Design of time-varying controllers for discrete-time linear systems with input saturation J.M. Gomes da Silva Jr., F. Lescher and D. Eckhard Abstract: A method for computing time-varying dynamic output feedback controllers for discrete- time linear systems subject to input saturation is proposed. The method is based on a locally valid polytopic representation of the saturation term. From this representation, it is shown that, at each sampling time, the matrices of the stabilising time-varying controller can be computed from the current system output and from constant matrices obtained as a solution of some matrix inequal- ities. Linear matrix inequality-based optimisation problems are therefore proposed in order to compute the controller aiming at the maximisation of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of L 2 disturbance tolerance and rejection. 1 Introduction The physical impossibility of applying unlimited control signals makes the actuator saturation an ubiquitous problem in control systems. In particular, it is well known that the input saturation is a source of performance degra- dation, limit cycles, different equilibrium points and even instability. Hence, there has been a great interest in studying these negative effects and also in proposing control design procedures taking directly into account the control bounds ([1–4] and references therein). Most of these works consider state feedback control laws. Although the proposition of state feedback methods allows a good insight into the problem, the practical applicability of these methods is limited. Considering output feedback solutions, fewer works are found in the literature. Most of them focus on the determination of global or semi-global stabilising controllers [5]. The main drawback of these results is that they can only be applied to null-controllable systems. Moreover, when performance or robustness requirements must be satisfied, it can be impossible to achieve global or semi-global stability. On the other hand, we found very few works dealing with the synthesis of local stabilising controllers via output feedback. Gomes da Silva Jr. et al. [6] have proposed observer-based control laws. The main problem is that the solutions consider particular quadratic Lyapunov functions (the P matrix should be block diagonal) that lead, in general, to very conservative solutions. Tyan and Bernstein [7] proposed a method for designing dynamic output controllers using the positive real lemma. The main objective pursued in that paper was the minimisation of a linear quadratic Gaussian (LQG) criterion. A region of stability is associated with the closed-loop system. However, it should be pointed out that the size and the shape of this region are not taken into account in the design procedure, which can lead to very con- servative domains of stability. Furthermore, the controller is computed from the solution of strongly-coupled Riccati equations which, in general, are not simple to solve. Nguyen and Jabbari [8] proposed a time-varying dynamic controller. As the proposed approach considers only continuous-time systems, its main drawback resides in the fact that the stability properties cannot be ensured if the controller is discretised for a digital implementation. Furthermore, in that paper, no explicit consideration was made about the region of attraction associated with the con- troller or about the internal stability of the system. Kiyama and Iwasaki [9] studied the conditions for the synthesis of stabilising dynamic feedback controllers based on classical sector conditions. These conditions lead to an indirect procedure for computing the controller that is not explicitly discussed here. It was also shown that in the considered case, the use of saturating control laws does not help in obtaining larger regions of stability. It is, however, very important to highlight the fact that no constraints on the performance were taken into account in that analysis. Gomes da Silva Jr. et al. [10] have shown that, although the optimal region of stability was obtained with a linear control law, the closed-loop poles associated with this solution can be very close to the imaginary axis, which implies a very slow behaviour. On the other hand, it should be pointed out that all of the above studies were concerned only with continuous-time systems. The aim of this work is the proposition of a technique for the design of stabilising dynamic output feedback control- lers for discrete-time linear systems in the presence of satur- ating actuators. First, the problem of computing a controller in order to ensure the local (regional) asymptotic stability of the closed-loop system is focused upon. The derived conditions can therefore be used with the aim of enlarging the region of attraction of the closed-loop system while ensuring a certain degree of time-domain performance for the system operation in a neighbourhood of the origin (equilibrium point). Secondly, considering that the system is subject to the action of L 2 bounded disturbances, the problems of disturbance tolerance and attenuation are addressed. In this case, conditions for ensuring the closed-loop system state trajectories are bounded, and the # The Institution of Engineering and Technology 2006 doi:10.1049/iet-cta:20050415 Paper first received 17th October 2005 and in revised form 27th February 2006 J.M. Gomes da Silva Jr. and D. Eckhard are with the Department of Electrical Engineering, UFRGS, Av. Osvaldo Aranha 103, Porto Alegre (RS) 90035-190, Brazil F. Lescher is with EIGSI-ERPA, 26 rue de Vaux le Foletier, La Rochelle 17000, France E-mail: jmgomes@ece.ufrgs.br IET Control Theory Appl., Vol. 1, No. 1, January 2007 155