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

• URL: http://arxiv.org/abs/2504.11555v1
• Date: Tue, 15 Apr 2025 18:53:51 GMT
• Title: Sub-optimality of the Separation Principle for Quadratic Control from Bilinear Observations
• Authors: Yahya Sattar, Sunmook Choi, Yassir Jedra, Maryam Fazel, Sarah Dean,
• Abstract summary: We consider the problem of controlling a numerical dynamical system from bilinear observations with minimal quadratic cost.<n>Despite the similarity of this problem to standard linear quadratic Gaussian (LQG) filters, we show that when neither is bilinear the Separation Principle model.<n>We illustrate our results through experiments in multiple synthetic settings.
• Score: 16.7109402673221
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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