14 September BE 2567

Summary

This lecture transcript focuses on control systems, emphasizing the importance of understanding the standard controller's proportional, integral, and derivative gains. It discusses deriving transfer functions, especially in the presence of time delays, and how to handle disturbances. The lecture also covers the ideal transfer function values for a robust control system, the significance of KP and KD gains, and the application of the final value theorem. It introduces concepts like natural frequency, damping ratio, rise time, and overshoot, providing formulas for calculating these parameters. The instructor encourages practice with transfer functions and system response plots, aiming to prepare students for exams and real-world applications.

Takeaways

• 📊 The lecture introduces KP (proportional gain), KI (integral gain), and KD (derivative gain) in control systems.
• 🔧 Students are advised to derive transfer functions from the provided slides and practice the concepts.
• ⚙️ The focus is on transfer functions, specifically how output is related to input and disturbance in a system.
• 💡 Understanding KP alone might not be sufficient to make system corrections, so additional controllers like PD (Proportional-Derivative) are introduced.
• 📈 Students must apply concepts like the final value theorem and Laplace domain calculations for system analysis.
• ⏳ Time constants such as rise time, settling time, and natural frequency are crucial in determining system performance.
• 🎯 The goal is to minimize error by fine-tuning parameters like KP, KD, and settling time, especially for second-order systems.
• 🧮 Practical exercises involve calculating peak time, overshoot, and using formulas provided during exams to solve problems.
• ⚖️ Robustness against disturbances is highlighted as a key goal for control systems, aiming for an output-to-disturbance transfer function of zero.
• 📋 Students are expected to sketch response plots and compare their system's behavior to standard models to predict performance.

Q & A

What is the significance of Lecture 6 in the context of the script?

Lecture 6 is significant as it discusses the Standard Controller, which is crucial for understanding the concepts of proportional gain (KP), integral gain, and derivative gain in control systems.

What is the definition of a transfer function as mentioned in the script?

The transfer function is defined as the ratio of the output to the input of a system, which is a fundamental concept in control systems for analyzing the relationship between these variables.

Why is it important to practice deriving the transfer function?

Practicing the derivation of the transfer function is important because it allows one to understand the dynamic behavior of a system and how it responds to different inputs, which is essential for control system design.

How does the presence of a time delay (TD) affect the system as discussed in the script?

The presence of a time delay (TD) in a system complicates the control process. The script suggests that when deriving the transfer function, the time delay is considered as part of the system's output when there is no input, which affects the system's response to disturbances.

What does the script suggest about the ideal transfer function relating output to input in a perfectly designed control system?

The script suggests that in an ideally designed control system, the transfer function relating output to input should be as close to one as possible, indicating a direct and efficient response from the system.

Why is it desirable for the transfer function relating output to disturbance to be zero?

A transfer function relating output to disturbance that is zero indicates that the system is robust against disturbances, meaning that disturbances have minimal impact on the system's output, which is a key goal in control system design.

What is the role of KP in the context of the script?

KP, or the proportional gain, plays a role in the control system by determining the system's response to the error between the set point and the actual output. The script suggests that sometimes KP alone might not be sufficient, and additional control elements like derivative gain (KD) might be needed.

What is the significance of the final value theorem mentioned in the script?

The final value theorem is significant as it allows one to determine the steady-state error in a control system. The script emphasizes the importance of being able to apply this theorem to obtain the exact value of the system's response over time.

Why is it important to understand the natural frequency and damping ratio of a system?

Understanding the natural frequency and damping ratio of a system is important because these parameters provide insights into the system's stability and responsiveness. The script suggests that these values can be derived from the characteristic equation of the closed-loop transfer function.

What is the significance of rise time, peak time, and setting time in the context of system response?

Rise time, peak time, and setting time are significant parameters in the context of system response as they describe how quickly the system reacts to a change, the maximum overshoot it experiences, and the time it takes to settle to a steady state, respectively. These parameters are crucial for evaluating the performance of a control system.

How can one compute the rise time, peak time, and setting time from the system's transfer function?

One can compute the rise time, peak time, and setting time from the system's transfer function by using the derived natural frequency and damping ratio. The script provides formulas for calculating these times, which involve using the values of omega_n (natural frequency) and zeta (damping ratio).

Outlines

00:00

🔍 Understanding Control Systems and Transfer Functions

This paragraph discusses the importance of understanding the concept of a standard controller in control systems. It introduces the proportional gain (KP), integral gain (Ki), and derivative gain (Kd). The main focus is on deriving the transfer function, which is the ratio of output to input. The paragraph also addresses how to deal with systems that have time delays (TD) by considering the output in relation to the input and disturbance. It emphasizes the need to practice deriving transfer functions and understanding how they relate to system behavior, especially in the presence of disturbances. The goal is to design a controller that minimizes the effect of disturbances on the system's output.

