Eigenstructure assignment

In LTI systems, the right eigenvectors fix the shape of the mode while the product of initial conditions and the left eigenvectors determines the amount each mode is excited in the response. By the use of multi-input feedback control, the closed-loop system cannot be uniquely determined by only assigning eigenvalues. That is, in addition to the eigenvalue assignment, there exists a freedom to assign eigenvectors. Eigenstructure assignment, simultaneous assignment of eigenvalues and eigenvectors, brings out design freedom beyond eigenvalue assignment. Moore [55] identified for the first time this freedom offered by state feedback beyond pole assignment in the case where the prescribed poles are distinct. Klein and Moore [56] presented an eigenstructure assignment method in the case of non-distinct closed-loop eigenvalues. Fahmy and O’Reilly [57] introduced a parametric approach of eigenstructure assignment using state feedback, in which the assignable set of eigenvectors is dependent upon arbitrarily chosen parameters. Srinathkumar [58] investigated the eigenstructure assignment problem using output feedback. He derived sufficient conditions for the maximum number of assignable eigenvalues via output feedback and also determined the maximum number of eigenvectors which can be partially

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assigned with certain number of entries arbitrarily chosen. To preserve the symmetry and sparseness of system matrices of second-order mechanic systems, some eigenstructure assignment techniques were developed for second-order mechanic systems. Inman and Kress [59] developed an eigenstructure assignment approach using inverse eigenvalue methods for mechanical systems represented by second- order systems of differential equations. Datta et al. [60] presented a partial eigenstructure assignment algorithm for systems modelled by a set of second order differential equations such that certain eigenpairs of a vibrating system may be assigned while the other eigenpairs remain unchanged. Ram and Mottershead [9] proposed a multi-input partial pole placement method based on measured receptances of open-loop systems, in which some modal constraints may be imposed on closed-loop right eigenvectors.

A judicious choice of a right eigenvector may be useful in reshaping closed-loop responses while a judicious choice of a left eigenvector may prevent a mode from being excited. Simultaneous assignment of eigenvalues and right eigenvectors is known as right eigenstructure assignment. By contrast, simultaneous assignment of eigenvalues and left eigenvectors is known as left eigenstructure assignment. Right and/or left eigenstructure assignment has been widely applied for active vibration control.

Disturbance decoupling, which aims to make the disturbance have no effects on controlled output, was considered in [61] by right eigenstructure assignment. Sobel et al. [62, 63] employed eigenstructure assignment to obtain some decoupled aircraft motions in flight control systems. Specifically, some modes are decoupled by specifying some entries of the corresponding right eigenvectors to be zero. Song and

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Jayasuriya [64] proposed an algorithm primarily for modal localisation by prescribing the entries associated with certain areas of closed-loop right eigenvectors to relatively small values. Therefore, the vibration of these areas is relative small and the vibrational energy is restricted to the other areas. Shelley and Clark [65] presented an eigenvector assignment algorithm for modal localisation in mass-spring systems. The closed-loop eigenvalues were kept the same as their open-loop values. The closed-loop eigenvectors were given by scaling the entries, related to isolated areas, of the complete set of open-loop eigenvectors with a small factor while the entries related to localised areas with a big factor. Therefore, the displacements in the isolated areas would be proportionally smaller than the displacements in the localised portions of the system. An experimental implementation of their algorithm was presented in [66]. Also, they generalised the idea of modal localisation to distributed parameters systems [67, 68]. It was also shown that the performance of modal localisation depends on the number of actuators.

Zhang et al. [69] used left eigenstructure assignment to reject undesired inputs to a vibrating flexible beam by orthogonalising left eigenvectors to the disturbance input matrix. Orthogonalising left eigenvectors to disturbance input matrix may degrade the controllability of the system. Both controllability and the disturbance rejection using left eigenstructure assignment were considered simultaneously in [70]. In [69, 70], zeros were prescribed to the entries of the desired left eigenvectors corresponding to nonzero entries of the forcing vector. Alternatively, Wu and Wang [71] minimised the inner product of each left eigenvector and each forcing vector such that each left eigenvector was as closely orthogonal to each forcing vector as possible. It is understandable that the simultaneous assignment of right and left eigenstructure may improve the control performance or achieve multiple control

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objectives. However, the assignment of left eigenstructure conflicts with that of right eigenstructure because they are coupled with each other. Choi [72] proposed an algorithm for simultaneous approximate assignment of left and right eigenstructure such that disturbance rejection and the disturbance decoupling are approximately achieved. Wu and Wang [73] presented a simultaneous left and right eigenstructure assignment method for active vibration isolation. Specifically, the left eigenvectors were prescribed to be as closely orthogonal to the forcing vector of the system as possible and the entries associated with the concerned region of the right eigenvectors are constrained to relatively small values. A performance index was minimised such that simultaneous assignment left and right eigenstructure was approximately achieved.