July Lecture Side Lecture by Suradet Tantrairatn Instructor and Researcher Chapter Two week2 January 2009 Analysis of Control Systems in State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Review Exercise Aircraft Roll-Dynamics Systems

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Review Exercise (Result)

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

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 State Space Representation of Transfer Function การแปลงระหว่างฟังก์ชันถ่ายโอนกับตัวแทนปริภูมิสเตต ในกรณีระบบ SISO ทั่ว ๆ ไป หากมีฟังก์ชันถ่ายโอน ดังนี้ สามารถแปลงฟังก์ชันถ่ายโอนให้เป็นระบบตัวแปรสเตตใน รูปแบบบัญญัติ (Canonical form) 1) รูปแบบควบคุมได้ ( Controllability form ) 2) รูปแบบสังเกตได้ ( Observability form ) 3) รูปแบบทแยง ( Diagonal form ) 3) รูปแบบจอร์แดน ( Jordan form )

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Controllable Canonical form จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Controllable Canonical form (Example) จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Homework 1

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Observable Canonical form จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Observable Canonical form (Example) จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Observer canonical form from the controller canonical form

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

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Exercise 1 จงแปลงรูปจาก Transfer Function ดังนี้ ให้อยู่ในรูปบัญญัติ ( Controllable and Observable Canonical Form ) ? Controllable Canonical Form

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Exercise (Const) จงแปลงรูปจาก Transfer Function ดังนี้ ให้อยู่ในรูปบัญญัติ ( Controllable and Observable Canonical Form ) ? Observable Canonical Form

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Diagonal Canonical form จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Diagonal Canonical form (Example) จากรูปแบบ Transfer Function เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Jordan Canonical form จากรูปแบบ Transfer Function แยก factor ได้ เป็นรูปแบบ State Space

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Diagonal canonical form from the Controllable canonical form First Step Find the eigenvalues of Matrix A in Controllable Canonical Form Example

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Diagonal canonical form from the Controllable canonical form Then Step Use Diagonalization of nxn Matrix will transform P -1 AP into the diagonal The transformation x = Pz where

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Jordan canonical form from the Controllable canonical form First Step Like as Diagonal form Then Step Use Diagonalization of nxn Matrix will transform S -1 AS into the diagonal The transformation x = Sz where

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example Find the eigenvalues of A matrix

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

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Example (Const) จะได้รูปสมการ state space ดังนี้ และ Output ดังนี้

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Exercise 2 จงแปลงรูปตามโจทย์ ให้อยู่ในรูปบัญญัติ ( Diagonal Canonical Form ) ( ทั้ง 2 วิธี ) ?

© AIRBUS UK LTD All rights reserved. Confidential and proprietary document. Month 200X2009 Subject Name Automotive Automatic ControlPage 2 Transformation of System Models with MATLAB Command in MATLAB Example