pendulumModelAndController.mat contains a SISO inverted pendulum transfer function model G and its associated PID controller C.

Load the inverted pendulum and controller model to the workspace.

load('pendulumModelAndController','G','C');
size(G)

Transfer function with 1 outputs and 1 inputs.

size(C)

PID controller with 1 output and 1 input.

Use feedback to create the negative feedback loop with G and C.

sys = feedback(G,C)

sys =
 
                        1.045e-05 s^2
  ---------------------------------------------------------
  2.3e-06 s^4 + 1.725e-06 s^3 - 4.035e-05 s^2 - 5.018e-06 s
 
Continuous-time transfer function.

sys is the resultant closed loop continuous-time transfer function obtained using negative feedback. feedback converts the PID controller model C to a transfer function before connecting it to the continuous-time transfer function model G. For more information, see Rules That Determine Model Type.

For this example, consider two transfer functions that describe a plant G and controller C respectively.

Create the plant and controller transfer functions.

G = tf([2 5 1],[1 2 3],'inputname',"torque",'outputname',"velocity");
C = tf([5,10],[1,10]); 

Use feedback to create the negative feedback loop using G and C.

sys = feedback(G,C,-1)

sys =
 
  From input "torque" to output "velocity":
  2 s^3 + 25 s^2 + 51 s + 10
  ---------------------------
  11 s^3 + 57 s^2 + 78 s + 40
 
Continuous-time transfer function.

sys is the resultant closed loop transfer function obtained using negative feedback with torque as the input and velocity as the output.

For this example, consider two transfer functions that describe a plant G and controller C respectively.

Create the plant and controller transfer functions.

G = tf([2 5 1],[1 2 3],'inputname',"torque",'outputname',"velocity");
C = tf([5,10],[1,10]); 

Use feedback to create the positive feedback loop using G and C.

sys = feedback(G,C,+1)

sys =
 
  From input "torque" to output "velocity":
  -2 s^3 - 25 s^2 - 51 s - 10
  ---------------------------
  9 s^3 + 33 s^2 + 32 s - 20
 
Continuous-time transfer function.

sys is the resultant closed loop transfer function obtained using positive feedback with torque as the input and velocity as the output.

Based on the figure below, consider connecting two MIMO transfer functions with two inputs and two outputs in a negative feedback loop.

For this example, create two random continuous state-space models using rss.

G = rss(4,2,2);
C = rss(2,2,2);
size(G)

State-space model with 2 outputs, 2 inputs, and 4 states.

size(C)

State-space model with 2 outputs, 2 inputs, and 2 states.

Use feedback to connect the two state-space models in a negative feedback loop according to the above figure.

sys = feedback(G,C,-1);
size(sys)

State-space model with 2 outputs, 2 inputs, and 6 states.

The resulting state-space model sys is a 2 input, 2 output model with 6 states. The negative feedback loop is completed such that,

mimoPlantAndController.mat contains a 2 input, 2 output transfer function plant model G and a 2 input, 2 output transfer function controller model C to be connected as follows:

First, load the plant and controller models to the workspace.

load('mimoPlantAndController.mat','G','C');
size(G)

Transfer function with 2 outputs and 2 inputs.

size(C)

Transfer function with 2 outputs and 2 inputs.

By default, feedback would connect the first output of G to the first input of C and the second output of G to the second input of C. In order to connect the plant and controller according to the figure, name the respective I/Os of the two systems to ensure the correct connections.

G.InputName 

ans = 2x1 cell
    {'torque'}
    {'angle' }

G.OutputName

ans = 2x1 cell
    {'velocity'}
    {'force'   }

C.InputName

ans = 2x1 cell
    {'force'   }
    {'velocity'}

C.OutputName

ans = 2x1 cell
    {'angle' }
    {'torque'}

Then use the 'name' flag with the feedback command to make the connections according to the I/O names.

sys = feedback(G,C,'name');

The resulting closed loop negative feedback transfer function sys has the feedback connections in the required order.

Consider a state-space plant G with five inputs and four outputs and a state-space feedback controller K with three inputs and two outputs. The outputs 1, 3, and 4 of the plant G must be connected the controller K inputs, and the controller outputs to inputs 2 and 4 of the plant.

For this example, generate randomized continuous-time state-space models using rss for both G and K.

G = rss(3,4,5);
K = rss(3,2,3);

Define the feedout and feedin vectors based on the inputs and outputs to be connected in a feedback loop.

feedin = [2 4];
feedout = [1 3 4];
sys = feedback(G,K,feedin,feedout,-1);
size(sys)

State-space model with 4 outputs, 5 inputs, and 6 states.

sys is the resultant closed loop state-space model obtained by connecting the specified inputs and outputs of G and K.