切换线性时滞系统的跟踪控制
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摘要
切换时滞系统作为一类重要的混杂系统,有着重要的理论研究价值和广泛的工程应用背景。由于在切换时滞系统中连续动态、离散动态和时滞的并存并相互作用,使系统的行为变得十分复杂,系统运行机制远未明晰,大量的分析与综合问题亟等解决。作为控制理论和工程中基本的控制问题之一,跟踪控制在切换时滞系统的研究中占有重要地位,在航空航天、机器人控制、电力系统和计算机网络控制中有着广阔的应用前景,但这方面研究结果目前还相当少。
本文在切换系统和时滞系统的稳定性分析及综合的基础上,将跟踪控制的理论和方法应用到切换线性时滞系统中去,初步建立了切换线性时滞系统跟踪控制的理论框架。针对切换时滞系统模型中具有不确定性、带有干扰、子系统有不可稳及状态不可测等情形,分别利用凸优化技术、单Lyapunov函数方法、多Lyapunov函数方法以及基于时间的平均驻留时间方法等,研究了渐近跟踪和具有H∞性能的跟踪等问题。
本文主要工作包括如下几方面：
(一)研究了切换线性时滞系统的状态跟踪控制问题,针对时滞为常数的情形,给出了时滞独立的跟踪控制问题可解的充分条件,通过利用凸组合技术和单Lyapunov函数方法,设计了跟踪控制器和状态依赖的滞后切换律,以使系统满足H∞模型参考的跟踪性能指标。
(二)采用基于平均驻留时间的切换方法来考虑切换时滞系统鲁棒H∞跟踪控制问题,且时变时滞的导数是可以不存在或任意的。通过将给定系统与参考模型联立而构成增广系统,采用误差状态的反馈控制,使增广系统内稳定且具有权重的H∞模型参考跟踪性能。由于采用自由权矩阵的方法,得出了时滞依赖的跟踪控制条件,这是比时滞独立具有更小保守性的条件。同时,由于所选用的Lyapunov-Krasovskii泛函不涉及时滞项导数的信息,在限制条件上有所减少,因此在设计上带来更大方便,降低了设计的困难和运算量。在处理不确定性方面,首先通过对标称系统的讨论,得出相应结论,再借助一个熟知的引理,自然过渡到含不确定项的系统上来,从而使所得结果具有鲁棒性。
(三)讨论了基于观测器的切换时滞系统H∞跟踪控制问题,主要针对系统的状态不可知或不易知时,使用系统的输出来设计反馈控制器,从而使系统具有H∞模型参考跟踪性能。为了处理估计误差和外界干扰带来的困难,引进常数变易法作为工具来处理切换时滞系统的跟踪控制问题。有了这一工具,我们可以顺利处理系统实际状态和参考模型状态间的误差,以及系统的实际状态和观测器状态间的误差,从而避免使用关于误差项的线性增长条件。
(四)针对一类不确定项满足某种匹配条件的切换时滞系统,采用基于平均驻留时间的方法,讨论其输出渐近跟踪给定参考模型的输出问题。当参考模型本身也是时滞系统时,借助于常数变易公式,对参考模型的状态给予了估计。通过讨论给定系统和参考模型之间的误差系统的指数稳定性,从而实现系统输出渐近跟踪参考模型输出。
(五)针对切换时滞系统中含有可稳与不可稳的子系统时,拓广平均驻留时间的思想,确定可稳的子系统与不可稳的子系统的激活时间比,从而实现切换系统的整体跟踪效果。在导出跟踪控制问题可解的条件时,采用了PI控制器,从而有效地消除稳态跟踪误差。
最后对全文所做的工作进行了总结,并讨论了下一步可能研究的工作。
As an important class of hybrid systems, switched time-delays systems are of great significance both in theory development and engineering applications. Due to the co-existence and interaction among the continuous dynamics, discrete dynamics and time-delays, the behavior of such systems is very complicated. The mechanism of such systems is far from clear. Many analysis and design problems deserve further investigation. As one of the basic problems in control theory and engineering, tracking control of switched time-delays systems has broad applications in navigation, spaceflight, robot control and networked control systems. However, up to now results on such issues have been rarely found.
