Automatica 38 (2002) 945 949 www.esevier.com/ocate/automatica Brief Paper Input-to-state stabiity for a cass of Lurie systems Murat Arcak a;, Andrew Tee b a Department of Eectrica, Computer and Systems Engineering, Rensseaer Poytechnic Institute, Troy, NY 280-3590, USA b Department of Eectrica and Computer Engineering, University of Caifornia, Santa Barbara, CA 9306-9560, USA Received 2 March 2002; accepted 30 May 2002 Abstract We anayze input-to-state stabiity (ISS) for the feedback interconnection of a inear bock and a noninear eement. This study is of importance for estabishing robustness against actuator noninearities and disturbances. In the absoute stabiity framework, we prove ISS from a positive rea property of the inear bock, by restricting the sector noninearity to grow unbounded as its argument tends to innity. When this growth condition is vioated, exampes show that the ISS property is ost. The resut is used to give a simpe proof of boundedness for negative resistance osciators, such as the van der Po osciator. In a separate appication, we reax the minimum phase assumption of an earier boundedness resut for systems with noninearities that grow faster than inear.? 2002 Esevier Science Ltd. A rights reserved. Keywords: Noninear systems; Absoute stabiity; Robustness to disturbances. Introduction The absoute stabiity probem, formuated by Lurie and coworkers in the 940s, has been a we-studied and fruitfu area of research, as presented in Aizerman and Gantmacher (964), and Narendra and Tayor (973). Severa stabiity criteria have been deveoped for the system in Fig. which consists of the inear bock H(s)=C(sI A) B in feedback with the noninearity ( ). These criteria make use of the input output properties of the inear bock H(s), and characterize casses of noninearities which ensure stabiity. The most we known among them is the circe criterion of Sandberg (964a, b) and Zames (966), which proves inear-gain L p -stabiity (p [; ]) from the disturbance d to the output y, when the noninearity ( ) satises the sector property y(y) 0 y 0 () and H(s) isstricty positive rea (SPR). However, the SPR property requires that H(s) be asymptoticay stabe and An abridged version of this paper was presented at the 2002 American Contro Conference, Anchorage, Aaska. This paper was recommended for pubication in revised form by Associate Editor Zhihua Qu under the direction of Editor Hassan Khai. Corresponding author. Te.: (58)276-6535; fax: (58)276-626. E-mai addresses: arcakm@rpi.edu (M. Arcak), tee@ece.ucsb.edu (A. Tee). minimum phase, and disaows a arge cass of systems with poes or zeros on the imaginary axis. With a ess restrictive positive rea (PR) condition Tee (2002) showed that inear-gain stabiity is ost, and instead, derived noninear gain functions for L p -stabiity with p. In this paper we study L -stabiity of the feedback system in Fig., within the input-to-state stabiity (ISS) framework of Sontag (989). The main resut (Theorem ) estabishes ISS when H(s) is PR and the sector noninearity ( ) grows unbounded as its argument goes to innity y (y) : (2) This growth condition is crucia for the ISS property because, with the saturation function sat(y):=sgn(y)min {; y }, which vioates (2), the soutions of the system ẏ = sat(y)+d (3) are driven to innity by a constant disturbance d. This means that system (3), which is as in Fig. with H(s)==s, is not ISS because the saturation function is not powerfu enough to counteract arge disturbances (see Liu, Chitour, & Sontag (996) for a detaied discussion on saturation functions, and an ISS property with respect to sma disturbances). Our ISS resut, presented in Section 2, is used in Section 3 to prove boundedness for a cass of systems that encompasses negative resistance osciators (Khai, 2002, 0005-098/02/$ - see front matter? 2002 Esevier Science Ltd. A rights reserved. PII: S0005-098(02)0000-0
