时滞随机控制系统的镇定问题及在趋同控制中的应用

The purpose of this project is to solve two challenging theoretical problems in stabilization for stochastic systems with delays, which is of great significance in both application and theory, and to investigate applications in consensus control. Firstly, a mean-square stabilization problem for discrete-time stochastic systems with multiple input delays will be considered. The aim is to present simple, functional and less conservative conditions for the mean-square stabilization by Riccati-like equations. To this end, an optimal control problem will be constructed and solved. Secondly, effects of delays on the mean-square stabilization for discrete-time stochastic systems with a single input delay will be discussed. Conditions for the existence of an exact delay margin will be given and the exact delay margin will be presented via algorithms. The approach is based on the necessary and sufficient condition for mean-square stabilization proposed by the applicant and co-workers, i.e., there exists a unique positive-definite solution to the algebraic Riccati-ZXL equation. Finally, a consensus control problem for multi-agents systems with diverse communication delays will be studied. Based on the closed relationship between stabilization and consensus control, the reduction technique, which is effective to deal with stabilization for time-delay systems, will be applied to the above problem and consensusability conditions and consensus protocols will be established.

时滞随机系统的镇定问题具有重要的应用价值和理论意义，本项目的目的旨在解决其中两个具有挑战性的基础理论问题，并探讨在趋同控制中的应用。首先，考虑多输入时滞离散时间随机系统的均方镇定问题，目标是通过Riccati类型的方程给出简洁实用且保守性低的均方镇定条件。拟通过构造并求解最优控制问题的方法建立均方镇定条件。其次，研究时滞对单输入时滞离散时间随机系统均方镇定性的影响，建立存在精确时滞界的条件，并通过算法给出精确时滞界。研究方法基于申请人及合作者前期建立的系统均方镇定的充要条件，即代数Riccati-ZXL方程存在唯一正定解。最后，考虑具有不同通信延迟的多智能体系统的趋同控制问题。基于镇定性和趋同控制的紧密关系，将研究时滞系统镇定性的有效方法，即退化技巧，应用到上述问题，给出可趋同的条件和趋同协议。

该项目研究了以下两个问题。第一，考虑了一般的多输入时滞离散时间随机系统的均方镇定问题。通过Riccati类型的方程，建立了系统均方镇定的充分条件，给出了镇定控制器的设计方法。采用的方法基于构造带约束的容许控制集和Lyapunov泛函，所得到的Riccati类型方程形式简洁，其变量个数和维数不随时滞增大而增大。第二，考虑了单输入时滞离散时间随机系统均方镇定的时滞界问题。证明了该系统一定存在时滞界M，使得当时滞取值小于等于M时，系统均方镇定，当时滞取值大于M时，系统不可均方镇定。采用的方法基于构造Lyapunov泛函。在几种特殊情况下，给出了时滞界的解析表达式。