# This file has been generated automatically on Sat 25 Jul 2020 23:48:30 AEST
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@InCollection{itoruffer2018-a-left-eigenvector-producing-a-smooth-lyapunov-function-of-iss-networks,
abstract = { For a class of monotone nonlinear systems, it is shown that
a continuously differentiable Lyapunov function can be
constructed implicitly from a left eigenvector of vector
fields. The left eigenvector which is a continuous function of
state variables is deduced from a right eigenvector which
represents a small gain condition. It is demonstrated that
rounding off the edges of n-orthotopes, which is the
maximization of state variables, yields level sets of the
Lyapunov function. Applying the development to comparison
systems gives continuously differentiable input-to-state
Lyapunov functions of networks},
address = {Cham},
author = {Ito, Hiroshi and R{\"u}ffer, Bj{\"o}rn S.},
booktitle = {Positive Systems: Theory and Applications (POSTA 2018)},
doi = {10.1007/978-3-030-04327-8_20},
editor = {Lam, James and Chen, Yun and Liu, Xingwen and Zhao, Xudong
and Zhang, Junfeng},
note = {Proceedings of POSTA 2018},
pages = {247--268},
publisher = {Springer},
title = {A Left Eigenvector Producing a Smooth {L}yapunov Function of
{ISS} Networks},
volume = {480},
year = {2019}
}
@Article{giladiruffer2019-a-lyapunov-function-construction-for-a-non-convex-douglas--rachford-iteration,
abstract = {While global convergence of the Douglas-Rachford iteration is
often observed in applications, proving it is still limited to
convex and a handful of other special cases. Lyapunov
functions for difference inclusions provide not only global or
local convergence certificates, but also imply robust
stability, which means that the convergence is still
guaranteed in the presence of persistent disturbances. In this
work, a global Lyapunov function is constructed by combining
known local Lyapunov functions for simpler, local sub-problems
via an explicit formula that depends on the problem
parameters. Specifically, we consider the scenario where one
set consists of the union of two lines and the other set is a
line, so that the two sets intersect in two distinct points.
Locally, near each intersection point, the problem reduces to
the intersection of just two lines, but globally the geometry
is non-convex and the Douglas-Rachford operator multi-valued.
Our approach is intended to be prototypical for addressing the
convergence analysis of the Douglas-Rachford iteration in more
complex geometries that can be approximated by polygonal sets
through the combination of local, simple Lyapunov functions.
},
arxiv = {1708.08697},
author = {Ohad Giladi and Bj{\"o}rn S. R{\"u}ffer},
doi = {10.1007/s10957-018-1405-3},
journal = {{J}. {O}ptim. {T}h. {A}ppl.},
number = {3},
pages = {729--750},
title = {A {L}yapunov function construction for a non-convex
{D}ouglas--{R}achford iteration},
volume = {180},
year = {2019}
}
@Article{guiverlogemannruffer2019-small-gain-stability-theorems-for-positive-lure-inclusions,
abstract = {Stability results are presented for a class of differential
and difference inclusions, so-called positive Lur'e inclusions
which arise, for example, as the feedback interconnection of a
linear positive system with a positive set-valued static
nonlinearity. We formulate sufficient conditions in terms of
weighted one-norms, reminiscent of the small-gain condition,
which ensure that the zero equilibrium enjoys various global
stability properties, including asymptotic and exponential
stability. We also consider input-to-state stability, familiar
from nonlinear control theory, in the context of forced
positive Lur'e inclusions. Typical for the study of positive
systems, our analysis benefits from comparison arguments and
linear Lyapunov functions. The theory is illustrated with
examples.},
author = {Chris Guiver and Hartmut Logemann and Bj{\"o}rn R{\"u}ffer},
doi = {10.1007/s11117-018-0605-2},
journal = {{P}ositivity},
number = {2},
pages = {249--289},
title = {Small-gain stability theorems for positive {L}ur'e
inclusions},
volume = {23},
year = {2019}
}
@Article{tranrufferkellett2019-convergence-properties-for-discrete-time-nonlinear-systems,
abstract = {Three similar convergence notions are considered. Two of them
are the long established notions of convergent dynamics and
incremental stability. The other is the more recent notion of
contraction analysis. All three convergence notions require
that all solutions of a system converge to each other. In this
paper, we investigate the differences between these
convergence properties for discrete-time, time-varying
nonlinear systems by comparing the properties in pairs and
using examples. We also demonstrate a time-varying smooth
Lyapunov function characterization for each of these
convergence notions. In addition, with appropriate
assumptions, we provide several sufficient conditions to
establish relationships between these properties in terms of
Lyapunov functions. },
arxiv = {1612.05327},
author = {Duc N. Tran and Bj{\"o}rn S. R{\"u}ffer and Christopher M.
Kellett},
doi = {10.1109/TAC.2018.2879951},
journal = {{{IEEE}} {T}rans. {A}utom. {C}ontrol},
number = {8},
pages = {3415--3422},
title = {Convergence Properties for Discrete-time Nonlinear Systems},
volume = {64},
year = {2019}
}
@InProceedings{tranrufferkellett2016-incremental-stability-properties-for-discrete-time-systems,
abstract = {Incremental stability describes the asymptotic behavior
between any two trajectories of a dynamical system. Such
properties are of interest, for example, in the study of
observers or synchronization of chaotic systems. In this
paper, we develop the notions of incremental stability and
incremental input-to-state stability (ISS) for discrete-time
systems. We derive Lyapunov function characterizations for
these properties as well as a useful summation-to-summation
formulation of the incremental stability property. },
author = {Duc N. Tran and Bj{\"o}rn S. R{\"u}ffer and Christopher M.
