Rémi Barraquand
email: remi.barraquand@inrialpes.fr

Patrick Reignier
email: patrick.reignier@inrialpes.fr

Investigating The Role Of Mutual Cognitive Environment For End-User Programming

Keywords: End-User Programming, Situation Models, Cognitive Environment

In this paper we present a situated end user programming approach where user co-constructs, in an iterative process, a mutual cognitive environment with the system. We argue that co-construction of a mutual cognitive environment, between both the human and the system, is a key toward social human-computer interaction. Preliminary results are illustrated with a step by step case study: a user teaches the system new perceptual and abstract concepts using hand gesture and an interactive learning table.

Introduction

Information and communication services pervade nearly all aspect of our daily life, however they have no ability to perceive and understand the social situation or affective reactions of human. Furthermore, they have no sense of social roles played by interacting humans: systems are unaware of human goals and intentions.

Such abilities are beyond systems constructed with current technologies for two main reasons. First, systems lack an ability to acquire the skills for polite interaction using the implicit feedback provided by humans during interaction. Second, systems even lack the most rudimentary understanding of the social order that structures normal human interaction.

Common sense is the collection of shared concepts and ideas that are accepted as correct by a community of people. Social common sense refers to the shared rules for polite, social interaction that implicitly rule behavior within a social group [START_REF] Barraquand | Learning polite behavior with situation models[END_REF]. To a large extent, such common sense is developed using implicit feedback during interaction between individuals. Reeves and Nass demonstrate convincingly in The Media Equation [START_REF] Reeves | The Media Equation: How People Treat Computers, Television, and New Media Like Real People and Places[END_REF] that interactions with computers are identical to real social relationships and they argue that a social interface may be the truly universal interface. Thus our goal in this research is to develop methods to endow an artificial agent with the ability to acquire social common sense using day to day interaction with people. Our approach focuses on a key aspect of social common sense: the ability to build an appropriate mental model of social situations. We believe that such methods can provide a foundation for socially polite man-machine interaction, and ultimately for other forms of cognitive abilities.

Common knowledge is a phenomenon which underwrites much of social life. In order to communicate or otherwise to collaborate, humans make use of socially shared conventions, common understanding or background knowledge. While theory of mutual knowledge has the characteristic to produce a regression at infinity, Sperber and Wilson [START_REF] Sperber | Relevance: Communication and Cognition[END_REF] developed a weaker but empirically more adequate concept, the mutual manifestness. In this paper we investigate the role that a mutual cognitive environment plays in collaborative social interaction and in the construction of situation models. We believe that building correct social situation models is a key toward designing context aware systems.

Related Work

Designing a context aware application is a complex task. There is no widely accepted definition of what is a context. For Dey et al. [START_REF] Dey | A conceptual framework and a toolkit for supporting the rapid prototyping of context-aware applications[END_REF], context is "any information that characterizes a situation related to the interaction between humans, applications and the surrounding environment." Dourish [START_REF] Dourish | What we talk about when we talk about context[END_REF], on a more conceptual approach, proposes two categories for context models : representational and interactional. In a representational approach, context and activity are separable. Context is information about the environment where the activity takes place. The main question is how context could be encoded and represented. This is a more "programmer" point of view. In the interactional approach, context is a relational property between objects or activities. It is a more user centered approach. Dourish opposes those two points of view. We believe that those two approaches can be complementary. A software programmer, can use a representational context model to specify a first "general" contextual behavior (not adapted for everyone). The end user then adapt this general contextual model to its own needs. This adaptation phase has to be done while (or after) using and interacting with the system. This is the interactional aspect and corresponds to End User Programmming.

End user programming can be seen as offering alternatives to extensive programming for building applications. The question is how the end user can transfer to the system its relevant knowledge so that it can accomplish the required task. In [START_REF] Korpip | Customizing user interaction in smart phones[END_REF], the mobile phone behavior is contextually specialized using feed forward rules. A specific interface allows to easily create new rules based on predefined concepts, brought by the programmer. The end user can specify how those concepts should be used to fulfill his needs. The evaluations showed that the end user appreciates to be able to enter its own knowledge in the system as long as the rules are not too complicated to write. The main problem is that Nokia's system is bringing its own predefined concepts which might not correspond to the user's concepts and interpretations.

