Direct Policy Optimization for Linear-Quadratic Gaussian Controllers

Core Concepts

Optimizing over the orbit space of controllers is crucial for direct LQG policy optimization.

Abstract

The content discusses direct policy optimization for linear-quadratic Gaussian (LQG) controllers. It introduces a geometric approach using Riemannian quotient manifolds for stabilizing full-order minimal output-feedback controllers. The paper proves local convergence guarantees with a linear rate for direct LQG policy optimization, showing improved performance over ordinary gradient descent. The analysis includes theoretical foundations, algorithm details, convergence proofs, and numerical experiments comparing different optimization methods.
I. Introduction

Direct policy optimization synthesizes controllers through constrained optimization over controller parameters.
PO bridges control synthesis and data-driven methods.
First-order methods update parameters for local convergence guarantees.
II. Preliminaries

Continuous-time linear system model with controllability and observability assumptions.
Output-feedback controller parameterization for LQG setting.
LQG cost function for controllers.
III. Our Algorithm

Introduction of Riemannian Gradient Descent (RGD) for LQG.
Algorithm 1 for RGD over minimal controllers.
Backtracking line-search procedure for step size determination.
IV. Orbit Space of Output-Feedback Controllers

Construction of Riemannian quotient manifolds for controllers.
Definition of orbit space modulo coordinate transformation.
Stability analysis and convergence properties.
V. Convergence Analysis

Assumption on non-degeneracy of LQG controller.
Theorem on local convergence of RGD with linear rate.
Conditions for convergence and stability certificates.
VI. Numerical Experiments and Results

Comparison of RGD with ordinary gradient descent.
Performance evaluation on representative systems.
Impact of different KM metrics on optimization results.
VII. Future Directions

Exploration of second-order PO methods for feedback synthesis.
Study of LQG PO over reduced-order controllers.
Investigation of discrete formulation and finite-horizon PO.
VIII. Acknowledgements

Acknowledgment of support from NSF and AFOSR grants.
Thanks to Shahriar Talebi for discussions on policy optimization.

Stats

"Over the past few years, it has been recognized that the landscape of stabilizing output-feedback controllers of relevance to LQG has an intricate geometry."
"The search space of full-order controllers is large with n2+nm+np dimensions."
"We prove a local convergence guarantee with linear rate and show the proposed approach exhibits significantly faster and more robust numerical performance as compared with ordinary gradient descent for LQG."

Quotes

"Optimizing over the orbit space of controllers is the right theoretical and computational setup for direct LQG policy optimization."
"In this paper, we present a geometric approach for resolving many of these issues."

Key Insights Distilled From

Output-feedback Synthesis Orbit Geometry

by Spencer Krai... at arxiv.org 03-27-2024

https://arxiv.org/pdf/2403.17157.pdf

Deeper Inquiries

How can second-order PO methods enhance feedback synthesis beyond first-order procedures?

Second-order Policy Optimization (PO) methods can enhance feedback synthesis beyond first-order procedures by incorporating information about the curvature of the optimization landscape. Unlike first-order methods like gradient descent, second-order methods take into account not only the gradient but also the Hessian matrix, providing a more accurate and efficient optimization process. By considering the second derivative of the cost function, these methods can navigate complex and non-convex spaces more effectively, leading to faster convergence and potentially better solutions. In the context of feedback synthesis, this means that second-order methods can handle more intricate control objectives, such as those involving non-strict saddle points or degenerate controllers, with greater precision and robustness. Overall, the use of second-order PO methods can offer improved convergence rates, better handling of complex optimization landscapes, and potentially superior performance in feedback synthesis tasks.

What are the implications of studying LQG PO over reduced-order controllers?

Studying Linear Quadratic Gaussian (LQG) Policy Optimization (PO) over reduced-order controllers has several implications for control synthesis and optimization. By focusing on reduced-order controllers, researchers can potentially simplify the optimization process by working in lower-dimensional spaces. This reduction in dimensionality can lead to faster computations, reduced complexity, and improved scalability of the optimization algorithms. Additionally, studying LQG PO over reduced-order controllers allows for a more targeted approach to control synthesis, focusing on the most critical components of the system while discarding less influential factors. This can lead to more interpretable and efficient control policies, especially in systems with high-dimensional state and output spaces. Furthermore, the study of reduced-order controllers can provide insights into the trade-offs between controller complexity and performance, helping to strike a balance between computational efficiency and control effectiveness in practical applications.

How does the discrete formulation of the setup impact the optimization process?

The discrete formulation of the setup in the context of Linear Quadratic Gaussian (LQG) Policy Optimization (PO) introduces specific considerations and challenges that can impact the optimization process. When transitioning from a continuous-time linear system to a discrete-time representation, the optimization problem may need to be reformulated to account for the discrete dynamics and sampling intervals. This discrete formulation can affect the stability and performance of the control system, as discretization introduces quantization errors and approximations that may influence the behavior of the optimized controller. Additionally, the discrete setup may require adjustments in the optimization algorithms to handle the discrete nature of the system dynamics and measurements. Techniques such as discretization methods, sampling strategies, and numerical integration schemes become crucial in ensuring the accuracy and effectiveness of the optimization process in the discrete domain. Overall, the discrete formulation of the setup necessitates careful consideration of the impact of discretization on the optimization process and the resulting control policies to ensure robust and reliable performance in practical applications.