05:18

📚 Exploring P and PD Controllers in Control Systems

The second paragraph delves into the details of P (proportional) and PD (proportional-derivative) controllers. It mentions that while P controllers are simple, PD controllers are more complex and may be necessary for certain systems. The paragraph stresses the importance of understanding the error in the Laplace domain and applying the final value theorem to obtain exact values. It also encourages practice with transfer functions and understanding how to derive the output in response to a reference input. The key takeaway is the ability to analyze system behavior using transfer functions and to apply control theories to design effective controllers.

10:19

📈 Sketching System Responses and Understanding Time Constants

This paragraph instructs on how to sketch the system's step response and emphasizes the need to understand the closed-loop transfer function. It discusses the comparison of the system's transfer function to the standard form of a second-order system to determine natural frequency and damping ratio. The paragraph provides formulas for calculating the rise time and peak time, which are critical parameters in evaluating system performance. It also touches on the concept of setting time, which is the time it takes for the system's response to reach a certain percentage of its final value. The goal is to understand how these time constants relate to the system's stability and responsiveness.

15:20

📊 Calculating and Interpreting System Response Metrics

The fourth paragraph focuses on calculating and interpreting various system response metrics such as rise time, peak time, and setting time. It provides formulas for these metrics and explains how to use them to analyze system performance. The paragraph also discusses the significance of the damping ratio in determining the system's stability and how to compute the maximum overshoot. It emphasizes the importance of being able to use a calculator to compute these values and to understand their physical meaning in the context of system behavior.

20:22

📝 Applying Control System Concepts to Exam Problems

The final paragraph provides a summary of what to expect in a midterm exam regarding control systems. It mentions that students will need to derive transfer functions, obtain error responses, and design gain values for controllers. The paragraph also suggests that students may need to apply stability criteria to ensure the designed controllers are effective. The key message is to apply the concepts learned throughout the course to solve practical problems and to demonstrate a deep understanding of control system dynamics.

Keywords

💡Error

In the context of the video, 'Error' refers to the difference between the desired output (set point) and the actual output of a system. This is a fundamental concept in control systems engineering, where the goal is often to minimize error to achieve precise control. The script mentions 'Error' in relation to the standard controller and the need to understand its behavior in the Laplace domain, indicating its importance in system analysis and control.

💡Transfer Function

A 'Transfer Function' is a mathematical representation that describes the relationship between the input and output of a system. It is typically used in control systems to model the system's behavior in the frequency domain. The script emphasizes the need to derive the transfer function, which is crucial for understanding how changes in the input affect the output, and for designing controllers that can achieve the desired system response.

💡Proportional Gain (KP)

The 'Proportional Gain' (KP) is a parameter in a proportional controller that determines the controller's response to the current error. A higher KP value results in a stronger response to error, which can lead to faster correction but may also cause overshoot or instability. The script discusses the introduction of KP and its role in the proportional, integral, and derivative (PID) controller, which is a common type of feedback controller used in industrial控制系统.

💡Integral Gain (Ki)

The 'Integral Gain' (Ki) is another parameter in a PID controller that addresses the accumulation of error over time. It helps to eliminate steady-state errors by integrating the error signal. The script mentions the introduction of Ki, suggesting that it is one of the components that students need to understand and practice when dealing with PID controllers and their tuning.

💡Derivative Gain (Kd)

The 'Derivative Gain' (Kd) is the final parameter in a PID controller that predicts the future trend of the error signal, based on its rate of change. It helps to reduce overshoot and provides a smoother response by anticipating changes. The script indicates that while Kd is not always necessary, it may be included to enhance the performance of the controller.

💡Laplacian

In the script, 'Laplacian' refers to the Laplace transform, a mathematical tool used to analyze the behavior of dynamic systems in the frequency domain. The Laplace transform is used to convert differential equations, which describe the dynamics of systems, into algebraic equations, known as transfer functions. The script mentions practicing deriving the transfer function in the Laplace domain, which is essential for control system analysis.

💡Disturbance

A 'Disturbance' in control systems is an undesired input that affects the system's output. The script discusses the physical meaning of the transfer function relating output to disturbance, emphasizing the importance of designing systems that are robust against disturbances. The goal is to minimize the impact of disturbances on the system's performance, as indicated by a transfer function value of zero.

💡Feedback

Feedback is a fundamental concept in control systems where the output or a function of the output is used to influence the control action. The script mentions considering the disturbance as feedback, which is a way to improve the system's response to undesired inputs. Proper feedback mechanisms help in stabilizing the system and achieving the desired performance.

