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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.

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Presentation on theme: "July 2004 2009 Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve week3 January 2009 Design of Control System in State."— Presentation transcript:

1 July Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Twelve week3 January 2009 Design of Control System in State Space

2 © 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.

3 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Review First Order:

4 © 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 )

5 © 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

6 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Design By Pole Placement

7 © 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

8 © 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

9 © 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

10 © 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

11 © 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)

12 © 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)

13 © 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)

14 © 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

15 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example

16 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example

17 © 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.

18 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example

19 © 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.

20 © 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)

21 © 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)

22 © 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)

23 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example

24 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

25 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Solving Pole-Placement Problems with MATLAB

26 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

27 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

28 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

29 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

30 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

31 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

32 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula

33 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula

34 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Ackermann’s Formula

35 © 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)

36 © 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)

37 © 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)

38 © 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)

39 © 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

40 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

41 © 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

42 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

43 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

44 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

45 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling Define state variables

46 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

47 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

48 © 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.

49 © 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

50 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

51 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling Use state-feedback control

52 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

53 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

54 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

55 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

56 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling The desired characteristic equation

57 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

58 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

59 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

60 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

61 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

62 © 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

63 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

64 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Mathematical Modeling

65 © 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

66 © 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

67 © 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

68 © 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

69 © 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

70 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

71 © 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.

72 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

73 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

74 © 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

75 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

76 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

77 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Obtaining System Response To Initial Condition

78 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program

79 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 MATLAB Program

80 © 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

81 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

82 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2

83 © AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2


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