zgrid

Generate z-plane grid of constant damping factors and natural frequencies

Syntax

zgrid

zgrid(T)

zgrid(zeta,wn)

zgrid(zeta,wn,T)

zgrid(___,'new')

zgrid(AX,___)

Description

zgrid generates a grid of constant damping factors from 0 to 1 in steps of 0.1 and natural frequencies from 0 to π/T in steps of 0.1*π/T for root locus and pole-zero maps. The default steps of 0.1*π/T represent fractions of the Nyquist frequencies. zgrid then plots the grid over the current axis. zgrid creates the grid over the plot without altering the current axis limits if the current axis contains a discrete z-plane root locus diagram or pole-zero map. Use this syntax to plot multiple systems with different sample times.

Alternatively, you can select Grid from the context menu in the plot window to generate the same z-plane grid.

zgrid(T) generates the z-plane grid by using default values for damping factor and natural frequency relative to the sample time T.

zgrid(zeta,wn) plots a grid of constant damping factor and natural frequency lines for the damping factors and normalized natural frequencies in the vectors zeta and wn, respectively. When the sample time is not specified, the frequency values in wn are interpreted as normalized values, that is, wn/T.

zgrid(zeta,wn,T) plots a grid of constant damping factor and natural frequency lines for the damping factors and natural frequencies in the vectors zeta and wn, relative to sample time T. zeta lines are independent for T but the wn lines depend on the sample time value. Use this syntax to create the z-plane grid with specific values of wn.

zgrid(___,'new') clears the current axes first and sets hold on.

zgrid(AX,___) plots the z-plane grid on the Axes or UIAxes object in the current figure with the handle AX. Use this syntax when creating apps with zgrid in the App Designer.

Examples

Plot z-plane Grid Lines on the Root Locus

To see the z-plane grid on the root locus plot, type

H = tf([2 -3.4 1.5],[1 -1.6 0.8],-1)

H =
 
  2 z^2 - 3.4 z + 1.5
  -------------------
   z^2 - 1.6 z + 0.8
 
Sample time: unspecified
Discrete-time transfer function.

rlocus(H)
zgrid
axis equal

Normalized and True z-plane Grid Lines on the Pole-Zero Map

For this example, consider a discrete-time transfer function sys with a sample time of 0.1s. Now plot the pole-zero map of sys and visualize the default z-plane grid without specifying the sample time.

sys = tf([2 -3.4 1.5],[1 -1.6 0.8],0.1);
Ts = 0.1;
figure()
pzmap(sys)
zgrid()
axis equal

Observe that the frequencies on the z-plane grid are normalized in terms of $$f\frac{\pi }{T}$$. To obtain the true frequency values on the grid, specify the sample time with the zgrid command.

figure()
pzmap(sys)
zgrid(Ts)
axis equal

Now, observe that the frequency values on the plot are true values, that is, they are non-normalized.

Input Arguments

zeta — Damping ratio

vector

Damping ratio, specified as a vector in the same order as wn.

wn — Natural frequency values

vector

Natural frequency values, specified as a vector. Natural frequencies are plotted as true values when T is specified. When the sample time is not specified, zgrid normalizes the values as wn/T.

T — Sample time

positive scalar | -1

Sample time, specified as:

• A positive scalar representing the sampling period of a discrete-time system. The actual frequency values are displayed on the frequency grid.

• -1 for a discrete-time system with an unspecified sample time. The frequency values are displayed as normalized values f*π/T for the default grid.

zeta lines are independent of T while wn lines are dependent on the sample time. You must specify T to plot specific values of wn. When the sample time T is not specified, the required wn values are interpreted as normalized values, that is, wn/T.

AX — Object handle

Axes object | UIAxes object

Object handle, specified as an Axes or UIAxes object. Use AX to create apps with zgrid in the App Designer.

Version History

Introduced before R2006a