Plotting/Representing function of complex variable in matlab

Question

I am studying the principle of argument in control system, and I have this function

$f(s) = \frac {1}{s^2-5s+6} $

I want to plot this function for $ |s| = 1 $ i.e. a unit circle in s domain.

I get the transfer function using

G = zpk([],[2,3],1);

I tried creating a (HALF)unit circle with

x=linspace(-1,1);
y=sqrt(1-x.^2);

I could represent s as x+iy, but then how would i put it in $ f(s) $, should I be using the TF ?

Kindly suggest me the proper method to do this.

Answer 1

This can be done by using polar representation of $s$, i.e. $s = re^{i\varphi}$ with $r = |s| = 1$. With this method you don't need the control toolbox.

Substituting $s$ with $e^{i\varphi}$ in your function gives

$$f(e^{i\varphi})=\frac1{(e^{i\varphi}-2)(e^{i\varphi}-3)}$$

Now just calculate values for $\varphi$ in the range $[0, 2\pi]$.

Note that the resulting values are complex, so you need to choose how to visualize them. You can for example plot the real and imaginary values or the modulus and arguments.

phi = linspace(0, 2*pi);
f = 1./((exp(i*phi)-2).*(exp(i*phi)-3));
figure
plot(phi,real(f)); hold on; plot(phi,imag(f),'r');
figure
plot(phi,abs(f)); hold on; plot(phi,angle(f),'r');

Answer 2

colormap(hsv(64));
z = cplxgrid(30);
cplxmap(z,1./(z.^2-5.*z+6));

see here for more details.