Reduced-order controllers for the discrete-time H∞ control problem with unstable invariant zeros

This paper addresses the existence and design methods of reduced-order controllers for the discrete time H_∞ control problem with unstable invariant zeros in the state-space realization of the transfer function matrix from the control input to the controlled error or from the exogenous input to the observation output, where the realization is induced from a stabilizable and detectable realization of the generalized plant. This paper presents new controller degree bounds for the H_∞ control problem and provides new solutions to designing of reduced-order controllers for the H_∞ control problem with invariant zeros located outside of the open unit disk. When such unstable invariant zero exists (even it is a zero on the unit circle), this paper shows that reduced-order controllers with strictly orders less than the generalized plant order exist if the H_∞ control problem is solvable. Two LMI based design methods for constructing the reduced-order controllers are given. Furthermore, this paper shows that the results obtained here unify previous special results of reduced-order controllers for the H_∞ control problems with infinite zeros, or unstable real transmission zeros, etc.