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Design of Control System in State Space

Design of Control System in State Space

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Design of Control System in State Space

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  1. 2009 Lecture Side January 2009 Lecture by Design of Control System in State Space Chapter Twelve week3 Control Engineering

  2. Introduction This Chapter we will learn about state-space design methods based on the pole-placement method and the quadratic optimal regulator method. Control Engineering 2009 Subject Name Automotive Automatic Control

  3. Review First Order: Control Engineering 2009 Subject Name Automotive Automatic Control

  4. Review Second Order: Back to review Chapter4 Transient Response Analysis( Ogata Book ) Control Engineering 2009 Subject Name Automotive Automatic Control

  5. Pole Placement Pole Placement ( วิธีการวางโพล ) คือ ตั้งข้อกำหนดสำหรับตำแหน่งโพลทั้งหมดของระบบวงปิด และออกแบบตัวควบคุมที่จะได้ตำแหน่งโพลตามข้อกำหนดนั้น เงื่อนไขจำเป็นของระบบหรือพลานต์ที่ทำให้สามารถทำการเคลื่อนย้ายโพลทั้งหมดไปยังตำแหน่งที่ต้องการได้ ในการออกแบบทั่วไปจะไม่ได้ต้องการให้ระบบมีเสถียรภาพอย่างเดียว แต่ยังต้องการสมรรถนะหรือผลตอบสนองตามต้องการด้วย ดังนั้นการกำหนดตำแหน่งของโพลระบบวงปิดจึงมิใช่เพียงแต่ว่าต้องการอยู่บนด้านซ้ายของระนาบเชิงซ้อนเท่านั้น แต่อาจจะต้องอยู่ในพื้นที่ที่จะให้ผลตอบสนองที่ดีด้วย เช่น ถ้าตำแหน่งโพลอยู่ใกล้แกนจินตภาพมากเกินไป ผลตอบสนองจะมีลักษณะแกว่ง ในระบบอันดัล n ทั่ว ๆ ไป ความสัมพันธ์ผลตอบสนองทางเวลาของระบบกับตำแหน่งของโพล มักมีความซับซ้อน จึงเป็นการยากที่จะกำหนดตำแหน่งโพลเพื่อให้ได้ผลตอบสนองที่ดี ดังนั้นวิธีการออกแบบนี้โดยทั่วไปอาศัยหลักการของระบบที่มีลักษณะเด่นเป็นอันดับสอง Control Engineering 2009 Subject Name Automotive Automatic Control

  6. Design By Pole Placement Control Engineering 2009 Subject Name Automotive Automatic Control

  7. Design By Pole Placement (a) Open-loop control system; (b) Closed-loop control sysytem Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  8. Design By Pole Placement Control signal The Solution is Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  9. Determination of Matrix K using Transformation Matrix T. ai are coefficients of the characteristic polynomial Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  10. Determination of Matrix K using Transformation Matrix T. (2) where Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  11. Determination of Matrix K using Transformation Matrix T. (3) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  12. Determination of Matrix K using Transformation Matrix T. (4) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  13. Determination of Matrix K using Transformation Matrix T. (5) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  14. 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 Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  15. Example Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  16. Example Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  17. Determination of Matrix K Using Direct Substitution Method. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  18. Example Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  19. Determination of Matrix K Using Ackerman’s Formula. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  20. Determination of Matrix K Using Ackerman’s Formula. (2) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  21. Determination of Matrix K Using Ackerman’s Formula. (3) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  22. Determination of Matrix K Using Ackerman’s Formula. (4) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  23. Example Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  24. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  25. Solving Pole-Placement Problems with MATLAB Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  26. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  27. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  28. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  29. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  30. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  31. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  32. Ackermann’s Formula Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  33. Ackermann’s Formula Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  34. Ackermann’s Formula Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  35. MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  36. MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  37. MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  38. MATLAB Program (Produce The Feedback Gain Matrix K by Using Ackermann’s Formula) Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  39. Design of Regulator-type Systems by Pole Placement Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  40. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  41. Mathematical Modeling We assume that the moment of inertia of the pendulum about its center of gravity is zero Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  42. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  43. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  44. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  45. Mathematical Modeling Define state variables Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  46. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  47. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  48. Mathematical Modeling In terms of vector-matrix equations. Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  49. Mathematical Modeling By substituting the given numerical values Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2

  50. Mathematical Modeling Control Engineering 2009 Subject Name Automotive Automatic Control Month 200X Page 2