946 M. Arcak, A. Tee / Automatica 38(2002) 945 949 thus; (8) hods with =. For mutivariabe noninearities; (7) does not impy (8). A counterexampe is (y):=y + y 3 Jy; (9) where J satises J + J T = 0 and J T J = I. In this exampe; y T (y)= y 2 and (y) = y 2 + y 8 ; which mean that (7) is satised with ( y )= y ; but (8) is vioated. Fig.. The Lurie system (4) (5), represented as the feedback interconnection of the inear bock H(s) and the noninearity ( ). Exampe 2.9). In Section 4, we study reative degree one systems in feedback with a noninearity that grows faster than inear. For such systems we prove boundedness with a PR condition on the zero dynamics, which reaxes an earier minimum phase assumption in Arcak, Larsen, and Kokotovic (2002). Concusions are given in Section 5. To ensure existence of soutions, henceforth the noninearity ( ) is assumed to be continuous, and the disturbance d is assumed to be a measurabe function of time. 2. Main resut We now study the system ẋ = Ax + B[ (y)+d]; (4) y = Cx; (5) which is as in Fig. with H(s)=C(sI A) B, and prove ISS with respect to d for a cass of mutivariabe noninearities ( ):R m R m dened by inequaities (7) and (8) beow. Theorem. Consider system (4) (5); where x R n ; ( ):R m R m ; and (C; A) is detectabe. If there exist a matrix P = P T 0 satisfying A T P + PA 6 0; PB= C T (6) a constant 0; and a cass-k function ( ); such that y ( y ) 6 y T (y) for a y R m ; (7) (y) 6 y T (y) when y (8) then the system is ISS with respect to d. Remark. When (A; B; C) is a minima reaization a straightforward modication of the Positive Rea Lemma (Khai; 2002; Lemma 6.2) for P 0 shows that our assumption (6) is equivaent to the PR property of H(s)=C(sI A) B. For a more genera resut; in Theorem we aow nonminima reaizations and ony restrict (C; A) to be detectabe. Remark 2. For scaar noninearities ( ):R R condition (7) is equivaent to the sector property () and the growth condition (2). For scaar noninearities condition (8) is redundant because (7) impies y T (y) = y (y) and; Proof of Theorem. Before proceeding with the proof of ISS; we caim from (7) and (8) that there exist a constant 0 and a cass-k function ( ); satisfying ( (y) + y ) 6 y T (y) when y ; (0) ( y ) y 2 + ( (y) ) (y) 2 6 y T (y) when y 6 : () Inequaity (0) hods with 6 2 min{; ()}; because; from (8) and (7); y impies (y) 6 2 yt (y); (2) y 6 2 () y 6 2 ( y ) y 6 2 yt (y): (3) To prove (); we et u ( ) be a cass-k function such that (y) 6 u ( y ) for a y R m ; and show that we can seect a cass-k function ( ) to satisfy ( y ) y 2 + ( u ( y )) u ( y ) 2 6 y ( y ) when y 6 : (4) To this end; we note that the eft-hand side of (4) satises ( y ) y 2 + ( u ( y )) u ( y ) 2 6 (( y )+( u ( y )))( y 2 + u ( y ) 2 ) 6 2(p( y ))q( y ); (5) where p( y ):= y + u ( y ) and q( y ):= y 2 + u ( y ) 2. Thus; by seecting ( ) such that; when y 6 ; 2(p( y ))q( y ) y ( y ) (6) we ensure from (5) that (4) hods. Finay; (4) impies () because of (7). To proceed with the proof of ISS, we et f(x; d) denote the right-hand side of (4); that is, ẋ = f(x; d):=ax B[(B T Px)+d] (7) and construct a positive denite, radiay unbounded ISS-Lyapunov function V (x) satisfying V (x);f(x; d) 6 ( x )+( d ) (8) for some cass-k functions ( ) and ( ). The rst component of our ISS-Lyapunov function is V 0 (x):=x T Px; (9) which, from (6), satises V 0 (x);f(x; d) 6 2y T (y)+2y T d (20)
