1. 1 Lecture 15: Stability and Control III — Control Philosophy of control: closed loop with feedback Ad hoc control thoughts Controllability Three link robot control Holonomic systems with N degrees of freedom and K distinct inputs

2. 2 PLANT CONTROL GOAL INVERSE PLANT + - + - Ideal output input feedback Actual output error Mechanical/electromechanical system disturbance + K dimensional vector

3. 3 SIMULATION LINEAR CONTROL GOAL EQUILIBRIUM + - + - Ideal output input feedback Actual output error disturbance + Q0Q0

4. 4 LINEAR CONTROL + - + - Ideal output input feedback Actual output error disturbance + Q0Q0

5. 5 Let’s think about control in the context of the simple inverted pendulum  add a small, variable torque at the pivot

6. 6 There’s a change of sign from the simple pendulum from last time because I have chosen a different definition of  We have equilibrium at  = 0, and Q = 0 there as well. We know that this will be unstable if it is perturbed with Q remaining zero Let’s see how this goes in a state space representation Hamilton’s equations

7. 7 (I’ve put in the  because Q is zero at equilibrium)

8. 8 If q starts to increase, we feel intuitively that we ought to add a torque to cancel it We can expand the feedback term

9. 9 multiply the column vector and the row vector combine the forced system into a single homogeneous system

10. 10 The characteristic polynomial for this new problem can be solved for and so I can make s 2 negative by applying some gain g 1. So this very simple feedback can make an unstable system marginally stable We can do better...

11. 11 Suppose we feed back the speed of the pendulum as well as its position?

12. 12 And now the characteristic polynomial comes from Combining everything again we get

13. 13 We can adjust this to get any real and imaginary parts we want If you are familiar with the idea of a natural frequency and a damping ratio then you might like to set the control problem up in that language The linear term is the key — the feedback from the derivative

14. 14 The real part is always negative. If  is less than unity, there is an imaginary part. If  equals unity the system is said to be critically damped can be made the same as the one degree of freedom mass-spring equation by setting giving

15. 15 ??

16. 16 This suggests a bunch of questions Is this generalizable to more complicated systems? Is there a nice ritual one can always employ? Is this always possible? Will the linear control control the nonlinear system? How much of this does it make sense to include in this course? YES SOMETIMES NO SOMETIMES ??

17. 17 The question of possibility is really important so I’m going to address that as soon as I can develop some more notation The general perturbation problem for control will be In the ad hoc example we worked, we used Q, the generalized force, as input We cannot do that in general, so I have put in f E as the actual external force vector

18. 18 For a single input system like the one we just saw B will be a column vector and f E a scalar and the equation is We’ll eventually see how we get from Euler-Lagrange to state space including B

19. 19 We want f E (or f E for one input) to be proportional to x We see that G has as many rows as there are inputs and as many columns as there are state variables G is a row vector for single input systems the minus sign is conventional

20. 20 Rename dummy indices (j —> m) to make it possible to combine terms We have for the single input case Our control characteristic polynomial will come from and the question is: is it always possible to find G such that the roots are where we want them?

21. 21 There are always at least as many gains as there are roots, so you’d think so But it isn’t. The controllability criterion, which I will state without proof, is that the rank of must be equal to the number of variables in the state There are as many terms in W as there are variables in the state

22. 22 W has as many rows as there are variables. The number of columns in W is equal to the number of variables times the number of inputs In the single input case W is a square matrix AND there is a nice simple way to figure out what the gains must be for stability We are not going to explore this — we haven’t the time — and it is covered in most decent books on control theory We can get by with guided intuition.

23. 23 ??

24. 24 Single input systems are much simpler than multi-input systems but we have need of multi-input systems frequently I will outline the intuitive approach to multi-input systems which works best (at least for me) through the Euler-Lagrange equations (This may be a bit hard to follow; we’ll do an example shortly.) We are working on the yellow box in the block diagram

25. 25 Euler-Lagrange equations which we can rewrite

26. 26 For a steady equilibrium, which is what we are learning how to do perturbation We can drop this term because of the  2.

27. 27 and we need to perturb the gradient of the Lagrangian to finish the linearization or

28. 28 We can use our old method of converting to first order odes on this and decide controllability (before we knock ourselves out trying to control it) The state vector is and the A matrix is straightforward

29. 29 The meaning of B is not immediately clear B is a matrix connecting the actual inputs to the system It is not the connection between the generalized forces and the system We get that by going back and remembering how the generalized forces are defined

30. 30 Recall The vector forces and torques are made of components and we arrange the components into the vector f E The velocities and rotation rates are linear functions of the The terms inare linear in the components of f E We can combine all this to write

31. 31 The coefficients will depend on the individual problem but in any case we will have We can substitute this into the momentum equation

32. 32 Now we can go back to the state picture with a better idea of what A and B are state vector state equations Explicit appearance of the actual forces in the differential equations

33. 33 and the B matrix is The A matrix is

34. 34 Can we make sense out of the whole thing — how do we do a problem?! Suppose we are given a mechanism and a task for it to do Consider only the simple task of going to some configuration and stopping there We can treat the final configuration as our equilibrium and design a control that will bring us to that equilibrium We want to make

35. 35 all the holonomic constraints generalized coordinates Lagrangian generalized forces Euler-Lagrange trackHamilton track goal equilibrium forceslinearization control design simulation (nonlinear) Start from square one using what we know so far

36. 36 LINEAR CONTROL + - + - Ideal output input feedback Actual output error disturbance + Q0Q0 Euler-Lagrange Hamilton

37. 37 ??