pzmap

Pole-zero map of dynamic system

Syntax

[p,z] = pzmap(sys)

pzmap(sys)

pzmap(sys1,sys2,...,sysN)

pzmap(sys1,ColorSpec1,...,sysN,ColorSpecN)

Description

[p,z] = pzmap(sys) returns the system poles and transmission zeros of the dynamic system model sys.

The following figure shows pole-zero maps for a continuous-time (left) and discrete-time (right) linear time-variant model.

• In continuous-time systems, 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 systems, 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.

pzmap(sys) plots a pole-zero map for sys. In the plot, x and o represent poles and zeros, respectively. For SISO systems, pzmap plots the system poles and zeros. For MIMO systems, pzmap plots the system poles and transmission zeros.

pzmap(sys1,sys2,...,sysN) plots poles and zeros for multiple models in a single plot. The models can have different numbers of inputs and outputs and can be a mix of continuous and discrete systems.

pzmap(sys1,ColorSpec1,...,sysN,ColorSpecN) sets the color for the plot of each system. For more plot customization options, use pzplot.

Examples

Pole-Zero Plot of Dynamic System

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

$$H\left(s\right)=\frac{2s^2+5s+1}{s^2+3s+5}.$$

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

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.

Pole-Zero Plot of Identified System

Plot the pole-zero map of a discrete time identified state-space (idss) model. In practice you can obtain an idss model 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)

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

Pole-Zero Map of Multiple Models

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

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

Poles and Zeros of Transfer Function

Use pzmap to calculate the poles and zeros of the following transfer function:

$$sys\left(s\right)=\frac{4.2s^2+0.25s-0.004}{s^2+9.6s+17}$$

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

Identify Near-Cancelling Pole-Zero Pairs

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)

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. pzmap is useful to visually identify such near-canceling pole-zero pairs to perform pole-zero simplification.

Input Arguments

sys — Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf (Control System Toolbox), zpk (Control System Toolbox), or ss (Control System Toolbox) models.

• Sparse state-space models, such as sparss (Control System Toolbox) or mechss (Control System Toolbox) models.

• Generalized or uncertain LTI models such as genss (Control System Toolbox) or uss (Robust Control Toolbox) models. Using uncertain models requires Robust Control Toolbox™ software.

  • For tunable control design blocks, the function evaluates the model at its current value to plot the response.

  • For uncertain control design blocks, the function plots the nominal value and random samples of the model.

• Identified LTI models, such as idtf, idss, or idproc models.

If sys is an array of models, the plot shows responses of all models in the array on the same axes.

ColorSpec — Color

"r" | "g" | "b" | ...

Color, specified as one of the following values.

Color	Description
"r"	red
"g"	green
"b"	blue
"c"	cyan
"m"	magenta
"y"	yellow
"k"	black
"w"	white

Output Arguments

p — Poles of the system

column vector

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

z — Transmission zeros of the system

column vector

Transmission zeros of the system, returned as a column vector. z is the same as the output of tzero(sys).

Tips

• Use the functions sgrid or zgrid to plot lines of constant damping ratio and natural frequency in the s- or z-plane on the pole-zero plot.

• For MIMO models, pzmap displays all system poles and transmission zeros on a single plot. To map poles and zeros for individual input-output pairs, use iopzmap.

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

Version History

Introduced before R2006a