Multiplicative Hankel norm approximation of linear multivariable systems

In control system design, specifications are commonly given in terms of log-magnitude quantities (i.e. decibels). This description induces a multiplicative or relative error criterion for plant model reduction. Techniques from the theory of additive Hankel norm approximation and spectral factorization are synthesized to produce a multiplicative approximant of a possibly unstable or non-square plant. It is shown that the reduced-order model preserves the unstable poles and right half-plane zeros of the original system. Explicit state-space formulae are provided for the construction of the reduced-order model and its error properties are discussed. The method is illustrated by a numerical example.