A Study on Robust Stability for Multi-Rate Model Predictive Control for Linear Systems with Additive Disturbance

In this paper, we discuss a model predictive control (MPC) for a multi-rate discrete-time system with additive unknown disturbance. In the multi-rate setting, it is assumed that state measurement and input update are performed less frequently than the system update, which may lead to instability of the overall system when a disturbance enters the system. We here propose a robust stability-guaranteed design for the multi-rate robust MPC (MR-RMPC), by extending the authors' previous work on disturbance-free version to cases of the uncertain systems. To this end, the lifting method is adopted to represent the multi-rate (and thus time-varying) system as an linear time-invariant system with a larger dimension, for which a robust positively invariant set is newly derived. The proposed MR-RMPC is constructed based on the Riccati equation for the lifted system, which directly implies that robust stability condition is carried out.