Safe Learning-Based Control of Stochastic Jump Linear Systems: a Distributionally Robust Approach
For control engineers dealing with uncertain jump linear systems, this work provides a tractable LMI-based method that guarantees stability with limited data, reducing conservatism as sample size grows.
The paper develops a distributionally robust control law for stochastic jump linear systems with unknown disturbance distributions estimated from finite samples, achieving mean-square stability guarantees with high probability. The method scales well with system dimensions and sample space size, and demonstrates superior sample complexity over stochastic approaches in a numerical example.
We consider the problem of designing control laws for stochastic jump linear systems where the disturbances are drawn randomly from a finite sample space according to an unknown distribution, which is estimated from a finite sample of i.i.d. observations. We adopt a distributionally robust approach to compute a mean-square stabilizing feedback gain with a given probability. The larger the sample size, the less conservative the controller, yet our methodology gives stability guarantees with high probability, for any number of samples. Using tools from statistical learning theory, we estimate confidence regions for the unknown probability distributions (ambiguity sets) which have the shape of total variation balls centered around the empirical distribution. We use these confidence regions in the design of appropriate distributionally robust controllers and show that the associated stability conditions can be cast as a tractable linear matrix inequality (LMI) by using conjugate duality. The resulting design procedure scales gracefully with the size of the probability space and the system dimensions. Through a numerical example, we illustrate the superior sample complexity of the proposed methodology over the stochastic approach.