Ahmed Ahmed
email: ahmed.ahmed@ensam.eu

Mathias Kleiner
email: mathias.kleiner@ensam.eu

Lionel Roucoules
email: lionel.roucoules@ensam.eu

Remy Gaudy

Rèmy Gaudy

Bertrand Larat
email: bertrand.larat@capgemini.com

Model-based Interoperability Solutions for the Supervision of Smart Gas Distribution Networks

Keywords: smart environment, smart gas, interoperability, data aggregation, model-based engineering

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I. INTRODUCTION

Distributed systems are widely spread in many industrial/enterprise environments. In a given context, different systems (hardware and/or software) may coexist and bring to the environment their own set of capabilities: sensors, actuators, data analyzers, data repositories, visualization, etc. The supervision and integration of these systems to achieve common goals leads to large complex systems often called Systems of Systems SoS [START_REF] Jamshidi | Systems of Systems Engineering: Principles and Applications[END_REF]. These SoS include emerging smart environments such as smart grids, smart gas, smart cities, etc. [START_REF] Su | Smart city and the applications[END_REF][START_REF]Smart Grid Reference Architecture[END_REF]. A number of challenges arise from the heterogeneity of such systems which need to manage different platforms, standards, semantics and communication mechanisms. This includes behavior, semantic and syntactic interoperability issues [START_REF] Kubicek | Organizational Interoperability in E-Government: Lessons from 77 European Good-Practice Cases[END_REF]. This work takes place in a French national project 1 for the real-time management of a smart gas distribution network. One of the main project's objectives is the development and integration of additional systems in the current supervision platform. In particular, gas quantity/quality sensors need to be associated to decision-making processes in order to drive the injection of "green gas" into the distribution network. The current architecture, mainly based on the use of a single standard (OPCUA [START_REF] Mahnke | OPC Unified Architecture[END_REF]), requires many ad hoc developments whenever new systems have to be integrated. Therefore the objective of this work is to develop and study the viability of a modern interoperability framework, based on modeling principles, in order to ease the integration of new components and operational requirements. The envisioned solution, still in the early phases of development, is sketched through its application on two industrial scenarios. This paper is structured as follows: section II introduces the context of the Gontrand project and the various systems that currently constitute the smart gas environment. A brief introduction to modeling techniques is also presented. In section III, we first present the foundations of our model-based hub and describe its application on two industrial scenarios from the project. Section IV discusses some related work, and finally section V concludes with some insights on ongoing and future work.

II. CONTEXT

In the coming years, the injection of "green gas" in the national gas distribution network will have a great influence on its supervision and management. One of the objectives of the Gontrand project is to manage in real-time the quantity and the quality of gas at each injection point of the gas network. To fulfill this objective, various components are to be developed and integrated: an intelligent gas analyzer, a machine-tomachine platform, a decision-making process, a global supervision system, etc. This project is conducted by GDF SUEZ and a consortium of 9 industrial and academic partners in the fields of engineering (gas analyzing), telecom (communications), and computer science (software engineering, supervision, and assisted decision-making). In this article, we will focus on the interoperable integration platform.

The current GDF architecture uses Supervisory Control and Data Acquisition (SCADA) [START_REF] Zhang | Advanced Industrial Control Technology[END_REF] systems. SCADA systems are widely used to monitor and control remote equipment. It includes the full chain from the equipment control module to the acquisition server and up to the human machine interface (HMI). The acquisition server aggregates the data from various devices and passes commands to this equipment. A HMI is directly connected to this server to monitor and command the equipment. However, for business and equipment reasons, various SCADA systems may coexist in a given environment.

One of the viable technical solutions for implementing SCADA is the OPC Unified Architecture (OPCUA) industry standard. It provides a wide range of capabilities and is often seen as a potential solution in the industrial evolution (Industry 4.0 [START_REF] Drath | Industrie 4.0: Hit or Hype? [Industry Forum[END_REF]), Internet of thing (IOT) and smart environments [START_REF]OPC Unified Architecture: Interoperability for Industrie 4.0 and the Internet of Things[END_REF]. OPCUA defines communication mechanisms and abstract information models that allow domain-specific data models creation. Using OPCUA, the acquisition server needs to be configured whenever new equipment is added to its pool.

Additionally, a number of other systems take part in the environment such as Geographic Information Systems (GIS), Computerized Maintenance Management Systems (CMMS), weather forecasters, computer-assisted decision-making systems, and others. A central dashboard room must be able to supervise these various components. Figure 1 illustrates an example of such a deployed environment. All of these systems may need to contribute and receive data in order to achieve global operational goals, and these components do not usually share the same communication and format standards for data exchange. Here are some examples of such scenarios:

 A new equipment is added in the field. This information may be provided by the CMMS component in a CSV file. The corresponding acquisition server must be reconfigured using the OPCUA standard.

 An alarm is generated by the SCADA indicating a problem with an equipment. Using the localization information from the CMMS, an indicator is to be displayed on the map provided by the GIS component.

 A simulation must be run on the gas network in order to evaluate the viability of green gas injection at some point. This requires the aggregation of various data sources: network topology (GIS and/or CMMS), weather conditions (weather forecaster), and current distribution status (SCADA components).

 An assisted decision-making process requires the visualization of global indicators from aggregated data.

These scenarios allow for a first important level distinction between data exchanges. On the one hand configuration data is to be shared whenever the system integrates new components. On the other hand operating data is exchanged at runtime for the achievement of global supervision objectives.

