具有状态和控制乘性噪声控制系统的可镇定条件

Conditions of Stabilizability for Control Systems with State and Control Multiplicative Noises

摘要: 本文研究了一类具有乘性噪声的离散时间线性时不变系统的镇定问题。这类乘性噪声同时存在于状态和控制输入通道，并且被假设为独立同分布的随机过程。为了确保开环不稳定系统能通过状态反馈实现均方意义下的镇定，我们推导出了均方可镇定的基本条件。通过求解非负矩阵的谱半径，首先给出了系统可镇定的充要条件。然后，运用拉格朗日乘数法得出了最优控制镇定系统的充分条件，这些条件由三个耦合的非标准代数Riccati方程决定。在一种特殊情形下，我们得出结论：状态和控制通道均存在乘性噪声的系统与状态通道无噪声系统共用同一个最优控制增益。最后，数值仿真验证了算法的有效性。

Abstract: This paper studies the stabilization problem for a class of discrete-time linear time-invariant systems with multiplicative noises. These multiplicative noises are assumed to be present in the state and control input channels and modeled as independent and identically distributed random processes. We develop fundamental conditions of mean-square stabilizability which ensure that an open-loop unstable system can be stabilized by state feedback in the mean-square sense. By solving the spectral radius of a non-negative matrix, a sufficient and necessary condition which stabilize system is first presented. And then, the Lagrange multiplier method is used to obtain the sufficient condition for optimal control stabilizing system, which is determined by three coupled non-standard algebraic Riccati equations. In a special case, we conclude that the system with state and control multiplicative noises can share a common optimal controller gain with the noise-free system for the state channel. Finally, numerical simulation illustrates the efficiency of the algorithm.