随机跳变系统的鲁棒控制与故障检测及其应用
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摘要
实际问题中,大多数控制系统不可避免的会受到各种不确定性因素的影响,包括系统本身的不确定性和外部干扰的不确定性,这些因素都具有一定的随机特性。因此,我们需要充分考虑这些随机因素对系统的影响,建立精确的随机模型。作为一类特殊的随机系统,马尔可夫(Markov)跳变系统的研究很好的推动了随机系统理论的发展,也极大的丰富了控制理论的研究内容。而且,此类系统在实际生产过程中应用相当广泛,很多实际过程,如制造系统、生物系统、经济系统、电力系统以及网络通信系统等,都可以描述为跳变系统模型。因此,研究随机跳变系统理论具有十分重要的理论意义和实际意义。
本文以随机跳变系统为研究对象,同时考虑到不确定性、时滞、非线性等因素的影响。研究的主要内容包括有限短时间控制器设计,基于观测器的鲁棒控制器设计,输出调节控制器设计以及系统故障检测分析等方面。所研究的控制器设计均是基于反馈控制方法,并基于随机Lyapunov-Krosovskii泛函理论,而得到的结果可以用对应的一组线性矩阵不等式(LMI)的进行处理。同时,由于内点算法的提出,相应的LMI求解十分方便。研究的问题切实可行,并具备一定的创新意义。全文共分六章,主要工作包括以下几个方面:
(1).讨论了跳变系统的有限短时间分析与综合问题。首先给出连续时间线性随机跳变系统有限短时间稳定和镇定的相关概念,并将结论推广到离散跳变系统情形。在此基础上,将跳变系统有限短时间稳定性的主要论点推广到不确定跳变系统,时滞跳变系统,离散跳变系统和非线性跳变系统中,并在研究中引入鲁棒控制,H∞控制和模糊控制等设计方法。最后通过仿真验证方法的可行性。
(2).讨论了一类线性不确定时滞跳变系统的H∞控制,无源控制和有限短时间H∞控制问题。针对能量有界的输入噪声,设计了基于观测器的优化鲁棒控制器。基于鲁棒控制理论,通过对重构的观测器系统的分析,提出了使得系统随机稳定且满足一定输入输出条件增益的模态依赖的控制器存在条件。结合构造的Lyapunov-Krosovskii泛函和LMI变换,给出了H∞控制器,无源控制器和有限时间H∞控制器的设计方法,并将其描述为一个优化问题。仿真示例显示,基于观测器设计的优化控制器使系统具有随机稳定性,状态的跟踪性能好,抑制干扰能力强,满足所给的范数指标。
(3).将基于解析模型的状态估计方法用于故障检测,分别研究了含未知扰动和故障的线性和非线性跳变系统故障检测问题。利用构造的Lyapunov-Krosovskii泛函和LMI技术,证明并给出了故障检测观测器或滤波器有解的充分条件,并提出了优化设计方法。理论证明显示,本文设计的观测器或滤波器使系统具有随机稳定性,抑制干扰的能力强,并能灵敏地检测出故障。
(4).应用状态反馈和误差反馈理论讨论了一类线性跳变系统的输出调节问题。针对连续时间跳变系统和离散时间跳变系统给出了满足输出调节的充分性条件。为了获得满足系统随机稳定,并能使得输出渐近跟踪的输出调节控制器,本文应用LMI技术设计出反馈控制器,同时,用半定规划(SDP)优化问题来逼近调节器方程,获取尽可能的近似解。仿真结论显示,所设计的输出调节器是满足要求的,可以在确保闭环系统随机稳定前提下使得输出渐近跟踪。
在论文最后,给出了概括总结和前景展望,并指出了在跳变系统研究中有待进一步解决和完善的问题。
In the practical process, many control systems inevitably encounter various uncertainties and external disturbances, and such factors usually impact the systems in a stochastic way. To establish the precise stochastic model, we should fully consider these affecting factors. As a special class of stochastic systems, the research of Markov jump systems gives an impetus to stochastic control, and these discussions enrich the research and control theory. Moreover, the application of Markov jump systems are more comprehensive, for instance, manufacturing systems, bio-systems, economic systems, electrical power systems and network communication systems, etc. Therefore, the study of stochastic jump systems is very important and can possess real significance.
