Controllability of a hovercraft model (and two general results)

Abstract

Modelling and controllability studies of a hovercraft system are undertaken. The system studied is a little more complicated than some in the literature in that the inertial dynamics of the thrust fan are taken into account. The system is shown to be representative of a large class of systems that are controllable only a set described by the zeros of a nontrivial analytic function. Recent results for controllability using vector-valued quadratic forms are useful in arriving at the stated conclusions. As part of the development, two new controllability results of independent interest are proved.

This paper was rejected for CDC'03 because of an erroneous review. We did, and still do, think this is an interesting paper. Therefore, to maintain the integrity of this online version, I am putting the review in question, and our response, online. No malice towards the reviewer is intended; we merely wish to reaffirm these results. Our response is in italics.

This was an interesting paper that I spent a lot of time on. It has many drawbacks in terms of explanations, typos, etc. but was interesting and at first blush appeared to be a nice use of the modern results of Lewis, Bullo, Leonard, Murray, and others. The claimed result, however, is counter-intuitive and caused me to dive into the proofs.

We are happy the reviewer found the results counter-intuitive. Indeed, one's intuition can be difficult to trust in these sorts of problems.

The primary problem, and the reason I had to reject the paper, is that the proof of the main result- Thm. 4.3- is incorrect. There are problems throughout the proof but the main issue is as follows. It is established that B_Y_q0 is NEVER definite- this is a fairly easy thing to show (although I have problems with how it is shown in the proof but the basic idea is right). In the final part of the proof is the line:

``It remains to show that when B_Y_q is indefinite only on a proper analytic subset S, then from q0 in Q/S the system is not STLCC. This, however, follows from Thm. 4.2.''

Unfortunately Thm 4.2 states that the system is not STLCC if B_Y_q is definite. Since we've aready shown B_Y_q is not definite everywhere that thm does not apply. Thiis invalidates the thm and prop. 4.7.

This completely misses the point of Theorem 4.2 which says that $B_{Y_q}$ need only be semidefinite in order to conclude lack of controllability, provided that the semidefiniteness holds, with a fixed rank, in a neighbourhood of $q$.

Also, in the cases for the two linear maps: what do you mean by dim(ker(L11(1))) = n-1? L11 maps R^(n-2) to R and thus the domain is only n-2 dim'l. How is the dimension of the kernel larger than that of the domain?

This is a typo. We were replacing in our mind $n-2$ with $n$.

Other problems throughout the paper to take a look at:

sec. 2.2, right at the start "...we note that Q=SE(2)xSO(2) is a principal fibre bundle..." No, it isn't- not until you give me the base space. I know it's obvious but the technical details should be correct. It is written correctly later on (pi:Q -> SO(2))

This is correct, in some sense. We might have said ``$Q=SE(2)\times SO(2)$ is the total space of a principal fibre bundle.'' However, the reviewer left off the remainder of the sentence which says what the group action is. This diminishes the technical offence to close to nothing, we think.

sec. 2.2, proof of Lemma 2.1 You state that G-invariance is given by DPhi(a,phi)'[g(a,phi)]DPhi(a,phi) = [g(abar.a,phi)]

This is incorrect- the inner product matrix on the LHS should be evaluated at the translated point and on the RHS at the original point. Perhaps a better expression would be g(Phi(a,phi))(TPhi.X,TPhi.Y) = g(a,phi)(X,Y) for all X,Y in TqQ The same error is made in the proof of Lemma 2.2. Note that the Lemmas are correct.

This is quite right. Another typo.