An analysis and design method for linear systems subject to actuator saturation and disturbance

Automatica 38 (2002) 35 359 Brief Paper www.elsevier.com/locate/automatica An analysis and design method for linear systems subject to actuator saturation and disturbance Tingshu Hu a;, Zongli Lin a; ;, Ben M. Chen b a Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA b Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 7576, Singapore Received 2 November 999; revised 6 November 2000; received in nal form 7 August 200 Abstract We present a method for estimating the domain of attraction of the origin for a system under a saturated linear feedback. A simple condition is derived in terms of an auxiliaryfeedback matrix for determining if a given ellipsoid is contractivelyinvariant. This condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. This condition is then extended to determine the invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stabilityrequirements. The eectiveness of the developed methods is illustrated with examples.? 200 Elsevier Science Ltd. All rights reserved. Keywords: Actuator saturation; Stability; Domain of attraction; Invariant set; Disturbance rejection. Introduction In this paper, we are interested in the control of linear systems subject to actuator saturation and persistent disturbances: ẋ = Ax + B(u)+Ew; x R n ; u R m ; w R q ; () where x is the state, u is the control, w is the disturbance and ( ) is the standard saturation function. Our rst concern is the closed-loop stability(when w = 0) under a given linear state feedback u = Fx. There has been a lot of work on this topic (see, e.g., Davison & Kurak, 97; Gilbert & Tan, 99; Hindi & Boyd, 998; Khalil, 996; Loparo & Blankenship, 978; Pittet, Tarbouriech, & Burgat, 997; Vanelli & Vidyasagar, 985; Weissenberger, 968 and the references therein). In This paper was not presented at anyifac meeting. This paper was recommended for publication in revised form byassociate Editor Andrew R. Teel under the direction of Editor Hassan Khalil. Corresponding author. Tel.: +-434-9246342; fax: +-434-924888. E-mail addresses: th7f@virginia.edu (T. Hu), zl5y@virginia.edu (Z. Lin), bmchen@nus.edu.sg (B.M. Chen). Supported in part bythe US Oce of Naval Research Young Investigator Program under grant N0004-99--0670. particular, various simple and general methods for estimating the domain of attraction have been developed by applying the absolute stability analysis tools, such as the circle and Popov criteria (see, e.g., Hindi & Boyd, 998; Khalil, 996; Pittet et al., 997 ; Weissenberger, 968), where the saturation is treated as a locallysector bounded nonlinearityand the domain of attraction is estimated by use of quadratic and Lur e type Lyapunov functions. In Hindi and Boyd (998) and Pittet et al. (997), the condition for local stabilityand some performance problems are expressed in terms of (nonlinear) matrix inequalities in system parameters and other auxiliary optimization parameters. Byxing some of the parameters, these matrix inequalities simplifyto linear matrix inequalities (LMIs) and can be treated with the LMI software. Since the circle criterion is applicable to general memoryless sector bounded nonlinearities, we can expect the conservatism in estimating the domain of attraction when it is applied to the saturation nonlinearity. In this paper, a less conservative estimation of the domain of attraction is obtained byusing a quadratic Lyapunov function. This is made possible byexploring the special propertyof saturation. Moreover, since this condition is given in terms of LMIs, it is veryeasyto handle in both analysis and design. 0005-098/02/$ - see front matter? 200 Elsevier Science Ltd. All rights reserved. PII: S 0005-098(0)00209-6

