# Sensitivity of Control System

In this blog we will discuss about the sensitivity of control system. We will also talk about sensitivity function. Later we will derive sensitivity expression for close loop control system as well as open loop control system for disturbances in forward path elements and feedback path elements. So, let’s start the discussion.

Whenever we come across the term sensitivity, it comes to our mind that sensitivity should be high. This statement is true for some individual elements like controllers, sensors and measuring instruments (Ammeter, voltmeter and wattmeter etc).

These devices should be highly sensitive because these devices should be able to detect even smallest of the fluctuations in their input and give output according to that.

But when we talk about the sensitivity of a control system the case is exactly opposite because sensitivity of a control system is different from sensitivity of individual elements.

As we know that a control system is the integration of various individual elements like controller, control element, plant, sensors etc. So, the sensitivity of these individual elements should be high but sensitivity of the whole integrated system (control system) should be low.

Therefore, we say that “a good control system should be less sensitive”. You will understand this statement by the end of the discussion.

We comment on the sensitivity of a control system when some disturbance occurs in it. Now these disturbances may occur due to some internal agencies (ex: change in system parameters) or some external agencies (ex: change in environmental conditions).

Suppose some disturbance occurs in a control system. In this case

• If the output of the system is getting affected to a very large extent due to that disturbance then the control system is said to be highly sensitive.
• If the output of the system is not getting affected to a large extent due to that disturbance then the control system is called as less sensitive.

Now out of these two control systems which one will you choose. Off course you will choose the control system which is less sensitive to disturbances. That’s why we have stated earlier that “a good control system should be less sensitive”.

On the basis of above discussion, one might say that we should build a control system that does not get affected to any kind of disturbance at all. In this case the control system is said to be an ideal control system.

But practically this is not possible because all the control systems that exists, do get affected to one or other disturbance up to some extent. Some of these control systems may be highly sensitive while others may be less sensitive.

We have drawn following conclusions on the basis of above discussion

1. A control system is said to be highly sensitive if its output gets affected to a very large extent due to disturbances.
2. A control system is said to be less sensitive if its output does not get affected to a large extent due to disturbances.
3. The requirement of a good control system is that it should always be less sensitive to any internal or external disturbances.

## Sensitivity Function

To perform the sensitivity analysis of a control system or to evaluate the sensitivity of any control system, we need a tool (function) which is called as “Sensitivity Function”. Using this function, we will perform sensitivity analysis of control system. So, lets see what is that function.

Now, to write sensitivity function we need to define two terms which are $$\alpha$$ and $$\beta$$.

• $$\alpha$$ : A variable that changes its value.

What does $$\alpha$$  represents?

$$\alpha$$ represents the output of the system or the system itself because whether the output of the system is getting affected or the system is getting affected, both are same.

• $$\beta$$ : A parameter that changes the value of $$\alpha$$

What does $$\beta$$ represents?

$$\beta$$ represents any kind of disturbances. It may be any internal disturbance in the forward path elements i.e. G(s) or feedback path elements i.e. H(s). $$\beta$$ may also represent external disturbances.

When some disturbance ($$\beta$$) occurs in the control system then output of the system or state of the system ($$\alpha$$) changes.

Sensitivity function is given by

$${ S }_{ \beta }^{ \alpha }$$ = Sensitivity of $$\alpha$$ w.r.t. $$\beta$$.

### Mathematical Definition:

Sensitivity function is defined as the ratio of percentage change in $$\alpha$$ to the percentage change in $$\beta$$ i.e.

$${ S }_{ \beta }^{ \alpha }=\frac { percentage\quad change\quad in\quad \alpha }{ percentage\quad change\quad in\quad \beta }$$

### Mathematical Representation:

$${ S }_{ \beta }^{ \alpha }=\frac { \partial \alpha /\alpha }{ \partial \beta /\beta }\\{ S }_{ \beta }^{ \alpha }=\frac { \beta }{ \alpha } \frac { \partial \alpha }{ \partial \beta }$$

## Sensitivity Analysis of Close Loop Control System:

In this section we will derive the expression of sensitivity for a close loop control system when some disturbance occurs in its forward path elements i.e. [G(s)] and feedback path elements i.e. [H(s)].

CASE: 1 When disturbance occurs in forward path elements i.e. [G(s)]

Before deriving sensitivity expression, we have to define $$\alpha$$ and $$\beta$$.

$$\alpha$$ = Close loop control system = M(s)

Here, M(s) = Mathematical form of a close loop control system

$$\beta$$ = Disturbances in forward path elements i.e. G(s)

Here, G(s) may represent controller, control element or plant etc.

