非线性与时滞不确定随机系统的鲁棒稳定性与控制研究
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摘要
现实系统中常常存在许多不确定的随机因素,当所考虑的系统对精度要求不高或随机因素可以忽略时,系统常被简化为确定性系统模型以便于分析和综合。考虑随机因素,把系统随机不确定性建模为随机过程,这样的随机系统能更真实、更准确地反映实际工程技术中的系统运动规律。由于动态随机系统用随机微分方程来描述,于是随机微分方程日益受到人们的重视,越来越广泛的应用于模型的建立和分析中。同时,随机控制理论也广泛地应用于经济、人口系统等社会领域以及航空航天、导航与控制、制造工程等工程领域。随机系统的研究已成为现代控制理论研究中的一个热点问题。
在实际系统中,非线性、时滞是普遍存在的,通常时滞是引起系统不稳定或产生振荡的根源。控制系统中时滞的存在,使理论分析和工程应用增加了特殊的难度。另一方面,被控系统往往受到一些诸如参数误差、未建模动态以及不确定的外界干扰等不确定因素的影响,系统模型具有某种不确定性。控制界针对不确定性对系统性能影响的研究产生了鲁棒控制理论。因此,非线性随机系统和不确定随机时滞系统的鲁棒稳定性和控制研究具有重要的理论研究意义和实际应用价值。本文利用随机Lyapunov稳定性理论,模型变换及自由权矩阵方法,借助于It(o|^)微分公式、Schur补引理,线性矩阵不等式、一些重要引理和不等式等工具,研究了非线性随机系统和不确定随机时滞系统的鲁棒稳定性、鲁棒镇定和一些控制问题。
本文主要内容和成果有以下几个方面:
1.介绍了随机系统的研究背景和意义,简述了非线性随机系统和不确定随机时滞系统的研究进展,也对随机系统的基本理论作了回顾,包括随机过程、Brown运动、It(o|^)随机系统的稳定性概念及Lyapunov随机稳定性定理等。
2.研究了具有离散时滞项和分布时滞项的不确定非线性随机系统的鲁棒指数稳定性。在过去一些年的研究中,分布时滞项常被看成所讨论系统的扰动,因此,所给出的稳定性判据也许无法运用也许在某种情形下具有保守性。本文中不再把分布时滞项看作扰动项,为得到时滞相关的稳定性条件,采用了保守性小的广义系统变换方法。不同于一般的广义系统变换只是对离散时滞项进行变换,而是对系统中的离散时滞项和分布时滞项均用广义系统变换重写。并构造出新的Lyapunov-krasovski泛函,结合积分不等式技巧,给出了基于线性矩阵不等式(LMI)的时滞相关的指数稳定的充分条件。
3.研究了凸多面体不确定变时滞非线性随机系统的参数相关的鲁棒稳定性问题。所讨论的系统模型更广泛即带有非线性项、同时具有分布时滞和离散时滞,并且不要求时变时滞的导数小于1。通过构造参数相关的Lyapunov-Krasovskii泛函,并运用自由权矩阵方法,得到了时滞相关及参数相关的鲁棒稳定性的充分条件,改进了以往文献的结果。
4.研究了两类非线性随机系统的控制问题。对一类无穷维非线性随机系统——It(o|^)随机KdVB方程,讨论了其边界自适应控制问题,给出了参数自适应控制律及边界反馈控制律的设计。其次研究了一类非线性随机时滞扰动系统的状态反馈控制问题,利用Razumikhin技巧和Backstepping方法设计出了与时滞无关的非线性的状态反馈控制器。
5.研究了范数界不确定随机时滞系统的时滞相关指数镇定问题。通过引入参数化的中立型模型变换,构造Lyapunov-Krasovskii泛函,得到了完全基于LMI的时滞相关的镇定条件。
6.研究了凸多面体不确定随机时滞系统的参数依赖鲁棒镇定问题。一些例子表明,有些系统不能用固定增益(参数无关的控制器)来镇定,但是可以用参数依赖的控制器镇定。本文把参数相关的Lyapunov-Krasovskii泛函方法和自由权矩阵方法相结合,得到完全基于LMI的时滞相关及参数相关的鲁棒镇定的充分条件。由于在引入自由权矩阵时,减少了所用的自由矩阵数目,使得给出的控制器更易于实现。
7.研究了凸多面体不确定随机时滞系统的非脆弱鲁棒镇定问题,其中控制器不确定性采用的是凸多面体不确定描述(也是比较自然的描述)。通过构造合适的与参数相关的Lyapunov-Krasovskii泛函,运用自由权矩阵方法,使得所得到的LMI结果中不存在Lyapunov矩阵变量和系统矩阵的乘积项,得到了完全基于LMI的时滞相关非脆弱鲁棒指数镇定的充分条件。所给出的非脆弱状态反馈控制器,可以通过求解LMI来获得。最后总结全文并提出了进一步研究的方向。
There are some uncertain random factors in real system. When accuracy equirement isn’t high or random factors can be ignored, the systems model is often simplified as deterministic systems model. The deterministic systems model is convenient for systems analysis and synthesis. When random uncertainty is modeled as stochastic processes, the stochastic systems can describe actual engineering systems more really and more accurately. The dynamic stochastic system is described by stochastic differential equation. So stochastic differential equation has been paid more attention and is widely used in the systems model and systems analysis. Meanwhile, stochastic control theory is also widely used in the economic, demographic and other social areas, as well as aviation and aerospace, navigation and control, manufacturing engineering and other engineering fields. Stochastic systems theory has become a popular research field of modern control theory.
