Boundary Realizations Method for Interval
Linear Dynamic Systems
SERGEY G. PUSHKOV and SVETLANA YU. KALINKINA
Biysk Technological Institute of Altay State Technical University, 27, Trophimova str., 659305 Biysk,
Altay Region, Russia, e-mail: email@example.com
(Received: 2 November 2004; accepted: 24 February 2005)
Abstract. In the paper, the state space realization problem for interval dynamic discrete-time systems
is considered. We propose methods for computing algebraic realizations of totally non-negative and
totally non-positive systems, and develop their modiﬁcation applicable for systems of “mixed” type.
Numerical examples illustrate our results.
The state space representation is one of the prevailing forms of the representa-
tion of control objects and control systems. For dynamic systems, the problem of
representing information about an object is tightly connected with the so-called
realization problem, which requires construction of a state space model for the
dynamic system with a known relation between its input and output signals.
When modelling any dynamic process, we often deal with uncertainties and
ambiguities in data, and their sources can be quite various: round-off errors, mea-
surement uncertainty caused by native imperfection of instruments, using approx-
imate numbers, etc. One of the techniques that allow us to take into account such
uncertainties is representation of parameters of the object as some sets. If these sets
are intervals, we deal with interval uncertainty.
Foraspecial class of linear stationary (time invariant) dynamic discrete-time
systems, the problem of construction of a ﬁnite-dimensional realization has been
successfully solved in the context of the so-called algebraic approach . However,
when investigating systems subject to noises as well as those functioning under
uncertainty, it is necessary to draw additional mathematical tools. The methods of
mathematical statistics and probability theory are often used for such purposes, but
in the recent decades fuzzy sets theory and interval analysis became more and more
popular for the analysis of systems with data uncertainties and ambiguities.
In particular, interval methods are widely used for both the analysis of static
systems and the solution of control problems for dynamic systems. Many problems
of the mathematical control theory allow natural “intervalization” through replacing
real-valued parameters and/or variables by the corresponding intervals. Most of such
“intervalizated problems” prove to be completely adequate to the original practical