On Maximizing the Convergence Rate for Linear Systems With Input Saturation

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY On Maxiizing the Convergence Rate for Linear Systes With Inut Saturation Tingshu Hu, Zongli Lin, Yacov Shaash Abstract In this note, we consider a few iortant issues related to the axiization of the convergence rate inside a given ellisoid for linear systes with inut saturation. or continuous-tie systes, the control that axiizes the convergence rate is sily a bang bang control. Through studying the syste under the axial convergence control, we reveal several fundaental results on set invariance. An iortant consequence of axiizing the convergence rate is that the axial invariant ellisoid is roduced. We rovide a sile ethod for finding the axial invariant ellisoid, we also study the deendence of the axial convergence rate on the Lyaunov function. Index Ters Convergence rate, invariant set, saturation, stability. I. INTRODUCTION ast resonse is always a desired roerty for control systes. The tie otial control roble was forulated for this urose (see, e.g., [10] [11]). Although it is well known that the tie otial control is a bang bang control, this control strategy is rarely ileented in real systes. The ain reason is that it is generally iossible to characterize the switching surface. A notion directly related to fast resonse is the convergence rate of the state trajectories. or a linear syste, the convergence rate is erined by the real art of the ole which is closest to the iaginary axis. or linear systes subject to actuator saturation, efforts have been ade to increase the convergence rate in various heuristic ways. or exale, the Q atrix in linear quadratic regulator (LQR) design can be increased iecewisely [12] as the state trajectory converges to the origin. or better understing of the convergence rate its related robles, we need a recise definition of the convergence rate for a general nonlinear syste. Consider a nonlinear syste _x = f (x): Assue that the syste is asytotically stable at the origin. Given a Lyaunov function V (x), let L V () be a level set L V () =fx 2 R n : V (x) g. Suose that _ V (x) < 0 for all x 2 L V () nf0g. Then, the overall convergence rate of V (x) on L V () can be defined as := 1 2 inf 0 _ V (x) V (x) : x 2 L V () nf0g : (1) In recent years, control systes with actuator saturation have been extensively studied (see the secial issue on this toic [1] the references therein). In this note, we will investigate issues related to the axiization of the convergence rate for a linear syste subject to actuator saturation. We will be interested in quadratic Lyaunov functions, whose level sets are ellisoids. A very iortant consequence of Manuscrit received January 19, 2001; revised Seteber 7, 2001 October 28, Recoended by Associate Editor P. A. Iglesias. This work was suorted in art by the Office of Naval Research Young Investigator Progra under Grant N T. Hu Z. Lin are with the Deartent of Electrical Couter Engineering, University of Virginia, Charlottesville, VA USA (e-ail: th7f@virginia.edu; zl5y@virginia.edu). Y. Shaash is with the Deartent of Electrical Engineering, State University of New York, Stony Brook, NY USA (e-ail: yshaash@notes.sunysb.edu). Digital Object Identifier /TAC axiizing the convergence rate is that the axial invariant ellisoid of a given shae is roduced. As ointed out in [2], set invariance is a very iortant notion a owerful tool in studying the stability other erforances of systes (see also [3], [5], the references therein). Recent years have witnessed a surge of interest in this toic. In [3], [4], [6], [8], [9], invariant ellisoids are used to estiate the doain of attraction to study disturbance rejection caability of the closed-loo syste. Various criteria have been derived for erining if an ellisoid is invariant under a given saturated linear feedback law efforts have been ade to design controllers