MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern...
Transcript of MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern...
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Control Systems
Dr. Assaad A. Al SahlaniDepartment of Mechanical Engineering
Michigan State University
Lecture 0Introduction
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Instructor Class Instructor: Dr. Assaad A. Alsahlani, Associate Professor at ME department, 2328J Engineering Building, Email: [email protected]
Office Hours 2328J EB, MWF 10-11am
Laboratory Instructor: Dr. Brian Feeny, 2430 Engineering Building Email: [email protected]
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Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H.
Bishop, Prentice Hall, 12th edition, 2010, ISBN-10: 0-13-602458-0
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Main components of the course Lectures (about 40 lectures) Old Math Quiz Midterm1, Midterm2 Final (Final exam period) Laboratory work Grading: Homework plus Math Quiz (10%), Exam 1 (20%),
Exam 2 (20%), Final Exam (comprehensive) (25%),Laboratory work (25%)
Homework will be due in one week from the day it isassigned
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Tips to pass this course Come to the lectures as many times as you can. Print out and bring lecture slides to the lecture. Do “Exercises” given at the end of each lecture. Do homework every week. Read the textbook and the slides. Make use of instructor’s office hours. If you want to get a very good grade… Read the textbook thoroughly. Read optional references too. Do more than given “Exercises”. Use and be familiar with Matlab.
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Math Prerequisites Complex Numbers Add, Subtract, Multiply, Divide
Linear Algebra Matrix Multiply, Inverse, Sets of Linear Eq.
Linear Ordinary Differential Equations Laplace Transform to Solve ODE’s Linearization Logarithms Modeling of Physical Systems Mechanical, Electrical, Thermal, Fluid
Dynamic Responses 1st and 2nd Order Systems of ODE’s
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Prerequisites: Complex Numbers Ordered pair of two real numbers
Conjugate Addition
Multiplication
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Complex Numbers Euler’s identity
Polar form Magnitude Phase
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Logarithm The logarithm of x to the base b is written The logarithm of 1000 to the base 10 is 3, i.e.,
Properties:
Why?
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Summary & Exercises Prerequisites Complex numbers, Linear Algebra, Logarithm,
Laplace transform Dynamics
Next Introduction
Exercises Buy the course textbook and keep it! Review today’s slides on complex numbers and
logarithm Read Chapter 1 and 2 of the textbook.
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2017-2018 11
What is “Control”? Make some object (called system, or plant)
behave as we desire. Imagine “control” around you! Room temperature control Car/bicycle driving Voice volume control “Control” (move) the position of the pointer Cruise control or speed control Process control etc.
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2017-2018 12
What is “Control Systems”? Why do we need control systems? Convenient (room temperature control, laundry
machine) Dangerous (hot/cold places, space, bomb removal) Impossible for human (nanometer scale precision
positioning, work inside the small space that humancannot enter)
It exists in nature. (human body temperature control) Lower cost, high efficiency, etc.
Many examples of control systems around us
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2017-2018 13
Open-Loop Control Open-loop Control System Toaster, microwave oven, shooting a basketball
Calibration is the key! Can be sensitive to disturbances
PlantController
(Actuator)
Signal Input input output
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2017-2018 14
Example: Toaster A toaster toasts bread, by setting timer.
Objective: make bread golden browned and crisp. A toaster does not measure the color of bread during the
toasting process. For a fixed setting, in winter, the toast can be white and in
summer, the toast can be black (Calibration!) A toaster would be more expensive with sensors to
measure the color and actuators to adjust the timer basedon the measured color.
ToasterSetting of timer Toasted bread
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2017-2018 15
Example: Laundry machine A laundry machine washes clothes, by setting a
program.
A laundry machine does not measure how cleanthe clothes become. Control without measuring devices (sensors) are
called open-loop control.
MachineProgram setting Washed clothes
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2017-2018 16
Closed-Loop (Feedback) Control Compare actual behavior with desired behavior Make corrections based on the error The sensor and the actuator are key elements of
a feedback loop Design control algorithm
Plant
Sensor
Signal Input Error output
+ -ActuatorController
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2017-2018 17
Attempts to change the direction of the automobile.