352 T. Hu et al. / Automatica 38 (2002) 35 359 In the presence of disturbance, we are interested in knowing if there exists a bounded invariant set such that all the trajectories starting from inside of it will remain in it. This problem was addressed byblanchini (990) and Blanchini (994). In this paper, we would further like to synthesize feedback laws that have the ability to reject the disturbance. Here disturbance rejection is in the sense that, there is a small (as small as possible) neighborhood of the origin such that all the trajectories starting from the origin will remain in it. This performance was analyzed byhindi and Boyd (998) for the class of disturbances with nite energy. In this paper, we will deal with persistent disturbances and propose a controller design method. Furthermore, we are also interested in the problem of asymptotic disturbance rejection with nonzero initial states. A related problem was addressed byhu and Lin (200b) and Saberi, Lin, and Teel (996), where the disturbances are input additive and enter the system before the saturating actuator, i.e., the system has a state equation: ẋ = Ax + B(u + w). It is shown in these papers that given anypositive number D, anycompact subset X 0 of the null controllable region and anyarbitrarilysmall neighborhood X of the origin, there is a feedback control such that anytrajectorystarting from within X 0 will enter X in a nite time for all disturbances w: w 6 D. We, however, could not expect to have this nice result for system (), where the disturbance enters the system after the saturating actuator. If w or E is sucientlylarge, it mayeven be impossible to keep the state bounded. What we can expect is to have a set X 0 (as large as we can get) and a set X (as small as we can get) such that all the trajectories starting from X 0 will enter X in a nite time and remain in it thereafter. This paper is organized as follows. Section 2 addresses the analysis of and design for closed-loop stability. Section 3 addresses issues related to disturbance rejection. A brief concluding remark is given in Section 4. 2. Stability analysis 2.. Problem statement Consider the open-loop system ẋ = Ax + B(u); x R n ; u R m ; (2) where ( ) is the standard saturation function of appropriate dimensions. In the above system, : R m R m, and (u)=[(u ) (u 2 ) (u m ) T, where (u i )= sgn(u i )min{; u i }. Here we have slightlyabused the notation byusing to denote both the scalar valued and the vector valued saturation functions. Suppose that a state feedback u = Fx has been designed such that A + BF is Hurwitz. We would like to know how the closed-loop system behaves in the presence of saturation nonlinearity, in particular, to what extent the stability is preserved. Our main objective in this section is to obtain an estimate of the domain of attraction of the origin for the closed-loop system ẋ = Ax + B(Fx): (3) Denote the ith column of B as b i and the ith row of F as f i. Then BF = b f + +b m f m. For a matrix F R m n, dene L(F):={x R n : f i x 6 ;i [;m}: If F is the feedback matrix, then L(F) is the region in the state space where the control is linear in x. For x(0) = x 0 R n, denote the state trajectoryof system (3) as (t; x 0 ). Then the domain of attraction of the origin is { } S := x 0 R n : lim (t; x 0 )=0 : t Let P R n n be a positive-denite matrix. Denote E(P; )={x R n : x T Px 6 }: Let V (x)=x T Px. The ellipsoid E(P; ) is said to be contractively invariant if V (x)=2x T P(Ax+B(Fx)) 0 for all x E(P; )\{0}. Clearly, if E(P; ) is contractivelyinvariant, then it is inside the domain of attraction. We will develop conditions under which E(P; ) is contractively invariant and hence obtain an estimate of the domain of attraction. 