基于切换理论的周期分段线性正系统分析与滤波研究

Due to their practical applications and theoretical importance, periodic piecewise linear systems have received much attention recently. Most existing results focus on periodic piecewise linear systems with fixed periodic switching laws. The stability and filtering problems of periodic piecewise linear systems with time-varying periodic switching laws are very difficult due to the time-varying switching signals and nonnegative property of subsystems. To solve these problems, this project will use convex cone invariant subspace theory to derive necessary conditions for the nonexistence of time-varying periodic switching laws that guarantee the stability of the considered systems. When this kind of periodic switching laws exists, time-varying periodic switching laws are designed so that one can derive sufficient conditions for the exponential stability of the considered systems; when the disturbance term exists, this project aims to consider the stability and robustness of periodic piecewise linear systems by using the discretized Lyapunov function approach, and time-varying periodic switching laws are designed so that one can obtain sufficient conditions for the stability and robustness of periodic piecewise linear systems composed of all unstable subsystems and periodic piecewise linear systems composed of some unstable subsystems. In addition, the project also aims to consider the robust filtering problem of periodic piecewise linear systems by using the discretized Lyapunov function approach and matrix theory, and propose methods to design the robust reduced-order positive filters. The outcomes of robust filtering will not only reduce filter execution difficulty, but also improve the flexibility of filtering design. The project will provide new technical and theoretical supports for the stability and filtering of actual periodic piecewise linear positive systems such as TCP-like congestion control systems, and mitigating viral escape control systems.

周期分段线性正系统由于其广泛的应用前景和极高的理论价值成为控制界研究的热点。然而，已有成果主要集中在已知周期切换规则的周期分段线性正系统的稳定性分析及鲁棒滤波问题，未知周期切换规则的系统稳定性分析及鲁棒滤波问题尚未解决，未知周期切换规则及子系统非负特征亦增加了此类问题的难度。为了解决这些问题，本项目拟利用凸锥不变子空间理论推导研究对象不存在确保系统稳定的周期切换规则的必要条件；设计周期切换规则，给出研究对象指数稳定的充分条件；当系统存在扰动时，本项目拟利用离散李雅普诺夫函数方法，设计时变周期切换规则，推导由部分及全部不稳定子系统组成的周期分段线性正系统指数稳定性及鲁棒性条件；并利用上述方法及矩阵理论考虑系统的鲁棒降阶滤波问题,提供执行难度较小的鲁棒降阶正滤波器设计方法，提高滤波器设计的灵活性。项目成果将为传染病控制，网络控制等实际周期分段线性正系统的稳定性分析和滤波提供理论和技术支撑。

周期分段线性正系统由于其广泛的应用前景和极高的理论价值成为控制界研究的热点。已有成果主要集中在周期切换规则已定的周期分段线性正系统的稳定性分析及鲁棒滤波问题，然而，周期切换规则未定的系统稳定性分析及鲁棒滤波问题尚未解决，未定周期切换规则及子系统非负特征亦增加了此类问题的难度。为了改进现有结果的不足，本项目利用凸锥不变子空间理论推导研究对象不存在稳定的周期切换规则的必要条件；设计了周期切换规则，给出研究对象稳定的充分条件；当系统存在扰动时，本项目利用离散李雅普诺夫函数方法，设计周期切换规则，推导由部分或全部不稳定子系统组成的周期分段线性正系统指数稳定性鲁棒性条件，并在此基础上考虑了周期分段系统控制器设计问题，给出了周期分段线性正系统计算复杂度较低的控制器设计方案；最后利用离散李雅普诺夫函数方法及矩阵理论考虑研究对象的鲁棒降阶滤波问题,提供执行难度较小的鲁棒降阶正滤波器设计方法，提高了滤波器设计的灵活性。项目成果将为传染病控制，网络控制等实际周期分段线性正系统的稳定性分析和滤波提供理论和技术支撑。