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.

翻译：本文研究了在二次成本最小化目标下，基于双线性观测的线性动态系统控制问题。尽管该问题与标准线性二次高斯（LQG）控制具有相似性，但我们证明当观测模型为双线性时，分离原理不再成立，且最优控制器也非估计状态的仿射函数。此外，代价函数关于控制输入是非凸的，因此通常难以获得最优反馈控制器的解析表达式。在某些特定条件下，我们发现标准LQG控制器会局部最大化而非最小化系统代价。进一步分析表明，通过解析推导得到的最优控制器并不唯一，且均为估计状态的非线性函数。本文还提出了输入依赖可观测性的概念，并推导了卡尔曼滤波器协方差保持有界的条件。最后，通过在多种仿真场景中的数值实验验证了理论结果。