Kellett},
booktitle = {{P}roc. 55th {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2016.7798314},
pages = {477--482},
title = {Incremental Stability Properties for Discrete-Time Systems},
year = {2016}
}
@Article{noroozigeiselhartgrunerufferwirth2018-nonconservative-discrete-time-iss-small-gain-conditions-for-closed-sets,
abstract = {This paper presents a unification and a generalization of the
small-gain theory subsuming a wide range of the existing
small-gain theorems. In particular, we introduce small-gain
conditions that are necessary and sufficient to ensure
input-to-state stability (ISS) with respect to
\emph{not}-necessarily compact sets. Toward this end, we first
develop a Lyapunov characterization of $\omega$ISS via
finite-step $\omega$ISS Lyapunov functions. Then, we provide
the small-gain conditions to guarantee $\omega$ISS of a
network of systems. Finally, applications of our results to
partial input-to-state stability, ISS of time-varying systems,
synchronization problems and decentralized observers are given.},
arxiv = {1612.03710},
author = {Navid Noroozi and Roman Geiselhart and Lars Gr{\"u}ne and
Bj{\"o}rn S. R{\"u}ffer and Fabian R. Wirth},
doi = {10.1109/TAC.2017.2735194},
journal = {{{IEEE}} {T}rans. {A}utom. {C}ontrol},
number = {5},
pages = {1231--1242},
title = {Nonconservative Discrete-Time {ISS} Small-Gain Conditions for
Closed Sets},
volume = {63},
year = {2018}
}
@InCollection{fernandoruffer2016-a-preliminary-model-for-understanding-how-life-experiences-generate-human-emotions-and-behavioural-responses,
abstract = {Whilst human emotional and behaviour responses are generated
via a complex mechanism, understanding this process is
important for a broader range of applications that span over
clinical disciplines including psychiatry and psychology, and
computer science. Even though there is a large body of
literature and established findings in clinical disciplines,
these are under-utilised in developing more realistic
computational models. This paper presents a preliminary model
based on the integration of a number of established theories
in clinical psychology and psychiatry through an
interdisciplinary research effort. },
author = {Fernando, D. A. Irosh P. and R{\"u}ffer, Bj{\"o}rn},
booktitle = {Neural Information Processing: 23rd International Conference,
{ICONIP} 2016, Kyoto, Japan, October 16--21, 2016,
Proceedings, Part III},
doi = {10.1007/978-3-319-46675-0_30},
editor = {Hirose, Akira and Ozawa, Seiichi and Doya, Kenji and Ikeda,
Kazushi and Lee, Minho and Liu, Derong},
pages = {269--278},
publisher = {Springer},
title = {A preliminary model for understanding how life experiences
generate human emotions and behavioural responses},
year = {2016}
}
@InCollection{ruffer-nonlinear-left-and-right-eigenvectors-for-max-preserving-maps,
abstract = { It is shown that max-preserving maps (or join-morphisms) on
the positive orthant in Euclidean $n$-space endowed with the
component-wise partial order give rise to a semiring. This
semiring admits a closure operation for maps that generate
stable dynamical systems. For these monotone maps, the closure
is used to define suitable notions of left and right
eigenvectors that are characterized by inequalities. Some
explicit examples are given and applications in the
construction of Lyapunov functions are described.},
author = {Bj{\"o}rn S. R{\"u}ffer},
booktitle = {Positive Systems},
doi = {10.1007/978-3-319-54211-9_18},
pages = {227--237},
publisher = {Springer, Cham},
series = {{L}ecture {N}otes in {C}ontrol and {I}nformation {S}ciences},
title = {Nonlinear left and right eigenvectors for max-preserving
maps},
volume = {471},
year = {2017}
}
@Article{giladiruffer2019-a-perron-frobenius-type-result-for-integer-maps-and-applications,
abstract = { It is shown that for certain maps, including concave maps,
on the $d$-dimensional lattice of positive integers points,
`approximate' eigenvectors can be found. Applications in
epidemiology as well as distributed resource allocation are
discussed as examples.},
arxiv = {1609.01393},
author = {Ohad Giladi and Bj{\"o}rn S. R{\"u}ffer},
doi = {10.1007/s11117-018-0624-z},
journal = {{P}ositivity},
number = {3},
pages = {545--570},
title = {A {P}erron-{F}robenius type result for integer maps and
applications},
volume = {23},
year = {2019}
}
@InProceedings{rufferito2015-sum-separable-lyapunov-functions-for-networks-of-iss-systems,
abstract = {It has been known for a decade now that for networks of
input-to-state stable (ISS) systems a Lyapunov function for
the network can be constructed as a maximum of re-scaled
Lyapunov functions of the subsystems. In this work it is shown
that ---under the same conditions--- instead one could as well
choose a sum of re-scaled Lyapunov functions, which is not
inherently non-smooth, thus answering a long open question.
Moreover, our approach is constructive, and explicit formulas
for the cases of two and three subsystems are stated.},
author = {R{\"u}ffer, Bj{\"o}rn S. and Ito, Hiroshi},
booktitle = {{P}roc. 54th {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2015.7402475},
pages = {1823--1828},
title = {Sum-separable {L}yapunov functions for networks of {ISS}
systems: A gain function approach},
year = {2015}
}
@Article{dirritorantzerruffer2015-separable-lyapunov-functions:-constructions-and-limitations,
abstract = {For monotone systems evolving on the positive orthant of
$\mathbb R^n_+$ two types of Lyapunov functions are
considered: Sum- and max-separable Lyapunov functions. One can
be written as a sum, the other as a maximum of functions of
scalar arguments. Several constructive existence results for
both types are given. Notably, one construction provides a
max-separable Lyapunov function that is defined at least on an
arbitrarily large compact set, based on little more than the
knowledge about one trajectory. Another construction for a
class of planar systems yields a global sum-separable Lyapunov
function, provided the right hand side satisfies a small-gain
type condition. A number of examples demonstrate these methods
and shed light on the relation between the shape of sublevel
sets and the right hand side of the system equation. Negative
examples show that there are indeed globally asymptotically
stable systems that do not admit either type of Lyapunov
function. },
author = {Gunther Dirr and Hiroshi Ito and Anders Rantzer and Bj{\"o}rn
S. R{\"u}ffer},
doi = {10.3934/dcdsb.2015.20.2497},
journal = {{D}iscrete {C}ontin. {D}yn. {S}yst. Ser. {B}},
number = {8},
pages = {2497--2526},
title = {Separable {L}yapunov functions: Constructions and
limitations},
volume = {20},
year = {2015}
}
@InProceedings{itorufferrantzer2014-max--and-sum-separable-lyapunov-functions-for-monotone-systems-and-their-level-sets,
abstract = {For interconnected systems and systems of large size,
aggregating information of subsystems studied individually is
useful for addressing the overall stability. In the
Lyapunov-based analysis, summation and maximization of
separately constructed functions are two typical approaches in
such a philosophy. This paper focuses on monotone systems
which are common in control applications, and elucidates some
fundamental limitations of max-separable Lyapunov functions in
estimating domains of attractions. This paper presents several
methods of constructing sum- and max-separable Lyapunov
functions for second order monotone systems, and some
comparative discussions are given through illustrative
examples. },
author = {Hiroshi Ito and Bj{\"o}rn S. R{\"u}ffer and Anders Rantzer},
booktitle = {{P}roc. 53rd {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2014.7039750},
pages = {2371--2377},
title = {Max- and sum-separable {L}yapunov functions for monotone
systems and their level sets},
year = {2014}
}
@InProceedings{norooziruffer2014-non-conservative-dissipativity-and-small-gain-theory-for-iss-networks,
abstract = {This paper addresses input-to-state stability (ISS) analysis
for discrete-time systems using the notion of finite-step ISS
Lyapunov functions. Here, finite-step Lyapunov functions are
energy functions that decay after a fixed but finite number of
steps, rather than at every time step. We establish
non-conservative dissipativity and small-gain conditions for
ISS of networks of discrete-time systems, by generalizing
results of Gielen and Lazar (DOI:10.1109/CDC.2012.6426469) and
Geiselhart, Lazar, and Wirth (DOI:10.1109/TAC.2014.2332691) to
the case of ISS. The effectiveness of the results is
illustrated through two examples. },
author = {Navid Noroozi and Bj{\"o}rn S. R{\"u}ffer},
booktitle = {{P}roc. 53rd {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2014.7039872},
pages = {3131--3136},
title = {Non-conservative dissipativity and small-gain theory for
{ISS} networks},
year = {2014}
}
@InProceedings{ruffersailer2014-input-to-state-stability-for-discrete-time-monotone-systems,
abstract = {It is well known that input-to-state stability admits an
astonishing number of equivalent characterizations. Here it is
shown that for monotone systems on $\mathbb R^n_+$ there are
some additional characterizations that are useful for network
stability analysis. These characterizations include system
theoretic properties, algebraic properties, as well as the
problem of finding simultaneous bounds on solutions to a
collection of inequalities. },
author = {Bj{\"o}rn S. R{\"u}ffer and Rudolf Sailer},
booktitle = {Proc. 21st Int. Symp. Mathematical Theory of Networks and
Systems (MTNS)},
pages = {96--102},
title = {Input-to-State Stability for Discrete-Time Monotone Systems},
year = {2014}
}
@InProceedings{rantzerrufferdirr2013-separable-lyapunov-functions-for-monotone-systems,
abstract = {Separable Lyapunov functions play vital roles, for example,
in stability analysis of large-scale systems. A Lyapunov
function is called max-separable if it can be decomposed into
a maximum of functions with one-dimensional arguments.
