Iago Felipe Trentin
email: iagofelipe.trentin@orange.com

Olivier Boissier

Fano Ramparany

Insights about user-centric contextual online adaptation of coordinated multi-agent systems in smart homes

Keywords: Systèmes multi-agent, Planification, Action, Coordination, Adaptation, Maison intelligente Multi-agent systems, Planning, Acting, Coordination, Adaptation, Smart home

Avec le développement des maisons intelligentes, il est de plus en plus nécessaire de déployer des systèmes capables de s'adapter à des contextes fortement dynamiques ainsi qu'aux besoins des utilisateurs. Ces maisons sont de plus en plus équipées d'objets connectés qui manquent souvent de l'intelligence nécessaire à leur adaptation et coordination en ligne. Cet article a pour objectif d'analyser des approches susceptibles de répondre à ces défis et d'identifier des directions possibles de travaux futurs pour développer des systèmes multi-agents capables de combiner planification et action, en étant sensibles au contexte et dirigés par des buts, et qui pourront inspirer les générations futures de maisons intelligentes.

Introduction

With the advance of ubiquitous technologies, the physical objects and environments of our daily life are becoming more intelligent, capable of perceiving and acting in the world. But for the time being, such technologies have difficulties adapting to highly dynamic environments such as smart homes, and users also struggle to make different objects work together. This paper is motivated by a current smart home challenge: the ability to adapt to context in order to satisfy user evolving needs. Such complex behavior can only be achieved by embedding Artificial Intelligence (AI) techniques into Internet of Things (IoT) objects.

To tackle the aforementioned challenge, there are solutions in the literature that allow embedding intelligent entities (a.k.a. agents) with: acting capabilities, i.e. act in the world and react to predicted changes; and also planning capabilities, i.e. reason upon current context to find new solutions to a given need. In a smart home, for maximum user satisfaction, agents should be able to timely adapt to user needs, with minimum conscious and direct input from occupants. This online adaptation may be possible by endowing agents with planning and acting functions designed to execute together at run-time.

In this paper, we aim to bring up and renew questionings about adaptation capabilities of IoT objects, specifically focusing on online adaptation of plans and actions in a usercentric manner that respects user goals and user values. We analyze current state-of-the-art solutions, organizing them according to common properties we find in the literature. The understanding and exploring of the state-of-the-art approaches presented may allow future work to create a decentralised and coordinated planning and acting system. It would be: user-centric, gravitating towards enhancing human experience; goal-driven, intelligently reasoning about and pursuing user objectives; data-driven, adapting at runtime to environmental and user inputs; and trustworthy, being consistent with user values and expectations.

Based on these challenges we analyze the state of the art as follows: Section 2 presents the challenges we are interested in, with special focus on the smart home domain; Section 3 presents well-established solutions, especially in the planning and acting domains, that have potential to help addressing our challenges; Section 4 shows the interest in uniting planning and acting to ultimately have a synergistic solution; Section 5 identifies complementary domains, the goal and value reasoning, that add valuation to such a solution; Section 6 expands the envisioned solution to consider interaction and coordination in a decentralised system.

Motivation

This section elaborates the above stated challenges and interest in developing a system capable of adapting itself to context and users, in IoT environments such as smart homes. We consider publicly available solutions, their current state and limitations, that partially fulfill identified smart home requirements. It also shows aspects that can be improved or redesigned to better address our requirements. Since it explores a large domain, this paper is not expected to be an exhaustive study of the available literature.

Current IoT solutions

Current smart homes are becoming increasingly more complex, permeated by ubiquitous and heterogeneous elements designed by different vendors [START_REF] Kovatsch | Practical semantics for the internet of things: Physical states, device mashups, and open questions[END_REF]. Some solutions help with discovery of objects and composition of new services mixing available ones [START_REF] Kovatsch | Practical semantics for the internet of things: Physical states, device mashups, and open questions[END_REF][START_REF] Lécué | Composition de services web: Une approche basée liens sémantiques[END_REF], other popular solutions allow the development of IoT mashups 1 . For a great smart home user experience, the composition and delivery of user-centric services are important.

Although there is interest in ad-hoc mashups, often temporary and non-scalable, for small tactical and specific needs [START_REF] Guinard | Towards physical mashups in the web of things[END_REF], such applications do not present reasoning nor adaptive behavior. They simply perform tasks previously programmed by humans. Also, with discovery and composition of services but no intelligent entities to demand and use them, user experience tends to become cumbersomedue to constant need for manual actions, like system reconfiguration, user input, etc.

An IoT sub-domain, the growing Web of Things (WoT), allows objects to connect by means of the existing Web infrastructure [START_REF] Blackstock | loT mashups with the WoTKit[END_REF]. Nevertheless, the resulting system has a priori no coordinating abilities, therefore provided services may not be optimal, or can even conflict with each other.

Intelligent agents

In the previous section, we discussed the benefits in embedding intelligence into objects, giving them adaptive, reasoning, coordinating and self-managing capacities, which can be done by the use of AI techniques. In the AI field, an intelligent agent is a goal-driven entity capable of perceiving and acting upon its environment [START_REF] Russell | Artificial intelligence: a modern approach[END_REF]. An agent must have acting capabilities in order to interact with its environment, and this connection may have many forms.

An agent with direct and tight coupling to its environment performs reactive acting, i.e. a data-driven behavior where actions follow predefined models and plans.

One with a more flexible coupling performs adaptive acting, i.e. contextual changes trigger refinement 2 of avail-1 Such as https://nodered.org, and https://ifttt.com 2 Process of finding alternative actions applicable in the current context able actions to find possible alternatives to continue current plans, individually or collectively with other agents [START_REF] Ciortea | Responsive decentralized composition of service mashups for the internet of things[END_REF][START_REF] Vallée | Un intergiciel multi-agent pour la composition flexible d'applications en intelligence ambiante[END_REF].

Going further, an agent with a freely and relaxed coupling has a proactive behavior, presenting planning and reasoning capabilities. It is then able to reason upon available context information in order to find new solutions to a given need, allowing it to create new plans to achieve goals, which improves its adaptability.

Numerous solutions for industrial scenarios, such as manufacturing lines [START_REF] Ciortea | Repurposing manufacturing lines on the fly with multiagent systems for the web of things[END_REF], use intelligent agents, but these environments are predictable by design, being composed of agents that perform regular, repetitive and controlled processes. In highly dynamic scenarios, like smart homes, reasoners need to be both timely and context-aware in order to answer user needs fast and autonomously.

For an agent to be context-aware, it needs to perceive and understand the world. In order to do so, context collection, modeling and reasoning techniques [START_REF] Sorici | CONSERT: Applying semantic web technologies to context modeling in ambient intelligence[END_REF] are needed, specially in the IoT domain where sensors generate enormous amounts of raw data [START_REF] Perera | Context aware computing for the internet of things: A survey[END_REF].

We are interested in dynamic context modeling techniques because they present open and expandable models that can fit wide ranges of IoT environments [START_REF] Euzenat | Dynamic context management for pervasive applications[END_REF], and we know smart homes are highly heterogeneous and dynamic ones. In this paper we will consider that context is given ready-to-use by an external function.

Based on the analysis of this section, the agent of interest in a smart home is one capable of adaptive acting and context-aware planning.

Challenges and requirements

An advanced smart home is expected to perform responsive contextual adaptation in a user-centric manner. The use of agents to tackle this challenge allows us to identify some smart home requirements.

As said previously, such environments are highly dynamic, and the system's main objective is to satisfy user needs. Its final architecture must also take into account heterogeneous objects, while allowing flexible interaction among autonomous agents, creating a social IoT ecosystem [START_REF] Ciortea | Weaving a social web of things : Enabling autonomous and flexible interaction in the internet of things[END_REF], which we can call a multi-agent based smart home.

To identify smart home requirements the following scenario is presented: a user watching a movie receives a call.

In this case, the smart home has to be context-aware to perceive and understand the situation, and adaptive to change its plan at run-time, allowing the user to answer the call. It may, for example, mute the TV and activate subtitles. In this case the smart home needed no user input, being autonomous to pursue goals to satisfy user needs.

In the beginning of the scenario, when the user asked to watch a movie, it is interesting that the smart home is capable to perform goal reasoning and infer secondary related goals, such as turning of lights and closing windows.

Another aspect we tackle is the fact that a smart home is constantly interacting with humans, therefore it must be responsible, i.e. in accordance with user values. In our scenario, if the user privileges their leisure time more than their work, the smart home may decide to ignore the call if it was from the user's boss. But if family is of utmost importance, the system should never ignore a call from user's mom. This property goes beyond defining rules and constraints for system behavior [START_REF] Rust | Using message-passing DCOP algorithms to solve energyefficient smart environment configuration problems[END_REF], it dynamically reasons on the impact of user values on actions in current context.

The last property is coordination. The TV and the phone presented in the scenario must interact in order to perform any of the described adaptations.

It is important that all above properties are preserved during run-time, because smart home and ubiquitous systems in general must be constantly online and responsive.

