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Reachability and optimal control for linear hybrid automata: a quantifier elimination approach

Pang, Y. and Spathopoulos, M.P. and Xia, H. (2007) Reachability and optimal control for linear hybrid automata: a quantifier elimination approach. International Journal of Control, 80 (5). pp. 731-748. ISSN 0020-7179

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Abstract

This paper considers an optimal control problem for linear hybrid automata (LHA). First, we present a controller synthesis algorithm based on reachability analysis. The algorithm computes the maximal initial set from which the controller drives the system to a given target set. It is shown that, using quantifier elimination (QE), an under-approximation of the maximal reachable set can be derived. Next, a weighted time-optimal control problem is solved by transforming it into a constrained optimization problem whose constraints are a set of inequalities with quantifiers. Quantifier elimination (QE) techniques are employed in order to derive the quantifier free inequalities that are shown to be linear. Thus, the optimal cost is obtained using linear programming. For any state belonging to the maximal initial set the optimal switching times and the optimal continuous control inputs are computed. These are used in order to derive a hybrid controller which is optimal with respect to the cost function. Our results are applied to an air traffic management example which is of practical interest.