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.

Wecomment on the sensitivity of a control system when some disturbance occurs init. Now these disturbances may occur due to some internal agencies (ex: changein system parameters) or some external agencies (ex: change in environmentalconditions).

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

- If the output of the system is gettingaffected to a very large extent due to that disturbance then the control systemis said to be
**highly sensitive.** - If the output of the system is not gettingaffected to a large extent due to that disturbance then the control system iscalled as
**less sensitive.**

Nowout of these two control systems which one will you choose. Off course you willchoose the control system which is less sensitive to disturbances. That’s whywe have stated earlier that **“a good control system should be lesssensitive”. **

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.

Wehave drawn following conclusions on the basis of above discussion

- Acontrol system is said to be
**highly sensitive**if its output gets affectedto a very large extent due to disturbances. - Acontrol system is said to be
**less sensitive**if its output does not getaffected to a large extent due to disturbances. - Therequirement of a good control system is that it should always be less sensitiveto any internal or external disturbances.

**Sensitivity Function**

Toperform the sensitivity analysis of a control system or to evaluate thesensitivity of any control system, we need a tool (function) which is called as**“Sensitivity Function”. **Using this function, we will perform sensitivityanalysis 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 threeelements then we say that there is some disturbance in forwardpath elements i.e. [G(s)]. If controller, control element or plant is notworking 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 whenthere 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 **Sensorsor Transducers.**

So, if sensors are not working properly then we saythat there is some disturbance in feedback path elementsi.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 whenthere 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 drawfollowing conclusions

- A close loopcontrol 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 affectedif there is any disturbance in the forward path but not to that extent up towhich it will get affected due to disturbances in feedback path.
- A small disturbance in feedback elementswill show a very large impact on the system or the output of the system.
- That is why, whenever there is someproblem in a close loop control system then we should always check for feedbackpath elements (sensors) first. If sensors are working properly then we shouldcheck forward path elements.

**Example:Automatic Door**

Wecan take automatic door as an example of close loop control system. When aperson walks towards an automatic door, it opens automatically because the dooris 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:

**SensitivityAnalysis 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 anopen loop control system is 100% sensitive to disturbance in its forward pathand this is a major disadvantage because disturbance in forward path of openloop 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.*

**Read More:**

Introduction to control system

Types of control systems

Difference between open loop and close loop control system

#### Sensitivity of Control Systems

As an expert in control systems, I can provide comprehensive insights into the sensitivity of control systems and sensitivity functions. The sensitivity of a control system refers to its responsiveness to disturbances, whether they are internal (e.g., changes in system parameters) or external (e.g., changes in environmental conditions). It is essential to understand that while individual elements of a control system, such as controllers, sensors, and measuring instruments, should be highly sensitive to detect even the smallest fluctuations, the integrated control system as a whole should be less sensitive to ensure stability and reliability.

The sensitivity function is a crucial tool for analyzing the sensitivity of a control system. It is defined as the ratio of the percentage change in the system's output to the percentage change in the disturbance. This function allows us to evaluate how the system responds to different disturbances and is expressed as ({ S }_{ \beta }^{ \alpha }=\frac { percentage\quad change\quad in\quad \alpha }{ percentage\quad change\quad in\quad \beta }).

#### Sensitivity Analysis of Close Loop Control System

In a close loop control system, the sensitivity to disturbances in the forward path elements (G(s)) and feedback path elements (H(s)) is of particular interest. When a disturbance occurs in the forward path elements, the sensitivity function ({ S }*{ G\left( s \right) }^{ M\left( s \right) }) is given by (\frac { 1 }{ \left[ 1+G\left( s \right) H\left( s \right) \right] }), where ([1+G(s)H(s)]) is known as the Noise Reduction Factor. On the other hand, when a disturbance occurs in the feedback path elements, the sensitivity function ({ S }*{ H\left( s \right) }^{ M\left( s \right) }) is (\frac { -G\left( s \right) H\left( s \right) }{ 1+G\left( s \right) H\left( s \right) }). Notably, a close loop control system is more sensitive to disturbances in the feedback path elements than in the forward path elements.

#### Sensitivity Analysis of Open Loop Control System

In contrast, an open loop control system exhibits different sensitivity characteristics. When a disturbance occurs in the forward path elements, the sensitivity function ({ S }*{ G\left( s \right) }^{ M\left( s \right) }) is 1, indicating that the system is 100% sensitive to disturbances in its forward path. However, in an open loop control system, disturbances in the feedback path elements do not affect the output, resulting in a sensitivity function ({ S }*{ H\left( s \right) }^{ M\left( s \right) }) of 0.

#### Conclusion

The sensitivity of control systems plays a critical role in their performance and stability. While close loop control systems offer the advantage of corrective action against disturbances, open loop control systems are highly sensitive to disturbances in their forward path. Understanding the sensitivity of control systems is essential for designing and implementing effective control strategies.

For further insights into the sensitivity of control systems and related topics, feel free to engage in a detailed discussion.