On Controller Reduction in Linear Quadratic Gaussian Control with Performance Bounds

Zhaolin Ren, Yang Zheng, Maryam Fazel, Na Li
Proceedings of The 5th Annual Learning for Dynamics and Control Conference, PMLR 211:1008-1019, 2023.

Abstract

The problem of controller reduction has a rich history in control theory. Yet, many questions remain open. In particular, there exist very few results on the order reduction of general non-observer based controllers and the subsequent quantification of the closed-loop performance. Recent developments in model-free policy optimization for Linear Quadratic Gaussian (LQG) control have highlighted the importance of this question. In this paper, we first propose a new set of sufficient conditions ensuring that a perturbed controller remains internally stabilizing. Based on this result, we illustrate how to perform order reduction of general (non-observer based) output feedback controllers using balanced truncation and modal truncation. We also provide explicit bounds on the LQG performance of the reduced-order controller. Furthermore, for single-input-single-output (SISO) systems, we introduce a new controller reduction technique by truncating unstable modes. We illustrate our theoretical results with numerical simulations. Our results will serve as valuable tools to design direct policy search algorithms for control problems with partial observations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v211-ren23a, title = {On Controller Reduction in Linear Quadratic Gaussian Control with Performance Bounds}, author = {Ren, Zhaolin and Zheng, Yang and Fazel, Maryam and Li, Na}, booktitle = {Proceedings of The 5th Annual Learning for Dynamics and Control Conference}, pages = {1008--1019}, year = {2023}, editor = {Matni, Nikolai and Morari, Manfred and Pappas, George J.}, volume = {211}, series = {Proceedings of Machine Learning Research}, month = {15--16 Jun}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v211/ren23a/ren23a.pdf}, url = {https://proceedings.mlr.press/v211/ren23a.html}, abstract = {The problem of controller reduction has a rich history in control theory. Yet, many questions remain open. In particular, there exist very few results on the order reduction of general non-observer based controllers and the subsequent quantification of the closed-loop performance. Recent developments in model-free policy optimization for Linear Quadratic Gaussian (LQG) control have highlighted the importance of this question. In this paper, we first propose a new set of sufficient conditions ensuring that a perturbed controller remains internally stabilizing. Based on this result, we illustrate how to perform order reduction of general (non-observer based) output feedback controllers using balanced truncation and modal truncation. We also provide explicit bounds on the LQG performance of the reduced-order controller. Furthermore, for single-input-single-output (SISO) systems, we introduce a new controller reduction technique by truncating unstable modes. We illustrate our theoretical results with numerical simulations. Our results will serve as valuable tools to design direct policy search algorithms for control problems with partial observations. } }
Endnote
%0 Conference Paper %T On Controller Reduction in Linear Quadratic Gaussian Control with Performance Bounds %A Zhaolin Ren %A Yang Zheng %A Maryam Fazel %A Na Li %B Proceedings of The 5th Annual Learning for Dynamics and Control Conference %C Proceedings of Machine Learning Research %D 2023 %E Nikolai Matni %E Manfred Morari %E George J. Pappas %F pmlr-v211-ren23a %I PMLR %P 1008--1019 %U https://proceedings.mlr.press/v211/ren23a.html %V 211 %X The problem of controller reduction has a rich history in control theory. Yet, many questions remain open. In particular, there exist very few results on the order reduction of general non-observer based controllers and the subsequent quantification of the closed-loop performance. Recent developments in model-free policy optimization for Linear Quadratic Gaussian (LQG) control have highlighted the importance of this question. In this paper, we first propose a new set of sufficient conditions ensuring that a perturbed controller remains internally stabilizing. Based on this result, we illustrate how to perform order reduction of general (non-observer based) output feedback controllers using balanced truncation and modal truncation. We also provide explicit bounds on the LQG performance of the reduced-order controller. Furthermore, for single-input-single-output (SISO) systems, we introduce a new controller reduction technique by truncating unstable modes. We illustrate our theoretical results with numerical simulations. Our results will serve as valuable tools to design direct policy search algorithms for control problems with partial observations.
APA
Ren, Z., Zheng, Y., Fazel, M. & Li, N.. (2023). On Controller Reduction in Linear Quadratic Gaussian Control with Performance Bounds. Proceedings of The 5th Annual Learning for Dynamics and Control Conference, in Proceedings of Machine Learning Research 211:1008-1019 Available from https://proceedings.mlr.press/v211/ren23a.html.

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