Measures of modal controllability and observability for first- and second-order linear systems

For a system described by the triple (A9B9C) where the matrix A has a set of distinct eigenvalues and a wellconditioned modal matrix, we propose measures of modal controllability and observability. The angles between the left eigenvectors of A and the columns of the matrix B are used to propose modal controllability measures and the angles between the rows of the matrix C, and the right eigenvectors of A are used to propose modal observability measures. Gross measures of controllabili ty of a mode from all inputs and its observability in all outputs are also proposed. These measures are related to other measures suggested in the literature. A closed-form relation between the norm of the residue and the proposed measures is given, thus linking the residue to the unobservability or uncontrollability of the mode. We finally show that the proposed measures can be applied directly to second-order models.