Planning and Acting Agents

The previous section presented a global overview of the main motivation of this article, with special attention to smart homes. In these environments, we expect IoT objects to be intelligent agents, capable of acting upon and perceiving a dynamic world. This section will investigate the deliberation functions necessary to plan and act, and some state-of-the-art implementations. It is important to note that the word "deliberation" in this paper has the same meaning as in [START_REF] Ghallab | Automated Planning and Acting[END_REF], which can sometimes differ from the common meaning we see in multi-agent systems literature.

The adaptability, context-awareness and autonomy of a smart home depends on the sophistication of acting and planning functions. The acting function receives the context from the external world (see Figure 1). The planning function above gives plans to be used by the acting one.

Acting

The "Acting" shown in Figure 1 is a deliberation function, the one that has direct contact to the external world, responsible for following plans (a set of ordered actions built to achieve goals), and react to context (with every input it checks if plans are still feasible and chooses appropriate available actions to be executed), all driven by previously chosen goals.

Among existing agent architectures, a popular one is the BDI (belief-desire-intention) agent model [START_REF] Bordini | Programming Multi-Agent Systems in AgentSpeak using Jason[END_REF]. BDI agents have 3 components: beliefs, i.e. information about the current context; desires, i.e. goals the agent would like to achieve; intentions, i.e. agent's chosen goals to be achieved by executing a plan. BDI agents have a basic deliberation function that may be mapped to the "Acting" function described. The procedural plans building the plan library of BDI agent may be mapped to the "Know-How" appearing in Figure 1 in the sense that they define methods with their preconditions that can be used by the "Acting" function to execute and refine its execution based in context [START_REF] Ghallab | Automated Planning and Acting[END_REF].

The BDI architecture inspired several agent-oriented programming languages, such as Jason [START_REF] Bordini | Programming Multi-Agent Systems in AgentSpeak using Jason[END_REF], an implementation of AgentSpeak [START_REF] Rao | AgentSpeak(l): BDI agents speak out in a logical computable language[END_REF], technologies actively studied and expanded [START_REF] Ricci | AgentSpeak(ER): Enhanced encapsulation in agent plans[END_REF]. The BDI model has also been extended in many ways. For example, the suspension and resuming of goals, not present in the original model, is presented in [START_REF] Thangarajah | Suspending and resuming tasks in BDI agents[END_REF]. Also, a common goal notation called CAN (Conceptual Agent Notation) is proposed in [START_REF] Winikoff | Declarative & procedural goals in intelligent agent systems[END_REF].

Planning capabilities are not part of the BDI model. In Figure 1, the "Planning" step is executed before the agent becomes active -it means the agent will have access to a pre-compiled library of plans, but will not be able to create new plans during execution.

This behavior is similar to the Refinement Acting Engine (RAE) [START_REF] Ghallab | Automated Planning and Acting[END_REF]. RAE-enabled agents present adaptive acting capabilities -they do not perform search algorithms to build new plans, but simply execute available plans applicable in the current context to achieve desired goals.

There are similar approaches in the AI field to create adaptive systems, such as [START_REF] Diaconescu | Selfintegrating organic control systems: From craysh to smart homes[END_REF], and others that use machine learning techniques, like neural networks, etc. We will focus on intelligent agents because they have an explainable process, whose reasoning is often easier to follow and comprehend than other AI techniques.

Planning

"Planning" is a deliberation function which complements "Acting" by providing it a set of plans customized to achieve previously defined user goals.

Several planning techniques exist, such as first-principles planning (FPP, informally, the creation of new plans based upon lower-level actions to achieve a goal) [START_REF] Xu | A formal approach to embedding first-principles planning in BDI agent systems[END_REF][START_REF] Silva | First principles planning in BDI systems[END_REF], and markov decision process (MDP) [START_REF] Bellman | A markovian decision process[END_REF]. Hierarchical Task Network (HTN) is a well-known planning technique in which abstract actions are refined into lower-level actions, until tangible actions that can be sent to execution are obtained [START_REF] Herzig | On hierarchical task networks[END_REF]. In Figure 1, the HTN planner would be the "Planning" function. Its inputs are a set of user goals and an operational model (the "Know-What"), composed of more abstract actions than the commands in the "Know-How" set, which will be combined by the planner to create plans capable of achieving goals given a context. Its outputs are new plans. In this process, goals are the guiding elements of the planning process.

The difference between "Know-What" (descriptive model in [START_REF] Ghallab | Automated Planning and Acting[END_REF]) and "Know-How" (operational model in [START_REF] Ghallab | Automated Planning and Acting[END_REF]) in Figure 1 is: the former is a set of abstract actions used by the "Planning" function to create plans to achieve goals; the latter is a set of low-level commands used by the "Acting" to execute the abstract actions present in plans.

Agent

Online Adaptation

Using planning and acting functions, described in the previous section, one can implement a basic agent, capable of planning by using known beliefs to create plans to react to predicted changes to achieve goals. But its "Planning" is not context-aware, therefore it provides plans compiled off-line, and does not adapt at execution since it has no feedback from "Acting" as shown in Figure 1.

To overcome the smart home challenge of being capable of adapting to context to satisfy user evolving needs, we thus seek to connect planning and acting capabilities. The expected synergistic result is an agent capable of performing online contextual and user-centric adaptation.

In our scenario from Section 2.3, if the smart home system never encountered the situation before (a call while playing a movie), it may need to perform planning in order to adapt to it. It may, for example, create a plan to pause the film and resume it when the call ends. By doing so, it performed online adaptation, a feature that highly improves system adaptability, context-awareness and autonomy.

Combining planning and acting

The original BDI model can be extended to perform lookahead planning. The integration of planning algorithms and agent reasoning was surveyed in [START_REF] Meneguzzi | Planning in BDI agents: A survey of the integration of planning algorithms and agent reasoning[END_REF]. An example is the embedding of FPP in BDI agents [START_REF] Xu | A formal approach to embedding first-principles planning in BDI agent systems[END_REF][START_REF] Silva | First principles planning in BDI systems[END_REF], giving it some planning capabilities. Another is the use of HTN libraries as guides for BDI reasoning [START_REF] Silva | BDI agent reasoning with guidance from HTN recipes[END_REF] aiming at combining HTN domain and BDI, allowing the latter to have access to the information contained in the former. A different approach fuses HTN planner with mental attitudes (beliefs, desires, and intentions) of the BDI architecture [START_REF] Wickler | Planning and choosing: Augmenting HTN-based agents with mental attitudes[END_REF].

In a smart home, agents should do their best to autonomously adapt at run-time to user needs and environmental changes. Therefore we have interest in merging planning and acting capabilities, and even further, in a solution that perform both in a synergistic and online manner.

HTN planning and BDI acting

BDI agents and HTN planners are both well-known solutions. The former is capable of adaptive acting by interleaving refinement and execution of actions, and the latter performs planning in order to create new plans to achieve goals. The previous section showed that BDI agents can use the outputs (plans) from HTN planners, but Figure 1 clearly shows there is no feedback from the "Acting" function to the "Planning" one. It will prevent "Planning" from being context-aware, and the resulting agent will not be able to adapt to non-predicted situations.

If we establish that connection, as we can see in Figure 2, we create a feedback loop that allows the agent to perform online adaptation, i.e. online planning and acting. Now "Planning" has access to the current context, and can provide new plans to the "Acting" whenever it encounters a new situation, performing planning at run-time.

The Refinement Engine for Acting and Planning (REAP) [START_REF] Ghallab | Automated Planning and Acting[END_REF] and the BDI agent programming language named CANPlan [START_REF] Sardiña | Hierarchical planning in BDI agent programming languages: A formal approach[END_REF][START_REF] Sardiña | A BDI agent programming language with failure handling, tive goals, and planning[END_REF] are solutions that aim to incorporate HTN-style planning into BDI-like agents, ultimately allowing agents to perform online planning and acting. Another solution is HTN Acting [START_REF] Silva | HTN acting: A formalism and an algorithm[END_REF][START_REF]Addendum to "HTN acting: A formalism and an algorithm[END_REF], whose approach also combines HTN planning with BDI behavior, i.e. performing interleaved deliberation, acting and failure recovery. By adapting HTN planning semantics, HTN Acting does the opposite of REAP and CANPlan, that is adapting BDI agents.

Agent

Responsible Goal-driven Adaptation

Agents in a smart home are subject to environmental changes, and the agent architecture presented in the previous section is capable of online adaptation to those changes. In this section, we identify that agent adaptation must also take into account user goals and values, two concepts that deserve specialized treatment during agent deliberation by the performing of goal-and value-reasoning.

These two reasoners are able to give a smart home the goal-driven and responsible properties we are looking for.

Beyond planning and acting

In the previously presented scenario in Section 2.3, we saw the system chose to achieve secondary related goals. The agent capabilities were beyond online planning and acting, its behavior were augmented to perform goalreasoning. It gave the system a higher level of autonomy, improving user experience since it needed minimum to no user input.

The smart home from our scenario also presents valuereasoning capabilities, which increased system autonomy by helping the choice to answer or ignore the call from user's boss or mom. This function directly influences the goal-reasoning, the planning and the acting processes.

