Robust-Guaranteed Approximation of Disturbance Invariant Set for Systems with Near-Unit-Disk Spectral Radius*

This study presents a practical algorithm for approximating the Robust Positively Invariant (RPI) set within the context of robust Tube Model Predictive Control (MPC) for discrete-time, linear time-invariant systems. When the stable matrix exhibits a spectral radius close to the unit disk, computing the RPI set becomes challenging, potentially rendering it infeasible. We first analyze the impact of the spectral radius on RPI set convergence, providing an insight into the problem. Subsequently, we propose an approach to integrate approximation into the RPI set computation while preserving the robustness of the corresponding tightened sets. This is achieved by enforcing the upper and lower dimensional bounds of the RPI set during computation. Additionally, we incorporate disturbance estimation error bounding into the Tube MPC framework to address substantial additive disturbances. These disturbances, if directly treated by Tube MPC, otherwise lead to over-conservative or empty tightened state and control sets. Throughout the study, we demonstrate the effectiveness of the proposed algorithm through numerical simulations of a car-following problem.