S(x) = Cx = 0. Obviously, the traditional existence and uniqueness conditions of the solution for the initial problem are no longer valid since the right-hand sides are discontinuous on the manifolds. Therefore, when the control laws are applied to equation (1), a suitable mathematical description of equation (1) on the manifolds needs to be found in order to analyze the dynamic behavior of motion in the sliding mode. A standard technique to consider such a problem is to use the equivalent control method (Utkin, 1992; Orlov and Utkin, 1987). In this section, we will prove that this method is also available for a class of DPS by using a novel analysis approach.
*Received 8 December 1992; revised 9 June 1993; received in final form 27 February 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Utkin under the direction of Editor Huibert Kwakernaak. Corresponding author Dr Yue-Ming Hu. Tel. +862711140; Fax +862 0551 6862. t The project was supported in part by the National Natural Science Foundation of China and the Croucher Foundation of Hong Kong. :~Department of Electronic Engineering, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong. § Department of Mechanical and Marine Engineering, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong. II Department of Automation, South China University of Technology, Guangzhou 510641, P.R.C. 1961
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Brief Papers
2.2. Definitions and lemmas. For the sake of brevity, the norm of each space in this paper will be designated by tt.11. Let V* be the dual space of Banach space V and V*(x) be the value of x * • V* at x • V. Defining the semi-inner product [., .] as follows [x, y] = inf {x*(x):x* • F(y)}
(2)
where F(y) is the duality set for y e V defined by
F(y) = {x*:x* • X and x * ( y ) = Ilxll2= IIY IF'}.
(3)
indicates that Cx = 0 if and only if Px = O, which will play an important role in the design of the sliding mode controls. 2.3. Mathematical description theorems o f sliding mode. To obtain the mathematical description equation of the sliding mode for equation (1), we will use the equivalent control method and the regularization principle (Orlov and Utkin, 1987). Suppose that the sliding motion does occur on the sliding manifold S(x) = 0. Then, using the equivalent control method, we can easily obtain the sliding mode equation as follows .f = n o x + (1 - P)f(t, x), (9)
From the H a h n - B a n a c h theorem it follows that F(y) is nonempty for every y • V. In fact, for any z • V, Ilzll = 1. By the Hahn-Banach theorem, f * • V* exists such that f*(z)=l and I I f * l l = l . Let x * = l l Y t l f * , then x * • V * , IIx*ll = ItYlI, x*(y) = IlYlIf*(Y) = IlYlIZf(Y/IlYlI) = ttYll2, that is, F ( y ) is nonempty. Evidently, for the defined x* • V*, we have lx*(x)l-< IIx*tl llxll = Ilxll IlYlI. From equations (2)-(3) we deduce that the semi-inner product [., .] has the following properties: (1) I[x,y]l -< tlxll IlYlI; (2) [x,x] = Ilxl12; and (3) [ x + a y , y ] = [ x , y ] + a IlYlle. Particularly, if V is a Hilbert space, then it is easy to prove that the semi-inner product defined as above is identical with the one given in V.
where Ao and P are defined by equation (6). However, as imperfect factors (e.g. switching delays, errors in the control laws, approximate description of the plant based on incomplete knowledge) exist for the practical systems, the real sliding motion does not occur on the manifold S ( x ) = 0 exactly but in a small a-vicinity of the manifold. To prove that the real sliding mode can be described approximately by equation (9), we will use the regularization principle. Let, in a a-vicinity of the sliding manifold S ( x ) = 0 , control u in equation (1) be replaced by the control ~ such that the solution of equation (1) corresponding to this control does exist for the time interval [to, q] and belongs to the boundary layer IIS(x)ll -< 8. Then, we have:
Definition 1. A linear operator A is called upper-bounded if there is a constant d such that
Theorem 1. Let the following conditions be satisfied. (i) Operator A generates a strongly continuous semigroup T(t) and has a nonempty resolvent set p(A). (ii) Operators A and P are commutable:
[Ax, x] <- d Ilxll2.
