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We present a new method for the automated synthesis of safe and robustProportional-Integral-Derivative (PID) controllers for stochastic hybridsystems. Despite their widespread use in industry, no automated methodcurrently exists for deriving a PID controller (or any other type ofcontroller, for that matter) with safety and performance guarantees for such ageneral class of systems. In particular, we consider hybrid systems withnonlinear dynamics (Lipschitz-continuous ordinary differential equations) andrandom parameters, and we synthesize PID controllers such that the resultingclosed-loop systems satisfy safety and performance constraints given asprobabilistic bounded reachability properties. Our technique leverages SMTsolvers over the reals and nonlinear differential equations to provide formalguarantees that the synthesized controllers satisfy such properties. Thesecontrollers are also robust by design since they minimize the probability ofreaching an unsafe state in the presence of random disturbances. We apply ourapproach to the problem of insulin regulation for type 1 diabetes, synthesizingcontrollers with robust responses to large random meal disturbances, therebyenabling them to maintain blood glucose levels within healthy, safe ranges.

Automated Synthesis of Safe and Robust PID Controllers for Stochastic Hybrid Systems