State Space Approach to Modal Expansions

The preceding discussion of modal synthesis was based primarily on fitting a sum of biquads to measured frequency-response peaks. A more general way of arriving at a modal representation is to first form a state space model of the system [452], and then convert to the modal representation by diagonalizing the state-space model. This approach has the advantage of preserving system behavior between the given inputs and outputs. Specifically, the similarity transform used to diagonalize the system provides new input and output gain vectors which properly excite and observe the system modes precisely as in the original system. This procedure is especially more convenient than the transfer-function based approach above when there are multiple inputs and outputs. For some mathematical details, see [452]^9.7For a related worked example, see §C.17.6.