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# 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

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 ωk = w(k).

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

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

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

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

## Examples

### Nyquist Plot of Dynamic System

Plot the Nyquist response of the system

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

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

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.

### Create Nyquist Plot of Identified Model With Response Uncertainty

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 (Control System Toolbox).

## Algorithms

See `bode`.

## See Also

#### Introduced before R2006a

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