Based on our industrial partners feedback, our first evaluation shows that these issues are currently handled using ad hoc developments centralized in the SCADA dashboard component (acting as the question marks on Figure 1). These solutions, which involve numerous human manual interventions, are time-consuming, error-prone, and lack both flexibility and generality.

III. RELATED WORK

Interoperability is an essential requirement for smart environments since it guarantees correct integration of systems and assets [START_REF] Uslar | Standardization in Smart Grids[END_REF]. Several workgroups and organization steer their efforts towards solving this issue of interoperability using a layered architectural approach such as the GridWise Architectural Council [START_REF]GridWise Interoperability Context-Setting Framework[END_REF] or the Smart Grid Architecture Model [START_REF]Smart Grid Reference Architecture[END_REF].

Within these architectures, domain-specific standards have been proposed at the syntactic and semantic layers to guarantee information interoperability. For instance, for the electricity domain, the Common Information Model (IEC 61970, 61968 and 62325) [START_REF] Uslar | The Common Information Model CIM: IEC 61968/61970 and 62325 -A Practical Introduction to the CIM[END_REF] defines components of the electrical power systems and their relationships. In the oil and gas industry several standards coexist, such as ISA-95 [START_REF]Enterprise-control system integration[END_REF], B2MML (Business to Manufacturing Markup Language), MIMOSA [START_REF]An Operations and Maintenance Information Open System Alliance[END_REF], ISO-15926 [START_REF]Industrial automation systems and integration-Integration of life-cycle data for process plants including oil and gas production facilities[END_REF], or PRODML [START_REF] Energistics | PRODML Standards[END_REF]. OPCUA [START_REF] Mahnke | OPC Unified Architecture[END_REF] is a generic standard for data integration platforms. Work has been done the electricity domain for the mapping of CIM to OPCUA [START_REF] Rohjans | UML-based modeling of OPC UA address spaces for power systems[END_REF]. However studies [START_REF] Veyber | Model-driven Platform for Oil and Gas Enterprise Data Integration[END_REF] show the limitations of these standards for smart gas networks. [START_REF] Veyber | Model-driven Platform for Oil and Gas Enterprise Data Integration[END_REF] also proposed the use of a model-based integration framework using PRODMML as the central metamodel. In contrast to these unification approaches, in our work, we seek federated interoperability solutions which do not impose the use of a centralized unified domain-specific standard. Model-based engineering (MBE) [START_REF] Daz | Advances and Applications in Model-Driven Engineering[END_REF], which naturally promotes separation of concerns, could support such a federated approach.

In MBE, everything is a model. The OMG proposes a 3layer architecture [START_REF] Omg | Meta Object Facility (MOF)[END_REF] . Its main principles are illustrated in Figure 2. Models are described through modeling languages (metamodels), themselves described by an auto-descriptive In the following, considering the specific scenarios and issues of our smart gas environment, we propose to study solutions, based on model-based engineering principles, to ease data exchange at both configuration and operating levels while preserving the decentralized nature of the global system.

IV. A MODEL-BASED HUB FOR DATA EXCHANGE

In this section, we sketch a solution based on modeling principles to realize a flexible mediator, dubbed "SmartHub", able to realize the various data exchanges required in our smart gas environment. We then describe its application on two industrial scenarios.

We have defined a number of features desirable for such a component:  It must be able to aggregate data originating from various sources which use different communication and data format standards. Symmetrically, it must be able to produce and communicate data for various components.

 It must be able to work at both the configuration and operating levels.

 It should not be a centralized mandatory component, i.e., it should not prevent direct communication between components when this is already possible.

 It should ease the integration of new components through a flexible interface definition

A. SmartHub main principles

Figure 3 illustrates the main principles that constitute the SmartHub:

 An adaptable layer manages various communication protocols in order to connect to the different systems. It currently supports various subscribe/request and publish/response standards.

 An extensible layer, based on a repository of supported data formats and syntactic/semantic rules, handles the projection of data between components technical spaces and the modelling environment.

 A global data model (GDM) allows the aggregation and fetching of specific data through a set of operational rules. Although the description and evaluation of this GDM is out of scope of the paper, it can be noted that it relies on an inner distinction between configuration data (meta-data, system components, etc.) and operating data. Although its generalization is considered, it is currently specific to the studied smart gas environments.

In the following, we describe its application on a configuration data exchange scenario and an operating data exchange scenario.

B. Configuring SCADA servers with equipment information

In this scenario, roughly sketched in Figure 4, the maintenance system (CMMS) holds the equipment information required to configure the OPCUA SCADA servers. Following the principles described in the previous subsection, a number of operations, illustrated in Figure 5, are performed by the SmartHub:

1. The CMMS sends the new configuration data (here, a CSV file) through a simple request/response mechanism.

2. The SmartHub looks into its projection repository to inject the data as a model.

3. The SmartHub applies the (predefined) operational rules to aggregate the data into the GDM.

4. Fetching rules are applied to obtain the data required by the target components from the GDM.

5. For each target component, considering its data format and the projection repository, a corresponding data file is extracted. In our case, a set of OPCUA files are generated. Although it is out of scope of this paper to describe it fully, it should be noted that OPCUA particular semantics require a complex chain of model operations.

6. Finally, this data is exchanged through the corresponding communication mechanism (here a simple publish/subscribe standard).

C. Displaying alarms from a SCADA server on a GIS

In this scenario, roughly sketched in Figure 6, runtime data must be aggregated from both an OPCUA SCADA server and a CMMS component and then delivered to a GIS component. Following the same modeling principles, a number of operations, illustrated in Figure 7, are performed by the SmartHub:

1. The SCADA server raises an OPCUA alarm obtained through a publish/subscribe mechanism.

2. Using the projection repository, the alarm data is injected into the modeling environment. The GDM might already have the localization data required for the GIS system from a previous configuration data exchange. If not, the SmartHub may request additional localization information from the CMMS, which is then also injected into the modeling environment.

