nyquist

Nyquist plot of frequency response

Syntax

nyquist(sys) nyquist(sys,w) nyquist(sys1,sys2,...,sysN) nyquist(sys1,sys2,...,sysN,w) nyquist(sys1,'PlotStyle1',...,sysN,'PlotStyleN') [re,im,w] = nyquist(sys) [re,im] = nyquist(sys,w) [re,im,w,sdre,sdim] = nyquist(sys)

Description

nyquist creates a Nyquist plot of the frequency response of a dynamic system model. When invoked without left-hand arguments, nyquist produces a Nyquist plot on the screen. Nyquist plots are used to analyze system properties including gain margin, phase margin, and stability.

nyquist(sys) creates a Nyquist plot of a dynamic system sys. This model can be continuous or discrete, and SISO or MIMO. In the MIMO case, nyquist produces an array of Nyquist plots, each plot showing the response of one particular I/O channel. The frequency points are chosen automatically based on the system poles and zeros.

nyquist(sys,w) explicitly specifies the frequency range or frequency points to be used for the plot. To focus on a particular frequency interval, set w = {wmin,wmax}. To use particular frequency points, set w to the vector of desired frequencies. Use logspace to generate logarithmically spaced frequency vectors. Frequencies must be in rad/TimeUnit, where TimeUnit is the time units of the input dynamic system, specified in the TimeUnit property of sys.

nyquist(sys1,sys2,...,sysN) or nyquist(sys1,sys2,...,sysN,w) superimposes the Nyquist plots of several LTI models on a single figure. All systems must have the same number of inputs and outputs, but may otherwise be a mix of continuous- and discrete-time systems. You can also specify a distinctive color, linestyle, and/or marker for each system plot with the syntax nyquist(sys1,'PlotStyle1',...,sysN,'PlotStyleN').

[re,im,w] = nyquist(sys) and [re,im] = nyquist(sys,w) return the real and imaginary parts of the frequency response at the frequencies w (in rad/TimeUnit). re and im are 3-D arrays (see "Arguments" below for details).

[re,im,w,sdre,sdim] = nyquist(sys) also returns the standard deviations of re and im for the identified system sys.

Arguments

The output arguments re and im are 3-D arrays with dimensions

$(number of outputs) \times (number of inputs) \times (length of w)$

For SISO systems, the scalars re(1,1,k) and im(1,1,k) are the real and imaginary parts of the response at the frequency ω_k = w(k).

$\begin{array}{l} re (1, 1, k) = Re (h (j ω_{k})) \\ im (1, 1, k) = Im (h (j w_{k})) \end{array}$

For MIMO systems with transfer function H(s), re(:,:,k) and im(:,:,k) give the real and imaginary parts of H(jω_k) (both arrays with as many rows as outputs and as many columns as inputs). Thus,

$\begin{array}{l} re (i, j, k) = Re (h_{i j} (j ω_{k})) \\ im (i, j, k) = Im (h_{i j} (j ω_{k})) \end{array}$

where h_ij is the transfer function from input j to output i.

Examples

collapse all

Nyquist Plot of Dynamic System

Open Live Script

Create the following transfer function and plot its Nyquist response.

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

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

The nyquist function can display a grid of M-circles, which are the contours of constant closed-loop magnitude. M-circles are defined as the locus of complex numbers where the following quantity is a constant value across frequency.

$T (j ω) = | \frac{G (j ω)}{1 + G (j ω)} |$ .

Here, ω is the frequency in radians/TimeUnit, where TimeUnit is the system time units, and G is the collection of complex numbers that satisfy the constant magnitude requirement.

To display the grid of M-circles, right-click in the plot and select Grid. Alternatively, use the grid command.

grid on

Create Nyquist Plot of Identified Model With Response Uncertainty

This example uses:

System Identification Toolbox

Open Live Script

Compute the standard deviations of the real and imaginary parts of the frequency response of an identified model. Use this data to create a 3σ plot of the response uncertainty.

Load estimation data z2.

load iddata2 z2;

Identify a transfer function model using the data.

sys_p = tfest(z2,2);

Obtain the standard deviations for the real and imaginary parts of the frequency response for a set of 512 frequencies, w.

w = linspace(-10*pi,10*pi,512);
[re,im,wout,sdre,sdim] = nyquist(sys_p,w);

Here re and im are the real and imaginary parts of the frequency response, and sdre and sdim are their standard deviations, respectively. The frequencies in wout are the same as the frequencies you specified in w.

Create a Nyquist plot showing the response and its 3σ uncertainty.

re = squeeze(re);
im = squeeze(im); 
sdre = squeeze(sdre);
sdim = squeeze(sdim);
plot(re,im,'b',re+3*sdre,im+3*sdim,'k:',re-3*sdre,im-3*sdim,'k:')
xlabel('Real Axis');
ylabel('Imaginary Axis');

Tips

You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots. For the most flexibility in customizing plot properties, use the nyquistplot command instead of nyquist.
There are two zoom options available from the right-click menu that apply specifically to Nyquist plots:
- Full View — Clips unbounded branches of the Nyquist plot, but still includes the critical point (–1, 0).
- Zoom on (-1,0) — Zooms around the critical point (–1,0). (To access critical-point zoom programmatically, use nyquistplot instead.)
To activate data markers that display the real and imaginary values at a given frequency, click anywhere on the curve. The following figure shows a nyquist plot with a data marker.

Algorithms

See bode.

Documentation