(4)
A linear operator A is called strictly dissipative if there is a positive number a such that
[Ax, x] <- - a Ilxll2.
(5)
Evidently, a strictly dissipative operator is upper-bounded. For a wide class of DPS, the linear operator A in state equation (1) does exist upper-bounded although it is generally unbounded. Well-known strictly dissipative operators are second-order partial derivative operators in a heat process (Orlov and Utkin, 1987) and fourth-order partial derivative operators in a flexible manipulator system (Hu, 1991; Chen and Yeung, 1991). Using the properties of semi-inner product [.,.] and operator A (Klaus, 1977; Pazy, 1983), we have:
Lemma 1. If A is an upper-bounded linear operator, then a strongly continuous semigroup T(t) (t->0) must exist such that (i) A is the infinitesimal generator of T(t); and (ii) liT(011 -
P=B(CB)
'C;
A,,=(1-P)A,
(6)
where operator CB is supposed to be continuously invertible.
Lemma 2. Operators P and A0 defined by equation (6) have the following properties: (4) p2 = p; P(I - P) = 0; (1 - p)2 = 1 - P; (5) PB = B; CP = C; PAo=O; and (6) let x = x ( t ) be any solution of equation (1), then S(x) - Cx ~ 0 (as t ~ ~) if and only if Px ~ 0 (as t ~ ~). Proof. Properties (4) and (5) can be obtained immediately via equation (6). Property (6) is obtained by the following inequalities IIPxll = IIB(CB) 'Cxll <-ItB(CB) 111 Ilfxll
(7)
IlCxll = IlfPxll <-ttCII Ilexll
(8)
and
PAx=APx,
(iii) f ( t , x ) is Lipschitzian with respect to x and measurable with respect to t. (iv) Control ff is bounded in any bounded region, and for any -to • D ( A ) , equation (1) exists as a unique bounded solution Y(t) under u = ff and this solution belongs to the boundary layer ilS(Y)II-< 8. Then, for the class of DPS (1) considered, we have lim ]]£(t) - x(t)t[ = 0.
uniformly for all t e [to, q],
g~O
where x(t) is the solution of the ideal sliding mode equation (9) under S(x(to)) = O, x(to) • D(Ao), and Pl£(to) -x(to)It-<
/3a (/3 > 0). The proof of the theorem is given in the Appendix. By Lemma 1 we obtain the following result which coincides with theorem 5.1 given in Utkin and Orlov (1990).
Corollary 1. If A is an upper-bounded operator and conditions (ii)-(iv) of Theorem 1 are satisfied, then the conclusion of Theorem 1 is true. Condition (i) in Theorem 1 is obviously essential for infinite-dimensional systems. Therefore, not only the previously known conditions (1) and (2) given in Orlov and Utkin (1987) are weakened considerably, but also the analysis approach is improved. Furthermore, the conclusion of the above theorem can be easily proved to be true on the infinite time interval [to, ,e) for the class of DPS with stable sliding modes. Theorem 2. Under conditions (ii)-(iv) of Theorem l, the conclusion of Theorem 1 is also true on the time interval [to, oc) if: (v) operator A has a nonempty resolvent set and generates a strongly continuous semigroup T(t) such that IIT(t)ll<-Mexp{-at},
t>-O;and
(l(I)
(vi) a positive measurable function g(t) exists such that (a)
Ilf(t,x)-f(t,y)ll<-g(t)llx-yll;
(11)
(b)
dz <- L(t - 10),
(12)
(c) Property (4) in l_emma 2 shows that operator P defined by equation (6) is a projective operator in V, while property (3)
x ED(A).
MLIll-PII
The proof of the theorem is given in the Appendix.
