%%% Bjoern Rueffer's BibTeX file, created on 02.04.2012 at 13:58 (local time zone) %%% -*-BibTeX-*- @preamble{ "\def\cprime{$'$} " } @article{DashkovskiyJiangRuffer:2012:Editorial:-Special-issue-on-robust-stabi:, author = {Dashkovskiy, Sergey N. and Jiang, Zhong-Ping and R{\"u}ffer, Bj{\"o}rn S.}, journal = {Math.\ Control Signals Syst.}, number = {1--2}, pages = {1--2}, title = {Editorial: Special issue on robust stability and control of large-scale nonlinear systems}, volume = {24}, year = {2012}, doi = {10.1007/s00498-012-0083-1}, } @techreport{RufferWouwMueller:2011:Convergent-Systems-vs.-Incremental-Stabi:, author = {R{\"u}ffer, Bj{\"o}rn S. and van de Wouw, Nathan and Mueller, Markus}, institution = {{U}niversit{\"a}t {P}aderborn}, journal = {Technical report, {U}niversit{\"a}t {P}aderborn}, title = {Convergent Systems vs. Incremental Stability}, year = {2011}, 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.}, } @article{Ruffer:2010:Discussion-of-On-a-small-gain-theorem-fo:, author = {R{\"u}ffer, Bj{\"o}rn S.}, journal = {Eur.\ J.\ Control}, number = {4}, pages = {366--367}, title = {{Discussion of ``{O}n a small gain theorem for {ISS} networks in dissipative {L}yapunov form''}}, volume = {17}, year = {2011}, } @inproceedings{ItoJiangDashkovskiy:2011:A-Small-Gain-Theorem-and-Construction-of:, author = {Ito, Hiroshi and Jiang, Zhong-Ping and Dashkovskiy, Sergey N. and R{\"u}ffer, Bj{\"o}rn S.}, booktitle = {Proc.\ American Contr.\ Conf.}, pages = {1971--1977}, title = {A Small-Gain Theorem and Construction of Sum-Type {L}yapunov Functions for Networks of {iISS} Systems}, year = {2011}, abstract = {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.}, } @article{RufferWirth:2010:Stability-verification-for-monotone-syst:, author = {R{\"u}ffer, B. S. and Wirth, Fabian R.}, journal = {Numer.\ Algorithms}, number = {4}, pages = {529--543}, title = {Stability verification for monotone systems using homotopy algorithms}, volume = {58}, year = {2011}, 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.}, doi = {10.1007/s11075-011-9468-3}, } @inproceedings{RufferItoDower:2010:Computing-asymptotic-gains-of-large-scal:, author = {R{\"u}ffer, B. S. and Ito, Hiroshi and Dower, Peter M.}, booktitle = {Proc.\ 49th\ IEEE Conf.\ Decis.\ Control}, pages = {7413--7418}, title = {Computing asymptotic gains of large-scale interconnections}, year = {2010}, 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.}, } @inproceedings{RufferDowerIto:2010:Applicable-comparison-principles-in-larg:, 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 = {March}, note = {(electronic)}, title = {Applicable comparison principles in large-scale system analysis}, year = {2010}, 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.}, } @inproceedings{RufferKellettDower:2010:On-copositive-Lyapunov-functions-for-a-c:, address = {Budapest, Hungary}, author = {R{\"u}ffer, B. S. and Kellett, C. M. and Dower, P. M.}, booktitle = {Proc.\ 19th\ Int.\ Symp.\ Math.\ Th.\ Networks Systems (MTNS)}, month = {July}, note = {(electronic)}, title = {On copositive {L}yapunov functions for a class of monotone systems}, year = {2010}, 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.}, } @inproceedings{RufferDowerKellett:2010:On-Robust-Stability-of-the-Belief-Propag:, address = {Budapest, Hungary}, author = {R{\"u}ffer, B. S. and Dower, P. M. and Kellett, C. M. and Weller, S. R.}, booktitle = {Proc.\ 19th\ Int.\ Symp.\ Math.\ Th.\ Networks Systems (MTNS)}, month = {July}, note = {(electronic)}, title = {On robust stability of the {B}elief {P}ropagation {A}lgorithm for {LDPC} decoding}, year = {2010}, 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.}, } @article{Ruffer:2009:Small-gain-conditions-and-the-comparison:, author = {B. S. R{\"u}ffer}, journal = {{IEEE} {T}rans.\ {A}utom.\ {C}ontrol}, month = {July}, number = {7}, pages = {1732--1736}, title = {Small-gain conditions and the comparison principle}, volume = {55}, year = {2010}, 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.}, doi = {10.1109/TAC.2010.2048053}, } @techreport{RufferKellett:2008:Implementing-the-Belief-Propagation-Algo:, author = {B. S. R{\"u}ffer and C. M. Kellett}, institution = {Department of Electrical Engineering and Computer Science, University of Newcastle, Australia}, month = {November}, title = {{Implementing the Belief Propagation Algorithm in MATLAB}}, year = {2008}, } @inproceedings{DashkovskiyRufferWirth:2008:Stability-of-interconnections-of-ISS-sys:, 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{RufferSailerWirth:2009:Comments-on-A-multichannel-IOS-Small-Gai:, author = {B. S. R{\"u}ffer and R. Sailer and F. R. Wirth}, journal = {{IEEE} {T}rans.