In the following sections we better elaborate the goaland value-reasoning.

Goal-reasoning automation

In the literature we find two main types of goals [START_REF] Riemsdijk | Goals in agent systems: A unifying framework[END_REF]: goals-to-be (a.k.a. declarative goals) -i.e. states the agent wants to achieve or maintain [START_REF] Duff | Maintenance goals in intelligent agents[END_REF]; and goals-to-do (a.k.a. procedural goals) -i.e. sets of actions the agent wants to successfully execute.

Goals are the guiding elements of the planning process. Most systems expect users to provide the goals to be pursued. In this case, they have automated planning and act-ing, but they do not deliberate about which goals should be pursued, i.e. they do not reason upon goals, nor create new goals, nor choose which ones to pursue, nor drop existing ones (except in cases of failure or inability to achieve).

In our example scenario, the creation and activation of "turn off lights" and "close window" goals can be performed by a goal-reasoner as shown in Figure 3, which using its current beliefs (e.g. lights are on, windows are open, user started watching a movie) would reason and decide to pursue those goals to maximize user comfort, bettering user experience.

The architecture presented in [START_REF] Cointe | Jugement éthique pour la décision et la coopération dans les systèmes multi-agents[END_REF][START_REF] Battaglino | Emotional range in value-sensitive deliberation[END_REF] is an example of a basic goal reasoning, which autonomously creates goals to redirect agent behavior if an user value is at stake. The work from [START_REF] Harland | Aborting, suspending, and resuming goals and plans in BDI agents[END_REF] gives operational semantics in the CAN [START_REF] Winikoff | Declarative & procedural goals in intelligent agent systems[END_REF] language that enables goal reasoning features necessary to manage goals and plans, such as abortion, suspension and resuming. Another approach presents a goal deliberation strategy called Easy Deliberation [START_REF] Pokahr | A goal deliberation strategy for BDI agent systems[END_REF], which allows activation and deactivation of goals in BDI agents, as well as defining relationships among goals to enable conflict free pursuing.

Value-driven responsibility

Smart home systems are designed for human users, and therefore they should be responsible systems by taking into account user values, which may include for example the user safety and privacy. The value-reasoning function shown in Figure 3 reasons based on current available beliefs, choosing to prioritize one value over another. In the scenario where a user receives a call while watching a movie, the system could prioritize work-related value over leisure-related value if, for example, the user calendar showed an important work meeting the next day, therefore if the boss was calling their call would not be ignored.

Figure 3 illustrates that the system reasoning process happens in the following order: first, it ranks the values based on the current context; second, it chooses goals that respect those values; third, it chooses or creates plans to achieve the chosen goals that respect the values; finally, it performs acting to realize those plans.

Value-based reasoning research have been highly influenced by The Theory of Basic Human Values [START_REF] Schwartz | Universals in the content and structure of values: Theoretical advances and empirical tests in 20 countries[END_REF], that identifies ten motivationally distinct values. For example, the article [START_REF] Cranefield | No pizza for you: Value-based plan selection in BDI agents[END_REF] performs value-reasoning by creating a constraint satisfaction problem in which all applicable plans are ranked using human values, and the plan that better suits user values is chosen. Note that the order of reasoning is the opposite of the one presented in Figure 3: here the agent first chose goals and create plans, then later ranks the available plans based in user values.

The architecture in Figure 3 has similarities to the emotional BDI agent architecture proposed in [START_REF] Cointe | Jugement éthique pour la décision et la coopération dans les systèmes multi-agents[END_REF][START_REF] Battaglino | Emotional range in value-sensitive deliberation[END_REF], which includes not only values but also emotional appraisal into the agent reasoning, but whose architecture works differently: its goal reasoning is not affected by the values, in-stead once the system interprets that the current context endangers user values, it will create a goal that tells the system to satisfy the endangered values.

Another approach use cultural values and rules to not only influence agent individual behavior, but to help coordinating it with other agents that, being in the same environment, have also similar cultural values and rules [START_REF] Vanhée | Using Cultures and Values to Support Flexible Coordination[END_REF]. An agent with the architecture presented in the previous section is capable of adapting itself in an online, goaldriven and value-based fashion. But its behavior is completely ignorant of the presence of other intelligent agents, ultimately simply sensing other agents actions in the world and considering them simple changes to be adapted to. Therefore we are interested in allowing interaction and coordination among decentralised agents, as shown in Figure 4. We discuss in this existing solutions, their features and limitations for this coordination among agents.

Agent

Multi-agent system

Multi-agent systems (MAS) are compositions of multiple interacting intelligent agents. A smart home can be modeled as one. As proposed in [START_REF] Boissier | Dimensions in programming multi-agent systems[END_REF], such systems are characterized by the presence of four dimensions: Agents, Environment, Interaction and Organisation.

The Agents dimension is composed of the intelligent agents themselves, i.e. entities capable of reasoning, perceiving and acting on their environment.

The Environment dimension is composed by artifacts the agents can interact with, i.e. real objects, digital services, etc. that can be used by agents in order to change the world or to have access to needed information.

The Interaction dimension constitutes the direct link among agents, allowing them to share messages.

The Organisation dimension establishes system norms and agents roles. The agents may or may not obey to system norms, but in a cooperative setting they would normally follow norms. Also, agents may have specific roles in the system, which gives them some responsibilities and consequently affects agents behaviors.

The exploration of these dimensions may help design a cooperative smart home comprising decentralised agents.

Coordination approaches

The architecture presented in [START_REF] Ciortea | Repurposing manufacturing lines on the fly with multiagent systems for the web of things[END_REF] shows a possible approach using the organisation dimension as a mean of coordinating agents: the organisation has social schemes (sets of collective goals), which are used to allocate available agents that are capable of executing such goals. The allocation is done by giving roles to agents. In their system, the environment is a first-class abstraction [START_REF] Weyns | Environment as a first class abstraction in multiagent systems[END_REF], i.e. a component designed and programmed with clear responsibilities, such as providing agents with mechanisms for interaction and coordination and with access to available real objects.

Another approach, the Decentralised Online Multi-Agent Planning (DOMAP) presented in [START_REF] Cardoso | A modular framework for decentralised multi-agent planning (extended abstract)[END_REF][START_REF] Cardoso | A decentralised online multi-agent planning framework for multi-agent systems[END_REF], uses the environment dimension to coordinate agents. They have three special artifacts: "task board", "contract net board" and "social laws". The goals are declared in the "task board" artifact, and for each goal a "contract net board" is created, in which agents bid their "cost" for executing the goal. In the end, the agent with the best cost is chosen. The "social laws" are special artifacts that impose rules in order to resolve conflicts at run-time, and that all agents obey to.

Both presented approaches use a multi-agent programming framework called JaCaMo [START_REF] Boissier | Multi-agent oriented programming with JaCaMo[END_REF]. This framework allows the deployment of multi-agent systems by combining three platforms: Jason for the agents dimension, CArtAgO for the environment dimension, and Moise for the organisation dimension. The presented state-of-the-art insights help delimiting the current level of sophistication of each smart home identified requirements.

We saw that there are theoretical and concrete solutions that allow agents to perform online adaptation, i.e. online planning and acting, to a greater or lesser extent. For instance, the cited Refinement Engine for Acting and Planning (REAP), an augmented version of the Refinement Acting Engine (RAE), is a lead that is worth exploring to build a context-aware, adaptive and autonomous agent that plans and acts at run-time.

The BDI model has been shown to be highly expandable, e.g. incorporating planning capabilities. In future work we aim to develop the presented architecture in Figure 3, expanding even further the BDI model with goal-driven and responsible behavior, allowing agents to interpret and take into account user goals and values.

However, as previously said, smart homes are composed of heterogeneous objects. Therefore, the next step following the design of an agent capable of responsible goaldriven online adaptation is to tackle the coordination challenge of decentralised agents, in a variety of IoT environments, including the smart home.

Such a multi-agent system may inspire future smart homes to be better designed to human users, by being not only intelligent, adaptive and coordinated systems, but also autonomous and trustworthy. The home of the future truly smart is hopefully just a few years away, and with help from the advancements in research, specially the emergence of AI and IoT solutions, the next human generation may start seeing their home as an integral part of their very own family.

Figure 1 :

 1 Figure 1: A planning and acting agent

Figure 2 :

 2 Figure 2: An online planning and acting agent

Figure 3 :

 3 Figure 3: A responsible goal-driven online planning and acting agent

Figure 4 :

 4 Figure 4: Decentralized agents coordinating, and perceiving and acting upon the world


Johannes C Ziegler
email: johannes.ziegler@univ-amu.fr

EYES WIDE SHUT: LITERACY, PHONOLOGY AND ADAPTIVE RESONANCE

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Eyes wide shut: literacy, phonology and adaptive resonance

Johannes C. Ziegler 1* 1 1 Aix Marseille Université, CNRS, Laboratoire de Psychologie Cognitive

ABSTRACT

In the present article, I question the claim that a literacy bias is responsible for the fact that major theoreticians underestimated or ignored the role of literacy in spoken language. Instead, I argue that the strongly modular, localist and symbolic information-processing approach to cognition that has dominated psychological science throughout the 20 th century has prevented cross-fertilization and the emergence of a unified theory of written and spoken language processing. I show that the recognition of the fundamental role of phonology in reading had suffered from exactly the same "bias", which had eventually been overcome not by breaking the "literacy glasses" but by shifting theoretical frameworks. I conclude by arguing that a marriage between cognitive science, evolutionary biology and neuroscience is needed more than ever to develop a unified, developmentally, biologically and evolutionary plausible theory of written and spoken language.