Similarly, it is called sum-separable if it is a sum of such
functions. In this paper it is shown that for a monotone
system on a compact state space, asymptotic stability implies
existence of a max-separable Lyapunov function. We also
construct two systems on a non-compact state space, for which
a max-separable Lyapunov function does not exist. One of them
has a sum-separable Lyapunov function. The other does not.},
author = {Anders Rantzer and Bj{\"o}rn S. R{\"u}ffer and Gunther Dirr},
booktitle = {{P}roc. 52nd {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2013.6760604},
pages = {4590--4594},
title = {Separable {L}yapunov functions for monotone systems},
year = {2013}
}
@InProceedings{pogromskymatveevchailletruffer2013-input-dependent-stability-analysis-of-systems-with-saturation-in-feedback,
abstract = {The paper deals with global stability analysis of linear
control systems with saturation in feedback driven by an
external input. Various new criteria based on non-quadratic
Lyapunov functions are proposed, that unlike many previous
results, offer better account for the role of the external
excitation by providing input-dependent conditions for
stability of solutions. For example, it is shown that even if
the system fails to satisfy the incremental version of the
circle criterion, the stability is guaranteed whenever the
uniform root mean square value of the input signal is less
than a computable threshold. The general theoretical results
are illustrated in the case of the double integrator closed by
a saturated linear feedback with an external excitation.},
author = {Alexander {Yu.} Pogromsky and Alexey S. Matveev and Antoine
Chaillet and Bj{\"o}rn S. R{\"u}ffer},
booktitle = {{P}roc. 52nd {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2013.6760820},
pages = {5903--5908},
title = {Input-dependent stability analysis of systems with saturation
in feedback},
year = {2013}
}
@InProceedings{chailletpogromskyruffer2013-a-razumikhin-approach-for-the-incremental-stability-of-delayed-nonlinear-systems,
abstract = {This paper provides sufficient conditions for the incremental
stability of time-delayed nonlinear systems. It relies on the
Razumikhin-Lyapunov approach, which consists in invoking
small-gain arguments by treating the delayed state as a
feedback perturbation. The results are valid for multiple
delays, as well as bounded time-varying delays. We provide
conditions under which the limit solution of a time-delayed
nonlinear system to a periodic (resp. constant) input is
itself periodic and of the same period (resp. constant). As an
illustration, a specific focus is given on a class of delayed
Lur'e systems.},
author = {Antoine Chaillet and Alexander {Yu.} Pogromsky and Bj{\"o}rn
S. R{\"u}ffer},
booktitle = {{P}roc. 52nd {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2013.6760110},
pages = {1596--1601},
title = {A {R}azumikhin approach for the incremental stability of
delayed nonlinear systems},
year = {2013}
}
@Article{itojiangdashkovskiyruffer2013-robust-stability-of-networks-of-iiss-systems:-construction-of-sum-type-lyapunov-functions,
abstract = {This paper gives a solution to the problem of verifying
stability of networks consisting of integral input-to-state
stable (iISS) subsystems. The iISS small-gain theorem
developed recently has been restricted to interconnections of
two subsystems. For large-scale systems, stability criteria
relying only on gain-type information that were previously
developed address only input-to-state stable (ISS) subsystems.
To address the stability problem involving iISS subsystems
interconnected in general structure, this paper shows how to
construct Lyapunov functions of the network by means of a sum
of nonlinearly rescaled individual Lyapunov functions of
subsystems under an appropriate small-gain condition.},
author = {Hiroshi Ito and Zhong-Ping Jiang and Sergey Dashkovskiy and
Bj{\"o}rn S. R{\"u}ffer},
doi = {10.1109/TAC.2012.2231552},
journal = {{{IEEE}} {T}rans. {A}utom. {C}ontrol},
month = {{M}ay},
number = {5},
pages = {1192--1207},
title = {Robust stability of networks of {iISS} systems: Construction
of sum-type {L}yapunov functions},
volume = {58},
year = {2013}
}
@InProceedings{itoruffer2013-a-two-phase-approach-to-stability-of-networks-given-in-iiss-framework:-utilization-of-a-matrix-like-criterion,
abstract = {This article is concerned with global asymptotic stability
(GAS) of dynamical networks. The case when subsystems are
integral input-to-state stable (iISS) has been recognized as
notoriously difficult to deal with in the literature. In fact,
the lack of energy dissipation for large input denies direct
application of the small-gain argument for input-to-state
stable (ISS) subsystems. Here for networks consisting of iISS
subsystems it is demonstrated that a two-phase approach allows
us to make use of the ISS small-gain argument by separating a
trajectory into a transient and a subsequent convergence. In
contrast to the previous iISS results, the two-phase approach
immediately leads to a sufficient criterion for GAS of general
nonlinear networks in a matrix-like form (order condition).},
author = {Ito, Hiroshi and R{\"u}ffer, Bj{\"o}rn S.},
booktitle = {{P}roc. {IEEE} {A}merican {C}ontr. {C}onf.},
doi = {10.1109/ACC.2013.6580587},
pages = {4838--4843},
title = {A Two-Phase Approach to Stability of Networks Given in {iISS}
Framework: Utilization of a Matrix-Like Criterion},
year = {2013}
}
@InProceedings{dashkovskiyrufferwirth2012-small-gain-theorems-for-large-scale-systems-and-construction-of-iss-lyapunov-functions,
abstract = {This invited paper is a significantly shortened excerpt of
the article S. N. DASHKOVSKIY, B. S. R{\"U}FFER, AND F. R.
WIRTH, Small gain theorems for large scale systems and
construction of ISS Lyapunov functions, SIAM J. Control
Optim., 48 (2010), pp. 4089--4118.
We consider interconnections of n nonlinear subsystems in the
input-to-state stability (ISS) framework. Foreach subsystem an
ISS Lyapunov function is given that treats the other
subsystems as independent inputs. A gain matrix is used to
encode the mutual dependencies of the systems in the network.
