Stabilization of switched affine systems via multiple shifted Lyapunov functions

Attractors and limit cycles of discrete-time switching affine systems: Nominal and uncertain cases

“…3 and tends to the presumed cycle after a small transient time as pointed out in item (iii) of Theorem 1. The results are interestingly very similar to the ones in [16] and [32]. However, the method provided in [32] is limited by the length of the cycle that needs to be equal to the number of mode.…”

“…The results are interestingly very similar to the ones in [16] and [32]. However, the method provided in [32] is limited by the length of the cycle that needs to be equal to the number of mode. i.e.…”

“…Finally, an extension to the case of uncertain systems is conducted, enhancing then the notion of robust limit cycles. To do so, we propose to follow our preliminary results provided in [32] where a different Lyapunov function inspired from [17] is included to better understand the attractor of the resulting control Lyapunov function. An important by-product of the following analysis is that the usual underlying necessary condition consisting in the existence of a Schur stable convex combination of matrices A i is not required anymore, which, apart from [16], has been rarely completed in the literature.…”

“…Stabilizing Switched Affine Systems (SAS's) to a desired position, that is not in the set of equilibrium points of the modes, has motivated a large collection of contributions, especially in the continuous-time framework: global quadratic stabilization via a min-switching strategy [10], use of dynamic programming to select the mode to activate and also the time to switch [31], globally stabilizing min-switching strategy taking into account a cost to minimize [15], local stabilization without requiring the existence of a Hurwitz combination of the linear parts [25], practical stabilization with minimum dwell-time guarantees [1], robust stabilization to an unknown equilibrium point [2,3]. We can emphasize also contributions in the discrete-time domain leading to the practical stabilization via different types of Lyapunov functions: quadratic forms [14], switched quadratic functions based on Lyapunov-Metzler inequalities [17] or thanks to multiple shifted quadratic functions [32]. Besides equilibrium points, dynamical systems may have as asymptotic behaviors, self-sustained oscillations, or limit cycles: that are closed and isolated trajectories [34,Section 7].…”

“…The main difficulty is to determine the switching times related to a limit cycle [20,22]. There are only few contributions on limit cycles for discrete-time SAS's: one case study is provided in [29], based on prac-tical stability [38] or clock-and state-dependent switched Lyapunov functions [16] and also the preliminary result on multiple shifted quadratic Lyapunov functions putting in evidence the notion of limit cycles [32]. This paper is focused on limit cycles for nominal and uncertain discrete-time SAS's, and their stabilizability thanks to state-dependent switching laws.…”