M. Arcak, A. Tee / Automatica 38(2002) 945 949 947 with y := B T Px = Cx: (2) Considering the two cases d 6 2 ( y ) and d 2 ( y ), and using (7), we obtain the inequaity 2y T d 6 2 y d 6 y ( y )+2 (2 d ) d 6 y T (y)+2 (2 d ) d ; (22) which resuts in V 0 (x);f(x; d) 6 y T (y)+2 (2 d ) d : (23) Because y T (y) is ony a semidenite function of x in genera, we wi add to V 0 (x) another term V (x), which renders the right-hand side negative denite in x. To this end, we rst rewrite (7) as ẋ =(A LC)x + Ly + B[ (y)+d]; (24) where L is chosen so that A LC is Hurwitz. Such a matrix exists since the pair (C; A) is detectabe. Next, we et V (x):=(x T P x); (25) where P = P T 0 is such that (A LC) T P + P (A LC) 6 I (26) and s { } (s):= min ; ;() d; (27) 0 where the constant 0 and the cass-k function ( ) are to be specied. By construction, V ( ) is positive denite and radiay unbounded. To prove that V (x):=v 0 (x)+v (x) (28) is an ISS-Lyapunov function as in (8), we et k 0be such that 2 max{ B T P x ; L T P x } 6 k x, and note from (24) that V (x);f(x; d) 6 (x T P x)[ x 2 + k x ( (y) + y + d )]: (29) Because () 6 =, we can nd a constant c 0, independent of, such that (x T P x)k x 6 c; x R n (30) and obtain V (x);f(x; d) 6 (x T P x) x 2 + c( (y) + y + d ): (3) Choosing 6 =c in (27), and using (0), we ensure that y V (x);f(x; d) 6 (x T P x) x 2 + y T (y)+ d : (32) To show that a simiar estimate hods when y 6, we denote by the maximum eigenvaue of P, and note that (x T P x)k x (y) 6 4 (x T P x) x 2 +4k 2 (y) 2 (6 k 2 (y) 2 ) (33) (consider the two cases (y) 6 =4k x and x 6 4k (y), and use () 6 () from (27)), and (x T P x)k x y 6 4 (x T P x) x 2 +4k 2 y 2 (6k 2 y 2 ) (34) (consider the two cases y 6 =4k x and x 6 4k y ). Thus, with the choice ():= ( ) 4k 2 6k (35) 2 substitution of (33) and (34) in (29) resuts in V (x);f(x; d) 6 2 (x T P x) x 2 + ( y ) y 2 + ( (y) ) (y) 2 + d ; (36) which, from (), impies y 6 V (x);f(x; d) 6 2 (x T P x) x 2 + y T (y)+ d : (37) Finay, noting from (27) that ()= = for sucienty arge, we can nd a cass-k function ( ) such that 2 (x T P x) x 2 ( x ) x R n : (38) Thus, it foows from (32) and (37) that, for a vaues of x and d, V (x);f(x; d) 6 ( x )+y T (y)+ d : (39) Adding inequaities (23) and (39), we concude that V (x)= V 0 (x)+v (x) is an ISS-Lyapunov function, because it satises (8) with ( d ):= d +2 (2 d ) d. In the absence of the disturbance d, when goba asymptotic stabiity of the origin of (4) (5) is of interest, the proof technique used above can be viewed as an aternative to using LaSae s invariance principe. The proof of Theorem expoits the detectabiity property that woud be used in LaSae s invariance theorem, instead to prove ISS in the presence of d. Indeed, the second component V (x) ofour ISS-Lyapunov function transates the detectabiity property into the Lyapunov inequaity (39), which renders V negative denite in x. (See Angei (999) for a reated approach to proving ISS from detectabiity.) 3. Boundedness of negative resistance osciators We now use Theorem to give a simpe proof of boundedness for the negative resistance osciator ẋ = x 2 ; ẋ 2 = x (x 2 ) (40) (Khai, 2002, Exampe 2.9), where the noninearity ( ) satises (0)=0, (0) 0, (y) as y and (y) as y : (4)