3. According to the operational rules, (both) data is aggregated into the GDM.

4. Fetching rules allow the production of a model containing the necessary target data.

5. Thanks to the projection repository, the data is formatted according to the target component requirements (here a KML file).

6. Finally, data is sent to the GIS through publish/subscribe.

D. Architecture Implementation

In this work, we have chosen the Eclipse Modeling Framework (EMF) [START_REF]Eclipse Modeling Framework[END_REF] to support the implementation of the SmartHub solution. EMF supports model and data interchange via the XML Metadata Interchange (XMI) format. Metamodels are defined using the ECORE language. Projections have been realized using a combination of the Acceleo tool [START_REF]Acceleo[END_REF] and EMF's built-in XSD/XML support. The aggregation and fetching rules have been implemented using a combination of the rule-based ATL tool [START_REF] Jouault | Transforming models with ATL[END_REF] and direct java manipulation. Indeed, our persisting framework requires incremental capabilities that are hardly handled by current transformation tools [START_REF] Czarnecki | Feature-based survey of model transformation approaches[END_REF]. Finally, communications are supported through specific Eclipse plugins. Evaluation and selection of the various available tools with regard to our whole set of industrial scenarios is under investigation. In this paper, we have studied interoperability solutions between heterogeneous systems for an existing industrial smart gas environment. In particular, in a bottom-up approach, we have exhibited various scenarios in which we distinguish between configuration and operating data exchange levels. Rather than relying on a unifed standard, we suggested a federated model-based software engineering solution able to handle various standards, presented its main principles, and described its application on two industrial scenarios.

CMMS

SCADA Server

The proposed SmartHub is still in its early phases of development. A number of theoretical and practical issues have arised which are subject to ongoing and future work. At the modeling level, we currently investigate the generality of our global data model. Indeed, our experience shows that though many concepts are independent from the application domain, some specifics still need to be considered. We currently use a two-level (configuration/operating) generic metamodel which is extended with smart gas networks considerations. We plan to fully describe and formalize this modeling architecture. At the implementation level, the smartHub requires model operations with advanced features such as incrementality and synchronization mechanisms. We thus wish to study the viability of existing tools to replace some of our ad hoc developments. Finally, a number of operational issues (hub data persistence and historization, of aggregation/fetching rules, performance) are still under investigation.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Example deployment for the supervision system of a smart gas distribution network

Fig. 4 .Fig. 5 .Fig. 3 .

 453 Fig. 4. Exchange of configuration data between CMMS and SCADA server

Fig. 6 .

 6 Fig. 6. Displaying alarm on GIS

Fig. 7 .

 7 Fig. 7. Displaying an alarm from SCADA on the GIS

Gontrand FUI project: www.advancity.eu/info-fui-17


Antoine Girard

Paolo Mason

Lyapunov Functions for Shuffle Asymptotic Stability of Discrete-Time Switched Systems

Keywords: Switched systems, Shuffle stability, Lyapunov methods

In this paper, we investigate stability of discretetime switched systems under shuffled switching signals. A switching signal is said to be shuffled if each mode of the switched system is activated infinitely often. We introduce the notion of shuffle Lyapunov functions and show that the existence of such a function is a sufficient condition for global uniform shuffle asymptotic stability. In the second part of the paper, we show that for a specific class of switched systems, with linear and invertible dynamics, existence of a shuffle Lyapunov function is also necessary, even for the weaker notion of global shuffle attractivity. Examples and numerical experiments are used to illustrate the main results of the paper.

I. INTRODUCTION

Switched systems consist of a collection of dynamics with a switching rule specifying at each time instant the dynamics describing the evolution of the system. The popularity of switched systems is due to the fact that despite the simplicity of their mathematical formulation, they can exhibit complex and non-intuitive behavior. Moreover, their relevance in many engineering applications has motivated abundant research for almost three decades.

Early work on stability of switched systems has focused on stability for switching signals that are arbitrary or that satisfy some (minimum or average) dwell-time condition (see e.g. [START_REF] Sun | Switched linear systems: control and design[END_REF], [START_REF] Liberzon | Switching in systems and control[END_REF], [START_REF] Lin | Stability and stabilizability of switched linear systems: a survey of recent results[END_REF] and the references therein). More recently, several works have considered the problem of proving stability for subsets of switching signals. In general, such switching signals are assumed to be generated by some finite state automaton and stability is characterized either in term of constrained joint spectral radius [START_REF] Dai | A Gel'fand-type spectral radius formula and stability of linear constrained switching systems[END_REF], [START_REF] Kozyakin | The Berger-Wang formula for the Markovian joint spectral radius[END_REF], [START_REF] Xu | Approximation of the constrained joint spectral radius via algebraic lifting[END_REF] or using Lyapunov functions [START_REF] Lee | Uniformly stabilizing sets of switching sequences for switched linear systems[END_REF], [START_REF] Athanasopoulos | Stability analysis of switched linear systems defined by graphs[END_REF], [START_REF] Philippe | Stability of discrete-time switching systems with constrained switching sequences[END_REF], [START_REF] Pepe | Converse Lyapunov theorems for discrete-time switching systems with given switches digraphs[END_REF]. However, there are some subsets of switching signals that cannot be specified using classical finite state automata. For instance, [START_REF] Kundu | On stability of discrete-time switched systems[END_REF] considers switching signals specified using weighted-automata.