This dissertation studies tracking control of switched linear systems with time-delays based on stability analysis and control synthesis of switched systems and time-delays systems. Some theories and methods of tracking control are applied and extended to establish a theory framework of tracking control for switched linear time-delays systems. By using convex optimization techniques, single Lyapunov functions, multiple Lyapunov functions and average dwell time methods, the dissertation studies the tracking control problems for the cases that the systems are with uncertainties, perturbations, and have stabilizable and unstabilizable subsystems, respectively. Meanwhile, the tracking control problems for switched time-delays systems are addressed for the output tracking control and the tracking control with H∞model reference tracking performance, respectively.
The main contributions of this thesis are as follows.
1. State tracking control for switched linear time-delay systems is studied with de-lay independent sufficient conditions given for the case that the delay is constant. By using convex optimization techniques and single Lyapunov function method, tracking controllers and a hysteresis switching law are designed such that the H∞model reference tracking performance is guaranteed.
2. Average dwell time approach is utilized to design robust tracking controllers and switching law for the switched linear systems with time-varying delays, and sufficient conditions for the solvability of robust tracking control problem are developed such that the weighted H∞model reference robust tracking performance is satisfied. Free weight- ing matrix scheme and a specific Lyapunov-Krasovskii functional are adopted to obtain the delay dependent criteria which are less conservative. The constrained conditions are reduced and simplified compared with existing results, and thus the difficulties and com-plexities are reduced.
3. When the system states are not available, observer based tracking control for switched linear systems with time-delays is considered. The possibility of designing switching control law based on measured output instead of the state information is in-vestigated to achieve H∞model reference tracking performance. In order to overcome the difficulties caused by the estimation error and exotic disturbance, the variation-of-constants formula is adopted. Delay independent and delay dependent criteria are given for the cases that the system is with constant delay and time-varying delays, respectively. The simulation examples show the effectiveness of the proposed method.
4. Robust tracking control is addressed for a class of switched time-delay systems with uncertainties that satisfy certain matching conditions. Since the reference models are still time-delay systems, the variation-of-constants formula is introduced to estimate the reference states. Asymptotic tracking is achieved by means of exponential stability of the error system.
5. For the switched time-delays systems with stabilizable and unstabilizable sub-systems, the tracking control problem is solvable if the activation time ratio between sta-bilizable subsystems and unstabilizable subsystems is no less than a specified constant. To this end, average dwell time approach is extended to the switching tracking control design, and "PI" (proportional-integral) controllers are adopted to the design which atten-tively eliminates the steady state tracking error.
Finally, the results of the dissertation are summarized and further research topics are pointed out.
本文在切换系统和时滞系统的稳定性分析及综合的基础上,将跟踪控制的理论和方法应用到切换线性时滞系统中去,初步建立了切换线性时滞系统跟踪控制的理论框架。针对切换时滞系统模型中具有不确定性、带有干扰、子系统有不可稳及状态不可测等情形,分别利用凸优化技术、单Lyapunov函数方法、多Lyapunov函数方法以及基于时间的平均驻留时间方法等,研究了渐近跟踪和具有H∞性能的跟踪等问题。
本文主要工作包括如下几方面：
(一)研究了切换线性时滞系统的状态跟踪控制问题,针对时滞为常数的情形,给出了时滞独立的跟踪控制问题可解的充分条件,通过利用凸组合技术和单Lyapunov函数方法,设计了跟踪控制器和状态依赖的滞后切换律,以使系统满足H∞模型参考的跟踪性能指标。
(二)采用基于平均驻留时间的切换方法来考虑切换时滞系统鲁棒H∞跟踪控制问题,且时变时滞的导数是可以不存在或任意的。通过将给定系统与参考模型联立而构成增广系统,采用误差状态的反馈控制,使增广系统内稳定且具有权重的H∞模型参考跟踪性能。由于采用自由权矩阵的方法,得出了时滞依赖的跟踪控制条件,这是比时滞独立具有更小保守性的条件。同时,由于所选用的Lyapunov-Krasovskii泛函不涉及时滞项导数的信息,在限制条件上有所减少,因此在设计上带来更大方便,降低了设计的困难和运算量。在处理不确定性方面,首先通过对标称系统的讨论,得出相应结论,再借助一个熟知的引理,自然过渡到含不确定项的系统上来,从而使所得结果具有鲁棒性。
(三)讨论了基于观测器的切换时滞系统H∞跟踪控制问题,主要针对系统的状态不可知或不易知时,使用系统的输出来设计反馈控制器,从而使系统具有H∞模型参考跟踪性能。为了处理估计误差和外界干扰带来的困难,引进常数变易法作为工具来处理切换时滞系统的跟踪控制问题。有了这一工具,我们可以顺利处理系统实际状态和参考模型状态间的误差,以及系统的实际状态和观测器状态间的误差,从而避免使用关于误差项的线性增长条件。
(四)针对一类不确定项满足某种匹配条件的切换时滞系统,采用基于平均驻留时间的方法,讨论其输出渐近跟踪给定参考模型的输出问题。当参考模型本身也是时滞系统时,借助于常数变易公式,对参考模型的状态给予了估计。通过讨论给定系统和参考模型之间的误差系统的指数稳定性,从而实现系统输出渐近跟踪参考模型输出。
