Algebraic design techniques for reliable stabilization

In this paper we study two problems in feedback stabilization. The first is the simultaneous stabilization problem, which can be stated as follows. Given plants
G_{0}, G_{1},..., G_{l}
, does there exist a single compensator
C
that stabilizes all of them? The second is that of stabilization by a stable compensator, or more generally, a "least unstable" compensator. Given a plant
G
, we would like to know whether or not there exists a stable compensator
C
that stabilizes
G
; if not, what is the smallest number of right half-place poles (counted according to their McMillan degree) that any stabilizing compensator must have? We show that the two problems are equivalent in the following sense. The problem of simultaneously stabilizing
l + 1
plants can be reduced to the problem of simultaneously stabilizing
l
plants using a stable compensator, which in turn can be stated as the following purely algebraic problem. Given
2l
matrices
A_{1}, ..., A_{l}, B_{1}, ..., B_{l}
, where
A_{i}, B_{i}
are right-coprime for all
i
, does there exist a matrix
M
such that
A_{i} + MB_{i}
, is unimodular for all
i?
Conversely, the problem of simultaneously stabilizing
l
plants using a stable compensator can be formulated as one of simultaneously stabilizing
l + 1
plants. The problem of determining whether or not there exists an
M
such that
A + BM
is unimodular, given a right-coprime pair (
A, B
), turns out to be a special case of a question concerning a matrix division algorithm in a proper Euclidean domain. We give an answer to this question, and we believe this result might be of some independent interest. We show that, given two
n \times m
plants
G_{0} and G_{1}
we can generically stabilize them simultaneously provided either
n
or
m
is greater than one. In contrast, simultaneous stabilizability, of two single-input-single-output plants, g
0
and g
1
, is not generic.