Rejecting an Unknown Matched Disturbance from an Infinite-Dimensional Control System

In this paper we address the problem of rejecting an unknown disturbance, which is matched with the input, from an infinite-dimensional plant belonging to the class of regular linear systems. The plant input and output are finite-dimensional and the time-derivative of the disturbance is assumed to be bounded with a known bound. In our solution approach to this problem, we drive a stable ODE using the output of the plant. Via a state transformation obtained by solving a Sylvester equation with possibly unbounded operators, we derive an auxiliary ODE in which the disturbance and the input are matched. We then build a nonlinear disturbance observer for the auxiliary ODE, based on the super-twisting sliding mode algorithm, to generate asymptotically accurate estimates for the unknown disturbance. By letting the input to the plant to be the negative of the disturbance estimate obtained, the matched disturbance in the plant can be rejected. In case the plant is unstable, including a stabilizing feedback signal in the input will ensure that the plant state converges to zero asymptotically. Our approach requires the state of the plant to be known. When only the plant output is known, our approach can be implemented using a state observer for the plant and then modifying the disturbance observer suitably. We demonstrate the efficacy of our approach in simulations by taking the plant to be an anti-stable 1D wave equation and assuming output measurement.

I. Introduction

Stabilizing plants while rejecting unknown matched disturbances, i.e. disturbances present in the input channel of the plant, is a problem of practical interest. Several papers in the literature have addressed this problem for finite-dimensional plants, see for instance [1], [17] and the references therein. However, for linear infinite-dimensional plants this problem is harder to solve and fewer works have addressed it. One of the earliest works to study this problem for infinite-dimensional plants in an abstract setting is [12], where a discontinuous control law is proposed based on the sliding mode technique. In that work it is assumed that the input operator is bounded, so plants with boundary control inputs cannot be considered, and that the full-state of the plant can be measured. Under similar assumptions [13] addressed this problem (and a tracking problem) for the 1D heat and wave equations. Subsequently, other works have addressed the above stabilization/disturbance rejection problem for particular boundary controlled infinite-dimensional plants such as the 1D heat and wave equations.