Under a small gain assumption on the monotone operator induced
by the gain matrix, a locally Lipschitz continuous ISS
Lyapunov function is obtained constructively for the entire
network by appropriately scaling the individualLyapunov
functions for the subsystems.},
address = {Maui, Hawaii, USA},
author = {Dashkovskiy, Sergey N. and R{\"u}ffer, Bj{\"o}rn S. and
Wirth, Fabian R.},
booktitle = {{P}roc. 51st {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2012.6426856},
pages = {4165--4170},
title = {Small gain theorems for large scale systems and construction
of {ISS} {L}yapunov functions},
year = {2012}
}
@InProceedings{itojiangdashkovskiyruffer2012-a-cyclic-small-gain-condition-and-an-equivalent-matrix-like-criterion-for-iiss-networks,
abstract = {This paper considers nonlinear dynamical networks consisting
of individually iISS (integral input-to-statestable)
subsystems which are not necessarily ISS
(input-to-statestable). Stability criteria for internal and
external stability of the networks are developed in view of
both necessity and sufficiency. For the sufficiency, we show
how we can construct a Lyapunov function of the network
explicitly under the assumption that a cyclic small-gain
condition is satisfied. The cyclic small-gain condition is
shown to be equivalent to a matrix-like condition. The two
conditions and their equivalence precisely generalize some
central ISS results in the literature. Moreover, the necessity
of the matrix-like condition is established. The allowable
number of non-ISS subsystems for stability of the network is
discussed through several necessity conditions.},
address = {Maui, Hawaii, USA},
author = {Ito, Hiroshi and Jiang, Zhong-Ping and Dashkovskiy, Sergey N.
and R{\"u}ffer, Bj{\"o}rn S.},
booktitle = {{P}roc. 51st {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2012.6426994},
pages = {4158--4164},
title = {A cyclic small-gain condition and an equivalent matrix-like
criterion for {iISS} networks},
year = {2012}
}
@InProceedings{rufferwouwmueller2012-from-convergent-dynamics-to-incremental-stability,
abstract = {This paper advocates that the convergent systems property and
incremental stability are two intimately related though
different properties. Sufficient conditions for the convergent
systems property usually rely upon first showing that a system
is incrementally stable, as e.g. in the celebrated Demidovich
condition. However, in the current paper it is shown that
incremental stability itself does not imply the convergence
property, or vice versa. Moreover, characterizations of both
properties in terms of Lyapunov functions are given. Based on
these characterizations, it is established that the
convergence property implies incremental stability for systems
evolving oncompact sets, and also when a suitable uniformity
condition is satisfied.},
address = {Maui, Hawaii, USA},
author = {R{\"u}ffer, Bj{\"o}rn S. and van de Wouw, Nathan and Mueller,
Markus},
booktitle = {{P}roc. 51st {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2012.6425916},
pages = {2958--2963},
title = {From convergent dynamics to incremental stability},
year = {2012}
}
@Article{dashkovskiyjiangruffer2012-editorial:-special-issue-on-robust-stability-and-control-of-large-scale-nonlinear-systems,
author = {Dashkovskiy, Sergey N. and Jiang, Zhong-Ping and R{\"u}ffer,
Bj{\"o}rn S.},
doi = {10.1007/s00498-012-0083-1},
entrysubtype = {miscellanous},
journal = {{M}ath. {C}ontrol {S}ignals {S}yst.},
number = {1--2},
pages = {1--2},
title = {Editorial: Special issue on robust stability and control of
large-scale nonlinear systems},
volume = {24},
year = {2012}
}
@Article{rufferwouwmueller2013-convergent-systems-vs.-incremental-stability,
abstract = {Two similar stability notions are considered; one is the long
established notion of convergent systems, the other is the
younger notion of incremental stability. Both notions require
that any two solutions of a system converge to each other. Yet
these stability concepts are different, in the sense that none
implies the other, as is shown in this paper using two
examples. It is shown under what additional assumptions one
property indeed implies the other. Furthermore, this paper
contains necessary and sufficient characterizations of both
properties in terms of Lyapunov functions.},
author = {R{\"u}ffer, Bj{\"o}rn S. and van de Wouw, Nathan and Mueller,
Markus},
doi = {10.1016/j.sysconle.2012.11.015},
journal = {{S}ystems {C}ontrol {L}ett.},
mrnumber = {3029818},
pages = {277--285},
title = {Convergent Systems vs. Incremental Stability},
volume = {62},
year = {2013}
}
@Article{ruffer2011-discussion-of-on-a-small-gain-theorem-for-iss-networks-in-dissipative-lyapunov-form,
author = {R{\"u}ffer, Bj{\"o}rn S.},
doi = {10.1016/S0947-3580(11)70601-X},
entrysubtype = {miscellanous},
journal = {{E}ur. {J}. {C}ontrol},
number = {4},
pages = {366--367},
title = {Discussion on ``{O}n a small gain theorem for {ISS} networks
in dissipative {L}yapunov form''},
volume = {17},
year = {2011}
}
@InProceedings{itojiangdashkovskiyruffer2011-a-small-gain-theorem-and-construction-of-sum-type-lyapunov-functions-for-networks-of-iiss-systems,
abstract = {This paper gives a solution to the problem of verifying
stability of networks consisting of integral input-to-state
stable (iISS) subsystems. The iISS small-gain theorem
developed recently has been restricted to interconnection of
two subsystems. For large-scale systems, stability criteria
relying only on gain-type information have been successful
only in dealing with input-to-state stable stable (ISS)
subsystems. To address the stability problem involving iISS
subsystems interconnected in general structure, this paper
shows how to construct Lyapunov functions of the network by
means of nonlinear sum of individual Lyapunov functions of
subsystems given in a dissipation formulation under an
appropriate small-gain condition.},
author = {Ito, Hiroshi and Jiang, Zhong-Ping and Dashkovskiy, Sergey N.
and R{\"u}ffer, Bj{\"o}rn S.},
booktitle = {{P}roc. {IEEE} {A}merican {C}ontr. {C}onf.},
doi = {10.1109/ACC.2011.5991069},
pages = {1971--1977},
title = {A Small-Gain Theorem and Construction of Sum-Type {L}yapunov
Functions for Networks of {iISS} Systems},
year = {2011}
}
@Article{rufferwirth2011-stability-verification-for-monotone-systems-using-homotopy-algorithms,
abstract = {A monotone self-mapping of the nonnegative orthant induces a
monotone discrete-time dynamical system which evolves on the
same orthant. If with respect to this system the origin is
attractive then there must exist points whose image under the
monotone map is strictly smaller than the original point, in
the component-wise partial ordering. Here it is shown how such
points can be found numerically, leading to a recipe to
compute order intervals that are contained in the region of
attraction and where the monotone map acts essentially as a
contraction. An important application is the numerical
verification of so-called generalized small-gain conditions
that appear in the stability theory of large-scale systems. },
arxiv = {1005.0741},
author = {R{\"u}ffer, B. S. and Wirth, Fabian R.},
doi = {10.1007/s11075-011-9468-3},
journal = {{N}umer. {A}lgorithms},
mrnumber = {2854205},
number = {4},
pages = {529--543},
title = {Stability verification for monotone systems using homotopy
algorithms},
volume = {58},
year = {2011}
}
@InProceedings{rufferitodower2010-computing-asymptotic-gains-of-large-scale-interconnections,
abstract = { This paper considers the problem of verifying stability of
large-scale nonlinear dynamical systems. Using a comparison
principle approach we present a numerical method of estimating
the asymptotic gain characterizing the effect of external
disturbances on the stability of a large-scale
interconnection. The unique idea is to make use of solely the
knowledge of one single trajectory of the comparison system
for estimating the behavior of all possible trajectories. It
is shown that an asymptotic gain can be obtained from just a
single trajectory of a disturbance-free comparison system. The
single-trajectory approach leads to a computationally cheap
implementation with which we can numerically check whether or
not a large-scale system is input-to-state practically stable.