In [START_REF] Dey | a CAPpella: programming by demonstration of context-aware applications[END_REF], the user is doing its task in the environment while the system is recording all the sensors. Using a dedicated human machine interface, the enduser can then annotate the recorded data and a context model is constructed using machine learning approaches. This annotation phase is an interesting way to transfer knowledge and concepts from the user to the system while letting the system building its own internal representation. The main drawback is that it can be rapidly painful and sensor data annotation is done offline (not during the interaction). Ganneau and al. [START_REF] Ganneau | Learning key contexts of use in the wild for driving plastic user interfaces engineering[END_REF] are directly learning (using bayesian networks) the situations' description without a post-annotation phase. The interaction for knowledge injection is more transparent for the end user.

On-line transparent interaction is more natural and corresponds to the interactional aspect of Dourish but can be ambiguous (see section 3). Off-line specification using a dedicated interface can allow the end user to specify more precisely but is painful. We propose a global approach based on both approaches: transparent user interaction and dedicated off-line interactions to disambiguate the injected knowledge.

Situated End User Programming

In an idealist scenario people should teach machines through a social and collaborative process. Learning to act appropriately in social situation is a first step toward a better interaction. In a previous work [START_REF] Barraquand | Learning polite behavior with situation models[END_REF] we have sought to train an association between behavior and social situation. However the success of this approach relies on getting both human and system to share a common understanding of social situation.

Mutual Intelligibility

Salembier et al. [START_REF] Salembier | Intelligibilité mutuelle et contexte partagé. Inspirations théoriques et réductions technologiques[END_REF] review research that point out the importance of the mutual access to contextual information in collaborative work. The better the mutual understanding the greater the collaboration. This mutual understanding is not necessarily achieved by sharing mutual knowledge but rather by the concept of mutual manifestness. Introduced by Sperber and Wilson this concept is weaker but empirically more adequate than the theory of mutual knowledge which has the characteristic to produce regression at infinity.

For Sperber and Wilson "a fact is manifest to an individual at a given time, if and only if, this individual is able at this time to represent this fact mentally and to accept his representation as being true or probably true" [START_REF] Sperber | Relevance: Communication and Cognition[END_REF]. Following this notion, Sperber and Wilson define the one of cognitive environment. A personal cognitive environment (PCA) is defined as whole facts which are manifest for a given individual. A shared cognitive environment (SCE) indicates all the facts which are manifest to several individuals. This simply means that they are able to perceive or deduce the same facts, and not that they share a belief, a knowledge, or a representation concerning those facts. The mutual cognitive environment (MCE) indicates a shared cognitive environment in which the identity of individuals who have access to this environment is manifest. As they share the same environment, they can establish an interaction in relation to their common perception of contextual events. Salembier points out that the notion of cognitive environment does not take into account the activity of individual. He [START_REF] Salembier | Intelligibilité mutuelle et contexte partagé. Inspirations théoriques et réductions technologiques[END_REF] then proposes the definition of shared context, which reduce mutual cognitive environment through activity filtering. Thus shared context is a set of contextual information or events mutually manifest for a set of actors, at a given time in a certain situation, taking into account their perception and cognitive abilities, their task, and current activity.

Considering social learning as a collaborative process and context as a key issue [START_REF] Coutaz | Context is key[END_REF] in interaction between human and computer, we argue that mutual intelligibility is the key toward a social human-computer interaction.