In this paper, we consider the class of discrete-time shuffled switching signals. A switching signal is said to be shuffled if each mode of the switched system is activated infinitely often. Since the sets of liveness and safety properties are disjoint [START_REF] Baier | Principles of model checking[END_REF], the set of shuffled switching signals, which is an example of liveness property, cannot be characterized using classical finite state automata that can only describe safety properties. Actually, shuffled switching signals may be seen as an example of ω-regular language firstname.lastname@l2s.centralesupelec.fr which can be characterized using either non-deterministic Büchi automata, deterministic Rabin automata, or deterministic Muller automata, which are finite state automata with various acceptance conditions (see e.g. [START_REF] Baier | Principles of model checking[END_REF]). Shuffled switching signals have first been introduced in [START_REF] Gurvits | Stability of discrete linear inclusion[END_REF] for switched linear systems, where a necessary and sufficient condition for shuffle asymptotic stability (i.e. asymptotic stability with respect to shuffled switching signals) has been proposed based on the maximum spectral radius of an infinite set of matrices. Shuffled switching signals are also considered in [START_REF] Wang | Stability analysis of switched linear systems defined by regular languages[END_REF], where it is shown that stability of a class of switched systems with switching signals constrained by an arbitrary ω-regular language is equivalent to shuffle asymptotic stability of an associated lifted switched system. In the same paper, it is shown that some robust notion of shuffle asymptotic stability is equivalent to asymptotic stability for arbitrary switching signals.

Shuffled switching signals can also be related to multiagent consensus with switching communication topologies as follows. Consider that at each time instant only one edge in the communication graph is active and is selected by a switching signal. If one uses a shuffled switching signal, then each edge is activated infinitely often so that the union of future communication graphs is connected at all time, and it is well-known (see e.g. [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF], [START_REF] Blondel | Convergence in multiagent coordination, consensus, and flocking[END_REF]) that the consensus is asymptotically reached. On the contrary, if some edges are never activated the union of future communication graphs may be disconnected and the consensus cannot be approached.

In this work, we develop for the first time a Lyapunov approach to stability analysis for shuffled switching signals.

In Section III, we introduce the notion of shuffle Lyapunov functions and show that the existence of such a function is a sufficient condition for global uniform shuffle asymptotic stability. In Section IV, we show that for a class of switched systems, with linear and invertible dynamics, existence of a shuffle Lyapunov function is also necessary, even for the weaker notion of global shuffle attractivity. Several examples are used to illustrate the main results of the paper. Finally, in Section V, a numerical example is provided.

Notations: In this paper, . denotes the usual Euclidean norm on R n . A function γ : R + 0 → R + 0 is a K function if it is continuous, strictly increasing and with γ(0) = 0; it is a K ∞ function if it is a K function and additionally lim r→+∞ γ(r) = +∞. A function β : R + 0 × R + 0 → R + 0 is a KL function if for all s ∈ R + 0 , β(., s) is a K function and for all r ∈ R + 0 , β(r, .) is non-increasing and lim s→+∞ β(r, s) = 0.

II. SHUFFLE ASYMPTOTIC STABILITY

We consider a discrete-time switched system of the following form:

x(t + 1) = f θ(t) (x(t)), t ∈ N (1)
where x : N → R n is the trajectory and θ : N → Σ is the switching signal. Σ = {1, . . . , m}, with m ≥ 2, is the finite set of modes and {f σ : R n → R n |σ ∈ Σ} is the collection of vector fields. Given an initial state x 0 ∈ R n , and a switching signal θ, the trajectory defined by (1) with x(0) = x 0 is unique and denoted x(., x 0 , θ).

In this paper, we are interested in a particular class of switching signals called shuffled. In words, a switching signal is said to be shuffled if each mode in Σ is activated infinitely often. This translates formally as below:

Definition 1. A switching signal θ : N → Σ is shuffled if ∀σ ∈ Σ, ∀T ∈ N, ∃t ≥ T : θ(t) = σ.
The sequence of shuffling instants (τ θ k ) k∈N is defined by τ θ 0 = 0 and for all k ∈ N,

τ θ k+1 = min t > τ θ k ∀σ ∈ Σ, ∃s ∈ N : τ θ k ≤ s < t and θ(s) = σ .
The shuffling index κ θ : N → N is given by

κ θ (t) = max{k ∈ N| τ θ k ≤ t}.
Intuitively, between the instants τ θ k and τ θ k+1 -1, each mode in Σ is activated at least once. For a shuffled switching signal, the sequence of shuffling instants is well-defined, strictly increasing and thus goes to +∞. Similarly, the shuffling index is well-defined, non-decreasing and goes to +∞. An illustration of these concepts is shown in Figure 1.

We study stability properties of switched system (1) under the effect of shuffled switching signals.

Definition 2. Switched system (1) is said to be • globally shuffle attractive (GSA) if for all x 0 ∈ R n , for all shuffled switching signal θ:

lim t→+∞ x(t, x 0 , θ) = 0;
• globally uniformly shuffle asymptotically stable (GUSAS) if there exists a KL function β such that for all x 0 ∈ R n , for all shuffled switching signal θ:

x(t, x 0 , θ) ≤ β( x 0 , κ θ (t)), ∀t ∈ N.
We first establish some relations between stability notions:

Proposition 1. If switched system (1) is GUSAS, then it is GSA and for all x 0 ∈ R n , for all switching signal θ:

x(t, x 0 , θ) ≤ β( x 0 , 0), ∀t ∈ N.
Proof. If switched system (1) is GUSAS, then for all x 0 ∈ R n , for all shuffled switching signal θ

lim t→+∞ x(t, x 0 , θ) ≤ lim t→+∞ β( x 0 , κ θ (t)) = 0. t θ(t) t κ θ (t) τ θ 0 τ θ 1 τ θ 2 τ θ 3
Fig. 1. A switching signal (with m = 2) and the associated shuffling instants and index.

