几类切换系统的滑模控制
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摘要
切换系统是由一组连续或离散动态子系统组成,并按某种切换规则在各子系统之间切换的混杂系统,是混杂系统理论与应用研究中非常活跃的一个分支。由于众多实际工程系统,比如化工过程、交通传输过程、计算机控制系统、通讯工业等可以建模成切换系统,因而在过去的几年内,切换系统已经受到国内外学者的广泛关注。另外,在许多智能控制器的设计上,为了克服传统的单一控制器的缺点,为提高系统性能,而采取多控制器切换的智能控制系统,这样其闭环系统也形成了切换系统。另一方面,滑模控制具有对参数摄动不敏感、抗干扰能力强和响应速度快等优点,近二十年滑模控制方法已经取得了许多重要的成果,并用于研究时变切换系统、不确定系统、时延系统和随机系统等,但有关切换系统的滑模控制很少有报道,有待进一步发展和完善。本文将在前人工作的基础上,针对切换系统中的参数摄动和非线性扰动,研究切换系统的滑模控制问题,并对一类Markov切换系统,进行了滑模控制器设计。论文的主要研究工作可以概括如下:
1.第1章首先简单介绍了问题的研究背景、切换系统的基本概念、分析方法、设计方法和当前国内外在该领域的研究现状。其次,介绍了滑模控制的基本知识、特性、设计方法和应用,为后续工作提供了充分的理论基础。最后介绍了本论文的主要研究内容。
2.第2章回顾了切换系统的稳定性、能控性和能观测性分析以及切换系统的滑模控制。首先给出了公共二次Lyapunov函数存在的条件和构造方法,并用公共二次Lyapunov函数分析了切换线性系统的稳定性。其次用多Lyapunov函数法分析了切换线性系统的稳定性。同时总结了以上两种方法各自的特点及在切换系统稳定性分析方面的保守性和适用性。最后分析了切换系统的控制以及滑模控制用于切换系统的必要性,总结了切换系统理想滑模面和滑模控制器的设计方法。
3.第3章研究了线性切换系统基于观测器的鲁棒控制问题。针对系统状态不易直接测量,或不可能实际上获得系统全部状态变量的切换系统,应用公共二次Lyapunov函数设计了切换系统的降阶观测器,使得观测误差系统在任意切换下是渐进稳定的。并基于所设计的观测器进一步获得切换系统的理想滑模面和滑模控制器。
4.第4章研究了切换系统的输出反馈滑模控制问题。首先,针对一类无扰动的线性切换系统,设计了输出反馈滑模函数,给出了滑模系数存在的充分必要条件,设计了输出反馈滑模控制器。其次,在第一节的基础上进一步研究了非线性扰动切换系统的降阶输出反馈滑模控制问题。给出了切换系统滑动模态可达条件,同时设计了各子系统的理想滑模面和滑模控制器,使闭环切换系统对于未知非线性扰动具有鲁棒性。
5.第5章研究离散线性切换系统的滑模控制问题。首先,基于多Lyapunov函数设计了任意切换下稳定的滑模面。采用滑模函数预测模型及滑模参考轨迹,结合预测控制中的反馈校正和滚动优化方法,设计了一组滑模控制器,使得闭环系统在任意切换下渐近稳定。其次,针对传统变结构控制应用于离散系统时易失去其鲁棒性的问题,提出了一种采用带调整参数递归式滑模函数的离散滑模控制策略,改进了离散时间滑模控制系统的趋近律方法,并针对不确定因素的影响设计了无须知道扰动上界的估计器,得到了可实现的离散滑模控制律。
6.第6章研究一类不确定Markov切换系统的滑模控制问题。引入线性变换将系统变换为标准型,设计了滑模面,应用线性矩阵不等式,给出了系统在滑模面上均方意义下指数稳定的充分条件。设计了确保系统运动轨迹在有限时间内到达滑模面,并且能够保持均方意义下指数稳定的滑模控制器。
最后是全文的总结以及展望。
Switched systems form a class of hybrid systems consisting of a family of subsystems described by continuous-time or discrete-time dynamics, and a rule specifying the switching among them. Switched systems have received increasing attentions in the past few years, since many real-world systems such as chemical processes, transportation systems, computer-controlled systems, and communica-tion industries can be modelled as switched systems. More importantly, many intelligent control strategies are designed based on the idea of controllers switching to overcome the shortcoming of the traditionally used single controller and to improve the performance, thus making the corresponding closed-loop sys-tems to become switched systems. On the other hand, sliding mode control (SMC) has proven to be an effective robust control strategy for incompletely modelled or nonlinear systems since its appearance in the 1950s. In the past two decades, sliding mode control has successfully been applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines. Sliding mode control utilizes a discontinuous control to force the system state trajectories to some predefined sliding surfaces on which the system has desired properties such as stability, disturbance rejection capabil-ity, and tracking ability. Many important results have been reported for this kind of control strategy, which includs some different systems, such as uncertain sys-tems, time-delay systems, and stochastic systems.
