Applying discrete-time proportional Integral observers for state and disturbance estimations

In this note, we apply the proportional integral observer to simultaneously estimate system states and unknown disturbances for discrete-time nonminimum phase systems. Conditions for providing the existence of such observer are given. When the disturbances do not vary too much between two consecutive sampling instances, the proposed method can render the estimation errors of system states and disturbances to be constrained in a small bounded region. Simulation results support the theoretical developments.

I. Introduction

Proportional integral observer (PIO) to be devised using an additionally introduced integral term of the output estimation error in observer design can offer certain degrees of freedom. Beale and Shafai [2] make use of this additional freedom in the observer-based controller design, as a result of which PIO becomes less sensitive to parameter variation of the system. Niemann et al. [13] used the PIO to study the loop transfer recovery problem in continuous time systems and Shafai et al. [16] studied the same problem in discrete time systems. Moreover, Busawon and Kabore [3] have demonstrated that the PIO design can effectively reduce the effect of measurement noises as opposed to the proportional observer. Out of necessity, some authors extended this technique to design the observer for descriptor systems [7] or nonlinear systems [1], [11]. In particular, Shafai et al. [17] applied the PIO in estimating actuator and sensor faults. Since the PIO has two feedback loops, a proportional loop and an integral loop of the output estimation error, i.e., the PIO presented in this note allows us to estimate not only the states but also the unknown inputs.