Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems Bahram Shafai, Hanai Sadaka and Rasoul Ghadami Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA. shafai@ece.neu.edu hsadaka@ece.neu.edu rghadami@ece.neu.edu Abstract: This paper revisits the problem of robust stability and stabilization of uncertain time-delay systems. We focus on the class of non-negative discrete-time delay systems and show that it is asymptotically stable if and only if an associated non-negative system without delay is asymptotically stable ⋆ . This fact allows one to establish strong result on robust stability and stability radius for this class of systems. An alternative representation of delay systems is also constructed whereby its system matrix is in block companion from. Under the assumption of non-negativity for delay systems, this alternative form represents a conventional non-negative system and similar strong robust stability results are derived. Finally, we consider the problem of constrained stabilization and provide a new LMI feasibility solution for it. This makes it possible to stabilize a general discrete-time delay system such that the closed-loop system admits non-negative structure with desirable properties. 1. INTRODUCTION The problem of stability and the control design for delay systems attracted many researchers for the past several decades and excellent books have been published in this area (Gu et al. [2003], Niculescu [2001] and Magdi [2000]). Literature reports various methods for stability and sta- bilization of delay systems which are either extending the classical transfer function approach (Olgac and Sipahi [2002]) or newly methods based on LMI (Magdi [2000]). Among the special types of systems, the class of Metz- lerian and nonnegative systems play important role for continuous-time and discrete-time systems (Berman et al. [1989], Kaczorek [2002], Farina and Rinaldi [2000], Haddad and Chellaboina [2004]). The robust stability and con- strained non-negative stabilization of regular systems were introduced by Shafai et al. in several papers ( [1991a,b], [1997], see also the references therein) and the results were also published in (Bhattacharyya et al. [1995]). The aim of this paper is to present techniques for robust stability anal- ysis and control of positive time delay systems. Motivated by the problem of observer design for time delay systems (Rami et al. [2007]) and simple stability result of positive delay systems (Buslowicz [2008]) we provide additional insights into the robust analysis and stabilization with positivity constraints. In a recent paper we established strong result for the class of continuous-time Metzlerian delay systems (Shafai and Sadaka [2012]) based on the fact that asymptotic stability with delay is equivalent to the stability of an associated Metzlerian system without delay. Consequently, important results have been derived ⋆ In this paper, it is important to distinguish two terminologies. The discrete-time delay systems in the absent of delay states is referred to as delay-free systems and when the delay arguments are absent, we use the notion “discrete-time systems without delay”. for stability, robust stability direct computation of stabil- ity radius, and constrained stabilization which resemble to the case of conventional positive systems without delay. In this paper, we develop a parallel strategy for establish- ing similar results for the non-negative discrete-time delay systems. In particular, we show that the non-negative discrete-time delay systems is asymptotically stable if and only if an associated non-negative discrete-time system without delay is asymptotically stable. This fact allows one to obtain strong results on robust stability and stability radius for this class of systems. Based on an alternative representation of delay systems, similar results are also derived. Finally, the problem of constrained stabilization is considered. We provide a new LMI feasibility solution for it which makes it possible to stabilize a general discrete- time delay system such that the closed-loop system admits non-negative structure with desirable properties. 2. PRELIMINARY RESULTS Consider the linear discrete-time systems with delays described by x(k + 1) = A 0 x(k)+ l  i=1 A i x(k − i)+ Bu(t) (1) y(k)= Cx(k)+ Du(k) (2) with the initial conditions x(−k) ∈R n ; k =1, 2,...,l, where l is a positive integer, x(k) ∈ R n , u(k) ∈ R m , y(k) ∈R p are the state, input and output vec- tors; respectively, and A i ∈R n×n ,i =0, 1,...,l; B ∈ R n×m ; C ∈R p×n ,D ∈R p×m are system matrices. The system (1) is asymptotically stable if and only if all roots of the characteristic equation Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University, Boston, USA. June 22-24, 2012 978-3-902823-04-5/12/$20.00 © 2012 IFAC 31 10.3182/20120622-3-US-4021.00058