This article focuses on the problem of linear quadratic Gaussian (LQG) control for discrete time‐varying system with input delay and state/control‐dependent noises. When the state variables can be exactly observed, first, we obtain the maximum principle by applying the method of variation. Second, a nonhomogeneous relationship between the state and the costate is developed in virtue of the obtained maximum principle and the mathematical induction. It is noted that the nonhomogeneous relationship is the solution to the forward and backward stochastic difference equations (FBSDEs). Finally, based on the solution to the FBSDEs, a necessary and sufficient condition for the optimal LQG control is derived in terms of coupled Riccati equations. Moreover, the analytical expression of the optimal controller is presented. The derived results are applied in networked control systems with packet dropout. Numerical examples are shown to illustrate the effectiveness of the proposed algorithm.

中文翻译：

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具有同时输入延迟和状态/控制相关噪声的离散时变系统的最优线性二次高斯控制

本文重点讨论具有输入延迟和状态/控制相关噪声的离散时变系统的线性二次高斯（LQG）控制问题。当可以精确观察状态变量时，首先，我们通过应用变分方法获得最大原理。其次，借助所获得的最大原理和数学归纳法，发展出国家与卑鄙者之间的不均匀关系。注意，非齐次关系是前向和后向随机差分方程（FBSDE）的解。最后，基于对FBSDE的解，根据耦合的Riccati方程推导了最优LQG控制的必要和充分条件。此外，给出了最优控制器的解析表达式。得出的结果将应用于具有数据包丢失功能的网络控制系统中。数值例子说明了所提算法的有效性。