$\mathcal{H}_{\infty}$ PID Control With Fading Measurements: The Output-Feedback Case

Abstract

This paper is concerned with the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> proportional-integral-derivative (PID) control problem for a class of linear discrete-time systems with fading measurements. The fading measurements are governed by the Rice fading model whose coefficients are hypothesized to be a series of independent and identically distributed Gaussian variables. By utilizing the received measurements subject to fading phenomena, a novel output-feedback PID controller is proposed where the integral-loop (accumulation sum-loop for the discrete-time case) is equipped with the limited time-window in order to reduce the computational burden. The main objective of the addressed problem is to design a desired PID controller such that both the exponentially mean-square stability and the prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance are guaranteed for the closed-loop system in the presence of fading measurements. With the help of Lyapunov stability theory, a sufficient condition is obtained to guarantee the desired performance and, on the basis of such a condition, the synthesis issue of the PID controller is subsequently discussed, where the orthogonal decomposition combined with a free matrix is introduced to facilitate the controller design. Finally, a numerical example is exploited to demonstrate the usefulness and effectiveness of the presented control scheme.