Integrated optimal actuator/sensor placement and robust control of uncertain transport-reaction processes

Abstract

This paper focuses on transport-reaction processes with unknown time-varying parameters and disturbances described by quasi-linear parabolic PDE systems, and addresses the problem of computing optimal actuator/sensor locations for robust nonlinear controllers. Initially, Galerkin's method is employed to derive finite-dimensional approximations of the PDE system which are used for the synthesis of robust nonlinear state feedback controllers via geometric and Lyapunov techniques and the computation of optimal actuator locations. The controllers enforce boundedness and uncertainty attenuation in the closed-loop system. The optimal actuator location problem is subsequently formulated as the one of minimizing a meaningful cost functional that includes penalty on the response of the closed-loop system and the control action. Owing to the boundedness of the state, the cost is defined over a finite-time interval (the final time is defined as the time needed for the process state to become smaller than the desired uncertainty attenuation limit), while the optimization is performed over a broad set of initial conditions and time-varying disturbance profiles. Subsequently, under the assumption that the number of measurement sensors is equal to the number of slow modes, we employ a standard procedure for obtaining estimates for the states of the approximate finite-dimensional model from the measurements. The optimal location of the measurement sensors is computed by minimizing a cost function of the estimation error in the closed-loop infinite-dimensional system. We show that the use of these estimates in the robust state feedback controller leads to a robust output feedback controller, which guarantees boundedness of the state and uncertainty attenuation in the infinite-dimensional closed-loop system, provided that the separation between the slow and the fast eigenvalues is sufficiently large. We also establish that the solution to the optimal actuator/sensor problem, which is obtained on the basis of the closed-loop finite-dimensional system, is near-optimal in the sense that it approaches the optimal solution for the infinite-dimensional system as the separation between the slow and fast eigenvalues increases. The theoretical results are successfully applied to a typical diffusion-reaction process with nonlinearites and uncertainty to design a robust nonlinear output feedback controller and compute the optimal actuator/sensor locations for robust stabilization of an unstable steady state.