Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems

Abstract

We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box or an ellipsoid, we provide a method to compute certified outer approximations of the reachable set.The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the Lebesgue measure on this set. Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials. For each fixed degree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite program. When the degree of the polynomial approximation tends to infinity, we provide strong convergence guarantees of the super level sets to the reachable set. We also present some application examples together with numerical results.

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“…On the other hand, ROA approaches usually consider a finite time for reaching the target set. To tackle this issue, we propose here to extend to continuous-time systems the work presented in [9], where occupation measures are used to formulate the infinite-time reachable set computation problem for discrete-time polynomial systems. For a given a > 0, we define the following linear programming problem:…”

Controller Synthesis for Discrete-time Hybrid Polynomial Systems via Occupation Measures

“…[40]). For discrete-time systems, it is realized by the push forward operator [45], [48]. In the following, we introduce the discrete-time controlled Liouville equation in the nonhybrid setting as proposed in [48], and in the next subsection we shall see how it enables us to formulate optimization problems for hybrid systems.…”

“…The general framework of the approach is to first formulate the problem as an infinitedimensional linear programming (LP) problem on measures and its dual on continuous functions, and to then approximate the LP by a hierarchy of finite-dimensional semidefinite programming (SDP) programs on moments of measures and their duals on sums-of-squares (SOS) polynomials. The approach has been applied to approximating the region of attraction, the maximum controllable set, or the forward/backward reachable set for discrete-time/continuoustime autonomous/controlled hybrid/non-hybrid polynomial systems [40]- [45]. It has also been applied to controller synthesis for those systems [42], [46]- [49].…”

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