The main contents can be stated as follows:
1. In Chapter 1, we first present the background, the basic conceptions, and some related approaches for the switched systems. Then, we sum-marize and analyze the basic conception, properties, design approaches and recent study status of SMC, from which we can find some desider-ated problems to be studied. Finally, we introduce the main contents and the background of several important systems and control problems in this thesis.
2. In Chapter 2, we introduce the analysis approaches of the stability, controllability and the observability for the switched systems. Firstly, the construction and the existence analysis of CQLF for switched sys-tems are discussed. Moreover, the stability conditions by using CQLF approach are established. Secondly, the stability conditions by using multiply Lyapunov function approach are also established. The switching law design approaches are summarized. The conservatism of the stability conditions obtained by using the above-mentioned two ap-proaches is compared. Finally, the controllability and the observability conditions for the switched systems are given respectively.
3. In Chapter 3, we investigate the SMC and the observer designs of a class of switched systems with a nonlinear disturbance based on the multiply Lyapunov function approach. The sliding mode controller and the switching law are designed by using the estimated states, which guarantee the asymptotic stability of the closed-loop systems.
4. In Chapter 4, we address the problem of output feedback SMC of switched systems. Firtly, for a class of disturbance-free linear switched systems, we design a sliding function by using only output information, and propose the existence condition of the sliding mode dynamics. Based on the obtained condition, we get the solution to the parameters of the designed sliding function. Also, we synthetize the output feed-back sliding mode controller and the switching law. Secondly, based on the results obtained in this chapter, we further investigate the problem of the reduced-order output feedback SMC of switched systems with nonlinear disturbance. The condition for the existence of sliding mode dynamics is proposed. Also, the controllers for all the subsystems and the switching law are designed, which guatantee the robustness of the closed -loop systems to the unknown disturbance.
5. In Chapter 5, our attention is focused on the studying of SMC of dis-crete-time switched systems. Firstly, by applying the feedback control and the rolling optimaization approaches in the predictive control, we design a class of predictive sliding mode controllers, which guarantee the asymptotic stability of the closed-loop system under arbitrary switching. Secondly, as for the robustness-loss of the traditional SMC of discrete-time systems, we proposed a discrete-time SMC strategy containing a recursive sliding surface function. Moreover, the reaching law approaches used in discrete-time SMC systems are improved. In addition, as for the systems with that its uncertainties and disturbances’upper bounds are unknown, an admissible discrete-time SMC law is proposed.
6. In Chapter 6, we study the SMC of a class of Markov switching sys-tems. A model transformation is introduced firstly, by which the origi-nal system becomes a“regular”form. A linear sliding surface is de-signed. A sufficient condition of the mean-square exponential stability is proposed for the sliding mode dynamics by using linear matrix ine-quality approach. Finally, a sliding mode controller is synthetized, which guarantees the reachability of the system’s trajectories to pre-defined sliding surface in a finite time.
The conclusion and perspective are given in the end of the paper.
1.第1章首先简单介绍了问题的研究背景、切换系统的基本概念、分析方法、设计方法和当前国内外在该领域的研究现状。其次,介绍了滑模控制的基本知识、特性、设计方法和应用,为后续工作提供了充分的理论基础。最后介绍了本论文的主要研究内容。
2.第2章回顾了切换系统的稳定性、能控性和能观测性分析以及切换系统的滑模控制。首先给出了公共二次Lyapunov函数存在的条件和构造方法,并用公共二次Lyapunov函数分析了切换线性系统的稳定性。其次用多Lyapunov函数法分析了切换线性系统的稳定性。同时总结了以上两种方法各自的特点及在切换系统稳定性分析方面的保守性和适用性。最后分析了切换系统的控制以及滑模控制用于切换系统的必要性,总结了切换系统理想滑模面和滑模控制器的设计方法。
3.第3章研究了线性切换系统基于观测器的鲁棒控制问题。针对系统状态不易直接测量,或不可能实际上获得系统全部状态变量的切换系统,应用公共二次Lyapunov函数设计了切换系统的降阶观测器,使得观测误差系统在任意切换下是渐进稳定的。并基于所设计的观测器进一步获得切换系统的理想滑模面和滑模控制器。
4.第4章研究了切换系统的输出反馈滑模控制问题。首先,针对一类无扰动的线性切换系统,设计了输出反馈滑模函数,给出了滑模系数存在的充分必要条件,设计了输出反馈滑模控制器。其次,在第一节的基础上进一步研究了非线性扰动切换系统的降阶输出反馈滑模控制问题。给出了切换系统滑动模态可达条件,同时设计了各子系统的理想滑模面和滑模控制器,使闭环切换系统对于未知非线性扰动具有鲁棒性。
5.第5章研究离散线性切换系统的滑模控制问题。首先,基于多Lyapunov函数设计了任意切换下稳定的滑模面。采用滑模函数预测模型及滑模参考轨迹,结合预测控制中的反馈校正和滚动优化方法,设计了一组滑模控制器,使得闭环系统在任意切换下渐近稳定。其次,针对传统变结构控制应用于离散系统时易失去其鲁棒性的问题,提出了一种采用带调整参数递归式滑模函数的离散滑模控制策略,改进了离散时间滑模控制系统的趋近律方法,并针对不确定因素的影响设计了无须知道扰动上界的估计器,得到了可实现的离散滑模控制律。
6.第6章研究一类不确定Markov切换系统的滑模控制问题。引入线性变换将系统变换为标准型,设计了滑模面,应用线性矩阵不等式,给出了系统在滑模面上均方意义下指数稳定的充分条件。设计了确保系统运动轨迹在有限时间内到达滑模面,并且能够保持均方意义下指数稳定的滑模控制器。
最后是全文的总结以及展望。
