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The Feedback Operator
Definition: The feedback operator is defined as:
g|h = g/(1+h*g)
The feedback operator returns the closed loop system for the following block diagram:
The variables g and h can be of any compatible type and the dimensions of g and h’ must be the
same.
Unity Feedback: For unity feedback systems:
compute the closed loop system by:
g|1
If g is multivariable then it must be square, and the unity gain in the feedback path expands to the
correct size identity matrix.
Precedence: The feedback operator has higher precedence than multiplication, so the closed loop
response of:
must be computed using parentheses:
(g*k)|h
Note that for multivariable systems g*k and k*g are quite a bit different.
Transfer Functions: For transfer functions g = ng/dg and h = nh/dh the feedback operator returns:
g|h=
ng d h
ng nh + d g d h
Due to cancellations that take place during transfer function algebra the same result is returned
whether g|h or g/(1+h*g) is used. The feedback operator is convenient but not necessary.
In the following example a closed loop system is computed and displayed:
CC>g=10*(s+1)/s/(s+2)
CC>h=3/(s+4)
CC>cl=g|h