that result in large invariant ellisoids (see, e.g., [4], [6], [8], [9], [12]). To exlore the full otential of saturating actuators, i.e., to design a controller that will roduce the largest invariant ellisoid, we need to answer the fundaental question: what is the largest ellisoid that can be ade invariant with the bounded control delivered by a saturating actuator? We will address this issue in this note through studying the syste under the axial convergence control. It turns out that the axial convergence rate is liited by the shae of the ellisoid, or, the P atrix in the Lyaunov function. We will develo a ethod to raise the liit by suitably choosing the P atrix. This note is organized as follows. Section II shows that the axial convergence control is a bang bang tye control with a sile switching schee that it roduces the axial invariant ellisoid of a given shae. A ethod for erining the largest invariant ellisoid is also given in this section. Section III reveals soe roerties liitations about the overall convergence rate rovides ethods to deal with these liitations. A brief concluding reark is ade in Section IV. Throughout this note, we will use stard notation. or a vector u 2 R, we use juj1 to denote the 1-nor. We use sat(1) to denote the stard saturation function, i.e., fsat(s)g i = sign(s i ) inf1; js i jg. We use sign(1) to denote the sign function which takes value +1or01. II. MAXIMAL CONVERGENCE RATE CONTROL AND MAXIMAL INVARIANT ELLIPSOID Consider a linear syste subject to actuator saturation _x = Ax + Bu x 2 R n u 2 R juj1 1: (2) Assue that the syste is stabilizable that B has a full-colun rank. Denote the ith colun of B as b i. In this note, we study the convergence rate of a quadratic Lyaunov function. Given a ositive definite atrix P > 0, let V (x) =x T Px. or a ositive nuber, the level set associated with V (x) is the ellisoid Along the trajectory of (2) E(P; ) =fx 2 R n : x T Px g: _V (x; u) =2x T P (Ax + Bu) = x T (A T P + PA)x +2 x T Pb i u i : Under the constraint that juj1 1, the control that axiizes the convergence rate, or iniizes _ V (x; u), is sily u i = 0sign bi T Px ; i =1; 2;...;: (3) Under this bang bang control, we have _V (x) =x T (A T P + PA)x 0 2 x T Pb i sign bi T Px : Now, consider the closed-loo syste _x = Ax 0 b i sign b T i Px : (4) /03$ IEEE

2 1250 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 The discontinuity of the bang bang control ay cause technical robles like nonexistence of solution. Since this roble can be hled by using a high gain saturated feedback to relace the bang bang control (see [7] for ore ail), in what follows, we use the bang bang control law to investigate the ossibility that an ellisoid can be ade invariant with a bounded control juj1 1. In the sequal, we sily say a bounded control. Recall that an ellisoid E(P; ) is invariant for a syste _x = f (x) if all the trajectories starting fro it will stay inside of it. It is contractively invariant if _V (x) =2x T Pf(x) < 0 8 x 2E(P; ) nf0g: Since the bang bang control (3) iniizes _ V (x; u) at each x,wehave the following obvious fact. act 1: An ellisoid E(P; ) can be ade contractively invariant for (2) with a bounded control if only if it is contractively invariant for (4), i.e., the following condition is satisfied: _V (x) =x T (A T P + PA)x 0 2 x T Pb i sign b T i Px < 0 8 x 2E(P; ) nf0g: (5) It is clear fro act 1 that the axial convergence rate control roduces the axial invariant ellisoid. or an arbitrary atrix P> 0, there ay exists no such that E(P; ) can be ade invariant. In what follows, we give a condition on P such that E(P; ) can be ade invariant for soe rovide a ethod for finding the largest. Proosition 1: or a given atrix P> 0, the following three stateents are equivalent. a) There exists a >0such that (5) is satisfied. b) There exists an 2 R 2n such that (A + B) T P + P (A + B) < 0: (6) c) There exists a k>0such that (A 0 kbb T P ) T P + P (A 0 kbb T