1 Detectability and Stabilizability of Discrete-Time Switched Linear Systems Ji-Woong Lee and Pramod P. Khargonekar Abstract For discrete-time switched linear systems under nondeterministic autonomous switching, the existence of causal finite-path-dependent stabilizing output injection and state feedback laws are characterized by increasing unions of linear matrix inequality conditions. These convex characterizations lead to the notions of causal finite-path-dependent detectability and stabilizability, which in turn yield a separation result for dynamic output feedback stabilization. By generalizing the standard duality concept to switched systems under arbitrary switching path constraints, we relate these notions to direct extensions of time-varying detectability and stabilizability requirements. Index Terms Causality, directed graph, discrete linear inclusion (DLI), duality, linear matrix inequality (LMI), separation principle. I. I NTRODUCTION A switched system consists of a finite number of subsystems that are switched according to the time variation of the system’s mode of operation. Switched systems can be used to model complex hybrid systems [1], [2], where continuous dynamics and discrete state transitions coexist and depend on each other. Switched systems, and hybrid systems in general, arise in many different contexts such as multi-rate sampled-data systems [3], nonlinear control [4]–[6], adaptive control [7]–[9], asynchronous systems [10], power systems [11], [12], fuzzy systems [13], signal processing [14], [15], supervisory control [16], networked control [17]–[19], distributed networks [20], [21], systems biology [22], [23], and so on. Due to their ubiquity in modern engineering problems, switched systems are receiving increasing interest and attention as the recent books [24]–[26] and survey articles [27]–[30] indicate. However, until the recent advance made in [19], [31], no efficient “control-oriented” algorithm to completely solve even the simplest problem of determining the stability of discrete linear inclusions (i.e., discrete-time switched linear systems under unconstrained mode switching) has been known: see [32]–[38] for non-control-oriented stability analysis results, and [39]–[43] for the associated computational issues; see [44]–[46] for some of the control-oriented but conservative results based on average dwell time, common Lyapunov function, and multiple Lyapunov functions approaches. J.-W. Lee is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA (e-mail: jiwoong@ufl.edu). P. P. Khargonekar is with the College of Engineering, University of Florida, Gainesville, FL 32611, USA (e-mail: ppk@ufl.edu). June 6, 2007 DRAFT