Design of an effective controlle via disturbance accommodating left eigenstructure assignment

The transient responses of a linear system having undesired disturbances are dominantly governed by the system's left eigenstructure(eigenvalues/left eigenvectors). In control system design problem of altering the transient response of the system, both the controllability and the disturbance suppressibility, should be considered simultaneously to obtain a robust, effective controller. The controllability of the system may be degraded if the left eigenstructure is chosen to suppress the disturbance, or vice versa. In this paper, first, proposed are a modal disturbance suppressibility measure and an improved version of the modal controllability measure suggested by Hamdan and Nayfeh. Second, a simple and general left eigenstructure assignment scheme, considering both the proposed modal disturbance suppressibility measure and the improved version of modal controllability measure, is suggested. When the previous works are applied to assign the left eigenvectors may differ from the desired ones. But the proposed left eigenstructure assignment scheme makes it possible to achieve the desired colsed-loop eigenvalues exactly, provided the desired left eigenvectors reside in the achievable subspace. In case the desired left eigenvectors do not reside in the achievable subspace, the closed-loop eigenvalues are achieved exactly and the left eigenvectors are assigned to the best possible set of eigenvectors in the least square sense. Finally, a numerical example is included to confirm and demonstrate the usefulness of our propositions and to illustrate the proposed design scheme.