(13)
Brief Papers Conditions (v) and (vi) in Theorem 2 are, in fact, sufficient conditions to guarantee the asymptotic stability of the sliding mode, which will be proved in the next section. Particularly, if A is a strictly dissipative operator, f ( t , x ) is globally Lipschitzian with respect to x with Lipschitz constant L such that L I l I - P l l < a , then the conclusion of Theorem 2 is obviously true. It should be noted that, by the above theorems, we can uniformly approximate any ideal sliding state by real states realizing only approximately the sliding conditions, regardless of the disturbances causing the sliding error. That is, all types of regularizations for system (1) lead to the same sliding mode equation (9) which may be obtained with the use of the equivalent control method. Therefore, equation (9) is the well-posed mathematical description of the sliding mode for DPS (1), 3. Design o f sliding mode controllers The design of the sliding mode control consists of two steps: firstly, a sliding mode is designed to have the prescribed dynamic properties by a proper choice of operator C; secondly, the discontinuous control law u(t, x) is designed to guarantee the existence of the sliding mode on manifold S(x) = Cx =0. A n invariance control system is then designed, as indicated by Orlov and Utkin (1987) and Hu (1991), if system uncertainties or parameter variations belong to the span of input operator B. 3.1. Existence conditions o f sliding mode. Although some existence conditions of the sliding mode have been proposed for a set of concrete DPS (Utkin, 1992; Orlov and Utkin, 1987; Hu, 1991), general existence conditions of sliding mode for DPS have not yet been obtained. In this section, we will give some existence conditions for the sliding mode in order to design the discontinuous control law u(t,x) straightforwardly. Definition 2. Let y(t) be any continuous function in V or I/2 defining the left derivative of IIy(t)ll as follows
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proper sliding manifold S(x) = Cx = 0 so that the motion in the sliding mode has prescribed properties. By Theorem 1, the sliding mode equation is given by equation (9). Theorem 4. The sliding motion is asymptotically stable if the conditions in Theorem 2 are satisfied. Proof Since A and P are commutable by the assumption, from equation (9), we deduce = A ( I - P)x + (1 - P)f(t, x).
x = (1 - P)x + Px = (! - P)x.
Ily(t)ll = lim sup Ily(t)ll - I l y ( t - h)ll r~o h E(0.r] h
( I - p ) e = I - P.
(21)
(1 - P ) ~ = A ( I - P ) x + (1 - P ) f ( t , (1 - P ) x )
(22)
It follows that
which is equivalent to the equation (I = A q + (1 - P)f(t, q),
Lemma 3. If y(t) is differentiable, then (i) D-[[y(t)[[ exists; (ii) Ily(t)ll D - I[y(t)ll --- [y(t), y(t)]; (15) (iii) Ily(t)ll is strictly decreasing if D Ily(t)ll < 0 ; and (iv) IIy(t)l[ is strictly decreasing if [~(t), y(t)] < 0.
The proof of the lemma is given in the Appendix. Let [., .] be a semi-inner product in V2 defined in the same way as the one defined in V. Then, we have: Theorem 3. Let x = x(t) be any solution of equation (1) if
(16)
or
[P£, ex] <- -/3 Ilexll,
then the solution x = x ( t ) S(x) = Cx = 0 in finite time.
must
reach
(17) the
manifold
Proof In fact, equation (16) [or (17)] together with equation (15) implies that
D - IIS(x)ll <- -/3,
(or D
Ilexll ~ -/3).
(23)
f0
r ( t - r)(1 - P)f(r, q ( r ) ) dr,
q(t) = T ( t - to)q(to) +
(24)
since A generates a strongly continuous semigroup T(t) and f(t, q) satisfies condition (iii) in the Theorem 1. Using the given conditions (v) and (vi), we obtain IIq(t)ll --- Me - ' ( ' - ' ° ) IIq(to)ll + M I l l - ell
f,
e -~(' ~)g(r) IIg(r)ll dt.
(25)
~t 0
Using equations (20) and (25) and Gronwall's inequality, we obtain
By equations (2) and (3), we have the following important conclusions.
