Control Design And Stability Analysis Of Timoshenko Beam Systems

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.