A Measure for the Decentralized Assignability of Eigenvalues

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. It is to be noted however, that if a system is unstable and if the unstable modes are not DFM, there may still be difficulty in stabilizing the system; this would occur for example if an unstable model is "close" to being a DFM. In order to recognize this problem, we introduce an assignability measure which is based on a continuous rather than a discrete metric. This is done in the spirit of [8] and [14] so that we can judge how close an eigenvalue is to being a DFM. Implications of this measure on controller design are also explored.