Manual closed-loop (feedback) control. Although the controlled system is “Automobile”, the
input and the output of the system can bedifferent, depending on control objectives!
Ex: Automobile direction control
Auto
Steeringwheelangle
DirectionDesireddirection
Eye
HandBrain
Error
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2017-2018 18
Attempts to maintain the speed of the automobile.
Cruise control can be both manual and automatic. Note the similarity of the diagram above to the
diagram in the previous slide!
Ex: Automobile cruise control
AutoAcceleration Speed
Desiredspeed
Sensor
ActuatorController
DisturbanceError
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2017-2018 19
Basic elements in feedback controlsystems
PlantInput OutputReference
Sensor
ActuatorController
Disturbance
Control system design objectiveTo design a controller s.t. the output follows
the reference in a “satisfactory” mannereven in the face of disturbances.
Error
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2017-2018 20
Systematic controller design process
PlantInput OutputReference
Sensor
ActuatorController
Disturbance
1. Modeling
Mathematical model
2. Analysis
Controller
3. Design
4. Implemenation
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2017-2018 21
Goals of this courseTo learn basics of feedback control systems Modeling as a transfer function and a block diagram
• Laplace transform (Mathematics!)• Mechanical, electrical, electromechanical systems
Analysis• Step response, frequency response• Stability: Routh-Hurwitz criterion, (Nyquist criterion)
Design• Root locus technique, frequency response technique,
PID control, lead/lag compensator Theory, (simulation with Matlab), practice in laboratories
![Page 22: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/22.jpg)
2017-2018 22
Course roadmap
Laplace transform
Transfer function
Models for systems• mechanical• electrical• electromechanical
Linearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• (Nyquist)
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
![Page 23: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/23.jpg)
2017-2018 23
Summary & Exercises Introduction Examples of control systems Open loop and closed loop (feedback) control Automatic control is a lot of fun!
Next Laplace transform
Exercises Buy the course textbook at the Bookstore. Read Chapter 1 and Apendix A, B of the textbook.
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Fall 2008 24
Control Systems
Lecture 1Introduction
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Fall 2008 25
Tips to pass this course Come to the lectures as many times as you can. Print out and bring lecture slides to the lecture. Do “Exercises” given at the end of each lecture. Read the textbook and the slides. Make use of instructor’s office hours. If you want to get a very good grade…
![Page 26: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/26.jpg)
Fall 2008 26
What is “Control”? Make some object (called system, or plant)
behave as we desire. Imagine “control” around you! Room temperature control Car/bicycle driving Voice volume control “Control” (move) the position of the pointer Cruise control or speed control Process control etc.
![Page 27: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/27.jpg)
Fall 2008 27
What is “Control Systems”? Why do we need control systems? Convenient (room temperature control, laundry
machine) Dangerous (hot/cold places, space, bomb removal) Impossible for human (nanometer scale precision
positioning, work inside the small space that humancannot enter)
It exists in nature. (human body temperature control) Lower cost, high efficiency, etc.
Many examples of control systems around us
![Page 28: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/28.jpg)
Fall 2008 28
Open-Loop Control Open-loop Control System Toaster, microwave oven, shooting a basketball
Calibration is the key! Can be sensitive to disturbances
PlantController
(Actuator)
Signal Input input output
![Page 29: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/29.jpg)
Fall 2008 29
Example: Toaster A toaster toasts bread, by setting timer.
Objective: make bread golden browned and crisp. A toaster does not measure the color of bread during the
toasting process. For a fixed setting, in winter, the toast can be white and in
summer, the toast can be black (Calibration!) A toaster would be more expensive with sensors to
measure the color and actuators to adjust the timer basedon the measured color.
ToasterSetting of timer Toasted bread
![Page 30: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/30.jpg)
Fall 2008 30
Example: Laundry machine A laundry machine washes clothes, by setting a
program.
