基于观测器的线性切换系统输出反馈设计
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摘要
切换系统是由若干个子系统和控制子系统激活与否的切换律组成的一类重要的混杂系统。对于切换系统而言，可镇定是最基本的要求，也是目前切换系统研究的热点及难点领域之一。制约切换系统可镇定性设计的主要原因在于：由于切换律的引入，使得切换系统在保留子系统部分性能的同时，也可能产生子系统所没有的复杂的动态特性。这一点在自治线性切换系统的可镇定设计中显得尤为突出，因为此时的可镇定设计问题就是寻找到能使系统渐近稳定的切换律。因此，非常有必要研究自治线性切换系统的可镇定问题。
对于自治线性切换系统的可镇定设计，如果系统满足一致稳定条件，周期切换可以使得该系统稳定。除此之外，切换镇定问题所寻找的切换律必然是和系统状态相关的。但在实际应用中，系统状态往往不可知或者不可直接测量，因此，就有必要研究切换系统输出反馈镇定问题，考虑设计状态观测器来逼近系统状态值，然后根据观测器的值来生成切换信号。但由于此时系统的切换律是基于观测器切换的，因此，切换律的设计和观测器的设计互相耦合，使得基于观测器驱动的输出反馈镇定问题成为一个非常具有挑战性的问题。
在本文中，我们研究了连续时间自治线性切换系统的输出反馈切换镇定问题。本文在前两章对切换系统的相关概念及基本结论做了介绍，在此基础上，本文主要研究了以下问题：
1.针对子系统均可观测的自治线性切换系统，提出了一种改进的低增益观测器设计方法。基于此，接下来我们研究了其输出反馈切换镇定问题，提出了一种构造性的基于切换路径的观测器驱动切换镇定设计方法。该方法的基本思路为：首先寻找一组切换路径，使其满足指定收敛要求的收敛域能构成状态空间的一个全覆盖；接下来，基于上述切换路径的最小驻留时间，设计观测器；最后，根据提出的基于路径的观测器驱动切换律，实现系统稳定。利用此方法，不仅能使得任意可镇定系统稳定，而且能使切换律的设计与观测器设计分开，得到一个弱化的分离原理。
2.把输出反馈切换镇定设计推广到更一般的系统，即线性切换系统能观测，但子系统不一定能观测的情况。为此，首先我们研究了能观测线性切换系统的观测器设计问题，基于能观测切换路径的概念，提出了几种观测器设计方法。接下来，利用前面介绍的基于切换路径的观测器驱动切换镇定设计方法，在将不可观测切换路径修改为可观测算法的基础上，根据切换路径的观测器驱动切换律，实现了任意可镇定系统的稳定，同时保持了弱化的分离原理。
3.考虑到实际系统不可避免出现扰动，探讨了上述基于路径的观测器驱动切换可镇定设计方法所得系统针对结构性、非结构性以及切换信号扰动的鲁棒性。针对线性切换系统切换信号的扰动，引入切换信号之间的相对距离来描述切换信号受扰的程度。
4.作为实例，研究了汽车半主动悬挂系统的输出反馈镇定问题。在其线性切换系统模型基础上，利用本文所介绍的基于路径的观测器驱动切换可镇定设计方法对其进行设计，实现了系统的稳定，而且系统的鲁棒性也得以保证，仿真分析进一步验证了我们的结论。
最后，总结全文，并对一些有待研究的问题进行了展望。
A switched system is a hybrid system which consists of a fnite number of linear sub-systems and a switching law governing the activeness of those subsystems. For switchedsystems, stabilization is not only an essential requirement, but also a hot research feldlasting for decades. A well-known characteristic of switched systems is that they includesboth continuous and discrete dynamics, which makes the system dynamics extremelycomplicated, and the stabilization design very tough. Especially for the stabilization de-sign in autonomous switched systems, for which the problem is to fnd, if possible, anappropriate switching law that makes the system asymptotically stable.
