Markov跳变系统的滑模控制
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摘要
Markov跳变系统代表一类重要的随机系统,许多工业和科学领域,如制造系统、生化系统、电力系统甚至经济系统,网络通信系统都可以抽象成Markov跳变系统模型。该系统特点是运行过程中常常遭受环境的突然变化、人为干预、大系统内部各子系统间联结方式的改变、工作范围的变化以及系统部件的损坏等随机突变影响。因此研究跳变系统的分析与综合问题可以为这些工程系统控制问题的解决提供理论基础。
滑模控制是一种十分有效的鲁棒控制策略。七十年代以来,滑模变结构系统以其独特的优点和特性引起了西方学者的广泛重视,许多学者对其进行了深入的研究,使得滑模控制理论逐渐发展成为一个相对独立分支。本文根据滑模控制理论的发展现状,以及实际应用对于滑模控制理论提出的新要求,以跳变系统为研究对象,对其滑动模态的稳定性、耗散性以及基于观测器的故障诊断进行了深入的研究和探讨,并给出了相应的研究结果。
本文的主要研究工作概括如下:
(1)针对广义跳变系统的H∞控制问题,提出基于模态跳变的无记忆状态反馈控制器的设计方法。首先利用反证法与线性变换技术保证系统的正则性和因果性,再利用构造的Lyapunov函数和线性矩阵不等式,证明并给出了系统容许性的充分条件,最后,提出了次优H∞控制设计方法。仿真示例表明所设计的控制器对于干扰有很强的抑制能力。
(2)考虑不确定跳变系统的滑模控制问题,其中控制的不确定性是不匹配的且范数有界。利用Lyapunov方法和线性矩阵不等式技术,给出了系统滑动模态渐近稳定的充分条件。利用该线性矩阵不等式的解可以得到所设计的滑模控制律和积分滑模面。从理论上能够证明滑模控制律能保证状态轨迹驱动到指定的切换面上,并给出了仿真验证方法的有效性。
(3)针对既含有连续时间演化,又含有马尔可夫事件驱动的一类非线性不确定混杂动态跳变系统,研究相异滑模面上的耗散控制问题。对于跳变系统所具有的匹配的不确定性,可根据变结构控制理论,设计比例-积分多滑模面及相应滑模控制器。与此同时利用耗散控制理论并结合线性矩阵不等式,得到存在非线性和非匹配不确定性滑模面的无记忆状态耗散控制器。如此,系统滑模耗散控制器便可给出。最后仿真事例验证了定理的正确性。
(4)针对一类具有Markov跳变参数的非线性不确定系统,设计一种新的鲁棒滑模观测器。通过奇异值分解技术对系统进行降阶处理,并提出优化滑模策略,使得对系统范数有解不确定性以及外界干扰具有鲁棒性,滑模观测器中的参数利用线性矩阵不等式技术获得。利用设计的滑模观测器实现故障的检测与估计。
最后是全文的总结与展望。
Markov jump linear systems comprise an important class of stochastic time-variant linear systems that have been in great evidence over the last years. Large number of real systems such as manufactory processes, biology and chemistry systems, power systems, network systems as well as economic systems can be properly described by Markov jump linear system model,which subject to sudden environment changes, changes in the interconnections of subsystems, modification of the operating point of a linearized model of a nonlinear system, random failures and repairs of the components and so on. So the study of Markov jump linear system can provide a theoretical foundation to solve the control problem of these systems.
Sliding mode control is an effective robust control strategy.After the 1970s,the sliding mode control system received extensive attention of the western researchers for its distinct merits and features and it has been deeply studied from the different aspects with many mathematic methods.Now,the sliding mode control already become a relative independence research branch.Based on the study status of Sliding mode control and the new requirements from the practice for the theory of sliding mode control,some problems are studied and discussed in this paper,also the corresponding results are given.the main contents are as follows:
(1) In order to solve the H∞control problem for a class of singular system with markov parameters, a design method based on the memoryless state feedback controller of mode-jumping is proposed. the causal and regular was guaranteed in terms of linear transformation and block matrix. By using the constructed Lyapunov function and linear matrix inequalities, a sufficient condition that the systems were admissible was given and proved, and a sub-optimal design approach is presented. The controller was designed and the prescribed H∞performance condition was satisfied. Simulation results demonstrate that the proposed method is valid and the system has strong restraint ability against disturbance.
(2) The problem of sliding mode control for a class of jumping system is considered.There are uncertainties in both state matrices and input matrix. Moreover,the uncertainties is mismatched norm bounded.By using Lyapunov theory,a sufficient condition in terms of linear matrix inequalities is derived such as the asympotical stability of the full order sliding mode dynamicsis guaranteed. When the inear matrix inequalities are feasible,the design of both sliding modecontrol law and the integral sliding mode surface can be easily obtain via convex optimization.State trajectories are attracted onto the specified sliding surface.A simulation is given to illustrate the effectiveness of the proposed method.
(3) The different sliding mode surface dissipative control problems are discussed for a class of Markov jump nonlinear systems with uncertainties. The systems involve both time-evolving and event-driven mechanisms. In terms of sliding mode theory, different sliding mode surface and controllers are designed for the matched uncertainties. Dissipative control theory and linear matrix inequality are applied to sliding mode surface with nonlinear and unmatched uncertainties. Thereafter, the memoryless state feedback controllers are given. Finally, a sliding mode dissipative controllers are gained. A numerical example demonstrates the effectof the proposed design approach.
(4) An sliding mode observers is presented is presented for a class of Markov jump nonlinear systems with uncertainties.By appling the singular value decomposition to decompose the dynamical sytem.An optimizing sliding mode strategy is given which guarantee robustness to uncertainties of systems and disturbance,A particular design of sliding mode observer is presented for which the parameters can be obtained using inear matrix inequalities techniques.Based on the sliding mode observers,fault detection and estimation issues for a class of nonlinear systems with uncertainty is considered.
