Diagonalizing a State-Space Model
To obtain the modal representation, we may diagonalize any state-space representation. This is accomplished by means of a particular similarity transformation specified by the eigenvectors of the state transition matrix . An eigenvector of the square matrix is any vector for which
To see how this works, suppose we are able to find linearly independent eigenvectors of , denoted , . Then we can form an matrix having these eigenvectors as columns. Since the eigenvectors are linearly independent, is full rank and can be used as a one-to-one linear transformation, or change-of-coordinates matrix. From Eq.(G.19), we have that the transformed state transition matrix is given by
We have incidentally shown that the eigenvalues of the state-transition matrix are the poles of the system transfer function. When it is diagonal, i.e., when diag, the state-space model may be called a modal representation of the system, because the poles appear explicitly along the diagonal of and the system's dynamic modes are decoupled.
Notice that the diagonalized state-space form is essentially equivalent to a partial-fraction expansion form (§6.8). In particular, the residue of the th pole is given by . When complex-conjugate poles are combined to form real, second-order blocks (in which case is block-diagonal with blocks along the diagonal), this is corresponds to a partial-fraction expansion into real, second-order, parallel filter sections.
Finding the Eigenvalues of A in Practice
Matlab State-Space Filter Conversion Example