Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control

Yujie Tang, Yang Zheng, Na Li
Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:599-610, 2021.

Abstract

This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: 1) connectivity of the set of stabilizing controllers Cn; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that 1) the set of stabilizing controllers Cn has at most two path-connected components and they are diffeomorphic under a mapping defined by a similarity transformation; 2) there might exist many \emph{strictly suboptimal stationary points} of the LQG cost function over Cn and these stationary points are always \emph{non-minimal}; 3) all \emph{minimal} stationary points are globally optimal and they are identical up to a similarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.

Cite this Paper

BibTeX
@InProceedings{pmlr-v144-tang21a, title = {Analysis of the Optimization Landscape of Linear Quadratic Gaussian ({LQG}) Control}, author = {Tang, Yujie and Zheng, Yang and Li, Na}, booktitle = {Proceedings of the 3rd Conference on Learning for Dynamics and Control}, pages = {599--610}, year = {2021}, editor = {Jadbabaie, Ali and Lygeros, John and Pappas, George J. and A. Parrilo, Pablo and Recht, Benjamin and Tomlin, Claire J. and Zeilinger, Melanie N.}, volume = {144}, series = {Proceedings of Machine Learning Research}, month = {07 -- 08 June}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v144/tang21a/tang21a.pdf}, url = {https://proceedings.mlr.press/v144/tang21a.html}, abstract = {This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: 1) connectivity of the set of stabilizing controllers $\mathcal{C}_n$; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that 1) the set of stabilizing controllers $\mathcal{C}_n$ has at most two path-connected components and they are diffeomorphic under a mapping defined by a similarity transformation; 2) there might exist many \emph{strictly suboptimal stationary points} of the LQG cost function over $\mathcal{C}_n$ and these stationary points are always \emph{non-minimal}; 3) all \emph{minimal} stationary points are globally optimal and they are identical up to a similarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.} }
Endnote
%0 Conference Paper %T Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control %A Yujie Tang %A Yang Zheng %A Na Li %B Proceedings of the 3rd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2021 %E Ali Jadbabaie %E John Lygeros %E George J. Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire J. Tomlin %E Melanie N. Zeilinger %F pmlr-v144-tang21a %I PMLR %P 599--610 %U https://proceedings.mlr.press/v144/tang21a.html %V 144 %X This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: 1) connectivity of the set of stabilizing controllers $\mathcal{C}_n$; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that 1) the set of stabilizing controllers $\mathcal{C}_n$ has at most two path-connected components and they are diffeomorphic under a mapping defined by a similarity transformation; 2) there might exist many \emph{strictly suboptimal stationary points} of the LQG cost function over $\mathcal{C}_n$ and these stationary points are always \emph{non-minimal}; 3) all \emph{minimal} stationary points are globally optimal and they are identical up to a similarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.
APA
Tang, Y., Zheng, Y. & Li, N.. (2021). Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control. Proceedings of the 3rd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 144:599-610 Available from https://proceedings.mlr.press/v144/tang21a.html.

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