P ) < 0: (7) Proof: b)! a). If (6) is satisfied, then there exists a >0such that E(P; ) fx 2 R n : jxj 1 1g : If x 0 2E(P; ), then under the control u = x, x(t) will stay in E(P; ) we also have juj 1 1 for all t 0. This eans that E(P; ) can be ade contractively invariant with a bounded control. Hence, by act 1, we have (5). c)! b). It is obvious. a)! c). Let us assue that PB = 0 R, where R is an 2 nonsingular atrix. If not so, we can use a state transforation, x = Tx, with T nonsingular such that P! P =(T 01 ) T PT 01 B! B = TB PB! P B =(T 01 ) T PB = 0 R : Recall that we have assued that B has a full-colun rank. Also, let us accordingly artition x as x x A T P + PA P as A T P + PA = Q 1 Q 12 Q T 12 Q 2 P = P 1 P 12 P T 12 P 2 : or all x = x 0 ), wehavex T PB =0. So, if a) is true, then (5) holds for soe >0, which ilies that x T 1 Q 1x 1 < 0, for all x 1 such that x T 1 P 1x 1 =. It follows that Q 1 < 0. Hence, there exists a k>0 such that (A0kBB T P ) T P +P (A 0 kbb T P )= Q1 Q12 Q T 12 Q 2 0 krr T < 0: This shows that c) is true. Now assue that we have a P> 0 such that the conditions in Proosition 1 are satisfied. Given a > 0, we would like to erine if E(P; ) is contractively invariant for the closed-loo syste (4). Let us start with the single inut case. In this case, (5) silifies to _V (x) =x T (A T P + PA)x 0 2x T PBsign(B T Px) < 0 8x 2E(P; ) nf0g: (8) We clai that (8) is equivalent to x T (A T P + PA)x 0 2x T PBsign(B T Px) < 0 8 x ): (9) To see this, we consider kx for k 2 (0; 1] x ). Suose that x T (A T P + PA)x 0 2x T PBsign(B T Px) < 0: Since 02x T PBsign(B T Px) 0, wehave x T (A T P + PA)x 0 2xT PB sign(b T Px) < 0; 8 k 2 (0; 1]: k Therefore (kx) T (A T P + PA)(kx) 0 2(kx) T PBsign(B T Pkx) = k 2 x T (A T P + PA)x 0 2xT PB sign(b T Px) k < 0 for all k 2 (0; 1]. This shows that (8) is equivalent to (9). Based on this equivalence relation, we have the following necessary sufficient condition for the contractive invariance of a given ellisoid. Theore 1: Assue that =1. Suose that E(P; ) can be ade contractively invariant for soe 0 > 0. Let 1 ; 2 ;...; J > 0 be real nubers such that jp 0 A T P 0 PA P 01 PBB T P j P 0 A T =0 (10) P 0 PA B T P (A T P + PA0 j P ) 01 PB > 0: (11) Then, E(P; ) is contractively invariant for (4) iff j 0 B T P (A T P +PA0 jp ) 01 PB < 0 8 j =1; 2;...; J : If there exists no j > 0 satisfying (10) (11), then E(P; ) is contractively invariant. Here, we note that all the j s satisfying (10) are the eigenvalues of the atrix P 0 A T P + P AP 0 0I 0 01 P BB T P P 0 A T P + P AP 0 : Hence, the condition of Theore 1 can be easily checked. In the roof of Theore 1, we will use the following algebraic fact. Suose that X 1 X 4 are square atrices are nonsingular, then X 1 X 2 = (X 1 ) (X 4 0 X 3 X 01 1 X 2 ) X 3 X 4 = (X 4) (X 1 0 X 2X 01 1 X 3): (12) Proof of Theore 1: Denote g(x) = x T (A T P + PA)x 0 2x T PB. By the equivalence of (8) (9), the contractive invariance of E(P; ) is equivalent to ax g(x) : B T Px 0;x T Px = < 0: (13)

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY Since E(P; ) can be ade contractively invariant for soe >0,we ust have g(x) < 0 for all B T Px =0. In this case, the contractive invariance of E(P; ) is equivalent to that all the extrea of g(x) in the surface x T Px =, B T Px > 0, if any, are less than zero. By the Lagrange ultilier ethod, an extreu of g(x) in the surface x T Px =, B T Px > 0, ust satisfy (A T P + PA0 P )x = PB; x T Px = ; x T PB > 0 (14) for soe real nuber. And at the extreu, we have g(x) = 0 x T PB.If 0, then g(x) < 0 since x T PB > 0. So, we only need to consider >0. Now, suose that >0. ro (A T P + PA 0 P )x = PB,we conclude that (A T P + PA0 P ) 6= 0. To show this, we assue, without loss of generality, that A T P + PA= Q 1 Q 12 Q T 12 q 2 ;P = P 1 P 12 P T 12 2 ;PB= 0 r as in the roof of Proosition 1, it follows that Q 1 < 0. Since >0, Q 1 < 0 P 1 > 0, Q 1 0 P 1 is nonsingular. Let x = x x, x 2 