948 M. Arcak, A. Tee / Automatica 38(2002) 945 949 This cass of noninearities diers from that considered in Theorem, because the sector condition () is vioated around the origin due to (0) 0. Athough the origin is unstabe, (4) ensures boundedness of trajectories, as proved by Levinson and Smith (942), Cartwright (950), LaSae and Lefschetz (96), Mier and Miche (982), and Khai (2002). We now use the ISS resut of Theorem to generaize this boundedness resut to the arger cass (4) (5), which incudes higher order systems and bounded disturbances: Theorem 2. Consider system (4) (5); where x R n ; ( ):R R; (C; A) is detectabe; and d is a bounded disturbance. If there exists a matrix P = P T 0 satisfying (6); and if the noninearity ( ) satises (4); then the trajectories are bounded. Proof. We rst note from (4) that we can nd a constant a 0 such that y a y(y) 0: (42) Then; we et (y) be a continuous function such that (y)= (y) when y a (43) and y (y) 0 for a y 0. It foows that () and (2) hod for ; the atter because of (4); so that; according to Remark 2; ( ) satises the assumptions of Theorem. Rewriting (4) (5) as ẋ = Ax B[ (y)+ d(y)+d]; (44) where d(y):=(y) (y) is bounded because d(y) =0 when y a; we concude from Theorem that (44) is ISS with respect to the bounded disturbance d(y)+ d and; hence; the trajectories are bounded. Theorem 2 encompasses the negative resistance osciator (40), which is of the form (4) (5) and (6), with d =0, 0 0 A = ; B= ; C =[0 ] and P = I: 0 (45) It is important to note that, by computing the ISS gain function ( ) from the proof of Theorem, we can aso obtain an asymptotic upper bound on x(t) ( ) im sup x(t) 6 t d 0 + im sup d(t) t d 0 := max { d(y) }: (46) y 6a Thus, our ISS approach not ony proves boundedness, but aso gives an estimate for the size of the imit cyce. 4. Boundedness in systems with stiening noninearities In this section we use Theorem 2 to extend an earier resut, (Arcak et a., 2002, Theorem ), which estabishes ; boundedness of trajectories for a reative degree one, minimum phase, inear bock, in feedback with a stiening noninearity ( ), dened by the property im y (y) y + : (47) Using the Isidori norma form Isidori (995) for reative degree one systems, this feedback interconnection is expressed as ż = A 0 z + B 0 y; (48) ẏ = C 0 z ay (y)+d; (49) where the z-subsystem represents the zero dynamics of the inear bock. The proof in Arcak et a. (2002) reies on the minimum phase property; that is, the eigenvaues of A 0 are assumed to be in the open eft haf-pane. We now prove boundedness for a cass of systems with imaginary axis zeros: Theorem 3. Consider system (48) (49); where d is a bounded disturbance; (C 0 ;A 0 ) is a detectabe pair; and there exists a matrix P 0 = P0 T 0 such that A T 0 P 0 + P 0 A 0 6 0; P 0 B 0 = C0 T : (50) If the noninearity ( ):R R satises the stiening property (47); then the trajectories are bounded. Proof. To appy Theorem 2; we rewrite (48) (49) as in (4) (5); with A0 B A = 0 0 ; B= ; C = [0 ] (5) C 0 0 and (y):=(y)+ay: (52) We note that [ ] P0 0 P = (53) 0 satises (6) because of (50); and (C; A) is detectabe because (C 0 ;A 0 ) is detectabe. Likewise; (y) satises (4) because; from (47) (y) im + : (54) y y The conditions of Theorem 2 being satised; we concude that the trajectories of system (48) (49) are bounded. Our positive rea assumption (50) for the zero dynamics is ess restrictive than the minimum phase assumption of Arcak et a. (2002), as iustrated in the foowing exampe: Exampe. Consider the feedback interconnection in Fig. with H(s)= s 2 + s 3 s 2 +2s (55)