So, if there is any disturbance in these three elements then we say that there is some disturbance in forward path elements i.e. [G(s)]. If controller, control element or plant is not working properly then this will affect the system or output of the system.

Now, we will see the expression of sensitivity for a close loop control system when there is some disturbance in its forward path. So, here we go.

As you know that

$${ S }_{ \beta }^{ \alpha }=\frac { \beta }{ \alpha } \frac { \partial \alpha }{ \partial \beta }$$

Here, $$\alpha$$ = M(s) and $$\beta$$ = G(s)

$${ S }_{ G\left( s \right) }^{ M\left( s \right) }=\frac { G\left( s \right) }{ M\left( s \right) } \frac { \partial M\left( s \right) }{ \partial G\left( s \right) }$$……………..(1)

Since, $$\ M\left( s \right) =\frac { G\left( s \right) }{ 1+G\left( s \right) H\left( s \right) }$$ $$\Rightarrow \frac { G\left( s \right) }{ M\left( s \right) } =1+G\left( s \right) H\left( s \right)$$ ……………..(2)

Partially differentiating M(s) w.r.t G(s), we will get

$$\frac { \partial M\left( s \right) }{ \partial G\left( s \right) } =\frac { \partial }{ \partial G\left( s \right) } \left[ \frac { G\left( s \right) }{ 1+G\left( s \right) H\left( s \right) } \right] \\ \left\{ { \left( \frac { u }{ v } \right) }^{ \prime }=\frac { { u }^{ \prime }v-u{ v }^{ \prime } }{ { v }^{ 2 } } \right\} \\ \Rightarrow \frac { \partial M\left( s \right) }{ \partial G\left( s \right) } =\frac { 1+G\left( s \right) H\left( s \right) -G\left( s \right) H\left( s \right) }{ { \left[ 1+G\left( s \right) H\left( s \right) \right] }^{ 2 } }\\ \Rightarrow \frac { \partial M\left( s \right) }{ \partial G\left( s \right) } =\frac { 1 }{ { \left[ 1+G\left( s \right) H\left( s \right) \right] }^{ 2 } } \quad \quad ……….\left( 3 \right)$$

Put values of $$\frac { G\left( s \right) }{ M\left( s \right) }$$ and $$\frac { \partial M\left( s \right) }{ \partial G\left( s \right) }$$ from eq(2) and eq(3) respectively into eq(1), we will get

$${ S }_{ G\left( s \right) }^{ M\left( s \right) }=\left[ 1+G\left( s \right) H\left( s \right) \right] \times \frac { 1 }{ { \left[ 1+G\left( s \right) H\left( s \right) \right] }^{ 2 } }\\ \Rightarrow { S }_{ G\left( s \right) }^{ M\left( s \right) }=\frac { 1 }{ \left[ 1+G\left( s \right) H\left( s \right) \right] }$$

Here, [1+G(s)H(s)] = Noise Reduction Factor

Note: [1+G(s)H(s)] is called as “Noise Reduction Factor” because it reduces the effect of disturbance on the output of the system.

CASE: 2 When disturbance occurs in feedback path elements i.e. [H(s)]

Before deriving sensitivity expression, we have to define $$\alpha$$ and $$\beta$$.

$$\alpha$$ = Close loop control system = M(s)

Here, M(s) = Mathematical form of a close loop control system

$$\beta$$ = Disturbances in feedback path elements i.e. H(s)

Here, H(s) may represent Sensors or Transducers.

So, if sensors are not working properly then we say that there is some disturbance in feedback path elements i.e. [H(s)] and this will affect the system or output of the system.

Now, we will see the expression of sensitivity for a close loop control system when there is some disturbance in its feedback path. So, here we go.

