Presentation is loading. Please wait.

Presentation is loading. Please wait.

Modern Control Systems1 Lecture 05 Analysis (I) Time Response and State Transition Matrix 5.1 State Transition Matrix 5.2 Modal decomposition --Diagonalization.

Similar presentations


Presentation on theme: "Modern Control Systems1 Lecture 05 Analysis (I) Time Response and State Transition Matrix 5.1 State Transition Matrix 5.2 Modal decomposition --Diagonalization."— Presentation transcript:

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:


Download ppt "Modern Control Systems1 Lecture 05 Analysis (I) Time Response and State Transition Matrix 5.1 State Transition Matrix 5.2 Modal decomposition --Diagonalization."

Similar presentations


Ads by Google