Les yeux grands fermés : littératie, phonologie et résonance adaptative

RÉSUMÉ

Dans le présent article, je remets en question l'idée selon laquelle un « biais de littératie » est responsable du fait que les principaux théoriciens ont sous-estimé ou ignoré le rôle de la littératie dans l'étude du langage oral. Je développe l'hypothèse selon laquelle l'approche modulaire, localiste et symbolique du traitement de l'information qui a dominé les sciences psychologiques tout au long du XX e siècle a empêché l'émergence d'une théorie unifiée du traitement du langage écrit et parlé. Je montre que le même « biais » explique pourquoi il fallait des dizaines d'années pour reconnaître le rôle fondamental du traitement phonologique lors de la lecture. Je conclus en affirmant qu'un mariage entre les sciences cognitives, la biologie évolutive et les neurosciences est plus que jamais nécessaire pour développer une théorie unifiée du langage écrit et parlé plausible sur le plan développemental et biologique.

In their target article, [START_REF] Kolinsky | The worries of wearing literate glasses[END_REF] elegantly demonstrate that most scholars in the fields of language, linguistics and cognition underestimate or fully ignore the impact of literacy on language processing despite the large number of inventive experimental demonstrations using a variety of paradigms, measures and languages which show the opposite (e.g., [START_REF] Pattamadilok | Unattentive speech processing is influenced by orthographic L'année psychologique/Topics in Cognitive Psychology[END_REF][START_REF] Perre | When beef primes reef more than leaf: Orthographic information affects phonological priming in spoken word recognition[END_REF][START_REF] Salverda | Tracking the time course of orthographic information in spoken-word recognition[END_REF][START_REF] Ventura | The locus of the orthographic consistency effect in auditory word recognition[END_REF][START_REF] Ziegler | Feedback consistency effects in visual and auditory word recognition: where do we stand after more than a decade[END_REF][START_REF] Zou | Orthographic facilitation in Chinese spoken word recognition: an ERP study[END_REF]. These findings have converged to suggest that literacy strongly affects the way our brains process spoken language [START_REF] Dehaene | Illiterate to literate: behavioural and cerebral changes induced Document téléchargé depuis www[END_REF][START_REF] Dehaene | How learning to read changes the cortical networks for vision and language[END_REF][START_REF] Montant | Orthographic contamination of Broca's area[END_REF][START_REF] Pattamadilok | How does learning to read affect speech perception?[END_REF][START_REF] Pattamadilok | On-line Orthographic Influences on Spoken Language in a Semantic Task[END_REF][START_REF] Perre | Orthographic effects in spoken language: On-line activation or phonological restructuring[END_REF][START_REF] Taft | Presses Universitaires de France activation of orthography in spoken word recognition: Pseudohomograph priming[END_REF]. In fact, these results had been anticipated many years before in an editorial in Brain entitled "Literally changing the brain", in which Uta Frith (1998) stated that "the acquisition of an alphabetic code is like catching a virus… this virus infects all speech processing… Language is never the same again" (p. 1051). However, contrary to the Kolinsky and Morais, I do not think that wearing "literate glasses" is responsible for the fact that major theoreticians ignored the role of literacy in spoken language. I also do not think that the "double marriage of cognitive science with evolutionary biology and neuroscience, on the one hand, and informatics and computational sciences, on the other hand" (p. 6) can be blamed for the situation. Instead, I believe that the strongly modular, localist, and symbolic information-processing approach to cognition that has dominated psychological science throughout the 20 th century has prevented cross-fertilization and the emergence of a unified theory of written and spoken language processing.

Let us go back in time to make this point more clearly. Before thinking about a potential role of orthography in spoken language, one had to recognize that spoken language (i.e., phonology) played an essential role in reading, a tough row to hoe back in the 90s (e.g., [START_REF] Van Orden | A ROWS is a ROSE: Spelling, sound, and reading[END_REF][START_REF] Van Orden | Word identification in reading proceeds from spelling to sound to meaning[END_REF]. Was it because we were wearing literate glasses that it took the field more than 20 years to recognize the role of phonology in visual word recognition? A look across the street to developmental psychology and theories of reading development would have been sufficient to appreciate its fundamental role in learning-to-read [START_REF] Goswami | Phonological Skills and Learning to Read: Hillsdale[END_REF][START_REF] Share | Phonological recoding and self-teaching: Sine qua non of reading acquisition[END_REF]. Yet, the dominant theories of visual word recognition remained "stubbornly nonphonological" [START_REF] Carello | Can theories of word recognition remain stubbornly nonphonological[END_REF]) for many years. [START_REF] Ziegler | Reading acquisition, developmental dyslexia, and skilled reading across languages: a psycholinguistic grain size theory[END_REF] suggested that such reluctance was possibly due to the fact that "cognitive psychology and developmental psychology have often been uncomfortable bedfellows" (p. 21) but we now believe that it was the modular theoretical framework that favored Occam's razor [START_REF] Jacobs | Models of visual word recognition: Sampling the state of the art[END_REF] over biological and developmental plausibility [START_REF] Goswami | A developmental perspective on the neural code for written words[END_REF]. How could we have thought for one second that the way children learn to read -through phonological decoding and access to spoken language representations [START_REF] Ziegler | Modelling reading development through phonological decoding and self-teaching: implications for dyslexia[END_REF]) -would only be a transient process to be rapidly replaced by an autonomous encapsulated module that makes sense of letter strings without reference to phonology? The death blow came from brain imaging, which, ironically, had often been used as a search light for finding isolated modules rather than interactive systems. Yet, a large number of brain imaging studies showed that learning-to-read merely added an orthographic interface to the spoken language system, a new pathway that connects an ancient neural systems that deals with visual features (lines, angles, intersections) with yet another ancient system that is in charge of spoken language. As it turned out, written words activate the entire spoken language system (with the exception of a small region around the auditory cortex) and this is true for both alphabetic and nonalphabetic writing systems [START_REF] Rueckl | Universal brain signature of proficient reading: Evidence from four contrasting languages[END_REF].

The framework that predicted such strong phonological effects was the resonance theory of word perception [START_REF] Stone | Building a resonance framework for word recognition using design and system principles[END_REF][START_REF] Van Orden | Interdependence of form and function in cognitive systems explains perception of printed words[END_REF], which argued that word perception is made possible through a dynamic bidirectional coupling between orthography, phonology and semantics. The coupling was thought to be stronger between orthography and phonology than between orthography and semantics because of the systematic relations between letters and sounds (e.g., knowing that a word starts with the letter T tells us nothing about is meaning but all about its sound). Interestingly, it was not our awareness of wearing literate glasses but the key idea of a bidirectional coupling (i.e., resonance) between orthography and phonology that motivated [START_REF] Ziegler | Orthography shapes the perception of L'année psychologique/Topics in Cognitive Psychology[END_REF] to look for orthographic consistency effects in spoken language, which have since been replicated in many languages and paradigms using a variety of methods (mouse tracking, eye movements, EEG, TMS, fMRI). While many of us initially thought that these effects would only occur at a meta-phonological level [START_REF] Morais | Does awareness of speech as a sequence of phones arise spontaneously?[END_REF] or in laboratory tasks that presented spoken words in isolation, subsequent research has clearly established that literacy effects on spoken language can be observed when people listen to spoken sentences [START_REF] Dehaene | How learning to read changes the cortical networks for vision and language[END_REF] or in nonlexical tasks [START_REF] Perre | Literacy affects spoken language in a non-linguistic task: An ERP study[END_REF]. They are automatic [START_REF] Taft | Presses Universitaires de France activation of orthography in spoken word recognition: Pseudohomograph priming[END_REF], pre-attentive [START_REF] Pattamadilok | Unattentive speech processing is influenced by orthographic L'année psychologique/Topics in Cognitive Psychology[END_REF] and affect even very basic auditory processes at the level of the neural representations of phonemes in the auditory cortex [START_REF] Dehaene | How learning to read changes the cortical networks for vision and language[END_REF]. We see this as a small but significant demonstration that "changing glasses" can sometimes improve scientific vision.

