State Variables and System Dynamics | Lecture 9

1. Lecture 9

2. State vector a listing of state variables in vector form Eastern Mediterranean University

3. State equations System dynamics State vector Input vector Measurement Read-out map Output vector Eastern Mediterranean University

4. x:n-vector (state vector) u:p-vector (input vector) y:m-vector (output vector) n A:nxn System matrix n p B:nxp Input (distribution) matrix n n C:mxn Output matrix m p D:mxp Direct-transmission matrix m Eastern Mediterranean University

5. Forced sol’n & Solution of state eq’ns Consists of: Free response (Homogenous sol’n) (particular sol’n) Eastern Mediterranean University

6. Homogenous solution Homogenous equation has the solution State transition matrix X(0) Eastern Mediterranean University

7. State transition matrix An nxn matrix (t), satisfying Eastern Mediterranean University

8. Determination of (t):transform method Laplace transform of the differential equation: Eastern Mediterranean University

9. Determination of (t):transform method Eastern Mediterranean University

10. Determination of (t):time-domain solution Scalar case  where Eastern Mediterranean University

11. Determination of (t):time-domain solution For vector case, by analogy  where Can be verified by substitution. Eastern Mediterranean University

12. t0 t1 t2 Properties of TM (0)=I Φ(t) Φ(-t) -1(t)= (-t) Φ(t2-t0) Φ(t1-t0) Φ(t2-t1) Ф(t2-t1)Φ(t1-t0)= Φ(t2-t0) Φ(t) Φ(kt) Φ(t) Φ(t) Φ(t) Φ(t) Φ(t) [Φ(t)]k= Φ(kt) Eastern Mediterranean University

13. General solution Scalar case Eastern Mediterranean University

14. General solution Vector case Eastern Mediterranean University

15. General solution: transform method L{ }   Eastern Mediterranean University

16. Inverse Laplace transform yields: Eastern Mediterranean University

17. For initial time at t=t0 Eastern Mediterranean University

18. Zero-input response Zero-state response The output y(t)=Cx(t)+Du(t) Eastern Mediterranean University

19. Example • Obtain the state transition matrix (t) of the following system. Obtain also the inverse of the state transition matrix -1(t) . For this system the state transition matrix (t) is given by since Eastern Mediterranean University

20. Example The inverse (sI-A) is given by Hence Noting that -1(t)= (-t), we obtain the inverse of transition matrix as: Eastern Mediterranean University

21. Exercise 1 Find x1(t) , x2(t) The initial condition Eastern Mediterranean University

22. Exercise 1 (Solution) Eastern Mediterranean University

23. Example 2 Eastern Mediterranean University

24. Exercise 2 Find x1(t) , x2(t) The initial condition Input is Unit Step Eastern Mediterranean University

25. Exercise 2 (Solution) Eastern Mediterranean University

26. Matrix Exponential eAt Eastern Mediterranean University

27. Matrix Exponential eAt Eastern Mediterranean University

28. The transformationwhere 1,2,…,n are distinct eigenvalues of A. This transformation will transform P-1AP into the diagonal matrix Eastern Mediterranean University

29. Example 3 Eastern Mediterranean University

30. Method 2: Eastern Mediterranean University

31. Matrix Exponential eAt Eastern Mediterranean University

32. Matrix Exponential eAt Eastern Mediterranean University

33. Example 4 Eastern Mediterranean University

34. Laplace Transform Eastern Mediterranean University

35. Eastern Mediterranean University

36. Eastern Mediterranean University