Pole-zero plot of dynamic system
pzmap( creates a pole-zero plot
of the continuous or discrete-time dynamic
indicates the poles and zeros respectively, as shown in the following
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.
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,
pzmap plots the system
poles and zeros. For MIMO systems,
pzmap plots the system
poles and transmission zeros.
Pole-Zero Plot of Dynamic System
Plot the poles and zeros of the continuous-time system represented by the following transfer function:
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 = ; sys = idss(A,B,C,D,'Ts',0.1);
Examine the pole-zero map.
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.
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.
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
pzmap to calculate the poles and zeros of the following transfer function:
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.
State-space model with 1 outputs, 1 inputs, and 48 states.
Plot the poles and zeros of the system.
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.
sys — Dynamic system
dynamic system model | model array
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 as
tf (Control System Toolbox),
zpk (Control System Toolbox), or
ss (Control System Toolbox) models.
sys is an array of models,
pzmap plots all the poles and zeros of every model in
the array on the same plot.
p — Poles of the system
Poles of the system, returned as a column vector, in order of its
increasing natural frequency.
p is the same as the
pole(sys), except for the order.
z — Transmission zeros of the system
Transmission zeros of the system, returned as a column vector.
z is the same as the output of
Use the functions
zgridto plot lines of constant damping ratio and natural frequency in the s- or z-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, use
For additional options to customize the appearance of the pole-zero plot, use
Introduced before R2006a