On Controller Reduction in Linear Quadratic Gaussian Control with Performance Bounds

TL;DR

This paper develops new methods for reducing the order of controllers in LQG systems, providing performance bounds and techniques applicable to general controllers, with implications for policy optimization.

Contribution

It introduces sufficient conditions for stable controller reduction, applies balanced and modal truncation, and offers explicit performance bounds for reduced controllers in LQG control.

Findings

Balanced truncation preserves stability under certain conditions.

Explicit bounds on LQG performance after reduction.

New reduction technique for SISO systems by truncating unstable modes.

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 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.