MODEL REDUCTION IN THE GAP METRIC

Jan C. Willems
ESAT
K.U. Leuven
Belgium
Email:   Jan.Willems@esat.kuleuven.be
URL:     http://www.esat.kuleuven.be/~jwillems

Many physical systems, for example electrical circuits, are 
undirectional, in the sense that they do not possess a preferred 
input/output partition of the system variables. In this case, the usual 
model reduction methods do not apply. The purpose of this presentation 
is to outline an approach to reduce the state space of such systems.

We discuss linear time-invariant (LTI) systems with external variables 
described by constant coefficient differential equations. The set of 
solutions is called the behavior. If the system is controllable, then 
the trajectories in the behavior that are square integrable specify the 
system uniquely. These square integrable trajectories form a closed 
linear subspace of the Hilbert space 
$\mathcal{L}_2\left(\mathbb{R},\mathbb{R}^{\mathtt{w}}\right)$. If we 
have two such LTI systems, then the gap between the corresponding closed 
linear subspaces is a good measure of the proximity of these two 
systems.

The behavior of a controllable LTI system can be represented as the 
image of a stable, norm preserving transfer function. This transfer 
function is called the symbol of the image representation. The McMillan 
degree of this transfer function corresponds to the dimension of the 
state space of the LTI system. By reducing the dimension of the state 
space, for example by applying balanced truncation on the symbol, leads 
to a system with smaller McMillan degree. Moreover, the gap between the 
original LTI system and the reduced model can be bound by twice the sum 
of the neglected singular values of the symbol of the image 
representation.  This leads to a model reduction method, with an error 
bound, that applies to undirectional systems and to unstable 
input/output systems.