How to make a transfer function minimum phase?

2023 5 16
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更新时间：2023 5 21
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Dear MATLAB Community,
I have a plant Transfer Function which is non minimum phase. I want to make it stable minimum phase system so that I can inverse it without instaability.
% Define the transfer function
H = tf([-4.8 16000 0 0],[4.8 16080 286800 51160000]);
isminphase([-4.8 16000 0 0], [4.8 16080 286800 51160000])
Thanks!

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Paul
Paul 2023-5-16
Hi Govind,
H is defined as a continous time transfer function (Control System Toolbox), but isminphase as used above is a Signal Processing Toolbox function that is only applicable to discrete time filters.
Assuming that H really is a continous time transfer function, it isn't minimum phase. But there is no recipe "to make it stable, minimum phase." Such a goal is too open ended. There would have to be some other criteria as well.
Sam Chak
Sam Chak 2023-5-21
Ran in:
Ran in:
Hi @Govind Narayan Sahu
What do you want to do after getting the inverse of this function?
H = tf([-4.8 16000 0 0], [4.8 16080 286800 51160000])
H = -4.8 s^3 + 16000 s^2 ----------------------------------------- 4.8 s^3 + 16080 s^2 + 286800 s + 5.116e07 Continuous-time transfer function.
step(H, 0.01)
Hi @Sam Chak,
I want to design a transfer function (Hinv) that will behave inversely to H so that the final magnitude of H*Hinv is (approximately) equal to one. This is because when I pass an input signal, it amplifies the output at about 10 Hz.
H will be invertible if it has a minimum phase.
I read this article: https://doi.org/10.1080/10910344.2012.698970
I am able to create a frequency response that has a minimum phase, but not able to get a correct transfer function with the use of "invfreqz" and "freqz".
% Define the transfer function
num = [-4.8 16000 0 0];
den = [4.8 1.608e04 2.868e05 5.116e07];
H = tf(num,den); % Inverse Transfer function Den/Num
w = 2*pi*(0.1:0.01:100);
% Calculate complex cepstrum
c_hat_n = ifft(log(abs(squeeze(freqresp(H,w)))));
N = length(c_hat_n);
Nst = round(N/2)+1;
c_hat_n(Nst:N) = 0;
c_hat_n = c_hat_n*2;
h_hat_min = c_hat_n;
Hmin=exp(fft(h_hat_min));
w11 = angle(Hmin);
[b,a] = invfreqz((Hmin),w11, 3,3, [], 1000);
[h,w1] = freqz(b,a,N);
subplot(211)
hold on;plot(w/2/pi, abs(squeeze(freqresp(H,w))));
plot(w/2/pi, abs(Hmin),'r')
plot(w/2/pi, abs(h),'g')
xlabel('Frequency [Hz]')
ylabel('Magnitude, [N/N]')
subplot(212)
hold on;plot(w/2/pi, phase(squeeze(freqresp(H,w)))+2*pi);
plot(w/2/pi, phase(Hmin),'r')
plot(w/2/pi, 2*pi-w1,'g')
xlabel('Frequency [Hz]')
ylabel('Phase, [degree]')

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