2-18-10: Deriving the Transfer Function for a Simple Closed-Loop System

I'm taking a course in Feedback and Control Systems as part of my graduate studies, and, while helping another student last night after class, I realized that it is indeed a very good thing to know how to derive a solution rather than to simply memorize an equation.

The question was simple: how does one derive the transfer function to a simple closed loop system?

Take the following system as an example:

The system components are the Reference input R(s), what you want the system's output to be, M(s), the Controller, G(s), the Plant or model of the system, Y(s), the output of the system, and H(s), the Sensor. In addition, I have added X(s), which is the output of the Sensor, and W(s), which is after the summing block.

Last night I "walked around the loop" with my fellow student to help him understand how to get the transfer function, which is Y(s) / R(s), or simply the ratio of the output to the reference input.

Let's start by finding W(s) and X(s) in terms of the system components.

(1) W(s) = R(s) – X(s) the output of the summing block

(2) X(s) = H(s) * Y(s) remember that Y(s) is fed back through H(s)

Now let's find Y(s) in terms of W(s):

(3) Y(s) = W(s) * M(s) * G(s)

Now insert Equation 1 to eliminate W(s) from Equation 3 above.

(4) Y(s) = [R(s) – X(s)] * M(s) * G(s)

Now insert Equation 2 to eliminate X(s)

(5) Y(s) = [R(s) – [H(s) * Y(s)] ] * M(s) * G(s)

Now we have an equation for Y(s) in terms of all the other system components. Now we just need to get Y(s) / R(s):

(6) Y(s) = R(s) * M(s) * G(s) – H(s) * Y(s) * M(s) * G(s)

(7) Y(s) + H(s) * Y(s) * M(s) * G(s) = R(s) * M(s) * G(s)

(8) Y(s) * [1 + H(s) * M(s) * G(s)] = R(s) * M(s) * G(s)

And the finale:

(9) Y(s) / R(s) = [ M(s) * G(s)] / [1 + H(s) * M(s) * G(s)]