Nonlinear and time-delay of system are commonly encountered in real systems. Time-delay is frequently a source of instability or oscillation. The existence of time-delay of control systems increases the difficulty of theoretical analysis and engineering applicationl. On the other hand, the controlled systems are often impacted by parameter error, unmodeled dynamics and uncertain external disturbances. The system model has some uncertainties. The studies on the impact of uncertainty for system performance produce robust control theory. Therefore, it is of a great importance in theoretical and practical application to research into robust stability and controlof nonlinear stochastic systems and uncertain stochastic time-delay systems. By using stochastic Lyapunov stability theory, model transform and free-weighting matrix method, and by means of It(o|^) differential formula, Schur complement lemma, linear matrix inequality, some important lemmas and inequalities, we study the robust stability, robust stabilization and control of nonlinear stochastic systems and uncertain stochastic time-delay systems.
The main contents and contribution of this dissertation are summarized as followings:
1. The first two chapters give an introduction to the background and significance of stochastic systems, and the latest progress in the stability and control of non-linear stochastic systems and uncertain stochastic time-delay systems. Then the basic theory of stochastic systems is reviewed, including the stochastic process, Brown motion, It(o|^) stochastic stability and Lyapunov stochastic stability theorem.
2. The robust exponential stability for a class of uncertain nonlinear stochastic systems with discrete and distributed delays is investigated. During the past years, distributed-delay term is often looked as a perturbation of the discussed systems. Therefore, the stability criteria given in these references may not work or may be conservative in some cases. In this paper, the distributed-delay term doesn't be treated as a perturbation. We ues descriptor model transformation with less conconservatism to obtain delay–dependent stability conditions, and different from the usual descriptor model transformation only used in the discrete-delay term, that is, the descriptor model transformation is not only used in discrete-delay term but also in distributed-delay term. Combined with a new type of Lyapunov-Krasovskii functional and integral inequality technique, delay-dependent robust exponential stability in mean square criteria are derived in terms of linear matrix inequalities (LMI).
3. The robust stability problem for nonlinear time-varying delay stochastic systems with polytopic-type uncertainties is discussed. Since nonlinear term, distributed delay and discrete delay term in the uncertain systems, the model becomes more general and the upper bound of derivative of the delay term needn’t less than 1. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stability conditions are presented in terms of linear matrix inequalities. The results in this paper improve the existing stability criteria.
4. The control problem for two nonlinear stochastic systems are considered. The adaptive boundary control for a class of infinite-dimensional nonlinear stochastic (It(o|^) stochastic KdVB equations) is discussed. A nonlinear boundary control law and an adaptation law are proposed. Secondly, the state feedback control for a class of stochastic nonlinear systems with time-delay disturbs is investigated. Based on the technique of Razumikhin and backstepping method, delay-independent state feedback controller is given.
5. The delay-dependent robust exponential stabilization for uncertain stochastic systems with time delay is investigated. Based on parameterized neutral model transformation and Lyapunov-Krasovskii functional approach, a sufficient condition of delay-dependent exponential stability in mean square for the closed-loop systems is derived in terms of linear matrix inequality.
6. The parameter-dependent state feedback control problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed. Some examples show that many systems can’t be stabilization by fixed gain matrix (parameter-independent controller), but can be stabilization by parameter-dependent controller. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stabilization conditions are presented in terms of linear matrix inequalities. Since the number of free-weighting matrices is reduced when introducing free-weighting matrix, the given parameter-dependent controller is easier to implement.