A laundry machine does not measure how cleanthe clothes become. Control without measuring devices (sensors) are
called open-loop control.
MachineProgram setting Washed clothes
![Page 31: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/31.jpg)
Fall 2008 31
Closed-Loop (Feedback) Control Compare actual behavior with desired behavior Make corrections based on the error The sensor and the actuator are key elements of
a feedback loop Design control algorithm
Plant
Sensor
Signal Input Error output
+ -ActuatorController
![Page 32: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/32.jpg)
Fall 2008 32
Attempts to change the direction of the automobile.
Manual closed-loop (feedback) control. Although the controlled system is “Automobile”, the
input and the output of the system can bedifferent, depending on control objectives!
Ex: Automobile direction control
Auto
Steeringwheelangle
DirectionDesireddirection
Eye
HandBrain
Error
![Page 33: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/33.jpg)
Fall 2008 33
Attempts to maintain the speed of the automobile.
Cruise control can be both manual and automatic. Note the similarity of the diagram above to the
diagram in the previous slide!
Ex: Automobile cruise control
AutoAcceleration Speed
Desiredspeed
Sensor
ActuatorController
DisturbanceError
![Page 34: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/34.jpg)
Fall 2008 34
Basic elements in feedback controlsystems
PlantInput OutputReference
Sensor
ActuatorController
Disturbance
Control system design objectiveTo design a controller s.t. the output follows
the reference in a “satisfactory” mannereven in the face of disturbances.
Error
![Page 35: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/35.jpg)
Fall 2008 35
Systematic controller design process
PlantInput OutputReference
Sensor
ActuatorController
Disturbance
1. Modeling
Mathematical model
2. Analysis
Controller
3. Design
4. Implemenation
![Page 36: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/36.jpg)
Fall 2008 36
Goals of this courseTo learn basics of feedback control systems Modeling as a transfer function and a block diagram
• Laplace transform (Mathematics!)• Mechanical, electrical, electromechanical systems
Analysis• Step response, frequency response• Stability: Routh-Hurwitz criterion, (Nyquist criterion)
Design• Root locus technique, frequency response technique,
PID control, lead/lag compensator Theory, (simulation with Matlab), practice in laboratories
![Page 37: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/37.jpg)
Fall 2008 37
Course roadmap
Laplace transform
Transfer function
Models for systems• mechanical• electrical• electromechanical
Linearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• (Nyquist)
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
![Page 38: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/38.jpg)
Fall 2008 38
Summary & Exercises Introduction Examples of control systems Open loop and closed loop (feedback) control Automatic control is a lot of fun!
Next Laplace transform
Exercises Buy the course textbook at the Bookstore. Read Chapter 1 and Apendix A, B of the textbook.
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Control Systems
Lecture 2Laplace transform
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Course roadmap
Laplace transformTransfer function
Models for systems• electrical• mechanical• electromechanical
Block diagrams
Linearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• (Nyquist)
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
![Page 41: MECH466: Automatic Control · 2020. 10. 27. · 3 Course information Required Textbook: Modern Control Systems, Richard C. Dorf and Robert H. Bishop, Prentice Hall, 12th edition,](https://reader035.fdocuments.in/reader035/viewer/2022070217/61236ccb36ec8e1523654e25/html5/thumbnails/41.jpg)
41
Laplace transform One of most important math tools in the course! Definition: For a function f(t) (f(t)=0 for t<0),
We denote Laplace transform of f(t) by F(s).
f(t)
t0F(s)
(s: complex variable)
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Examples of Laplace transform Unit step function
Unit ramp function
f(t)
t0
1
f(t)
t0
(Memorize this!)
(Integration by parts)
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Integration by parts
EX.
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Examples of Laplace transform (cont’d) Unit impulse function
Exponential function
f(t)
t0
f(t)
t0
1
(Memorize this!)
Width = 0Height = infArea = 1
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Examples of Laplace transform (cont’d)
Sine function
Cosine function(Memorize these!)
