Robust Invariant Sets Computation for Switched Discrete-Time Polynomial Systems

Bai Xue, Naijun Zhan

In this paper we systematically study the (maximal) robust invariant set generation problem for switched discrete-time polynomial systems subject to state constraints and perturbation inputs from theoretical and computational perspectives. The (maximal) robust invariant set of interest in this paper is a set of (all) states such that every possible trajectory starting from it stay inside a specified state constraint set forever regardless of actions of perturbations. We show that the maximal robust invariant set can be characterized as the zero level set of the unique bounded solution to a modified Bellman equation. Especially when there is only one subsystem in the system, the solution to the derived equation can be Lipschitz continuous. The uniqueness property of bounded solutions renders possible the gain of the maximal robust invariant set via value iteration methods. We further relax the equation into a system of inequalities and encode these inequality constraints using sum of squares decomposition for polynomials. This results in the gain of robust invariant sets by solving a single semi-definite program, which can be addressed via interior point methods in polynomial times. In this way we could obtain tractable solutions to problems more efficiently. Finally, we demonstrate our computational approaches on two illustrative examples.

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