Vibration Confinement by Minimum Modal Energy Eigenstructure Assignment

A novel Eigenstructure Assignment (ESA) method for vibration confinement of flexible structures has been developed. This method is an output feedback control and determines the closed-loop systems that their eigenvectors are orthogonalized to the open-loop eigenvectors. This method is a numerical method and used Singular Value Decomposition (SVD) to find the null space of the closed-loop eigenvectors. The matrix that spans the null space can be used to regenerate the open-loop system as well as the systems that have orthogonal eigenvectors to the regenerated open-loop system. As a result the isolation of vibration is independent of the type of the disturbance. Also in this method, the energy of the closed-loop system is minimized. As an important outcome, the proposed method needs neither to specify the closed-loop eigenvalues nor to define a desired set of eigenvectors. INTRODUCTION The idea of Eigenstructure Assignment (ESA) is given by Moore [1]. He characterized the class of all eigenvector sets related to a distinct set of closed-loop eigenvalues using state feedback [2]. Therefore, a control problem of eigenvalue placement for a MIMO system which had been introduced earlier by Wonham [3], had been redefined to both placement of eigenvalues in desired locations and choosing a set of the associated eigenvectors from a class of possible eigenvectors. It was Cunningham who first used Singular Value Decomposition (SVD) to find the null space for the achievable eigenvector sub-space. In his output feedback control method, the basis vectors were optimally combined to minimize the error between achievable and desirable eigenvectors. This method was the first practical method of eigenstructure assignment in order to have a desirable transient response behavior [4]. Using SVD, a finite number of actuators are needed to shape the eigenvectors of the system [5]. Shelly et al studied the absolute displacement in first and second order systems, because the existing eigenstructure assignment would not have a control on them. They showed that it is not possible to tell if the absolute displacements in a system are increased, decreased or remained intact just by changing the system’s eigenvectors [6]. Furthermore, they introduced a mode localization technique called eigenvector scaling while studying the time domain response of the system. This method changes specific elements of each eigenvector in order to uniformly decrease the relative displacement of the corresponding areas in the system [7]. They showed analytically that absolute displacements in isolated areas can be reduced by eigenvector shaping, regardless of the type of the disturbance. Some experimental results of eigenvector shaping have been reported in [2, 8, 9]. The eigenstructure shaping method is an active control method and is basically regenerating the behavior of the system when passive mode localization happens, by scaling and reforming part or all of the system mode shapes. Since all the shape modes are scaled in the same way, vibration confinement of the system is not affected by the type of disturbance. An application of this method is also reported in [10]. One of the drawbacks of the uniform scaling is that the number of needed actuator/sensor that has to be equivalent to the number of coupled modes of the system. It means that the action between neighboring systems has the key role in the