This paper revisits the design of compensator-based controller for a class of infinite dimensional systems. In order to save computational time, a functional observer is employed to reconstruct a functional of the state which coincides with the full state feedback control signal. Such a full-state feedback corresponds to an idealized case wherein the state is available. Instead of reconstructing the entire state via a state-observer and then use this state estimate in a controller expression, a functional observer is used to estimate the product of the state and the feedback operator, thus resulting in a significant reduction in computational load. This observer design is subsequently integrated with a sensor selection in order to improve controller performance. An appropriate metric is used to optimize the sensor location resulting in improved performance of the functional observer-based compensator. The integrated design is further extended to include a controller with an unknown input functional observer. The results are applied to 2D partial differential equations and detailed numerical studies are included to provide an appreciation in the significant savings in both operational and computational costs.
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Finite Dimensional Functional Observer Design for Parabolic Systems
This paper combines two control design aspects for a class of infinite dimensional systems, and each of the designs aims at significantly reducing the implementation complexity and computational load. A functional observer, and its extension of an unknown input functional observer, aims to reconstruct a functional of the infinite dimensional state. The resulting compensator only requires the solution to an operator Sylvester equation plus one differential equation for each dimension of the control signal, as opposed to an infinite dimensional filter evolution equation and an associated operator Riccati equation for the filter operator covariance. When the functional to be estimated coincides with the expression of a full state feedback control signal, then the functional observer becomes the minimum order compensator. When the parabolic system admits a decomposition whereby the system is decomposed into a lower finite dimensional subspace comprising the unstable eigenspectrum and an infinite stable subspace, then the functional observer-based compensator design becomes the minimum order compensator for the finite dimensional subsystem. This approach dramatically reduces the computation for solving the ARE needed for the full state controller and the associated Sylvester equation needed for the functional observer. Numerical results for a parabolic PDE in one and two spatial dimensions are included.
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- Award ID(s):
- 1825546
- PAR ID:
- 10385851
- Date Published:
- Journal Name:
- 2021 60th IEEE Conference on Decision and Control
- Page Range / eLocation ID:
- 1155 to 1160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CDC45484.2021.9682848
