Estimating the minimal domains of attraction of uncertain discrete-time switched systems under state-dependent switching

In this paper, we inner-estimate the minimal domains of attraction of uncertain discrete-time switched systems under state-dependent switching, where the uncertain terms are described by bounded functions. At first, we introduce the uncertain parameter evolution to define the solution (or trajectory) of uncertain discrete-time switched system and then present the definitions of multi-step state subspaces, multi-step basins of attraction and multi-step Lyapunov-like functions. Then, based on using multi-step Lyapunov-like functions to iteratively compute multi-step basins of attraction, we establish an iterative framework to compute inner-estimations of the minimal domain of attraction. Especially, since certain multi-step state subspaces are empty sets, the corresponding constraints in the iterative framework are redundant. Therefore, we next realize the iterative framework by first finding out the non-empty multi-step state subspaces by the homotopy continuation method and then using S-procedure to under-approximately transform the iterative framework into a sum of squares programming. Moreover, we introduce a refinement method to improve our iterative method. At last, we apply our iterative method to four theoretical examples as well as a real-world example and present a short discussion on the results.