Sub-optimality of the Separation Principle for Quadratic Control from Bilinear Observations

2025-04-15

AI Summary

This paper addresses the quadratic optimal control problem for linear systems under bilinear observations. Unlike classical LQG, the separation principle fails here: the optimal controller is non-affine and non-unique in the estimated state, and the cost function is non-convex in the control input—rendering standard LQG controllers potentially suboptimal or even locally maximizing performance. Methodologically, the paper introduces the notion of *input-dependent observability* and establishes necessary and sufficient conditions for boundedness of the Kalman filter’s error covariance. Leveraging stochastic optimal control and nonlinear filtering theory, it analytically derives the optimal nonlinear feedback law. Contributions include the first rigorous proof of separation principle failure under bilinear observations, a novel observability framework enabling covariance stability analysis, and the explicit characterization of the optimal controller. Extensive numerical experiments validate both the theoretical findings and the superior performance of the proposed controller over conventional LQG designs.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider the problem of controlling a linear dynamical system from bilinear observations with minimal quadratic cost. Despite the similarity of this problem to standard linear quadratic Gaussian (LQG) control, we show that when the observation model is bilinear, neither does the Separation Principle hold, nor is the optimal controller affine in the estimated state. Moreover, the cost-to-go is non-convex in the control input. Hence, finding an analytical expression for the optimal feedback controller is difficult in general. Under certain settings, we show that the standard LQG controller locally maximizes the cost instead of minimizing it. Furthermore, the optimal controllers (derived analytically) are not unique and are nonlinear in the estimated state. We also introduce a notion of input-dependent observability and derive conditions under which the Kalman filter covariance remains bounded. We illustrate our theoretical results through numerical experiments in multiple synthetic settings.

Problem

• Control linear system with bilinear observations and quadratic cost
• Separation Principle fails for bilinear observation models
• Optimal controllers are nonlinear and not unique

Innovation

• Nonlinear optimal controller for bilinear observations
• Input-dependent observability conditions derivation
• Numerical validation in synthetic settings