While resonance theory focused much on the statistical properties of associations between form and function, the theory remained somewhat detached from the neurobiological, developmental and evolutionary aspects of how words and their meanings might be represented in the brain [START_REF] Pulvermuller | Words in the brain's language[END_REF]. That is, children come to the task of learning to read with an almost fully developed spoken language system, in which spoken words are associated with their meanings. Thus, it is a developmentally and evolutionary optimal strategy to connect the visual word form system to the primary spoken language system regardless of whether or not there is a tight mapping of letters onto sounds. Thus, the reading network is not created from scratch but its development relies heavily on the re-use of the spoken language network [START_REF] Goswami | A developmental perspective on the neural code for written words[END_REF][START_REF] Ziegler | Reading acquisition, developmental dyslexia, and skilled reading across languages: a psycholinguistic grain size theory[END_REF]. Neural reuse is indeed a fundamental developmental and evolutionary strategy for realizing cognitive functions (see Grasso & Montant, 2019, this issue). Together then, the time seems ripe for developmentally, biologically and evolutionary plausible theories that go beyond modules [START_REF] Van Orden | The module mistake[END_REF]. More than ever, a marriage between cognitive science, evolutionary biology and neuroscience is needed to make further progress.

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L'année psychologique/Topics in Cognitive

Psychology, 2018, 118, 397-402

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B Lossouarn
email: boris.lossouarn@lecnam.net

J.-F Deü

G Kerschen

A fully passive nonlinear piezoelectric vibration absorber

Keywords: Vibration damping, Nonlinear absorber, Piezoelectric shunt, Passive inductor

The objective this study is to develop the first fully passive nonlinear piezoelectric tuned vibration absorber (NPTVA). The NPTVA is designed to mitigate a specific resonance of a nonlinear host structure. To avoid the use of synthetic inductors which require external power, closed magnetic circuits in ferrite material realize the large inductance values required by vibration mitigation at low frequencies. The saturation of an additional passive inductor is then exploited to build the nonlinearity in the NPTVA. The performance of the proposed device is demonstrated both numerically and experimentally.

Introduction

Piezoelectric tuned vibration absorbers can effectively mitigate specific resonances [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF][START_REF] Hogsberg | Balanced calibration of resonant shunt circuits for piezoelectric vibration control[END_REF][START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Soltani | Piezoelectric vibration damping using resonant shunt circuits: an exact solution[END_REF][START_REF] Lossouarn | Design of inductors with high inductance values for resonant piezoelectric damping[END_REF] or multiple resonances [START_REF] Wu | Method for multiple mode piezoelectric shunting with single PZT transducer for vibration control[END_REF][START_REF] Porfiri | Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers[END_REF][START_REF] Lossouarn | Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network[END_REF][START_REF] Lossouarn | Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network[END_REF] of linear mechanical systems. Yet, mechanical nonlinearities which can originate from, e.g., materials, large displacements and boundary conditions, are a frequent occurrence in engineering applications. These nonlinearities introduce mistuning that can significantly reduce vibration mitigation performance. Nonlinear vibration absorbers, among which we can cite the autoparametric vibration absorber [START_REF] Oueini | Analysis and application of a nonlinear vibration absorber[END_REF][START_REF] Vyas | Dynamics of autoparametric vibration absorbers using multiple pendulums[END_REF], the nonlinear energy sink [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II: resonance capture[END_REF][START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Gourc | Chatter control in turning process with a nonlinear energy sink[END_REF] and other variants [START_REF] Shaw | On the response of the non-linear vibration absorber[END_REF][START_REF] Ema | A fundamental study on impact dampers[END_REF][START_REF] Collette | A combined tuned absorber and pendulum impact damper under random excitation[END_REF][START_REF] Poovarodom | Application of non-linear multiple tuned mass dampers to suppress man-induced vibrations of a pedestrian Bridge[END_REF][START_REF] Alexander | Exploring the performance of a nonlinear tuned mass damper[END_REF][START_REF] Lacarbonara | Flutter control of a lifting surface via visco-hysteretic vibration absorbers[END_REF][START_REF] Ducarne | Structural vibration reduction by switch shunting of piezoelectric elements: modeling and optimization[END_REF], represent an interesting alternative to linear vibration absorbers in view of their increased bandwidth.

In [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF], Habib et al. developed the nonlinear tuned vibration absorber (NLTVA), which presents the unique feature that its nonlinearity possesses the same mathematical form as that of the primary system. Based on this absorber and its theoretical extension to piezoelectric absorbers [START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF], the present study aims at designing the first fully passive resonant shunt that synthesizes a cubic-like nonlinearity.

Instead of using a nonlinear capacitor, a physical inductor subjected to magnetic saturation is considered. An increase of the electrical current flowing through the shunt leads to a decrease of the equivalent inductance, which generates a hardening nonlinearity in the electrical domain. Through an adequate magnetic circuit, it becomes possible to approximate the principle of similarity [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF] with purely passive electrical components.

The paper is organized as follows. The next section recalls the main features of the linear piezoelectric tuned vibration absorber and shows its limits for vibration mitigation of nonlinear structures. The concept of a nonlinear piezoelectric tuned vibration absorber (NPTVA) is introduced through a principle of nonlinear similarity with the host mechanical structure. While a NPTVA can be realized with a nonlinear capacitance, we show that a variable inductance can serve the same objective. Section 3 proposes to design a suitable fully passive nonlinear inductor using magnetic saturation. For demonstration, Section 4 considers an experimental beam possessing a cubic nonlinearity and covered with piezoelectric patches. A linear shunt with a single passive inductor is first built for vibration mitigation at low forcing amplitudes. For higher amplitudes, the designed saturable inductor is connected in series with a linear inductor to form the nonlinear piezoelectric shunt. Finally, conclusions are drawn in Section 5.

The nonlinear piezoelectric tuned vibration absorber (NPTVA)

The linear resonant shunt

A piezoelectric tuned vibration absorber consists of a piezoelectric element shunted with an inductor and a resistor in order to mitigate mechanical vibrations [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF][START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Soltani | Piezoelectric vibration damping using resonant shunt circuits: an exact solution[END_REF]. The shunt generates an electrical resonance that is tuned to the target mechanical resonance frequency. Considering a single-degree-of-freedom mechanical model, the resulting coupled system of equations can be written as follows,

mü + c u + K E u = f + e 2 C ε q e -u , Lq + R q = 1 C ε (eu -q) , (1)
where u represents the mechanical displacement, q is the electric charge displacement and f is the excitation force. The constants m, c, K E , L, R, C ε and e are the mass, damping coefficient, stiffness in short circuit, inductance, resistance, piezoelectric capacitance and coupling coefficient, respectively. Those constants allow the calculation of a coupling factor

k c = ω 2 O -ω 2 S ω 2 S
, where

ω S = K E m and ω O = K E + e 2 C ε m (2)
are the short-circuit and open-circuit natural frequencies, respectively. The natural frequency ω O is actually deduced from Eq. ( 1) with q = 0, which corresponds to a zero-current condition. A minimization of the maximum displacement at constant forcing amplitude [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF] then leads to an approximation for the optimal inductance and resistance values:

L = 1 C ε ω 2 O and R = 3 2 k c C ε ω O , (3)
where we remark that the electrical resonance 1/ √ LC ε has to be tuned to the open-circuit natural frequency ω O . For vibration mitigation of a linear structure, the use of a linear resonant shunt tuned to its optimal inductance and resistance values offers an effective damping solution, as shown in Figure 1(a) which is computed from the numerical values in Table 1.

The resulting frequency response function, and thus the damping performance, does not depend on the excitation amplitude for a linear system. However, when a nonlinear stiffness is present, e.g., K E = K E 0 +K NL u 2 , the linear resonant shunt looses its effectiveness. Still based on the numerical values in Table 1, Figure 1(b) is obtained from time simulations involving a slow frequency-sweep excitation force. We observe a significant detuning of the linear absorber for increasing forcing amplitudes. Specifically, an increase in the amplitude of the second resonance peak occurs because of the varying mechanical resonance, while the electrical resonance frequency is kept constant. As shown in [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF][START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF], one can even observe at greater forcing amplitudes the merging of a detached resonance curve with the second peak, which leads to unacceptable vibration amplitudes. This paper develops a fully passive nonlinear piezoelectric tuned vibration absorber (NPTVA) to address the detuning of the linear device.

