Introduction of Control System

     CONTROL SYSTEM  

*What is a system?

---A system is a collection of an element which can take input and processed it to give the output.

#Block diagram of System:

*Its drawbacks:

---System has no control on its input which may cause damage to the system.

Due to this region, we need a control system.

*Control system:

---Control system is a system in which the input is controlled as per required.

#Block diagram of the control system:

*Types of control system:-

---There are two types of control system.

1. Open loop control system.

2. Closed loop control system.

1. Open loop control system:

--- Open loop control system is a type of control system in which the output is dependent on the input of the system. But the controlling action of the system is not dependent on the output of the system.

* Block diagram of the open loop control system:

---Block diagram of the open loop control system is given as:

*Example of the open loop control system:

---System Engine.

---Motor vehicle.

*Advantages:

---The advantages of the open loop system are as follows:

1. Simple construction.

2. Economical.

3. Easy to maintain.

4. Stable.

*Drawbacks of the open loop control system:

---Drawbacks of the open loop control system is that in open loop control system it is not possible to automatic control the output of the system, therefore, it is inaccurate.

This type of drawback can be removed by the closed loop control system.

2. Closed loop control system:

---Closed loop control system is a control system in which the controlling action is dependent upon the output or the change in the output of the system.

#Block diagram of the closed loop control system.

---The block diagram of the closed loop control system is given as:

Feedback of the system is the circuit consists of the inductor, resistor, and capacitor.

*Example of the closed loop control system:

1. Refrigerator.

2. Air conditioner.

3. Automatic electric iron. 

*Advantages of the closed loop control systems.

1. It is more accurate than open loop control system.

2. It has the noise reduction ability.

*Disadvantages of the closed loop control system:

1. Complex construction.

2. It is less stable.

3. It has less gain as compared to the open loop control system.

@Transfer function:

--- Transfer function of the control system is given as the ratio of the output to the input in its Laplace domain with zero (or no) initial condition.

It is given as:

*Transfer function of an open system is given as:

IF,

Input=R(s).

Output=C(s).

Transfer function=G(s).

Then,

Or it is also given as

 * Transfer function of a closed loop system:

---It is done for both the negative and positive feedback closed-loop system.

*Transfer function of a closed loop system (For the negative feedback):

---The block diagram for the transfer function of a closed loop control system for the negative feedback are as follow:

In this block diagram, the input is R(s), an output is C(s), error is E(s) and I(s) is the 
output of the feedback.

From the block diagram:

E(s) = R(s) - I(s)………………………………..(i)

And I(s) is given as,

I(s) = C(s) * H(s)…………………………….(ii)

And from the definition of the transfer function,

C(s) = G(s) * E(s)…………………………..(iii)

Put the value of E(s) from eq.(i) to the eq. of the transfer function[eq.(iii)].

We get,

C(s) = G(s)[R(s) – I(s)]

C(s) = G(s) * R(s) – G(s)*I(s).

C(s) + G(s) * I(s) = G(s) * R(s).

From eq (ii),

C(s) + G(s)*C(s)*H(s) = G(s) * R(s).

 

Is the required transfer function for a closed loop system(For a negative feedback). 

*Transfer function of a closed loop control system(For positive feedback).

---The block diagram for the transfer function of a closed loop control system (for positive feedback) are as follows:

In this case,

E(s) = R(s) + I(s).

Therefore, the transfer function is given as,

*Procedure for the calculation of the transfer function:

----The procedure for the calculation of the transfer function is given as follows.

(i). Write the differential equation of the given system.

(ii). Convert the differential equation into its Laplace domain.

(iii). Calculate the ratio of the output upon input in its Laplace domain.

*Reason for using the Laplace transform in the transfer function:

---By using the Laplace transform it became possible to convert both the input and the output in the same form which is required for the mathematical calculation of the transfer function.

EXAMPLE:

Where,

x(t) = Input of the system.

y(t) = Output of the system.

Transform it into the Laplace domain.

Then we get,

Is the required transfer function.

Illustration of the Block Diagram Reduction Techniques for Shifting of Take off Point And Shifting Of Summing Point Operation.

Illustration of the Block Diagram Reduction Techniques for Shifting of Take off Point And Shifting Of Summing Point Operation Are Given As Follows:

ILLUSTRATION I.