7. The non-fragile stabilization problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed, and the perturbed matrix in the actual implemented controller is assumed satisfying polytopic-type uncertainties (a sort of more natural description). By using parameter-dependent Lyapunov-Krasovskii functional method and free-weighting matrix method, the product terms for Lyapunov matrix and system matrix are separated. Then a non-fragile robust exponential stabilization condition for stochastic delay-varying systems with polytopic-type uncertainties is proposed in terms of linear matrix inequalities. The non-fragile state feedback controller can be obtained by solving LMI.
Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.
在实际系统中,非线性、时滞是普遍存在的,通常时滞是引起系统不稳定或产生振荡的根源。控制系统中时滞的存在,使理论分析和工程应用增加了特殊的难度。另一方面,被控系统往往受到一些诸如参数误差、未建模动态以及不确定的外界干扰等不确定因素的影响,系统模型具有某种不确定性。控制界针对不确定性对系统性能影响的研究产生了鲁棒控制理论。因此,非线性随机系统和不确定随机时滞系统的鲁棒稳定性和控制研究具有重要的理论研究意义和实际应用价值。本文利用随机Lyapunov稳定性理论,模型变换及自由权矩阵方法,借助于It(o|^)微分公式、Schur补引理,线性矩阵不等式、一些重要引理和不等式等工具,研究了非线性随机系统和不确定随机时滞系统的鲁棒稳定性、鲁棒镇定和一些控制问题。
本文主要内容和成果有以下几个方面:
1.介绍了随机系统的研究背景和意义,简述了非线性随机系统和不确定随机时滞系统的研究进展,也对随机系统的基本理论作了回顾,包括随机过程、Brown运动、It(o|^)随机系统的稳定性概念及Lyapunov随机稳定性定理等。
2.研究了具有离散时滞项和分布时滞项的不确定非线性随机系统的鲁棒指数稳定性。在过去一些年的研究中,分布时滞项常被看成所讨论系统的扰动,因此,所给出的稳定性判据也许无法运用也许在某种情形下具有保守性。本文中不再把分布时滞项看作扰动项,为得到时滞相关的稳定性条件,采用了保守性小的广义系统变换方法。不同于一般的广义系统变换只是对离散时滞项进行变换,而是对系统中的离散时滞项和分布时滞项均用广义系统变换重写。并构造出新的Lyapunov-krasovski泛函,结合积分不等式技巧,给出了基于线性矩阵不等式(LMI)的时滞相关的指数稳定的充分条件。
3.研究了凸多面体不确定变时滞非线性随机系统的参数相关的鲁棒稳定性问题。所讨论的系统模型更广泛即带有非线性项、同时具有分布时滞和离散时滞,并且不要求时变时滞的导数小于1。通过构造参数相关的Lyapunov-Krasovskii泛函,并运用自由权矩阵方法,得到了时滞相关及参数相关的鲁棒稳定性的充分条件,改进了以往文献的结果。
4.研究了两类非线性随机系统的控制问题。对一类无穷维非线性随机系统——It(o|^)随机KdVB方程,讨论了其边界自适应控制问题,给出了参数自适应控制律及边界反馈控制律的设计。其次研究了一类非线性随机时滞扰动系统的状态反馈控制问题,利用Razumikhin技巧和Backstepping方法设计出了与时滞无关的非线性的状态反馈控制器。
5.研究了范数界不确定随机时滞系统的时滞相关指数镇定问题。通过引入参数化的中立型模型变换,构造Lyapunov-Krasovskii泛函,得到了完全基于LMI的时滞相关的镇定条件。
6.研究了凸多面体不确定随机时滞系统的参数依赖鲁棒镇定问题。一些例子表明,有些系统不能用固定增益(参数无关的控制器)来镇定,但是可以用参数依赖的控制器镇定。本文把参数相关的Lyapunov-Krasovskii泛函方法和自由权矩阵方法相结合,得到完全基于LMI的时滞相关及参数相关的鲁棒镇定的充分条件。由于在引入自由权矩阵时,减少了所用的自由矩阵数目,使得给出的控制器更易于实现。
7.研究了凸多面体不确定随机时滞系统的非脆弱鲁棒镇定问题,其中控制器不确定性采用的是凸多面体不确定描述(也是比较自然的描述)。通过构造合适的与参数相关的Lyapunov-Krasovskii泛函,运用自由权矩阵方法,使得所得到的LMI结果中不存在Lyapunov矩阵变量和系统矩阵的乘积项,得到了完全基于LMI的时滞相关非脆弱鲁棒指数镇定的充分条件。所给出的非脆弱状态反馈控制器,可以通过求解LMI来获得。最后总结全文并提出了进一步研究的方向。
There are some uncertain random factors in real system. When accuracy equirement isn’t high or random factors can be ignored, the systems model is often simplified as deterministic systems model. The deterministic systems model is convenient for systems analysis and synthesis. When random uncertainty is modeled as stochastic processes, the stochastic systems can describe actual engineering systems more really and more accurately. The dynamic stochastic system is described by stochastic differential equation. So stochastic differential equation has been paid more attention and is widely used in the systems model and systems analysis. Meanwhile, stochastic control theory is also widely used in the economic, demographic and other social areas, as well as aviation and aerospace, navigation and control, manufacturing engineering and other engineering fields. Stochastic systems theory has become a popular research field of modern control theory.
