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.

Function | Description | ||
---|---|---|---|

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. |