july 2004 2009 lecture side lecture by suradet tantrairatn instructor and researcher chapter twelve...
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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
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
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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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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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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Mathematical Modeling
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Mathematical Modeling
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Mathematical Modeling
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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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Mathematical Modeling
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Mathematical Modeling
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Mathematical Modeling
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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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Mathematical Modeling
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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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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
Obtaining System Response To Initial Condition
State equation
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Control equation
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Substitute the numerical values.
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
Initial condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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Month 200X 2009 Subject Name Automotive Automatic Control Page 2
Obtaining System Response To Initial Condition
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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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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
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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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