1. Solution of Linear State- Space Equations

2. Outline Laplace solution of linear state-space equations. Leverrier algorithm. Systematic manipulation of matrices to obtain the solution. 2

3. Linear State-Space Equations 1. Laplace transform to obtain their solution x( t ). 2. Substitute in the output equation to obtain the output y( t ). 3

4. Laplace Transformation Multiplication by a scalar (each matrix entry). Integration (each matrix entry). 4

5. State Equation 5

6. Matrix Exponential 6

7. Zero-input Response 7

8. Zero-state Response 8

9. Solution of State Equation 9

10. State-transition Matrix LTI case φ ( t − t 0 ) = matrix exponential Zero-input response: multiply by state transition matrix to change the system state from x(0) to x(t). State-transition matrix for time-varying systems φ ( t, t 0 ) – Not a matrix exponential (in general). – Depends on initial & final time (not difference between them). 10

11. Output 11

12. Example 7.7 12 x 1 = angular position, x 2 = angular velocity x 3 = armature current. Find: a)The state transition matrix. b)The response due to an initial current of 10 mA. c)The response due to a unit step input. d)The response due to the initial condition of (b) together with the input of (c)

13. a) The State-transition Matrix 13

14. State-transition Matrix 14

15. Matrix Exponential 15

16. b) Response: initial current =10 mA. 16

17. c) Response due to unit step input. 17

18. Zero-state Response 18

19. d) Complete Solution 19

20. The Leverrier Algorithm 20

21. Algorithm 21

22. Remarks Operations available in hand-held calculators (matrix addition & multiplication, matrix scalar multiplication). Trace operation ( not available) can be easily 22 p ) y programmed using a single repetition loop. Initialization and backward iteration starts with: P n-2 = A + a n-1 I n a n-2 = − ½ tr{P n-2 A} 22

23. Partial Fraction Expansion 23

24. Resolvent Matrix 24

25. Example 7.8 Calculate the matrix exponential for the state matrix of Example 7.7 using the Leverrier algorithm. 25

26. Solution 26

27. (ii) k = 0 27

28. Check and Results 28

29. Partial Fraction Expansion 29

30. Constituent Matrices 30

31. Matrix Exponential 31

32. Properties of Constituent Matrices 32