2 R, suose that x 6= 0satisfies then (A T P + PA0 P )x = PB x 1 = 0(Q 1 0 P 1 ) 01 (Q 12 0 P 12 )x 2 0(Q T 12 0 P T 12)(Q 1 0 P 1) 01 2 (Q 12 0 P 12) +q x 2 = r: Multilying both sides with (Q 1 0 P 1) alying (12), we obtain (A T P + PA0 P )x 2 = (Q 1 0 P 1 )r: Since r 6= 0 (Q 1 0 P 1) 6= 0, we ust have (A T P + PA 0 P ) 6= 0. Thus, for all >0 x satisfying (14), we have x =(A T P + PA 0 P ) 01 PB, fro x T Px =, we obtain B T P (A T P + PA0 P ) 01 P (A T P + P AP ) 01 PB = : (15) Denote 8=P 0 A T P 0 PA, then the (15) can be written as B T P 8 01 P 8 01 PB = : By invoking (12), we obtain PB 0B T P 8 01 P 01 =0 0 0 P 01 0 BT P I 0 I PB 0 = B T P PB 0 0 P 0 I =0 P 0 A T P 0 PA P 01 PBB T P P 0 A T P 0 PA =0: This last equation is (10). Also, at the extreu, we have x T PB > 0. This is equivalent to (11) inally, at the extreu B T P (A T P + PA0 P ) 01 PB > 0: g(x) =x T (A T P + PA)x 0 2x T PB = 0 B T P (A T P + PA0 P ) 01 PB: Hence, the result of the theore follows. Recall that (8) is equivalent to (9). This ilies that there is a 3 > 0 such that E(P; ) is contractively invariant if only if < 3. Therefore, the axiu value 3 can be obtained by checking the condition of Theore 1 using bisection ethod. or systes with ultile inuts, we ay divide the ) into subsets. or exale, consider = 2, the surface of E(P; ) can be divided into the following subsets: S 1 = x 2 R n : b T 1 Px =0;b T 2 Px 0; x T Px = ; 0S 1 S 2 = x 2 R n : b T 1 Px 0; b T 2 Px =0;x T Px = ; 0S 2 S 3 = x 2 R n : b T 1 Px > 0; b T 2 Px > 0; x T Px = ; 0S 3 S 4 = x 2 R n : b T 1 Px > 0; b T 2 Px < 0; x T Px = ; 0S 4 : With this artition, E(P; ) is contractively invariant iff ax _V (x) < 0 ax _V (x) < 0 (16) all the local extrea of _ V (x) in S 3 S 4 are negative. In S 1, _ V (x) =x T (A T P + PA)x 0 2x T Pb 2. Let N 2 R n2(n01) be a atrix of rank n01 such that b T 1 PN =0, i.e., fny : y 2 R n01 g is the kernel of b T 1 P. The constraint b T 1 Px = 0can be relaced by x = Ny, y 2 R n01. Thus ax _V (x) = ax y T N T (A T P + PA)Ny 02y T N T Pb 2 : b T 2 PNy 0;y T N T PNy = : This is siilar to the otiization roble (13) in the roof of Theore 1 excet with a reduced order. The second otiization roble in (16) can be hled in the sae way. In S 3, _ V (x) =x T (A T P + PA)x 0 2x T P (b 1 + b 2 ). All the local extrea of _ V (x) in S 3 ( in S 4) can be obtained like those of g(x) in the roof of Theore 1. or systes with ore than two inuts, we need to divide the surface into closed sets like S 1 S 2 2 sets like S 3 S 4 (with all strict inequalities excet for x T Px = ). This indicates that the coutational burden increases exonentially as increases. III. OVERALL CONVERGENCE RATE We now consider (4) under the axial convergence control _x = Ax 0 b i sign(bi T Px): (17) Assue that E(P; ) is contractively invariant for (17). We would like to know the overall convergence rate in E(P; ). We will see later that as decreases (note that a trajectory goes into saller E(P; ) as tie goes by), the overall convergence rate increases but is liited by the shae of E(P; ). This liit can be raised by choosing P roerly. The overall convergence rate, denoted by, is defined by (1) in Section I. Here, we would like to exaine its deendence on, so we write () := 1 _ 2 in 0 V (x) : x 2E(P; ) nf0g : V (x) The ain results of this section are contained in the following theore. Theore 2: a) () = 1 _ 2 in 0 V (x) b) () increases as decreases. c) Let then : x T Px = : (18) 0 = in 0x T (A T P + PA)x : x T Px =1;x T PB =0 li!0 () = 0 2 : (19)