\ {A}utom.\ {C}ontrol}, month = {July}, 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}, 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.}, doi = {10.1109/TAC.2010.2048938}, } @inproceedings{Scholz-ReiterWirthFreitag:2005:Some-remarks-on-the-stability-of-manufac:, 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 = {Proc.\ Int.\ Scientific Annual Conference on Operations Research}, pages = {91--96}, publisher = {Springer}, title = {Some remarks on the stability of manufacturing logistic networks. {S}tability margins}, year = {2005}, } @article{RufferKellettWeller:2009:Connection-between-cooperative-positive-:, author = {R{\"u}ffer, B. S. and Kellett, C. M. and Weller, S. R.}, journal = {Automatica J.\ IFAC}, 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}, 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.}, doi = {10.1016/j.automatica.2010.03.012}, } @article{RufferDashkovskiy:2009:Local-ISS-of-large-scale-interconnection:, author = {R{\"u}ffer, B. S. and Dashkovskiy, S. N.}, journal = {Systems Control\ Lett.}, number = {3--4}, pages = {241--247}, title = {Local {ISS} of large-scale interconnections and estimates for stability regions}, volume = {59}, year = {2010}, 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.}, doi = {10.1016/j.sysconle.2010.02.001}, } @techreport{DashkovskiyRufferWirth:2009:Construction-of-ISS-Lyapunov-functions-f:, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, institution = {ZeTeM, Universit{\"a}t Bremen, Germany}, month = {July 19th}, title = {Construction of {ISS} {L}yapunov functions for networks}, year = {2006}, 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.}, } @inproceedings{DashkovskiyRufferWirth:2007:A-Lyapunov-small-gain-theorem-for-strong:, address = {Pretoria, South Africa}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ 7th\ IFAC Symp.\ Nonlinear Control Systems}, month = {August 22--24}, pages = {283--288}, title = {A {L}yapunov small-gain theorem for strongly connected networks}, year = {2007}, 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 $R^n$ 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.}, } @phdthesis{Ruffer:2007:Monotone-dynamical-systems-graphs-and-st:, author = {B. S. R{\"u}ffer}, month = {October}, note = {Available online at \verb|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{DashkovskiyRufferWirth:2007:Application-of-small-gain-type-theorems-:, 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}, month = {August 28--30}, pages = {359-366}, publisher = {Springer}, title = {Application of small gain type theorems in logistics of autonomous processes}, year = {2007}, 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.}, } @inproceedings{DashkovskiyRufferWirth:2007:Numerical-verification-of-local-input-to:, address = {New Orleans, LA, USA}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ 46th\ IEEE Conf.\ Decis.\ Control}, pages = {4471--4476}, title = {Numerical verification of local input-to-state stability for large networks}, year = {2007}, 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.}, } @inproceedings{DashkovskiyRufferWirth:2006:Discrete-time-monotone-systems:-Criteria:, address = {Kyoto, Japan}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ 17th\ Int.\ Symp.\ Math.\ Th.\ Networks Systems (MTNS)}, pages = {89--97}, title = {Discrete time monotone systems: Criteria for global asymptotic stability and applications}, year = {2006}, abstract = {For two classes of monotone maps on the \mbox{$\mathbf{n}$-dimensional} positive orthant we show that for a discrete dynamical system induced by a map the origin of $\mathbf{\R^n_+}$ is globally asymptotically stable, if and only if the map $\mathbf{\Gamma}$ is such that for any point in $\mathbf{s\in\R^n_+}$, $\mathbf{s\ne0}$, the image-vector $\mathbf{\Gamma(s)}$ is such that at least one component is strictly less than the corresponding component of $\mathbf{s}$. One class is the set of $\mathbf{n\times n}$ matrices of class $\mathbf{\mathcal{K}_\infty}$ functions; these induce monotone operators on $\mathbf{\R^n_+}$. Maps of the other class satisfy some geometric property for an invariant set.}, } @inproceedings{DashkovskiyRufferWirth:2006:An-ISS-Lyapunov-function-for-networks-of:, address = {Kyoto, Japan}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ 17th\ Int.