时滞正系统的分析与结构化控制

Positive systems are a special category of systems possessing the characteristic that their inputs, states, and outputs are intrinsically nonnegative. Applications of such systems have been found in modeling compartmental networks, population evolution, chemical reactors, and gene regulation. In this project, a variety of analysis and control problems will be tackled for positive delay systems. The stability and performance analysis of nonlinear positive delay systems will be first carried out, with extensions to monotone systems which possess the property of cone invariance. Meanwhile, in the light of the theory of coupled differential-difference equations and the comparison principles, the stability of neutral delay systems will also be exploited. With the aid of the afore-established analysis results, some classes of structured optimal control problems for positive delay systems that can be directly cast as convex optimization will be identified, and linear-programming-based iterative algorithms will be proposed for other more general situations. Moreover, networked interconnected positive systems will be considered with emphasis on their delay robustness analysis and distributed control. In addition, the proposed control strategies will be further applied to the consensus and rendezvous problems of multi-agent systems with communication delays. It is anticipated that this project will not only develop control techniques with improved solvability and lower cost for large-scale positive delay systems, but also provide new insights into the modeling and analysis of other related areas, including cooperative control of multi-agent systems, coordination of multiple robots, and power control of wireless sensor networks.

正系统是一类特殊的动力系统，其输入、状态与输出均具有非负的特性。这类系统在房室网络、人口演化、化学反应、基因调控的建模中有着广泛应用。本项目将解决时滞正系统的几类分析与控制问题。首先，项目将探讨带有时滞的非线性正系统的稳定性与性能分析，并推广至更一般的具有锥不变性的单调系统。同时，利用比较原理与耦合微分-差分方程理论，研究中立型系统的稳定性。基于建立的时滞正系统分析结果，我们将寻找几类可直接转化为凸优化的时滞正系统结构化最优控制问题，并对其他更一般情形建立基于线性规划的迭代算法。进一步，项目将讨论网络化互联正系统的时滞鲁棒性以及分布式控制。此外，所得控制算法将应用于带有通讯时滞的多智能体系统一致性与集结问题中。本项目将对大规模时滞正系统建立可解性高且实现成本低的控制算法，并将为多智能体系统合作控制、多机器人协调、无线传感器网络功率控制等其他相关领域的建模与分析提供崭新的视角。

正系统是一类内部状态及输入输出具有非负特性的特殊系统。许多表征实际模型的动力系统状态变量都自然带有非负约束。这类系统在房室网络、逻辑网络、拥塞控制、交通流量控制、基因调控网络的建模中有着广泛应用。本项目基于线性规划对偶理论等理论工具，给出了正系统输入输出信号空间上的系统增益的等价刻画，提出的控制设计方法主要由线性规划或者锥规划的形式给出，具有较低的计算复杂度，可用于大规模网络化系统。对于时滞单调系统，本项目系统地建立了基于微分方程比较原理的分析方法。基于此方法，针对具有无界离散时滞的单调系统，给出了系统时滞与其收敛速率之间的显式关系。对于单调系统，基于锥上定义的偏序，以锥规划为基本工具，解决了其输入输出增益的刻画。其应用并不局限于系统本身具有单调性，例如随机系统中亦可以用本项目建立的基于锥规划的单调系统的分析方法。进一步，本项目基于对称锥理论建立了状态空间定义在对称锥上的单调系统的有界实引理。最后，本项目给出了单调系统理论在多智能体包含控制、基于区间观测器的正系统故障检测等多个实际问题中的应用。