Sparsity-Promoting Reachability Analysis and Optimization of Constrained Zonotopes

The constrained zonotope is a polytopic set representation widely used for set-based analysis and control of dynamic systems. This paper develops methods to formulate and solve optimization problems for dynamic systems in real time using constrained zonotope reachability analysis. An alternating direction method of multipliers (ADMM) algorithm is presented that makes efficient use of the constrained zonotope structure. To increase the efficiency of the ADMM iterations, reachability calculations are presented that increase the sparsity of the matrices used to define a constrained zonotope when compared to typical methods. The developed methods are used to formulate and solve predictive control, state estimation, and safety verification problems. Numerical results show that optimization times using the proposed approach are competitive with state-of-the-art QP solvers and conventional problem formulations. A combined set-valued state estimation and moving horizon estimation algorithm is presented and experimentally demonstrated in the context of robot localization.