In linear systems theory, the problem of constructing an observer \hat
x for the system
\align \dot x =&\ Ax + Bu\\ y =&\ Cx\endalign
reduces to the problem of finding a matrix P such that all the
eigenvalues of A - PC'C have negative real part.
This is achieved by choosing a non-negative number \sigma such that
the eigenvalues of M = \sigma I + A all have positive real part, and
then considering the positive definite matrix R given by
R = \int_0^{\infty}\exp(-M't)C'C\exp(-Mt)dt\ .
R satisfies the algebraic equation
M'R + RM = C'C \tag 1
which represents an n^2\times n^2 system \footnote{There are only
n(n+1)/2 unknowns, but it is structurally convenient to ignore the
symmetry of R.} of simultaneous linear equations.
The required matrix P is given by P = R^{-1}. P satisfies
PM' + MP = PC'CP\ ,\tag 2
so that an alternative procedure is to calculate P directly as the
limiting solution of a matrix Ricatti differential equation.
To show that P is appropriate, equation (2) is rewritten as
P(M' - C'CP) + (M - PC'C)P = -PC'CP \tag 3
which is the standard form for Liapunov stability theory.
Since P is positive definite, and the right-hand side of (3) is at
least negative semi-definite, the eigenvalues of
M - PC'C = \sigma I + A - PC'C
have real parts \le 0, and it is easy to show that they are in
fact strictly < 0. Hence the eigenvalues of A - PC'C have real parts
< -\sigma.
The above is the standard analysis of this problem. Because of the difficulty in computing `nice' answers, illustrations, if provided, are restricted to very simple 2\times 2 cases. However, since matlab is now available on our system, I decided to try a slightly more ambitious example.
The matrix A was chosen as
A = \pmatrix \format\ \r &\ \r &\ \r &\ \r \\
1 & -1 & 0 & -1\\6 & 6 & 6 & 5\\ -1 & 1 & 2 & 0\\-7 & -7
& -9 & -5 \endpmatrix
whose eigenvalues were known to be -1, 1, 2, 2.
The matrix C was chosen rather arbitrarily as
C = \pmatrix 2 & 2 & -1 & -3\\1 & 2 & -1 & -2\endpmatrix\ .
Using these values and \sigma = 2, matlab calculated R as
R = \pmatrix 63.1670 & 55.3435 & 67.0518 & 64.5729\\
55.3435 & 49.0015 & 57.9966 & 55.8093\\
67.0518 & 57.9966 & 73.3071 & 71.1342\\
64.5729 & 55.8093 & 71.1342 & 69.3246 \endpmatrix
the sort of result which would normally lead to abandonment of further
calculation.
Nothing daunted, matlab proceded to produce
P \left(= R^{-1}\right) = \pmatrix \format\ \r &\ \r &\ \r &\ \r \\
19.1273 & -12.2417 & -19.7175 & 12.2711\\
-12.2417 & 8.2543 & 11.6502 & -7.1968\\
-19.7175 & 11.6502 & 25.7302 & -17.4147\\
12.2711 & -7.1968 & -17.4147 & 12.2475 \endpmatrix\ ,
and
A - PC'C = \pmatrix \format\ \r &\ \r &\ \r &\ \r \\
17.8304 & 26.0113 & -13.5057 & -31.3360\\
-4.9403 & -11.9504 & 14.9752 & 24.9155\\
-34.4415 & -45.1238 & 25.0619 & 56.5033\\
20.5604 & 29.7630 & -27.3815 & -50.9419 \endpmatrix\ .
As a check, I asked the computer to calculate the eigenvalues of this final matrix and was surprised to receive the answer -6, -6, -5, -3, a result described by a colleague as `inherently unlikely'.
Given this result, I hypothesised that
\text eig (A - PC'C) = -2\sigma - \text eig (A)\ ,
or equivalently
The proof of this result follows immediately from equation (3) which can
be rewritten as
\gather P(M - PC'C)'P^{-1} = -M \\ \sigma I + P(A - PC'C)'P^{-1} =
-\sigma I - A\\
P(A - PC'C)'P^{-1} = -2\sigma I - A\ . \endgather
Since a matrix is similar to its transpose, the result is proved.
A far as I know, this result is new. Had it been known, it should have supplanted the standard proof outlined above, since it provides a sharper result with less effort. I can only ascribe its prior non-discovery to the fact that no-one had done the sort of calculation described above.
