Automatica 49 (2013) 1–8 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Bode-like integral for stochastic switched systems in the presence of limited information ✩ Dapeng Li a,1 , Naira Hovakimyan b a United Technologies Research Center, 411 Silver Lane, East Hartford, CT, USA b Department of Mechanical Science and Engineering, UIUC, Urbana, IL 61801, USA article info Article history: Received 12 October 2010 Received in revised form 1 May 2012 Accepted 16 July 2012 Available online 1 October 2012 Keywords: Networked control systems Performance limits abstract In this paper, we establish a Bode sensitivity integral formula for a class of feedback closed-loop systems with stochastic switched plants and controllers. Using information theory, we study the information conservation law, based on which a log integral theorem is obtained for the closed loops of interest. Furthermore we develop several algebraic conditions to explicitly capture the performance limitations. Application of this theoretical framework to Networked Control Systems (NCS) is used as an illustrative example. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction Recent results on fundamental limitations of feedback con- trol in the presence of communication channels presented a fairly general and complete approach, in discrete-time setting, towards unification of information theory and control theory, (Martins & Dahleh, 2008; Martins, Dahleh, & Doyle, 2007). Entropy rate in- equalities corresponding to the information flux in a typical causal closed-loop configuration were derived towards obtaining a Bode- like integral formula. Prior to this, extensions of Bode’s theorem have been claimed for certain discrete-time nonlinear systems and linear time-varying systems respectively, (Iglesias, 2001a,b). In this paper, we extend the framework from Martins and Dahleh (2008) to closed loops with stochastic switched plants. While switched control systems have been studied from various perspectives (Liberzon, 2003), it is still not clear how to charac- terize their fundamental limitations within an appropriate frame- work similar to the Bode’s formula. This work is the first attempt, to the best of the authors’ knowledge, to address this unsolved problem. Throughout the paper, we restrict the switching pattern that governs the transitions among (finite) subsystems to a finite ✩ Research is supported by AFOSR under Grant FA9550-09-1-0265. The material in this paper was presented at the American Control Conference (ACC2011), June 29 – July 1, 2011, San Francisco, California, USA. This paper was recommended for publication in revised form under the direction of Editor Berç Rüstem. E-mail addresses: dapeng.ustc@gmail.com (D. Li), nhovakim@illinois.edu (N. Hovakimyan). 1 Tel.: +1 217 377 0767; fax: +1 860 998 9264. state homogenous Markov chain, which is further assumed to ad- mit an invariant measure that characterizes its long-term average behavior. This class of stochastic switched systems (or variations with similar probabilistic switching laws) is important on its own, and has been attracting continuous attention since its inception for its theoretical richness and deep rooting in practice. Fundamen- tal control problems such as stochastic stability (Costa & Fragoso, 1993; Feng, Loparo, Ji, & Chizeck, 1992), optimal control (Costa & Fragoso, 1995) and filtering (Shi, Boukas, & Agarwal, 1999) have been extensively studied in both continuous-time and discrete- time settings. On the more practical side, among other applications ranging from regime changing models in macro economics (Hamil- ton, 1990) to single-event aversion in high-altitude aerospace sys- tems (Zhang, Gray, & González, 2008), its importance has been recently rediscovered as an appropriate model for networked con- trol systems (NCS) with random data-outage caused by either packet drop-outs or communication delays (Hespanha, 2004; Ishii, 2008; Ling & Lemmon, 2004). The development of the paper begins with some necessary technical assumptions that allow the system of interest to be studied by information theory. The methods from information theory are intensively employed towards deriving a conservation law, in the form of an entropy rate inequality, that fundamentally governs the disturbance attenuation ability of the closed loop regardless the choice of the controller. To obtain the information conservation law, arguments similar to Kim (2010) and Martins and Dahleh (2008) are adopted to explore the probabilistic dependencies among signals residing in the closed loop. Those sequential relations, as will be revealed in the main result, are accounted for by the closed loop topology and causality. What 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.09.001