Principle of similarity through nonlinear capacitance

As shown theoretically and numerically for mechanical absorbers by Habib et al. [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF][START_REF] Habib | Passive linearization of nonlinear resonances[END_REF], the introduction of a nonlinear component into the tuned vibration absorber counteracts the effect of the nonlinearity of the primary system. According to the so-called "principle of similarity", the absorber should be governed by equations similar to those of the primary system. It means that the added nonlinearity should possess the same mathematical form as that of the original nonlinear system. For instance, if the nonlinearity in the host system is cubic or quintic, the absorber should possess a cubic or a quintic spring, respectively. Reference [24] also provides a tuning rule that depends on ε, the mass ratio between the absorber and the primary structure:

k 2i k 1i = (2ε) i-1 2 ε 1 + 3.5 × 1.5 i-3 2 ε for i ≥ 2, (4)
where k 1i and k 2i are the nonlinear spring coefficients of the primary system and of the absorber, respectively. Considering a cubic nonlinearity, i.e. i = 3, and a mass ratio ε 1, we obtain k 23 /k 13 2ε 2 . Based on this principle of similarity, a NPTVA possessing a nonlinear capacitance was proposed in [START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF]. For a primary system with cubic nonlinearity, we thus require a NPTVA with an electrical component providing a voltage proportional to q 3 :

mü + c u + K E 0 u + K NL u 3 = f + e 2 C ε q e -u , L 0 q + R q + 1 C NL q 3 = 1 C ε (eu -q) .
(

) 5
where C NL is the constant characterizing the nonlinear capacitor. While Eq. ( 4) does not strictly apply to a NPTVA, this tuning rule gives a suitable approximation for cases involving a nonlinear resonant shunt [START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF]. Here, the nonlinear spring coefficient of the primary system is k 13 = K NL and the one of the nonlinear absorber is k 23 = e 4 /C NL in order to ensure dimensional homogeneity. For the same reason, the mass ratio is ε = e 2 L 0 /m, which gives 1

C NL 2 L 0 m 2 K NL , (6)
when e 2 L 0 /m 1. Considering a NPTVA that satisfies Eq. ( 6), Figure 2(a) depicts a behavior close to the fully linear case, i.e., equal peaks can be maintained, which demonstrates the effectiveness of the NPTVA. Similar results can be obtained for other types of nonlinearity as long as the principle of similarity is considered [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF][START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF].

For a single resonance, a one-term harmonic balance approximation is usually enough [START_REF] Soltani | The nonlinear piezoelectric tuned vibration absorber[END_REF]. We thus assume q = Q sin(ωt + δ q ) and sin 3 (ωt + δ q ) = 3 4 sin(ωt + δ q ), [START_REF] Porfiri | Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers[END_REF] where the symbol refers to the one-term harmonic balance approximation, Q is the electric charge displacement amplitude, ω is the angular frequency and δ q is the phase constant. From Eqs. ( 5) and ( 7), we define the nonlinear voltage

v NL = 1 C NL q 3 = 3 4C NL Q 3 sin(ωt + δ q ), (8)
which refers to the nonlinear electrical component that needs to be incorporated into the resonant shunt in order to compensate the mechanical nonlinearity through a principle of similarity.

Principle of similarity through nonlinear inductance

Capacitors offering a nonlinear relation between voltage and current can be found, but they exhibit a non-negligible linear contribution [START_REF] Gluskin | The use of non-linear capacitors[END_REF], which makes the practical implementation of the nonlinear voltage in Eq. ( 8) difficult. Another solution could be to use a synthetic impedance, but it requires a power supply, which may be bulky and does not prevent from potential stability issues. The purely passive solution considered herein exploits an inductor whose equivalent inductance value decreases with an increase of the electrical current q as

L( q) = L 0 1 -α q2 . (9)
The coupled system in Eq. ( 1) becomes

mü + c u + K E 0 u + K NL u 3 = f + e 2 C ε q e -u , L 0 q + R q -L 0 α q2 q = 1 C ε (eu -q) . (10)
Adopting the one-term harmonic balance approximation in Eq. ( 7), the first harmonic of the nonlinear voltage is

v NL = -L 0 α q2 q = L 0 αω 4 4 Q 3 sin(ωt + δ q ). (11)
A cubic relation similar to Eq. ( 8) is obtained if we ensure that

L 0 αω 4 4 = 3 4C NL so α = 6 L 0 K NL m 2 ω 4 . (12)
It thus turns out that a NPTVA involving an inductor with quadratic decrease can effectively mitigate the vibrations of a hardening system with cubic mechanical nonlinearity. This is confirmed by the simulations in Figure 2(b), which show that equal peaks are maintained for forcing amplitudes that would lead to a strong detuning of the corresponding linear shunt.

The equivalent inductance L eq around the first harmonics is defined as

L eq q = L 0 q + v NL . (13)
Knowing that q = -Qω 2 sin(ωt + δ q ), Eq. ( 11) gives a law for the equivalent variable inductance

L eq = L 0 1 - α 4 Q2 , (14)
where Q = ωQ. The objective is then to design an inductor that approximates this variable inductance law for the implementation of a NPTVA.

Design of a nonlinear inductor based on magnetic saturation

Considering a physical inductor, magnetic saturation generates a decrease in its equivalent inductance value. In order to benefit from this property and to design a variable inductor dedicated to a NPTVA, it is first required to clarify the link between magnetic flux and equivalent inductance value. Then, material data provided by magnetic core manufacturers gives access to an estimation of the inductance nonlinearity.

From magnetic properties to equivalent inductance

As shown in Eq. ( 14), a decrease in the inductance value with respect to the current amplitude is required. Magnetic saturation can actually offer such a nonlinear dependence because the inductance is directly related to the magnetic flux Φ. While Φ = L q in a linear case, as a more general rule

Φ( q, qmax ) = L 0 ( qmax ) q + L 2 q3 + L 4 q5 + ... (15)
The constants L i are used to build an odd Taylor series approximating the nonlinear behavior of a magnetic field subjected to saturation. Also note, in L 0 ( qmax ), the dependence to qmax which refers to the maximum amplitude of the current q. This represents the fact that, even before saturation, we observe a variation of the inductance that does not directly depend on q but on its maximum value. In practice, inductors made of ferrite magnetic material generate an equivalent inductance which first increases before saturation occurs. At such a low current, the relation between Φ and q remains linear but the slope increases with the amplitude of the electrical current [START_REF] Legg | Magnetic measurements at low flux densities using the alternating current bridge[END_REF]. This effect is due to a physical phenomenon called "wall movements" at the scale of the magnetic domains [START_REF] Ducharne | Characterization and modeling of magnetic domain wall dynamics using reconstituted hysteresis loops from Barkhausen noise[END_REF]. For this reason, the initial inductance L 0 can be described by a second Taylor series which depends on the maximum current through

L 0 ( qmax ) = L 00 + L 01 qmax + L 02 q2 max + ... , (16)
a monotonically increasing function from L 00 to a maximum value L 0 ( qmax → ∞) = L ∞ 0 . If only the first harmonic of the current q = Q cos(ωtδ q ) is retained, we have q max = Q so

Φ = L 00 Q + L 01 Q2 + L 02 Q3 + ... + L 2 3 4 Q3 + L 4 5 8 Q5 + ... cos(ωt -δ q ). ( 17
)
As the equivalent inductance is defined from Φ = L eq q dt = L eq q ,

L eq = L 00 + L 01 Q + L 02 + 3 4 L 2 Q2 + L 03 Q3 + L 04 + 5 8 L 4 Q4 + ... (18)
Consequently, the nonlinear behavior of an inductor can be predicted from the description of the magnetic field as in Eq. ( 15) and the variable inductance in Eq. ( 16).

Exploitation of material data

Magnetic hysteresis curves are usually provided by core manufacturers. For example, Figure 3(a) shows the B-H relation in a T38 ferrite material (EPCOS), where B m is the magnetic flux density and H m is the magnetic field strength in the magnetic material. When neglecting the hysteresis behavior related to damping effects, the relation between B m and H m can be described by a monotonically increasing odd function B m = f (H m ). This function is not linear, unlike the B-H relation in the air B a = µ 0 H a , where µ 0 is the vacuum permeability. As stated previously, we want to reconstruct the total magnetic flux in an inductor with n turns and an equivalent cross-section is A e , which is defined by

Φ = nBA e where B = B m = B a (19)
because of the continuity of the magnetic field density. Moreover, Ampere's law gives

n q = Hdl = H m l e + H a s (20)
for a core of equivalent magnetic length l e including an air gap s l e . For any value of the magnetic field strength H m , the magnetic flux Φ and the current q can thus be computed from material and geometrical data through

Φ = nA e f (H m ) and q = 1 n H m l e + s µ 0 f (H m ) . (21)
B-H curves such as the one in Figure 3(a) present a distinct magnetic saturation, which means that L 0 ( qmax ) is close to its maximum value L ∞ 0 . So, the magnetic flux computed from Eq. ( 21) does not depend on qmax anymore and Eq. [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF] gives

Φ( q) = L ∞ 0 q + L 2 q3 + L 4 q5 + ... (22)
Those constants are thus obtained from a polynomial approximation of the magnetic flux as a function of the electrical current.

On the other hand, material data generally gives access to the amplitude permeability µ a = B max /H max . An example amplitude permeability curve is shown in Figure 3(b) for the T38 material. After replacing H m by H m max in Eq. ( 21), the maximum magnetic flux is expressed as

Φ max = nA e µ a H m max and qmax = 1 n l e + s µ a µ 0 H m max ( 23
)
while Eqs. ( 15) and ( 16) lead to

Φ( qmax ) = L 00 qmax + L 01 q2 max + [L 02 + L 2 ] q3 max + L 03 q4 max + [L 04 + L 4 ] q5 max + ... (24)
So, with Φ max = Φ( qmax ), all the constants in Eq. ( 24) can be obtained from a polynomial approximation of the maximum magnetic flux computed with the amplitude permeability. In the end, the nonlinear coefficient in Eqs. ( 22) and ( 24) leads to the equivalent inductance value in Eq. [START_REF] Ema | A fundamental study on impact dampers[END_REF]. As stated previously, this inductance needs to approach the variable inductance law in Eq. ( 14) for the implementation of a NPTVA. Note that a saturable inductor can still be completed with a linear inductor in series in order to reach an adequate low level inductance L 0 , as performed in the following experiments.

