Singular perturbation approximation of balanced systems

The singular perturbation approximation technique for model reduction is related to the direct truncation technique if the system model to be reduced is stable, minimal, and internally balanced. It is shown that these two methods constitute two fully compatible model reduction techniques for a continuous-time system, and both methods yield a stable, minimal, and internally balanced reduced-order system with the same L_∞-norm error bound on the reduction. Although the upper bound for both reductions is the same, the direct truncation method tends to have smaller errors at high frequencies and larger errors at low frequencies, whereas the singular perturbation approximation method will display the opposite character. It is also shown that a certain bilinear mapping not only preserves the balanced structure between a continuous-time system and an associated discrete-time system, but also preserves the slow singular perturbation approximation structure. Hence, the continuous-time results on the singular perturbation approximation of balanced systems are easily extended to the discrete-time case. Examples are used to show the compatibility of and the differences in the two reduction techniques for a balanced system.