Observer-based Multirate Feedback Control Design for Two-time-scale System

The use of a lower sampling rate for designing a discrete-time state feedback-based controller fails to capture information of fast states in a two-time-scale system, while the use of a higher sampling rate increases the amount of computation considerably. Thus, the use of single-rate sampling for systems with slow and fast states has evident limitations. In this paper, multirate state feedback (MRSF) control for a linear time-invariant two-time-scale system is proposed. Here, multirate sampling refers to the sampling of slow and fast states at different sampling rates. Firstly, a block-triangular form of the original continuous two-time-scale system is constructed. Then, it is discretized with a smaller sampling period and feedback control is designed for the fast subsystem. Later, the system is block-diagonalized and equivalently represented into a system with a higher sampling period. Subsequently, feedback control is designed for the slow subsystem and overall MRSF control is derived. It is proved that the derived MRSF control stabilizes the full-order system. Being the transformed states of the original system, slow and fast states need to be estimated for the MRSF control realization. Hence, a sequential two-stage observer is formulated to estimate these states. Finally, the applicability of the design method is demonstrated with a numerical example and simulation results are compared with the single-rate sampling method. It is found that the proposed MRSF control and observer designs reduce computations without compromising closed-loop performance.