Pole and Zero Plots

Supported Models

You can create pole-zero plots of linear input-output polynomial, state-space, and grey-box models.

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What Does a Pole-Zero Plot Show?

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:

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How to Plot Model Poles and Zeros Using the GUI

To create a pole-zero plot for parametric linear models in the System Identification Tool GUI, select the Zeros and poles check box in the Model Views area. For general information about creating and working with plots, see Working with Plots in the System Identification Tool GUI.

To include or exclude a model on the plot, click the corresponding model icon in the System Identification Tool GUI. Active models display a thick line inside the Model Board icon.

In addition, you can display a confidence interval for each pole and zero on the plot. 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.

The following table summarizes the Zeros and Poles plot settings.

Zeros and Poles Plot Settings

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How to Plot Poles and Zeros at the Command Line

You can create a pole-zero plot for linear polynomial, linear state-space, and linear grey-box models using the pzmap command. pzmap lets you include several models on a plot.

To display confidence intervals for a specified number of standard deviations, use the following syntax:

pzmap(model,'sd',sd)

where sd is the number of standard deviations of a Gaussian distribution. For example, a confidence value of 99% for the nominal model curve corresponds to 2.58 standard deviations.

pzmap(mod)

For detailed information about pzmap, see the corresponding reference page.

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Reducing Model Order Using Pole-Zero Plots

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.

pzmap(model,'sd',1)

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.

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Noise Spectrum Plots

Nonlinear ARX Model Plots

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