2.2. A set invariance condition based on circle criterion A multivariable circle criterion is presented in Khalil (996, Theorem 0:) and is applied to estimate the domain of attraction for system (3), with a given feedback gain F, in Hindi and Boyd (998) and Pittet et al. (997). Proposition (Khalil; 996; Pittet et al:; 997). Assume that (F; A; B) is controllable and observable. Given an ellipsoid E(P; ); if there exist positive diagonal matrices K ;K 2 R n n with K I; K + K 2 I such that (A + BK F) T P + P(A + BK F) + 2 (F T K 2 + PB)(K 2 F + B T P) 0 (4) and E(P; ) L(K F); then E(P; ) is a contractively invariant set and hence inside the domain of attraction. A similar condition based on circle criterion is given in Hindi and Boyd (998). These conditions are then used for stabilityand performance analysis with LMI software in Hindi and Boyd (998) and Pittet et al. (997). Since inequality(4) is not jointlyconvex in K ;K 2 and P, these

T. Hu et al. / Automatica 38 (2002) 35 359 353 parameters need to be optimized separatelyand there is no guarantee that the global optimal solutions can be obtained for related problems. 2.3. An improved condition for set invariance We will develop a less conservative set invariance condition byexploring the special propertyof the saturation nonlinearity. It is based on direct Lyapunov function analysis in terms of an auxiliary feedback matrix H R m n. This condition turns out to be equivalent to some LMIs. Denote the ith row of H as h i. For two matrices F; H R m n and a vector R m, denote f +( )h M(; F; H)=. : (5) m f m +( m )h m Let V = { R m : i = or 0}. There are 2 m elements in V. We will use a V to choose from the rows of F and H to form a new matrix M(; F; H): if i =, then the ith row of M(; F; H) isf i and if i = 0, then the ith row of M(; F; H) ish i. For example, suppose m = 2, then {M(; F; H): V} = { H; [ h f 2 ; [ f h 2 ; F } : Theorem. Given an ellipsoid E(P; ); if there exists an H R m n such that (A + BM(; F; H)) T P + P(A + BM(; F; H)) 0 (6) for all V and E(P; ) L(H); i.e.; h i x 6 for all x E(P; ); i [;m; then E(P; ) is a contractively invariant set. Proof. Let V (x)=x T Px, we need to show that V (x)=2x T P(Ax + B(Fx)) 0 x E(P; )\{0}: Here we have V (x)=2x T A T Px +2x T PB(Fx) =2x T A T Px + m 2x T Pb i (f i x): i= For each term 2x T Pb i (f i x), () If x T Pb i 0 and f i x 6, then 2x T Pb i (f i x)= 2x T Pb i 6 2x T Pb i h i x. Here we note that 6 h i x x E(P; ). (2) If x T Pb i 0 and f i x, then (f i x) 6 f i x and 2x T Pb i (f i x) 6 2x T Pb i f i x. (3) If x T Pb i 6 0 and f i x, then 2x T Pb i (f i x)=2x T Pb i 6 2x T Pb i h i x. Here we note that h i x x E(P; ). (4) If x T Pb i 6 0 and f i x 6, then (f i x) f i x and 2x T Pb i (f i x) 6 2x T Pb i f i x: Combining all the four cases, we have 2x T Pb i (f i x) 6 max{2x T Pb i h i x; 2x T Pb i f i x} for every x E(P; ) and each i [;m. Therefore, for every x E(P; ), m V (x) 6 2x T A T Px + max{2x T Pb i h i x; 2x T Pb i f i x}: i= Now we associate every x E(P; ) with a vector (x) V as follows: if 2x T Pb i h i x 2x T Pb i f i x, then we set i =, otherwise we set i = 0. It follows that m V (x)62x T A T Px +2 ( i x T Pb i f i x +( i )x T Pb i h i x) i= ( m ) =2x T A T Px +2x T P b i ( i f i +( i )h i ) x i= =2x T (A + BM(; F; H)) T Px: In view of (6), we have that E(P; )\{0}. V (x) 0 for all x A geometric interpretation of Theorem can be found in Hu and Lin (200a). If we restrict H to be K F, where K is the same as that in Proposition, then we have Corollary. Given an ellipsoid E(P; ); if there exists a positive diagonal matrix K R n n such that (A + BM(; F; K F)) T P + P(A + BM(; F; K F)) 0 for all V and E(P; ) L(K F); then E(P; ) is a contractively invariant set. This corollaryis equivalent to Theorem 0:4 in Khalil (996) when applied to saturation nonlinearity. Obviously, the condition in Corollary is more conservative than that in Theorem because the latter provides more freedom in choosing the H matrix. However, it is evident from Khalil (996) that the condition in Proposition is even more conservative than that in Corollary. Computations show that in general, for a xed P, Theorem allows for a larger than Corollary. Therefore, Theorem oers a wider choice of invariant ellipsoids for optimization and will lead to less conservative estimation of the domain of attraction. 