},
author = {R{\"u}ffer, B. S. and Ito, Hiroshi and Dower, Peter M.},
booktitle = {{P}roc. 49th {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2010.5717387},
pages = {7413--7418},
title = {Computing asymptotic gains of large-scale interconnections},
year = {2010}
}
@InProceedings{rufferdowerito2010-computational-comparison-principles-for-large-scale-system-stability-analysis,
abstract = {Stability analysis of complex and large-scale systems is
often aided by some form of model reduction, ideally down to a
one-dimensional system via a Lyapunov function. In this
context comparison principles arise very naturally. If the
comparison system can be shown to be monotone, then an
extension of a homotopical fixed point algorithm can be used
to verify practical quasi-global asymptotic stability of the
composite nominal system. This method is applied to a class of
nonlinear examples. },
address = {Kumamoto, Japan},
author = {R{\"u}ffer, B. S. and Dower, P. M. and Ito, Hiroshi},
booktitle = {Proc. of the 10th SICE Annual Conference on Control Systems},
month = {{M}arch},
note = {(electronic)},
title = {Computational comparison principles for large-scale system
stability analysis},
year = {2010}
}
@InProceedings{rufferkellettdower2010-on-copositive-lyapunov-functions-for-a-class-of-monotone-systems,
abstract = {This paper considers several explicit formulas for the
construction of copositive Lyapunov functions for global
asymptotic stability with respect to monotone systems evolving
in either discrete or continuous time. Such monotone systems
arise as comparison systems in the study of interconnected
large-scale nominal systems. A copositive Lyapunov function
for such a comparison system can then serve as a prototype
Lyapunov functions for the nominal system. We discuss several
constructions from the literature in a unified framework and
provide sufficiency criteria for the existence of such
constructions. },
address = {Budapest, Hungary},
author = {R{\"u}ffer, B. S. and Kellett, C. M. and Dower, P. M.},
booktitle = {{P}roc. 19th {I}nt. {S}ymp. {M}ath. {T}h. {N}etworks
{S}ystems {({MTNS})}},
month = {{J}uly},
note = {(electronic)},
title = {On copositive {L}yapunov functions for a class of monotone
systems},
year = {2010}
}
@InProceedings{rufferdowerkellettweller2010-on-robust-stability-of-the-belief-propagation-algorithm-for-ldpc-decoding,
abstract = {The exact nonlinear loop gain of the belief propagation
algorithm (BPA) in its log-likelihood ratio (LLR) formulation
is computed. The nonlinear gains for regular low-density
parity-check (LDPC) error correcting codes can be computed
exactly using a simple formula. It is shown that in some
neighborhood of the origin this gain is actually much smaller
than the identity. Using a small-gain argument, this implies
that the BPA is in fact locally input-to-state stable and
produces bounded outputs for small-in-norm input LLR vectors.
In a larger domain the algorithm produces at least bounded
trajectories. Further it is shown that, as the block length
increases, these regions exponentially shrink. },
address = {Budapest, Hungary},
author = {R{\"u}ffer, B. S. and Dower, P. M. and Kellett, C. M. and
Weller, S. R.},
booktitle = {{P}roc. 19th {I}nt. {S}ymp. {M}ath. {T}h. {N}etworks
{S}ystems {({MTNS})}},
month = {{J}uly},
note = {(electronic)},
title = {On robust stability of the {B}elief {P}ropagation {A}lgorithm
for {LDPC} decoding},
year = {2010}
}
@Article{ruffer2010-small-gain-conditions-and-the-comparison-principle,
abstract = { The general input-to-state stability (ISS) small-gain
condition for networks in a trajectory formulation is shown to
be equivalent to the requirement that a discrete-time
comparison system induced by the gain matrix of the network is
ISS. },
author = {B. S. R{\"u}ffer},
doi = {10.1109/TAC.2010.2048053},
journal = {{{IEEE}} {T}rans. {A}utom. {C}ontrol},
month = {{J}uly},
mrnumber = {2675841},
number = {7},
pages = {1732--1736},
title = {Small-gain conditions and the comparison principle},
volume = {55},
year = {2010}
}
@TechReport{rufferkellett2008-implementing-the-belief-propagation-algorithm-in-matlab,
abstract = {We provide some example Matlab code as a supplement to the
paper [6]. This technical report is not intended as a
standalone introduction to the belief propagation algorithm,
but instead only aims to provide some technical material,
which didn't fit into the paper.},
author = {B. S. R{\"u}ffer and C. M. Kellett},
entrysubtype = {miscellanous},
institution = {Department of Electrical Engineering and Computer Science,
University of Newcastle, Australia},
month = {{N}ovember},
title = {Implementing the {B}elief {P}ropagation {A}lgorithm in
{MATLAB}},
year = {2008}
}
@InProceedings{dashkovskiyrufferwirth2008-stability-of-interconnections-of-iss-systems,
abstract = {Interconnections of several nonlinear systems with inputs are
considered in this paper. Each system is assumed to be
input-to-state stable (ISS). However an interconnection of
such systems is not stable in general. We provide a stability
condition for interconnections of such systems. Some
interpretations of this condition are given. Moreover we show
how an ISS-Lyapunov function for such a network can be
constructed explicitly if this condition is satisfied. Further
we give the idea of numerical verification of the small gain
condition for networks.},
address = {Kyoto, Japan},
author = {Dashkovskiy, S. N. and R{\"u}ffer, B. S. and Wirth, Fabian
R.},
booktitle = {Proc. of the 8th SICE Annual Conference on Control Systems},
pages = {52431--52434},
title = {Stability of interconnections of {ISS} systems},
year = {2008}
}
@Article{ruffersailerwirth2010-comments-on-a-multichannel-ios-small-gain-theorem-for-systems-with-multiple-time-varying-communication-delays.,
abstract = {The small-gain condition presented by Polushin et al. may be
replaced by a strictly weaker one to obtain essentially the
same result. The necessary minor modifications of the proof
are given. Using essentially the same arguments, a global
version of the result is also presented. },
author = {B. S. R{\"u}ffer and R. Sailer and F. R. Wirth},
doi = {10.1109/TAC.2010.2048938},
journal = {{{IEEE}} {T}rans. {A}utom. {C}ontrol},
month = {{J}uly},
mrnumber = {2675839},
number = {7},
pages = {1722--1725},
title = {Comments on ``{A} multichannel {IOS} Small Gain Theorem for
Systems With Multiple Time-Varying Communication Delays.''},
volume = {55},
year = {2010}
}
@InProceedings{scholz-reiterwirthfreitagdashkovskiyjagalski2006-some-remarks-on-the-stability-of-production-networks,
abstract = {The increasing complexity of production and logistics
networks and the requirement of higher flexibility lead to a
change of paradigm in control: Autonomously controlled systems
where decisions are taken by parts or goods themselves become
more attractive. The question of stability is an important
issue for the dynamics of such systems. In this paper we are
going to touch this question for a special production network
with autonomous control. The stability region for a
corresponding fluid model is found empirically. We point out
that further mathematical investigations have to be undertaken
to develop some stability criteria for autonomous systems. },
address = {Bremen, Germany},
author = {B. Scholz-Reiter and F. R. Wirth and M. Freitag and S. N.
