How to calculate the norm of the transfer function in frequency domain?

To calculate the norm of the transfer function by substituting s=jω is troublesome, especially for some complicated transfer functions. Is there a way to calculate the norm directly? Thanks!

For example, transfer funciton:

Substituting s=jω，

then,

Thus we can plot the figure in frequency domain,

Matlab code:

omega=0:0.01:10;

G1_N=0.25e-2 .* omega .^ 2 + 0.1e1;

G1_D=((2 .* omega) - 0.15e0 .* (omega .^ 3)) .^ 2 + (0.1e1 + 0.5e-2 .* (omega .^ 4) - (omega .^ 2)) .^ 2;

G1=sqrt(G1_N ./ G1_D);

plot(omega,G1)

Answer 1

[William Rose]

@Cola,

You don't need to multiply the function by its complex conjugate to get a purely real denominator. Just divide complex numerator by the complex denominator, to get a new complex number, and take the abs() of it.

[William Rose]

@Cola, in this case you have

Therefore I can write matlab code (without doing any more algebraic manipulaitons):

omega=0:0.01:10;
s=1i*omega;
G1=(-0.05*s-1)./(.005*s.^4+.15*s.^3+s.^2+2*s+1); 
semilogx(omega,abs(G1),'-bx'); grid on

Try it.

[Cola]

@William Rose Thank you so much.

[William Rose]

@Cola, you're welcome, and good luck with your work.

Answer 2

[Paul]

Check out

doc tf

to learn how to create a transfer function (tf) object. Once you have G(s) defined as a tf object use bode() to compute its magnitude (and phase if desired)

doc bode

[William Rose]

@Cola,

@Paul is right that bode() and other functions in the system identification toolbox have great capabilities.

Iy you want a quick phase plot without using the toolbox, you can use the same simple approach that I mentioned for the magnitude plot:

omega=0.01:0.01:100;
s=1i*omega;
G=(-0.05*s-1)./(.005*s.^4+.15*s.^3+s.^2+2*s+1);  %transfer function
subplot(211), semilogx(omega,abs(G),'-bx'); grid on
ylabel('Magnitude |G(\omega)|');
subplot(212), semilogx(omega,angle(G),'-bx'); grid on
xlabel('Frequency \omega'); ylabel('Phase \Phi(G(\omega)) (radians)');

[Paul]

Actually, I was thinking of the Control System Toolbox. I'm not famiiar with the System ID toolbox.

If just wanting to use base Matlab, I'd probably use polyval().

[Cola]

@Paul @William Rose Thanks to you. And we also can use the order 'freqs' to calculate the norm.