For the second part of the proposition, let x 0 ∈ R n , and θ be a switching signal (not necessarily shuffled). Let t ∈ N, it is clear that there exists a shuffled switching signal θ t , which coincides with θ up to time t. Then, we get

x(t, x 0 , θ) = x(t, x 0 , θ t ) ≤ β( x 0 , κ θt (t)) ≤ β( x 0 , 0).
It is important to remark that the second part of Proposition 1 holds for all switching signals (not only shuffled). Then, if a switched system is GUSAS, it is also Lyapunov stable for all arbitrary switching signals. It is also clear that if a switched system is globally uniformly asymptotically stable for all switching signals then it is GUSAS. However, the converse is not true (see Example 1). In addition, a GSA switched system need not be GUSAS (see Example 2).

III. SUFFICIENT CONDITIONS BASED ON LYAPUNOV FUNCTIONS

In this section, we establish sufficient conditions for a switched system to be GUSAS based on the following notion of Lyapunov function:

Definition 3. A function V : Σ × R n → R + 0 is called a shuffle Lyapunov function if there exists K ∞ functions α 1 , α 2 ,
and a K function ρ with ρ(r) < r for all r > 0, such that for all x ∈ R n , the following hold:

α 1 ( x ) ≤ V (σ, x) ≤ α 2 ( x ), σ ∈ Σ (2) V (σ, f σ (x)) ≤ V (σ, x), σ, σ ∈ Σ, σ = σ (3) V (σ + 1, f σ (x)) ≤ V (σ, x), σ ∈ Σ \ {m} (4) V (1, f m (x)) ≤ ρ(V (m, x)) (5)
Theorem 1. If there exists a shuffle Lyapunov function, then the switched system (1) is GUSAS.

Proof. Let us consider an initial condition x 0 ∈ R n and a shuffled switching signal θ, let us denote x(.) = x(., x 0 , θ). Let η : N → Σ be defined by η(0) = 1 and the following rules:

η = 1 η = 2 η = m θ = 1 θ = 1 θ = 2 θ = 2 θ = m -1 θ = m θ = m
     if θ(t) = η(t), then η(t + 1) = η(t); if θ(t) = η(t) and η(t) = m, then η(t + 1) = η(t) + 1; if θ(t) = η(t) and η(t) = m, then η(t + 1) = 1. (6)
Equivalently, the evolution of η can be described by the finite state automaton shown on Figure 2. Now, let us consider

W : N → R + 0 , defined by W (t) = V (η(t), x(t))
, for all t ∈ N. According to the dynamics of x and η, it follows from (3), ( 4) and ( 5) that for all t ∈ N, W (t + 1) ≤ W (t).

Let k ∈ N, between the shuffling instants τ θ k and τ θ k+1 -1, there exists an instant t such that θ(t) = η(τ θ k ). Then, it follows that either η(τ θ k+1 ) ≥ η(τ θ k ) + 1 or there exists an instant t between τ θ k and τ θ k+1 -1 such that η(t) = m and η(t + 1) = 1. From the previous statement, we can conclude that between the instants τ θ k and τ θ k+m -1, there exists an instant t such that η(t) = m and η(t + 1) = 1, which from [START_REF] Kozyakin | The Berger-Wang formula for the Markovian joint spectral radius[END_REF] gives that W (t + 1) ≤ ρ(W (t)). Using the fact that W is non-increasing the rest of the time, we get for all k ∈ N,

W (τ θ k+m ) ≤ ρ(W (τ θ k )). Hence, it follows that ∀i ∈ N, W (τ θ i×m ) ≤ ψ(W (τ θ 0 ), i) = ψ(W (0), i) (7)
where ψ : R + 0 × N → R + 0 is given for all r ∈ R + 0 by ψ(r, 0) = r and ψ(r, i + 1) = ρ(ψ(r, i)), for i ∈ N. Since ρ is a K function, ψ(., i) is also a K function for all i ∈ N. Since 0 ≤ ρ(s) < s for all s > 0, we get that for all r > 0, ψ(r, .) is strictly decreasing and bounded below by 0. By continuity of ρ we get that lim i→+∞ ψ(r, i) = 0. Then, there exists a KL function ψ :

R + 0 × R + 0 → R + 0 such that for all r ∈ R + 0 and i ∈ N, ψ(r, i + 1) = ψ(r, i). Let t ∈ N, and let i ∈ N such that t ∈ [τ θ i×m , τ θ (i+1)×m ), we get that i×m ≤ κ θ (t) < (i+1)×m.
Using monotonicity properties of W and ψ and by [START_REF] Lee | Uniformly stabilizing sets of switching sequences for switched linear systems[END_REF], we have

W (t) ≤ W (τ θ i×m ) ≤ ψ(W (0), i) ≤ ψ(W (0), i + 1) ≤ ψ(W (0), κ θ (t)/m)
Then, by (2), we get that for all t ∈ N,

x(t) ≤ α -1 1 ψ α 2 ( x(0) ), κ θ (t)/m
Let β : R + 0 ×R + 0 → R + 0 , be given for r, s ∈ R + 0 by β(r, s) = α -1 1 ψ α 2 (r), s/m . Using the properties of α 1 , α 2 and ψ, we get that β is a KL function.

