Minimum Modal-State Realizations from MIMO Transfer-Function Realizations

Transfer-function realizations (TFR's) of multi-input, multi-output (MIMO) linear dynamic systems can be converted into minimum modal state variable realizations (MSR's) by the use of singular-value decompositions (SVD's) of the residue matrices for each pole. Each residue matrix is decomposed into the sum of outer products of vectors that form columns of the output matrix and rows of the input matrix of the MSR. If the rank of the residue matrix is r, there are r independent modes with the same eigenvalue. A MSR using only real numbers can be obtained by using a block diagonal dynamics matrix. When coupled repeated poles occur, the MSR is of the Jordan form. Starting from the residue matrix associated with the highest power of the repeated eigenvalue, the Jordan form is constructed by a series of singular value decompositions. The number of states in the MSR depends on the ranks of the residue matrices as determined by the SVD. The SVD quantifies the importance of each state by describing the error that results if states associated with small singular values are deleted. This method of converting TFR's into MSR's can be used on approximate TFR descriptions such as those determined from experimental data.