Dashkovskiy and T. Jagalski and C. de Beer and B. S.
R{\"u}ffer},
booktitle = {Operations Research Proceedings},
doi = {10.1007/3-540-32539-5_15},
pages = {91--96},
publisher = {Springer},
title = {Some Remarks on the Stability of Production Networks},
volume = {2005},
year = {2006}
}
@Article{rufferkellettweller2010-connection-between-cooperative-positive-systems-and-integral-input-to-state-stability-of-large-scale-systems,
abstract = {We consider a class of continuous-time cooperative systems
evolving on the positive orthant. We show that if the origin
is globally attractive, then it is also globally stable and,
furthermore, there exists an unbounded invariant manifold
where trajectories strictly decay. This leads to a
small-gain-type condition which is sufficient for global
asymptotic stability (GAS) of the origin. We establish the
following connection to large-scale interconnections of
(integral) input-to-state stable (ISS) subsystems: If the
cooperative system is (integral) ISS, and arises as a
comparison system associated with a large-scale
interconnection of (i)ISS systems, then the composite nominal
system is also (i)ISS. We provide a criterion in terms of a
Lyapunov function for (integral) input-to-state stability of
the comparison system. Furthermore, we show that if a
small-gain condition holds then the classes of systems
participating in the large-scale interconnection are
restricted in the sense that certain iISS systems cannot
occur. Moreover, this small-gain condition is essentially the
same as the one obtained previously by Dashkovskiy,
R{\"u}ffer, and Wirth (2007, in press) for large-scale
interconnections of ISS systems. },
author = {R{\"u}ffer, B. S. and Kellett, C. M. and Weller, S. R.},
doi = {10.1016/j.automatica.2010.03.012},
journal = {{A}utomatica {J}. {IFAC}},
mrnumber = {2877182},
number = {6},
pages = {1019--1027},
title = {Connection between cooperative positive systems and integral
input-to-state stability of large-scale systems},
volume = {46},
year = {2010}
}
@Article{rufferdashkovskiy2010-local-iss-of-large-scale-interconnections-and-estimates-for-stability-regions,
abstract = {We consider interconnections of locally input-to-state stable
(LISS) systems. The class of LISS systems is quite large, in
particular it contains input-to-state stable (ISS) and
integral input-to-state stable (iISS) systems. Local
small-gain conditions both for LISS tra jectory and Lyapunov
formulations guaranteeing LISS of the composite system are
provided in this paper. Notably, estimates for the resulting
stability region of the composite system are also given. This
in particular provides an advantage over the linearization
approach, as will be discussed. },
author = {R{\"u}ffer, B. S. and Dashkovskiy, S. N.},
doi = {10.1016/j.sysconle.2010.02.001},
journal = {{S}ystems {C}ontrol {L}ett.},
mrnumber = {2642263},
number = {3--4},
pages = {241--247},
title = {Local {ISS} of large-scale interconnections and estimates for
stability regions},
volume = {59},
year = {2010}
}
@TechReport{dashkovskiyrufferwirth2006-construction-of-iss-lyapunov-functions-for-networks,
abstract = {The construction of an input-to-state stability (ISS)
Lyapunov function for networks of ISS system will be
presented. First we construct ISS Lyapunov functions for each
strongly connected component, then what remains is a cas- cade
(or disconnected aggregation) of these strongly connected
components. Using known results the constructed Lyapunov
functions can be aggregated to one single ISS Lyapunov
function for the whole network. The Lyapunov function
construction for the strongly connected compo- nents basically
depends on two steps: The construction of a function to the
positive orthant in Rn and the combination of the given ISS
Lyapunov functions of the subsystems to a common ISS Lyapunov
function for the composite system. },
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
entrysubtype = {miscellanous},
institution = {ZeTeM, Universit{\"a}t Bremen, Germany},
month = {{J}uly 19th},
title = {Construction of {ISS} {L}yapunov functions for networks},
year = {2006}
}
@InProceedings{dashkovskiyrufferwirth2007-a-lyapunov-iss-small-gain-theorem-for-strongly-connected-networks,
abstract = {Abstract: We consider strongly connected networks of
input-to-state stable (ISS) systems. Provided a small gain
condition holds it is shown how to construct an ISS Lyapunov
function using ISS Lyapunov functions of the subsystems. The
construction relies on two steps: The construction of a
strictly increasing path in a region defined on the positive
orthant in $\mathbb R^n$ by the gain matrix and the
combination of the given ISS Lyapunov functions of the
subsystems to a ISS Lyapunov function for the composite
system. Novelties are the explicit path construction and that
all the involved Lyapunov functions are nonsmooth, i.e., they
are only required to be locally Lipschitz continuous. The
existence of a nonsmooth ISS Lyapunov function is
qualitatively equivalent to ISS. },
address = {Pretoria, South Africa},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. 7th {IFAC} {S}ymp. {N}onlinear {C}ontrol {S}ystems},
month = {{A}ugust 22--24},
pages = {283--288},
title = {A {L}yapunov {ISS} small-gain theorem for strongly connected
networks},
year = {2007}
}
@PhDThesis{ruffer2007-monotone-dynamical-systems-graphs-and-stability-of-large-scale-interconnected-systems,
author = {B. S. R{\"u}ffer},
entrysubtype = {miscellanous},
month = {{O}ctober},
note = {Available online at
\texttt{http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000109058}},
school = {Universit{\"a}t Bremen, Germany},
title = {Monotone dynamical systems, graphs, and stability of
large-scale interconnected systems},
year = {2007}
}
@InProceedings{dashkovskiyrufferwirth2008-application-of-small-gain-type-theorems-in-logistics-of-autonomous-processes,
abstract = {In this paper we consider stability of logistic networks. We
give a stability criterion for a general situation and show
how it can be applied in special cases. For this purpose two
examples are considered. },
address = {Bremen, Germany},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {Proc. 1st Int. Conference Dynamics in Logistics},
doi = {10.1007/978-3-540-76862-3_36},
month = {{A}ugust 28--30},
pages = {359--366},
publisher = {Springer},
title = {Application of small gain type theorems in logistics of
autonomous processes},
year = {2008}
}
@InProceedings{dashkovskiyrufferwirth2007-numerical-verification-of-local-input-to-state-stability-for-large-networks,
abstract = {We consider networks of locally input-to-state stable (LISS)
systems. Under a small gain condition the entire network is
again LISS. An efficient numerical test to check the small
gain condition is presented in this paper. An example from
applications serves as a demonstration for quantitative
results. },
address = {New Orleans, LA, USA},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. 46th {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2007.4434626},
pages = {4471--4476},
title = {Numerical verification of local input-to-state stability for
large networks},
year = {2007}
}
@InProceedings{dashkovskiyrufferwirth2006-discrete-time-monotone-systems:-criteria-for-global-asymptotic-stability-and-applications,
abstract = {For two classes of monotone maps on the $n$-dimensional
positive orthant we show that for a discrete dynamical system
induced by a map the origin of $\mathbb R^n_+$ is globally
asymptotically stable, if and only if the map $\Gamma$ is such
that for any point in $s\in\mathbb R^n_+$, $s\ne0$, the
image-vector $\Gamma(s)$ is such that at least one component
is strictly less than the corresponding component of $s$.
