Model Reduction and Dynamics

A realizable transfer function has infinitely many realizations. Realizations with the smallest possible dimension are called minimal realizations. A state model is a minimal realization of a proper rational transfer function G(s) if and only if all the states are controllable and observable. The output system of minimal realization and pole-zero cancellation below has minimal order and the same response characteristics as the original system by eliminating uncontrollable or unobservable states in state-space models or cancelling pole-zero pairs with same value in transfer functions or zero-pole-gain models.

The location of closed-loop poles in the s-plane affects the transient-response feature and stability of the system. The system response can be predicted by observing the pole-zero map of the system. The damping ration and undamped natural frequency locate the poles in s-plane. As a result, their values determine the system dynamic and steady-state performance. Bandwidth is a specification for system performance in terms of frequency response. It is a measure of speed of response. The DC gain is the ratio of the output of a system to its input after all transients have decayed. Functions below provide ways to calculate all these parameters and specification.