Control to facet for polynomial systems

Abstract

This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems.

Kinodynamic planning for robotic manipulators using set-based methods

“…This admissible exit facet represents the area of the boundary where the trajectory T f (x 0 , λ(x)) will escape in finite time, under appropriate control λ(x), emanating from x 0 ∈ R ϵ (X T ). Several existing works on control-to-facet strategies focus on non-linear hybrid systems with sets defined by simplices or other simple polynomial descriptions of the set boundaries [32]- [34].…”

Controller Synthesis for Nonlinear Systems with Reachability Specifications Using Monotonicity

“…. , N be basic controllers for the controllable intervals X − s (t) = [x s (t), x s (t)] and let V − s (t, x) be the corresponding functions of type (12). Then u(t, x) can be formally defined by…”

“…[8], [9]). The problem of polytope-to-polytope control for nonlinear systems in relation with symbolic control has also been studied using algebraic properties of multi-affine and polynomial vector fields in [10], [11], [12], or a combination of linear approximations with robust control in [13].…”

Abstraction of Continuous-Time Systems Based on Feedback Controllers and Mixed Monotonicity

“…The problem of polytope-to-polytope control for nonlinear control systems in relation with symbolic control has been considered extensively in the literature (see [16], [17], [18], [19], [20], [21]). It has been shown (see e.g.…”