As you know that

$${ S }_{ \beta }^{ \alpha }=\frac { \beta }{ \alpha } \frac { \partial \alpha }{ \partial \beta }$$

Here, $$\alpha$$ = M(s) and $$\beta$$ = H(s)

$${ S }_{ H\left( s \right) }^{ M\left( s \right) }=\frac { H\left( s \right) }{ M\left( s \right) } \frac { \partial M\left( s \right) }{ \partial H\left( s \right) } \quad \quad \quad …….\left( 1 \right)$$

Since, $$M\left( s \right) =\frac { G\left( s \right) }{ 1+G\left( s \right) H\left( s \right) }$$

Multiplying both sides by 1/H(s), we will get

$$\frac { M\left( s \right) }{ H\left( s \right) } =\frac { G\left( s \right) }{ H\left( s \right) \left[ 1+G\left( s \right) H\left( s \right) \right] }\\ \Rightarrow \frac { H\left( s \right) }{ M\left( s \right) } =\frac { H\left( s \right) \left[ 1+G\left( s \right) H\left( s \right) \right] }{ G\left( s \right) } \quad \quad \quad …….\left( 2 \right)$$

Partially differentiating M(s) w.r.t H(s), we will get

$$\frac { \partial M\left( s \right) }{ \partial H\left( s \right) } =\frac { \partial }{ \partial H\left( s \right) } \left[ \frac { G\left( s \right) }{ 1+G\left( s \right) H\left( s \right) } \right] \\ \left\{ { \left( \frac { u }{ v } \right) }^{ \prime }=\frac { { u }^{ \prime }v-u{ v }^{ \prime } }{ { v }^{ 2 } } \right\} \\ \Rightarrow \frac { \partial M\left( s \right) }{ \partial H\left( s \right) } =\frac { -{ \left[ G\left( s \right) \right] }^{ 2 } }{ { { \left[ 1+G\left( s \right) H\left( s \right) \right] } }^{ 2 } } \quad \quad \quad …….\left( 3 \right)$$

Put values from eq(2) and eq(3) in eq(1), we will get

$${ S }_{ H\left( s \right) }^{ M\left( s \right) }=\frac { H\left( s \right) \left[ 1+G\left( s \right) H\left( s \right) \right] }{ G\left( s \right) } \times \frac { -{ \left[ G\left( s \right) \right] }^{ 2 } }{ { { \left[ 1+G\left( s \right) H\left( s \right) \right] } }^{ 2 } }\\ \Rightarrow { S }_{ H\left( s \right) }^{ M\left( s \right) }=\frac { -G\left( s \right) H\left( s \right) }{ 1+G\left( s \right) H\left( s \right) }$$

Now we want to know whether a close loop control system is more sensitive to disturbances in forward path elements i.e. G(s) or to disturbances in feedback path elements i.e. H(s). For that we have to compare the two sensitivity expressions that we derived for disturbances in forward and feedback path respectively.

Note: For comparison we will take mod value of $${ S }_{ H\left( s \right) }^{ M\left( s \right) }$$ but while solving a numerical negative sign is taken into consideration.

$${ S }_{ G\left( s \right) }^{ M\left( s \right) }=\frac { 1 }{ \left[ 1+G\left( s \right) H\left( s \right) \right] }$$ and $${ S }_{ H\left( s \right) }^{ M\left( s \right) }=\frac { -G\left( s \right) H\left( s \right) }{ 1+G\left( s \right) H\left( s \right) }$$

On comparing the above two expression we can draw following conclusions

• A close loop control system is more sensitive to disturbances in feedback path elements [H(s)] than forward path elements [G(s)].
• No doubt that the system will get affected if there is any disturbance in the forward path but not to that extent up to which it will get affected due to disturbances in feedback path.
• A small disturbance in feedback elements will show a very large impact on the system or the output of the system.
• That is why, whenever there is some problem in a close loop control system then we should always check for feedback path elements (sensors) first. If sensors are working properly then we should check forward path elements.

#### Example: Automatic Door

We can take automatic door as an example of close loop control system. When a person walks towards an automatic door, it opens automatically because the door is employed with sensors which senses the approach of a person.

Now suppose, you walk towards an automatic door but it does not open. It means that there is some disturbance in the system. In this case what should we check first?

As we know that a close loop control system is more sensitive to disturbances in its feedback path, hence, there is a high probability that sensors are not working properly. Therefor, we should always check for disturbances in sensors first and if the sensors are working properly then we should check for disturbance in forward path elements.

For better understanding of sensitivity of close loop control system, watch this video:

## Sensitivity Analysis of Open Loop Control System:

In this section we will derive the sensitivity expression for an open loop control system when some disturbance occurs in its forward path elements i.e. [G(s)] and feedback path elements i.e. [H(s)].

CASE: 1 When disturbance occurs in feedback path elements i.e. [H(s)]

As we know that in an open loop control system, feedback is not connected to the forward path. That’s why, even if some disturbance occurs in feedback path elements, it will not affect the output of the system at all. It means that sensors might not work properly but this will not affected working of the system.

