Block Diagrams of Systems
Sources:
- B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 386-401). Oxford University Press.
Figure 4.18a shows a block diagram of a system with a transfer function \(H(s)\) and its input and output \(X(s)\) and \(Y(s)\). respectively.
Large systems can be represented by suitably interconnected subsystems, each of which can be readily analyzed. Each subsystem can be characterized in terms of its input–output relationships.
Subsystems may be interconnected by using:
- cascade,
- parallel, and
- feedback
interconnections.
The cascade case
When transfer functions appear in cascade, as depicted in Fig. 4.18b, the transfer function of the overall system is the product of the two transfer functions. \[
\frac{Y(s)}{X(s)}=\frac{W(s)}{X(s)} \frac{Y(s)}{W(s)}=H_1(s) H_2(s)
\] 
The parallel case
Similarly, when two transfer functions, \(H_1(s)\) and \(H_2(s)\), appear in parallel, as illustrated in Fig. 4.18c, the overall transfer function is given by \(H_1(s)+H_2(s)\).
The feadback case
When the output is fed back to the input, as shown in Fig. 4.18d, the overall transfer function \(Y(s) / X(s)\) can be computed as follows. The inputs to the adder are \(X(s)\) and \(-H(s) Y(s)\). Therefore, \(E(s)\), the output of the adder, is \[ E(s)=X(s)-H(s) Y(s) \]
But \[ \begin{aligned} Y(s) & =G(s) E(s) \\ & =G(s)[X(s)-H(s) Y(s)] \end{aligned} \]
Therefore, \[ Y(s)[1+G(s) H(s)]=G(s) X(s) \] so that \[ \frac{Y(s)}{X(s)}=\frac{G(s)}{1+G(s) H(s)} \]
The feedback system is also called the close-loop system, as shown in Fig. 4.33b, whereas the system without feedback is called the open-loop system, as shown in Fig. 4.33a.
