Comparison between continuous and discrete transfer function s, z, to w

I was trying to do a Bode/Nyquist Plot of a transfer function in both continuous and discrete mode

My system has a sampling frequency of 15 kHz. I am trying to compare Bode Plots of continuous and discrete functions. I transformed my function from s to z and then to w.

My function is a resonant controller with damping.

Gv6(nn)=tf([0 kp],[0 1])+tf([k1*nn 0],[1,2*zeta*w0,(w0)^2])+tf([k5*nn 0],[1,2*zeta*5*w0,(5*w0)^2])+tf([k7*nn 0],[1,2*zeta*7*w0,(7*w0)^2])+tf([k11*nn 0],[1,2*zeta*11*w0,(11*w0)^2])+tf([k13*nn 0],[1,2*zeta*13*w0,(13*w0)^2])+tf([k17*nn 0],[1,2*zeta*17*w0,(17*w0)^2])+tf([k19*nn 0],[1,2*zeta*19*w0,(19*w0)^2])+tf([k23*nn 0],[1,2*zeta*23*w0,(23*w0)^2]);

nn=1 in this case zeta=0.03, w0=2*pi*50, kp=1, all other kn=250*2*zeta*n*w0, n harmonic number

I am using the following functions:

sys=Gv6(1);
opt = c2dOptions('Method','tustin','PrewarpFrequency',2*pi*f);
sysd = c2d(sys,Tsamp,opt);
[num,den] = tfdata(sysd);
syms z
t_sym = poly2sym(cell2mat(num),z)/poly2sym(cell2mat(den),z);
syms z w
Tsamp=1/15000;
G=t_sym;
GN=subs(G,z,(2+Tsamp*w)/(2-Tsamp*w));
[n,d]=numden(GN);
eq1 = n;
eq2 = d;
s = tf('s');
eq1_s = strrep(char(eq1),'w','s');
eq2_s = strrep(char(eq2),'w','s');
G = eval(eq1_s) / eval(eq2_s);
bode(Gv6(1),sysd); hold on;grid on;
bode(G); hold on;grid on;

So far the continues and discrete plots does not look similar in the lower frequencies, but look OK at high frequencies, What is wrong with my code?

Answer 1

[TED MOSBY]

Hi,

The problem in lower frequency arose because when in your manual “inverse” step you wrote:

z  = (2 + Ts*w)/(2 - Ts*w);      % <-- K is missing here

That assumed K=1. Dropping K changes the dc (low-frequency) gain, which is why the two Bode plots only matched at higher frequencies (they cross exactly at the pre-warp frequency where the scale error is zero).

To fix this you can let MATLAB undo the mapping for you:

sysc_from_d = d2c(sysd,'tustin',wp);   % use SAME wp
bode(sys, sysc_from_d), grid on

Now 'd2c' knows about the K factor, so the curves sit on top of each other everywhere.

Here is more information on 'd2c' :

https://www.mathworks.com/help/control/ref/dynamicsystem.d2c.html?s_tid=srchtitle_support_results_1_d2c

Hope this helps!