The intuition of the proof of Theorem 1 is the following. The function W is always non-increasing and is strictly decreasing when the value of η transitions from m to 1. By construction of the dynamics of η, this happens an infinite number of times if and only if θ is a shuffled switching signal. Thus W asymptotically converges towards 0.

Remark 1. If α 1 and α 2 are quadratic functions (α 1 (r) = γ 1 r 2 , α 2 (r) = γ 2 r 2 , with 0 < α 1 < α 2 ) and if ρ is linear (ρ(r) = λr with 0 < λ < 1), then we get that β is given by the closed form expression β(r, s) = γ2 γ1λ λ s 2m r.

Example 1. Let us consider a switched system in R 3 with 2 modes, where the vector fields are given by f σ (x) = A σ x where

A 1 = 1 0 0 0 λ1 cos(φ1) -λ1 sin(φ1) 0 λ1 sin(φ1) λ1 cos(φ1) , φ 1 ∈ [0, 2π], λ 1 ∈ (0, 1) A 2 = λ2 cos(φ2) -λ2 sin(φ2) 0 λ2 sin(φ2) λ2 cos(φ2) 0 0 0 1 , φ 2 ∈ [0, 2π], λ 2 ∈ (0, 1).
It is clear that for unshuffled switching signals (i.e., in this case, switching signals that remain constant after some time), the state of the switched system will not converge to 0 asymptotically. To verify that the switched system is GUSAS, we use Theorem 1 with

V (1, x) = x 2 1 + x 2 2 + x 2 3 and V (2, x) = x 2 1 + x 2 2 + x 2 3
λ1 . To simplify the exposition, we introduce the auxiliary function

U (x) = x 2 1 + x 2 2 λ1 + x 2 3
λ1 . It is clear that (2) holds with α 1 (r) = r and α 2 (r) = r λ1 . Then, for (3), we check that

V (1, A 2 x) = λ 2 x 2 1 + λ 2 x 2 2 + x 2 3 ≤ V (1, x)
and

V (2, A 1 x) ≤ U (A 1 x) = x 2 1 + x 2 2 + x 3 3 ≤ V (2, x).
From above, we also have that V (2, A 1 x) ≤ V (1, x) and thus (4) holds. Finally, we also have

V (1, A 2 x) ≤ max(λ 1 , λ 2 )V (2, x).
Then, (5) holds with ρ(r) = max(λ 1 , λ 2 )r. It follows that the switched system is GUSAS.

IV. A CONVERSE LYAPUNOV RESULT

In this section, we identify a class of switched systems for which the sufficient conditions exposed in the previous section are also necessary for GUSAS. For this class of systems, we actually show an even stronger result, since we prove that if the switched system is GSA then there exists a shuffle Lyapunov function. This class is characterized by the following assumption: Assumption 1. For all σ ∈ Σ, f σ (x) = A σ x where A σ is an invertible matrix.

Before stating our main theorem, let us establish instrumental preliminary results.

A. Preliminary results

Let I ∈ R n×n denote the identity matrix, given a finite sequence of matrices

M i ∈ R n×n , i = N 1 , . . . , N 2 with N 1 , N 2 ∈ N, N 1 ≤ N 2 , we denote N2 i=N1 M i = M N2 × • • • × M N1 .
Then, we define several sets of matrices:

• M consists of all finite products of matrices in {A σ | σ ∈ Σ}, together with the identity matrix; • M σ consists of all finite products of matrices in {A σ | σ ∈ Σ \ {σ}}, for σ ∈ Σ, together with the identity matrix; • M s consists of all finite products of matrices in {A σ | σ ∈ Σ} where each matrix appears at least once. Formally, we have:

M = N i=0 A σi σ i ∈ Σ, i = 0, . . . , N, N ∈ N ∪ I , M σ = N i=0 A σi σ i ∈ Σ \ {σ}, i = 0, . . . , N, N ∈ N ∪ I , M s = N i=0 A σi σ i ∈ Σ, i = 0, . . . , N, N ∈ N and ∀σ ∈ Σ, ∃i : σ i = σ .
Clearly, we have M σ ⊆ M for all σ ∈ Σ and M s ⊆ M.

The first two lemmas show that the convergence of trajectories towards 0 is equivalent to the convergence of the product of matrices towards 0: Lemma 1. Under Assumption 1, if the switched system (1) is GSA, then for all shuffled switching signal θ:

lim N →+∞ N i=0 A θ(i) = 0.
Proof. Let θ be a shuffled switching signal, then

∀t ≥ 1, ∀x 0 ∈ R n , x(t, x 0 , θ) = t-1 i=0 A θ(i) x 0 . (8)
Let us consider δ > 0 and let e 1 , . . . , e n denote the vectors of the canonical basis of R n . Since the system is GSA, there exists T j , j = 1, . . . , n such that ∀j = 1, . . . , n, ∀t ≥ T j , x(t, e j , θ) ≤ δ n .

Then, let x 0 ∈ R n with coordinates x 0,1 , . . . , x 0,n , let T = max(T 1 , . . . , T n ), by linearity, it follows from above that

∀t ≥ T, x(t, x 0 , θ) ≤ n i=1 |x 0,j | x(t, e j , θ) ≤ x 0 δ.
Then by [START_REF] Athanasopoulos | Stability analysis of switched linear systems defined by graphs[END_REF] we get that ∀t ≥ T,

t-1 i=0 A θ(i) ≤ δ,
which leads directly to the statement of Lemma 1.