4 1252 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 Proof: a) Consider x ) k 2 (0; 1], _V (kx) 0 V (kx) = 0 k2 x T (A T P + PA)x 0 2k xt Pb i sign(bi T Pkx) k 2 x T Px = 0xT (A T P + PA)x + 2 k xt Pb i sign(bi T Px) x T : (20) Px Since x T Pb i sign(b T i Px) 0 0( _ V (kx))=(v (kx)) increases as k decreases. It follows that the inial value of 0( _ V (x))=(v (x)) is obtained on the boundary of E(P; ), which ilies (18). b) This follows fro the roof of a). c) ro a), we see that 2() = in 0xT (A T P + PA)x +2 : x T Px = = in 0x T (A T P + PA)x + 2 in + 2 x T Pb i sign(b T i Px):x T Px =1 0x T (A T P + PA)x xt Pb i sign(b T i Px) x T Pb i sign(b T i Px):x T Px =1; x T PB =0 = in 0x T (A T P + PA)x : x T Px =1; x T PB =0 = 0 : It then follows that 2() 0 for all >0. To rove (19), it suffices to show that given any ">0, there exists a >0 such that 2() 0 0 ". Denote X 0 = x 2 R n : x T Px =1; x T PB =0 X () = x 2 R n : x T Px =1; jx T PBj1 : It is clear that li!0 dist(x (); X 0 ) = 0, where dist(1; 1) is the Hausdorff distance. 1 By the unifor continuity of x T (A T P + PA)x on the surface fx 2 R n : x T Px = 1g, we have that, given any ", there exists a >0 such that in 0x T (A T P +PA)x : x T Px=1; jx T PBj1 0 0 ": (21) 1 Let X X be two bounded subsets of R. Their Hausdorff distance is defined as where dist(x 1; X 2):=ax ~ d(x 1; X 2); ~ d(x 2; X 1) ~d(x 1; X 2)= su x 2X Here, the vector nor used is arbitrary. inf x 2X jx1 0 x2j: Since xt Pb isign(b T i Px) 0, wehave in 0x T (A T P + PA)x + 2 for all >0. Let x T Pb i sign(b T i Px): x T Px =1; jx T PBj1 0 0 " (22) 1 = in 0x T (A T P + PA)x : x T Px=1; jx T PBj1 : If ", then for all >0 in 0x T (A T P +PA)x+ 2 Cobining this with (22), we have x T Pb i sign(b T i Px) : x T Px =1; jx T PBj ": 2() = in 0x T (A T P + PA)x x T Pb i sign(bi T Px):x T Px =1 0 0 ": (23) for all >0. This shows that 2() 0 0 ", 8 >0. If 1 < 0 0 ", then for <((2)=( ")) 2,wehave in 0x T (A T P +PA)x+ 2 x T Px =1; jx T PBj1 x T Pb i sign(b T i Px): > 0 0 ": Cobining this with (22), we also obtain (23) 2() 0 0 " for <(2=( ")) 2. This coletes the roof. Theore 2 says that the convergence rate increases as the ellisoid becoes saller it aroaches an uer bound 0 =2 as goes to 0. Hence, the convergence rate is bounded by 0 =2 for all. or a given, () can be obtained by couting the axiu of _ V (x) ). or the single inut case, Theore 1 rovides a ethod for erining if this axiu is negative. The exact value of () can be couted with a rocedure siilar to the roof of Theore 1. Since the overall convergence rate is liited by 0 =2, we would like 0 not to be too sall. The following roosition will lead to an LMI aroach to choosing P for axiizing 0. Proosition 2: 0 = su s.t. (A + B) T P + P (A + B) 0P: (24) Proof: Notice that, for any,wehave x T (A + B) T P + P (A + B) x = x T (A T P + PA)x It follows that, hence 0 = in 0x T (A + B) T P +P (A + B)) x : x T Px =1; x T PB =0 in 0x T (A + B) T P +P (A + B)) x : x T Px =1 8 x T PB =0: 0 su in 0x T (A + B) T P + P (A + B) x : x T Px =1 : (25)

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY We clai that 0 =suin 0x T (A + B) T P + P (A + B) x : x T Px =1 : (26) In view of (25), it suffices to show that for any ">0, there exists an = 0kB T P, with k>0, such that in 0x T (A + B) T P + P (A + B) x : x T Px =1 0 0 ": (27) This can be roven by exloiting the sae idea used in the roof of Theore 2 c). Denote ( ) = in 0x T (A + B) T P + P (A + B) x : ro (26), we have 0 = su ( ). It can be shown that x T Px =1 : ( )=ax :(A + B) T P + P (A + B) 0P : This brings us to (24). The atrix inequality constraint in Proosition 2 indicates that the ellisoid E(P; ) has a convergence rate =2 under the linear state feedback u = x(for any >0). ro Proosition 2 Theore 2, we see that the convergence rate is liited by the axial value which can be achieved with a linear state feedback can actually aroach this axial value as goes to 0. or a fixed P, 0 is a finite value. Assue that (A, B) is controllable, then the eigenvalues of (A + B) can be arbitrarily assigned. If we also take P as a variable, then 0 0=2 can be ade equal to the largest real art of the eigenvalues of A +B (see the definition, as given in Section I, of the overall convergence rate for a linear syste). This eans that 0 can be ade arbitrarily large. But generally, as 0 becoes very large, the atrix P will be badly