Model Reduction

Because real systems are highly complex and may involve physical phenomena beyond mathematical description, plant models are likely to be of high order. In addition system models are obtained either from identification based on experimental data or from modeling based on physics principles which add complexity to state-space descriptions. Even in the case when such models do notinvolve pole and zero cancelation, redundancies may still exist in the state-space model owing to its being nearly unreachable or nearly unobservable or both. Hence being minimal for realization is not adequate. High order state-space models will lead to high order controllers and increase overhead to analysis, design, and implementation of the feedback control systems. For this reason there is a strong incentive to reduce the order of the system model. The real issue is how to quantify and remove redundancies in the original high order model so that the reduced order model admits high fidelity in representation of the physical process. In the past a few decades, several techniques are emerged for order reduction of state-space models. In this chapter, methods of balanced realization and optimal Hankel-norm approximation are presented. For a given state-space realization, its Hankel singular values will be shown to provide a suitable measure of the model redundancy. Specifically under the balanced realization, the subsystem corresponding to small Hankel singular values contributes little to the system behavior which can thus be truncated directly. This method can be further improved to obtain reduced order models of higher fidelity. More importantly upper bounds will be derived to quantify the approximation error between the reduced model and the original high order model. In addition the method of inverse balanced truncation (IBT) is introduced which results in reduction errors multiplicative and relative form. The contents of this chapter include error measures, balanced truncations, and optimal Hankel-norm approximation, which rely heavily on basic concepts and mathematical analysis from system theory in the previous chapter.