Switched systems form a class of hybrid systems consisting of a family of subsystems described by continuous-time or discrete-time dynamics, and a rule specifying the switching among them. Switched systems have received increasing attentions in the past few years, since many real-world systems such as chemical processes, transportation systems, computer-controlled systems, and communica-tion industries can be modelled as switched systems. More importantly, many intelligent control strategies are designed based on the idea of controllers switching to overcome the shortcoming of the traditionally used single controller and to improve the performance, thus making the corresponding closed-loop sys-tems to become switched systems. On the other hand, sliding mode control (SMC) has proven to be an effective robust control strategy for incompletely modelled or nonlinear systems since its appearance in the 1950s. In the past two decades, sliding mode control has successfully been applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines. Sliding mode control utilizes a discontinuous control to force the system state trajectories to some predefined sliding surfaces on which the system has desired properties such as stability, disturbance rejection capabil-ity, and tracking ability. Many important results have been reported for this kind of control strategy, which includs some different systems, such as uncertain sys-tems, time-delay systems, and stochastic systems.
The main contents can be stated as follows:
1. In Chapter 1, we first present the background, the basic conceptions, and some related approaches for the switched systems. Then, we sum-marize and analyze the basic conception, properties, design approaches and recent study status of SMC, from which we can find some desider-ated problems to be studied. Finally, we introduce the main contents and the background of several important systems and control problems in this thesis.
2. In Chapter 2, we introduce the analysis approaches of the stability, controllability and the observability for the switched systems. Firstly, the construction and the existence analysis of CQLF for switched sys-tems are discussed. Moreover, the stability conditions by using CQLF approach are established. Secondly, the stability conditions by using multiply Lyapunov function approach are also established. The switching law design approaches are summarized. The conservatism of the stability conditions obtained by using the above-mentioned two ap-proaches is compared. Finally, the controllability and the observability conditions for the switched systems are given respectively.
3. In Chapter 3, we investigate the SMC and the observer designs of a class of switched systems with a nonlinear disturbance based on the multiply Lyapunov function approach. The sliding mode controller and the switching law are designed by using the estimated states, which guarantee the asymptotic stability of the closed-loop systems.
4. In Chapter 4, we address the problem of output feedback SMC of switched systems. Firtly, for a class of disturbance-free linear switched systems, we design a sliding function by using only output information, and propose the existence condition of the sliding mode dynamics. Based on the obtained condition, we get the solution to the parameters of the designed sliding function. Also, we synthetize the output feed-back sliding mode controller and the switching law. Secondly, based on the results obtained in this chapter, we further investigate the problem of the reduced-order output feedback SMC of switched systems with nonlinear disturbance. The condition for the existence of sliding mode dynamics is proposed. Also, the controllers for all the subsystems and the switching law are designed, which guatantee the robustness of the closed -loop systems to the unknown disturbance.
5. In Chapter 5, our attention is focused on the studying of SMC of dis-crete-time switched systems. Firstly, by applying the feedback control and the rolling optimaization approaches in the predictive control, we design a class of predictive sliding mode controllers, which guarantee the asymptotic stability of the closed-loop system under arbitrary switching. Secondly, as for the robustness-loss of the traditional SMC of discrete-time systems, we proposed a discrete-time SMC strategy containing a recursive sliding surface function. Moreover, the reaching law approaches used in discrete-time SMC systems are improved. In addition, as for the systems with that its uncertainties and disturbances’upper bounds are unknown, an admissible discrete-time SMC law is proposed.
6. In Chapter 6, we study the SMC of a class of Markov switching sys-tems. A model transformation is introduced firstly, by which the origi-nal system becomes a“regular”form. A linear sliding surface is de-signed. A sufficient condition of the mean-square exponential stability is proposed for the sliding mode dynamics by using linear matrix ine-quality approach. Finally, a sliding mode controller is synthetized, which guarantees the reachability of the system’s trajectories to pre-defined sliding surface in a finite time.
The conclusion and perspective are given in the end of the paper.
引文
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