One class is the set of $n\times n$ matrices of class
$\mathcal K_\infty$ functions; these induce monotone operators
on $\mathbb R^n_+$. Maps of the other class satisfy some
geometric property for an invariant set. },
address = {Kyoto, Japan},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. 17th {I}nt. {S}ymp. {M}ath. {T}h. {N}etworks
{S}ystems {({MTNS})}},
pages = {89--97},
title = {Discrete time monotone systems: Criteria for global
asymptotic stability and applications},
year = {2006}
}
@InProceedings{dashkovskiyrufferwirth2006-on-the-construction-of-iss-lyapunov-functions-for-networks-of-iss-systems,
abstract = {We consider a finite number of nonlinear systems
interconnected in an arbitrary way. Under the assumption that
each subsystem is input-to-state stable (ISS) regarding the
states of the other subsystems as inputs we are looking for
conditions that guarantee input-to-state stability of the
overall system. To this end we aim to construct an
ISS-Lyapunov function for the interconnection using the
knowledge of ISS-Lyapunov functions of the subsystems in the
network. Sufficient conditions of a small gain type are
obtained under which an ISS Lyapunov function can be
constructed. The ISS-Lyapunov function is then given
explicitly, and guarantees that the network is ISS. },
address = {Kyoto, Japan},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. 17th {I}nt. {S}ymp. {M}ath. {T}h. {N}etworks
{S}ystems {({MTNS})}},
pages = {77--82},
title = {On the construction of {ISS} {L}yapunov functions for
networks of {ISS} systems},
year = {2006}
}
@InProceedings{dashkovskiyrufferwirth2005-a-small-gain-type-stability-criterion-for-large-scale-networks-of-iss-systems,
abstract = {We provide a generalized version of the nonlinear small-gain
theorem for the case of more than two coupled input-to-state
stable systems. For this result the interconnection gains are
described in a nonlinear gain matrix and the small gain
condition requires bounds on the image of this gain matrix.
The condition may be interpreted as a nonlinear generalization
of the requirement that the spectral radius of the gain matrix
is less than one. We give some interpretations of the
condition in special cases covering linear gains and linear
systems. },
address = {Seville, Spain},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. {J}oint 44th {IEEE} {C}onf. {D}ecis. {C}ontrol
and {E}urop. {C}ontr. {C}onf.},
doi = {10.1109/CDC.2005.1583060},
pages = {5633--5638},
title = {A small-gain type stability criterion for large scale
networks of {ISS} systems},
year = {2005}
}
@InProceedings{dashkovskiyrufferwirth2008-stability-of-autonomous-vehicle-formations-using-an-iss-small-gain-theorem-for-networks,
abstract = {We consider a formation of vehicles moving on the two
dimensional plane. The movement of each vehicle is described
by a system of ordinary differential equations with inputs.
The formation is maintained using autonomous controls that are
designed to maintain fixed relative distances and orientations
between vehicles. Moreover this formation should track a given
trajectory on the plane. The vehicles can measure the relative
distances and angles to their neighbors. These values are the
inputs from one system to another. With the help of a general
ISS small-gain theorem for networks we will show that the
dynamics of such a formation is stable for the given controls.
The notion of local input-to- state stability (local ISS) will
be used for this purpose. },
address = {Bremen, Germany},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {PAMM, Special Issue: 79th Annual Meeting of the International
Association of Applied Mathematics and Mechanics (GAMM)},
doi = {10.1002/pamm.200810911},
month = {{M}arch},
number = {1},
pages = {10911--10912},
title = {Stability of autonomous vehicle formations using an {ISS}
small-gain theorem for networks},
volume = {8},
year = {2008}
}
@InProceedings{dashkovskiyrufferwirth2008-applications-of-the-general-lyapunov-iss-small-gain-theorem-for-networks,
abstract = {We recall the definitions of input-to-state-stability
Lyapunov functions and general small gain theorems. These are
then exemplarily used to prove input-to-state stability of and
to construct ISS Lyapunov functions for four areas of
applications: Linear systems, a Cohen-Grossberg neuronal
network, error dynamics in formation control, as well as
nonlinear transistor-linear resistor circuits. },
address = {Cancun, Mexico},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
booktitle = {{P}roc. 47th {IEEE} {C}onf. {D}ecis. {C}ontrol},
doi = {10.1109/CDC.2008.4739299},
month = {{D}ecember 9--11},
pages = {25--30},
title = {Applications of the general {L}yapunov {ISS} small-gain
theorem for networks},
year = {2008}
}
@InProceedings{rufferkellettweller2009-integral-input-to-state-stability-of-interconnected-iiss-systems-by-means-of-a-lower-dimensional-comparison-system,
abstract = {We consider arbitrarily many interconnected integral
Input-to-State Stable (iISS) systems in an arbitrary
interconnection topology and provide an (i)ISS comparison
principle for networks. We show that global asymptotic
stability of the origin (GAS) of a lower-dimensional system
termed the comparison system, which is based on the individual
dissipative Lyapunov iISS inequalities, together with a
scaling condition implies the existence of an iISS Lyapunov
function of the composite system. A sufficient (but not
necessary) condition for 0-GAS of the interconnection is shown
in this paper to be the generalized small-gain condition
derived by Dashkovskiy et al., but this time in a dissipative
Lyapunov setting. We also provide geometric intuition behind
growth rate conditions for the stability of cascaded iISS
systems. },
address = {Shanghai, P.R.China},
author = {B. S. R{\"u}ffer and C. M. Kellett and S. R. Weller},
booktitle = {{P}roc. {J}oint 48th {IEEE} {C}onf. {D}ecis. {C}ontrol
and 28th {C}hinese {C}ontr. {C}onf.},
doi = {10.1109/CDC.2009.5400344},
pages = {638--643},
title = {Integral input-to-state stability of interconnected {iISS}
systems by means of a lower-dimensional comparison system},
year = {2009}
}
@Article{ruffer2010-monotone-inequalities-dynamical-systems-and-paths-in-the-positive-orthant-of-euclidean-n-space,
abstract = {Given monotone operators on the positive orthant in
n-dimensional Euclidean space, we explore the relation between
inequalities involving those operators, and induced monotone
dynamical systems. Attractivity of the origin implies
stability for these systems, as well as a certain inequality.
