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

TL;DR

This paper introduces an automated method for synthesizing safe and robust PID controllers for stochastic hybrid systems with nonlinear dynamics, providing formal safety guarantees and robustness against disturbances.

Contribution

It presents a novel SMT-based approach for automatically synthesizing PID controllers with safety and performance guarantees for complex stochastic hybrid systems.

Findings

Successfully applied to insulin regulation in type 1 diabetes

Controllers maintain safe blood glucose levels under disturbances

Provides formal safety guarantees using SMT solvers

Abstract

We present a new method for the automated synthesis of safe and robust Proportional-Integral-Derivative (PID) controllers for stochastic hybrid systems. Despite their widespread use in industry, no automated method currently exists for deriving a PID controller (or any other type of controller, for that matter) with safety and performance guarantees for such a general class of systems. In particular, we consider hybrid systems with nonlinear dynamics (Lipschitz-continuous ordinary differential equations) and random parameters, and we synthesize PID controllers such that the resulting closed-loop systems satisfy safety and performance constraints given as probabilistic bounded reachability properties. Our technique leverages SMT solvers over the reals and nonlinear differential equations to provide formal guarantees that the synthesized controllers satisfy such properties. These controllers are also robust by design since they minimize the probability of reaching an unsafe state in the presence of random disturbances. We apply our approach to the problem of insulin regulation for type 1 diabetes, synthesizing controllers with robust responses to large random meal disturbances, thereby enabling them to maintain blood glucose levels within healthy, safe ranges.