On the quantitative characterization of approximate decentralized fixed modes using transmission zeros

In [16], it is shown that the spectrum of a linear time-invariant multivariable system, using decentralized linear time-invariant controllers, can only be assigned to a symmetric set of complex numbers that include the decentralized fixed modes (DFM). This is of fundamental importance to the decentralized stabilization problem. A measure of how close a mode is from being a DFM is introduced in [14]; this measure is used to quantify the difficulty of shifting such a mode. Following [4], modes which are "hard" to shift are considered to be approximate decentralized fixed modes. In this paper, we generalize the results of [5] to derive two new assignability measures. The first is coordinate invariant and is based on the distance between a mode and a set of transmission zeros. The second is coordinate dependent and is based on the minimum singular value of a set of transmission zero matrices.