💡Reference Value

The 'Reference Value' is the desired set point or target value that a control system aims to achieve. In the script, it is mentioned in the context of how the system behaves when the reference value is zero, which is a scenario used to analyze the system's response to disturbances. Understanding the system's behavior under different reference values is crucial for controller design.

💡Stability Criteria

Stability criteria are rules or conditions that must be met for a control system to be stable, meaning it will respond predictably to inputs and disturbances. The script hints at applying Routh-Hurwitz stability criteria, which is a mathematical method used to determine the stability of a system by analyzing its characteristic equation. This is an important aspect of control system design to ensure that the system does not oscillate uncontrollably or fail to reach the desired state.

💡Response Time

In the context of the script, 'Response Time' refers to how quickly a system reaches a certain percentage of its final value in response to a change in input or a disturbance. The script discusses the concept of rise time, which is the time it takes for the system's output to rise from 0% to 100% (or as mentioned, from 10% to 90%) of its final value. This is an important performance metric in control systems, as it indicates how fast the system can react to changes.

Highlights

Introduction to the concept of the Standard Controller, including proportional gain (KP), integral gain (KI), and derivative gain (KD).

Emphasis on practicing the derivation of the transfer function, which is crucial for understanding system dynamics.

Explanation of how to handle systems with time delays (TD) in the transfer function derivation.

The importance of considering the physical meaning of the transfer function, especially in relation to disturbances.

Discussion on the ideal transfer function values for a well-tuned control system, highlighting the preference for zero sensitivity to disturbances.

Guidance on determining the value of KP for a controller, questioning whether a KP less than 5 is sufficient.

Mention of the need to introduce different types of controllers, such as PD, in the mid-term exam.

Stress on understanding the error in the Laplace Domain and the application of the Final Value Theorem.

Advice on sketching the system's response plot based on the given transfer function.

Introduction of the concept of natural frequency and damping ratio in the context of second-order systems.

Instructions on how to compute the rise time and setting time using the formulas provided.

Explanation of the significance of the peak time and how to compute it using the damping ratio.

Discussion on the setting time and its practical implications for system stability and performance.

Highlight of the need to apply Routh stability criteria in problem-solving.

Emphasis on the importance of being able to derive the transfer function from given system parameters.

The necessity of obtaining the characteristic equation from the closed-loop transfer function for system analysis.

Advice on how to use a calculator effectively to compute various system response times and parameters.

Transcripts

Error The Error and We Also

introduce dist To The System so

you need to look at the Lecture

6 that we talk about the Standard

Controller If you understand The

Concept for the Standard

Controller It is

in

in

KP We introduce KP proportional

gain We introduce K integral gain

or derivative gain but what you

need to Practice what you need to

Practice is to derive the transfer

function to derive the transfer

function from this slide from this

slide So The System

KP

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function is define by output div

by input you are Okay with it We

Practice many time but When The

System has TD how can we de with

it Good News is good News is

When We derive transfer function

That relate output to

input We Take TD

equ We Only The

Output with to

the

the

and output

The What about

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We

output

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TD physical Meaning Of relate

output to TD is When we don't

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equ Z input Z But We Have How

System

behave for

That

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to zer but Bad New is al this

is

Zero We need to consider this

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new Reference you need to The

System

and

relationship a Quick question for

you a Quick question for you a

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the control ideally for the

control

if your Controller is good Control

System is Fin finely Tune transfer

function output divide by input

what is your preference what is

your prefer Value output divide by

input If The System is perfect

One How about transfer function

That relate output to disturbance

if your System is very Clean

robust against disturbance output

divide by disturbance What should

be the Value We

prefer zer Zero If The System is

good is robust Again dist B

although dist B B Small We don't

care Because transfer function

always be

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Value of KP sometimes KP is not

enough We need to include kd

But We ask you what should be

the Value of KP

That Less

5

2 KP is sufficient or not to

make refer by

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introduce typ of Controller

in mid exam We Only have Control

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ignore but

PD P

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PD

PD PD PD control

for

not for this Two Important

SL

E

Error E Error in the Lap Domain

E Error in the Lap

Domain If We Have diagram Of The

System Please Make You can der E

in the Lap Domain and please make

sure You can Apply Final Value

theorem to

obtain the Exact Value

by and

You can Practice from this slide

If you want to prepare for

exam We Have System transfer

function We Have System transfer

function your System transfer

function May be the Same or

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sy

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resp this

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to I say not much relate to

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point from

problem We gna Give you a

transfer function of System for

example GS equ to 1 div by 1 1

you know for

example If The System transfer

function If The System transfer

function is One dy by T

S I Mean this transfer function

If The System is this for your

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it may be 1 per ms S + TS + K

Something like

that you need to

sketch this plot in the exam

GR you don't have L you don't

have L how can you sketch this

SP how can you sketch this SP

I

think This is the close Loop

transfer function

and If you compare to the

Standard form of Second Order

System we know what is

Jumping What is the Natural

frequency

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I don't want you to Remember The

