# nyquist

nyquist plot

### Syntax

nyquist(sl) nyquist(sl, fmin, fmax) nyquist(sl, fmin, fmax, step) nyquist(sl, frq) nyquist(frq, db, phi) nyquist(frq, repf) nyquist(.., comments) nyquist(.., symmetry)

### Arguments

- sl
A siso or simo linear dynamical system, in state space, transfer function or zpk representations, in continuous or discrete time.

- fmin,fmax
real scalars (frequency lower and upper bounds (in Hz)).

- step
real (logarithmic discretization step), if not given an adaptative discretization is used.

- comments
string vector (captions).

- symmetry
a boolean, default value is %t.

- frq
vector or matrix of frequencies (in Hz) (one row for each output of

`sl`

).- db,phi
real matrices of modulus (in dB) and phases (in degree) (one row for each output of

`sl`

).- repf
matrix of complex numbers. Frequency response (one row for each output of

`sl`

)

### Description

Nyquist plot i.e Imaginary part versus Real part of the
frequency response of `sl`

. If the
`symmetry`

argument is true or omitted the
Nyquist plot displays the symmetric graph (positive and negative
frequencies).

For continuous time systems `sl(2*%i*%pi*w)`

is
plotted. For discrete time system or discretized systems
`sl(exp(2*%i*%pi*w*fd)`

is used ( `fd=1`

for discrete time systems and `fd=sl('dt')`

for
discretized systems )

`sl`

can be a continuous-time or
discrete-time SIMO system given by its state space,
rational transfer function (see syslin) or zpk representation. In case of
multi-output the outputs are plotted with different
colors.

The frequencies are given by the bounds `fmin,fmax`

(in Hz) or by a row-vector (or a matrix for multi-output)
`frq`

.

`step`

is the ( logarithmic ) discretization step.
(see `calfrq`

for the choice of default value).

`comments`

is a vector of character strings
(captions).

`db,phi`

are the matrices of modulus (in Db) and
phases (in degrees). (One row for each response).

`repf`

is a matrix of complex numbers. One row for
each response.

Default values for `fmin`

and
`fmax`

are `1.d-3`

,
`1.d+3`

if `sl`

is continuous-time or
`1.d-3`

, `0.5`

/sl.dt (nyquist frequency)
if `sl`

is discrete-time.

Automatic discretization of frequencies is made by calfrq.

To obtain the value of the frequency at a selected point(s) you can activate the datatips manager and click the desired point on the nyquist curve(s).

### Graphics entities organization

The `nyquist`

function creates a compound
object for each SISO system. The following piece of code allows
to get the handle on the compound object of the ith system:

ax=gca();//handle on current axes hi=ax.children($+i-1)// the handle on the compound object of the ith system

This compound object has two children: a compound object that defines the small arrows (a compound of small polylines) and the curve labels (a compound of texts) and a polyline which is the curve itself. The following piece of code shows how one can customize a particular nyquist curve display.

hi.children(1).visible='off'; //hides the arrows and labels hi.children(2).thickness=2; //make the curve thicker

### Examples

//Nyquist curve s=poly(0,'s') h=syslin('c',(s^2+2*0.9*10*s+100)/(s^2+2*0.3*10.1*s+102.01)); h1=h*syslin('c',(s^2+2*0.1*15.1*s+228.01)/(s^2+2*0.9*15*s+225)) clf(); nyquist(h1) // add a datatip ax=gca(); h_h=ax.children($).children(2);//handle on Nyquist curve of h tip=datatipCreate(h_h,[1.331,0.684]); datatipSetOrientation(tip,"upper left");

//Hall chart as a grid for nyquist s=poly(0,'s'); Plant=syslin('c',16000/((s+1)*(s+10)*(s+100))); //two degree of freedom PID tau=0.2;xsi=1.2; PID=syslin('c',(1/(2*xsi*tau*s))*(1+2*xsi*tau*s+tau^2*s^2)); clf(); nyquist([Plant;Plant*PID],0.5,100,["Plant";"Plant and PID corrector"]); hallchart(colors=color('light gray')*[1 1]) //move the caption in the lower right corner ax=gca();Leg=ax.children(1); Leg.legend_location="in_upper_left";

### See also

- syslin — linear system definition
- bode — Bode plot
- black — Black-Nichols diagram of a linear dynamical system
- nyquistfrequencybounds — Computes the frequencies for which the nyquist locus enters and leaves a given rectangle.
- calfrq — frequency response discretization
- freq — frequency response
- repfreq — frequency response
- phasemag — phase and magnitude computation
- datatips — Tool for placing and editing tips along the plotted curves

### History

Version | Description |

6.0 | handling zpk representation |

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