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2009 Lecture Side JanuaryJuly 2009 2004 Lecture by Suradet Tantrairatn Instructor and Researcher Analysis of Control Systems in State Space Chapter Two week2 Review Exercise © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Aircraft Roll-Dynamics Systems 2009 Subject Name Automotive Automatic Control Month 200X Page 2 © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Review Exercise (Result) 2009 Subject Name Automotive Automatic Control Month 200X Page 2 © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Practice in SIMULINK 2009 Subject Name Automotive Automatic Control Month 200X Page 2 State Space Representation of Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. การแปลงระหว่างฟั งก์ชนั ถ่ายโอนกับตัวแทนปริ ภมู ิสเตต ในกรณีระบบ SISO ทัว่ ๆ ไป หากมีฟังก์ชนั ถ่ายโอน ดังนี ้ สามารถแปลงฟั งก์ชนั ถ่ายโอนให้ เป็ นระบบตัวแปรสเตตใน รูปแบบบัญญัติ (Canonical form) 1) รูปแบบควบคุมได้ ( Controllability form ) 2) รูปแบบสังเกตได้ ( Observability form ) 3) รูปแบบทแยง ( Diagonal form ) 3) รูปแบบจอร์ แดน ( Jordan form ) 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Controllable Canonical form จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Controllable Canonical form (Example) จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Homework 1 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Observable Canonical form จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Observable Canonical form (Example) จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Observer canonical form from the controller canonical form Aob A © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. T cont Bob C T cont Cob Bcont T 2009 Subject Name Automotive Automatic Control Month 200X Page 2 From Example T Aob Acont T Bob Ccont T © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Cob Bcont 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Exercise 1 จงแปลงรูปจาก Transfer Function ดังนี ้ © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. ให้ อยู่ในรูปบัญญัติ ( Controllable and Observable Canonical Form ) ? Controllable Canonical Form 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Exercise (Const) จงแปลงรูปจาก Transfer Function ดังนี ้ © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. ให้ อยู่ในรูปบัญญัติ ( Controllable and Observable Canonical Form ) ? Observable Canonical Form 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Diagonal Canonical form จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Diagonal Canonical form (Example) จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Jordan Canonical form จากรูปแบบ Transfer Function © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. แยก factor ได้ เป็ นรูปแบบ State Space 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Diagonal canonical form from the Controllable canonical form First Step Find the eigenvalues of Matrix A in Controllable Canonical Form © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Example 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Diagonal canonical form from the Controllable canonical form © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Then Step Use Diagonalization of nxn Matrix will transform P-1AP into the diagonal The transformation x = Pz where 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Jordan canonical form from the Controllable canonical form © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. First Step Like as Diagonal form Then Step Use Diagonalization of nxn Matrix will transform S-1AS into the diagonal The transformation x = Sz where 2009 Subject Name Automotive Automatic Control Month 200X Page 2 © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Example Find the eigenvalues of A matrix 2009 Subject Name Automotive Automatic Control Month 200X Page 2 © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Example (Const) 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Example (Const) จะได้ รูปสมการ state spaceดังนี ้ © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. และ Output ดังนี ้ 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Exercise 2 จงแปลงรูปตามโจทย์ © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. ให้ อยู่ในรูปบัญญัติ ( Diagonal Canonical Form ) ( ทัง้ 2 วิธี ) ? 2009 Subject Name Automotive Automatic Control Month 200X Page 2 Transformation of System Models with MATLAB Command in MATLAB © AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document. Example 2009 Subject Name Automotive Automatic Control Month 200X Page 2