Hybrid control synthesis for eventuality specifications using level set methods

Abstract

This paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications. Given a hybrid system modelled by a hybrid automaton and a target set of states, the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time. Due to the existence of set-valued disturbance inputs, the problem is studied in a game-theoretic framework. Having shown that a least restrictive solution does not exist, we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property. To implement the algorithm, reachable sets of pursuit-evasion differential games need to be computed. For that reason level set methods are employed, where the boundary of the reachable set is characterized as the zero level set of a Hamilton-Jacobi equation. The procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology. Finally, to demonstrate the practical character of our results, a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

Time-optimal control for discrete-time hybrid automata

“…The backward reachability of the continuous system is a game (Trontis and Spathopoulos 2003), where the controller wins if it can eventually drive the system to the given target set within a time T f in spite of disturbances. Let F X & 2 X be the target set.…”

“…Time-optimal control for hybrid systems 855 (Trontis and Spathopoulos 2003). Then, the zero sublevel set of V c describes W c (t) as W c ðtÞ ¼ fx 2 XjV c ðx, tÞ 0g ð 26Þ…”

Synthesis of controllers for target problems of hybrid systems using approximate computation

“…On the other hand, the game theoretic approach in combination with level set methods (Tomlin 1998, Lygeros et al 1999, Tomlin et al 2000, Trontis and Spathopoulos 2003a and the viability theory approach using set valued analysis (Aubin 1991) have produced numerical and analytical results for the computation of reachable sets. Unfortunately, the above techniques result in obtaining neither over-approximation nor under-approximation of the reachable sets.…”

“…The target control problem for hybrid automata has been studied in and , 2003a, 2003b and the solution is derived using reachability analysis. The basic idea here, instead of using the whole state space for reachability, is to find the shortest discrete path that has the minimal discrete cost from the initial set to the target set.…”

Synthesis of Supervisory Controllers for Hybrid Systems Using Abstraction Refinement

“…Most of these approaches establish extensions of the supervisory control theory for discrete systems (Ramadge and Wonham, 1989) to hybrid models, and involve the step of determining which hybrid states are reachable under the influence of a discrete control action. Techniques to accomplish this step include level set methods (Trontis and Spathopoulos, 2003), viability methods (Bayen et al, 2002), and computing ellipsoidal or polyhedral inclusions of reach sets . However, these techniques are known to be computationally expensive, and to scale badly with the dimension of the continuous state space -an objective for improving the efficiency of controller synthesis is thus to reduce reach set computation as much as possible.…”