Alan Jones
Department of Mathematics
The University of Queensland
In a previous issue of the Gazette , E.~Tuck~[2] raised a couple of interesting questions about the quadratic functional
I(f)=\int_{-1}^1 f(x)\pvint_{-1}^1{f(y)\over x-y}dydx,
where PV indicates that the integral with respect to~y is a Cauchy principal value. We will show that
I(f)=0\quad\hbox{for all f\in L_2(-1,1),} \eqno(1)
as one would expect from the skew symmetry of the kernel~1/(x-y). However, the example in~[2] demonstrates that I(f)\ne0 is possible if f is not square-integrable on the interval~(-1,1). In fact, let
v(x)=\sqrt{1-x\over 1+x}and w(x)=\sqrt{1+x\over 1-x} for -1 < x < 1.\eqno(2)
We will show that if f has the form
f(x)=g(x)+{a\over\sqrt{2}}v(x)+{b\over\sqrt2}w(x) for -1 < x < 1,} \eqno(3)
where g is any function satisfying g\in L_p(-1,1) for some~p>2, then
I(f)={\pi^2\over2}(a^2-b^2). \eqno(4)
The representation~(3) means that when g is (say) bounded,
f(x)\sim{a\over\sqrt{1+x}}\quad\hbox{as x\downarrow-1,}
and
f(x)\sim{b\over\sqrt{1-x}}\quad\hbox{as x\uparrow1.}
The foregoing claims are consequences of some standard properties of the truncated Hilbert transform:
Tf(x)=\pvint_{-1}^1{f(y)\over x-y}dy\quad\hbox{for -1< x< 1.}
If 1< p< \infty and 1/p+1/p'=1, then T:L_p(-1,1)\to L_p(-1,1) is a bounded linear operator, and
\pair{Tf,g}=-\pair{f,Tg}\quad \hbox{for f\in L_p(-1,1) and g\in L_{p'}(-1,1),}
where \pair{f,g}=\int_{-1}^1 f(x)g(x)dx. Both of these facts follow immediately from the corresponding properties of the Hilbert transform on the whole real axis; see Tricomi~[1, \S4.3].
Thus, if f\in L_2(-1,1), then I(f)=\pair{f,Tf}=-\pair{Tf,f}=-I(f), proving~(1).
If g\in L_p(-1,1) and 2< p < \infty, then the two functions in~(2) belong to~L_{p'}(-1,1) because 1 < p' < 2, so \pair{g,Tv}=-\pair{Tg,v} and \pair{g,Tw}=-\pair{Tg,w}. Moreover, as a special case of~(5) below,
Tv(x)=\pi\and Tw(x)=-\pi\quad\hbox{for -1 < x < 1,}
and then because
\int_{-1}^1 v(x)dx=\pi=\int_{-1}^1 w(x)dx,
we see that \pair{v,Tw}=-\pair{Tv,w}. Thus, if f has the form~(3), then
\eqalign{ I(f)&=\biggl\langle g+{av\over\sqrt2}+{bw\over\sqrt2}, T\biggl(g+{av\over\sqrt2}+{bw\over\sqrt2}\biggr)\biggr\rangle\cr &=I(g)+{a^2\over2}I(v)+{b^2\over2}I(w) =0+{a^2\pi^2\over2}-{b^2\pi^2\over2},\cr}
proving (4).
It is not possible to allow stronger singularities in~(3), as one sees by considering the function
u_\alpha(x)=\biggl({1-x\over1+x}\biggr)^\alpha\quad\hbox{for -1 < x < 1,}
where -1 < \alpha < 1. If |\alpha|<1/2, then I(u_\alpha)=0 by~(1). If 1/2 < |\alpha| < 1, then the outer integral in~I(u_\alpha) is divergent because
Tu_\alpha(x)=\pi\csc(\alpha\pi)-\pi\cot(\alpha\pi) \biggl({1-x\over1+x}\biggr)^\alpha \quad\hbox{for -1 < x < 1.} \eqno(5)
Exercise: derive (5) by applying Cauchy's integral formula and the Sokhotski-Plemelj formulae to the function
hi(z)=\biggl({z-1\over z+1}\biggr)^\alpha-1 \quad\hbox{for z\notin[-1,1].}
Bill McLean
School of Mathematics
The University of New South Wales
Professor van der Poorten has suggested to our President of the Technion Society of Australia --- NSW Chapter Professor Graham de Vahl Davis, that the Australian Mathematical Society may be interested in receiving notification about the recipients of the Kurt Mahler Mathematics Prize which is awarded annually at the Technion - Israel Institute of Technology, Haifa, Israel.
Hence I am pleased to announce that the Kurt Mahler Mathematics Prize for 1995 was recently awarded to Professor Jack Sivan and Dr Eli Alchadaf from the Faculty of Mathematics at the Technion. Prof. Sivan and Dr Alchadaf received the award for their works entitled ``On the Projective Schur Group of a Field'', and, ``Projective Schur Algebras Have Abelian Splitting Fields''.
The Mahler Prize was established in 1973 with a fund donated by the Australian number theorist, Professor Kurt Mahler, for the support of research in Pure and Applied Mathematics at the Technion. This award is made annually to a member or members of the Technion's Faculty of Mathematics by a committee chaired by the Institute's Vice President for Research. The committee also includes the Dean of the Faculty of Mathematics, a senior faculty member of the Mathematics Faculty and a recognized expert in mathematics from a non-Technion body appointed by the committee chairman.
The Technion, established in 1924, is Israel's oldest university and its only institute of higher learning devoted fully to the education of engineers, applied scientists, and physicians. Situated in the scenic Carmel Mountain Range, Technion City overlooks the Haifa Bay and the lower Galilee. During the 1994-95 academic year, it has an enrolment of 10,600 students, 8,000 undergraduates and 2,600 graduates, studying in its 19 faculties and departments and several dozen research institutes. The faculty of more than 700 teaches a curriculum covering the full spectrum of engineering and physical science, and medicine.
I hope that you will share this information with your colleagues and in this way help us to continue the important tradition of this Prize, which meant so much to Kurt Mahler.
Caryn Yaacov
Division of Public Affairs and Resource Development
Technion - Israel Institute of Technology