2009 Lecture Side
JanuaryJuly
2009
2004
Lecture by
Suradet Tantrairatn
Instructor and Researcher
Analysis of Control Systems
in State Space
Chapter Eleven
week2
Review Exercise
จงแปลง Transfer Function
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
ให้ อยู่ในรูปบัญญัติ ( Controllable และ Jordan Canonical Form ) ?
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
Solving the time-invariant state equation
review the solution of the scalar differential equation
by assume a solution x(t) of the form
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Als;
t=0:
2009 Subject Name Aircraft Materials
Month 200X
Page 2
Solving the time-invariant state equation (2)
now solve the vector-matrix differential equation
assume that the solution in the form of vector power series in t,
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
also,
2009 Subject Name Aircraft Materials
Month 200X
Page 2
Solving the time-invariant state equation (3)
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
t = 0;
2009 Subject Name Aircraft Materials
Month 200X
Page 2
Laplace Transform Approach to the Solution of
Homogeneous State Equation
Take Laplace transform
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Take inverse Laplace transform
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Laplace Transform Approach to the Solution of
Homogeneous State Equation (2)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
State - Transition Matrix
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Example
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
Exercise 1
Find x1(t) , x2(t)
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
The initial condition
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Exercise 1 (Solution)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Solution of Nonhomogeneous State Equation
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Solution of Nonhomogeneous State Equation (2)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Solution of Nonhomogeneous State Equation (3)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Example 2
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
Exercise 2
Find x1(t) , x2(t)
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
The initial condition
Input is Unit Step
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Exercise 2 (Solution)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Matrix Exponential eAt
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
Matrix Exponential eAt
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Method 1 : Solution
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Example 3
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Example 3
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Matrix Exponential eAt
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Matrix Exponential eAt
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Example 4
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
Exercise 4
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
( โดยการหา state-transition matrix ใช้ ทงั ้ 3 วิธี)
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Exercise 4
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2
© AIRBUS UK LTD 2002. All rights reserved. Confidential and proprietary document.
Laplace Transform
2009 Subject Name Automotive Automatic Control
Month 200X
Page 2