Design of Observer-Based Feedback Controller for Multi-Rate Systems With Various Sampling Periods Using Cyclic Reformulation

Signal sensing periods typically vary depending on the sensor used and may differ even within a single control system that involves multiple sensors. Likewise, input periods can vary based on the actuator used. This paper discusses the design of observer-based feedback controllers for linear, time-invariant, discrete-time systems operating in a multi-rate sensing and actuating environment. The observation and control periods of the sensors and actuators in the plant are assumed to have mutually rational ratios. First, we reduce the multi-rate system to a periodically time-varying system and provide a linear matrix inequality (LMI) condition for analyzing the $l_{2}$ performance using cyclic reformulation, which is a type of time-invariant reformulation for periodic systems. Next, we extend the analysis method to design an observer-based feedback controller for the multi-rate system. This allows us to obtain multi-rate observer gains and feedback gains based on the $l_{2}$ -induced norm from disturbances to outputs. Finally, we present numerical results to demonstrate the effectiveness of the observer-based feedback system in the multi-rate environment.