Experimental setup based on a nonlinear piezoelectric beam

An experimental setup is proposed for validation of the NPTVA. The determination of the mechanical nonlinearity is first performed and the structure is coupled to a resonant piezoelectric shunt. While the linear shunt shows a strong detuning at high forcing amplitudes, the introduction of a properly designed saturable inductor increases the vibration mitigation performance.

Mechanical nonlinearity

The experimental setup in Figure 4(a) is made of a 700 mm long cantilever beam of cross-section area 14 mm × 14 mm [START_REF] Habib | Passive linearization of nonlinear resonances[END_REF][START_REF] Thouverez | Presentation of the ECL benchmark[END_REF][START_REF] Grappasonni | Experimental demonstration of a 3D-printed nonlinear tuned vibration absorber[END_REF]. This beam has been covered with an array of 2 mm thick PZT-5A bimorph patches for the implementation of a NPTVA. As represented in Figure 5, the twenty patches are placed all along the structure but only half of them are used in the following experiments in order to maximize piezoelectric coupling for the first mode of the cantilever beam. The mechanical nonlinearity is due to a thin lamina located at the free end of the beam, as shown in Figure 4(b). To determine the mechanical nonlinearity, the beam is excited with an electrodynamic shaker while an impedance head measures the input force and the resulting acceleration. Measurements are first performed with short-circuited piezoelectric patches in order to consider a purely mechanical problem around the first resonance of the beam. Indeed, Eq. ( 1) shows that a short circuit, i.e., L = R = 0, gives q = eu and thus

mü + c u + K E 0 u + f NL = f, (25)
where f NL represents the nonlinear force contribution due to the thin lamina and K E 0 is the linear stiffness. The experimental results in Figure 6 are obtained from a stepped sine excitation with controlled forcing amplitude. They reveal a hardening nonlinearity that becomes significant when the displacement at the end of the beam exceeds 0.1 mm. Indeed, an hysteresis is observed through a jump phenomenon: the path followed by the displacement response is different for increasing or decreasing frequency of the sine excitation. Below 0.1 mm, however, the beam is almost linear and exhibits a resonance frequency around 29.6 Hz.

A low-level forcing frequency-sweep is then used to excite the beam around its first resonance with a sweep rate equal to 0.2 Hz/s. The acceleration signal is integrated in order to compute the velocity u and the displacement u. All the time signals are then filtered with a passband whose cutoff frequencies are 20 Hz and 40 Hz. The high-pass filtering aims at removing constant terms that appear through the numerical integration process and the low-pass filtering is used for being consistent with the theoretical part that focuses on a one-term harmonic balance approximation. This provides ü , u , u and f representing the first harmonic approximations of the primary time signals. Because the maximum displacement is around 0.1 mm, the nonlinear term in Eq. ( 1) is neglected and the constants are computed from the least-square solution of the system

  m c K E 0   = [ü u u ] -1 f . ( 26
)
The modal parameters m = 0.3872 kg, c = 0.3237 N.s/m and K E 0 = 13380 N/m are found with a very low normalized mean-squared error [START_REF] Kerschen | VTT benchmark: application of the restoring force surface method[END_REF] equal to 0.04 %. The mass and stiffness then lead to a natural frequency with short-circuited patches equal to K E 0 /m/2π = 29.59 Hz. The restoring force surface method is used to characterize the mechanical nonlinearity at higher excitation levels. Compared to the previous measurements, the forcing amplitude is multiplied by 10 so as to approach 1 mm for the maximum displacement at the end of the beam. The nonlinear restoring force f NL is available as it can be computed from the filtered temporal series through

-1 0 1 x 10 -3 -0.2 0 0.2 -2 0 2 u ⋆ (m) u⋆ (m/s) f ⋆ N L (N) (a) -1 -0.5 0 0.5 1 x 10 -3 -2 -1 0 1 2 u ⋆ (m) f ⋆ N L (N) (b)
f NL = f -mü -c u -K E 0 u (27)
and its phase space representation is given in Figure 7(a). The two-dimensional section corresponding to u = 0 represented in Figure 7(b) highlights the cubic nonlinearity induced by the thin lamina:

f NL = K NL u 3
. Actually, a one-term harmonic balance approximation u u = U sin(ωt + δ u ) means that

f NL = K NL u 3 . (28)
A nonlinear coefficient K NL = 2.507 × 10 9 N/m 3 is then obtained by filtering the signal u 3 and solving Eq. ( 28) from a least-square method, which provides an excellent fit with a normalized mean-squared error equal to 0.4 %. Furthermore, from Eq. ( 7),

f NL ( u = 0) = 3 4 K NL u 3 , (29)
which is the cubic approximation represented in Figures 7(a) and 7(b). This approximation offers a good correlation with the experimental measurements, which definitely validates the cubic form used to characterize the nonlinearity at the end of the beam.

Piezoelectric damping with a linear resonant shunt

Before considering a nonlinear shunt, a first step is to design a passive inductor that realizes a linear tuned vibration absorber when connected to the piezoelectric patches. To this end, it is required to determine the natural frequency of the beam when the piezoelectric patches are open-circuited. Low-level measurements similar to those with short-circuited patches lead to an open-circuit angular frequency ω O = K D /m where K D = K E + e 2 /C ε as shown in Eq. ( 2). From the total piezoelectric capacitance which is around C ε = 268 nF, the piezoelectric constant e can be calculated. Then, from the short-circuit and open-circuit resonance frequencies, ω S /2π = 29.59 Hz and ω O /2π = 29.82 Hz, respectively, the coupling ratio in Eq. ( 2) is k c = 0.125. Finally, from Eq. ( 3), the optimal shunt inductance and resistance are L 0 = 106 H and R = 3050 Ω. All the numerical values related to the primary mechanical structure, the electrical circuit and the coupling terms are summarized in Table 1.

A passive inductor with a magnetic circuit in ferrite material can satisfy the previous requirements [START_REF] Lossouarn | Design of inductors with high inductance values for resonant piezoelectric damping[END_REF]. For the present application, a RM14 magnetic core in N87 ferrite material (EPCOS) is selected. As it offers a permeance A L = L 0 /n 2 = 1000 nH, n = 10 5 turns of copper wire give access to an inductance value equal to 100 H at low excitation level. This is slightly below the required 108 H but the final tuning at low forcing amplitude is performed by adding a 20 nF capacitor in parallel with the piezoelectric patches in order to approach the angular frequency ω O for the electrical resonance. A coil is thus made with 3 wires of 0.5 mm diameter in parallel, each of them representing 10 5 turns. The corresponding direct current resistance due to Joule effect in the copper wire is equal to 2100 Ω, which is below the optimal 3050 Ω. However, no external resistor was added to the shunt in order to prevent from any overdamped configuration that could result from additional damping in the magnetic material or in the piezoelectric patches. The inductor in Figure 8(a) is finally obtained and connected to the piezoelectric patches. Figure 8(b) shows a 23 dB vibration reduction compared to the cases with short-or open-circuited patches. Those results shows the efficiency of the piezoelectric tuned vibration absorber when the electrical resonance is correctly tuned to the mechanical one. However, the results in Figure 9(a) show the limits of the linear shunt, which was tuned in order to approximate an equal-peak configuration at low forcing amplitude. In accordance with Figure 1(b), increasing forcing amplitudes generate a strong detuning of the absorber. Actually, the detuning is even more significant in the present case because of a slight increase of both the piezoelectric capacitance and the inductance value. For the inductance, recall that our passive component is made of magnetic material that does not behave purely linearly, as shown in Eq. ( 16). Concerning the piezoelectric capacitance, it was experimentally noticed that it is also not constant when reaching high voltage amplitudes. Over the voltage range 0 ≤ V ≤ 100 V, the capacitance variation can be approximated by

C ε = C ε 0 (1 + βV ), ( 30
)
where β is evaluated at 6.5 × 10 -4 V -1 . While this variation is negligible below 10 V, the following experiments with a functional nonlinear shunt generate voltages up to 100 V. Equation ( 30) thus needs to be considered when designing the variable inductor for the implementation of a NPTVA. Furthermore, both the capacitance and inductance variations can be introduced into the electromechanical model developed in Section 2. This leads to the numerical results in Figure 9(b) that offer a good correlation with the experiments.

Variable inductance from material and experimental data

As the slight capacitance variation influences the electrical resonance, it needs to be introduced in Eq. ( 14) that represents the variable inductor law. Because we want to maintain L 0 = 1/(C ε ω 2 0 ), from βV 1, the corresponding Taylor series truncated to the first non-constant term is

L eq = L 0 1 -βL 0 ω Q - α 4 Q2 , (31)
where the voltage V has been replaced by L 0 ω Q after neglecting the resistance and nonlinear inductance contributions. All the numerical values are recalled in Table 1, where the constant α has been computed from Eq. [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators, Part II: resonance capture[END_REF]. Represented in Figure 10(a), Eq. ( 31) gives the corrected variable inductance law for the design of a nonlinear inductor intended for the implementation of a NPTVA on the considered piezoelectric beam.