Formula I Give you

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Formula So what you need to do

is what you need to do

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is I Give you the transfer

function of System you need to

derive the close Loop transfer

function you need to obtain

the the characteristic equation

from denominator of CL transfer

function you need to be Able To

achieve Natural frequency and

dumping ratio Natural frequency and

dumping ratio by compare that

characteristic equation to the

Standard form of Second Order

System After You Know Omega n

dumping OM D I will Give you the

Formula So That You can compute

the time Right Time Is Easy You

can Just Take One

Point by OM n that is time The

physical Meaning Of Time Is The

Time That The System Take to

increase The

resp from Zero No from 10 to 90%

right Time If you compute the

right Time by Calculator If you

get Two If you get for the right

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sketch The resp it Take Time To

increase Two and you say that

This is right Time

Time Time

Time Formula For

Time

Ma and

โอ I'm Sorry That SL is not enough

but I think This is the Better

the

time big and

If the SM Time In The System Go

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from 10 to 90% We define 10 to

90 in of zer to 100 Because in

some System that is Over damp

some Over System Go very

slow If you need to Wait Until

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time By The Way If you define it

from

maxum If the Value By The Go

Value But the go to

1 maxum be

20 setting Time

def the time that

the the vibration in 2 per in 1%

or in

5% If you say setting Time 5% If

you say setting Time

5% Around here setting time 5%

but If you define setting Time 2%

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setting 1% it may Around There

Until The

resp by the Error of that

percentage we cannot Say setting

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but 2% is the Popular one 2% is

the Popular

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in the exam gna be

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the exam is equal to 1 x dy by

omeg n Which means if you can

compute oma n You can compute the

right Time 10% To 90% and you

need to take this information To

suet Your plot to your

plot time Time simple and We Give

this Formula to

you by Omega

B is a peck time but you need

to sketch this in your exam

Answer Maximum

overshoot Formula is not

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easy Maximum overshoot is equal to

E Power Min Sin Pi DIY by squ

RO 1 - Sin Square We Give this

Value To You I Mean We Give this

Formula to you but you need to

know How to use you need to know

what is damping

ratio how to use Calculator to

compute E Power Min Sin Pi divide

by this Practice How to use

Calculator and when you already

compute Peak Time What Is The

Meaning Of Peak

Time เ้ยแก๊งเสื้อ LINE และเสื้อ

นิสิตเสื้อนิสิตนี่ชื่ออะไรวะลืมคุณชื่อ

อะไรนะไม่ใชเสืิเสื้อ

ขาวเอาชื่อที่อาจารย์จำ

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จริงอกล้องพบโอเคกล้องพบพ Time ในรูปนี้

อยู่ตรง

ไหนบอกมาเป็นตัวเลขเลยก็ได้คุณ

เก่งเก่งก

ดามใ

Where Is The

System 1 2 3

4 1 2 3 is

Number What Is The

Time

399

around compute This spe

Time if you don't have Man Lab

can you use a Simple Calculator

to get this

time in exam you need to compute

this PE

Time If you know oma D You Just

x dy by OM D and then you get

Time You Know that you know That

in my opinion the Fun Thing Is

You Know This Formula That G Give

you in the exam and you

have if You Have

Time You write a

Code You Find the Thing in the

and then you use Calculator and

then you compare The Value it

Should approximately Be the Same

setting

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Time

a is

Time ช่ a where isle

If You

Remember Men about setting Time

2 setting 5% setting

1 magn

is9 and After

Point deviation Less 2

this

indicate The setting time

2% in exam

room How Can You compute This

setting

2% มี

If you know dumping ratio If you

know Omega n 4 di by this you

get setting Time

2% If you need setting Time

5% div by this Lucky าร Give this

Formula for

you and then You Just need to

know That when you compute the

numerical Value Where G Be in the

Where Be in

the do you think This is S or

not That setting Time

5% numerical Value is Less than

2% Is It Make

Sense or make

S do you think you need shorter

Time or longer Time Until The

System Stop vibrate from 5% Error

to 2%

Error The smaller Error need

longer Time

The smaller

Error

โอสรุปเนาะ

สรุป conclusion

conclusion The First midterm exam

problem you need to derive the

transfer

function of อะไรนะเฮ้ยอะไรนะเฮ้ยเปิด

ข้อสอบอยู่ไง

[เพลง]

The Position

V

Second you

the

and CP

obtain

Error the third problem you Bring

The transfer function from the

first problem

again and

then you try to Design gain

KP So That you have the the

Design Range of Error Range of

Error

[ปรบมือ]

and problem 4 May need to Apply

The R stability

criteria So That