摘要
近几十年来,柔性系统的振动控制问题受到广泛关注,其中如何设计一个好的反馈控制器以及相应闭环系统的稳定性分析是研究该问题的重点.本文以能精确描述柔性杆件动力学行为的弹性梁模型–Timoshenko梁模型为基本研究对象,对分布参数系统控制中的几个难点问题:时滞控制、异位控制、弹性网络系统控制及闭环系统的稳定性作了初步研究.具体如下:
1.非同位系统往往不是极小相位的,该性质易造成异位控制器下的系统非耗散,相应的系统适定性和稳定性分析也变得很难处理,这使得异位控制一直是分布参数系统控制中的一个困难问题.本文研究了一类Timoshenko梁弹性混杂系统的异位反馈控制问题.设计了一类带有异位反馈项的边界反馈控制器.在设定的反馈控制器作用下,采用Riesz基方法结合频谱渐近分析的技巧推导出闭环系统满足谱确定增长条件,并在此基础上通过选择适当增益系数,证明了闭环系统的指数稳定性.
2.在无穷维系统中,时滞会破坏系统稳定性,因此分布参数系统的时滞控制设计是一个难点.本文考察了Timoshenko悬臂梁系统具有输入时滞的反馈控制问题.设计了一类带有部分输入时滞的边界反馈控制器来镇定系统,并讨论了时间延迟反馈输入对于该类Timoshenko弹性梁系统稳定性的影响.此外,本文根据系统频谱的分布状况,给出了相应的闭环系统不稳定、渐近稳定以及指数稳定性条件.
3.柔性结构网络系统在工程中尤其是航空航天科技中有着重要的应用价值,关于其控制问题的研究有着实际意义.本文考察了星形和树形Timoshenko弹性梁网络系统的反馈镇定问题.通过在节点处设计耗散反馈控制器来镇定这两类网络系统.我们采用矩阵形式来描述相应的闭环系统,将Riesz基方法和乘子方法推广到研究Timoshenko弹性梁网络系统的稳定性中,运用谱分析和渐近分析的技巧,分别证明了这两类网络系统的渐近稳定性与指数稳定性.该分析方法可以进一步推广到更为复杂的Timoshenko梁网络系统的稳定性研究中.
In recent years, the control problems on ?exible system, as an hot issue,is widely studied. The central problem is that how to design a nice feedbackcontroller and its stability analysis. In this thesis, the feedback control de-signs of Timoshenko beam which is one of the most realistic 1-d beam model,is studied. The stability of the corresponding closed loop systems is discussed.Some di?cult problems in distributed parameter system controls are consid-ered, those are time-delay feedback controls, non-collocated feedback controls,networks system feedback controls. some primary results are obtained. Thedetails are given as follows:
1. Since the non-collocated systems are usually not minimum-phase, thesystems easily become non-dissipative. The design of non-collocated feedbackcontrols and the well-posedness and stability analysis of the correspondingclosed loop systems are di?cult to do. Hence the non-collocated control prob-lem is always a tough problem in distributed parameter system controls. Inthis thesis, a kind of Timoshenko beam hybrid system with non-collocatedfeedback control is considered. A kind of non-collocated feedback controller isdesigned to stabilize this system. By the Riesz basis approach, together withthe asymptotic analysis of the frequency of the system, the spectrum deter-mined growth condition of the system is gotten. Based on this result, togetherwith the distribution of the spectrum, the stability of this kind of systemis discussed. By simulation, it is shown that this system can be stabilizedexponentially under certain choice of the feedback gains.
2. Since time delays can always destabilize the infinite dimensional sys-tem, the control problem involving time delays is di?cult for distributed pa-rameter system. In this thesis, a kind of time-delay boundary feedback con-troller is designed for the Timoshenko cantilever beam. The e?ect of thetime-delay feedback inputs on the stability of this kind of Timoshenko beam is discussed. The spectrum determined growth condition is gotten by the Rieszbasis property of the (generalized) eigenfunctions of the system operator. Thenthe conditions of instability, asymptotic stability, exponential stability of thiskind of system are obtained based on the spectral distribution. Some simula-tions are given to support these results.
3. The stabilization of star-shape and tree-shape networks of Timoshenkobeams system is considered. Since the flexible structure system has wideapplications, especially in aircraft and space vehicles, its control problem hasimportant significations in engineering. In this thesis, the dissipative joints’feedback controllers are designed to stabilize these two networks. The matricesare used to denote the corresponding closed loop systems. The Riesz basisapproach and multiplier method are extended to study the stability of thenetworks of Timoshenko beams. By the techniques of spectral analysis andasymptotic analysis, the asymptotic stability and exponential stability of thesetwo kinds of networks systems are proved. This method we used here can beextended to the more complex networks of Timoshenko beams systems.
引文
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