We study stability properties of interconnected hybrid systems with application to large-scale logistics networks.
Hybrid systems are dynamical systems that combine two types of dynamics: continuous and discrete. Such behaviour occurs in wide range of applications. Logistics networks are one of such applications, where the continuous dynamics occurs in the production
and processing of material and the discrete one in the picking up and delivering of material. Stability of logistics networks characterizes their robustness to the changes occurring in the network. However, the hybrid dynamics and the large size of the network lead to complexity of the stability analysis.
In this thesis we show how the behaviour of a logistics networks can be described by interconnected hybrid systems. Then we recall the small gain conditions used in the stability analysis of continuous and discrete systems and extend them to establish input-to-
state stability (ISS) of interconnected hybrid systems. We give the mixed small gain condition in a matrix form, where one matrix describes the interconnection structure of the system and the second diagonal matrix takes into account whether ISS condition for a subsystem is formulated in the maximization or the summation sense. The small gain condition is sufficient for ISS of an interconnected hybrid system and can be applied to an interconnection of an arbitrary finite number of ISS subsystems. We also show an application of this condition to particular subclasses of hybrid systems: impulsive systems, comparison systems and the systems with stability of only a part of the state.
Furthermore, we introduce an approach for structure-preserving model reduction for large-scale logistics networks. This approach supposes to aggregate typical interconnection patterns (motifs) of the network graph. Such reduction allows to decrease the number of computations needed to verify the small gain condition.
Contents
1 Mathematical models of logistics networks
1.1 Notation
1.1.1 Logistics network
1.1.2 Vectors and matrices
1.1.3 Graphs
1.1.4 Notions from control theory
1.1.5 Dynamical systems and their stability
1.2 Review of the known modelling approaches
1.2.1 Discrete deterministic systems
1.2.2 Continuous deterministic systems
1.2.3 Hybrid deterministic systems
1.2.4 Stochastic models
1.3 Comparison of the modelling approaches
2 Stability of interconnected hybrid systems
2.1 Interconnected hybrid systems
2.2 Stability notions
2.2.1 Input-to-state stability (ISS)
2.2.2 ISS in terms of Lyapunov functions
2.3 Gains
2.3.1 Gain operator
2.3.2 Mixed small gain condition
2.3.3 From summation to maximization
2.4 Stability conditions
2.4.1 Small gain theorems in terms of trajectories
2.4.2 Construction of ISS-Lyapunov functions for interconnected hybrid systems
2.4.3 Systems with stability of only a part of the state
2.4.4 Impulsive dynamical systems
2.4.5 Comparison systems
3 Model reduction approach for large-scale networks
3.1 Gain model
3.2 Aggregation rules
3.2.1 Aggregation of sequentially connected nodes
3.2.2 Aggregation of nodes connected in parallel
3.2.3 Aggregation of almost disconnected subgraphs
3.2.4 Notes on application of the aggregation rules
4 Conclusion and outlook
Research Objectives and Themes
This thesis investigates the stability properties of interconnected hybrid systems, with a specific focus on their application to large-scale logistics networks. The research aims to develop robust stability criteria, specifically Input-to-State Stability (ISS), and to introduce efficient model reduction techniques that preserve the structural integrity of these complex networks.
- Mathematical modeling of logistics networks as interconnected hybrid systems.
- Extension of small-gain theorems to hybrid systems in a mixed (summation and maximization) formulation.
- Construction of ISS-Lyapunov functions for interconnected hybrid systems.
- Development of structure-preserving model reduction techniques for large-scale networks based on interconnection motifs.
Excerpt from the Book
1.1.1 Logistics network
The main activities of a logistics network include production, inventory control, storing and processing. Thus, the network consists of different objects: suppliers, production facilities, distributors, retailers, customers, machines at a production facility. We call such objects locations. We denote by n the number of locations and we number all the locations by i = 1,...,n. The decision, a location takes, on handling the orders relies on a certain policy. By x we understand the state of a location. Usually, it is the stock level (inventory level) of a location or a work content to be performed. The variable q denotes a length of queue, e.g., the queue of customer orders at a location or products to be processed by a machine. The external input denoted by u, describes usually the flow of customer orders or the flow of raw material from the external suppliers. The output is denoted by y. A typical output is consumption. The customer demand is described by the variable d. An example of a logistics network that illustrates our notation is shown in Figure 1.1.
Summary of Chapters
1 Mathematical models of logistics networks: This chapter provides a comprehensive survey of eleven different modeling approaches for logistics networks, covering discrete, continuous, hybrid, and stochastic dynamics.
2 Stability of interconnected hybrid systems: This chapter establishes Input-to-State Stability (ISS) for interconnected hybrid systems using a mixed small-gain theorem and Lyapunov methods.
3 Model reduction approach for large-scale networks: This chapter introduces structure-preserving aggregation rules to reduce the size of gain matrices, thereby simplifying stability analysis for large-scale logistics networks.
4 Conclusion and outlook: This final chapter synthesizes the main contributions of the thesis and discusses potential future research directions, such as control design and non-convex analysis.
Keywords
Hybrid systems, Logistics networks, Input-to-state stability (ISS), Lyapunov functions, Small-gain theorem, Model reduction, Structure-preserving aggregation, Bullwhip effect, Impulsive systems, Nonlinear systems, Supply chain management, Interconnected systems.
Frequently Asked Questions
What is the primary focus of this thesis?
The thesis focuses on the stability analysis of large-scale logistics networks modeled as interconnected hybrid dynamical systems, particularly aiming to establish Input-to-State Stability (ISS) through small-gain conditions.
What types of dynamics are considered in the logistics models?
The work considers four types of dynamics: discrete, continuous, hybrid (combining both), and stochastic systems.
What is the main goal of the stability analysis presented?
The primary goal is to ensure the persistence and robustness of logistics networks against perturbations (such as fluctuations in demand) by applying the Input-to-State Stability (ISS) framework.
Which mathematical methodology is central to the stability proofs?
The thesis utilizes small-gain theorems, extended to hybrid systems in a mixed (summation and maximization) formulation, and constructs ISS-Lyapunov functions.
What is the motivation behind the model reduction approach?
Large-scale logistics networks result in high-dimensional models that make analytical stability verification computationally demanding. Model reduction allows for simpler verification while preserving the crucial structure of the network.
What are the key keywords characterizing this research?
Key terms include Hybrid systems, Logistics networks, Input-to-state stability, Lyapunov functions, Small-gain theorem, and Model reduction.
How does the thesis handle the complexity of large-scale networks?
It introduces "aggregation rules" based on network motifs (parallel, sequential, and almost disconnected subgraphs) to reduce the dimensionality of the gain matrix used in stability conditions.
Are the proposed model reduction methods structure-preserving?
Yes, a key contribution of the thesis is that the proposed aggregation rules are designed to preserve the main physical and structural features of the logistics network during the reduction process.
Does the work address systems that are only partially stable?
Yes, Section 2.4.3 specifically addresses systems where stability is required for only a portion of the state space, which is common in applications involving time or counter variables.
- Quote paper
- Mykhaylo Kosmykov (Author), 2011, Hybrid dynamics in large-scale logistics networks, Munich, GRIN Verlag, https://www.grin.com/document/198846
