The **block diagram** is to represent a control system in diagram form. In other words, practical representation of a control system is its block diagram. It is not always convenient to derive the entire transfer function of a complex control system in a single function. It is easier and better to derive the transfer function of the control element connected to the system, separately. The transfer function of each element is then represented by a block and they are then connected together with the path of signal flow. For simplifying a complex control system, block diagrams are used. Each element of the control system is represented with a block and the block is the symbolic representation of the transfer function of that element. A complete control system can be represented with a required number of interconnected blocks.

In the figure below, there are two elements with transfer function G_{one}(s) and G_{two}(s). Where G_{one}(s) is the transfer function of first element and G_{two}(s) is the transfer function of the second element of the system.

In addition to that, the diagram also shows there is a feedback path through which output signal C(s) is fed back and compared with the input R(s) and the difference between input and output is , which is acting as the actuating signal or error signal.

In each block of the diagram, the output and input are related together by a transfer function. Where the transfer function is:

Where C(s) is the output and R(s) is the input of that particular block.

A complex control system consists of several blocks. Each of them has its own transfer function. But overall transfer function of the system is the ratio of transfer function of final output to transfer function of initial input of the system. This overall transfer function of the system can be obtained by simplifying the control system by combining this individual blocks, one by one.

The technique of combining of these blocks is referred to as **block diagram reduction technique**. For successful implementation of this technique, some rules for block diagram reduction to be followed. Let us discuss these rules, one by one for reduction of **control system block diagram**. If you’re looking to do some control systems study, check out our control systems MCQs.

If the transfer function of input of control system is R(s) and the corresponding output is C(s), and the overall transfer function of the control system is G(s), then the control system can be represented as:

### Take off Point of Block Diagram

When we need to apply one or the same input to more than one blocks, we use what’s known as the **take off point**. This point is where the input has more than one path to propagate. Note that the input does not get divided at a point, but instead the input propagates through all the paths connected to that point without affecting its value. Hence, by having a take off point the same input signals can be applied to more than one systems or blocks. Representation of a common input signal to more than one blocks of a control system is done by a common point as shown in the figure below with point X.

#### Cascade Blocks

When several systems or control blocks are connected in cascaded manner, the transfer function of the entire system will be the product of transfer function of all individual blocks. Here it also to be remembered that the output of any block will not be affected by the presence of other blocks in the cascaded system.

Now, from the diagram it is seen that,

Where, G(s) is the overall transfer function of cascaded control system.

### Summing Point of Block Diagram

Instead of applying single input signal to different blocks as in the previous case, there may be such situation where different input signals are applied to same block. Here, resultant input signal is the summation of all input signals applied. Summation of input signals is represented by a point called summing point which is shown in the figure below by crossed circle. Here R(s), X(s) and Y(s) are the input signals. It is necessary to indicate the fine specifying the input signal entering a summing point in the **block diagram of control system**.

#### Consecutive Summing Point

A summing point with more than two inputs can be divided into two or more consecutive summing points, where alteration of the position of consecutive summing points does not affect the output of the signal. In other words – if there are more than one summing points directly inter associated, and then they can be easily interchanged from their position without affecting the final output of the summing system.

#### Parallel Blocks

When same input signal is applied different blocks and the output from each of them are added in a summing point for taking final output of the system then over all transfer function of the system will be the algebraic sum of transfer function of all individual blocks.

If C_{one}, C_{two} and C_{three} are the outputs of the blocks with transfer function G_{one}, G_{two} and G_{three}, then

#### Shifting of Take off Point

If same signal is applied to more than one system, then the signal is represented in the system by a point called take off point. Principle of **shifting of take off point** is that, it may be shifted either side of a block but final output of the branches connected to the take off point must be un-changed. The take off point can be shifted either sides of the block.

In the figure above the take off point is shifted from position A to B. The signal R(s) at take off point A will become G(s)R(s) at point B. Hence another block of inverse of transfer function G(s) is to be put on that path to get R(s) again.

Now let us examine the situation when take off point is shifted before the block which was previously after the block.

Here the output is C(s) and input is R(s) and hence

Here, we have to put one block of transfer function G(s) on the path so that output again comes as C(s).

#### Shifting of Summing Point

Let us examine the shifting of summing point from a position before a block to a position after a block. There are two input signals R(s) and ± X(s) entering in a summing point at position A. The output of the summing point is R(s) ± X(s). The resultant signal is the input of a control system block of transfer function G(s) and the final output of the system is

Hence, a summing point can be redrawn with input signals R(s)G(s) and ± X(s)G(s)

In the above block diagrams of control system output can be rewritten as

The above equation can be represented by a block of transfer function G(s) and input R(s) ± X(s)/G(s) again R(s)±X(s)/G(s) can be represented with a summing point of input signal R(s) and ± X(s)/G(s) and finally it can be drawn as below.

## Block Diagram of Closed Loop Control System

In a **closed loop control system**, a fraction of output is fed-back and added to input of the system. If H (s) is the transfer function of feedback path, then the transfer function of feedback signal will be B(s) = C(s)H(s). At summing point, the input signal R(s) will be added to B(s) and produces actual input signal or error signal of the system and it is denoted by E(s).