Transfer funtion in control system

 CONTROL SYSTEM - Part 2

*Methods for obtaining the transfer  function:

---There are two methods which the transfer function get obtained:-

1 . Block Diagram Method

2 . Signal Flow Graph Method

1 . Block diagram method:

*Block diagram:

---A block diagram is a pictorial representation of a system in which the function is represented by blocks which are connected by the lines in a systematic manner.

Block diagram of the control system is represented as:

*Requirements for block diagram representation in control system:

1 . Block.

2 . Summing point.

3 . Take off point.

Now, it is illustrated one by one as follows:

1 . Block: Block is pictorial representation in which a function is embedded in it.It is represented as:

2 . Summing point: Summing point is a point in which two or more than two signal is added or subtracted.

It is represented as:

3 . Take-off point: Take off point is a point in which a new signal is connected.

It is represented as:

*In a block diagram system it is represented as:

@Block diagram reduction techniques:

---Block diagram reduction techniques are the techniques of the simplification of the combination of two or more than two blocks into a single block.

---Blocks can be combined in the two manners:

(i). Cascade manner.

(ii). Parallel manner.

(i). Cascade manner: In the cascade manner the blocks can be combined in cascade form. And it is represented as:

It simplifies as:

Where,

R2(s) = G1(s)*G2(s)*R1(s).

(ii). Parallel manner: In a parallel manner the blocks can be combined in the parallel form. And it is represented as:

It simplifies as,

Where,

*Illustration of the block diagram reduction techniques:

Illustration 1:

First, consider the cascade form,

Now, consider the parallel form,

Now we get, the transfer function as,

*Operation in block diagram reduction techniques:

(i). Elimination of feedback.

(ii). Shifting of summing point.

(iii). Shifting of take-off point.

Now, it is illustrated one by one as follows:

(i). Elimination of feedback: In the elimination of the feedback the feedback path gets eliminated without disturbing the output.

Let us consider an example.

Now, simplifying it for elimination of the feedback loop we get,

Therefore, the transfer function is,

(ii). Shifting of summing point:

In the operation of the shifting of the summing point the summing point can be shifted either side of the block without disturbing the output of the block.

Now,

Let us take an example for an illustration purpose.

Illustration 1.

Now shift the summing point to the left side of the block such that the output will not change.

Now observe the output of the summing point which is

Now, for shift the summing point such that we get the same output,

By the adjustment of the input of the summing point we get,

Illustration 2.

Now, shift the summing point to the right side of the block such that the output will not change.

It is easily observed that the output of the summing point get multiplied by G1(s) i.e.,

By the adjustment  of the input of the summing point we get,

(iii). Shifting of take-off point:

In the operation of the shifting of the takeoff point, the take off point can be shifted either side of the block without disturbing the output of the block.

Now, let us take an example on the shifting of the take-off point for the illustration purpose.

Illustration 1.

Now, shift the take-off point to the left side of the block such that the output will not change.

Now, it is observed that the output of the system is R1*G1.

Then, shift the take off point such that we get the same output.

Therefore, by the adjustment we get,

Illustration 2.

Now, shift the take off point to the right side of the block such that the output will not change.

Therefore,  by the adjustment of it we get,

*Swapping between the summing points:

It is possible to swap the two summing point with each other such that the output remains unaffected.

It can be explained by the illustration which is given below:

Illustration 1.

After the swapping it we get,

*Splitting of the inputs at the summing points:

---Splitting of the input at the summing points can be done without disturbing the output.

It can be illustrated by the following examples:

After the splitting it we get,