Download presentation

Presentation is loading. Please wait.

Published byKiara Wingate Modified about 1 year ago

1
Modern Control Systems1 Lecture 05 Analysis (I) Time Response and State Transition Matrix 5.1 State Transition Matrix 5.2 Modal decomposition --Diagonalization 5.3 Cayley-Hamilton Theorem

2
Modern Control Systems2 1.Homogeneous solution of x(t) 2.Non-homogeneous solution of x(t) The behavior of x(t) et y(t) :

3
Modern Control Systems3 Homogeneous solution State transition matrix

4
Modern Control Systems4 Properties

5
Modern Control Systems5 Non-homogeneous solution Convolution Homogeneous

6
Modern Control Systems6 Zero-input responseZero-state response

7
Modern Control Systems7 Example 1 Ans:

8
Modern Control Systems8 Using Maison’s gain formula

9
Modern Control Systems9 How to findState transition matrix Methode 1: Methode 3: Cayley-Hamilton Theorem Methode 2:

10
Modern Control Systems10 Methode 1:

11
Modern Control Systems11 Method 2: Diagonalization diagonal matrix Example 4.5

12
Modern Control Systems12 Eigenvalue of A: Coordinate transformation matrix are independent. Then eigenvectors, Assume that all the eigenvalues of A are distinct, i.e. Diagonalization via Coordinate Transformation Plant:

13
Modern Control Systems13 where

14
Modern Control Systems14 Hence, system asy. stable ⇔ all the eigenvales of A lie in LHP New coordinate: Solution of (4.1): (4.1) The above expansion of x(t) is called modal decomposition.

15
Modern Control Systems15 Example Find eigenvector

16
Modern Control Systems16 Solution of (4.1): (4.2)

17
Modern Control Systems17 In the case of A matrix is phase-variable form and Vandermonde matrix for phase-variable form

18
Modern Control Systems18 Example: depend

19
Modern Control Systems19

20
Modern Control Systems20 Case 3:Jordan form Generalized eigenvectors

21
Modern Control Systems21 Example:

22
Modern Control Systems22 Method 3: Cayley-Hamilton Theorem Theorem: Every square matrix satisfies its char. equation. Given a square matrix A,. Let f (λ) be the char. polynomial of A. Char. Equation: By Caley-Hamilton Theorem

23
Modern Control Systems23 any

24
Modern Control Systems24 Example:

25
Modern Control Systems25 Example:

26
Modern Control Systems26

27
Modern Control Systems27

28
Modern Control Systems28

29
Modern Control Systems29

30
Modern Control Systems30

31
Modern Control Systems31

32
Modern Control Systems32 Example:

33
Modern Control Systems33 Example:

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google