(/3 > 0)
q(to) = (1 - P)x(to),
where q = ( 1 - P)x. Therefore, by a similar proof to the global existence theorem given in Alain (1981), we know that the solution of equation (23) does exist and satisfies the integral equation
(14)
if the limit exists.
IS(x), S(x)]-< -/3 IIS(x)ll,
(20)
On the other hand, by Lemma 2, we have
× D
(19)
Evidently, Px = 0 if S ( x ) = Cx =0. Therefore, for the sliding motion x = x(t), we have
(18)
The conclusion is, therefore, true by equation (18) and Lemma 3. Theorem 3 shows that either equation (16) or (17) can guarantee the existence of a sliding mode on S ( x ) = 0 for DPS (1). It will be seen that condition (4) is more convenient than condition (3) for the design of SM controls. Evidently, we can obtain similar existence conditions by using the right derivative of IIS(x)ll. 3.2. Design o f sliding manifold. The first step is to design a
IIq(t)ll -< M e -'~t'-'°) IIq(t0)ll + M II1 - ell x
f,
e - ° t ' - ~ ) g ( r ) IIg(r)ll dt.
(25)
~t 0
Using equations (20) and (25) and Gronwall's inequality, we obtain IIx(t)ll = IIq(t)ll <- M IIq(to)ll exp
{f0
[ - ~ + M I1! - PII g(r)] d r
<- M II1 - PII IIX(to)tl exp { - ( ~ - M L II1 - PII)(t - to)}.
(26) The conclusion follows by equations (13) and (26). By Theorem 4, the sliding motion will tend to the equilibrium point x = 0 as t--* ~ if we select an operator C e L(V, V2) such that equation (11) is satisfied. Note that for the sliding mode, the order of system (9) is naturally reduced since S(x) = Cx = 0. Therefore, we can use the pole assignment technique or the optimal control method under the quadratic cost functional to design a proper operator C so that the sliding mode has prescribed properties. This problem has been treated in Hu (1991). 3.3. Design o f sliding mode controllers. The second step is to design a discontinuous control law so that the sliding motion occurs on the manifold S(x) = 0. Theorem 5. If .4 satisfies the conditions in Corollary 1, then the following SM control law u = - a t ( C B ) - t C x - (CB)-~Cf(t, x) - az(CB)-1Cx/IIPx II,
(27) guarantees the existence of a sliding mode on manifold S(x) = Cx = 0 for DPS, where al -> a and a2 > 0.
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Brief Papers
Proof In fact, by equations (1), (4) and (27), we have [P.f, Px] = [PAx + Pf(t, x) + PBu. Px] = [APx - a , P x -a2Px/llPxll, Px] -<(a - a , ) I I P x l l 2 - a 2 IIPxll-< - a 2 IIPxll. The result follows from Theorem 3. For the actual system, parameter variations and disturbances exist on the system. In order to ensure that the system possesses a robustness property, we only need to select a~ and a2 in equation (27) to be suitably large. In fact, let h(t, x) be the function of system uncertainties or parameter variations which are not accessible for measurement. Then, using the equivalent control method, we know that the sliding mode does not depend on h(t, x) if it belongs to the span of operator B, and the sliding motion can also occur on manifold S(x) = 0 in such a case.
Theorem 6. If the bounds of system uncertainties or parameter variations are known: IIh(t, x)ll -< H(t, x), then the SM control law (27) guarantees the existence of a sliding mode on manifold S(x) = Cx = 0 for the disturbed system .i = Ax + f(t, x) + Bu + h(t, x),
(28)
if A, f(t,x) satisfies the conditions in Corollary 1, and constants a~ and a2 in equation (27) are chosen such that al>a;
a2>llPllH(t,x)+ao;
a0=const>0.
(29)
Proof In fact, using the above assumptions, we have [PSc,Px] = [PAx + Pf(t, x) + PBu + Ph(t, x), Px] <-(a-al)
liP.rile + IIPll Ilh(t,x)ll IIPxll-a2 IIPxll
<- - a 0 Ilexll.
