Finite Gain l p Stabilization of Discrete-Time Linear Systems Subject to Actuator Saturation: the Case of p =1 † Yacine Chitour ‡ Zongli Lin † D´ epartement de math´ ematiques, Universit´ e de Paris-Sud, Orsay-91405, France. Email: Yacine.Chitour@math.u-psud.fr ‡ Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743 Charlottesville, VA 22904-4743, U.S.A. Email: zlin@virginia.edu Abstract— It has been established by Bao, Lin and Sontag (2000) that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain l p stabilization can be achieved by linear output feedback, for every p ∈ (1, ∞] except p =1. An explicit construction of the corresponding feedback laws was given. The feedback laws constructed also resulted in a closed-loop system that is globally asymptoti- cally stable. This note complements the results of Bao, Lin and Sontag (2000) by showing that they also hold for the case of p =1. I. I NTRODUCTION This short note revisits the problem of simultaneous global asymptotic stabilization and finite gain L p (l p ) stabilization of a linear system in the presence of actuator saturation and measurement and actuator noises (see [8], [1] and the references therein). More specifically, we would like to construct a controller C so that the operator (u 1 ,u 2 ) → (y 1 ,y 2 ) as defined by the following standard systems interconnection (see Fig. 1)  y 1 = P (u 1 + y 2 ) y 2 = C (u 2 + y 1 ) (I. 1) is well-defined and finite gain stable. This problem was ❥ ✲ ❄ ✛ ✛ ✻ ❥ ✲ u 1 u 2 y 1 y 2 P C Fig. 1. Standard closed-loop connection. first studied for continuous-time systems and various results have been established for such systems. It was shown in [8] that, for neutrally stable open loop systems, linear feedback laws can be used to achieve finite gain L p stabilization, for all p ∈ [1, ∞]. Various continuity and incremental-gain properties of the resulting closed-loop 1 The work of Z. Lin was supported in part by NSF grant CMS- 0324329. system were discussed in detail in [4]. For a neutrally stable system, all open loop poles are located in the closed left-half plane, with those on the jω axis having Jordan blocks of size one. In the case that full state is available for feedback (i.e., y 1 = x and u 2 =0), it was shown in [7] that, if the external input signal is uniformly bounded, then finite-gain L p -stabilization, for any p ∈ (1, ∞] except p =1, and local asymptotic stabilization can always be achieved simultaneously by linear feedback, no matter where the poles of the open loop system are. The uniform boundedness condition of [7] was later removed for the case p =2 by resorting to nonlinear feedback [6]. More recently, the problem of L p stabilization for a double integrator system subject to input saturation feedback and disturbances that are not additive was investigated in [3], where it is considered the control system (DI ), ¨ x = σ(−x − ˙ x + u)+ v, with x ∈ IR and disturbances (u, v). For v =0 and zero initial conditions, it was established, among other results, that the L 2 -gain from u to the output of the saturation nonlinearity was finite. This partially solved Problem 36 as defined in [2]. For nonzero v, an L ∞ -bound is of course necessary to get any positive result regarding L p -stabilization. It was shown that (DI ) is L p -stable (see [3] for the precise definition of L p -stability) if and only if p ≤ 2. In other words, for p> 2, one can construct a disturbance v with finite L p -norm and arbitrarily small L ∞ -norm that results in an unbounded trajectory of (DI ). Examples of other works related to the topic are [5], [9], [10] and the references therein. The extension of the results of [8] to discrete-time systems was made in [1]. In particular, it was shown in [1] that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain l p stabilization can be achieved by linear output feedback, for every p ∈ (1, ∞] except p =1. An explicit construction of the corresponding feedback laws was given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable. The objective of this note is to complement the results of [1] by showing that they also hold true for the case of p =1. Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 FrP02-2 0-7803-7924-1/03/$17.00 ©2003 IEEE 5663