Situation Models

In the previous section, we emphasize the importance of mutual intelligibility in social learning and particularly the needs for a MCE between both human and system. However it is not clear what defined the facts that compose cognitive environment. Johnson-Laird in [START_REF] Johnson-Laird | Mental models: towards a cognitive science of language, inference, and consciousness[END_REF] introduced situation model as cognitive theory for human mental models. A situation model is a mental representation of a described or experienced situation in a real or imaginary world [START_REF] Zwaan | Situation models in language comprehension and memory[END_REF]. It is commonly defined as consisting of entities and of relations between those entities. While this model, as well as much of the subsequent literature in this area, has been concerned with spatial reasoning or linguistic understanding, these concepts can be adopted for the construction of software systems and services for understanding social interaction. In [START_REF] Crowley | Situation models: A tool for observing and understanding activity[END_REF], we describe the use of situation models for observing and understanding activity in order to provide context aware services. Situations are defined as a set of relations (predicate) between entities (agents, objects or abstract concepts) where entities are sets of properties.

Using this formalism we define a fact as an entity or a relation that can be either observed or deduced by an agent. A cognitive environment is then defined by the set of relations between entities that are manifest by this agent. Thus the shared context between agents (human or machine) is defined by the intersection of their situation models taking into account their perception and cognitive abilities, their task, and current activity. A fact (entity or relation) has the property to exist and is associated to an abstract representation that we call concept. Two facts can be instance of the same concept. Concept is for fact what a class is for object. Observed facts are any instance of concept that can be observed and represented as an entity or relation. Such entities include time of the day, weather, agent, object. Observed relation include spatial relation. Inferred facts are any instance of concept that can be inferred from other facts (observed or inferred). Inferred entities can include group, agent's activity, agent's task. These facts are not directly observable properties but rather interpretation.

Socially Guided Environment

The concept of mutual manifestness as part of the relevance theory has been concerned with human communication. In the previous section, we argued for the need of such an ability in human-machine interaction. Considering the main differences in term of cognition and perception between human and computer, this theory gives us a clear understanding of why perceiving social context from a human perspective is a difficult problem. Although this let us to propose our approach of end-user programming. In Relevance Theory [START_REF] Sperber | Relevance: Communication and Cognition[END_REF] Sperber and Wilson present communication as the process of making certain facts more manifest than others. Thus in our approach both agents are provided with tools allowing them to alter each other's cognitive environment. Because of the fundamental difference between human an machine, we need to introduce the following condition. A fact is manifest for the system if and only if the related concept is also manifest. Then by definition a concept is manifest if and only if it belongs to the cognitive environment. A concept is then mutually manifest if and only if it has been acknowledged by the user. Therefore, the first step toward making a fact mutually manifest is to teach the system new concepts.

The first tools provides the user with a method to teach the system whatever perceptual concept he thinks is relevant. This concept is tokenized from the environment using bounding volumes such as blob or frustum constructed by user hand gestures. This tool is well adapted to make manifest spatial concept such as a chair, a desk or an area. At the same time, the system is able to display different views of its cognitive environment by displaying the relevant facts it is able to perceive or deduce from the environment. This second tools presents two benefits: first it provides transparency to the user, then it allows the user to respond whenever he did not agree. One of this view is provided by an interactive table. The table provides a window through the system's mind which lets the user browse and interact with the contained concepts and facts. Different actions are then proposed to the user such as teaching a new abstract concept, adding tags to a already learned concept, adding or removing a concept from the mutual cognitive environment. Tags are a simple way for the user to give interpretation to newly extracted concepts (eg. Chair, Desk, Table ). The user can also see the current and previous system's situation models and related action, thus enabling the user to tell the system the action it should perform in a given situation. The interface provide also a tools for the correct assignment of reward in the case of social reinforcement learning [START_REF] Barraquand | Learning polite behavior with situation models[END_REF]. Finally we integrated an attentional model to control the manifestness of a fact. The greater the attention of the user toward a fact the more manifest the fact. That way entity closest to the user for example will be more manifest than entity outside the environment.

Experimental Evaluation

In this section, we demonstrate our preliminary results with a step by step case study which illustrates a standard scenario in a context aware environment. The goal is to show how a user and a system can build conjointly a mutual cognitive environment.

Experimental Setup

The experiment takes place in the PRIMA SmartRoom, an environment specifically designed to observe (using cameras and microphones) and interact with users as they go through normal day-life activity such as work, meeting, rest, etc. The system is capable of tracking the position of persons as they evolve through the environment and also provides a rough estimate of the personâ ȂŹ s posture. The system is also equipped with a steerable projector with automatic image rectification able, given a specific region, to transform a planar surfaces in a projection area.

