Finite spectrum assignment and observer for multivariable systems with commensurate delays

This paper is concerned with control of multivariable systems with commensurate delays. The purpose of this paper is to enlarge the class of finite spectrum assignable systems to spectrally controllable multivariable systems with commensurate delays. By overcoming an infinite number of eigenvalues which are contained in the system and make control very complex, it is proved that if the system is spectrally controllable, there is a delayed feedback matrix such that the closed-loop system is spectrally controllable through a single input. Combining this result with previous ones about single-input systems, it is verified that spectral controllability is equivalent to finite spectrum assignability for multivariable systems with commensurate delays. An observer for the system with delayed outputs is presented. By using these results, the multivariable system can be regulated as desired without preliminary knowledge of open-loop eigenvalues and a stability test of transcendental functions whenever the system is spectrally controllable and spectrally observable.