Remark: Instead of computing Laplacetransform for each function, and/ormemorizing complicated Laplace transform,use the Laplace transform table !
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Laplace transform table
Inverse Laplace Transform
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Properties of Laplace transform1. Linearity
Ex.
Proof.
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48
Properties of Laplace transform2.Time delay
Ex.
Proof.f(t)
0 T
f(t-T)
t-domain s-domain
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Properties of Laplace transform3. Differentiation
Ex.
Proof.
t-domain s-domain
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Properties of Laplace transform4. Integration
Proof.
t-domain s-domain
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Properties of Laplace transform5. Final value theorem
Ex.
if all the poles of sF(s) are inthe left half plane (LHP)
Poles of sF(s) are in LHP, so final value thm applies.
Ex.
Some poles of sF(s) are not in LHP, so final valuethm does NOT apply.
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Properties of Laplace transform6. Initial value theorem
Ex.
Remark: In this theorem, it does not matter ifpole location is in LHS or not.
if the limits exist.
Ex.
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Properties of Laplace transform7. Convolution
IMPORTANT REMARK
Convolution
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Summary & Exercises Laplace transform (Important math tool!) Definition Laplace transform table Properties of Laplace transform
Next Solution to ODEs via Laplace transform
Exercises Read Chapters 1 and 2. Solve Quiz Problems.
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Control Systems
Lecture 3Solution to ODEs via Laplace transform
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Course roadmap
Laplace transformTransfer function
Models for systems• electrical• mechanical• electromechanical
Block diagramsLinearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• Nyquist
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
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Laplace transform (review) One of most important math tools in the course! Definition: For a function f(t) (f(t)=0 for t<0),
We denote Laplace transform of f(t) by F(s).
f(t)
t0F(s)
(s: complex variable)
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An advantage of Laplace transform We can transform an ordinary differential
equation (ODE) into an algebraic equation (AE).
ODE AE
Partial fractionexpansionSolution to ODE
t-domain s-domain
12
3
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Example 1ODE with initial conditions (ICs)
1. Laplace transform
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Properties of Laplace transformDifferentiation (review)
t-domain
s-domain
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2. Partial fraction expansion
Multiply both sides by s & let s go to zero:
Similarly,
unknowns
Example 1 (cont’d)
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3. Inverse Laplace transform
Example 1 (cont’d)
If we are interested in only the final value of y(t), applyFinal Value Theorem:
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Example 2
S1
S2
S3
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In this way, we can find a rathercomplicated solution to ODEs easily by
using Laplace transform table!
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Example: Newton’s law
We want to know the trajectory of x(t). By Laplace transform,
M
(Total response) = (Forced response) + (Initial condition response)
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EX. Air bag and accelerometer Tiny MEMS accelerometer Microelectromechanical systems (MEMS)
(Pictures from various websites)
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Ex: Mechanical accelerometer
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Ex: Mechanical accelerometer (cont’d) We would like to know how y(t) moves when unit
step f(t) is applied with zero ICs. By Newton’s law
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Ex: Mechanical accelerometer (cont’d) Suppose that b/M=3, k/M=2 and Ms=1. Partial fraction expansion
Inverse Laplace transform
0 2 4 6 8 10-0.5
-0.4
-0.3
-0.2
-0.1
0
Time [sec]
Am
plit
ude
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Summary & Exercises Solution procedure to ODEs
1. Laplace transform2. Partial fraction expansion3. Inverse Laplace transform
Next, modeling of physical systems usingLaplace transform
Exercises Derive the solution to the accelerometer problem. E2.4 of the textbook in page 135.
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Control Systems
Lecture 4Modeling of electrical systems
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Course roadmap
Laplace transformTransfer function
Models for systems• electrical• mechanical• electromechanical
Block diagrams
Linearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• Nyquist
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
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Controller design procedure (review)
plantInput OutputRef.
