## 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.