(五)针对切换时滞系统中含有可稳与不可稳的子系统时,拓广平均驻留时间的思想,确定可稳的子系统与不可稳的子系统的激活时间比,从而实现切换系统的整体跟踪效果。在导出跟踪控制问题可解的条件时,采用了PI控制器,从而有效地消除稳态跟踪误差。
最后对全文所做的工作进行了总结,并讨论了下一步可能研究的工作。
As an important class of hybrid systems, switched time-delays systems are of great significance both in theory development and engineering applications. Due to the co-existence and interaction among the continuous dynamics, discrete dynamics and time-delays, the behavior of such systems is very complicated. The mechanism of such systems is far from clear. Many analysis and design problems deserve further investigation. As one of the basic problems in control theory and engineering, tracking control of switched time-delays systems has broad applications in navigation, spaceflight, robot control and networked control systems. However, up to now results on such issues have been rarely found.
This dissertation studies tracking control of switched linear systems with time-delays based on stability analysis and control synthesis of switched systems and time-delays systems. Some theories and methods of tracking control are applied and extended to establish a theory framework of tracking control for switched linear time-delays systems. By using convex optimization techniques, single Lyapunov functions, multiple Lyapunov functions and average dwell time methods, the dissertation studies the tracking control problems for the cases that the systems are with uncertainties, perturbations, and have stabilizable and unstabilizable subsystems, respectively. Meanwhile, the tracking control problems for switched time-delays systems are addressed for the output tracking control and the tracking control with H∞model reference tracking performance, respectively.
The main contributions of this thesis are as follows.
1. State tracking control for switched linear time-delay systems is studied with de-lay independent sufficient conditions given for the case that the delay is constant. By using convex optimization techniques and single Lyapunov function method, tracking controllers and a hysteresis switching law are designed such that the H∞model reference tracking performance is guaranteed.
2. Average dwell time approach is utilized to design robust tracking controllers and switching law for the switched linear systems with time-varying delays, and sufficient conditions for the solvability of robust tracking control problem are developed such that the weighted H∞model reference robust tracking performance is satisfied. Free weight- ing matrix scheme and a specific Lyapunov-Krasovskii functional are adopted to obtain the delay dependent criteria which are less conservative. The constrained conditions are reduced and simplified compared with existing results, and thus the difficulties and com-plexities are reduced.
3. When the system states are not available, observer based tracking control for switched linear systems with time-delays is considered. The possibility of designing switching control law based on measured output instead of the state information is in-vestigated to achieve H∞model reference tracking performance. In order to overcome the difficulties caused by the estimation error and exotic disturbance, the variation-of-constants formula is adopted. Delay independent and delay dependent criteria are given for the cases that the system is with constant delay and time-varying delays, respectively. The simulation examples show the effectiveness of the proposed method.
4. Robust tracking control is addressed for a class of switched time-delay systems with uncertainties that satisfy certain matching conditions. Since the reference models are still time-delay systems, the variation-of-constants formula is introduced to estimate the reference states. Asymptotic tracking is achieved by means of exponential stability of the error system.
5. For the switched time-delays systems with stabilizable and unstabilizable sub-systems, the tracking control problem is solvable if the activation time ratio between sta-bilizable subsystems and unstabilizable subsystems is no less than a specified constant. To this end, average dwell time approach is extended to the switching tracking control design, and "PI" (proportional-integral) controllers are adopted to the design which atten-tively eliminates the steady state tracking error.
Finally, the results of the dissertation are summarized and further research topics are pointed out.
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