For autonomous switched linear systems, when the system is consistently stabilizable,it has been established that a periodic switching signal could be found to solve thestabilization problem. Otherwise, state-driven switching mechanisms have to be soughtfor addressing the problem. Consider the fact that the state is usually unknown andcannot be measured directly in practice, an observer should be introduced. However, thedesign of the observer and the design of the stabilizing state-driven switching law are ingeneral coupled with each other as a common switching law is sought for both the originalsystem and the observer. This makes the observer-driven switching stabilization problemvery challenging.
In this dissertation, we focus on the observer-driven switching stabilization problem.For this, we frst introduce some basic knowledge about the switched systems. On thisbasis, the main contents of this dissertation are summarized in the following:
1. For switched autonomous linear systems whose subsystems are completely ob-servable, we frst introduce an improved less-overshooting observer design method, andthen consider the observer-driven switching stabilizing problem. A constructive path-wiseobserver-driven switching stabilization mechanism is proposed, by means of which, notonly the augmented system is stable, but also the design of the switching law and thedesign of the observer can be separated into two steps, thus a weak version of separationprinciple is established. Generally, to achieve stability with the observer-driven switchingstabilization scheme, we should implement the following steps. First, fnd a set of well- defned switching paths, whose corresponding contractive cones are a fnite partition ofthe state space. Second, place an observer based on the switching paths’ minimal dwelltime. Finally, a path-wise observer-driven switching law is introduced.
2. Relaxing the observability assumption to the observability of the switched linearsystem. For this, we frst study the observer design problem for observable switchedlinear systems, and proposed some observer design methods. Then by modifying theunobservable switching path to be observable, based on the path-wise observer-drivenswitching mechanism introduced before, an observer-driven switching law is given, whichmakes the augmented system stable. It is remarkable that the weak version of separationprinciple is also preserved.
3. Consider the fact that perturbations can enter into any system, the robustnessis analyzed for system under structural, unstructural and switching perturbations. Tocapture the sensitivity of perturbed switching signals, the relative distance between thenominal and perturbed switching signals is introduced.
4. As an illustrative example, the output-feedback stabilization problem for semi-active suspension vehicle is explored. Based on its switched linear system model, bymeans of the observer-driven switching stabilization mechanism, the problem is solved,and its robustness is proved.
Finally, the conclusions and the prospects of future research are given at the end ofthis dissertation.
对于自治线性切换系统的可镇定设计，如果系统满足一致稳定条件，周期切换可以使得该系统稳定。除此之外，切换镇定问题所寻找的切换律必然是和系统状态相关的。但在实际应用中，系统状态往往不可知或者不可直接测量，因此，就有必要研究切换系统输出反馈镇定问题，考虑设计状态观测器来逼近系统状态值，然后根据观测器的值来生成切换信号。但由于此时系统的切换律是基于观测器切换的，因此，切换律的设计和观测器的设计互相耦合，使得基于观测器驱动的输出反馈镇定问题成为一个非常具有挑战性的问题。
在本文中，我们研究了连续时间自治线性切换系统的输出反馈切换镇定问题。本文在前两章对切换系统的相关概念及基本结论做了介绍，在此基础上，本文主要研究了以下问题：
1.针对子系统均可观测的自治线性切换系统，提出了一种改进的低增益观测器设计方法。基于此，接下来我们研究了其输出反馈切换镇定问题，提出了一种构造性的基于切换路径的观测器驱动切换镇定设计方法。该方法的基本思路为：首先寻找一组切换路径，使其满足指定收敛要求的收敛域能构成状态空间的一个全覆盖；接下来，基于上述切换路径的最小驻留时间，设计观测器；最后，根据提出的基于路径的观测器驱动切换律，实现系统稳定。利用此方法，不仅能使得任意可镇定系统稳定，而且能使切换律的设计与观测器设计分开，得到一个弱化的分离原理。
2.把输出反馈切换镇定设计推广到更一般的系统，即线性切换系统能观测，但子系统不一定能观测的情况。为此，首先我们研究了能观测线性切换系统的观测器设计问题，基于能观测切换路径的概念，提出了几种观测器设计方法。接下来，利用前面介绍的基于切换路径的观测器驱动切换镇定设计方法，在将不可观测切换路径修改为可观测算法的基础上，根据切换路径的观测器驱动切换律，实现了任意可镇定系统的稳定，同时保持了弱化的分离原理。
3.考虑到实际系统不可避免出现扰动，探讨了上述基于路径的观测器驱动切换可镇定设计方法所得系统针对结构性、非结构性以及切换信号扰动的鲁棒性。针对线性切换系统切换信号的扰动，引入切换信号之间的相对距离来描述切换信号受扰的程度。
