1. 1 Analyze the non-autonomous systems by adapting (extending) the tools we developed for autonomous systems.

2. 2 time (t) t 0 =0t 0 =3 Response not affected by t 0 x(t) versus time (t) t 0 =6t 0 =9 x(t 0 )=3 t 0 =3 t 0 =6 t 0 =9 t 0 =0 x(t 0 )=3 x(t) versus time (t) time (t) x(t) Response depends on t 0 Autonomous vs. Nonautonomous Autonomous Nonautonomous

3. 3 Motivation: The Tracking Problem Up until now we have only talked about regulation (stabilizing the system to an equilibrium point). That problem yielded an autonomous system even with state feedback. Looking at the closed-loop system we see that it is non-autonomous. Do the tools for autonomous systems support this result??? Based on what we already know a solution may look like this

4. 4 Difference from autonomous: added this constraint on the initial condition t 0

5. 5 Evolution of the state from same initial state but different starting times t 01 (Modified from Chapter 3)

6. 6 Difference from autonomous: added this constraint on the initial condition t 0

7. 7 Bound on initial condition is independent of the initial time. Independent of t 0

8. 8

9. 9

10. 10

11. 11

12. 12

13. 13 1 (cont)

14. 14

15. 15

16. 16

17. 17

18. 18

19. 19

20. 20

21. 21

22. 22

23. 23

24. 24 ?

25. 25

26. 26

27. 27 Example 1 of Perturbation Analysis

28. 28 Example 1 of Perturbation Analysis (cont)

29. 29 Example 1 of Perturbation Analysis (cont) Perturbed system Unperturbed system Time

30. 30 Example 2 of Perturbation Analysis +0 g(x,t)

31. 31 Example 2 of Perturbation Analysis (cont)

32. 32 Example 2 of Perturbation Analysis (cont)

33. 33

34. 34 Discrete Cont.

35. 35 x(t) t Example: Digital Motor Controller Motor Encoder (A/D) Micro Controller (control algorithm) D/A + Amplifier Normally we make the assumption that if the sample rate is high then it acts like a continuous system. voltage position

36. 36 Approximation of slope at t=kT System at kT

37. 37

38. 38

39. 39

40. 40

41. 41

42. 42

43. 43 1 Example 7 (cont) Original System

44. 44 Example 7 (cont)

45. 45 Extra – Barbalat’s Lemma (Slotine page 124-128)

46. 46

47. Conclusion Have extended idea of Lyapunov function analysis to nonautonomous systems. Perturbation analysis examined the effect of disturbances (of a specific form). Have extended idea of Lyapunov function analysis to discrete time systems. Extended the Lyapunov analysis using Barbalat’s Lemma to make a statements about the derivative of the Lyapunov function.

48. Homework 4.1 Marquez Problems 4.4, 4.5

49. Homework 4.1 (sol) Doesn’t exist

50. Homework 4.1 (sol)

53. 53 Homework 4.2

54. 54 Homework 4.2 (sol)

55. 55 Homework 4.2 (sol)