Sensor
ActuatorController
Disturbance
1. Modeling
Mathematical model2. Analysis
Controller
3. Design
4. Implemenation
What is the “mathematical model”? Transfer function Modeling of electrical circuits
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Representation of the input-output (signal)relation of a physical system
A model is used for the analysis and design ofcontrol systems.
Mathematical model
Physicalsystem
Model
Modeling
Input Output
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Modeling is the most important and difficult taskin control system design. No mathematical model exactly represents a
physical system.
Do not confuse models with physical systems! In this course, we may use the term “system” to
mean a mathematical model.
Important remarks on models
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Transfer function A transfer function is defined by
A system is assumed to be at rest. (Zero initialcondition)
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Impulse response Suppose that u(t) is the unit impulse function
and system is at rest.
The output g(t) for the unit impulse input is calledimpulse response. Since U(s)=1, the transfer function can also be
defined as the Laplace transform of impulseresponse:
System
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Models of electrical elements:(constitutive equations)
v(t)
i(t)
R
Resistance CapacitanceInductance
v(t)
i(t)
L v(t)
i(t)
C
Laplacetransform
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Impedance Generalized resistance to a sinusoidal
alternating current (AC) I(s) Z(s): V(s)=Z(s)I(s)
V(s)I(s)Z(s)
Time domain Impedance Z(s)
Resistance
Capacitance
Inductance
Element
Memorize!
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Kirchhoff’s Voltage Law (KVL) The algebraic sum of voltage drops around any
loop is =0.
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Kirchhoff’s Current Law (KCL) The algebraic sum of currents into any junction
is zero.
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Impedance computation Series connection
Proof (Ohm’s law)
V(s)I(s)
Z1(s) Z2(s)
V1(s) V2(s)
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Impedance computation Parallel connection
Proof (Ohm’s law)
KCL
V(s)
I(s) Z1(s)
Z2(s)
I1(s)
I2(s)
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Modeling example
Kirchhoff voltage law (with zero initial conditions)
By Laplace transform,
v1(t)
i(t)
R2Input
R1
v2(t) OutputC
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Modeling example (cont’d)
Transfer function
v1(t)
i(t)
R2Input
R1
v2(t) OutputC
(first-order system)
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vd
Example: Modeling of op amp
Impedance Z(s): V(s)=Z(s)I(s) Transfer function of the above op amp:
Vi(s)
I(s)
InputZi(s)
Vo(s) Output
-+
i- Rule2: vd=0Rule1: i-=0Zf(s)
If(s)
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Modeling example: op amp
By the formula in previous two pages,
vi(t)
i(t)
R2
Input
R1
vo(t) Output
C
-+
i-
vd
Vd=0
i-=0
(first-order system)
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vd
Modeling exercise: op amp
Find the transfer function!
vi(t)
R2
Input
R1
vo(t) Output
C2
-+
i- Vd=0
i-=0
C1
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More exercises Find a transfer function from v1 to v2.
Find a transfer function from vi to vo.
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Summary & Exercises Modeling Modeling is an important task! Mathematical model Transfer function Modeling of electrical systems
Next, modeling of mechanical systems Exercises Do the problems in page 19 of this lecture note.
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Control Systems
Lecture 5Modeling of mechanical systems
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Course roadmap
Laplace transformTransfer function
Models for systems• electrical• mechanical• electromechanical
Block diagrams
Linearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• Nyquist
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
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Time-invariant & time-varying A system is called time-invariant (time-varying)
if system parameters do not (do) change in time. Example: Mx’’(t)=f(t) & M(t)x’’(t)=f(t) For time-invariant systems:
This course deals with time-invariant systems.
SysTime shift Time shift
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Newton’s laws of motion 1st law: A particle remains at rest or continues to move in a
straight line with a constant velocity if there is nounbalancing force acting on it.
