Modal Computing Method Research Of Output Feedback Control Gain Matrix In Pole Assignment Of Structure With Repeated Eigenvalues

With the development of the science and technology, structure vibration control has been an important research field due to its urgent application need in actual engineering structures. Active vibration control has received considerable attention in recent years at homes and overseas because of its excellent advantages. The pole assignment method in the active vibration control method has been studied widely, but the past research works mainly focus on the discrete eigenvalues structure, the feedback control research in pole assignment of structure with repeated eigenvalues deserves the more attention.This dissertation deals with the modal control method of the pole assignment of the repeated eigenvalues,the control for local vibration method and related issues in a more in-depth theory and research, mainly composed of two parts :1.The two pole assignment control methods: state feedback control and output feedback control. When the control system doesn't satisfy the request, we can take the feedback control methods to change the original system's response, make the closed-loop system have the better control characteristic, which is the start and research contents of the active control in the vibration control system. So in the structure of the state space, under two control methods, studying the relationship between the pole assignment and the controllability and observability of the vibration control system, i.e. the relationship between the complete controllability and observability of the vibration control system and all the poles with corresponding eigenvector dispensability or some poles with corresponding eigenvector dispensability.2.In the assignment control of a part of eigenvalues and their eigenvectors of the vibration control system, for the discrete eigenvalues structure with single input , when only control force distribution matrix is not perpendicular to all eigenvalues, the system is completely controllable, at the same time any eigenvalue of the system is assignable. But when the system has multiple eigenvalues, it is almost uncontrollable to single input. So this dissertation studied pole assignment in output feedback control of the perfect system with repeated eigenvalues. Through this, the method is got, which is simple, enforceable output feedback control gain matrix account method, i.e. output feedback control gain matrix modal arithmetic, make the part vibration of the structure controlled, This is the main part of this research.The research work of this thesis consists of seven chapters:1) On the basis of reading extensive literatures, summarize the development of the modal control and the method application in some areas.2) In the state space, under original coordinate and modal coordinate; establish state space equation, base on space equation, and study the state space of the structure vibration analysis and state space method; the controllability and observability analysis of vibration system.3) Researches on the two feedback control methods of the structure vibration control, i.e. output feedback control and state feedback control in the state space of the vibration control system; relations between the two control methods and the controllability and observability of the vibration control system.4) Researches on the pole assignment theory in the state space of the vibration control system under the state feedback control and output feedback control. Relations between the pole assignment under the control of the state feedback and the controllability and observability of the vibration control system, when the vibration system is controllable and observable, the computational method of the state feedback control gain matrix which need to be researched; Relations between the pole assignment under the control of the output feedback and the controllability and observability of the vibration control system, and the computational method of the output feedback control gain matrix which need to be researched.5) In the state space of the control system, the structural state equation under the modal coordinates is set up according to the structural finite element equation; choosing methods according to the eigenvalue and eigenvector which is controlled; researches on the method of assigning the poles under the output feedback control based on the original coordinates and the modal coordinates, using the open-loop system vector projecting method and the least energy distribution method.6) Take a spring-mass system with repeated eigenvalues for example, according to the structural finite element equation set up the dynamics model, in the state space of structural vibration control system, building the state space equation, analysis controllability and observability of control system, accounting modal's all eigenvalues and all eigenvectors, using method of output feedback control in pole assignment of vibration system. The deformation of a structure is caused mainly by the response of the lower modes, and the amplitude responses of higher modes are very small in contrast to the lower modes. So we mainly study the control of the lower modes. Firstly build finite element equation of structure and respective state space equation, compute some lower order modes. The goal is to control some lower modes and almost not to disturb the other modes. Secondly select control force distribution matrix and sensor distribution matrix and guarantee the system is controllable and observable. In order to keep the structure stable and protect from spill over, the actuators and sensors are collocated. Thirdly build state space equation under part of modal coordinates, Utilizing two important methods of modal computing method of output feedback control gain of projection method of open-loop system vectors of reassigned poles and minimum energy assignment to compute output feedback gains, and then apply the feedback gains to original state space equation of structure. From the control effects we obtain the results which we want. Under two control methods, the output feedback gains are computed by using original coordinates, and by using model coordinates the output feedback gains of three-order, four-order and five-order are computed, respectively, and then apply those gains obtained under modal coordinates to original structural state space equation, comparing the control effect of modal gains with one of original gain. For seeing the result of pole assignment under the output feedback control clearly, the Bode diagrams of the vibration control system are drawed by Matlab, it is verified that modal computing method is available.7) The research work of this thesis is summarized, and future work on passive/active vibration control with modal control is proposed.