MATLAB: Using feedback() with time delay

Question

Dear Matlab community,

I would expect the same step response in the plot in the following code, but the step response of G_y_r is unstable, while T_CL is stable.

The only difference is that T_CL is built using the feedback() function, while G_y_r is built manually.

Does someone have a hint why G_y_r has two internal delays ("Internal delays (seconds): 2.5e-05 2.5e-05"), while T_CL has only one?

It seems that feedback() is doing something special here?

Also, G_y_r has more internal state variables than T_CL.

It would be great if someone could clarify this!

Thanks and best regards,

Robert

%


C = tf([1.5, 47.12], [1.0, 0.0]);
G_delay = tf([1], [1], 'OutputDelay', 1.0 / 40.0e3);
P = tf([6773, 5.131e7], [1.0, 1.123e4, 5.193e7]);
H = tf([5.988e19], [1.0, 348900, 4.591e10, 2.7e15, 5.988e19]);
% build closed-loop using feedback()
T_CL = minreal(feedback(P * C * G_delay, H));
% build closed-loop manually
G_y_r = minreal((P * C * G_delay) / (1.0 + P * C * G_delay * H));
%
T_CL, G_y_r
%
figure;
hold on;
step(T_CL, '-b');
step(G_y_r, '*r');
grid on;
legend('T_{CL}', 'G_{y, r}', ...
  'Location', 'best');

Answer 1

Hi Robert,

Your code is just fine except for one tiny but a curicial point that is internally built-in solver step size related issue of the control toolbox. In fact, there is only one time delay. Here is the correct code that addresses solver step size in computing the systems' step responses. Now, your simulation results of the both systems converge perfectly well.

clc; clear variables
%


C = tf([1.5, 47.12], [1.0, 0.0]);
G_delay = tf(1, 1, 'OutputDelay', 1.0 / 40.0e3);
P = tf([6773, 5.131e7], [1.0, 1.123e4, 5.193e7]);
H = tf(5.988e19, [1.0, 348900, 4.591e10, 2.7e15, 5.988e19]);
% build closed-loop using feedback()
T_CL = minreal(feedback(P * C * G_delay, H));
% build closed-loop manually
G_y_r = minreal((P * C * G_delay) / (1.0 + P * C * G_delay * H));
%
disp('System T_CL:  '); T_CL  %#ok

disp('System G_y_r: '); G_y_r %#ok
%
figure;
hold on;
dt = 1/1000;  % Time step
t=0:dt:.5;    % Simulation time space
step(T_CL, t, '-b');
step(G_y_r, t, '*r');
grid on;
legend('T_{CL}', 'G_{y, r}', 'Location', 'best');

Good luck.