Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes

Abstract This paper proposes a general and practical methodology for the integration of nonlinear output feedback control with optimal placement of control actuators and measurement sensors for transport-reaction processes described by a broad class of quasi-linear parabolic partial differential equations (PDEs) for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Initially, Galerkin's method is employed to derive finite-dimensional approximations of the PDE system which are used for the synthesis of stabilizing nonlinear state feedback controllers via geometric techniques. 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 and is solved by using standard unconstrained optimization techniques. Then, under the assumption that the number of measurement sensors is equal to the number of slow modes, we employ a procedure proposed in Christofides and Baker (1999) for obtaining estimates for the states of the approximate finite-dimensional model from the measurements. The estimates are combined with the state feedback controllers to derive output feedback controllers. 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. It is rigorously established that the proposed output feedback controllers enforce stability in the closed-loop infinite-dimensional system and 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 of the slow and fast eigenmodes increases. The proposed methodology is successfully applied to a representative diffusion–reaction process and a nonisothermal tubular reactor with recycle to derive nonlinear output feedback controllers and compute optimal actuator/sensor locations for stabilization of unstable steady states.

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