2nd law: : translational
: rotational
3rd law: For every action has an equal and opposite reaction
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Translational mechanical elements:(constitutive equations)
Mass DamperSpring
M
f(t)x(t) f(t) x1(t)
K x2(t)
f(t)
f(t) x1(t)
B x2(t)
f(t)
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Mass-spring-damper system
M
x(t)
K B
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Free body diagram
Newton’s law: F=ma
M
K B
Direction of actual force will beautomatically determined by therelative values!
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Mass-spring-damper system
Equation of motion
By Laplace transform (with zero initial conditions),
M
x(t)
K B
(2nd order system)
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Gravity?
At rest, y coordinate: x coordinate:
K
M
KM
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Automobile suspension system
M2
f(t)x2(t)
K1 B
K2
M1x1(t)
automobile
suspension
wheel
tire
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Automobile suspension system
Laplace transform with zero ICs
G2 G1
G3
F X2 X1
Block diagram
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Rotational mechanical elements(constitutive equations)
Moment of inertia FrictionRotational spring
JK
Btorque
rotation angle
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Torsional pendulum system Ex.2.12
J
K
B
friction betweenbob and air
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Torsional pendulum system
Equation of Motion
By Laplace transform (with zero ICs),
J
K
B
friction betweenbob and air
(2nd order system)
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Example
By Newton’s law
By Laplace transform (with zero ICs),
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Example (cont’d) From second equation:
From first equation:
G2Block diagram
G1
(2nd order system)
(4th order system)
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Rigid satellite Ex. 2.13
Thrustor
Doubleintegrator
• Broadcasting• Weather forecast• Communication• GPS, etc.
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Summary & Exercises Modeling of mechanical systems Translational Rotational
Next, block diagrams. Exercises Derive equations for the automobile suspension
problem.
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Exercises (Franklin et al.) Quarter car model: Obtain a transfer function
from R(s) to Y(s).
Road surface
M1x (t)
Ks B
Kw
M2 y(t) Answer
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Control Systems
Lecture 7Linearization, time delays
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Course roadmap
Laplace transform
Transfer function
Models for systems• electrical• mechanical• electromechanicalBlock diagramsLinearization
Modeling Analysis Design
Time response• Transient• Steady state
Frequency response• Bode plot
Stability• Routh-Hurwitz• Nyquist
Design specs
Root locus
Frequency domain
PID & Lead-lag
Design examples
(Matlab simulations &) laboratories
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What is a linear system? A system having Principle of Superposition
A nonlinear system does not satisfythe principle of superposition.
System
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Linear systems Easier to understand and obtain solutions Linear ordinary differential equations (ODEs), Homogeneous solution and particular solution Transient solution and steady state solution Solution caused by initial values, and forced solution
Add many simple solutions to get more complexones (use superposition!) Easy to check the Stability of stationary states
(Laplace Transform)
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Why linearization? Real systems are inherently nonlinear. (Linear
systems do not exist!) Ex. f(t)=Kx(t), v(t)=Ri(t) TF models are only for linear time-invariant (LTI)
systems. Many control analysis/design techniques are
available for linear systems. Nonlinear systems are difficult to deal with
mathematically. Often we linearize nonlinear systems before
analysis and design. How?
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How to linearize it? Nonlinearity can be approximated by a linear
function for small deviations around anoperating point Use a Taylor series expansion
Linear approximation
Nonlinear function
Operating point
Old coordinate
New coordinate
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Linearization Nonlinear system: Let u0 be a nominal input and let the resultant
state be x0
Perturbation: Resultant perturb: Taylor series expansion:
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Linearization (cont.)
notice that ; hence
L. sys.N. sys
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Motion of the pendulum
Linearize it at Find u0
New coordinates:
Linearization of a pendulum model
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Linearization of a pendulum model (cont’)
Taylor series expansion of
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Time delay transfer function TF derivation
The more time delay is, the more difficult tocontrol (Imagine that you are controlling thetemperature of your shower with a very longhose. You will either get burned or frozen!)
(Memorize this!)
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Summary and Exercises Modeling of Nonlinear systems Systems with time delay
Next Modeling of DC motors
Exercises Linearize the pendulum model at /4