2.4. Estimation of the domain of attraction With all the ellipsoids satisfying the set invariance condition, we would like to choose from among them the largest one to get a least conservative estimation of the domain of attraction. In the literature (see, e.g., Boyd, El Ghaoui, Feron, & Balakrishnan, 994; Davison &

354 T. Hu et al. / Automatica 38 (2002) 35 359 Kurak, 97; Pittet et al., 997), the largeness of an ellipsoid is usuallymeasured byits volume. Here, we will take its shape into consideration. Let X R R n be a prescribed bounded convex set. For a set S R n, dene R (S):=sup{ 0: X R S}: If R (S), then X R S. Two typical types of X R are the ellipsoids X R = E(R; ) = {x R n : x T Rx 6 }; R 0 and the polyhedrons X R =co{x ;x 2 ;:::;x l }; where co denotes the convex hull. Theorem gives a condition for an ellipsoid to be inside the domain of attraction. Now we would like to choose from all the E(P; ) s that satisfythe condition such that the quantity R (E(P; )) is maximized. This problem can be formulated as sup P 0;;H s:t: (a) X R E(P; ); (b) (A + BM(; F; H)) T P +P(A + BM(; F; H)) 0 V; (c) E(P; ) L(H): (7) If we replace with log det(p=) and remove constraint (a), then we obtain the problem of maximizing the volume of E(P; ). Similar modication can be made to other optimization problems to be formulated in this paper. Moreover, the following procedure to transform (7) into a convex optimization problem with LMI constraints can be adapted to the corresponding volume maximization (or minimization) problems. Now we transform the constraints of (7) into LMIs. If X R is a polyhedron, then by Schur complement, (a) is equivalent to ( ) P x 2 xi T x i 6 2 i T ( ) P 0 (8) x i for all i [;l. If X R is an ellipsoid E(R; ), then (a) is equivalent to ( ) R P 2 R I 2 ( ) P 0: (9) I Constraint (b) is equivalent to ( ) P (A + BM(; F; H)) T ( ) P +(A + BM(; F; H)) 0 V: (0) From Hindi and Boyd (998), constraint (c) is equivalent to ( ) P h i h i P h T i 6 ( ) ( ) P P 0 () h T i for all i [;m. Let == 2 ;Q=(P=) and G = H(P=). Also let the ith row of G be g i, i.e., g i = h i (P=). Note that M(; F; H)Q = M(; FQ; HQ)= M(; FQ; G). If X R is a polyhedron, then from (8),(0) and () optimization problem (7) can be rewritten as Q 0;G [ x T s:t: (a) i x i Q 0; i [;l; (b) QA T + AQ + M(; FQ; G) T B T +BM(; FQ; G) 0 V; [ gi (c) 0; i [;m; Q g T i (2) where all the constraints are given in LMIs. If X R is an ellipsoid, we just need to replace (a) with [ R I (a2) 0: I Q Note that there are 2 m matrix inequalities in constraint (b) corresponding to all V. Example. We use an example of Pittet et al. (997) to illustrate our results. The system is described by (3) with [ [ 0 0 A = ; B= ; F=[ 2 : 0 5 With x =[ 0:8 T and X R =co{x ; x }, we solve (2) and get ==( ) =2 =4:37. The maximal ellipsoid is E(P ; ); [ 0:70 0:0627 P = 0:0627 0:0558 (see the solid ellipsoid in Fig. ). The inner dashed ellipsoid is an invariant set obtained bythe circle criterion method in Pittet et al. (997) and the region bounded by the dash dotted curve is obtained bythe Popov method, also in Pittet et al. (997). We see that both the regions obtained bythe circle criterion and bythe Popov method can be actuallyenclosed in a single invariant ellipsoid. To get a better estimation, we vary x over a unit circle, and solve (2) for each x. Let the optimal be (x ). The outermost dotted boundaryin Fig. is formed bythe points (x )x as x varies along the unit circle.

