July Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve week3 January 2009 Design of Control System in State Space
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Introduction This Chapter we will learn about state-space design methods based on the pole-placement method and the quadratic optimal regulator method.
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Review First Order:
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Review Second Order: Back to review Chapter4 Transient Response Analysis( Ogata Book )
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Pole Placement Pole Placement ( วิธีการวางโพล ) คือ ตั้งข้อกำหนดสำหรับ ตำแหน่งโพลทั้งหมดของระบบวงปิด และออกแบบตัวควบคุมที่ จะได้ตำแหน่งโพลตามข้อกำหนดนั้น เงื่อนไขจำเป็นของระบบ หรือพลานต์ที่ทำให้สามารถทำการเคลื่อนย้ายโพลทั้งหมดไปยัง ตำแหน่งที่ต้องการได้ ในการออกแบบทั่วไปจะไม่ได้ต้องการให้ระบบมีเสถียรภาพ อย่างเดียว แต่ยังต้องการสมรรถนะหรือผลตอบสนองตาม ต้องการด้วย ดังนั้นการกำหนดตำแหน่งของโพลระบบวงปิดจึง มิใช่เพียงแต่ว่าต้องการอยู่บนด้านซ้ายของระนาบเชิงซ้อน เท่านั้น แต่อาจจะต้องอยู่ในพื้นที่ที่จะให้ผลตอบสนองที่ดีด้วย เช่น ถ้าตำแหน่งโพลอยู่ใกล้แกนจินตภาพมากเกินไป ผลตอบสนองจะมีลักษณะแกว่ง ในระบบอันดัล n ทั่ว ๆ ไป ความสัมพันธ์ผลตอบสนองทาง เวลาของระบบกับตำแหน่งของโพล มักมีความซับซ้อน จึงเป็น การยากที่จะกำหนดตำแหน่งโพลเพื่อให้ได้ผลตอบสนองที่ดี ดังนั้นวิธีการออกแบบนี้โดยทั่วไปอาศัยหลักการของระบบที่มี ลักษณะเด่นเป็นอันดับสอง
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Design By Pole Placement
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Design By Pole Placement (a) Open-loop control system; (b) Closed-loop control sysytem
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Design By Pole Placement Control signal The Solution is
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K using Transformation Matrix T. a i are coefficients of the characteristic polynomial
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K using Transformation Matrix T. (2) where
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K using Transformation Matrix T. (3)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K using Transformation Matrix T. (4)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K using Transformation Matrix T. (5)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 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
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K Using Direct Substitution Method.
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K Using Ackerman’s Formula.
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K Using Ackerman’s Formula. (2)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K Using Ackerman’s Formula. (3)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of Matrix K Using Ackerman’s Formula. (4)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Solving Pole-Placement Problems with MATLAB
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula)
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Design of Regulator-type Systems by Pole Placement
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling We assume that the moment of inertia of the pendulum about its center of gravity is zero
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling Define state variables
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling In terms of vector-matrix equations.
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling By substituting the given numerical values
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling Use state-feedback control
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling The desired characteristic equation
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling Inverted-pendulum system with state-feedback control
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of state-feedback gain matrix K with MATLAB
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of state-feedback gain matrix K with MATLAB
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Determination of state-feedback gain matrix K with MATLAB
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition State equation
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition Control equation
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition Substitute the numerical values.
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition Initial condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Response Of Inverted Pendulum System Subjected To Initial Condition
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2
© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2