For the present application, two inductors are used to build the nonlinear shunt. A suitable saturable inductor is first selected and completed with an additional quasi-linear inductor. The considered magnetic core for the saturable inductor is a RM10 type in T38 material (A L = 1.6×10 4 nH). The coil is made of 1100 turns of wire with a 0.15 mm diameter. For the linear part, the same core as in the previous section is used but the number of turns is reduced to n = 9130. Following the method presented in Section 3, the expected equivalent inductance value of both inductors in series can be computed, which is represented in Figure 10(a). Even if the current dependence does not offer a quadratic decrease, inductors with magnetic saturation can definitely approximate the variable inductance law in Eq. ( 31) over a specific range of excitation.

The actual nonlinearity of the inductors is measured experimentally by applying an excitation voltage v and measuring the current q flowing through the components. The considered excitation is an harmonic voltage with varying amplitude from 0 to 100 V at a constant frequency equal to 30 Hz. After passband filtering between 20 Hz and 40 Hz, the equivalent inductance and resistance values are defined from

v = L eq q + R eq q , ( 32
)
where the filtered time signals are v = V cos(ωt + δ v ), q = Q cos(ωt + δ q ) and q = -Q sin(ωt + δ q ). (33)

Consequently, the equivalent resistance and inductance values are Those are computed by extracting the amplitudes V , Q and Q as well as the phase shifts δ v and δ q . The result is a dependence of the equivalent values on the electrical current. The equivalent resistance R eq varies from 2000 to 2500 Ω while the equivalent inductance is given in Figure 10(a) that validates the theoretical prediction from material data. As mentioned previously, even if the inductance does not offer a pure quadratic nonlinearity, the experimental results reveal that they are a sufficiently good approximation over the considered range of excitation. The equivalent inductance value of the inductors gives access to the angular frequency of the resonant shunt ω exp that is also influenced by the varying piezoelectric capacitance C ε . Besides, the theoretical electrical angular frequency ω th is defined from the variable inductance law in Eq. ( 14) which involves a pure quadratic decrease of the inductance value. Consequently,

L eq = V Q sin(δ v -δ q ) and R eq = V Q cos(δ v -δ q ). (34)
ω 2 th = 1 L 0 C ε 0 1 -α 4 Q2 and ω 2 exp = 1 L eq C ε 0 (1 + βZ eq Q) , (35)
where Z eq = R 2 eq + (ωL eq ) 2 is calculated at ω O . From Eq. (1) at ω O , Q = ωOC ε 0 e F , so that both theoretical and experimental frequencies can be plotted as functions of the input force. This is given in Figure 10(b), where crossings between the two curves predict optimal tuning of the electrical resonance, i.e. an equal peak condition on the mechanical response.

Performance of the nonlinear shunt

In the presence of the saturable inductor in the piezoelectric shunt, Figure 11(a) depicts that the NPTVA is able to maintain equal peaks over the considered range of forcing amplitudes. This is due to the decrease in the equivalent inductance value for increasing forcing amplitudes, hence the electrical resonance frequency is able to follow the increase in the mechanical resonance frequency as indicated in Figure 10(b). This is confirmed by the local minima of the frequency response functions which are moved toward higher frequencies. Furthermore, we note that depending on the forcing amplitude, the global maximum is either the left or the right peak. This phenomenon is explained by Figure 10(b), which shows the comparison between the actual electrical resonance and the theoretical objective that refers to the variable mechanical resonance. As mentioned previously, equal peaks are only expected when the two curves cross each other. This happens a first time between 0.2 N and 0.4 N, and a second time between 0.6 N and 0.8 N, which is verified in those slight variations, the amplitudes of the peaks in the frequency response functions do not present a clear dependence on forcing amplitude, as if the coupled system were obeying the superposition principle. Those experiments validate the concept of a NPTVA and more specifically its implementation with a saturable inductor. Moreover, the numerical results in Figure 11(b) show that integrating, into the electromechanical model, the inductance identified in Figure 10(a) provides a decent estimate of the frequency response functions at various amplitudes.

One still has to mention some limitations highlighted by the results in Figure 12. In the experiments, forcing excitations above 0.8 N lead to detuning of the NPTVA. A first reason is that the saturable inductor was designed for a specific range of excitation, i.e., from 0.1 N to 0.8 N. Beyond these values, the equivalent inductance deviates away from the variable inductor law in Eq. [START_REF] Grappasonni | Experimental demonstration of a 3D-printed nonlinear tuned vibration absorber[END_REF]. This explains the increasing mistuning between the electrical and mechanical resonances that starts even before 0.8 N, as shown in Figure 10(b). A second reason is the occurrence of a merging between the second resonance peak and a detached resonance curve, which was illustrated in [START_REF] Habib | A principle of similarity for nonlinear vibration absorbers[END_REF]. As a consequence, the NPTVA cannot maintain equal peaks over an infinite range of forcing amplitudes. Still, the NPTVA represents a substantial improvement over the linear absorber, as confirmed by the comparison between Figures 9 and 11.

Conclusions

The main objective of this study is to realize the first fully passive nonlinear piezoelectric tuned vibration absorber (NPTVA). Relying on the principle of nonlinear similarity, the NPTVA comprises a saturable inductor whose equivalent inductance value depends on the electrical current flowing through the shunt. This results in a variable electrical resonance frequency that compensates the mechanical nonlinearity. Compared to a classical linear shunt, the nonlinear shunt is able to maintain an adequate tuning over a much wider range of forcing amplitudes. Eventually, the proposed NPTVA improves vibration mitigation without the need of power supply or external controller.

Figure 1 :

 1 Figure 1: Frequency response functions with a linear piezoelectric tuned vibration absorber -(a) linear structure in shortcircuit configuration (--) and linear structure with an optimal linear shunt (-), (b) nonlinear structure with the linear shunt for various forcing amplitudes, F = 0.2 N (-), F = 0.4 N (--), F = 0.6 N (-• -) and F = 0.8 N (• • • ).

Figure 2 :

 2 Figure 2: Simulated frequency response functions for various forcing amplitudes, F = 0.2 N (-), F = 0.4 N (--), F = 0.6 N (-• -) and F = 0.8 N (• • • ) -(a) NPTVA with a nonlinear capacitor, (b) NPTVA with a nonlinear inductor.

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Material data from EPCOS -(a) magnetic hysteresis loop (-) and its polynomial approximation (--), (b) amplitude permeability (-) and its polynomial approximation (--).

Figure 6 :

 6 Figure 6: Displacement at the end of the nonlinear beam for various forcing amplitudes : F = 0.01 N (-), F = 0.05 N (--), F = 0.2 N (• • • ).

Figure 7 :

 7 Figure 7: Experimental nonlinear restoring force (-) and cubic approximation (--) -(a) three-dimensional representation as a function of u and u , (b) restoring force when u = 0.

Figure 8 :

 8 Figure 8: Passive piezoelectric tuned vibration absorber -(a) physical inductor with a magnetic core in ferrite material, (b) experimental frequency response function around the first mechanical resonance, with short circuit (• • • ), with open circuit (• • • ) or with an optimal linear shunt for F = 0.03 N (-).

Figure 9 :

 9 Figure 9: Frequency response functions for various forcing amplitudes with a linear resonant shunt, F = 0.2 N (-), F = 0.4 N (--), F = 0.6 N (-• -) and F = 0.8 N (• • • ) -(a) experiments, (b) simulations.

Figure 10 :

 10 Figure 10: Equivalent inductance values and corresponding electrical resonance frequency -(a) inductance value from material data (--), from experimental data (-) and variable inductance law (• • • ), (b) objective (• • • ) and experimental resonance frequency (-).

Figure 11 :

 11 Figure 11: Frequency response functions for various forcing amplitudes with NPTVA based on a saturable inductor, F = 0.2 N (-), F = 0.4 N (--), F = 0.6 N (-• -) and F = 0.8 N (• • • ) -(a) experiments, (b) simulations.

Figure 12 :

 12 Figure 12: Experimental frequency response functions at higher levels causing adverse dynamics, F = 0.8 N (• • • ), F = 1 N (-• -) and F = 1.2 N (--).

Table 1 :

 1 Numerical values for the mechanical, electrical and coupling parameters

	Mechanical parameters		Electrical parameters
	m	0.3872 kg	L 0	106 H
	c	0.3237 N.s/m	R	3050 Ω
	K E 0 K NL ω S /2π ω O /2π Coupling parameters	13380 N/m 2.507 × 10 9 N/m 3 29.59 Hz 29.82 Hz	C ε C NL α β	268 nF 2.65 × 10 -15 C 3 /V 8.65 × 10 3 A -2 6.50 × 10 -4 V -1
	e k c	7.55 × 10 -3 C/m 12.5 %
		3