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

arXiv: https://arxiv.org/abs/2504.11555 · HTML: https://mlsanity.com/paper/2504.11555/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 7 of 7)

- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Semidefinite Programming Approach to Gaussian Sequential Rate-Distortion Trade-offs](https://mlsanity.com/paper/1411.7632/) (2014)
- [System Level Synthesis](https://mlsanity.com/paper/1904.01634/) (2019)
- [Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution](https://mlsanity.com/paper/1503.05029/) (2015)
- [Minimax Linear Optimal Control of Positive Systems](https://mlsanity.com/paper/2312.04990/) (2023)
- [Probabilistic Formulations for System Identification of Linear Dynamics with Bilinear Observation Models](https://mlsanity.com/paper/2502.15667/) (2025)

## Built on by (most cited first, 2 of 2)

- [Explore-then-Commit for Nonstationary Linear Bandits with Latent Dynamics](https://mlsanity.com/paper/2510.16208/) (2025)
- [Dual Control of Linear Systems from Bilinear Observations with Belief Space Model Predictive Control](https://mlsanity.com/paper/2604.24663/) (2026)

## Abstract (arXiv)

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.