conditioned, i.e., the ellisoid E(P; ) will becoe very thin in certain direction, hence very sall, with resect to a fixed shae reference set. On the other h, as entioned in [8] [9], if our only objective is to enlarge the doain of attraction with resect to a reference set, soe eigenvalues of A + B will be very close to the iaginary axis, resulting in very sall 0. These two conflicting objectives can be balanced, for exale, by resecifying a lower bound on 0 then axiizing the invariant ellisoid with resect to soe shae reference set. This ixed roble can be described as follows: with two feedback atrices H. We ay actually discard both H but instead use the bang bang control u i = 0sign(b T i Px), i = 1; 2;...;, or the saturated high-gain control u = 0sat(kB T Px). The final outcoe is that under these control laws, the closed-loo syste will have a contractively invariant set E(P; ) a guaranteed liit of the convergence rate 0 =2 =2. IV. CONCLUSION We have studied several issues related to the axiization of the convergence rate for continuous-tie linear systes with inut saturation. These issues include the axial convergence control, the axial ellisoid that can be ade invariant with a bounded control, the control laws that can roduce the axial invariant ellisoid the deendence of the axial convergence rate on the Lyaunov function. The counterart of these issues for discrete-tie syste are studied in [7], where soe of the results are quite different fro those in this note. or exale, the axial convergence control is continuous in the state is erined in a quite different way. REERENCES [1] D. S. Bernstein A. N. Michel, Eds., Int. J. Robust Nonlinear Control: Secial Issue on Saturating Actuators, 1995, vol. 5. [2]. Blanchini, Set invariance in control a survey, Autoatica, vol. 35, no. 11, , [3] S. Boyd, L. El Ghaoui, E. eron, V. Balakrishnan, Linear Matrix Inequalities in Systes Control Theory. Philadelhia, PA: SIAM, [4] J. M. Goes da Silva Jr. S. Tarbouriech, Local stabilization of discrete-tie linear systes with saturating controls: an LMI aroach, IEEE Trans. Autoat. Contr., vol. 46, , Jan [5] P. O. Gutan M. Cwikel, Adissible sets feedback control for discrete-tie linear dynaical systes with bounded control dynaics, IEEE Trans. Autoat. Contr., vol. AC-31, , Ar [6] H. Hindi S. Boyd, Analysis of linear systes with saturating using convex otiization, in Proc. 37th IEEE Conf. Decision Control, Orlo, L, 1998, [7] T. Hu Z. Lin, Control Systes With Actuator Saturation: Analysis Design. Boston, MA: Birkhäuser, [8], On enlarging the basin of attraction for linear systes under saturated linear feedback, Syst. Control Lett., vol. 40, no. 1, , May [9] T. Hu, Z. Lin, B. M. Chen, An analysis design ethod for linear systes subject to actuator saturation disturbance, Autoatica, vol. 38, no. 2, , [10] E. B. Lee L. Markus, oundations of Otial Control. New York: Wiley, [11] J. Macki M. Strauss, Introduction to Otial Control. New York: Sringer-Verlag, [12] G.. Wredenhagen P. R. Belanger, Piecewise-linear LQ control for systes with inut constraints, Autoatica, vol. 30, , su P>0;;;H s.t. a) X R E(P; ) b) (A + B) T P + P (A + B) < 0 c) E(P; ) fx 2 R n : jxj1 1g d) (A + BH) T P + P (A + BH) 0P (28) where X R is a shae reference set (see [7] [9]). The constraint a) eans that E(P; ) contains X R. By axiizing, E(P; ) will be axiized with resect to X R. The constraints b) c) guarantee that E(P; ) can be ade contractively invariant the constraint d) guarantees a lower bound on the convergence rate. This otiization roble can be transfored into one with LMI constraints as those in [7] [9]. By solving (28), we obtain the otial ellisoid E(P; ) along

Control and Stability of the Time-delay Linear Systems

ISSN 746-7659, England, UK Journal of Inforation and Couting Science Vol., No. 4, 206,.29-297 Control and Stability of the Tie-delay Linear Systes Negras Tahasbi *, Hojjat Ahsani Tehrani Deartent of Matheatics,