The main work of this dissertation investigates stochastic jump systems, including such systems subject to uncertainties, time delays and nonlinearities. By using the Lyapunov-Krosovskii functional theory, the research contents mainly relate to finite-time controller design, observer-based robust controller design, output regulator design and fault detection, etc. The controller design approaches are based on feedback control theory and all results can be reduced to a feasible problem of linear matrix inequalities (LMIs). With the aid of the interior-point algorithm, the solutions of LMIs can be easily obtained. The studied problems are feasible and are of innovation. The research works of this dissertation are divided into six chapters, and the main contents are as follows:
(1). The problems for the analysis and synthesis of jump systems are considered. For the continuous system and discrete system, sufficient conditions that the solution of finite time stable and stabilization controller is existed are respectively given and proved by using the constructed Lyapunov-Krasovskii functional approaches and LMIs techniques. And the main results are extended to jump systems with uncertainties, time-delays and nonlinearities, and the designed approaches include robust control, H∞control and fuzzy control. Simulation results illustrate the effectiveness of the developed approaches.
(2). The H∞control, passive control and finite-time H∞control problem of a class of uncertain time-delay jump linear systems are respectively studied. An observer-based optimized robust controller is designed for a given system with energy bounded noise inputs. Based on the robust control theory, the sufficient conditions for the existence of mode-dependent H∞controller, passive controller and finite-time H∞controller are respectively given by analyzing the reconstructed observer systems. By constructing proper Lyapunov-Krasovskii functional and applying LMIs, the design problem of the controllers are derived and described as optimization ones. Simulation results demonstrate that the presented observer-based optimized robust controller makes the systems stochastically stable, have better ability of tracking state and restraining disturbances, and satisfies the presented norm index.
(3). Using the analytic model-based state estimation approach, the fault detection schemes of linear and nonlinear jump system with external disturbances and unknown faults are respectively studied. By constructing proper Lyapunov-Krasovskii functional and applying LMIs technique, sufficient conditions for the solvability of the fault detection problem and the optimized design approach are presented and proved. The designed observer and filter make the systems stochastically stable, have better ability of restraining disturbances, and detect the faults sensitively.
(4). The output regulation problems of jump systems are respectively considered by applying state-feedback and error feedback schemes. With the extension of output regulation to stochastic control, sufficient conditions are obtained continuous-time and discrete-time jump system based on stochastic Lyapunov-Krasovskii functional. In order to ensure the relaxed solutions of the regulation equations, we described the problems as semi-definite programming (SDP) approaches via disciplined convex optimization. The resulting closed-loop system is guaranteed to be stochastically stable and the output tracking is achieved almost asymptotically. Moreover, the output regulation error almost asymptotically converges to zero. Simulation result is also given to illustrate the performance and effectiveness of the proposed approach.
Finally, the conclusions and research prospects are given. Furthermore, some further research work and existing issues for Markov jump systems are also pointed out.
本文以随机跳变系统为研究对象,同时考虑到不确定性、时滞、非线性等因素的影响。研究的主要内容包括有限短时间控制器设计,基于观测器的鲁棒控制器设计,输出调节控制器设计以及系统故障检测分析等方面。所研究的控制器设计均是基于反馈控制方法,并基于随机Lyapunov-Krosovskii泛函理论,而得到的结果可以用对应的一组线性矩阵不等式(LMI)的进行处理。同时,由于内点算法的提出,相应的LMI求解十分方便。研究的问题切实可行,并具备一定的创新意义。全文共分六章,主要工作包括以下几个方面:
(1).讨论了跳变系统的有限短时间分析与综合问题。首先给出连续时间线性随机跳变系统有限短时间稳定和镇定的相关概念,并将结论推广到离散跳变系统情形。在此基础上,将跳变系统有限短时间稳定性的主要论点推广到不确定跳变系统,时滞跳变系统,离散跳变系统和非线性跳变系统中,并在研究中引入鲁棒控制,H∞控制和模糊控制等设计方法。最后通过仿真验证方法的可行性。
(2).讨论了一类线性不确定时滞跳变系统的H∞控制,无源控制和有限短时间H∞控制问题。针对能量有界的输入噪声,设计了基于观测器的优化鲁棒控制器。基于鲁棒控制理论,通过对重构的观测器系统的分析,提出了使得系统随机稳定且满足一定输入输出条件增益的模态依赖的控制器存在条件。结合构造的Lyapunov-Krosovskii泛函和LMI变换,给出了H∞控制器,无源控制器和有限时间H∞控制器的设计方法,并将其描述为一个优化问题。仿真示例显示,基于观测器设计的优化控制器使系统具有随机稳定性,状态的跟踪性能好,抑制干扰能力强,满足所给的范数指标。
(3).将基于解析模型的状态估计方法用于故障检测,分别研究了含未知扰动和故障的线性和非线性跳变系统故障检测问题。利用构造的Lyapunov-Krosovskii泛函和LMI技术,证明并给出了故障检测观测器或滤波器有解的充分条件,并提出了优化设计方法。理论证明显示,本文设计的观测器或滤波器使系统具有随机稳定性,抑制干扰的能力强,并能灵敏地检测出故障。
(4).应用状态反馈和误差反馈理论讨论了一类线性跳变系统的输出调节问题。针对连续时间跳变系统和离散时间跳变系统给出了满足输出调节的充分性条件。为了获得满足系统随机稳定,并能使得输出渐近跟踪的输出调节控制器,本文应用LMI技术设计出反馈控制器,同时,用半定规划(SDP)优化问题来逼近调节器方程,获取尽可能的近似解。仿真结论显示,所设计的输出调节器是满足要求的,可以在确保闭环系统随机稳定前提下使得输出渐近跟踪。
