An Online Algorithm for Optimizing Invariant Conditions for Procedural Nonlinear Constrained Hybrid Systems

Previous work on hybrid system optimization presume fixed invariant conditions between modes and mostly focus on obtaining the optimal control law. However, invariant conditions affect time durations of each mode, and their active value directly influences the system's performance. The optimization methodology suggested in this paper covers this effect and embodies invariant conditions as a part of the optimization problem. Nevertheless, there exists a major difficulty in optimizing such structures: The optimal solution is directly dependent on the initial condition of the system, and so, it is difficult to find a single optimal solution that is valid for all possible initial conditions. In addition, the problem is a general form optimization problem, making it impossible to cast it as a linear or convex program. This paper suggests a two part optimization procedure to address this problem. First, for a finite amount of initial conditions, a black-box optimization algorithm is applied to compute a database of solutions. Then, while the system is running online with an arbitrary initial condition, a second algorithm optimizes the structure by applying parameters of the closest initial entry stored in the database and optimizing the parameters backwards from the final discrete state to initial discrete state. The competence of the discussed algorithms have been tested on three different domains. It is seen that runtime and recoverability improvements are achieved with the proposed method, compared to using an offline-only approach or using a single set of fixed parameters.