Case Study

One scenario that comes to mind when working with smart environment is the meeting scenario. In such scenario, smart environments could provide different services such as projecting content of the meeting at the correct place. In the following, we will illustrate the different steps that lead to the construction of a mutual cognitive environment intended to provide our smart meeting system with the needed concepts to characterize/recognize the relevant situations.

This construction process is organized in two stages. The first step is a usual top down reflection where the user builds a mental representation of the situation he wants to explain by selecting relevant assumptions. A possible situation model could be a "lecturer is close to an audience". Then a lecturer is a "person close to a projection area" and so on. The second step is an iterative process where both human and system interact to make relevant information mutually manifest. Using different modalities the system acknowledges the user of the different facts and related concepts that are manifest in the environment. The user on its side, introduces the system with missing concept. For this experiment the environment already knows some basic concepts, provided by software components, such as the entity-concept person, audience, region and the relational-concept close to and sit on. The following concepts are "mutually manifest" however chair and lecturer are not. We could only expect the environment to interpret a correct situation model once chair and lecturer will be mutually manifest. Then, the steerable projector will project accordingly the content of the meeting at the correct place in the environment.

When the user enters the room, the only facts belonging to the MCE is the user itself. The system is able to perceive the user position and the user knows it, thus this fact is mutually manifest. The first action performed by the user in the environment is to defined the concept of projection area which is then integrated to the MCE. This step is done by performing a hand gesture surrounding this area and tagging the extracted region on the interactive table as projection area). Then the user pursues by extracting the different sitting areas (see Figure 1) to which he will associate the tag chair.

This far, the system is able to observe new facts from its environment. For example any person sitting on the extracted chair or sofa will now be manifest to the system as the related concepts, person, chair and sit on are known. The last missing concept is the one of lecturer which is a deduced-like concept. In order to do that, the user interacts with an interface displayed on an interactive table. The interface shows the different concepts mutually manifest as well as observed and deduced facts. Abstract concepts are defined from other set of concepts This operation is achieved by different drag&drop operation. In our example, the diagram containing a person close to a projection area is labeled as lecturer (see right part of Figure 1).

During the real meeting presentation (see Figure 2) the user can, at any time, teach the system news action according to its interpretation of the current situation. The right part of Figure 2 presents the current interpretation of the situation by the system (remark, relation are not present). System can be taught to trigger the steerable projector service, and so the topic of the meeting will be displayed onto the correct projection area.

Conclusions

In an attempt to design context aware systems able to acquire social common sense from day to day interaction with non-expert user, a key challenge is to provide systems with methods to perceive social situation from a human perspective. We cannot expect developer to handle all possible social situations a system can encounter in its working life. An alternative inspired by end user programing offers better perspectives.

Our proposed approach provides transparent interaction and learning tools to build a mutual cognitive environment between the user and the system. We argue that mutual cognitive environment plays a significant role in the construction of situation models for social situation. Because system and user do not have the same perceptive and cognitive capabilities we propose a method where user will interactively (when the needs comes) teach to the system the concept that he thinks is relevant. Concepts are learnt by segmenting the environment using simple hand gestures and using an interactive table as a cognitive window through the system mind. The preliminary results are very encouraging and validate our approach.

Fig. 1 .

 1 Fig. 1. Teaching concepts (left) and learning abstract concept (right). in relation, thus user has to construct a diagram relating sub-concepts together.This operation is achieved by different drag&drop operation. In our example, the diagram containing a person close to a projection area is labeled as lecturer (see right part of Figure1).During the real meeting presentation (see Figure2) the user can, at any time, teach the system news action according to its interpretation of the current situation. The right part of Figure2presents the current interpretation of the situation by the system (remark, relation are not present). System can be taught to trigger the steerable projector service, and so the topic of the meeting will be displayed onto the correct projection area.

Fig. 2 .