So, we can say that sensitivity of an open loop control system w.r.t any disturbance in feedback path elements i.e. H(s) is zero.

$$\Rightarrow { S }_{ H\left( s \right) }^{ M\left( s \right) }=0$$

Hence, an open loop control system is not sensitive to any disturbance that occurs in its feedback path. This is a major advantage of open loop control systems.

CASE: 2 When disturbance occurs in forward path elements i.e. [G(s)]

Before deriving sensitivity expression, we have to define $$\alpha$$ and $$\beta$$.

$$\alpha$$ = Open loop control system = M(s)

Here, M(s) = Mathematical form of an open loop control system

$$\beta$$ = Disturbances in forward path elements i.e. G(s)

Here, G(s) may represent controller, control element or plant etc.

Now, we will see the sensitivity expression for an open loop control system when there is some disturbance in its forward path. So, here we go.

As you know that

$${ S }_{ \beta }^{ \alpha }=\frac { \beta }{ \alpha } \frac { \partial \alpha }{ \partial \beta }$$

Here, $$\alpha$$ = M(s) and $$\beta$$ = G(s)

$${ S }_{ G\left( s \right) }^{ M\left( s \right) }=\frac { G\left( s \right) }{ M\left( s \right) } \frac { \partial M\left( s \right) }{ \partial G\left( s \right) }$$……………..(1)

Since, mathematical form of open loop control systems is given by

$$M\left( s \right) =G\left( s \right) H\left( s \right) \\ \Rightarrow \frac { G\left( s \right) }{ M\left( s \right) } =\frac { 1 }{ H\left( s \right) } \quad \quad \quad …….\left( 2 \right)$$

Partially differentiating M(s) w.r.t G(s), we will get

$$\frac { \partial M\left( s \right) }{ \partial G\left( s \right) } =\frac { \partial }{ \partial G\left( s \right) } \left[ G\left( s \right) H\left( s \right) \right] // \Rightarrow \frac { \partial M\left( s \right) }{ \partial G\left( s \right) } =H\left( s \right) \quad \quad \quad …….\left( 3 \right)$$

Put values from eq(2) and eq(3) in eq(1), we will get

$${ S }_{ G\left( s \right) }^{ M\left( s \right) }=\frac { 1 }{ H\left( s \right) } \times H\left( s \right) \\ \Rightarrow { S }_{ G\left( s \right) }^{ M\left( s \right) }=1$$

­­The above expression suggests that an open loop control system is 100% sensitive to any disturbance that occurs in its forward path.

If there is some disturbance in the forward path of an open loop control system, the entire effect of that disturbance will be seen on the output of the system.

For example, if there is 10% disturbance in the forward path of an open loop control system then the entire 1X10 = 10% effect will be seen on the output of the system. This is a major disadvantage of open loop control systems.

#### Example: Washing Machine

We can take washing machine as an example of an open loop control system. Washing machines comes in different weight categories like 5kg, 10kg, etc. Suppose you purchase a 5kg washing machine. Now you can not dump 10kg clothes in a 5kg washing machine and even if you do so somehow, it means that you are disturbing the forward path of the machine.

Now, washing machine being an open loop control system, is 100% sensitive to disturbance in its forward path. So, if you dump 10kg clothes in a 5kg washing machine, forget about washing, the machine will not even rotate and that is 100% effect on the output of the system.

Now a days, washing machines comes with a display (as you can see in the image) which shows the information about time of washing and temperature of water for a particular type of cloth (Cotton, silk, woollen, etc). These displays are part of feedback mechanism.

Now suppose, some disturbance occurs in feedback mechanism then what will happen? Will this affect the working of machine?

As we know that washing machine is an open loop control system and an open loop control system is not sensitive to any disturbance in its feedback path.

So, even if the display is not working properly, this will not affect the working of machine. The washing machine will work as earlier. Its just that information will not be displayed anymore.

#### CONCLUSION

From the entire discussion, now we know that an open loop control system is 100% sensitive to disturbance in its forward path and this is a major disadvantage because disturbance in forward path of open loop control systems can cause severe damage to the system.

That’s why most of the control systems that we use today, are close loop control systems.

A close loop control system is capable of taking corrective action against disturbances that occurs in the system and for that feedback must be connected to the forward path. But as soon as feedback gets connected to the forward path, the system now becomes more sensitive to disturbances in feedback path.

But this shortcoming can be tolerated because close loop control systems offers very high accuracy as compared to open loop control systems.

Watch Sensitivity of control system in Hindi.