Approximate robust dynamic programming and robustly stable MPC

We present a technique for approximate robust dynamic programming that is suitable for linearly constrained polytopic systems with piecewise affine cost functions. The approximation method uses polyhedral representations of the cost-to-go function and feasible set, and can considerably reduce the computational burden compared to recently proposed methods for exact robust dynamic programming [Bemporad, A., Borrelli, F., & Morari, M. (2003). Min–max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600–1606; Diehl, M., & Björnberg, J. (2004). Robust dynamic programming for min–max model predictive control of constrained uncertain systems. IEEE Transactions on Automatic Control, 49(12), 2253–2257]. We show how to apply the method to robust MPC, and give conditions that guarantee closed-loop stability. We finish by applying the method to a state constrained tutorial example, a parking car with uncertain mass.