http://bjoern.rueffer.info/publications/

last updated on 10 February 2018

## Björn Rüffer/Publications

last updated on 10 February 2018

- Non-conservative small-gain conditions for input-to-state stability with respect to not-necessarily compact setsAccepted by IEEE Trans. Autom. Control on July 19th 2017.

arXiv:1612.03710*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 \emphnot-necessarily compact sets. Toward this end, we first develop a Lyapunov characterization of \omegaISS via finite-step \omegaISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee \omegaISS 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. - Separable Lyapunov functions: Constructions and limitationsDiscrete Contin. Dyn. Syst. Ser. B 20(8): 2497–2526, 2015.
*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. - Convergent Systems vs. Incremental StabilitySystems Control Lett. 62: 277–285, 2013.
*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. - Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functionsIEEE Trans. Autom. Control 58(5): 1192–1207, May 2013.
*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. - Stability verification for monotone systems using homotopy algorithmsNumer. Algorithms 58(4): 529–543, 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. - Belief Propagation as a Dynamical System: The Linear Case and Open ProblemsIET Control Theory Appl. 4(7): 1188–1200, July 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. - Small gain theorems for large scale systems and construction of ISS Lyapunov functionsSIAM J. Control Optim. 48(6): 4089–4118, 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. - Connection between cooperative positive systems and integral input-to-state stability of large-scale systemsAutomatica J. IFAC 46(6): 1019–1027, 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, Rüffer, and Wirth (2007, in press) for large-scale interconnections of ISS systems. - Small-gain conditions and the comparison principleIEEE Trans. Autom. Control 55(7): 1732–1736, July 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. - Comments on “A multichannel IOS Small Gain Theorem for Systems With Multiple Time-Varying Communication Delays.”IEEE Trans. Autom. Control 55(7): 1722–1725, July 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. - Local ISS of large-scale interconnections and estimates for stability regionsSystems Control Lett. 59(3–4): 241–247, 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. - Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-spacePositivity 14(2): 257–283, June 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 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. - An ISS small gain theorem for general networksMath. Control Signals Syst. 19(2): 93–122, May 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. - Routing in dynamischen NetzenLogistik Management 9(1): 25–36, 2007.
*Abstract.*Eine klassische Aufgabe in der Transportlogistik ist die Bestimmung einer kürzesten oder kostenoptimalen Route durch ein Netzwerk für Transportfahrzeuge auf der einen oder für die zu transportierenden Güter auf der anderen Seite. Diese Aufgabenstellung, auch Shortest Path Problem (SPP) genannt, ist für statische Netzwerke mittlerweile erschöpfend untersucht. Moderne und gerade auch selbststeuernde Transportnetzwerke weisen jedoch einen so hohen Grad an Dynamik auf, dass Lösungen und Algorithmen für statische Netze in diesen Bereichen zu keiner sinnvollen Lösung führen. Unabhängig vom eigentlich verwendeten Algorithmus kann man der Dynamik auf verschiedene Weisen entgegentreten, z. B. durch eine regelmäßige Neuplanung des Weges (Reaktives Routing). Eine noch nicht sehr gut untersuchte Möglichkeit, mit der Dynamik solcher Netze umzugehen, ist die Schätzung der zukünftigen Zustände. Dies kann unter gewissen Umständen Vorteile haben, z.B. bei großen und sehr dynamischen Netzen, wenn der Schätzaufwand die Verbesserungen rechtfertigt. Daher werden in dieser Arbeit drei grundsätzlich verschiedene Routingverfahren verglichen: statische, reaktive und schätzungsbasierte Routingverfahren. Hierzu wurde für eine beispielhafte Netztopologie untersucht, welchen Einfluss Netzgröße und die -dynamik auf die Leistungsfähigkeit der einzelnen Verfahren hat.

- Mathematical Models of Autonomous Logistic ProcessesUnderstanding Autonomous Cooperation and Control in Logistics, pages 121–138, Springer, 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.

- Nonlinear left and right eigenvectors for max-preserving mapsPositive Systems: Theory and Applications (POSTA 2016) Rome, Italy, September 14-16, 2016, pages 227–237, Springer International Publishing, 2017.

