In this article, we present a framework for the numerical solution of optimal control problems, constrained by ordinary differential equations which can run in (finitely many) different modes, where a change of modes leads to additional switching cost in the cost function, and whenever the system changes its mode, jumps in the differential states are possible. In addition, for each mode there are certain constraints which shall only hold as long as the system stays in the respective mode. We present the problem class and represent the problem as a mixed-integer optimal control problem. We re- formulate and relax the problem and discretize the control functions in the resulting problem. We present three different approaches for the treatment of switching costs and compare them with each other, whereat only one of them is suitable for the treatment of jumps in a general setting. We then take a direct approach (“first discretize, then optimize”) to solve the resulting control-discretized problem numerically, where a direct method based on hp–adaptive collocation is used for the discretization. The resulting finite dimensional optimization problems are mathematical programs with vanishing constraints, and we discuss numerical techniques to solve sequences of this challenging problem class. In the end of the article, we present two examples: first an academic one concerning switching costs only, and second an example from mechanics, where also jumps occur. In the latter example, we generate a walking-like motion and discuss the modeling as well as the configuration of our solver in detail.
Citation
Heidelberg University, INF 205, 69120 Heidelberg, 10/2018