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.

Bandwidth Calculate the bandwidth of a SISO system.
Pole-zero map Plot the pole-zero map of an LTI model.
Damping factors and frequencies Compute the damping factors and natural frequencies of system poles.
DC gain Compute low frequency(DC) gain of the system.
Sort poles  Sort the poles of systems.
Minimal realization Find a minimal realization of an LTI model.
Pole-zero cancellation Cancel the pole-zero pairs with same value of a system.

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