Problem when utilizing the bode function to plot the multiplication of several transfer functions

Question

Tsai Han Hao 2022-11-18

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1855493-problem-when-utilizing-the-bode-function-to-plot-the-multiplication-of-several-transfer-functions

编辑： Tsai Han Hao 2022-11-18

Hi all,

I'm right now trying to plot the bode plot of a system that contains multiple transfer fucntions, for example: G1*G2*G3.

(all the tranfer functions are declared by using tf() )

However, it just occurred to me that the bode plot of G1*G2*G3 is different from G2*G1*G3. And I felt quite confused about the results since mathematically they should be the same?

==> This is the results of the code below, as you can see, the bode plots are different.

The example code is just similar to:

%% Initialization

omega_D_Val = 50;

M_Val = 4;

Kf_Val = 40;

Mn_Val = 4;

Kfn_Val = 40;

Mn_head_Val = 4/1.5;

Kfn_head_Val = 40;

Ts_Val = 0.0005;

wR_Larger = 3000;

Cf_Val = 1;

Pure_gain = (M_Val/70) * wR_Larger * (1/M_Val);

D_env_Val = 1.5;

K_env_Val = 2500;

Integrator = tf(Ts_Val,[1 -1],Ts_Val);

%% Plotting Part

figure;

set(gcf,'Units','centimeters','position',[0 0 25 30]);

[~, Tenv_None, Nenv_None, ~, OSenv_None]...

= OuterLoop_TransferFcn_withEnv(omega_D_Val, wR_Larger, Ts_Val, (M_Val/70), Mn_Val, M_Val, Kfn_head_Val, Kfn_Val, Kf_Val, D_env_Val, K_env_Val, Cf_Val);

[Gain, ~, Frequency] = bode(Pure_gain * Integrator * Tenv_None * OSenv_None, {1e-4,6283.19});

semilogx(reshape(Frequency,[],1), 20*log10(reshape(Gain,[],1)), "-", 'Color', 'k', 'LineWidth',1.5);

hold on

[Gain, ~, Frequency] = bode(Integrator * Tenv_None * OSenv_None * Pure_gain, {1e-4,6283.19});

semilogx(reshape(Frequency,[],1), 20*log10(reshape(Gain,[],1)), "--", 'Color', 'k', 'LineWidth',1.5);

%% Function for tranfer functions build up

function [G_env, T_env, N_env, Compensator_env, OuterLoop_Sensitivity_Function_env]...

= OuterLoop_TransferFcn_withEnv(omega_D_Val, omega_R_Val, Ts_Val, Mn_head_Val, Mn_Val, M_Val, Kfn_head_Val, Kfn_Val, Kf_Val, D_env_Val, K_env_Val, Cf_Val)

syms omega_R omega_D Ts Mn_head Mn M Kfn_head Kfn Kf D_env K_env

syms z

Cf = Cf_Val;

% Alpha and Beta Define

alpha = (Mn*Kf)/(M*Kfn);

beta = (Mn*Kfn_head)/(Mn_head*Kfn);

Assigned_Parameters = [omega_D, omega_R, Ts, Mn_head, Mn, M, Kfn_head, Kfn, Kf, D_env, K_env];

Assigned_Values = [omega_D_Val, omega_R_Val, Ts_Val, Mn_head_Val, Mn_Val, M_Val, Kfn_head_Val, Kfn_Val, Kf_Val, D_env_Val, K_env_Val];

Denominator = (2*(z - 1)*((2 + omega_R*Ts)*z + (-2 + omega_R*Ts))*(M*z^2 + (M*alpha*omega_D*Ts + D_env*Ts - 2*M)*z + (M + K_env*Ts^2 - D_env*Ts - M*alpha*omega_D*Ts)))...

+ (Mn_head*Cf*omega_R*Ts*((2 + omega_D*Ts)*z + (-2 + omega_D*Ts))*(M*beta*z^3 + (D_env*Ts*beta - 2*M*alpha - M*beta)*z^2 + (K_env*beta*Ts^2 + 4*M*alpha -M*beta)*z + (K_env*beta*Ts^2 - D_env*beta*Ts - 2*M*alpha + M*beta)));

Denominator = subs(Denominator, Assigned_Parameters, Assigned_Values);

G_tf_Numerator = (Mn_head*omega_R*Ts*((2 + omega_D*Ts)*z + (-2 + omega_D*Ts))*(M*beta*z^3 + (D_env*Ts*beta - 2*M*alpha - M*beta)*z^2 + (K_env*beta*Ts^2 + 4*M*alpha -M*beta)*z + (K_env*beta*Ts^2 - D_env*beta*Ts - 2*M*alpha + M*beta)));

G_tf_Denominator = (2*(z - 1)*((2 + omega_R*Ts)*z + (-2 + omega_R*Ts))*(M*z^2 + (M*alpha*omega_D*Ts + D_env*Ts - 2*M)*z + (M + K_env*Ts^2 - D_env*Ts - M*alpha*omega_D*Ts)));

G_tf_Numerator = subs(G_tf_Numerator, Assigned_Parameters, Assigned_Values);

