State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm

Since the sensing period of a signal generally depends on the used sensor, it may be different from another, even in a single control system if it involves multiple sensors. This paper investigates a design problem of state observers for linear time-invariant discrete-time plants under a multi-rate sensing environment. The sensing periods of the sensors in the plant are assumed as mutually rational ratios. First, we characterize a state observer for a plant with multi-rate sensing as a periodically time-varying state observer. Then, we discuss the
$l_{2}$
performance analysis of a state estimation error with the given periodically time-varying state observer. A linear matrix inequality (LMI) condition is provided for the analysis. By extending the LMI condition for analysis, we also provide that for multi-rate observer synthesis. Finally, we numerically illustrate the effectiveness of the proposed multi-rate state observer. Even if all the sensors have the same period, the sensing timing is not unique. Therefore, we numerically analyze whether the performance changes when the observation timing between multiple sensors is different.