Lectures 35-37 Tracking

1. Lectures 35-37 Tracking 35. Some review and clarifications Fundamentals of tracking (§6.4) Example(s) 36. More review: problem set 9 37. More on tracking. The overhead crane

2. Lecture 35: Some review and clarification, and the essence of tracking What does state space mean? What does T do? What are gains?

3. review The state vector describes the state of the system. Figuratively speaking, this tells us the position and velocity of every element The state equations then tell us how each element of the state evolves in time We can think of state space as having k “axes” that serve as a basis for the state vector The transformation between x and z spaces is simply a rotation of the sets of axes — the bases of the x and z systems The x and z vectors represent the same state viewed from different systems of axes That’s why the eigenvalues do not change

4. review review The x space is given to us by the physics, and our choice of state vector We can find various z spaces to serve our purposes The diagonal transformation uses the eigenvectors as a basis which is why we need k independent eigenvectors for that to work Remember that T-1 has the eigenvectors for its columns The companion transformation has basis vectors in its T-1, which is where the controllability condition comes from (not obvious!)

5. review Bottom line: The state space is a way of representing the state of a system and we can pick what we like — limited by the physics!

6. review What are gains? We agree that we are using feedback control We feed back the output in an attempt to control it. The gains are the amount of each element of the output that we feed back. They can be assembled in a row vector, which transform when the coordinates transform which is why we have different gains in z space and x space

7. On to tracking The general idea We have a single input system and we want some reference state We can split x into two parts: the reference state and an error and then we can ask that e go to zero

8. The reference state is not totally arbitrary; it must be consistent with the physics. Friedland tackles the reference state and disturbances at the same time I’m going to tackle them separately, looking at tracking for the nonce We apply what we had on the previous slide:

9. looks like a simple problem with some extra forcing terms terms that we in some sense know, because we’ve picked xr Find some ur such that We want to get rid of the time derivative of xr for reasons that will become clear (This is not an absolute necessity.) Find some Arsuch that Two options present themselves to us: The latter is the approach taken in the book and that is what we will do this week. The nice thing about it is that we can fix the problem for a class of reference states

10. with that choice we can write We compare with Friedland equation (6.37) The difference is that Friedland includes a disturbance, which I am not doing this week When we follow Friedland’s discussion, we can use his E and x0 but remember that we mean A – Ar and xr, respectively

11. What about Ar? Clearly the class of constant reference states has Ar = 0 which is the case for the example he works Suppose we have a pair of second order equations with the state If we are to look for harmonic reference states

13. Back to the main stream We want e to go to zero We know how to make e go to zero in the absence of tracking We simply check for controllability, and then find relevant gains for a full state feedback control u = Gxe= Ge We can extend this idea by choosing u to depend on e and xr

14. We have to do the G part first Then we can move on to the Gr part

15. As it happens we cannot always make this give us a complete vector zero for e We settle for: The closed loop system must be asymptotically stable Some components of e can be held at zero The eigenvalues have negative real parts/lie in the left half plane. The closed loop system means the system without xr The following argument is not the be all and end all — sometimes we can do better, but let’s look at it anyway

16. This is the argument I was hoping you could decode for PS 9

17. We can ask that the derivative of the error goes to zero, from which is stable, hence invertible and we can write the latter form being equivalent to equation 6.40 If xr is as big as the space, with k components, then e has kcomponents There are k components of Gr but that turns out not to mean that I can fix this We’ll see this best in terms of examples.

18. We can make some of the error vanish — the output, as defined by C We get the nicest result when the output has as many elements as the input In our case we want to look at single input-single output (SISO) systems put in the matrix dimensions to help us understand The left hand side is a scalar — that part of e that we make disappear

19. rearrange we want this to work for any xr

20. C is not a square matrix, so it doesn’t have an inverse; we have to do the whole thing at once which is equivalent to equation 6.45 (the final A – Ar) is equivalent to E in Friedland Again , let’s put in the dimensions for a SISO system This term is a scalar, so its inverse is particularly simple

21. Example 6E

22. This is a really complicated problem, and it isn’t explained well I will defer discussion of this to another day Let’s look at 6F (which starts with 4F) instead.

23. We are going to have a desired (reference) normal acceleration which we will compare to the actual normal acceleration in terms of an error We can write the normal acceleration in term of the angle a

24. He assumes that the command signal is slowly varying, so that its time derivative can be neglected He keeps all the rest of this, and the algebra gets very intense.

25. He eventually winds up with a third order system for the state vector with the single input u, and the equations, after some algebra are control/ disturbance

26. The first thing he needs to do is see about the undisturbed system He moves the poles to add damping and keep a quick response time He does this using the Bass-Gura procedure, which is equivalent to what we do, though we haven’t looked at it I will do it our way, and get slightly different results because I don’t quite match what he is doing. We can discuss this another time.

27. The next thing he does is to apply the new stuff — Find a G0 to cancel errors. If we follow him we’ll get the same result We can now go to Mathematica and look at this problem