4.作为实例，研究了汽车半主动悬挂系统的输出反馈镇定问题。在其线性切换系统模型基础上，利用本文所介绍的基于路径的观测器驱动切换可镇定设计方法对其进行设计，实现了系统的稳定，而且系统的鲁棒性也得以保证，仿真分析进一步验证了我们的结论。
最后，总结全文，并对一些有待研究的问题进行了展望。
A switched system is a hybrid system which consists of a fnite number of linear sub-systems and a switching law governing the activeness of those subsystems. For switchedsystems, stabilization is not only an essential requirement, but also a hot research feldlasting for decades. A well-known characteristic of switched systems is that they includesboth continuous and discrete dynamics, which makes the system dynamics extremelycomplicated, and the stabilization design very tough. Especially for the stabilization de-sign in autonomous switched systems, for which the problem is to fnd, if possible, anappropriate switching law that makes the system asymptotically stable.
For autonomous switched linear systems, when the system is consistently stabilizable,it has been established that a periodic switching signal could be found to solve thestabilization problem. Otherwise, state-driven switching mechanisms have to be soughtfor addressing the problem. Consider the fact that the state is usually unknown andcannot be measured directly in practice, an observer should be introduced. However, thedesign of the observer and the design of the stabilizing state-driven switching law are ingeneral coupled with each other as a common switching law is sought for both the originalsystem and the observer. This makes the observer-driven switching stabilization problemvery challenging.
In this dissertation, we focus on the observer-driven switching stabilization problem.For this, we frst introduce some basic knowledge about the switched systems. On thisbasis, the main contents of this dissertation are summarized in the following:
1. For switched autonomous linear systems whose subsystems are completely ob-servable, we frst introduce an improved less-overshooting observer design method, andthen consider the observer-driven switching stabilizing problem. A constructive path-wiseobserver-driven switching stabilization mechanism is proposed, by means of which, notonly the augmented system is stable, but also the design of the switching law and thedesign of the observer can be separated into two steps, thus a weak version of separationprinciple is established. Generally, to achieve stability with the observer-driven switchingstabilization scheme, we should implement the following steps. First, fnd a set of well- defned switching paths, whose corresponding contractive cones are a fnite partition ofthe state space. Second, place an observer based on the switching paths’ minimal dwelltime. Finally, a path-wise observer-driven switching law is introduced.
2. Relaxing the observability assumption to the observability of the switched linearsystem. For this, we frst study the observer design problem for observable switchedlinear systems, and proposed some observer design methods. Then by modifying theunobservable switching path to be observable, based on the path-wise observer-drivenswitching mechanism introduced before, an observer-driven switching law is given, whichmakes the augmented system stable. It is remarkable that the weak version of separationprinciple is also preserved.
3. Consider the fact that perturbations can enter into any system, the robustnessis analyzed for system under structural, unstructural and switching perturbations. Tocapture the sensitivity of perturbed switching signals, the relative distance between thenominal and perturbed switching signals is introduced.
4. As an illustrative example, the output-feedback stabilization problem for semi-active suspension vehicle is explored. Based on its switched linear system model, bymeans of the observer-driven switching stabilization mechanism, the problem is solved,and its robustness is proved.
Finally, the conclusions and the prospects of future research are given at the end ofthis dissertation.
引文
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