Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity

In [1], [2], [3], [4] necessary conditions were obtained for hybrid optimal control problems (HOCPs) which resulted in a general Hybrid Maximum Principle (HMP); further, in [4], [5], a class of effficient, provably convergent Hybrid Maximum Principle (HMP) algorithms were obtained based up on the HMP. In [3], [4] the notion of optimality zones (OZs) was introduced as a theoretical framework for the computation of optimal location (i.e. discrete state) sequence for HOCPs (i.e. discrete state sequences with the associated switching times and states). This paper presents the algorithm HMPZ which fully integrates the prior computation of the OZs into the HMP algorithms of [4], [5]. Adding (a) the computational investment in the construction of the OZs for a given HOCP, and (b) the complexity of (i) the computation of the optimal schedule, (ii) the optimal switching time and optimal switching state sequence, and (iii) the optimal continuous control input, yields a complexity estimate for the algorithm (HMPZ) which is linear (i.e.O(L)) in the number of switching times L; this is to be compared with the geometric (i.e. O(|Q|^L)) growth of a direct combinatoric search over the set of location sequence, where Q denotes the discrete state set of the hybrid system.