Iterative Risk Allocation: A new approach to robust Model Predictive Control with a joint chance constraint

This paper proposes a novel two-stage optimization method for robust Model Predictive Control (RMPC) with Gaussian disturbance and state estimation error. Since the disturbance is unbounded, it is impossible to achieve zero probability of violating constraints. Our goal is to optimize the expected value of an objective function while limiting the probability of violating any constraints over the planning horizon (joint chance constraint). Prior arts include ellipsoidal relaxation approach [1] and Particle Control [2], but the former yields very conservative result and the latter is computationally intensive. Our approach divide the optimization problem into two stages; the upper-stage that optimizes risk allocation, and the lower-stage that optimizes control sequence with tightened constraints. The lower-stage is a regular convex optimization, such as Linear Programming or Quadratic Programming. The upper-stage is also convex, but the objective function is not always differentiable. We developed a fast descent algorithm for the upper-stage called Iterative Risk Allocation (IRA), which yield much smaller suboptimality than ellipsoidal relaxation method while achieving a substantial speedup compared to and Particle Control.