LMIs in Control/Matrix and LMI Properties and Tools/Small Gain Theorem
The Small Gain Theorem provides a sufficient condition for the stability of a feedback connection.
Suppose
is a Banach Algebra and
. If
, then
exists, and furthermore,
Assuming we have an interconnected system
:
and,
The above equations can be represented in matrix form as
Making
the subject, we then have:
If
is well-behaved, then the interconnection is stable.
For
to be well-behaved,
must be finite.
Hence, we have
and
for the higher exponents of
to converge to
If
, then this implies stability, since the higher exponents of
in the summation of
will converge to
, instead of blowing up to infinity.
A list of references documenting and validating the LMI.