Locally convexified rigid tube MPC

This paper proposes a locally convexified rigid tube model predictive control for obstacle avoidance for linear, discrete-time, systems subject to additive, bounded disturbances and state and control constraints. The inherently nonconvex obstacle avoidance constraints are locally convexified and converted into closed polyhedral constraints by deploying recent notions of safe sets and tubes. The proposed method leverages these recent notions of safe sets and tubes to reformulate the original nonconvex optimization problem for obstacle avoidance within the context of disturbances into a computationally highly efficient, strictly convex quadratic programming problem. In order to analyze the separation between a predicted state set and each obstacle avoidance constraint set, we employ the Euclidean distance between these two sets and the radius of the Chebyshev ball of their intersection. In the strict separation case, the separating hyperplanes are constructed by utilizing the separation theorem for convex sets, while in the weak separation case, the construction of separating hyperplanes is achieved through the use of the separation theorem and the properties of convex cones. The proposed method provides guarantees of strong system theoretic properties such as robust recursive feasibility, robust positive invariance and robust stability of the controlled uncertain system. A simulation example is also provided to numerically verify the effectiveness of the proposed method.