Controllability analysis of uncertain polytopic systems with time-varying state delay D. Famularo, G. Franz` e and F. Tedesco 1 1 DEIS - Universit` a degli Studi della Calabria Rende (CS), 87036, ITALY {franze,famularo,ftedesco}@deis.unical.it Abstract— In this paper reachability/controllability proper- ties for networked systems described by uncertain polytopic linear plants subject to time-varying state delays and input constraints are analyzed. Up to our best acknowledge, previous technical contributions on the matter were not been capable to properly address time-delay occurrences. To this end a new result making it possible to efficiently compute the set of states that can be robustly steered in a finite number of steps, via state feedback control, to a given target set is here proposed. A final simple example is used to show the applicability of the proposed results. I. I NTRODUCTION Establishing safety properties of hybrid and networked systems is one of the most interesting, but equally chal- lenging, problems for formal methods. With the increased embedding of digital controllers inside physical devices, such as automobiles and aircraft, the need for automated tools and techniques for formal analysis and verification has become more pressing. Exhaustive testing via simulation is, in most cases, neither possible nor practical. Reachability and safety concepts are now in fact being recognized as central problems in designing controllers. From a methodological point of view, Reachabil- ity/Controllability analysis is a relevant research issue in control theory because of its strict relationship with set invariance, set-membership state estimation, constrained con- trol, fault tolerant control schemes etc., see the books [4], [7], papers [3], [17], [19] and references therein. Starting ideas on reachability analysis and guaranteed state estimation can be traced back to the pioneering control literature (see the seminal papers [22], [6], [21], [14]). The common denominator behind these early approaches is that the exact reachability concept is a high computationally demanding task and it is necessary to resort to approximation schemes in order to properly attack the problem. Approximate reach- ability paradigms are several in literature exploiting essen- tially, ellipsoidal, polytopic or even zonotopic calculus to obtain guaranteed inner and/or outer estimates of the exact reachable sets/tubes or sets of possible states consistent with acquired information, system dynamics and uncertainty features (see [16], [15], [1] and references therein). A common feature of these approximate reachabil- ity/controllability sets computational approaches is to resort to the so-called predecessor operator algorithm by computing the set of states that can be robustly steered (using an admissible control input) to a given target set in a single step. The predecessor operator is then used in a recursive fashion in order to compute the set of states that can be robustly steered to the given target set in a finite number of steps (see [15] and [2]). As previously stated, the advent of computational improvements and the development of efficient software tools for classes of constrained systems of practical interest, such as hybrid and networked systems have renewed interest in these problems (see [7], [19], [5], [17] and [12]). To the best of authors’ knowledge, constrained networked systems have received relatively little attention. A networked system can in fact be regarded as a system subject to time-varying delays and such a paradigm is described via Functional Differential Equations (FDE), which differ from Ordinary Differential Equations (ODE) because they do not admit in general a finite dimensional state representation. As a consequence, performance analysis and control design for such systems suffer therefore from unavoidable structural complications and only conservative results and related ap- proximations are achievable, see the survey paper [20] and references therein. Moving from the previous considerations, the main aim of this paper is to present novel results and related com- putational schemes for controllability computations using ellipsoidal calculus for input constrained uncertain polytopic time-delay systems. We would like to point out that the proposed extension is not trivial because adequate relaxations are needed even if all the relevant sets are convex and the system is linear. The key point is then to give proper inner approximations of the one-step controllable sets by acting on both actual and delayed states. The contribution is twofold: first, Delay-Dependent (DD) closed-loop stability is guaranteed for input-constrained time-delay systems. Then, one-step controllable ellipsoidal regions are computed by resorting to ellipsoidal calculus for- mulas which are based on a series of convex approximations of the original problem obtained by means of set-invariance concepts and of the descriptor approach [11]. Finally, to validate the proposed results a numerical ex- 51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA 978-1-4673-2064-1/12/$31.00 ©2012 IEEE 2352