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What Does a Pole-Zero Plot Show? |

You can create pole-zero plots of linear identified models.
To study the poles and zeros of the noise component of an input-output
model or a time series model, use `noise2meas` to
first extract the noise model as an independent input-output model,
whose inputs are the noise channels of the original model.

The following figure shows a sample pole-zero plot of the model
with confidence intervals. `x` indicate poles and `o` indicate
zeros.

The general equation of a linear dynamic system is given by:

In this equation, *G* is an operator that
takes the input to the output and captures the system dynamics, and *v* is
the additive noise term.

The *poles* of a linear system are the roots
of the denominator of the transfer function *G*.
The poles have a direct influence on the dynamic properties of the
system. The *zeros* are the roots of the numerator
of *G*. If you estimated a noise model *H* in
addition to the dynamic model *G*, you can also
view the poles and zeros of the noise model.

Zeros and the poles are equivalent ways of describing the coefficients of a linear difference equation, such as the ARX model. Poles are associated with the output side of the difference equation, and zeros are associated with the input side of the equation. The number of poles is equal to the number of sampling intervals between the most-delayed and least-delayed output. The number of zeros) is equal to the number of sampling intervals between the most-delayed and least-delayed input. For example, there two poles and one zero in the following ARX model:

You can use pole-zero plots to evaluate whether it might be useful to reduce model order. When confidence intervals for a pole-zero pair overlap, this overlap indicates a possible pole-zero cancelation.

For example, you can use the following syntax to plot a 1-standard-deviation confidence interval around model poles and zeros.

showConfidence(iopzplot(model))

If poles and zeros overlap, try estimating a lower order model.

Always validate model output and residuals to see if the quality
of the fit changes after reducing model order. If the plot indicates
pole-zero cancellations, but reducing model order degrades the fit,
then the extra poles probably describe noise. In this case, you can
choose a different model structure that decouples system dynamics
and noise. For example, try ARMAX, Output-Error, or Box-Jenkins polynomial
model structures with an *A* or *F* polynomial
of an order equal to that of the number of uncanceled poles. For more
information about estimating linear polynomial models, see Identifying Input-Output Polynomial Models.

In addition, you can display a confidence interval for each pole and zero on the plot. To learn how to show or hide confidence interval, see How to Plot Model Poles and Zeros Using the GUI.

The *confidence interval* corresponds to
the range of pole or zero values with a specific probability of being
the actual pole or zero of the system. The toolbox uses the estimated
uncertainty in the model parameters to calculate confidence intervals
and assumes the estimates have a Gaussian distribution.

For example, for a 95% confidence interval, the region around the nominal pole or zero value represents the range of values that have a 95% probability of being the true system pole or zero value. You can specify the confidence interval as a probability (between 0 and 1) or as the number of standard deviations of a Gaussian distribution. For example, a probability of 0.99 (99%) corresponds to 2.58 standard deviations.

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