(30)
The conclusion follows from equation (30) and Theorem 3. Obviously, by using existence condition (17), we can use the upper boundedness of operator A directly and thus design the sliding mode controller straightforwardly. Hence, the controller in the form of equation (27) has bounded gain in any bounded region, which is important for the actual implementation. If we use the existence condition (16), which is naturally a generalization of the one provided for lumped parameter systems (Utkin, 1992), then, formally, we have
[S(x),S(x)]=[CAx+Cf(t,x)+CBu, Cx]
(31)
the term [CAx, Cx] will be difficult to estimate since C and A are not commutable generally. The example given in Orlov and Utkin (1987) showed that the condition (16) was inapplicable to the general multi-dimensional heat process. However, the problem proposed in Orlov and Utkin (1987) can easily be solved by using condition (17). In fact, let A = aoe/oy 2, D(A) = {Q E H l (0, 1); Q'(0) - Q'(1) = 0}, then operator A satisfies
[AQ, Q] <- --An,in IIQII 2, where h.mi, = min {h.,: 1 -< i -< n}, II. II is the norm of Sobolev space Hi(0, 1), operator C is chosen as a m × n matrix. Therefore, the conditions of Theorems 1 and 5 are satisfied, and hence the sliding mode controller can be designed directly using the proposed method. For details please refer to Hu and Zhou (1991). Morever, it is not difficult to prove that the real sliding mode obtained from the above-designed approach does belong to the boundary layer IIS(x)II <- 6 after some T0.
4. Conclusions Mathematical descriptions and the existence conditions of sliding modes are crucial problems in the design of sliding mode control systems. However, for general DPS, there is, as yet, no efficient approach to solve them since unbounded operators are often involved in such systems. Based on the operator properties presented in this paper, some essential problems in the SM control theory of DPS are solved perfectly. The major contribution of the present work is fourfold. Firstly, the equivalent control method is proved to
be applicable to a wide class of DPS under general assumptions; secondly, sufficient conditions which guarantee the asymptotic stability of sliding mode are obtained and the equivalent control method is proved to be available in the finite time interval for the class of DPS which satisfy the proposed stability conditions of the sliding mode: thirdly, general existence conditions of the sliding mode are also presented: and fourthly, SM controllers are designed straightforwardly for the class of DPS with upper-bounded operators. It should be noted that the sliding mode control strategy, in this paper, is discussed only in state space. Obviously, discontinuous control (27) requires that the state should be directly accessible for measurement, which is far too strong for DPS. However, this problem can be solved in practical implementations by two standard methods. One of them is to discuss the SM control problem in output space using the same idea as above, if the output system is available for control. Another one is to design the control laws using various approximation techniques. For results related to this problem see Hu (1991) and Chen and Yeung (1991).
