Piecewise Semi-Ellipsoidal Control Invariant Sets

Abstract

Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semiellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family.

Geometric control of hybrid systems

“…In [10], the authors study the computation of piecewise quadratic Lyapunov functions for continuous-time autonomous piecewise affine systems. In [15], the authors present a convex programming approach to compute piecewise semi-ellipsoidal controlled invariant sets for discrete-time control systems. A similar approach is developed in [17] for continuous-time control system.…”

“…This paper genralizes [16], [15] and [17] into a framework for computing convex controlled invariant sets for linear hybrid control systems. In [16], the authors treat the particular case where the continuous dynamic at each mode (see definition 1) is trivial, i.e., ẋ = 0.…”

“…In [16], the authors treat the particular case where the continuous dynamic at each mode (see definition 1) is trivial, i.e., ẋ = 0. In [15], the authors extends [16] to piecewise semi-ellipsoids. In [17], the authors handle the particular case where there is only one mode and no transitions (see definition 1).…”

“…In [17], the authors handle the particular case where there is only one mode and no transitions (see definition 1). While [16,15] covers discrete-time systems and [17] covers continuous-time systems, we show in this paper that the two methods can be combined to compute controlled invariant sets for hybrid systems, exhibiting both discrete-time and continuous-time dynamics. Using the set programming framework [14], this compatibility can be understood as a consequence of the fact that the controlled invariance conditions require the sets to be represented with their support functions (see definition 6) both in discrete-time and continuoustime.…”