MODAL OPTIMAL CONTROL PROCEDURE FOR NEAR DEFECTIVE SYSTEMS

This study discusses a modal optimal control procedure for defective systems with repeated eigenvalues. From the view point of mathematics, although near defective close eigenvalues are distinct, the characteristic of the system is also defective. Therefore, we have to transform near defective systems into the defective one, and then modal optimal control procedure for the defective systems can be extended to deal with the corresponding problems for near defective systems with close eigenvalues. Because of the defective characteristic of the system, we have to use an invariant sub-space recursive method with numerical stability to calculate the generalized modes of the defective and near defective systems. The Potterʹs approach is extended to solve the Riccati equations in the generalized model subspace of the defective system. Because the order of the Jordan block matrix of the defective eigenvalues, m, is much smaller than that of the state matrix, n, i.e.,m ⪡n, the present modal optimal control procedure is very simple and reduces the computing effort for the complex system with large number of degrees of freedom. A numerical example is given to illustrate and verify the validity of the procedure.