Computer methods for stability analysis of the Roesser type model of 2D continuous-discrete linear systems

In continuous-discrete systems both continuous-time and discrete-time components are relevant and interacting and cannot be separated. Such systems are called hybrid systems. Examples of these can be found in the works of Gałkowski et al. (2003), Hespanha (2004), Johanson et al. (2004) and Liberzon (2003). The problems of dynamics and control of hybrid systems were studied by Dymkov (2005), Dymkov et al. (2003; 2004), Gałkowski et al. (2003), Rogers et al., (2007) and Liberzon (2003). In this paper we consider continuous-discrete linear systems whose models have a structure similar to that of models of 2D discrete-time linear systems. Such models, called 2D continuous-discrete models or 2D hybrid models, were considered by Kaczorek (2002) in the case of positive systems. A new model of positive 2D hybrid linear systems, similar to the Roesser model of 2D systems, was introduced for standard and for fractional systems by Kaczorek (2007; 2008a). The realization and solvability problems of positive 2D hybrid linear systems were considered by Kaczorek (2002; 2008b) as well as Kaczorek et al. (2008) and Sajewski (2009), respectively (see also Kaczorek, 2011, Chapter 12). The problems of stability and robust stability of 2D continuous-discrete linear systems were investigated by Bistritz (2003; 2004), Xiao (2001), Busłowicz, (2010a; 2010b; 2011a; 2011b) as well as Busłowicz and Ruszewski (2011) (see also Kaczorek, 2011, Chapter 12). The problem of stability of solutions of a class of hybrid difference-difference systems was considered by Marchenko and Loiseau (2009). The main purpose of this paper is to present computational methods for investigation of asymptotic stability of the Roesser type model of 2D continuous-discrete linear systems. The following notation will be used: R is the set of real numbers, R+ = [0,∞], Z+ is the set of non-negative integers, Rn×m is the set of real n×m matrices, λi{X} is the i-th eigenvalue of matrix X .