T. Hu et al. / Automatica 38 (2002) 35 359 355 3. Disturbance rejection 3.. Problem statement Fig.. The invariant sets obtained with dierent methods. 2.5. Controller design Now our objective is to choose a feedback matrix F R m n such that the estimated domain of attraction as obtained bythe method of the last subsection is maximized with respect to X R. This can be simplydone by taking the F in (2) as an extra optimization parameter. To make the optimization easy, we use a new parameter Y to replace FQ in (2) and the resulting LMI problem is Q 0;Y;G s:t: (2a); (2c) and (b) QA T + AQ + M(; Y; G) T B T +BM(; Y; G) 0; V: (3) The optimal F will be recovered from YQ. Consider a simpler optimization problem s:t: (2a); (2c) and (b) QA T + AQ + G T B T + BG 0: Q 0;G (4) If Y = G, then all the 2 m inequalities in (3b) are the same as (4b). Hence the new problem (4) can be considered as a result from forcing Y = G in (3). On this account, (4) should have an inmum no less than (3). On the other hand, since the 2 m inequalities in (3b) include (4b), problem (4) can also be considered as a result from discarding 2 m inequalityconstraints of (3b). Because of this, (4) should have an inmum no larger than (3). These arguments show that the optimal values of (4) and (3) must be the same. From the above analysis, we see that, if our only purpose is to enlarge the domain of attraction, we might as well solve the simpler optimization problem (4). The freedom in choosing Y can be used to improve other performances beyond large domain of attraction. Consider the open-loop system ẋ = Ax + B(u)+Ew; x R n ; u R m ; w R q ; (5) where, without loss of generality, we assume that the bounded disturbance w belongs to the set W := {w: w(t) T w(t) 6 t 0}: Let the state feedback be u = Fx. The closed-loop system is ẋ = Ax + B(Fx)+Ew: (6) For an initial state x(0) = x 0, denote the state trajectory of the closed-loop system under w as (t; x 0 ;w). A set in R n is said to be invariant if all the trajectories starting from it will remain in it regardless of w W. An ellipsoid E(P; ) is said to be strictly invariant if V =2x T P(Ax + B(Fx)+Ew) 0 for all w such that w T w 6 and all x @E(P; ), the boundaryof E(P; ). The notion of invariant set plays an important role in studying the stability and other performances of a system (see, e.g., Blanchini, 994; Boyd et al., 994 and the references therein). Our primaryconcern is the boundedness of the trajectories for some set of initial states (maybe as large as possible). This requires a large invariant set. On the other hand, for the purpose of disturbance rejection, we would also like to have a small invariant set containing the origin in its interior so that a trajectorystarting from the origin will stayclose to the origin. To formallystate the objectives of this section, we need to extend the notion of the domain of attraction as follows. Denition. Let B be a bounded invariant set of (6). The domain of attraction of B is { } S(B):= x 0 R n : lim d( (t; x 0 ;w); B)=0 w W ; t where d( (t; x 0 ;w); B) = x B (t; x 0 ;w) x is the distance from (t; x 0 ;w)tob. The problems we are to address in this section are given as follows: Problem (Set invariance analysis). Let F be known. Given an ellipsoid E(P; ), determine if E(P; ) is (strictly) invariant. Problem 2 (Invariant set enlargement). Given a bounded set X 0 R n, design F such that the closed-loop system has an invariant set E(P; ) 2 X 0 with 2 maximized.

356 T. Hu et al. / Automatica 38 (2002) 35 359 Problem 3 (Disturbance rejection). Given a set X R n, design F such that the closed-loop system has an invariant set E(P; ) 3 X with 3 minimized. Problem 4 (Disturbance rejection with guaranteed domain of attraction). Given two reference sets, X and X 0, design F such that the closed-loop system has an invariant set E(P; ) X 0, and for all x 0 E(P; ), (t; x 0 ;w) will enter a smaller invariant set E(P; ) 4 X with 4 minimized. 