Given a process model P(s) with size 3×2 (3 outputs, 2 inputs) and a controller K(s) with size 2×3 (2 outputs, 3 inputs), the negative feedback interconnection is valid since:
- Controller inputs match plant outputs (both 3)
- Controller outputs match plant inputs (both 2)
In MATLAB, you can use the “feedback” function directly as shown below:
T = feedback(P, K);
This returns the closed-loop transfer function from external inputs to plant outputs, resulting in a system of size 3×2.
Here is an example for the “feedback” function:
% Define Laplace variable
s = tf('s');
% Define 3x2 plant P (3 outputs, 2 inputs)
P = [1/(s+1), 2/(s+2);
3/(s+3), 4/(s+4);
5/(s+5), 6/(s+6)];
% Define 2x3 controller K (2 outputs, 3 inputs)
K = [1/(s+7), 2/(s+8), 3/(s+9);
4/(s+10), 5/(s+11), 6/(s+12)];
% Check dimensions
disp('Size of P (plant):'); disp(size(P)) % Should be [3 2]
disp('Size of K (controller):'); disp(size(K)) % Should be [2 3]
% Form closed-loop negative feedback system
T = feedback(P, K);
% Display size of closed-loop system
disp('Size of closed-loop system T:');
disp(size(T)) % Should be [3 2] because feedback assumes the external input comes before the plant input (which is 2-dimensional).
% Simulate step response
figure;
step(T)
title('Step Response of Closed-Loop System T');
xlabel('Time (seconds)');
ylabel('Output');
You can use feedback(P, K, +1) for positive feedback if needed.
For more information on the “feedback” function, please refer to the documentation:
The documentation also includes an example on negative feedback for MIMO systems, which can be accessed in MATLAB by running:
openExample('control/NegativeFeedbackLoopWithMIMOSystemsExample')
I hope you find this helpful.
