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Nonlinear feedback control of parabolic partial differential difference equation systems. (English) Zbl 1001.93036

Authors’ abstract: This paper proposes a general method for the synthesis of nonlinear output feedback controllers for single-input single-output quasi-linear parabolic partial differential difference equation (PDDE) systems, 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, a nonlinear model reduction scheme, which is based on a combination of Galerkin’s method with the concept of approximate inertial manifold, is employed for the derivation of differential difference equation (DDE) systems that describe the dominant dynamics of the PDDE system. Then, these DDE systems are used as the basis for the explicit construction of nonlinear output feedback controllers through a combination of geometric and Lyapunov techniques. The controllers guarantee stability and enforce output tracking in the closed-loop parabolic PDDE system independently of the size of the state delay, provided that the separation of the slow and fast eigenvalues of the spatial differential operator is sufficiently large and an appropriate matrix is positive definite. The methodology is successfully employed to stabilize the temperature profile of a tubular reactor with recycle at a spatially non-uniform unstable steady-state.

MSC:

93C23 Control/observation systems governed by functional-differential equations
93C20 Control/observation systems governed by partial differential equations
35B42 Inertial manifolds
93C70 Time-scale analysis and singular perturbations in control/observation systems
93B11 System structure simplification
93C95 Application models in control theory
93C10 Nonlinear systems in control theory
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