The proof of the following result is straightforward from the previous lemma: Lemma 2. Under Assumption 1, if the switched system (1) is GSA, then for all sequences of matrices (M i ) i∈N with M i ∈ M s we have

lim N →+∞ N i=0 M i = 0.
Lemma 1 with Theorem 20 in [START_REF] Wang | Stability analysis of switched linear systems defined by regular languages[END_REF] gives the following: Lemma 3. Under Assumption 1, if the switched system (1) is GSA, then there exists B > 0 such that for all M ∈ M, M ≤ B.

We finally get the following property: Lemma 4. Under Assumption 1, if the switched system (1) is GSA, then there exist λ ∈ (0, 1) and a norm of R n denoted . * such that for all M ∈ M s , M * ≤ λ.

Proof. By Lemma 3, we get that all matrices in M s are bounded by B. Together with Lemma 2, we get from Theorem 1 in [START_REF] Berger | Bounded semigroups of matrices[END_REF], that the joint spectral radius of M s is strictly less than 1. Then, the conclusion is obtained by Lemma 2 in [START_REF] Berger | Bounded semigroups of matrices[END_REF].

B. Converse Lyapunov theorem

We can now state the main result of the section:

Theorem 2. Under Assumption 1, the switched system ( 1) is GSA if and only if there exists a shuffle Lyapunov function.

Proof. The "if" part is a consequence of Theorem 1 and Proposition 1. The proof of the "only if" part is as follows.

Let . * be as in Lemma 4. Since all norms on R n are equivalent, there exist 0 < κ 1 < κ 2 such that for all x ∈ R n ,

κ 1 x ≤ x * ≤ κ 2 x . Let V : Σ × R n → R + 0 be defined for σ ∈ Σ \ {m} by V (σ, x) = sup Mm∈Mm, Mm-1∈Mm-1, ..., Mσ∈Mσ M m m-1 i=σ A i M i x * , (9)
and

V (m, x) = sup Mm∈Mm M m x * . ( 10
)
Let us prove that V is a shuffle Lyapunov function.

Since I belongs to M σ for all σ ∈ Σ we get for all σ ∈ Σ \ {m},

V (σ, x) ≥ m-1 i=σ A i x * ≥ κ 1 m-1 i=σ A i x and V (m, x) ≥ x * ≥ κ 1
x . Since all matrices A σ are invertible it follows that there exists γ 1 > 0 such that V (σ, x) ≥ γ 1 x for all σ ∈ Σ, for all x ∈ R n . From ( 9) and [START_REF] Pepe | Converse Lyapunov theorems for discrete-time switching systems with given switches digraphs[END_REF], and from Lemma 3, we get that for all σ ∈ Σ, for all

x ∈ R n V (σ, x) ≤ sup M ∈M M x * ≤ κ 2 sup M ∈M M x ≤ κ 2 B x . Hence, (2) holds. Let σ, σ ∈ Σ with σ = σ . If σ = m, V (σ, A σ x) = sup Mm∈Mm, M m-1 ∈M m-1 , ..., Mσ ∈Mσ M m m-1 i=σ A i M i A σ x * . For all M σ ∈ M σ , M σ A σ ∈ M σ and therefore it follows that V (σ, A σ x) ≤ V (σ, x). If σ = m, V (m, A σ x) = sup Mm∈Mm M m A σ x * . For all M m ∈ M m , M m A σ ∈ M m and therefore it follows that V (m, A σ x) ≤ V (m, x). Hence (3) holds. Let σ ∈ Σ \ {m}. If σ = m -1, then V (σ + 1, A σ x) = sup Mm ∈Mm, M m-1 ∈M m-1 , ..., M σ+1 ∈M σ+1 M m m-1 i=σ+1 A i M i A σ x * . Since I ∈ M σ , it follows that V (σ + 1, A σ x) ≤ V (σ, x). If σ = m -1, then V (σ + 1, A σ x) = sup Mm∈Mm M m A m-1 x * . Since I ∈ M m-1 , it follows that V (σ + 1, A σ x) ≤ V (σ, x). Hence (4) holds.
Finally, from Lemma 4,

V (1, A m x) = sup Mm∈Mm, M m-1 ∈M m-1 , ..., M 1 ∈M 1 M m m-1 i=1 A i M i A m x * ≤ sup M ∈Ms M x * ≤ λ x * ≤ λV (m, x).
Hence (5) holds and V is a shuffle Lyapunov function.

Corollary 1. Under Assumption 1, the switched system (1) is GSA if and only if it is GUSAS. Proof. The "if" part is a consequence of Proposition 1. For the "only if" part, if the switched system (1) is GSA then from Theorem 2, there exists a shuffle Lyapunov function, and from Theorem 1, the switched system is GUSAS.

Example 2. The following example shows that by removing Assumption 1, the converse Lyapunov result stated in Theorem 2 does not hold anymore. Let us consider a switched system in R 2 with 2 modes, where the vector fields are given by f σ (x) = A σ x where A 1 = ( 0 0 0 2 ) , A 2 = ( 2 0 0 0 ) . Clearly, the switched system is GSA since after both modes have been activated, the state of the system is equal to 0 for all time. We proceed by contradiction. Let us assume that there exists a shuffle Lyapunov function, then by Theorem 1 the switched system is GUSAS. Hence, there exists a KL function β such that for all x 0 ∈ R 2 , for all shuffled switching signal θ:

x(t, x 0 , θ) ≤ β( x 0 , κ θ (t)), ∀t ∈ N.

