Robust Stability Analysis of Nonlinear Switched Systems with Filippov Solutions Mohamadreza Ahmadi ∗ Hamed Mojallali ∗ Rafael Wisniewski ∗∗ Roozbeh Izadi-Zamanabadi ∗∗ ∗ Department of Electrical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran, P.O. Box: 41635-3756, Postal Code: 41996-13769, Tel: +98 131 6690270, Fax: +98 131 6690271, (email: mrezaahmadi@aol.com; mojallali@guilan.ac.ir). ∗∗ Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, 9220, Aalborg East, Denmark, Tel: +45 9940 8762, Fax: +45 9815 1739, (email: raf@es.aau.dk; riz@es.aau.dk). Abstract: This paper addresses the stability problem of a class of nonlinear switched systems with partitioned state-space and state-dependent switching. In lieu of the Carath´ eodory solutions, the general Filippov solutions are considered. This encapsulates solutions with infinite switching in finite time. Based on the theory of differential inclusions, a Lyapunov stability theorem is brought forward. These results are also extended to switched systems subject to polytopic uncertainty. Furthermore, the proposed stability theorems are reformulated using the sum of squares decomposition method which provides sufficient means to construct the corresponding Lyapunov functions via available semi-definite programming techniques. 1. INTRODUCTION A large group of engineering applications give rise to systems which encompass both discrete and continuous dynamics. Mathematically, these systems are character- ized by a collection of indexed differential or difference equations describing each subsystem and a switching rule between them. This rich family of systems is referred to switched or more generally hybrid systems. Examples of such systems in real world have been studied in open literature e.g. Wisniewski and Larsen [2008] and Larsen et al. [2007]. However despite numerous applications, their stability analysis has not been covered completely (Lin and An- taklis [2009]). Several interesting phenomena arise when dealing with such systems; namely, even if all the sub- systems are exponentially stable, one cannot guarantee the stability of the overall system (Branicky [1998]). Con- versely, an appropriate switching law may contribute to stability even when all subsystems are unstable (Liberzon [2003]). Still, depending on the considered type of solutions (Carath´ eodory, Filippov, and etc.), discrepant stability phenomenon may follow. As an illustration, a switched system with stable Carath´ eodory solutions, may possess divergent Filippov solutions (see Leth and Wisniewski [2012]). This research is mainly motivated by two contribu- tions (Pranja and Papachristodoulou [2003]) and (Leth and Wisniewski [2012]). Leth and Wisniewski [2012] ex- ploited the theory of differential inclusions (DI) and sug- gested Lyapunov-like stability theorem for piece-wise lin- ear switched systems defined on polyhedral sets with Fillipov solutions. Pranja and Papachristodoulou [2003] proposed sum of squares based stability analysis tools for a class of hybrid systems; however, a unified stabil- ity theorem has not been suggested or fully established. Additionally, the solutions considered in the latter article are in the sense of Carath´ eodory which connotate the exclusion of solutions with infinite switching in finite time from the analysis. In the present paper, firstly, we extend the results reported in (Leth and Wisniewski [2012]) to the general nonlinear switched systems defined on regular sets by incorporating the theoretical notions of differential in- clusions. Secondly, the robust stability problem of switched systems with polytopic uncertainty and Filippov solutions is addressed. Lastly, we propose sufficient conditions based on sum of squares (SOS) decomposition for the suggested stability theorems. This ensures computationally efficient means to investigate the stability of switched systems. This paper is organized as follows. In the next section, the notations and some mathematical concepts adopted in this study are limned. The main results of this paper are brought forward in section 3. Section 4 demonstrates the accuracy of the proposed methodology via an example. Finally, section 5 concludes the paper. 2. MATHEMATICAL PRELIMINARIES The notations employed in this paper are relatively straightforward. R ≥0 denotes the set [0, ∞). ‖·‖ denotes the Euclidean vector norm on R n , 〈·〉 the inner product, and B n ǫ the closed ball of radius ǫ in R n centered at origin. P accounts for the set of polynomial functions p : R n → R and P sos ⊂P is the subset of polynomials with an SOS decomposition; i.e, p ∈P sos if and only if there are p i ∈P ,i ∈{1,...,k} such that p = p 2 i + ··· + p 2 k . In this study, we consider a class of n-dimensional Preprints of the 7th IFAC Symposium on Robust Control Design The International Federation of Automatic Control Aalborg, Denmark, June 20-22, 2012 © IFAC, 2012. All rights reserved. 282