Eigenstructure Assignment Algorithm for Mechanical Second-Order Systems

Introduction E IGENSTRUCTURE assignment algorithms are widely used to design control systems.Most of the available eigenstructure assignment algorithms assign all of the eigenvalues to the desired values.The eigenstructureassignmentalgorithmcan be dividedinto two groups, that is, the null space approach and Sylvester equation approach.The null spaceapproachis used to solvemode decoupling problems by assigning a certain set of eigenvectors to the desired values, whereas the Sylvester equationapproach is used to design a controller for the vibration suppressionof  exible structures.3;4 It is known that the dynamics of a large class of mechanical systems can be represented by second-order systems of differential equations. Because the dimension of aerospace structural dynamic systems is usually large, one often encounters an uncomfortably high computational burden to design a controller, especially when an optimizationprocess is involved.4 In thisNote,we proposea specialized version of an eigenstructure assignment algorithm to take advantages of the structure of second-order differential equations for mechanical vibrating systems. Therefore, when we deal with the n second-order differential equation of mechanical vibrating systems,it is notnecessaryto determinethe2n£ 2n eigenvectormatrix for the corresponding2n Ž rst-order state-space equations. The proposed algorithm is more efŽ cient than the conventional eigenstructure assignment algorithm in the sense that it uses less memory and operations. The numerical accuracy and efŽ ciency of the proposed algorithmresults because the conventionaleigenstructure assignment algorithm must solve the Sylvester equation numerically, whereas, in this Note, an analytic formulation for solving the Sylvester equation is derived that takes full advantageof the mathematical structureof mechanical second-ordersystems. It can also be anticipated that numerical accuracywill be superior for most highdimensioned applications; the results of our calculations strongly support this supposition.