Consider x 0 = ( 0 1 ), and for t ∈ N let θ t be a shuffled switching signal such that for all s = 0, . . . , t, θ t (s) = 1. Let us remark that x(t + 1, x 0 , θ t ) = 0 2 t while x 0 = 1 and κ θt (t + 1) = 0. Hence, we should have for all t ∈ N, 2 t ≤ β(1, 0), which yields a contradiction. Remark 2. While the invertibility of the matrices A σ , σ ∈ Σ is not used in the proofs for Lemmas 1 and 2, it is crucial for Lemmas 3 and 4 (see the proof of Theorem 20 in [START_REF] Wang | Stability analysis of switched linear systems defined by regular languages[END_REF]). However, when the switched system is GUSAS, the conclusion of Lemma 3 (and thus of Lemma 4 and Theorem 2) holds without the invertibility assumption. Thus, it follows from Theorem 1, that a switched linear system is GUSAS if and only if it admits a shuffle Lyapunov function.

V. NUMERICAL EXAMPLE As an illustration of the results of the paper, we consider a synchronization problem. We consider a multi-agent system consisting of three discrete-time oscillators whose dynamics is given by

z i (t + 1) = Rz i (t) + u i (t), i = 1, . . . , 3
where z i (t) ∈ R 2 , u i (t) ∈ R 2 and R = cos(φ) -sin(φ) sin(φ) cos(φ) with φ = π 6 . The input u i (t) is used for synchronization purpose and is based on the available information at time t. There exist two communication channels: channel 1 between agents 1 and 2 and channel 2 between agents 2 and 3. At each instant, only one of these channels is active and the active channel is selected by a switching signal θ : N → Σ = {1, 2}. Then, the input value is given as follows

u 1 (t) = γ(z 2 (t) -z 1 (t)), if θ(t) = 1 0 if θ(t) = 2 u 2 (t) = γ(z 1 (t) -z 2 (t)), if θ(t) = 1 γ(z 3 (t) -z 2 (t)), if θ(t) = 2 u 3 (t) = 0, if θ(t) = 1 γ(z 2 (t) -z 3 (t)), if θ(t) = 2
where γ = 0.1. Denoting the vector of synchronization errors as x(t) = z2(t)-z1(t) z3(t)-z2(t) , the error dynamics is described by a 4-dimensional switched linear system with 2 modes given by the following matrices where I 2 is the 2 dimensional identity matrix:

A 1 = R-2γI2 0 γI2 R , A 2 = R γI2 0 R-2γI2
. We want to show that the agents synchronize if both communication channels are activated infinitely often. This can be done by verifying that the error system is GUSAS. We then look for a shuffle Lyapunov function of the form V (σ, x) = x Q σ x where Q σ is a positive definite symmetric matrix. For this form of function, for a linear function ρ given by ρ(r) = λr with λ ∈ (0, 1), the conditions in Theorem 1 translate directly into the following linear matrix inequalities: Solving these numerically, we find for λ = 0.88:

I ≤ Q σ , σ ∈ Σ A σ Q σ A σ ≤ Q σ σ, σ ∈ Σ, σ = σ A σ Q σ+1 A σ ≤ Q σ , σ ∈ Σ \ {m} A m Q 1 A m ≤ λQ m .
Q 1 = 2.28I 2 1.14I 2 1.14I 2 2.01I 2 , Q 2 = 2.00I 2 1.02I 2 1.02I 2 2.05I 2 .
Hence, the system is GUSAS.

For illustration, we consider the following scenario: for the first 50 time units, communication channel 2 is constantly activated, then at t = 50 a switch occurs and for the next 50 time units communication channel 1 is constantly activated, after t = 100 the communication are activated randomly with equal probability so that the switching signal is shuffled with probability 1. The simulation results are shown in Figure 3. It is interesting to remark that when the switching signal remains constant the synchronization error does not go to zero. Indeed, before t = 50, agents 2 and 3 synchronize but not agents 1 and 2; and between t = 50 and t = 100, agents 1 and 2 synchronize but agents 2 and 3 desynchronize. After t = 100, the switching signal starts to shuffle and one can see that all agents achieve synchronization. Similar observations can be made on the evolution of the shuffle Lyapunov function V (η(t), x(t)) where η(t) is defined as in the proof of Theorem 1. As expected, this function is always non-increasing. When the switching signal remains constant, between t = 0 and t = 49, and between t = 50 and t = 99, one can see that the Lyapunov function reaches a constant value, which is not zero. As soon as the switching signal starts to shuffle, the Lyapunov function approaches 0.

VI. CONCLUSION

In this paper, we have established several stability results for discrete-time switched systems with shuffled switching signals. The main contribution of the paper consists in a sufficient condition for global uniform asymptotic stability given by the existence of a shuffle Lyapunov function. For a class of switched linear systems, this condition is also shown to be necessary, even for the weaker notion of global shuffle attractivity. We have shown on a numerical example, how shuffle Lyapunov functions can be effectively computed using linear matrix inequalities.

The current work opens several research directions for the future. First, the development of numerical techniques to compute shuffle Lyapunov functions is necessary for cases where the simple linear matrix inequalities approach used in the paper is not successful. Secondly, for switched linear systems, it should be possible to define a joint spectral radius related to shuffled switching signals. It would be interesting to investigate its properties and to develop algorithms for its approximation. Finally, as shuffle stability seems to be related to stability with ω-regular switching sequences, it would be interesting to develop a Lyapunov approach for switched systems with switching signals specified by some ω-regular language.

Fig. 2 .

 2 Fig. 2. Automaton describing the dynamics of η.

The authors are with Laboratoire des signaux et systèmes (L2S) CNRS, CentraleSupélec, Université Paris-Sud, Université Paris-Saclay. 3, rue Joliot-Curie, 91192 Gif-sur-Yvette, cedex, France