In this post we are sharing about Transfer Function and Sensitivity in Control Systems subject. In the GATE exam we can expect questions from these concepts. We explained them clearly. Go through them. It may help you for your GATE preparation.
Hi guys, in this article we are sharing some basics in Control Systems. This subject has good percentage of marks in GATE Exam. GATE examination tests only basic knowledge in GATE Candidates. So if you are strong at basics then you can solve complex problems. So having basics is must for GATE Exam. In this post we are sharing about Transfer Function and Sensitivity in Control Systems subject. In the GATE exam we can expect questions from these concepts. We explained them clearly. Go through them. It may help you for your GATE preparation.

Control System: Control System is a system which controls the output quantity is called a control system. For example let’s say we have 1V as source and we want 5V as output. To achieve this we build a system with gain of 5 to get 5V. This system is called as control system. This system controls the input to get desired output.
In technical terms control system is an arrangement of different physical components such that it gives the desired output for the given input by means of regulate or control either direct or indirect method.
Control Systems
Control System Block Diagram
Control system includes (1) Input (2) Output (3) Control action and etc.

Control systems are two types.
(1) Open loop control systems(OLCS)
(2) Closed loop control systems(CLCS)

(1) Open Loop Control Systems:
Open Loop Control System is a system which can’t correct or control automatically the variation in its output. There will be no feedback present in this system. See the below figure.
Open Loop Control Systems
Open Loop Control System Block Diagram
Let us see what are the advantages of Control systems and also Disadvantages of Control Systems.
1. Open Loop Systems are simple.
2. Easier to construct.
3. Stable when compare to closed loop systems.
1. Inaccurate and unreliable.
2. Sensitivity will be very high.

(2) Closed Loop Control Systems:
Closed Loop Control System is a system in which output has an effect on the input quantity in order to maintain the desired output value. It checks how output is varying with respect to input. For that it has connection from output to input. And then it controls the output.
There will be feedback present in this type of systems. So this type of systems are also called as feedback systems.
There are two types of feedback present in feedback systems.
1. Positive Feedback (Output added to the input)
2. Negative Feedback (Output is subtracted from the input)

See the below figure.
Feedback control systems
Feedback Control Systems Block Diagram
In the above figure + symbol indicates the positive feedback and - symbol indicates the negative feedback.
Error signal e(t)=r(t) ± f(t)
f(t) is feedback signal.

Let's see the Advantages and Disadvantages of the Feedback control systems.
1. Accurate even in the presence of non-linearities, disturbances and noise.
2. Less affected by noise.
3. Sensitivity will be less.
4. Increases the bandwidth of the system.

1. Complex and costly.
2. Reduces the overall gain of the system.
3. There are Chances to a system to become unstable in feedback systems. So care is needed to design a stable feedback system.

Transfer Function:
We can characterize any control system by mathematically using Transfer Function (TF). This is simple and very important concept. You need to focus more on this concept to get good command over Control Systems subject. Always just remember Transfer Function is equal to output by input. 

Transfer function = Laplace Transform of Output/ Laplace Transform of Input
T.F= LT(c(t)) / LT(r(t)) = C(s)/R(s)
C(s) is LT of Output.
R(s) is LT of input.
c(t) is inverse LT of C(s).
r(t) is inverse LT of R(s).
LT => Laplace Transform
Note:: This Transfer Function formula applicable only when initial conditions are zero.

Transfer Function is also called as impulse response. Impulse response means the input given to the system is impulse function(δ(t)).
Transfer Function is nothing but output when the given input is unit impulse function. See below figure.
Impulse Response
Impulse Response
As we all know Laplace Transform of unit impulse function is 1. LT[δ(t)]=1.
T.F= LT of output/LT of input.
T.F=LT of output/1=LT of output.
So Transfer Function is equal to Laplace Transform of output when input is unit impulse function. That is why Transfer Function also called as Impulse Response. Remember this point. While solving problems you will get this kind of information.

We can find one quantity among input, TF, output when other two are known. True right.
We can use below formula to find TF.
Transfer function = LT of Output / LT of Input
TF= LT[c(t)] / LT[r(t)] = C(s)/R(s)
If we know R(s) & TF
then C(s) = TF × R(s).
c(t) = inverse LT of C(s)
Here TF is also in Laplace Transform. While solving TF problems please convert all the quantities to s-domain(Laplace Transforms) and then solve the problem and at the end you can convert the required parameter into time domain. If you follow the above procedure you will get all the answers correct.

In general a good control system should be insensitive to parameter variations but highly sensitive to input command. For a good control system sensitivity should be less then only the changes in output will be less w.r.t parameter variations. 

We know that in feedback systems sensitivity is less compared to open loop systems. Because feedback systems consider changes in the output also. So according to that it will give the new output.  In the feedback systems sensitivity is less means we are considering sensitivity w.r.t open loop parameters only. Here we are using feedback to make output less sensitive w.r.t to open loop parameters. In Feedback systems output is less sensitive to open loop parameter and highly sensitive to feedback parameters. So we have to use accurate valued components in feedback path. See below figure.
Sensitivity w.r.t Parameters
Sensitivity w.r.t Path Parameters
We can define sensitivity as  how much change is there in output for x% change in path parameter. We can define sensitivity mathematically as ratio of percentage change in overall gain to percentage change in any parameter of control system.
Sensitivity Formula
Sensitivity Formula
Here M = Overall Gain ( Consider both open loop and feedback paths)
P = Parameter (Any parameter from open loop and feedback paths)

Remember this formula and follow the below mentioned procedure while solving Sensitivity GATE problems.
First derivate the overall gain w.r.t parameter and then we have to substitute this derivated value, parameter value and overall gain value in the formula. In this way we can find sensitivity of the system easily. We will try to share all other gate material. Keep visiting this site for more posts. Hope this post helps for your GATE preparation. If you any doubts let us know through comments.

Thank you. Happy Reading.

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