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Reduced order controllers for spatially distributed systems via proper orthogonal decomposition. (English) Zbl 1075.65090

Summary: A method for reducing controllers for systems described by partial differential equations (PDEs) is presented. This approach differs from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale finite element approximation of the PDE controller onto low order bases that are computed using the proper orthogonal decomposition (POD).
Two methods for constructing input collections for POD, and hence low order bases, are discussed and computational results are included. The first uses the method of snapshots found in POD literature. The second is a new idea that uses an integral representation of the feedback control law. Specifically, the kernels, or functional gains, are used as data for POD. A low order controller derived by applying the POD process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably.

MSC:

65K10 Numerical optimization and variational techniques
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
93A15 Large-scale systems
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
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