--- THESE ARE THE FOLLOWING STEPS FOR SOLVE THIS.

* STEP 1: SHIFT THE TAKE OFF POINT BEFORE THE BLOCK G3.

* STEP 2: SOLVE FOR FEED BACK LOOP.

* STEP 3: SOLVE FOR FEEDBACK LOOP.

* STEP 3: SLOVE FOR CASCADE.

* STEP 4: TRANSFER FUNCTION IS GIVEN BY:

ILLUSTRATION 2.

--- THESE ARE THE FOLLOWING STEPS FOR SOLVE THIS:

* STEP 1: SHIFT THE TAKE OFF POINT BEFORE THE BLOCK G1.

* STEP 2: ADJUST THE SHIFTED LOOP.

* STEP 3: SOLVING FOR PARALLEL PATH.

* STEP 4: SOLVING FOR PARALLEL PATH.

* STEP 5: SOLVING FOR PARALLEL PATH.

* STEP 6: SOLVING FOR CASCADE.

* STEP 7: TRANSFER FUNCTION IS GIVEN AS,

Illustration of the Block Diagram Reduction Techniques for Cascade, Parallel and Feedback operation.

* Illustration of the Block Diagram Reduction Techniques for Cascade, Parallel and Feedback operation are given as follows :

ILLUSTRATION I.

ILLUSTRATION II.

An Introduction To Signal Flow Graph .

* Draw back of the block diagram reduction techniques :

--- Draw back of the block diagram reduction techniques is that it is a lengthy method.

And this type of drawback is not present in the another method of obtaining the transfer function which is known signal flow graph method.

* Signal flow graph method :

Signal flow graph method is discovered by S.T. MASON .

--- Signal flow graph method is also regarded as a modification of the block diagram.

* Basic modification in the signal flow graph method :

--- In signal flow graph there are following modification from block diagram reduction techniques :

I . Take off point: Take off point in signal flow graph method is represented as :

II . Block: Block in the signal flow graph method is represented as :

III . Summing point: Summing point in the signal flow graph method is represented as :

* Terms used in the signal flow graph method :

I . Loop gain: Loop gain is defined as’ the multiplication of the branch transmittance of the different loops'.

II . Forward path: Forward path is defined as ‘ path which is starting from one node and end to another node in the forward direction without crossing any node.

III . Forward path - gain : Forward path gain is defined as ' the multiplication of the branch transmittance of the different forward path '.

* Mason ‘ s gain formula :

--- Mason ‘s gain formula is written as :

Where,

- T = Transfer function .

.

- Mk = Gain of the forward-path.

-   🔺 = Signal flow graph determinant .

-🔺k = The value of the signal flow graph determinant ‘🔺 ‘ for which part of the graph not touching the forward path .

---  🔺  is given by the formula :

  🔺 = 1 - ( sum of gain of the individual loop ) + ( sum of gain product of two non - touching loops ) - ( sum of gain product of four non - touching loops ) - ……..  .

* Procedure to solve the signal flow graph by mason ‘s gain formula :

--- The procedure to solve the signal flow graph by mason ‘s gain formula is given as :

I . Find out the total no. of the forward path .

II . Find the product of the gain of the different forward path .

III . Find the total no. of individual loop ( closed path ) .

IV . Find out the total gain of the individual loops .

V . Find out the total no. of the non - touching loops ( Those loops which are not in touch with the other loop i.e., which are not joint to any common node with different loop) .

VI . Find out the total gain of the non - touching loop ( product of the gain of the non - touching loop ).

VII . Find out the’  🔺 k ‘ of the non - touching loop.

Where,

 

 🔺1 = 1 - ( sum of the gain of the loop non - touching the 1^st forward path ).

Similarly ,

  

  🔺 k = 1 - ( sum of the gain of the loop of the non - touching kth forward path ).

VIII . Find out the value of 🔺.

Where ,

 🔺 = 1 + ( sum of the gain of all the non - touching loops ) - ( sum of gain of all single loop ) .

XII . Find out the transfer function where transfer function is given as ,

Mk = Gain of the forward path .

 🔺 K = The value of the signal flow graph determinant ‘🔺  ‘ for which part of the graph which is non - touching the forward path .