Measures for controllability, observability and fixed modes

Measures are introduced that provide quantitative information about relative controllability and observability of a linear multivariable system. The proposed measures are directly related to frequency-domain characteristics such as zeros and residues, and to time-domain characteristics such as eigenvectors, controllability and observability Gramians and eigenvalue sensitivity. A simple and computationally efficient formula involving a quadratic form in the input (output) matrix and the left (right) eigenvector is presented for the calculation of controllability (observability) measure. For systems with repeated eigenvalues, the measures are shown to be proportional to the distances of transmission zeros of certain subsystems and the eigenvalues. The results are extended to decentralized systems, where a measure for fixed modes is proposed.>