arXiv:1609.06217*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. - Incremental Stability Properties for Discrete-Time SystemsProc. 55th IEEE Conf. Decis. Control, pages 477–482, 2016.
*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. - A preliminary model for understanding how life experiences generate human emotions and behavioural responsesNeural Information Processing: 23rd International Conference, ICONIP 2016, Kyoto, Japan, October 16–21, 2016, Proceedings, Part III, pages 269–278, Springer, 2016.
- Sum-separable Lyapunov functions for networks of ISS systems: A gain function approachProc. 54th IEEE Conf. Decis. Control, pages 1823–1828, 2015.
*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. - Max- and sum-separable Lyapunov functions for monotone systems and their level setsProc. 53rd IEEE Conf. Decis. Control, pages 2371–2377, 2014.
*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. - Non-conservative dissipativity and small-gain theory for ISS networksProc. 53rd IEEE Conf. Decis. Control, pages 3131–3136, 2014.
*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. - Input-to-State Stability for Discrete-Time Monotone SystemsProc. 21st Int. Symp. Mathematical Theory of Networks and Systems (MTNS), pages 96–102, 2014.
*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. - Separable Lyapunov functions for monotone systemsProc. 52nd IEEE Conf. Decis. Control, pages 4590–4594, 2013.
*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. - Input-dependent stability analysis of systems with saturation in feedbackProc. 52nd IEEE Conf. Decis. Control, pages 5903–5908, 2013.
*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. - A Razumikhin approach for the incremental stability of delayed nonlinear systemsProc. 52nd IEEE Conf. Decis. Control, pages 1596–1601, 2013.
*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. - A Two-Phase Approach to Stability of Networks Given in iISS Framework: Utilization of a Matrix-Like CriterionProc. IEEE American Contr. Conf., pages 4838–4843, 2013.
*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). - Small gain theorems for large scale systems and construction of ISS Lyapunov functionsProc. 51st IEEE Conf. Decis. Control, pages 4165–4170, 2012.
*Abstract.*This invited paper is a significantly shortened excerpt of the article S. N. DASHKOVSKIY, B. S. RÜFFER, 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. - A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networksProc. 51st IEEE Conf. Decis. Control, pages 4158–4164, 2012.
*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. - From convergent dynamics to incremental stabilityProc. 51st IEEE Conf. Decis. Control, pages 2958–2963, 2012.
*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. - A Small-Gain Theorem and Construction of Sum-Type Lyapunov Functions for Networks of iISS SystemsProc. IEEE American Contr. Conf., pages 1971–1977, 2011.
*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. - Computing asymptotic gains of large-scale interconnectionsProc. 49th IEEE Conf. Decis. Control, pages 7413–7418, 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. - Computational comparison principles for large-scale system stability analysisProc. of the 10th SICE Annual Conference on Control Systems, March 2010.

(electronic)*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. - On copositive Lyapunov functions for a class of monotone systemsProc. 19th Int. Symp. Math. Th. Networks Systems (MTNS), July 2010.

(electronic)*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. - On robust stability of the Belief Propagation Algorithm for LDPC decodingProc. 19th Int. Symp. Math. Th. Networks Systems (MTNS), July 2010.

(electronic)*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. - Integral input-to-state stability of interconnected iISS systems by means of a lower-dimensional comparison systemProc. Joint 48th IEEE Conf. Decis. Control and 28th Chinese Contr. Conf., pages 638–643, 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. - Stability of interconnections of ISS systemsProc. of the 8th SICE Annual Conference on Control Systems, pages 52431–52434, 2008.
- Application of small gain type theorems in logistics of autonomous processesProc. 1st Int. Conference Dynamics in Logistics, pages 359–366, Springer, August 28–30 2008.
*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. - Stability of autonomous vehicle formations using an ISS small-gain theorem for networksPAMM, Special Issue: 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), pages 10911–10912, March 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. - Applications of the general Lyapunov ISS small-gain theorem for networksProc. 47th IEEE Conf. Decis. Control, pages 25–30, December 9–11 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. - A Lyapunov ISS small-gain theorem for strongly connected networksProc. 7th IFAC Symp. Nonlinear Control Systems, pages 283–288, August 22–24 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 \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. - Numerical verification of local input-to-state stability for large networksProc. 46th IEEE Conf. Decis. Control, pages 4471–4476, 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. - Discrete time monotone systems: Criteria for global asymptotic stability and applicationsProc. 17th Int. Symp. Math. Th. Networks Systems (MTNS), pages 89–97, 2006.
*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 Γis such that for any point in s∈\mathbb R^n_+, s\ne0, the image-vector Γ(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_∞functions; these induce monotone operators on \mathbb R^n_+. Maps of the other class satisfy some geometric property for an invariant set. - On the construction of ISS Lyapunov functions for networks of ISS systemsProc. 17th Int. Symp. Math. Th. Networks Systems (MTNS), pages 77–82, 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. - Some Remarks on the Stability of Production NetworksOperations Research Proceedings, pages 91–96, Springer, 2006.
- A small-gain type stability criterion for large scale networks of ISS systemsProc. Joint 44th IEEE Conf. Decis. Control and Europ. Contr. Conf., pages 5633–5638, 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.

- Editorial: Special issue on robust stability and control of large-scale nonlinear systemsMath. Control Signals Syst. 24(1–2): 1–2, 2012.
- Discussion of “On a small gain theorem for ISS networks in dissipative Lyapunov form”Eur. J. Control 17(4): 366–367, 2011.
- Implementing the Belief Propagation Algorithm in MATLABTechnical report. Department of Electrical Engineering and Computer Science, University of Newcastle, Australia, November 2008.
- Monotone dynamical systems, graphs, and stability of large-scale interconnected systemsPhD Thesis. Universität Bremen, Germany, October 2007.

Available online at http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000109058 - Construction of ISS Lyapunov functions for networksTechnical report. ZeTeM, Universität Bremen, Germany, July 19th 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.

- A Lyapunov function construction for the Douglas-Rachford operator in a non-convex settingPreprint.

arXiv:1708.08697*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. - Convergence Properties for Discrete-time Nonlinear SystemsPreprint.

arXiv:1612.05327*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. - A Perron-Frobenius type result for integer maps and applicationsPreprint.

arXiv:1609.01393*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.