On H_∞ model reduction using LMIs

In this note, we deal with the problem of approximating a given nth-order linear time-invariant system G by an rth-order system G_r where r < n. It is shown that lower bounds of the H_∞ norm of the associated error system can be analyzed by using linear matrix ineqaulity (LMI)-related techniques. These lower bounds are given in terms of the Hankel singular values of the system G and coincide with those obtained in the previous studies where the analysis of the Hankel operators plays a central role. Thus, this note provides an alternative proof for those lower bounds via simple algebraic manipulations related to LMIs. Moreover, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the problem is essentially convex and the optimal reduced-order models can be constructed via LMI optimization.