Computation of invariant sets for complex systems

Abstract

In the analysis and control of complex systems such as hybrid or switched systems, the computation of sets satisfying given properties is paramount. Let us mention for instance safety analysis, model predictive control and Information-Theoretic bounds for communication networks. Mathematically, the computation of such sets is best handled by the formulation of a corresponding optimization problem. In numerical optimization, the introduction of a modeling layer on top of optimization solvers has been a key enabler, allowing different communities to share emergent technologies more easily. When the purpose is to compute sets rather than numbers (a.k.a. ‘set programming’), the numerical complexity is amplified, yet there does not currently exist such a clearly defined interface. Even the meaning of ‘computing a set’ is not clear and actually depends on the intended end-use. The purpose of this chapter is first to unify many control-theoretic problems under the umbrella of set programming. Then, we survey diverse numerical and theoretic techniques, that constitute a rich toolbox for handling such problems. We finally point out some current challenges.