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pzmap

Pole-zero plot of dynamic system

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Syntax

pzmap(sys)
pzmap(sys1,sys2,...,sysN)
[p,z] = pzmap(sys)

Description

example

pzmap(sys)creates a pole-zero plot of the continuous or discrete-timedynamic system modelsys.xandoindicates the poles and zeros respectively, as shown in the following figure.

From the figure above, an open-loop linear time-invariant system is stable if:

  • In continuous-time, all the poles on the complex s-plane must be in the left-half plane (blue region) to ensure stability. The system is marginally stable if distinct poles lie on the imaginary axis, that is, the real parts of the poles are zero.

  • In discrete-time, all the poles in the complex z-plane must lie inside the unit circle (blue region). The system is marginally stable if it has one or more poles lying on the unit circle.

example

pzmap(sys1,sys2……sysN)creates the pole-zero plot of multiple models on a single figure. The models can have different numbers of inputs and outputs and can be a mix of continuous and discrete systems. For SISO systems,pzmapplots the system poles and zeros. For MIMO systems,pzmapplots the system poles and transmission zeros.

example

[p,z] = pzmap(sys)returns the system poles and transmission zeros as column vectorspandz. The pole-zero plot is not displayed on the screen.

Examples

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Open Live Script

Plot the poles and zeros of the continuous-time system represented by the following transfer function:

H ( s ) = 2 s 2 + 5 s + 1 s 2 + 3 s + 5 . 

H = tf([2 5 1],[1 3 5]); pzmap(H) gridon

Figure contains an axes object. The axes object contains 2 objects of type line. This object represents H.

Turning on the grid displays lines of constant damping ratio (zeta) and lines of constant natural frequency (wn). This system has two real zeros, marked by o on the plot. The system also has a pair of complex poles, marked by x.

This example uses:

Open Live Script

Plot the pole-zero map of a discrete time identified state-space (idss) model. In practice you can obtain anidssmodel by estimation based on input-output measurements of a system. For this example, create one from state-space data.

A = [0.1 0; 0.2 -0.9]; B = [.1 ; 0.1]; C = [10 5]; D = [0]; sys = idss(A,B,C,D,'Ts',0.1);

Examine the pole-zero map.

pzmap(sys)

Figure contains an axes object. The axes object contains 2 objects of type line. This object represents sys.

System poles are marked by x, and zeros are marked by o.

Open Live Script

For this example, load a 3-by-1 array of transfer function models.

load('tfArray.mat','sys'); size(sys)
3x1 array of transfer functions. Each model has 1 outputs and 1 inputs.

Plot the poles and zeros of each model in the array with distinct colors. For this example, use red for the first model, green for the second and blue for the third model in the array.

pzmap(sys(:,:,1),'r',sys(:,:,2),'g',sys(:,:,3),'b') sgrid

Figure contains an axes object. The axes object contains 6 objects of type line. These objects represent untitled1, untitled2, untitled3.

sgridplots lines of constant damping ratio and natural frequency in the s-plane of the pole-zero plot.

Open Live Script

使用pzmapto calculate the poles and zeros of the following transfer function:

s y s ( s ) = 4 . 2 s 2 + 0 . 2 5 s - 0 . 0 0 4 s 2 + 9 . 6 s + 1 7 

sys = tf([4.2,0.25,-0.004],[1,9.6,17]); [p,z] = pzmap(sys)
p =2×1-7.2576 -2.3424
z =2×1-0.0726 0.0131
Open Live Script

This example uses a model of a building with eight floors, each with three degrees of freedom: two displacements and one rotation. The I/O relationship for any one of these displacements is represented as a 48-state model, where each state represents a displacement or its rate of change (velocity).

Load the building model.

load('building.mat'); size(G)
State-space model with 1 outputs, 1 inputs, and 48 states.

Plot the poles and zeros of the system.

pzmap(G)

Figure contains an axes object. The axes object contains 2 objects of type line. This object represents G.

From the plot, observe that there are numerous near-canceling pole-zero pairs that could be potentially eliminated to simplify the model, with no effect on the overall model response.pzmapis useful to visually identify such near-canceling pole-zero pairs to perform pole-zero simplification.

Input Arguments

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Dynamic system, specified as a dynamic system model or model array. Dynamic systems that you can use include continuous-time or discrete-time numeric LTI models such astf,zpk, orssmodels.

Ifsysis an array of models,pzmapplots all the poles and zeros of every model in the array on the same plot.

Output Arguments

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Poles of the system, returned as a column vector, in order of its increasing natural frequency.pis the same as the output ofpole(sys), except for the order.

Transmission zeros of the system, returned as a column vector.zis the same as the output oftzero(sys).

Tips

  • 使用the functionssgridorzgridto plot lines of constant damping ratio and natural frequency in thes- orz-plane on the pole-zero plot.

  • For MIMO models,pzmapdisplays all system poles and transmission zeros on a single plot. To map poles and zeros for individual I/O pairs, useiopzmap.

  • For additional options to customize the appearance of the pole-zero plot, usepzplot.

Version History

Introduced before R2006a

See Also

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