A homotopy method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem. Given an interconnected system
M
composed of
N
subsystems of the form
A_{i} + B_{i}K_{i}
,
i = 1,..., N , (A_{i}, B_{i})
, a controllable pair, and where the off diagonal blocks of
M
lie in the image of the appropriate B
i
, then is it possible to arbitrarily assign the characteristic polynomial of
M
by a suitable selection of the characteristic polynomials of
A_{i} + B_{i}K_{i}
? Moreover, is it possible to compute the appropriate characteristic polynomials of the
A_{i} + B_{i}K_{i}
(or equivalently construct the K
i
) needed to do so? The first question is answered by constructing a mapping
F: R^{n} \rightarrow R^{n}
which maps a prescribed set of
n
of the feedback gains (elements of
K_{i}, i=1,...,N
) to the
n
coefficients of the characteristic polynomial of
M
. The question then becomes, given a
p \in R^{n}
, does
F(x) = p
have a solution? The answer is found by constructing a homotopy
H: R^{n}x[O.1] \rightarrow R^{n}
where
H(x,1)= F(x)
and
H(x,0)
is some "trivial" function. Degree theory is then applied to guarantee that there exists an
x(t)
such that
H(x(t), t) = p
for all
t
in [0,1]. The parameterized Sard's theorem is then utilized to prove that (with probability 1)
x(t)
is a "smooth" curve, and hence can be followed numerically from
x(0)
to
x(1)
by the solution of a differential equation (Davidenko's method).