随机线性二次控制中策略迭代的稳定器设计：一种频谱分配方法
Stabilizer Design for Policy Iteration in Stochastic Linear Quadratic Control: A Spectrum-Assignment Approach
AI总结：
针对连续时间不定随机线性二次控制中无模型策略迭代的初始化难题，提出频谱分配方法设计初始稳定器，构建无模型算法实现最优控制，仿真验证了方法有效性。
AI中文摘要：
策略迭代（Policy Iteration, PI）是求解最优控制问题的重要强化学习工具，包含初始化阶段，即寻找初始稳定控制器。但该初始化阶段通常依赖完整模型信息，对无模型PI的初始化造成极大限制。对于依赖状态与控制的乘性噪声随机系统，其稳定性无法像确定性系统那样由Hurwitz条件保证，而是由结合漂移项与扩散项的Lyapunov型不等式保证，因此对应的无模型PI初始化问题更具挑战性。为此，针对连续时间不定随机线性二次控制，提出一种新颖的频谱分配方法，以获取PI的初始稳定器。借助Lyapunov型算子的频谱，通过调整累积因子，从稳定辅助系统逐步逼近原系统，从而得到稳定控制增益。此外，利用系统数据并调整累积因子，设计出一种不依赖初始稳定策略、可实现最优控制的无模型算法。最后，通过仿真结果验证所提方法的有效性。
英文摘要：
Policy iteration (PI) is an important reinforcement learning tool for solving optimal control problems which includes an initialization stage, i.e., the search for an initial stabilizing controller. However, the initialization stage typically relies on complete model information, thereby imposing substantial constraints on the initialization of model-free PI. For stochastic systems with multiplicative noise dependent on state and control, the stability is not ensured by Hurwitz conditions as in the deterministic case, but rather by a Lyapunov-type inequality that incorporates both drift and diffusion terms. Therefore, the corresponding model-free PI initialization problem is more challenging. To this end, a novel spectrum assignment method is proposed to obtain an initial stabilizer for PI in continuous-time indefinite stochastic linear quadratic control. With the help of the Lyapunov-type operator's spectrum, the original system is gradually approximated from the stable auxiliary system by adjusting a cumulative factor, thereby obtaining a stabilizing control gain. Furthermore, by leveraging system data and adjusting the cumulative factor, we design a model-free algorithm that does not rely on an initial stabilizing policy and can achieve optimal control. Finally, simulation results are provided to validate the effectiveness of the proposed methods.
