How to visualise feedback function?

numA = [0.1];
denomA = [1 0];
C = tf(numA, denomA)

C =
 
  0.1
  ---
   s
 
Continuous-time transfer function.

numB = [1];
denomB = [1 1];
G = tf(numB, denomB)

G =
 
    1
  -----
  s + 1
 
Continuous-time transfer function.

s = tf('s');
R = 10/s

R =
 
  10
  --
  s
 
Continuous-time transfer function.

My task is to write an M-file to find and plot the response of the following system to an input signal of R(s) = 10/s.

However, I am unable to understand the function of different parameters in the FUNCTION called feedback().

One thing I understood is that it is a closed negative loop, therefore I won't have to add 1 at the last parameter which is for positive feedback loop.

Hence, I am unsure whether to use one of these for the final output:

Given that R(s) = 10/s.

I am thinking between:

system1 = feedback(C,G)

system1 =
 
   0.1 s + 0.1
  -------------
  s^2 + s + 0.1
 
Continuous-time transfer function.

step(R*system1)

system2 = feedback(C*G,1,-1)

system2 =
 
       0.1
  -------------
  s^2 + s + 0.1
 
Continuous-time transfer function.

step(R*system2)

What is the difference between system1 and system 2? Both are closed loops. In system 1, C and G are two different models. In system 2, C and G are treated as one model by mutliplying with unity being the other model.

What is the visual difference between system1 and system2? What am I doing wrong?

Answer 1

[Sam Chak]

Hi @Rubayet

This is how you can use the 'feedback()' function to obtain the closed-loop system. You can compare the two approaches presented below:

numA    = [0.1];
denomA  = [1 0];
C       = tf(numA, denomA)

C =
 
  0.1
  ---
   s
 
Continuous-time transfer function.

numB    = [1];
denomB  = [1 1];
G       = tf(numB, denomB)

G =
 
    1
  -----
  s + 1
 
Continuous-time transfer function.

Approach #1: Use the built-in 'feedback()' function

%% closed-loop system
clsys   = feedback(C*G, 1)

clsys =
 
       0.1
  -------------
  s^2 + s + 0.1
 
Continuous-time transfer function.

%% input signal (a constant)
R       = 10;   % R(s) = 10/s --> R(t) = 10

%% System response
step(R*clsys), grid on

Approach #2: Use the direct formula

s       = tf('s');
clsys2  = (C*G)/(1 + (C*G))

clsys2 =
 
         0.1 s^2 + 0.1 s
  -----------------------------
  s^4 + 2 s^3 + 1.1 s^2 + 0.1 s
 
Continuous-time transfer function.

clsys2  = minreal(clsys2)       % simplification (minimal realization)

clsys2 =
 
       0.1
  -------------
  s^2 + s + 0.1
 
Continuous-time transfer function.

%% System response
step(R*clsys), grid on