6.3 Application to Classic Control 363FIGURE 6.4
Negative-feedback connection of
systems with transfer functionH 1 (s)
andH 2 (s). The input and the output
arex(t)andy(t), respectively, ande(t)
is the error signal.
H 2 (s)x(t) e(t) H 1 (s) y(t)
+
−Feedback Connection of LTI Systems
In control,feedbackconnections are more appropriate than cascade or parallel connections. In the
feedback connection, the output of the first system is fed back through the second system into
the input (see Figure 6.4). In this case, like in the parallel connection, beside the blocks representing
the systems we useaddersto add/subtract two signals.
It is possible to havepositive-ornegative-feedbacksystems depending on whether we add or subtract the
signal being fed back to the input. Typically, negative feedback is used, as positive feedback can greatly
increase the gain of the system. (Think of the screeching sound created by an open microphone near
a loud-speaker: the microphone continuously picks up the amplified sound from the loud-speaker,
increasing the volume of the produced signal. This is caused by positive feedback.) For negative
feedback, the connection of two systems is done by putting one in the feedforward loop,H 1 (s),
and the other in the feedback loop,H 2 (s)(there are other possible connections). To find the overall
transfer function we consider the Laplace transforms of the error signale(t),E(s), and of the output
y(t),Y(s), in terms of the Laplace transform of the inputx(t),X(s), and the transfer functionsH 1 (s)
andH 2 (s)of the systems:
E(s)=X(s)−H 2 (s)Y(s)Y(s)=H 1 (s)E(s)ReplacingE(s)in the second equation gives
Y(s)[1+H 1 (s)H 2 (s)]=H 1 (s)X(s)and the transfer function of the feedback system is then
H(s)=Y(s)
X(s)=
H 1 (s)
1 +H 1 (s)H 2 (s)(6.4)
As you recall, in Chapter 2 we were not able to find an explicit expression for the impulse response
of the overall system and now you can understand why.
6.3 Application to Classic Control...............................................................
Because of different approaches, the theory of control systems can be divided into classic and modern
control. Classic control uses frequency-domain methods, while modern control uses time-domain
methods. In classic linear control, the transfer function of the plant we wish to control is available;