Lecture 24. Assorted Topics

1. Lecture 24. Assorted Topics (We are sneaking up on control) DC motors and their behavior: a state space example The meaning of exp(At) Block diagrams The inverted pendulum on a cart (introduction) Cruise control

2. I want to take a brief segue into the subject of dc motors We are going to want to drive and control mechanisms and one way to do that is through motors You can also look at Friedland pp. 18-20

3. voltage in torque and rotation out

4. Simple dc motor equations torque rotation rate current back emf We will use, K1 = K = K2 input voltage - back emf = current xresistance (this assumes that i varies slowly enough that inductance is negligible)

5. We can do a little physics torque = rate of change of angular momentum plug in the motor torque The maximum torque occurs at zero speed The maximum speed occurs at zero torque starting torque no-load speed

6. Dynamics This is a first order system for the speed You may recall how to solve this in closed form (or you may not)

7. Note Rewrite Integrate

8. I left out the initial condition. I can add that in and get the final speed for any voltage. We have Put in a constant of integration Set T = 0 and divide by and we’ll have

9. If we care about the position of the shaft, not just its speed then we have a second order problem or

10. This is simpler than the suspension problem we just worked We can look at homogeneous and particular solutions as before We can find the eigenvalues almost by inspection

11. The eigenvectors are also easy The homogeneous solution

12. The particular solution can be found by an extension of the first order method recall and recall how we do that retrace those steps using matrices

13. Note Rewrite Integrate Put in the initial condition

14. We don’t need to use this fancy method for this problem at the level of what we are used to I’ll address that a bit when we’ve finished this problem Suppose that the input voltage is harmonic in line with the sorts of particular solutions we are used to

15. Invert Where

16. Expand Go back to the physical world

17. Take a look at the initial conditions Here’s the homogeneous solution in its vector form The vector form of the particular solution

18. Add the two and write the initial condition from which we can find a0 and a1 which I’m not actually going to do today

19. Note that we can modify all this to get a motor to drive a cart, which we’ll be doing as we go on (remember the overhead crane) force on the cart the motor torque wheel radius The dynamics

20. The left hand side is stable (its homogeneous solution decays) so a constant voltage leads to a constant speed.

21. QUESTIONS?

22. Go back to the weird formula The analysis is outlined in generality in Friedland §3.2, which we will look at later in the course The question that should leap to mind is: What is meant by I don’t want to make too much of this now we’ll tackle it at more length later, but this one is simple enough that we can pursue it a bit

23. can be defined by the Taylor series We can write our A in the form And write out the first few products

25. We can combine all of this to write We can recognize these series and write

26. QUESTIONS?

27. BLOCK DIAGRAMS Friedland uses a lot of these. Sometimes they are helpful sometimes they are not. He gives symbols for multiply, integrate and add We can see how this works by looking at the usual first order system block diagram

28. second order ode representation state space representation or simple block diagram representation - + a v y -

29. integrate v to get y “feedback” loops integrate the acceleration to get v - + a u y -

30. Here are some examples from Friedland

31. translated Friedland’s equations two inputs u two outputs x

32. All the lines in this figure represent vectors

33. The equations for that diagram are dynamics output We won’t be concerned much, if at all, with vector inputs We will not look at systems for which the output depends directly on the input — for us D = 0

34. QUESTIONS?

35. Our control efforts are going to be to find some feedback that will drive an error to zero Let’s take a look at setting up a couple of cases

36. Inverted pendulum on a cart which problem we will work much later m The errors we want to make vanish will be the angle q some cart position error q l u M y

37. Energies Constraints The Lagrangian Euler-Lagrange equations

38. Linearize Solve for the second derivatives Convert to state space

40. y output B input q output internal “feedback”

41. QUESTIONS?

42. CRUISE CONTROL desired speed INVERSE PLANT GOAL: SPEED nominal fuel flow Actual speed + + PLANT: DRIVE TRAIN Input: fuel flow - - error disturbance Feedback: fuel flow CONTROL

43. We have some open loop control — a guess as to the fuel flow We have some closed loop control — correct the fuel flow if the speed is wrong

44. disturbance simple first order model divide the force open loop part linearize and the goal is to make v’ go to zero

45. Let’s say a little about possible disturbances hills are probably the easiest to deal with analytically mgsinf f I’ll say more as we go on

46. The open loop picture + + f v 1/m -

47. I’m not in a position to simply ask f to cancel h(t) (because I don’t know what it is!) I want some feedback mechanism to give me more fuel when I am going too slow and less fuel when I am going too fast This system is already stable in the sense that the air drag acts in the same way as the artificial gain I don’t really need the primes anymore, so I will drop them

48. The closed loop picture The open loop picture + + - f v 1/m - control feedback

49. From the last lecture So we have Solution

50. We do not need this whole apparatus to get a sense of how this works Consider a hill, for which s(t) is constant, call it s0 We can find the particular solution by inspection The homogeneous solution decays, and we see that we have a permanent error in the speed The bigger K, the smaller the error, but we can’t make it go away (and K will be limited by physical considerations in any case)