Systems & Control Letters 45 (2002) 97–112 www.elsevier.com/locate/sysconle Analysis and design for discrete-time linear systems subject to actuator saturation  Tingshu Hu a ; ∗ , Zongli Lin a , Ben M. Chen b a Department of Electrical & Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA b Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576, Singapore Received 14 February 2001; received in revised form 7 July 2001; accepted 24 August 2001 Abstract We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. 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. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback which makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements. c  2002 Elsevier Science B.V. All rights reserved. Keywords: Actuator saturation; Stability analysis; Disturbance rejection; Set invariance 1. Introduction In this paper, we are interested in the control of linear systems subject to actuator saturation and persistent disturbances, x(k +1)= Ax(k )+ Bsat(u(k ))+ Ew(k ); x ∈ R n ;u ∈ R m ;w ∈ R q ; (1) where x is the state, u is the control, w is the disturbance and sat(·) is the standard saturation function. First, we will consider the closed-loop stability under a given linear state feedback u = Fx in the absence of the disturbance. There has been a lot of work on this topic (see, e.g. [3–5,9–14] and the references therein). For the continuous-time case, 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. [5,9,10,12], where the saturation is treated as a locally sector bounded nonlinearity and the domain of attraction is estimated by use of quadratic and Lur’e type Lyapunov functions). The multivariable circle criterion in [9]  This work was supported in part by the US Oce of Naval Research Young Investigator Program under grant N00014-99-1-0670. * Corresponding author. E-mail addresses: th7f@virginia.edu (T. Hu), zl5y@virginia.edu (Z. Lin), bmchen@nus.edu.sg (B.M. Chen). 0167-6911/02/$-see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0167-6911(01)00168-2