Nonlinear and time-delay of system are commonly encountered in real systems. Time-delay is frequently a source of instability or oscillation. The existence of time-delay of control systems increases the difficulty of theoretical analysis and engineering applicationl. On the other hand, the controlled systems are often impacted by parameter error, unmodeled dynamics and uncertain external disturbances. The system model has some uncertainties. The studies on the impact of uncertainty for system performance produce robust control theory. Therefore, it is of a great importance in theoretical and practical application to research into robust stability and controlof nonlinear stochastic systems and uncertain stochastic time-delay systems. By using stochastic Lyapunov stability theory, model transform and free-weighting matrix method, and by means of It(o|^) differential formula, Schur complement lemma, linear matrix inequality, some important lemmas and inequalities, we study the robust stability, robust stabilization and control of nonlinear stochastic systems and uncertain stochastic time-delay systems.
The main contents and contribution of this dissertation are summarized as followings:
1. The first two chapters give an introduction to the background and significance of stochastic systems, and the latest progress in the stability and control of non-linear stochastic systems and uncertain stochastic time-delay systems. Then the basic theory of stochastic systems is reviewed, including the stochastic process, Brown motion, It(o|^) stochastic stability and Lyapunov stochastic stability theorem.
2. The robust exponential stability for a class of uncertain nonlinear stochastic systems with discrete and distributed delays is investigated. During the past years, distributed-delay term is often looked as a perturbation of the discussed systems. Therefore, the stability criteria given in these references may not work or may be conservative in some cases. In this paper, the distributed-delay term doesn't be treated as a perturbation. We ues descriptor model transformation with less conconservatism to obtain delay–dependent stability conditions, and different from the usual descriptor model transformation only used in the discrete-delay term, that is, the descriptor model transformation is not only used in discrete-delay term but also in distributed-delay term. Combined with a new type of Lyapunov-Krasovskii functional and integral inequality technique, delay-dependent robust exponential stability in mean square criteria are derived in terms of linear matrix inequalities (LMI).
3. The robust stability problem for nonlinear time-varying delay stochastic systems with polytopic-type uncertainties is discussed. Since nonlinear term, distributed delay and discrete delay term in the uncertain systems, the model becomes more general and the upper bound of derivative of the delay term needn’t less than 1. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stability conditions are presented in terms of linear matrix inequalities. The results in this paper improve the existing stability criteria.
4. The control problem for two nonlinear stochastic systems are considered. The adaptive boundary control for a class of infinite-dimensional nonlinear stochastic (It(o|^) stochastic KdVB equations) is discussed. A nonlinear boundary control law and an adaptation law are proposed. Secondly, the state feedback control for a class of stochastic nonlinear systems with time-delay disturbs is investigated. Based on the technique of Razumikhin and backstepping method, delay-independent state feedback controller is given.
5. The delay-dependent robust exponential stabilization for uncertain stochastic systems with time delay is investigated. Based on parameterized neutral model transformation and Lyapunov-Krasovskii functional approach, a sufficient condition of delay-dependent exponential stability in mean square for the closed-loop systems is derived in terms of linear matrix inequality.
6. The parameter-dependent state feedback control problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed. Some examples show that many systems can’t be stabilization by fixed gain matrix (parameter-independent controller), but can be stabilization by parameter-dependent controller. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stabilization conditions are presented in terms of linear matrix inequalities. Since the number of free-weighting matrices is reduced when introducing free-weighting matrix, the given parameter-dependent controller is easier to implement.
7. The non-fragile stabilization problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed, and the perturbed matrix in the actual implemented controller is assumed satisfying polytopic-type uncertainties (a sort of more natural description). By using parameter-dependent Lyapunov-Krasovskii functional method and free-weighting matrix method, the product terms for Lyapunov matrix and system matrix are separated. Then a non-fragile robust exponential stabilization condition for stochastic delay-varying systems with polytopic-type uncertainties is proposed in terms of linear matrix inequalities. The non-fragile state feedback controller can be obtained by solving LMI.
Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.
引文
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