3.2. Condition for set invariance We consider closed-loop system (6) with a given F. The following theorem gives a sucient condition for the invariance of a set E(P; ). Theorem 2. For a given set E(P; ); if there exist an H R m n and a positive number such that (A + BM(; F; H)) T P + P(A + BM(; F; H)) + PEET P + P 6 0( 0) V (7) and E(P; ) L(H); then E(P; ) is a (strictly) invariant set for system (6). Proof. We prove the strict invariance. That is, for V (x)=x T Px, we will show that V =2x T P(Ax + B(Fx)+Ew) 0 for all x @E(P; ) and all w such that w T w 6. Following the procedure of the proof of Theorem, we can show that for every x E(P; ), there exists a V such that 2x T P(Ax + B(Fx)) 6 2x T (A + BM(; F; H)) T Px: Since 2x T PEw 6 xt PEE T Px + w T w 6 xt PEE T Px + we have V 6 x (2(A T + BM(; F; H)) T P + ) PEET P x + : It follows from (7) that for all x E(P; ), V xt Px + : Observing that on the boundaryof E(P; );x T Px =, hence V 0. It follows that E(P; ) is a strictlyinvariant set. Theorem 2 deals with Problem and can be easilyused for controller design in Problems 2 and 3. For Problem 2; we can solve the following optimization problem: sup P 0; ; 0;F;H s:t: 2 2 X 0 E(P; ); E(P; ) L(H) and (7): (8) Let Q =(P=) ;Y= FQ and G = HQ, then (7) is equivalent to QA T + AQ + M(; Y; G) T B T + BM(; Y; G) + EET + Q 0 V: If we x =, then the original optimization constraints can be transformed into LMIs as with (7). The global maximum of 2 will be obtained byrunning = from 0 to. For Problem 3; we have P 0; ; 0;F;H 3 s:t: E(P; ) 3 X ; E(P; ) L(H) and (7); which can be solved similarlyas Problem 2. (9) 3.3. Disturbance rejection with guaranteed domain of attraction Given X 0 R n, if the optimal solution of Problem 2 is 2, then there are innitelymanychoices of the feedback matrices F s such that X 0 is contained in some invariant ellipsoid. We will use this extra freedom for disturbance rejection, that is, to construct another invariant set E(P; ) which is as small as possible with respect to some X. Moreover, X 0 is inside the domain of attraction of E(P; ). In this way, all the trajectories starting from X 0 will enter E(P; ) 4 X for some 4 0. Here the number 4 is a measure of the degree of disturbance rejection. Before addressing Problem 4; we need to answer the following question: Suppose that for given F and P, both E(P; ) and E(P; 2 ); 2 are strictlyinvariant, then under what conditions will the other ellipsoids E(P; ); ( ; 2 ) also be strictlyinvariant? If theyare, then all the trajectories starting from within E(P; 2 ) will enter E(P; ) and remain inside it. Theorem 3. Given two ellipsoids; E(P; ) and E(P; 2 ); 2 0; if there exist H ;H 2 R m n and a positive number such that (A + BM(; F; H )) T P + P(A + BM(; F; H )) + PEET P + P 0 V; (20)

T. Hu et al. / Automatica 38 (2002) 35 359 357 (A + BM(; F; H 2 )) T P + P(A + BM(; F; H 2 )) + PEET P + 2 P 0 V (2) and E(P; ) L(H ); E(P; 2 ) L(H 2 ); then for every [ ; 2 ; there exists an H R m n such that (A + BM(; F; H)) T P + P(A + BM(; F; H)) + PEET P + P 0 V (22) and E(P; ) L(H). This implies that E(P; ) is also strictly invariant. Proof. Let h ;i and h 2;i be the ith row of H and H 2 respectively. The conditions E(P; ) L(H ) and E(P; 2 ) L(H 2 ) are equivalent to h ;i h T ;i P 0; 2 h 2;i h T 2;i P 0; i [;m: Since [ ; 2, there exists a [0; such that = = (= )+( )= 2. Let H = H +( )H 2. Clearly h T i h i 0: P From (20) and (2), and byconvexity, we have (22). In view of Theorem 3, to solve Problem 4; we onlyneed to construct two invariant ellipsoids E(P; ) and E(P; 2 ) satisfying the condition of Theorem 3 such that X 0 E(P; 2 ) and E(P; ) 4 X with 4 minimized. Since 2 can be absorbed into other parameters, we assume for simplicitythat 2 = and. Problem 4 can then be formulated as P 0;0 ; 0;F;H ;H 2; 4 s:t: (a) X 0 E(P; ); E(P; ) 4 X ; (b) (20); (2) (c) E(P; ) L(H ); (d) E(P; ) L(H 2 ): (23) If we x and, then (23) can also be transformed into a convex optimization problem with LMI constraints. To obtain the global inmum, we mayvary from 0 to and from 0 to. Fig. 2. The invariant ellipsoids and the null controllable region. Example 2. The open-loop system is described by (5) with [ [ [ 0:6 0:8 2 0: A = ; B= ; E= : 0:8 0:6 4 0: The system has a pair of unstable