 2 Fig. 2. Presentation meeting (left) and interpretation of the presentation meeting by the system.


Yuming Wu

Cungen Cao

Shi Wang

Dongsheng Wang

A Laplacian Eigenmaps Based Semantic Similarity Measure between Words

The measurement of semantic similarity between words is very important in many applicaitons. In this paper, we propose a method based on Laplacian eigenmaps to measure semantic similarity between words. First, we attach semantic features to each word. Second, a similarity matrix ,which semantic features are encoded into, is calculated in the original high-dimensional space. Finally, with the aid of Laplacian eigenmaps, we recalculate the similarities in the target low-dimensional space. The experiment on the Miller-Charles benchmark shows that the similarity measurement in the low-dimensional space achieves a correlation coefficient of 0.812, in contrast with the correlation coefficient of 0.683 calculated in the high-dimensional space, implying a significant improvement of 18.9%.

Introduction

Similarity measurement plays an important role in many areas, especially in semantic related applications [START_REF] Liu | Word Similarity Computing Based on How-net[END_REF]. So, the objective similarity measurement has to take more features of semantic level into consideration. For the purpose of attaching semantic features to words, we should have a knowledge source, from which we get semantic features and provide a flexible way to represent them, which can be extended to other knowledge sources without much modification.

In this paper, we propose a new method based on Laplacian eigenmaps [START_REF] Belkin | Laplacian Eigenmaps for Dimensionality Reduction and Data Representation[END_REF] to define the semantic similarity between words. First, we use an online dictionary as a knowledge source, which is in semi-structured text format. Several types of interpretations can be extracted directly from the webpages. Then, these interpretations are transformed into a set of attribute-value pairs. These attributevalue pairs are used as semantic features, which are represented in a high dimensional space. After that, the Laplacian eigenmaps based method is adopted to find the intrinsic coordinates in low dimensional space. Finally the similarities are recalculated under the intrinsic coordinates in the low-dimensional space.

The remainder of the paper is organized as follows. In section 2 we describe the background. Section 3 makes further analysis to knowledge sources and features representation. In section 4 we describe the materials and methods. Section 5 gives the experimental result and compares against other methods on Miller-Charles benchmark dataset. Finally, we discuss the shortcomings of the proposed method and conclude this paper.

Background and Related Work

There has been a great deal of work on semantic similarity measurement. Some of them incorporate semantic features in the definition of similarity measure [START_REF] Lin | An information-theoretic definition of similarity[END_REF] [9] [START_REF] Chen | A study on word similarity using context vector models[END_REF]. In Lins paper [START_REF] Lin | An information-theoretic definition of similarity[END_REF], an information-theoretic definition of similarity was proposed, the similarity of two objects, i.e. A and B only depends on the distributions of common(A; B) and description(A; B). This method is generic, but for some specific similarity, such as semantic similarity, Lins method may be unsuitable. As there is no an unified approach to encode the various semantic information in knowledge sources into the distribution, and when the data is sparse, the real distribution is hard to approximate. Resnik [START_REF] Resnik | Using information content to evaluate semantic similarity[END_REF] presented a measure of semantic similarity in an IS-A taxonomy, based on the notion of information content. This measure depends on the structure taxonomy. In Chens [START_REF] Chen | A study on word similarity using context vector models[END_REF] work, a context vector model to measure word similarity was adopted. The syntactic and semantic similarity is balanced by using related syntactic contexts only. Two major differences in these methods are the approach of using knowledge and representing semantic features. A more detail analysis on them is given in next section.

Knowledge Sources and Features Representation

Knowledge sources vary in their format, either structured or unstructured. Word-Net [START_REF]WordNet: An Electronic Lexical Database[END_REF]and HowNet [START_REF] Liu | Word Similarity Computing Based on How-net[END_REF] are structured knowledge sources, in which concepts are related in some manners. For example, the concepts in WordNet are related through several types of semantic relations, such as hypernym, antonym, holonym and meronym. The context around a word also provides strong and consistent clues to the sense of it [START_REF] Yarowsky | Unsupervised word sense disambiguation rivalling supervised method[END_REF]. Today, much text format materials are available on the Web. The context of a word can be relatively reliably extracted from those materials.

