1. Introduction to Block Diagrams
Block diagrams are used to represent the components of a system in graphical demonstration.
We can give the definitions on the block diagram of a closed loop system.
Closed loop system

2. Building Block Diagrams
X(s)
G1(s)
Y(s)
Breaking
point u1
Transfer function: + u1 u1 + - +
Breaking
point u1
Sum (Substracter)
Sum (Adder)
Connections algebra :
Series:
G1(s)
Y(s)
U(s)
G2(s)
G1(s) G2(s)
Y(s)
U(s)
Parallel:
Feedback: (Negative)
G5(s)
G6(s) + -
Y(s)
U(s)
G2(s) G3 +

3. Block Diagrams:
Let’s consider an example of block diagrams used in control systems.
Example 8.4:
As shown in the figure, a closed loop block diagram whose they have different transfer functions, consists of the blocks connected to each other.
A(s)
G1(s) Kt + - K
G2(s)
C(s)
R(s)
Here, K and Kt are the constants which are shown in the blocks. G1(s) and G2(s) are the transfer functions. R(s) is the Laplace transform of the reference or system’s input. C(s) is the Laplace transform of the response or output.

4. H(s) is the closed loop transfer function
H(s) is the closed loop transfer function. In closed loop systems, the input is a targeted quantity and the output is a measured quantity. A system is controlled by an error signal. The error signal is obtained by the difference of the input and output. It is required that the system should follow the targeted value.
Our aim is to find a transfer function of a whole system. Let be the Laplace transform of A(s) at a point shown in the block diagram. The difference of R(s) and C(s) is multiplied by K. Then, the difference of the preceding term and A(s)Kt, which is found by the product of A(s) and Kt, and is multiplied by G1(s) and then equaled to A(s).
Also, the product of A(s) and Kt equals to C(s).

5. A(s) obtained from the last equation substitutes into the first equation and rearranging the equation yields as
Taking the parenthesis of C(s), and then dividing the multiplier of C(s) into term at the left hand side gives as
The multiplier of R(s) becomes H(s) and rearranging the preceding equation founds H(s) as

6. Substituting the given transfer functions of G1(s) and G2(s) into H(s) yields as
Different eigenvalues of the closed loop system can be found depending on the values of K and Kt. The step response of closed loop systems can be analyzed. In control systems, the step response is an important indication for a closed loop system. The response must follow such a step input to achieve the control action.