complex poles. We rst ignore the disturbance and solve (3) for a feedback with the objective of maximizing the domain of attraction with respect to the unit ball, X R = E(I; ). The result is, ==( ) =2 =2:447; [ 0:0752 0:0566 P = ; 0:0566 0:33 F =[ 0:0025 0:2987 and the invariant ellipsoid is E(P ; ) (see the larger ellipsoid in Fig. 2). As a comparison, we also plotted the boundaryof the null controllable region (Hu & Lin, 200a) of the open-loop system (see the dashed outer curve). We next deal with Problem 2. Bysolving (8) with X 0 being a unit ball, we obtain 2 =2:395, with 2 =0:09. The resulting invariant ellipsoid is E(P 2 ; ), with P 2 = [ 0:0835 0:0639 0:0639 0:460 (see the inner dash dotted ellipsoid in Fig. 2). To deal with Problem 3; we solve (9), with X also being a unit ball. We obtain 3 =0:0606, which shows that the disturbance can be rejected to a verysmall level. Now we turn to Problem 4. The optimization result by solving Problem 2 gives us some guide in choosing X 0. Here we choose X 0 = E(I; 2 2 ), X = E(I; ). The optimal solution is 4 =0:9725; =0:006; =0:0489,

358 T. Hu et al. / Automatica 38 (2002) 35 359 F 4 =[0:2844 P 4 = Fig. 3. The invariant ellipsoids and a trajectory. :4430, and [ 0:45 0:0922 0:0922 0:872 : In Fig. 3, the larger ellipsoid is E(P4 ; ), the smaller ellipsoid is E(P4 ; ) and the outermost dashed closed curve is the boundaryof the null controllable region. A trajectory is plotted with x 0 @E(P4 ; ) and w = sign(sin(0:3t)). 4. Conclusions We considered linear systems subject to actuator saturation and disturbance. A condition for determining if a given ellipsoid is contractivelyinvariant was derived and shown to be less conservative than the existing conditions that are based on the circle criterion or the vertex analysis. With the aid of this condition, we developed analysis and design methods, both for closed-loop stabilityand disturbance rejection. Examples were used to demonstrate the eectiveness of these methods. References Blanchini, F. (990). Feedback control for LTI systems with state and control bounds in the presence of disturbances. IEEE Transaction on Automatic Control, AC-35, 23 243. Blanchini, F. (994). Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov function. IEEE Transaction on Automatic Control, AC-39, 428 433. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (994). Linear matrix inequalities in systems and control theory. SIAM Studies in Appl. Mathematics, Philadelphia. Davison, E. J., & Kurak, E. M. (97). A computational method for determining quadratic Lyapunov functions for non-linear systems. Automatica, 7, 627 636. Gilbert, E. G., & Tan, K. T. (99). Linear systems with state and control constraints: The theoryand application of maximal output admissible sets. IEEE Transaction on Automatic Control, AC-36, 008 020. Hindi, H., & Boyd, S. (998). Analysis of linear systems with saturating using convex optimization. Proceedings of the 37th IEEE Conference on Decision and Control, Florida (pp. 903 908). Hu, T., & Lin, Z. (200a). Control systems with actuator saturation: Analysis and design. Boston: Birkhauser. Hu, T., & Lin, Z. (200b). Practical stabilization of exponentially unstable linear systems subject to actuator saturation nonlinearity and disturbance. International Journal of Robust Nonlinear Control,, 555 588. Khalil, H. (996). Nonlinear systems. Upper Saddle River, NJ: Prentice-Hall. Loparo, K. A., & Blankenship, G. L. (978). Estimating the domain of attraction of nonlinear feedback systems. IEEE Transaction on Automatic Control, AC-23, 602 607. Pittet, C., Tarbouriech, S., & Burgat, C. (997). Stabilityregions for linear systems with saturating controls via circle and Popov criteria. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA (pp. 458 4523). Saberi, A., Lin, Z., & Teel, A. R. (996). Control of linear systems with saturating actuators. IEEE Transaction on Automatic Control, AC-4, 368 378. Vanelli, A., & Vidyasagar, M. (985). Maximal Lyapunov functions and domain of attraction for autonomous nonlinear systems. Automatica, 2, 69 80. Weissenberger, S. (968). Application of results from the absolute stabilityto the computation of nite stabilitydomains. IEEE Transactions on Automatic Control, AC-3, 24 25. Tingshu Hu was born in Sichuan, China in 966. She received her B.S. and M.S. degrees in Electrical Engineering from Shanghai Jiao Tong University, Shanghai, China, in 985 and 988, respectively, and a Ph.D degree in Electrical Engineering from Universityof Virginia, USA, in May200. Her research interests include systems with saturation nonlineari ties and robust control theory. She has published several papers in these areas. She is also a co-author (wiht Zongli Lin) of the book Control Systems with Actuator Saturation: Analysis and Design (Birkhauser, Boston, 200). She is currentlyan associate editor on the Conference Editorial Board of the IEEE Control Systems Society. Zongli Lin was born in Fuqing, Fujian, China on February24, 964. He received his B.S. degree in Mathematics and Computer Science from Amoy University, Xiamen, China, in 983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 989, and his Ph.D. degree in Electrical and Computer Engineering from Washington State University, Pullman, Washington, in May994. From July983 to July986, Dr. Lin worked as a control engineer at Chinese Academyof Space Technology. In January994, he joined the Department of Applied Mathematics and Statistics, State Universityof New York at StonyBrook as a visiting assistant professor, where he became an assistant professor in September 994. Since July997, he has been with the Department of Electrical and Computer Engineering at Universityof Virginia, where he is currently an associate professor. His current research interests include nonlinear control, robust control, and control of systems with saturating actuators. He has published several papers in these areas. He is also the author of the

T. Hu et al. / Automatica 38 (2002) 35 359 359 book, Low Gain Feedback (Springer-Verlag, London, 998) and a co-author (with Tingshu Hu) of the recent book Control Systems with Actuator Saturation: Analysis and Design (Birkhauser, Boston, 200). A senior member of IEEE, Dr. Lin was an associate editor on the Conference Editorial Board of the IEEE Control Systems Society and currentlyserves as an Associate Editor of IEEE Transactions on Automatic Control. He is also a member of the IEEE Control Systems Society s Technical Committee on Nonlinear Systems and Control and heads its Working Group on Control with Constraints. He is the recipient of a US Oce of Naval Research Young Investigator Award. Ben M. Chen, born on November 25, 963, in Fuqing, Fujian, China, received his B.S. degree in mathematics and computer science from AmoyUniversity, Xiamen, China, in 983, an M.S. degree in electrical engineering from Gonzaga University, Spokane, Washington, in 988, and a Ph.D. degree in Electrical and Computer Engineering from Washington State University, Pullman, Washington, in 99. He worked as a software engineer from 983 to 986 in South-China Computer Corporation, China, and was an assistant professor from 992 to 993 in the Electrical Engineering Department of State Universityof New York at StonyBrook. Since 993, he has been with the Department of Electrical and Computer Engineering, National Universityof Singapore, where he is currentlyan associate professor. His current research interests are in linear control and system theory, control applications, development of internet-based virtual laboratories and internet securitysystems. He is an author or co-author of ve monographs, Hard Disk Drive Servo Systems (London: Springer, 200), Robust and H Control (London: Springer, 2000), H Control and Its Applications (London: Springer, 998), H 2 Optimal Control (London: Prentice Hall, 995), Loop Transfer Recovery: Analysis and Design (London: Springer, 993), and one textbook, Basic Circuit Analysis (Singapore: Prentice Hall, st Ed., 996; 2nd Ed., 998). He was an associate editor in 997 998 on the Conference Editorial Board of IEEE Control Systems Society. He currently serves as an associate editor of IEEE Transactions on Automatic Control.