Under the right perspective the converse is also true. In
addition we construct an unbounded path in the set where tra
jectories of the dynamical system decay monotonically, i.e.,
we solve a positive continuous selection problem. },
author = {B. S. R{\"u}ffer},
doi = {10.1007/s11117-009-0016-5},
journal = {{P}ositivity},
month = {{J}une},
mrnumber = {2657634},
number = {2},
pages = {257--283},
title = {Monotone inequalities, dynamical systems, and paths in the
positive orthant of {E}uclidean $n$-space},
volume = {14},
year = {2010}
}
@Article{rufferkellettdowerweller2010-belief-propagation-as-a-dynamical-system:-the-linear-case-and-open-problems,
abstract = {Systems and control theory have found wide application in the
analysis and design of numerical algorithms. We present a
discrete-time dynamical system interpretation of an algorithm
commonly used in information theory called Belief Propagation.
Belief Propagation (BP) is one instance of the so-called
Sum-Product Algorithm and arises, e.g., in the context of
iterative decoding of Low-Density Parity-Check codes. We
review a few known results from information theory in the
language of dynamical systems and show that the typically very
high dimensional, nonlinear dynamical system corresponding to
BP has interesting structural properties. For the linear case
we completely characterize the behavior of this dynamical
system in terms of its asymptotic input-output map. Finally,
we state some of the open problems concerning BP in terms of
the dynamical system presented. },
author = {B. S. R{\"u}ffer and C. M. Kellett and P. M. Dower and S. R.
Weller},
doi = {10.1049/iet-cta.2009.0233},
journal = {{IET} {C}ontrol {T}heory {A}ppl.},
month = {{J}uly},
number = {7},
pages = {1188--1200},
title = {{B}elief {P}ropagation as a Dynamical System: The Linear Case
and Open Problems},
volume = {4},
year = {2010}
}
@Article{dashkovskiyrufferwirth2010-small-gain-theorems-for-large-scale-systems-and-construction-of-iss-lyapunov-functions,
abstract = {We consider a network consisting of n interconnected
nonlinear subsystems. For each subsystem an ISS Lyapunov
function is given that treats the other subsystems as
independent inputs. We use a gain matrix to encode the mutual
dependencies of the systems in the network. Under a small gain
assumption on the monotone operator induced by the gain
matrix, we construct a locally Lipschitz continuous ISS
Lyapunov function for the entire network by appropriately
scaling the individual Lyapunov functions for the subsystems.
},
arxiv = {0901.1842},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
doi = {10.1137/090746483},
journal = {{SIAM} {J}. {C}ontrol {O}ptim.},
mrnumber = {2645475},
number = {6},
pages = {4089--4118},
title = {Small gain theorems for large scale systems and construction
of {ISS} {L}yapunov functions},
volume = {48},
year = {2010}
}
@Article{dashkovskiyrufferwirth2007-an-iss-small-gain-theorem-for-general-networks,
abstract = {We provide a generalized version of the nonlinear small gain
theorem for the case of more than two coupled input-to-state
stable systems. For this result the interconnection gains are
described in a nonlinear gain matrix, and the small gain
condition requires bounds on the image of this gain matrix.
The condition may be interpreted as a nonlinear generalization
of the requirement that the spectral radius of the gain matrix
is less than 1. We give some interpretations of the condition
in special cases covering two subsystems, linear gains, linear
systems and an associated lower-dimensional discrete time
dynamical system. },
arxiv = {math/0506434},
author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth},
doi = {10.1007/s00498-007-0014-8},
journal = {{M}ath. {C}ontrol {S}ignals {S}yst.},
month = {{M}ay},
mrnumber = {2317822},
number = {2},
pages = {93--122},
title = {An {ISS} small gain theorem for general networks},
volume = {19},
year = {2007}
}
@Article{rekersbrinkrufferwenningscholz-reitergorg2007-routing-in-dynamischen-netzen,
abstract = {Eine klassische Aufgabe in der Transportlogistik ist die
Bestimmung einer k{\"u}rzesten oder kostenoptimalen Route
durch ein Netzwerk f{\"u}r Transportfahrzeuge auf der einen
oder f{\"u}r die zu transportierenden G{\"u}ter auf der
anderen Seite. Diese Aufgabenstellung, auch Shortest Path
Problem (SPP) genannt, ist f{\"u}r statische Netzwerke
mittlerweile ersch{\"o}pfend untersucht. Moderne und gerade
auch selbststeuernde Transportnetzwerke weisen jedoch einen so
hohen Grad an Dynamik auf, dass L{\"o}sungen und Algorithmen
f{\"u}r statische Netze in diesen Bereichen zu keiner
sinnvollen L{\"o}sung f{\"u}hren. Unabh{\"a}ngig vom
eigentlich verwendeten Algorithmus kann man der Dynamik auf
verschiedene Weisen entgegentreten, z. B. durch eine
regelm{\"a}{\ss}ige Neuplanung des Weges (Reaktives Routing).
Eine noch nicht sehr gut untersuchte M{\"o}glichkeit, mit der
Dynamik solcher Netze umzugehen, ist die Sch{\"a}tzung der
zuk{\"u}nftigen Zust{\"a}nde. Dies kann unter gewissen
Umst{\"a}nden Vorteile haben, z.B. bei gro{\ss}en und sehr
dynamischen Netzen, wenn der Sch{\"a}tzaufwand die
Verbesserungen rechtfertigt. Daher werden in dieser Arbeit
drei grunds{\"a}tzlich verschiedene Routingverfahren
verglichen: statische, reaktive und sch{\"a}tzungsbasierte
Routingverfahren. Hierzu wurde f{\"u}r eine beispielhafte
Netztopologie untersucht, welchen Einfluss Netzgr{\"o}{\ss}e
und die -dynamik auf die Leistungsf{\"a}higkeit der einzelnen Verfahren hat. },
author = {H. Rekersbrink and B. S. R{\"u}ffer and {B.-L.} Wenning and
B. Scholz-Reiter and C. G{\"o}rg},
journal = {{L}ogistik {M}anagement},
number = {1},
pages = {25--36},
title = {Routing in dynamischen Netzen},
volume = {9},
year = {2007}
}
@InCollection{scholz-reiterwirthfreitagdashkovskiyjagalski2007-mathematical-models-of-autonomous-logistic-processes,
abstract = {(Abstract of the book:) Autonomous co-operation addresses the
control problem of logistic processes characterized by
dynamical changing parameters and complex system behaviour.
During control procedures erratic, non-predictable changes of
parameters can occur. Therefore, future planning and control
has to face severe and vital uncertainties. Conventional
hierarchical systems are amplifying these difficulties because
of the additional time delay of information transfer and
additional calculation time. On the other hand, autonomous
co-operation enables logistic objects (e.g. a single
container) in decentralized structures to collect and evaluate
information simultaneously to any event of change, so that
they can render and execute decisions on their own. Therefore,
this book aims to give a profound understanding of autonomous
co-operation and to examine its potentials to increase the
robustness and positive emergence of logistic processes
substantially.},
author = {B. Scholz-Reiter and F. R. Wirth and M. Freitag and S. N.
Dashkovskiy and T. Jagalski and C. de Beer and B. S.
R{\"u}ffer},
booktitle = {Understanding Autonomous Cooperation and Control in
Logistics},
doi = {10.1007/978-3-540-47450-0_9},
editor = {H{\"u}lsmann, Michael and Windt, Katja},
pages = {121--138},
publisher = {Springer},
title = {Mathematical Models of Autonomous Logistic Processes},
year = {2007}
}