On H∞ model reduction for discrete-time linear time-invariant systems using linear matrix inequalities

In this paper, we address the H∞ model reduction problem for linear time-invariant discrete-time systems. We revisit this problem by means of linear matrix inequality (LMI) approaches and first show a concise proof for the well-known lower bounds on the approximation error, which is given in terms of the Hankel singular values of the system to be reduced. In addition, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the H∞ optimal reduced-order model can readily be constructed via LMI optimization. These results can be regarded as complete counterparts of those recently obtained in the continuous-time system setting.