References Alain, H. (1981). Nonlinear Evolution Equations--Global Behavior of Solutions. Springer, New York. Breger, A. M., et al. (1980). Sliding modes in control of distributed plants subjected to a mobile multicycle signal. Aut. Remote Control, 41, 72-83. Chen, Y. P. and K. S. Yeung (1991). Sliding mode control of multi-link manipulators. Int. J. Control, 54, 257-278. Curtain, R. F. and A. J. Pritchard (1978). Infinite Dimensional Linear Systems Theory. Springer, New York. Elmali, H. and N. Olgac (1992). Robust output tracking control of nonliner MIMO systems via sliding mode technique. Automatica, 28, 145-151. Franke, D. (1981). Control of distributed parameter systems with independent linear and bilinear modes. In Proc. 3rd IMA Conf. on Control Theory, 827-841. Academic Press, London. Hale, J. K. (1980). Ordinary Differential Equations. Krieger, Huntington. Hu, Y. M. (1991). Variable Structure Control of Distributed Parameter Systems: Theory and Applications, Ph.D thesis, South China University of Technology, Guangzhou, P.R.C. and also published by the National Defence Industry Press, Beijing, P.R.C., 1994 (in Chinese). Hu, Y. M. and Q. J. Zhou (1991). Variable structure control of distributed parameter systems. Control Theory and Applications, 8, 38-43. Hung, J. Y., W. Gao and J. C. Hung (t993). Variable structure control: a survey. IEEE Trans. Industr. Electron. 40, 2-22. Ktaus, D. (1977). Ordinary Differential Equations in Banach Space. Springer, New York. Matthews, G. P. and R. A. DeCarlo (1988). Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica, 24, 187-193. Orlov, Yu. V. and V. I. Utkin (1987). Sliding mode control in indefinite-dimensional systems. Automatica, 23, 753-757. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York. Tzafestas, S. G. (1982). Distributed Parameter Control Systems. Pergamon, Oxford. Utkin, V. I. (1992). Sliding Modes in Control and Optimizations. Springer, New York. Utkin, V. I. and Yu, V. Orlov (1990). Sliding Mode Control in Infinite-dimensional Systems. Nauka, Moscow.
Appendix Proof of Theorem 1. In the boundary layer IlS(x)ll <-& for any real sliding mode x(t), we have S(2) = C A 2 + Cf(t, £) + CB5
(A.I)
Brief which implies
Papers
1965
Consequently, since
if= -(CB)-IC[AZ+f(t,i)]+(CB)-1CZ
(A.2)
The real sliding mode equation is therefore obtained by substituting equation (A.2) into equation (1) as follows (A.3)
= (1 - P ) A E + (1 - P ) f ( t , £) + P~
E(t) - x(t) = P(£(t) - x(t)) + e(t) = P£(t) + e(t),
and IlP£(t)ll <-IIB(CB)-lll IIC.fll <-IIB(CB)-lll 8 it follows that
which is equivalent to
liE(t) -x(t)I1 - (/33 + IIB(CB) -111)8,
~. = A ~ + (I - P ) f ( t , E),
(A.4)
~(to) = (I - P)E(to),
since A and P are commutable, here ~ = (I-P)JT. h(t) = (1 - P ) f ( t , E), then
Let
and the conclusion is obtained. P r o o f o f Theorem 2. In fact, by equations (A.6) and (A.9)
and the given conditions, we have Ile(t)ll -< M/3 Ill - PII 8 exp { - o t ( t - to)} (A.5)
where L is the local Lipschitz constant. Equation (A.5) implies h e L([to, tl], V) since E is assumed to be bounded and f ( t , 0) is measurable. Therefore, the solution of equation (A.4) can be represented in the integral form
f(t)
= T(t -
to)z(to) +
f,i
T(t - t)(1 - P)f(t, ~ ( t ) ) d t
(A.6)
+
L
M Ill-PII g(r)liE(t) -x(t)ll
× exp { - t r ( t - z)} dr.
(A.14)
Since liE(t)- x(t)ll = II~(t) - z ( t ) l l <-IIe2?(t)ll + Ile(t)H, follows that
it
liE(t) - x(t)11 -< IIB(CB)-III 8 + lie(t)II <(M/3 II1-PII + IIB(CB)-IlI)8
by the corollary 4.2.2 given in Pazy (1983), since the solution of equation (A.4) exists by assumption (4) and h ~ L([to, tll, V). Where r ( t ) is the strongly continuous semigroup generated by operator .4 such that IIT(t)II -< M exp {jot}.
(A.7)
On the other hand, for the ideal sliding mode x(t), we have P x = 0. From equation (9) it follows that (A.8)
(1 - P):~ = A ( I - P ) x + (1 - P ) f ( t , (1 - P)x),
+
f,i
M II1-PII g ( t ) l i E ( r ) - x ( r ) l l
x exp { - a ( t - z)} dr.
