线性系统控制中，双线性观测使分离原理失效，控制器非线性且难求解。
Sub-optimality of the Separation Principle for Quadratic Control from Bilinear Observations
- 双线性观测下分离原理不成立，最优控制器非状态估计的线性函数。
- 代价函数对控制输入非凸，标准LQG控制器可能反而最大化代价。
- 提出输入相关可观测性概念，给出滤波器协方差有界的条件，适合控制理论研究者。
我们研究从双线性观测中以最小二次代价控制线性动力系统的问题。尽管该问题与标准线性二次高斯（LQG）控制相似，但我们证明：当观测模型为双线性时，分离原理不成立，且最优控制器并非状态估计的仿射函数。此外，代价函数对控制输入非凸，因此一般情况下难以获得最优反馈控制器的解析表达式。在某些设定下，标准LQG控制器实际上局部最大化代价而非最小化。同时，解析导出的最优控制器不唯一，且在状态估计上表现为非线性。我们还引入了输入依赖可观测性的概念，并推导出卡尔曼滤波器协方差保持有界的条件。通过多个合成场景的数值实验验证了理论结果。
原文摘要 · Abstract (English)
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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