Notes on the Computation of the Incremental Gain of Linear Systems
with Saturation
To be presented at the 34th CDC (in a shorter 2 page version).
B.G. Romanchuk
Abstract
Conditions for the non-existence of the incremental l_p gain
of single input discrete time linear systems with saturation is
presented, as well as algebraic tests for bounding the induced l_2 and L_2
gains. These algebraic tests are found to be the same as those which would arise
if one were to conservatively use a quadratic dissipation function to bound
the induced 2-norm.
Introduction
In this paper, the incremental gain properties of a class of single
input linear systems with saturation are examined. The main
justification of the study of the incremental gain is that it allows
some stronger conclusions than can be drawn from the use of the induced
norm in the analysis of nonlinear systems.
Unfortunately, it appears that having a finite incremental gain is
too strong a condition for analysis, a fact which has been noted in
the recent paper by Chitour, Liu and Sontag. That paper
examines not only finite incremental gain but also more general
continuity and differentiability properties of a specific class of
systems (the plant must have a neutrally stable dynamics matrix) in
continuous time. This present work amplifies those conclusions, as it is
shown that the incremental gain does not exist for open loop unstable
discrete time
plants (even over some bounded signal spaces for which the induced norm exists),
and an algebraic technique for bounding the l_2 and L_2 incremental gain
for some subset of the open loop stable plants is given. The inequalities
which result are the same as those which would result from attempting to
force a global quadratic solution to the test for the induced norm given
in the recent paper by the author, which indicates that such algebraic tests
are telling us about incremental gain magnitudes.
The class of plants for which the
results are applicable has been generalised at the cost of restricting our
attention to the standard saturation function (as opposed to the generic
class of nonlinearities studied in the paper of Chitour, et al.).