Michael Kosmykov

A Small-Gain Condition for Interconnections of ISS Systems With Mixed ISS Characterizations

We consider interconnected nonlinear systems with external inputs, where each of the subsystems is assumed to be input-to-state stable (ISS). Sufficient conditions of small-gain type are provided guaranteeing that the interconnection is ISS with respect to the external input. To this end we extend recently obtained small-gain theorems to a more general type of interconnections. The small-gain theorem provided here is applicable to situations where the ISS conditions are formulated differently for each subsystem and are either given in the maximization or the summation sense. Furthermore, it is shown that the conditions are compatible in the sense that it is always possible to transform sum formulations to maximum formulations without destroying a given small-gain condition. An example shows the advantages of our results in comparison with the known ones.

Input-to-state stability of interconnected hybrid systems

We consider the interconnections of arbitrary topology of a finite number of ISS hybrid systems and study whether the ISS property is maintained for the overall system. We show that if the small gain condition is satisfied, then the whole network is ISS and show how a non-smooth ISS-Lyapunov function can be explicitly constructed in this case.

A Lyapunov–Razumikhin approach for stability analysis of logistics networks with time-delays

Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. Complexity of the system, time-delays and perturbations in a customer demand may cause unstable behaviour of the network. This leads to the loss of the customers and high inventory costs. Thus the investigation of the network on stability is desired during its design. In this article we consider local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. We are looking for verifiable conditions that guarantee stability of the network under consideration. Lyapunov–Razumikhin functions and the local small gain condition are utilised to obtain such conditions. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate the proposed approach.

Stability of networks of hybrid ISS systems

Interconnection of several hybrid input-to-state stable (ISS) systems is considered in this paper. We ask under what condition is such an interconnection stable and how an ISS-Lyapunov function can be constructed for the whole interconnection. Small-gain condition to assure stability is given. A construction of an ISS-Lyapunov function for the whole system is provided under the small-gain condition.

Stability analysis of logistics networks with time-delays

Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. In this paper, we study local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. An appropriate Lyapunov–Razumikhin function and the small gain condition are utilized to establish some conditions for stability analysis of the network under consideration. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate an application of the proposed approach.

Some Remarks on Stability and Robustness of Production Networks Based on Fluid Models

The dynamics of complex, large-scale production networks present an important issue not only for the management of such networks but also for scientific research. This problem is usually investigated either by numerical simulations or by mathematical analysis of the associated queueing model. One major property of stable production networks is the robustness of these networks with respect to perturbations of the arrival process of jobs from outside. Given a generic structure of the considered network with several production locations, different products and re-entrant material flows the determination of robustness is non trivial. In this paper we use a fluid model approach to analyse the robustness of queueing networks. First conditions for stability of a fluid network are introduced. These conditions allow to assess the dynamic behavior of the production network. Second the obtained results are investigated with the help of a simulation of the fluid and queueing network. Simulations of a test case scenario accompany the results of the theoretical analysis.

A comparison of mathematical modelling approaches for stability analysis of supply chains

Production and transportation processes along a supply chain are dynamic. In particular they are subject to perturbations (e.g., breakdown of a resource) that can destabilise the network. Stability is a major property of a supply chain that is essential for a sustainable relationship to its customers. In order to verify the stability of a given supply chain different criteria have been developed. This paper addresses the problem of choosing a proper mathematical modelling approach for a real world network in order to investigate stability. For this reason we discuss different modelling approaches. Each of these approaches can model different characteristics of a supply chain and features a specific stability criterion. By comparing these approaches the paper supports choosing a proper modelling approach for a real world supply chain.

Application of the LISS Lyapunov-Krasovskii small-gain theorem to autonomously controlled production networks with time-delays

In this paper we consider general autonomously controlled production networks. A production network consists of geographically distributed plants, which are connected by transport routes such that transportation times (time-delays) have to be taken into account. In autonomous controlled production networks logistic objects (e.g., parts, orders) route themselves through a network based on local information. In this paper these kinds of logistic networks are investigated in view of stability to avoid negative outcomes such as high inventory costs or loss of customers. We use the local input-to-state stability (LISS) property and the tool of an LISS Lyapunov-Krasovskii functional for the stability investigation. By the application of the LISS Lyapunov-Krasovskii small-gain theorem we derive conditions, which guarantee stability of the production network.

ISS of Interconnected Impulsive Systems with and without Time-Delays

Abstract We consider networks of impulsive systems with and without time-delays and investigate under which conditions the whole network is input-to-state stable (ISS). We provide conditions on the size of the time intervals between impulses and on the interconnection structure that guarantee ISS of the overall system, where Lyapunov and Lyapunov-Razumikhin functions for the subsystems are used. The condition on the interconnection allows to construct a Lyapunov (-Razumikhin) function and a corresponding gain for the whole system.

An Approach to Model Reduction of Logistic Networks Based on Ranking

Simulations or mathematical analysis of a real-world logistic network require a model. In this context two challenges occur for modelling: First, the model should represent the real-world logistic network in an accurate way. Second, it should foster simulations or analytical analysis to be conducted in a reasonable time. A large size is often a drawback of many models. In the case of logistic networks this drawback can be overcome by reducing the number of locations and transportation links of the graph model. In this paper we present an approach to model reduction of a logistic network based on ranking. The rank of a given location states the importance of the location for the whole network. In order to calculate the importance of a location we introduce an adaptation of the PageRank algorithm for logistic networks. The information about the rank and the structural relations between the locations are used for our approach to model reduction. Depending on the structural relation between locations we suggest three different approaches to obtain a model of lower size.