Beyond Ellipsoids: Semi-Algebraic Tightening for Chance Constraints Under Actuator Saturation

Abstract

Motivated by stochastic model predictive control applications, we present a semi-algebraic approach to constraint tightening for chance-constrained systems with unbounded additive disturbances and saturated inputs. The saturated error dynamics are handled via their exact piecewise-affine structure, which naturally accommodates asymmetric saturation bounds. A polynomial Lyapunov function satisfying a drift condition is then designed using sum-of-squares optimization, yielding finite-time probabilistic reachable sets and a probabilistic ultimate bound. The set geometry is explicitly optimized for constraint tightening, further reducing conservatism. A numerical example demonstrates the effectiveness of the design.