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Nyquist plot of frequency response

`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)

`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`

.

The output arguments `re`

and `im`

are
3-D arrays with dimensions

$$(\text{numberofoutputs)}\times \text{(numberofinputs)}\times \text{(lengthofw)}$$

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).

$$\begin{array}{l}\text{re}(1,1,k)=\mathrm{Re}\left(h(j{\omega}_{k})\right)\\ \text{im}(1,1,k)=\mathrm{Im}\left(h(j{w}_{k})\right)\end{array}$$

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,

$$\begin{array}{l}\text{re}\text{\hspace{0.17em}}\text{(i,}\text{\hspace{0.17em}}\text{j,}\text{\hspace{0.17em}}\text{k)}=\mathrm{Re}\left({h}_{ij}(j{\omega}_{k})\right)\\ \text{im}\text{\hspace{0.17em}}\text{(i,}\text{\hspace{0.17em}}\text{j,}\text{\hspace{0.17em}}\text{k)}=\mathrm{Im}\left({h}_{ij}(j{\omega}_{k})\right)\end{array}$$

where *h _{ij}* is the transfer
function from input

Plot the Nyquist response of the system

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

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

The nyquist function has support for M-circles, which are the contours of the constant closed-loop magnitude. M-circles are defined as the locus of complex numbers where

$$T(j\omega )=\left|\frac{G(j\omega )}{1+G(j\omega )}\right|$$

is a constant value. In this equation, 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 activate the grid, select **Grid** from
the right-click menu or type

grid

at the MATLAB^{®} prompt. This figure shows the M circles for
transfer function *H*.

You have two zoom options available from the right-click menu that apply specifically to Nyquist plots:

**Tight**—Clips unbounded branches of the Nyquist plot, but still includes the critical point (-1, 0)**On (-1,0)**— Zooms around the critical point (-1,0)

Also, click anywhere on the curve to activate data markers that display the real and imaginary values at a given frequency. This figure shows the nyquist plot with a data marker.

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');

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 (Control System Toolbox).

See `bode`

.

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