【作者】 赵燕；

【导师】 孟凡伟；

【作者基本信息】 曲阜师范大学 ， 数学， 2020， 博士

【摘要】 正系统是指当初始条件非负时,输入、输出和状态变量都被约束为非负的动态系统.这种系统已经被广泛应用于许多实际过程的建模,如经济学、生物学、生态学和通信.由于实际控制系统的性能或多或少会受到未建模动态、参数摄动、外源干扰、测量误差等不确定性的影响,因此控制系统鲁棒性研究在控制理论和技术的发展中始终具有重要的地位.针对非线性控制系统的鲁棒性分析,Sontag等学者提出了输入状态稳定性(Input-to-state stability,ISS)及其各种扩展形式,并在各类系统中得到广泛地应用和研究.但是,ISS作为一种有效的非线性系统鲁棒分析工具,还没有应用于正态系统的稳定性分析和综合.本文将ISS应用于连续时间和离散时间的非线性正系统.对于连续时间和离散时间正系统,引入了ISS的一些新的定义.与一般非线性系统的ISS定义不同,我们的ISS定义能够充分反映正系统的状态和输入的正性要求.通过引入最大可分ISS-Lyapunov函数,给出了一般非线性正系统的ISS准则.在此基础上,给出了线性正系统和仿射非线性齐次系统ISS的判定准则.此外,我们也对次齐次协同系统的ISS特性进行了分析.给出了非线性正系统有限时间输入状态稳定性的一个新的定义.稳定性分析的工具方面,除了最大可分ISS-Lyapunov函数,我们还引入了最小类Lyapunov函数.利用它们,对次齐次正系统的解进行了估计,给出了很好的结果.从我们的结果来看,系统的ISS特性仅从系统的微分和代数特征就可以得到判断.仿真实例验证了我们结果的正确性.更多还原

【Abstract】 Positive systems are dynamical systems in which,input,output and state variables are constrained to be nonnegative for all time whenever the initial conditions are nonnegative.This kind of systems have been widely used to model many practical processes such as in economics,biology,ecology and communications.Since the performance of a real control system is affected more or less by uncertainties such as unmodelled dynamics,parameter perturbations,exogenous disturbances,measurement errors etc.,the research on robustness of control systems do always have a vital status in the development of control theory and technology.Aiming at robustness analysis of nonlinear control systems,Sontag’s input-to-state stability(ISS)and its various extensions are developed.And they have been widely applied and studied in various systems.But,as a useful tool of robust analysis for nonlinear systems,ISS has not been used to positive systems for stability analysis and synthesis.In this paper,ISS is applied to continuous-time and discrete-time nonlinear positive systems.For continuous-time and discrete-time positive systems,some new definitions of ISS are introduced.Different to the usual ISS definitions for nonlinear systems,our ISS definitions can fully reflect the positiveness requirements of states and inputs of the positive systems.By introducing the max-separable ISS Lyapunov functions,some ISS criterions are given for general nonlinear positive systems.Based on that,the ISS criterions for lin-ear positive systems and affine nonlinear homogeneous systems are given.Moreover,ISS properties are also analyzed for subhomogeneous cooperative systems.A new definition of finite-time input-to-state stability(FISS)for nonlinear positive systems is given.As useful tools for stability analysis,except the max-separable Lyapunov function,the min-separable Lyapunov-like function is first introduced.Using them,our main results on the solution estimates of subhomogeneous positive systems are provided.From our results,the ISS prop-erties of the systems can be judged only from the differential and algebraic characteristics of the systems.Simulation examples are given to verify the validity of our results.更多还原