nyquist

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 — définition d'un système dynamique linéaire
• 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