Control System Analysis, Transfer function

Open loop and closed loop transfer function – examples

To understand the concept of Transfer Function in open loop, or on the contrary, closed loop, we use a block diagram of a closed loop system, Figure 1:

null

Figure 1

Where G (s) is the transfer function of the plant and H (s) is the transfer function of the sensor. The sensor generates a signal B (s) that is fed back to the summing point, where it is compared with the reference signal R (s), generating a signal called the error signal E(s). Applying block algebra to Figure 1 we can clearly see that the output signal C (s) can be obtained by multiplying E (s) by G (s):

nullThat is to say:null

The function G (s) of equation (1) is known as the direct path transfer function (quotient between the output and the error signal):

null

Again applying block algebra to Figure 1 we can see that the feedback signal B (s) can be obtained by multiplying C (s) by H (s), that is:

nullThat is:null

The product G(s)H (s) from equation (2) is known as the open-loop transfer function (quotient between the feedback signal and the error signal):

null

Important notes:

  • If the transfer function H (s) of the feedback path (FT of the sensor) is equal to one, H(s)=1, only in this case, the closed-loop transfer function is equal to the transfer function direct path;
  • The direct path transfer function G (s) is also known simply as the Direct Transfer Function.

That is, if the system is represented by the DB in Figure 2:

null

Figure 2

Then the direct transfer function G(s) is also the open-loop function.

Once again, applying block algebra to Figure 1 we can see that the output signal C (s) can be obtained by multiplying G (s) by E (s), that is:

nullSolving for C (s), we obtain that:

nullFrom where:

null

The function C (s) / R (s) of equation (3) is known as the closed-loop transfer function (quotient between the output signal and the input signal):

null

Important note: Equation (3) allows us to obtain the Laplace transform of the output for any input, once we know what the closed-loop transfer function is, by:

null

Example:

 

I suggest to visit: Effect of adding a zero to a control system

Source:

  1. Katsuhiko Ogata, Ingeniería de Control Moderno, páginas 65-66.

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