具有随机乘性噪声的多变量线性网络控制系统的分析和综合

This project focuses on the stabilizability and output regulation problems of multi-input and multi-output (MIMO) linear networked control systems under stochastic multiplicative noises. The stochastic multiplicative noise can be applied to describe a number of data-loss network phenomena, such as packet drops, quantization error, etc. We assess the stabilizability and output regulation problems under mean-square criterion. This project will discuss a series of issues from three aspects: first, for MIMO continues time linear time-invariant (LTI) systems, we want to establish a general, explicit stabilizability condition, both necessary and sufficient, which provides a fundamental limit imposed by the system’s unstable poles, nonminimum phase zeros, their directions and statistical characteristics of multiplicative noises, furthermore, we will find the optimal output feedback controller to guarantee the systems stabilized; the performance goal of output regulation problem is measured by the energy or power of system’s output in mean-square sense, we want to obtain the explicit analytic expression of regulation performance limitation and the optimal output feedback controller to achieve the performance limits. Second, for the MIMO systems under stochastic multiplicative noise and time-delay simultaneously, we also want to find an explicit necessary and sufficient stabilizability condition, the explicit expression of the regulation performance goal as some as that in the first aspect, and the optimal output feedback controllers for stabilization problem and output regulation problem, respectively.Third, we want to expand the results alluded to above from network control systems with only up-link or down-link channel to the network control systems, in which both up-link and down-link channels are taken into account.The results in this project will lead solutions applicable to networked control problems addressing these issues.

本项目研究具有随机乘性噪声的连续时间多变量线性网路控制系统的可稳定问题及输出调节问题。采用随机乘性噪声描述网络中如丢包、量化误差等。本项目将从输入输出角度、在均方意义下从三个层面进行渐进式的研究：其一，建立多变量连续线性系统的可稳定充要条件与乘性噪声数理统计特性及系统特征的解析关系，并给出具有最大噪声容忍能力的线性输出反馈控制器设计方法，以均方意义下输出信号能量或功率为输出调节问题目标，给出性能极限解析表达式及最优输出反馈控制器设计方法；其二，若同时考虑网络中的时滞约束，分别阐明系统可稳定充要条件及输出调节性能极限与乘性噪声数理统计特性、时滞及系统特征的解关系、分别给出具有最大噪声容忍能力以及最大抗时滞能力的输出反馈可稳定控制器、实现输出调节性能极限的最优输出反馈控制器设计方法；其三，将上述两方面结论推广至双网络框架下网络控制系统。通过本项目的实施，为网络控制系统分析和设计提供理论支持。

经过课题组成员的努力，本项目完成了研究计划所列的大部分内容并略有拓展，在网络控制系统领域取得了一些重要进展，建立了随机乘性噪声以及网络时滞影响下的线性系统输出反馈均方镇定、均方可镇定以及输出反馈最优跟踪等优化控制问题相关结论，并将研究内容由单一网络控制系统拓展至复杂网络系统——多智能体系统。特别地，①建立了随机时滞影响下的多变量连续时间线性定长系统均方镇定的小增益定理，该小增益定理表现为确定性系统传递函数与信噪比矩阵乘积的谱半径的不等式形式；②给出多变量连续时间线性定长系统均方可镇定的充分必要条件，对于一般的最小相位系统，得到可计算的系统输出反馈均方镇定充分必要条件，该条件可转化为矩阵广义特征值问题，并通过线性矩阵不等式优化方法有效求解，对于非最小相位被控对象，证明非最小相位系统的输出反馈均方镇定问题是一类非凸优化问题，无法得到（解析的或可计算的）输出反馈均方可镇定充分必要条件；③建立了带整数步长随机延时与丢包通信信道的单变量线性定常离散系统的均方稳定与均方可镇定的充分必要条件，该条件与通道信噪比及系统不稳定零极点有解析关系；④在均方意义下，给出线性定常系统跟踪阶跃信号的跟踪误差能量最小值，该最小值可由系统非最小相位零点、与系统传递函数有关的积分运算以及随机乘性噪声信噪比解析描述。于此同时，我们给出了基于状态反馈的最优输出反馈控制器的设计方案，并证明在某种意义上分离原理依然成立；⑤复杂网络结构下，分别考虑离散时间、连续时间线性定常多智能体系统，建立系统趋同能力与时滞长度、网络拓扑以及智能体零极点之间的解析关系，构造基于相对输出信息的分布式控制器。..项目负责人在Springer出版社出版了英文专著一本《Limits of Stability and Stabilization of Time-Delay Systems: A Small-Gain Approach》，在控制领域顶级学术期刊《IEEE Transactions on Automatic Control》接受待发表1篇，在控制领域权威学术期刊《SIAM Journal on Control and Optimization》接收待发表1篇。