Optimization of hybrid systems

An overview of dynamic optimization problems with hybrid systems embedded is presented. The control parameterization approach is examined in detail, where the possibility of mode switching within the hybrid system causes nonsmoothness in the Master problem, thus invalidating gradient based solvers. To handle this, a decomposition approach is proposed that divides the general problem into smooth subproblems. The subproblem of determining the optimal mode sequence given fixed transition times is considered for continuous time linear time varying hybrid systems. A mixed-integer reformulation of the problem is proposed that retains the linear structure of the embedded hybrid system. A deterministic branch-and-cut framework incorporating outer approximation cuts and a dynamic bounds tightening heuristic is then employed to solve the nonconvex mixed-integer problem to guaranteed global optimality with a finite number of iterations. It is shown that the dynamic bounds tightening heuristic can have a dramatic effect on accelerating convergence of the algorithm. Future directions for research are also discussed, including the application of recent developments in semi-infinite programming for the formal safety verification of processes under uncertainty.