c2d Tustin not matching analytical method
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I have a second order lowpass filter. Discrete transter function using c2d with tustin method does not match the analytical solution (see the code below with bode comparison). Analytical solution was obtained by subing s = 2/Ts * (z-1)/(z+1) in continuous transfer function which is apparantly consistent with Matlab c2d tsting option with the difference that the Matlab code is first converting TF to PZK and then fomulating c2d (see reference folders below) which is not an exact match to my fomulation but I did not expect this big of a difference. Thoughts appreciated.
/Applications/MATLAB_R2024b.app/toolbox/shared/controllib/engine/+ltipack/@tfdata/c2d.m % ltipack.tfdata method
/Applications/MATLAB_R2024b.app/toolbox/shared/controllib/engine/+ltipack/@zpkdata/c2d.m % ltipack.zpkdata method
%% Filter Specs
Ts = 0.01;
zeta = 1;
wn_rps = 30 *2*pi;
%% C2D (tustin)
s = tf('s');
tf_cont = wn_rps^2 / (s^2 + 2*zeta*wn_rps*s + wn_rps^2);
tf_disc = c2d(tf_cont, Ts, 'Method','tustin');
%% Analytical (tustin)
% s = 2/Ts * (z-1)/(z+1)
K = 2/Ts;
num = wn_rps^2 * [1 2 1];
den = [ K^2 + 2*zeta*wn_rps*K + wn_rps^2, ...
-2*K^2 + 2*wn_rps^2, ...
K^2 - 2*zeta*wn_rps*K + wn_rps^2 ];
tf_disc_analytical = tf(num, den, Ts);
%% Plot
figure, bode(tf_disc), hold on, bode(tf_disc_analytical)
采纳的回答
Hi Alborz,
%% Filter Specs
Ts = 0.01;
zeta = 1;
wn_rps = 30 *2*pi;
%% C2D (tustin)
s = tf('s');
tf_cont = wn_rps^2 / (s^2 + 2*zeta*wn_rps*s + wn_rps^2);
tf_disc = c2d(tf_cont, Ts, 'Method','tustin')
A little-known "feature" of c2d is that it only looks at the first character of the 'method' argument, which is the third argument, and matches that to the first character of one of the allowable methods. In this case, the 'M' corresponds to 'matched' (and it appears that the fourth argument, in this case 'tustin', is ignored.
tf_disc = c2d(tf_cont, Ts, 'matched')
The correct conversion for Tustin would be
tf_disc = c2d(tf_cont, Ts, 'tustin')
%% Analytical (tustin)
% s = 2/Ts * (z-1)/(z+1)
K = 2/Ts;
num = wn_rps^2 * [1 2 1];
den = [ K^2 + 2*zeta*wn_rps*K + wn_rps^2, ...
-2*K^2 + 2*wn_rps^2, ...
K^2 - 2*zeta*wn_rps*K + wn_rps^2 ];
tf_disc_analytical = tf(num, den, Ts)
minreal normalizes the leading coefficient of the denominator to unity to match the computed result above.
tf_disc_analytical = minreal(tf_disc_analytical)
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