在论文最后,给出了概括总结和前景展望,并指出了在跳变系统研究中有待进一步解决和完善的问题。
In the practical process, many control systems inevitably encounter various uncertainties and external disturbances, and such factors usually impact the systems in a stochastic way. To establish the precise stochastic model, we should fully consider these affecting factors. As a special class of stochastic systems, the research of Markov jump systems gives an impetus to stochastic control, and these discussions enrich the research and control theory. Moreover, the application of Markov jump systems are more comprehensive, for instance, manufacturing systems, bio-systems, economic systems, electrical power systems and network communication systems, etc. Therefore, the study of stochastic jump systems is very important and can possess real significance.
The main work of this dissertation investigates stochastic jump systems, including such systems subject to uncertainties, time delays and nonlinearities. By using the Lyapunov-Krosovskii functional theory, the research contents mainly relate to finite-time controller design, observer-based robust controller design, output regulator design and fault detection, etc. The controller design approaches are based on feedback control theory and all results can be reduced to a feasible problem of linear matrix inequalities (LMIs). With the aid of the interior-point algorithm, the solutions of LMIs can be easily obtained. The studied problems are feasible and are of innovation. The research works of this dissertation are divided into six chapters, and the main contents are as follows:
(1). The problems for the analysis and synthesis of jump systems are considered. For the continuous system and discrete system, sufficient conditions that the solution of finite time stable and stabilization controller is existed are respectively given and proved by using the constructed Lyapunov-Krasovskii functional approaches and LMIs techniques. And the main results are extended to jump systems with uncertainties, time-delays and nonlinearities, and the designed approaches include robust control, H∞control and fuzzy control. Simulation results illustrate the effectiveness of the developed approaches.
(2). The H∞control, passive control and finite-time H∞control problem of a class of uncertain time-delay jump linear systems are respectively studied. An observer-based optimized robust controller is designed for a given system with energy bounded noise inputs. Based on the robust control theory, the sufficient conditions for the existence of mode-dependent H∞controller, passive controller and finite-time H∞controller are respectively given by analyzing the reconstructed observer systems. By constructing proper Lyapunov-Krasovskii functional and applying LMIs, the design problem of the controllers are derived and described as optimization ones. Simulation results demonstrate that the presented observer-based optimized robust controller makes the systems stochastically stable, have better ability of tracking state and restraining disturbances, and satisfies the presented norm index.
(3). Using the analytic model-based state estimation approach, the fault detection schemes of linear and nonlinear jump system with external disturbances and unknown faults are respectively studied. By constructing proper Lyapunov-Krasovskii functional and applying LMIs technique, sufficient conditions for the solvability of the fault detection problem and the optimized design approach are presented and proved. The designed observer and filter make the systems stochastically stable, have better ability of restraining disturbances, and detect the faults sensitively.
(4). The output regulation problems of jump systems are respectively considered by applying state-feedback and error feedback schemes. With the extension of output regulation to stochastic control, sufficient conditions are obtained continuous-time and discrete-time jump system based on stochastic Lyapunov-Krasovskii functional. In order to ensure the relaxed solutions of the regulation equations, we described the problems as semi-definite programming (SDP) approaches via disciplined convex optimization. The resulting closed-loop system is guaranteed to be stochastically stable and the output tracking is achieved almost asymptotically. Moreover, the output regulation error almost asymptotically converges to zero. Simulation result is also given to illustrate the performance and effectiveness of the proposed approach.
Finally, the conclusions and research prospects are given. Furthermore, some further research work and existing issues for Markov jump systems are also pointed out.
引文
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