July 2004 2009 Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve...
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July 2004 2009 Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve week3 January 2009 Design of Control System in State Space
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Transcript of July 2004 2009 Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve...
July 20042009 Lecture Side
Lecture by
Suradet Tantrairatn
Instructor and Researcher
Chapter Twelve
week3
January 2009
Design of Control System in State Space
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Introduction
This Chapter we will learn about state-space design
methods based on the pole-placement method and the quadratic optimal regulator method.
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Review
First Order:
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Review
Second Order:
Back to review Chapter4 Transient Response Analysis( Ogata Book )
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Pole Placement
Pole Placement ( วิ�ธี�การวิางโพล ) คื�อ ตั้��งข้�อก�าหนดสำ�าหร�บตั้�าแหน�ง
โพลทั้��งหมดข้องระบบวิงปิ!ด และออกแบบตั้�วิคืวิบคื"มทั้�#จะได�ตั้�าแหน�งโพลตั้ามข้�อก�าหนดน��น เง�#อนไข้จ�าเปิ'นข้องระบบหร�อพลานตั้(ทั้�#ทั้�าให�สำามารถทั้�าการเคืล�#อนย้�าย้โพลทั้��งหมดไปิย้�งตั้�าแหน�งทั้�#ตั้�องการได�
ในการออกแบบทั้�#วิไปิจะไม�ได�ตั้�องการให�ระบบม�เสำถ�ย้รภาพอย้�างเด�ย้วิ แตั้�ย้�งตั้�องการสำมรรถนะหร�อผลตั้อบสำนองตั้ามตั้�องการด�วิย้ ด�งน��นการก�าหนดตั้�าแหน�งข้องโพลระบบวิงปิ!ดจ.งม�ใช่�เพ�ย้งแตั้�วิ�าตั้�องการอย้0�บนด�านซ้�าย้ข้องระนาบเช่�งซ้�อนเทั้�าน��น แตั้�อาจจะตั้�องอย้0�ในพ��นทั้�#ทั้�#จะให�ผลตั้อบสำนองทั้�#ด�ด�วิย้ เช่�น ถ�าตั้�าแหน�งโพลอย้0�ใกล�แกนจ�นตั้ภาพมากเก�นไปิ ผลตั้อบสำนองจะม�ล�กษณะแกวิ�ง
ในระบบอ�นด�ล n ทั้�#วิ ๆ ไปิ คืวิามสำ�มพ�นธี(ผลตั้อบสำนองทั้างเวิลาข้องระบบก�บตั้�าแหน�งข้องโพล ม�กม�คืวิามซ้�บซ้�อน จ.งเปิ'นการย้ากทั้�#จะก�าหนดตั้�าแหน�งโพลเพ�#อให�ได�ผลตั้อบสำนองทั้�#ด� ด�งน��นวิ�ธี�การออกแบบน��โดย้ทั้�#วิไปิอาศั�ย้หล�กการข้องระบบทั้�#ม�ลั�กษณะเด่นเป็�นอั�นด่�บสอัง
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Design By Pole Placement
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Design By Pole Placement
(a) Open-loop control system; (b) Closed-loop control sysytem
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Design By Pole Placement
Control signal
The Solution is
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K using Transformation Matrix T.
ai are coefficients of the characteristic polynomial
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K using Transformation Matrix T. (2)
where
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K using Transformation Matrix T. (3)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K using Transformation Matrix T. (4)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K using Transformation Matrix T. (5)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Summary to Find Matrix K Using Transformation Matrix T
Step1: Check the controllability condition Step2: From the characteristic polynomial for matrix A
Step3: Determine the transformation Matrix T
Step4: Using the desired eigenvalues
Final Step : Calculate K from
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Example
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Example
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K Using Direct Substitution Method.
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Example
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K Using Ackerman’s Formula.
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K Using Ackerman’s Formula. (2)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K Using Ackerman’s Formula. (3)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of Matrix K Using Ackerman’s Formula. (4)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Example
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Solving Pole-Placement Problems with MATLAB
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Ackermann’s Formula
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Ackermann’s Formula
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Ackermann’s Formula
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Design of Regulator-type Systems by Pole Placement
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
We assume that the moment of inertia of the pendulum about its center of gravity is zero
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
Define state variables
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
In terms of vector-matrix equations.
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
By substituting the given numerical values
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
Use state-feedback control
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
The desired characteristic equation
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
Inverted-pendulum system with state-feedback control
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Mathematical Modeling
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of state-feedback gain matrix K with MATLAB
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of state-feedback gain matrix K with MATLAB
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002.
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ts r
ese
rved
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fide
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pro
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tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Determination of state-feedback gain matrix K with MATLAB
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prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
State equation
© A
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UK
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002.
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rved
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prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Control equation
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UK
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002.
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rved
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fide
ntia
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pro
prie
tary
do
cum
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
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ts r
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rved
. Con
fide
ntia
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pro
prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Substitute the numerical values.
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002.
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rved
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fide
ntia
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prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
righ
ts r
ese
rved
. Con
fide
ntia
l and
pro
prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
righ
ts r
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rved
. Con
fide
ntia
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pro
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tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Initial condition
© A
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002.
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rved
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tary
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
righ
ts r
ese
rved
. Con
fide
ntia
l and
pro
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tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
righ
ts r
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rved
. Con
fide
ntia
l and
pro
prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
© A
IRB
US
UK
LT
D 2
002.
All
righ
ts r
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rved
. Con
fide
ntia
l and
pro
prie
tary
do
cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
MATLAB Program
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rved
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tary
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Response Of Inverted Pendulum System Subjected To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
© A
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002.
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rved
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fide
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tary
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cum
ent.
Month 200X 2009 Subject Name Automotive Automatic Control Page 2
© A
IRB
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002.
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tary
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cum
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