The conclusion and perspective are given in the end of the thesis.
滑模控制是一种十分有效的鲁棒控制策略。七十年代以来,滑模变结构系统以其独特的优点和特性引起了西方学者的广泛重视,许多学者对其进行了深入的研究,使得滑模控制理论逐渐发展成为一个相对独立分支。本文根据滑模控制理论的发展现状,以及实际应用对于滑模控制理论提出的新要求,以跳变系统为研究对象,对其滑动模态的稳定性、耗散性以及基于观测器的故障诊断进行了深入的研究和探讨,并给出了相应的研究结果。
本文的主要研究工作概括如下:
(1)针对广义跳变系统的H∞控制问题,提出基于模态跳变的无记忆状态反馈控制器的设计方法。首先利用反证法与线性变换技术保证系统的正则性和因果性,再利用构造的Lyapunov函数和线性矩阵不等式,证明并给出了系统容许性的充分条件,最后,提出了次优H∞控制设计方法。仿真示例表明所设计的控制器对于干扰有很强的抑制能力。
(2)考虑不确定跳变系统的滑模控制问题,其中控制的不确定性是不匹配的且范数有界。利用Lyapunov方法和线性矩阵不等式技术,给出了系统滑动模态渐近稳定的充分条件。利用该线性矩阵不等式的解可以得到所设计的滑模控制律和积分滑模面。从理论上能够证明滑模控制律能保证状态轨迹驱动到指定的切换面上,并给出了仿真验证方法的有效性。
(3)针对既含有连续时间演化,又含有马尔可夫事件驱动的一类非线性不确定混杂动态跳变系统,研究相异滑模面上的耗散控制问题。对于跳变系统所具有的匹配的不确定性,可根据变结构控制理论,设计比例-积分多滑模面及相应滑模控制器。与此同时利用耗散控制理论并结合线性矩阵不等式,得到存在非线性和非匹配不确定性滑模面的无记忆状态耗散控制器。如此,系统滑模耗散控制器便可给出。最后仿真事例验证了定理的正确性。
(4)针对一类具有Markov跳变参数的非线性不确定系统,设计一种新的鲁棒滑模观测器。通过奇异值分解技术对系统进行降阶处理,并提出优化滑模策略,使得对系统范数有解不确定性以及外界干扰具有鲁棒性,滑模观测器中的参数利用线性矩阵不等式技术获得。利用设计的滑模观测器实现故障的检测与估计。
最后是全文的总结与展望。
Markov jump linear systems comprise an important class of stochastic time-variant linear systems that have been in great evidence over the last years. Large number of real systems such as manufactory processes, biology and chemistry systems, power systems, network systems as well as economic systems can be properly described by Markov jump linear system model,which subject to sudden environment changes, changes in the interconnections of subsystems, modification of the operating point of a linearized model of a nonlinear system, random failures and repairs of the components and so on. So the study of Markov jump linear system can provide a theoretical foundation to solve the control problem of these systems.
Sliding mode control is an effective robust control strategy.After the 1970s,the sliding mode control system received extensive attention of the western researchers for its distinct merits and features and it has been deeply studied from the different aspects with many mathematic methods.Now,the sliding mode control already become a relative independence research branch.Based on the study status of Sliding mode control and the new requirements from the practice for the theory of sliding mode control,some problems are studied and discussed in this paper,also the corresponding results are given.the main contents are as follows:
(1) In order to solve the H∞control problem for a class of singular system with markov parameters, a design method based on the memoryless state feedback controller of mode-jumping is proposed. the causal and regular was guaranteed in terms of linear transformation and block matrix. By using the constructed Lyapunov function and linear matrix inequalities, a sufficient condition that the systems were admissible was given and proved, and a sub-optimal design approach is presented. The controller was designed and the prescribed H∞performance condition was satisfied. Simulation results demonstrate that the proposed method is valid and the system has strong restraint ability against disturbance.
(2) The problem of sliding mode control for a class of jumping system is considered.There are uncertainties in both state matrices and input matrix. Moreover,the uncertainties is mismatched norm bounded.By using Lyapunov theory,a sufficient condition in terms of linear matrix inequalities is derived such as the asympotical stability of the full order sliding mode dynamicsis guaranteed. When the inear matrix inequalities are feasible,the design of both sliding modecontrol law and the integral sliding mode surface can be easily obtain via convex optimization.State trajectories are attracted onto the specified sliding surface.A simulation is given to illustrate the effectiveness of the proposed method.
(3) The different sliding mode surface dissipative control problems are discussed for a class of Markov jump nonlinear systems with uncertainties. The systems involve both time-evolving and event-driven mechanisms. In terms of sliding mode theory, different sliding mode surface and controllers are designed for the matched uncertainties. Dissipative control theory and linear matrix inequality are applied to sliding mode surface with nonlinear and unmatched uncertainties. Thereafter, the memoryless state feedback controllers are given. Finally, a sliding mode dissipative controllers are gained. A numerical example demonstrates the effectof the proposed design approach.
(4) An sliding mode observers is presented is presented for a class of Markov jump nonlinear systems with uncertainties.By appling the singular value decomposition to decompose the dynamical sytem.An optimizing sliding mode strategy is given which guarantee robustness to uncertainties of systems and disturbance,A particular design of sliding mode observer is presented for which the parameters can be obtained using inear matrix inequalities techniques.Based on the sliding mode observers,fault detection and estimation issues for a class of nonlinear systems with uncertainty is considered.
The conclusion and perspective are given in the end of the thesis.
引文
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