非线性随机脉冲时滞系统的稳定性分析、控制与滤波及应用

In this project, we study the stability, control and filtering of the nonlinear stochastic impulse systems with interval time-varying delays and its application on population dynamics. In order to avoid solving coupled Hamilton-Jacobi equations and Hamilton-Jacobi inequality, a possible scheme is to adopt the global linearization approach and the T-S fuzzy linearized method for the original system. i.e., the nonlinear stochastic system with interval time-varying delays can be interpolated by the convex combination of finite linearized stochastic systems with some approximation errors. Based on Lyapunov-Krasovskii functional theory, free-weighting matrices methoed, integal inequality approach, delay compartmentalization technique, backstepping methoed, It? formula, Sontag's formula, martingale inequalities, Gronwall inequality and linear matrix inequalities (LMIs) methoed, we are to investigate the stability, control and filtering of the equivalent systems resulting from the original systems, including all kinds of stability analysis,robust H_∞ control and filtering, mixed H_2/H_∞control and filtering,output feedback control, passivity and dissipation control. In addition, stochastically comparative theorem is to be established between the nonlinear stochastic time-delay systems with impulses and the nonlinear stochastic time-delay systems. All the results to be obtained are delay-dependent and easy to be veried. This project will enrich the theory and methodology on the nonlinear stochastic time-delay systems with impulses. As an application, impulse stabilization, management and persistence of stochastic time-delay Volterra systems with impulses are also considered. Therefore, this project is important in both theory and practical applications.

本项目拟开展非线性随机脉冲时滞系统稳定性、控制与滤波及应用的研究。拟运用全局线性化方法和T-S模糊模型线性近似方法，通过将拟研究的非线性随机时滞系统表示为有限个线性随机时滞系统的凸组合加上近似误差，基于Lyapunov-Krasovskii泛函理论，自由权矩阵方法、积分不等式方法、时滞划分方法、反推设计法、It?微分公式、Sontag公式、鞅不等式、Gronwall不等式和线性矩阵不等式方法，研究转化后的等价线性系统的稳定性，鲁棒H_∞控制与滤波，混合H_2/H_∞控制与滤波，输出反馈控制和无源性、耗散性控制。拟应用随机比较原理建立随机脉冲时滞系统与随机时滞系统的对应关系。预期所得成果保守性低，且易于验证。作为应用，本项目拟结合随机脉冲时滞种群竞争系统，进行脉冲镇定、生态系统管理和持续性研究。因此，开展本项目的研究具有重要的理论价值和实际意义。

本项目开展了非线性随机时滞系统稳定性、控制与滤波的研究。在全局线性化框架下，通过将非线性随机时滞系统转化为有限个线性随机时滞系统的凸组合加上近似误差，基于Lyapunov-Krasovskii泛函理论，综合运用自由权矩阵方法、积分不等式方法、Ito微分公式、广义Finsler引理，结合鞅不等式、Gronwall不等式、Burkholder-Davis-Gundy 不等式、Chebyshev不等式、Borel-Cantelli 引理和线性矩阵不等式方法等，研究转化后的等价系统的依概率全局渐近稳定性、均方指数稳定性和几乎必然指数稳定性，鲁棒H∞控制与滤波，混合H2/H∞控制与滤波。本项目所采用的研究方法克服了现有方法中二阶（耦合）随机偏微分Hamilton–Jacoby方程或者不等式难以求解解析解甚至数值解的缺陷；与现有文献相比，所得结果不仅显著地降低了保守性，而且较大幅度地简化了计算复杂度。