M. Arcak, A. Tee / Automatica 38(2002) 945 949 949 Fig. 2. Limit cyce in system (55) (56), projected onto the (z ;z 2 )-pane (eft), and the (z 2 ;y)-pane (right). and the stiening noninearity (y)=y 3 : (56) The boundedness resut of Arcak et a. (2002) is not appicabe because H(s) has a pair of zeros on the imaginary axis. To appy Theorem 3; we note that H(s) is reative degree one; and rewrite system (55) (56) as in (48) (49); with d = 0 and 0 0 A 0 = ; B 0 0 = ; C 0 =[0 ]; a= : (57) The origin is unstabe from the Jacobian inearization. However; because (50) hods with P 0 = I; Theorem 3 ensures boundedness. Numerica simuations indicate that the trajectories converge to one of the two stabe equiibria (z ;z 2 ;y)= (; 0; ); or the imit cyce in Fig. 2; projected onto the (z ;z 2 )-pane (eft), and the (z 2 ;y)-pane (right). 5. Concusion We have estabished ISS for the Lurie system in Fig., by restricting the inear bock to be PR, and the sector noninearity in the feedback oop to grow unbounded. A usefu extension of our resut woud be to further reax the PR assumption using mutipier methods, such as the Popov and Zames Fab mutipiers. These extensions woud aso enarge the casses of systems for our boundedness resuts, Theorems 2 and 3. Angei, D. (999). Input-to-state stabiity of PD-controed robotic systems. Automatica, 35, 285 290. Arcak, M., Larsen, M., & Kokotovic, P. (2002). Boundedness without absoute stabiity in systems with stiening noninearities. European Journa of Contro, 8(3). Cartwright, M. L. (950). Forced osciations in noninear systems. In S. Lefschetz (Ed.), Contributions to the theory of noninear osciations (pp. 49 24). Princeton: Princeton University Press. Isidori, A. (995). Noninear contro systems (3rd ed.). Berin: Springer. Khai, H. K. (2002). Noninear systems (3rd ed.). Upper Sadde River, NJ: Prentice-Ha. LaSae, J., & Lefschetz, S. (96). Stabiity by Liapunov s direct method with appications. New York: Academic Press. Levinson, N., & Smith, O. K. (942). A genera equation for reaxation osciations. Duke Mathematica Journa. Liu, W., Chitour, Y., & Sontag, E. (996). On nite-gain stabiizabiity of inear systems subject to input saturation. SIAM Journa of Contro and Optimization, 34, 90 29. Mier, R. K., & Miche, A. N. (982). Ordinary dierentia equations. New York: Academic Press. Narendra, K. S., & Tayor, J. (973). Frequency domain methods in absoute stabiity. New York: Academic Press. Sandberg, I. W. (964a). A frequency domain condition for the stabiity of systems containing a singe time-varying noninear eement. The Be System Technica Journa, 43, 60 638. Sandberg, I. W. (964b). On the L 2 -boundedness of soutions of noninear functiona equations. The Be System Technica Journa, 43, 58 599. Sontag, E. D. (989). Smooth stabiization impies coprime factorization. IEEE Transactions on Automatic Contro, 34, 435 443. Tee, A. R. (2002). On absoute L p stabiity with noninear gain. Systems and Contro Letters, submitted for pubication. Zames, G. (966). On the input output stabiity of time-varying noninear feedback systems Parts I and II. IEEE Transactions on Automatic Contro,, 228 238 and 465 476. References Aizerman, M. A., & Gantmacher, F. R. (964). Absoute stabiity of reguator systems. San Francisco: Hoden-Day. (Transated from the Russian origina, Moscow: Akademiya Nauk SSSR) 963.