G_tf_Denominator = subs(G_tf_Denominator, Assigned_Parameters, Assigned_Values);

[G_tf_Numerator, G_tf_Denominator] = numden(G_tf_Numerator/G_tf_Denominator);

[bG,~] = coeffs(G_tf_Numerator, z, 'All');

[bG_D,~] = coeffs(G_tf_Denominator, z, 'All');

T_env_Numerator = M*z^2 + (alpha*omega_D*Ts*M - 2*M)*z + (M - alpha*omega_D*Ts*M);

T_env_Denominator = M*z^2 + (D_env*Ts - 2*M + alpha*omega_D*Ts*M)*z + (K_env*Ts^2 - D_env*Ts + M - alpha*omega_D*Ts*M);

T_env_Numerator = subs(T_env_Numerator, Assigned_Parameters, Assigned_Values);

T_env_Denominator = subs(T_env_Denominator, Assigned_Parameters, Assigned_Values);

[T_env_Numerator, T_env_Denominator] = numden(T_env_Numerator/T_env_Denominator);

[bT,~] = coeffs(T_env_Numerator, z, 'All');

[bT_D,~] = coeffs(T_env_Denominator, z, 'All');

N_env_Numerator = M*z^2 + (D_env*Ts - 2*M)*z + (M + K_env*Ts^2 - D_env*Ts);

N_env_Denominator = M*z^2 + (D_env*Ts - 2*M + M*alpha*omega_D*Ts)*z + (M + K_env*Ts^2 - D_env*Ts - M*alpha*omega_D*Ts);

N_env_Numerator = subs(N_env_Numerator, Assigned_Parameters, Assigned_Values);

N_env_Denominator = subs(N_env_Denominator, Assigned_Parameters, Assigned_Values);

[N_env_Numerator, N_env_Denominator] = numden(N_env_Numerator/N_env_Denominator);

[bN,~] = coeffs(N_env_Numerator, z, 'All');

[bN_D,~] = coeffs(N_env_Denominator, z, 'All');

C_tf_Numerator = (Mn_head*Cf*omega_R*Ts*((2 + omega_D*Ts)*z + (-2 + omega_D*Ts))*(M*beta*z^3 + (D_env*Ts*beta - 2*M*alpha - M*beta)*z^2 + (K_env*beta*Ts^2 + 4*M*alpha -M*beta)*z + (K_env*beta*Ts^2 - D_env*beta*Ts - 2*M*alpha + M*beta)));

C_tf_Numerator = subs(C_tf_Numerator, Assigned_Parameters, Assigned_Values);

[C_tf_Numerator, C_tf_Denominator] = numden(C_tf_Numerator/Denominator);

[bC,~] = coeffs(C_tf_Numerator, z, 'All');

[bC_D,~] = coeffs(C_tf_Denominator, z, 'All');

OS_tf_Numerator = (2*(z - 1)*((2 + omega_R*Ts)*z + (-2 + omega_R*Ts))*(M*z^2 + (M*alpha*omega_D*Ts + D_env*Ts - 2*M)*z + (M + K_env*Ts^2 - D_env*Ts - M*alpha*omega_D*Ts)));

OS_tf_Numerator = subs(OS_tf_Numerator, Assigned_Parameters, Assigned_Values);

[OS_tf_Numerator, OS_tf_Denominator] = numden(OS_tf_Numerator/Denominator);

[bOS,~] = coeffs(OS_tf_Numerator, z, 'All');

[bOS_D,~] = coeffs(OS_tf_Denominator, z, 'All');

% Transfer Function Declarations

G_env = tf(double(bG)/double(bG_D(1)), double(bG_D)/double(bG_D(1)), Ts_Val);

T_env = tf(double(bT)/double(bT_D(1)), double(bT_D)/double(bT_D(1)), Ts_Val);

N_env = tf(double(bN)/double(bN_D(1)), double(bN_D)/double(bN_D(1)), Ts_Val);

Compensator_env = tf(double(bC)/double(bC_D(1)), double(bC_D)/double(bC_D(1)), Ts_Val); % (G*Cf)/(1 + G*Cf)

OuterLoop_Sensitivity_Function_env = tf(double(bOS)/double(bOS_D(1)), double(bOS_D)/double(bOS_D(1)), Ts_Val); % 1/(1 + G*Cf)

end

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Answer 1

Tsai Han Hao 2022-11-18

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1855493-problem-when-utilizing-the-bode-function-to-plot-the-multiplication-of-several-transfer-functions#answer_1104403

编辑：Tsai Han Hao 2022-11-18

Hi all,

I've found out that the tf data type is the main problem that causes the difference between the Bode plot of bode(G1*G2*G3) and bode(G2*G3*G1).

If you look through the Algorithms on how Matlab do the bode calculations, you can find out that zpk is the data type that Matlad do the Bode plot.

https://www.mathworks.com/help/ident/ref/lti.bode.html

Therefore, by utilizing bode(zpk(G1)*zpk(G2)*zpk(G3)), the results of the Bode plot will be correct no matter how you arrange the tranfer functions.

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