\ Symp.\ Math.\ Th.\ Networks Systems (MTNS)}, pages = {77--82}, title = {An {ISS} {L}yapunov function for networks of {ISS} systems}, year = {2006}, 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.}, } @inproceedings{DashkovskiyRufferWirth:2005:A-small-gain-type-stability-criterion-fo:, address = {Seville, Spain}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ Joint 44th\ IEEE Conf.\ Decis.\ Control and Europ.\ Contr.\ Conf.}, pages = {5633--5638}, title = {A small-gain type stability criterion for large scale networks of {ISS} systems}, year = {2005}, 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.}, } @inproceedings{DashkovskiyRufferWirth:2008:Stability-of-autonomous-vehicle-formatio:, 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)}, month = {March}, number = {1}, pages = {10911--10912}, title = {Stability of autonomous vehicle formations using an {ISS} small-gain theorem for networks}, volume = {8}, year = {2008}, 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.}, doi = {10.1002/pamm.200810911}, } @inproceedings{DashkovskiyRufferWirth:2008:Applications-of-the-general-Lyapunov-ISS:, address = {Cancun, Mexico}, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, booktitle = {Proc.\ 47th\ IEEE Conf.\ Decis.\ Control}, month = {December 9--11}, pages = {25--30}, title = {Applications of the general {L}yapunov {ISS} small-gain theorem for networks}, year = {2008}, 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.}, } @inproceedings{RufferKellettWeller:2009:Integral-input-to-state-stability-of-int:, address = {Shanghai, P.R.China}, author = {B. S. R{\"u}ffer and C. M. Kellett and S. R. Weller}, booktitle = {Proc.\ Joint 48th\ IEEE Conf.\ Decis.\ Control and 28th {C}hinese {C}ontr.\ {C}onf.}, pages = {638--643}, title = {Integral input-to-state stability of interconnected {iISS} systems by means of a lower-dimensional comparison system}, year = {2009}, 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.}, } @article{Ruffer:2009:Monotone-inequalities-dynamical-systems-:, author = {B. S. R{\"u}ffer}, journal = {Positivity}, month = {June}, 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}, 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 ori- gin 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.}, doi = {10.1007/s11117-009-0016-5}, } @article{RufferKellettDower:2009:Belief-Propagation-as-a-Dynamical-System:, author = {B. S. R{\"u}ffer and C. M. Kellett and P. M. Dower and S. R. Weller}, journal = {IET Control Theory Appl.}, month = {July}, number = {7}, pages = {1188--1200}, title = {{Belief Propagation as a Dynamical System: The Linear Case and Open Problems}}, volume = {4}, year = {2010}, 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.}, doi = {10.1049/iet-cta.2009.0233}, } @article{DashkovskiyRufferWirth:2009:Small-gain-theorems-for-large-scale-syst:, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, journal = {SIAM J.\ Control Optim.}, number = {6}, pages = {4089--4118}, title = {{Small gain theorems for large scale systems and construction of ISS Lyapunov functions}}, volume = {48}, year = {2010}, 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.}, doi = {10.1137/090746483}, } @article{DashkovskiyRufferWirth:2007:An-ISS-small-gain-theorem-for-general-ne:, author = {S. N. Dashkovskiy and B. S. R{\"u}ffer and F. R. Wirth}, journal = {Math.\ Control Signals Syst.}, month = {May}, number = {2}, pages = {93--122}, title = {{An ISS small-gain theorem for general networks}}, volume = {19}, year = {2007}, 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.}, doi = {10.1007/s00498-007-0014-8}, } @article{RekersbrinkRufferWenning:2007:Routing-in-dynamischen-Netzen:, author = {H. Rekersbrink and B. S. R{\"u}ffer and {B.-L.} Wenning and B. Scholz-Reiter and C. G{\"o}rg}, journal = {Logistik Management}, number = {1}, pages = {25--36}, title = {{Routing in dynamischen Netzen}}, volume = {9}, year = {2007}, 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 Einfl uss Netzgr{\"o}{\ss}e und die -dynamik auf die Leistungsf{\"a}higkeit der einzelnen Verfahren hat.}, } @incollection{Scholz-ReiterWirthFreitag:2007:Mathematical-Models-of-Autonomous-Logist:, 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}, editor = {H{\"u}lsmann, Michael and Windt, Katja}, pages = {121--138}, publisher = {Springer}, title = {{Mathematical Models of Autonomous Logistic Processes}}, year = {2007}, 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.}, }