(A.15)
Using Gronwall's inequality, we obtain liE(t) - x(t)II <- dl 8 + d2B[1 - exp { - ( a - M L Ill - Pll)(t - to)}], where
(A.16)
d~ = IIB(CB) -~ II + Mfl I l l - ell;
since A and P are commutable and x = ( l - P ) x . Let z = (1 - P)x, then equation (A.8) is equivalent to the form
d2 = c l M L III - P l l / ( a - M L I11 - Pll).
(A.9)
The results now follow directly from equations (13) and (A.16).
By a similar proof to the global existence theorem given in Alain, (1981), we know that the solution of equation (A.9) does exist and satisfies the integral equation
P r o o f o f L e m m a 3. (1) For any h > 0 and 0 < 0 -< 1, we have
= A z + (1 - P ) f ( t , z),
Z(t) = T ( t - to)Z(to) +
Z(to) = (1 - P)x(to).
f,0
IlY - OhY~ll = I1(1 - O)y + Oy - Oh fll -< 0 IlY - h)ll + (1 - O) IlYlI.
T ( t - "¢)(1 - P ) f ( t , z(z)) dz.
This implies (A.10)
Let e ( t ) = ~ . ( t ) - z(t). Then by equations (A.6), (A.7) and
IIY II - l[Y - Ohp II ~ I1YII - IIY - h))II hO
(A.10), we have
Therefore Ill Y II - IIY - hy II]/h is increasing with respect to h and upper-bounded, since
lie(011 -< M exp {p(t - to)} Ile(to)ll +Mlll-PII
h
L
IIf(z,£(r))-f(z,z(r))ll
Ilyll - Ily - h3~ll < [[y[[- (llyll - IIh~ll) h h
II.~ll.
x exp {p(t - z)} d r This implies that
<- M/3 Ill - P]I 6 exp {p(t - to)} + ML Ill-PII
lim sup IIy II - IIy - h3~ll r ~ O O--
[lle(r)l[ + IIB(CB)-'II 8]
× exp {p(t - z)} dr.
(A.11)
Using Gronwall's inequality, we obtain
exists. On the other hand, we have I(lly(t)ll- [ly(t - h )[I) - (lly(t)[I - Ily(t) - hp(t)lDI
lie(t) II < 131B exp {(p + ~2)(t - to)},
(A.12)
= I Ily(t) - h.9(t)ll - Ily(t - h)lll - IIy(t) - y ( t - h) - h.O(t)l I
where
=o(h).
131 = M/3 Ill - PII + M L Ill - PII IIB(CB)-lll x sup
f,
exp{-p(z-to)}dr;
~2=MLII1-PII.
to~t~tl ~10
Therefore,/33 > 0 exists such that Ile(t)ll -
Therefore, D - Ily(t)ll does exist. (2) For any h > 0 , using the properties (1)-(3) of the semi-inner product [., .], we obtain [y(t) - y ( t - h), y(t)] = [y(t), y(t)] - [y(t - h), y(t)]
t ~ [to, td.
(A.13)
> I[y(t)l] 2 - Ily(t)ll [ly(t - h)l[
1966
Brief Papers which is concluded by the following inequality:
which implies
- h )"v(t) l >-[[y(t)H Ily(t)ll - Ily(t- h)H
[y(t)-Y(th
[y(t)- y(t-h), y(t)] _[5,(t), y(t)] = [y(t)-y(t-h)-~(t)'y(t)]h
Let h ~ 0 It follows that
<- Y(t)-h--(t-h) 9(t)
(ly(t)ll
[)(t), y(t)] -> Ily(t)ll D - Ity(t)ll since
ft~/y't'-y(t-h) y(t)l]=[.9(O,y(t)],
lim h--o [
h
'
J
and the assumption that ~(t) exists. The second conclusion is therefore true, Conclusions (3) and (4) are obviously true by the properties of left derivative (Hale, 1980) and the second conclusion.