Synthesizing Switching Logic to Minimize Long-Run Cost

Given a multi-modal dynamical system, optimal switching logic synthesis involves generating the conditions for switching between the system modes such that the resulting hybrid system satisfies a quantitative specification. We formalize and solve the problem of optimal switching logic synthesis for quantitative specifications over long run behavior. Our paper generalizes earlier work on synthesis for safety. We present an approach for specifying quantitative measures using reward and penalty functions, and illustrate its effectiveness using several examples. Each trajectory of the system, and each state of the system, is associated with a cost. Our goal is to synthesize a system that minimizes this cost from each initial state. We present an automated technique to synthesize switching logic for such quantitative measures. Our algorithm works in two steps. For a single initial state, we reduce the synthesis problem to an unconstrained numerical optimization problem which can be solved by any offthe-shelf numerical optimization engines. In the next step, optimal switching condition is learnt as a generalization of the optimal switching states discovered for each initial state. We prove the correctness of our technique and demonstrate the effectiveness of this approach with experimental results.