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    Controllability of a hovercraft model (and two general results)

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    2003b_letter.pdf (437.7Kb)
    Date
    2003
    Author
    Lewis, Andrew D.
    Tyner, David R.
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    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.
    URI for this record
    http://hdl.handle.net/1974/60
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