Authors:
Jan Willem Polderman,
Volume: 1, Page 2484 Paper number 3501
Abstract:
We want to design a compensator for a behavior through an appropriate
behavioral interconnection. The problem is that the behavior that we
want to control is not known. All that is given is a desired interconnected
behavior and the prior information that this desired behavior can indeed
be achieved by means of regular interconnection. This problem calls
for an adaptive flavored strategy. The strategy that we propose is
as follows. Measurements are taken during successive time intervals
of unit length. Each time a measurement is taken the Most Powerful
Unfalsified Model of that measurement and the desired behavior is determined.
Since this model contains the desired behavior it is possible to find
additional constraints such that the desired behavior is achieved.
Moreover these additional constraints can be chosen such that the corresponding
interconnection is regular relative to the true unknown behavior. This
regularity property makes it possible to invoke the additional constraints
in the next time interval by incorporating a transient period. The
new measurement therefore satisfies these additional constraints. The
procedure is repeated for the new measurement and so on. The main result
is that within a finite, though unknown, number of measurements the
new measurements are constrained to the desired behavior.
Authors:
Margreta Kuijper,
Volume: 1, Page 2488 Paper number 3502
Abstract:
It is well known that classical decoding of Reed-Solomon error-correcting
block codes is equivalent to system-theoretic minimal partial realization.
In the first part of the paper we show how this type of decoding can
also be formulated as minimal polynomial interpolation. We compare
this type of interpolation with system-theoretic interpolation techniques
that are used for control applications. We then present a procedure
that achieves minimal polynomial interpolation by iteratively constructing
a row reduced representation of an interpolating behavior. Motivated
by the need for improved decoding techniques, in particular soft-decision
decoding, we turn to ``list decoding'' in the second part of the paper.
Here the aim is to construct a list of all codewords that are within
a pre-specified Hamming distance from the received word. A connection
is made with recent work in the coding-theoretic literature that performs
list decoding by using bivariate interpolating polynomials. We point
out that this new development opens up yet another connection between
coding theory and system theory, namely the connection between list
decoding and minimal multivariable interpolation.
Authors:
Mauro Bisiacco,
Maria Elena Valcher,
Volume: 1, Page 2494 Paper number 3503
Abstract:
In the last decade, the behavioral approach to dynamic systems has
been fruitfully applied to the multidimensional, in particular two-dimensional
(2D), context. Within the behavioral setting, several classic results
that hold true for 2D state space models have found a natural generalization,
among them, the autonomous/controllable decomposition, that holds true
for every (linear shift-invariant) complete 2D behavior. This decomposition
constitutes the straightforward, even though nontrivial, extension
of the well-known "free evolution"/"forced evolution" decomposition,
that holds true for every trajectory of a (linear and shift-invariant)
2D state space model. The relevance of this decomposition, which
has been intensively investigated also in the context of 1D behaviors,
is immediately apparent. However, while in the 1D case it is always
possible to express a behavior B as a direct sum of its (uniquely determined)
controllable part and of some autonomous behavior, in the two-dimensional
case this is not always feasible. Nevertheless, it is always possible
to obtain a decomposition in which the autonomous part has a finite-dimensional
intersection with the controllable one. The aim of this paper is that
of extending these results, and, in particular, those presented in
a recent paper and concerned with the direct sum decomposition, to
a more general setting. In fact, given a 2D complete behavior B and
one of its sub-behaviors B1 we aim to investigate under what conditions
a further complete behavior B2 can be found, such that B is the direct
sum of B1 and B2, and the intersection of B1 and B2 is finite-dimensional
autonomous. This constitutes a complete generalization of the decomposition
theorem, as it represents a decomposition with "minimal intersection",
in which one of the two terms is a priori fixed. Significantly enough,
the possibility of obtaining such a decomposition is related to the
algebraic properties of certain matrix pairs, involved in the kernel
description of B and B1.
Authors:
Harish K. Pillai,
Jan C. Willems,
Volume: 1, Page 2500 Paper number 3504
Abstract:
This paper deals with systems described by constant coefficient linear
partial differential equations. We define dissipativity with respect
to a quadratic differential form, i.e., a quadratic functional in the
system variables and their partial derivatives. The main result states
the equivalence of dissipativity and the existence of a storage function
or of a dissipation rate.
Authors:
Ralf Peeters,
Paolo Rapisarda,
Volume: 1, Page 2504 Paper number 3505
Abstract:
A two-variable polynomial approach to solve the one-variable polynomial
Lyapunov equation is put forward. Such approach yields an iterative
solution method based on the method of Faddeev for the computation
of matrix resolvents. The resulting algorithm is especially suitable
for applications requiring symbolic or exact computation.
Authors:
Harry L. Trentelman,
Paolo Rapisarda,
Volume: 1, Page 2510 Paper number 3506
Abstract:
In this short paper we study the existence of positive and negative
semidefinite solutions of the algebraic Riccati equation corresponding
to linear quadratic problems with an indefinite cost functional. An
important role is played by certain two-variable polynomial matrices
associated with the algebraic Riccati equation. We characterize all
unmixed solutions in terms of the Pick matrices associated with these
two-variable polynomial matrices. As a corollary it turns out that
the signatures of the extremal solutions are determined by the signatures
of particular Pick matrices.
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