In this paper, we use an online dictionary3 as knowledge source . In the online dictionary, there are plenty of items to interpret the sense of a given word. These items interpret the word from different aspects, such as interpretations in another language, sample sentences to describe the usage of the word, and further interpretations in same language and synonyms. After extractions, for each word, a collection of attribute-value pairs are obtained. For instance, the word "car" has an attribute-value pair < hasSynonym, "automobile" >, which stands for that "car" and "automobile" is synonymous to each other. These attribute-value pairs are taken as the semantic features. The concrete examples are show in section 4.

Materials and Methods

Interpretations in Online Dictionary

In a dictionary, a word or phrase is interpreted from multiple aspects in detail. These interpretations or descriptions are rich in semantic information. For the word "car", for example, we list some interpretations and descriptions, as depicted in Figure 1, to illustrate how the semantic features appear in an online dictionary.

Word Car

Word Senses: A wheeled vehicle adapted to the rails of railroad. Sample Sentences: The car in front of me stopped suddenly and I had to brake .

Synonyms: auto, automobile, machine, motorcar, gondola. As shown in figure 1,there are three subtitles the webpage, i.e. "word senses", "sample sentences" and synonyms. The word "car" is interpreted with several meanings, word senses, sample sentences, synonyms and so on. The words "wheel", and "vehicle", under the subtitle of "Word Senses", have close links to the word "car". We take the subtitle as attribute name, and take each word under a subtitle as an attribute value of the corresponding attribute. Note that the word itself can be also used as an attribute value. Sample attribute-value pairs for the word "car" are listed as Table 1 The context is a rich semantic source for a specific word, while there are also some stop words , such as "of"," but", and "in" in the context, which have little contribution to similarity between two words. Therefore, we leave out all these stop words in the process of measuring similarity.

Definition of Similarity Measure

Let W be the set of words, and f is a mapping from a word to an attribute-value pairs set.

f (w i ) = {< attribute k , value j >} (1)
W (w i , w j ) = |f (w i ) ∩ f (w i )| |f (w i ) ∪ f (w i )| (2)
where | . | refers cardinality of a set. As we take these semantic features independently, the dimension of space, in which these words are represented, is identical to the number of independent semantic features.

The number of attribute-value pairs is necessarily large. So the words are represented as points in the high dimensional space. Because the quantity of the words is limited, the words are very sparse in the high-dimensional space. Intuitively, there should exist a low dimensional embedding for the set of all the words.

A natural problem arises: how to find the optimal embedding in a lowdimensional space? Due to [START_REF] Belkin | Laplacian Eigenmaps for Dimensionality Reduction and Data Representation[END_REF][5], the optimal embedding can be found by solve the following generalized eigenvalue problem.

Ly = λDy (3)
where L = D -W is called Laplacian matrix and D(D ii = j W ji ) is the diagonal weight matrix , Give the dimension of the target space, let M be the matrix with column as the first m eigenvectors which satisfy the formula (3).

Then, the optimal map should map x i to be y i , which is the ith row of matrix M .

Given the representations in a low-dimensional Euclidean space, the improved semantic similarity between y i and y j can be calculated as:

Sim improved (y i , y j ) = e -yi-yi 2 (4)
where . refers to the 2-norm in Euclidean space. Now, we give the full process of how to measure the semantic similarity between two words. A more formal description of this process will be shown in the algorithm "semantic similarity based on Laplacian eigenmaps". The main ideas behind this algorithm are as follows. Firstly, encode the local relevance into a similarity matrix, in which each element is the basic similarity calculated by formula (2). Then we use Laplacian eigenmaps to find another representation in low dimensional space. Finally, we recalculate the semantic similarity in low dimensional space. Figure 2 show the algorithm of "Semantic Similarity based on Laplacian eigenmaps" .

Experiments and Discussions

We take Miller-Charles dataset [START_REF] Miller | Contextual correlates of semantic similarity[END_REF] as a benchmark dataset. The dataset has been used in several works [6] [9]. The Miller-Charles dataset can be considered as a reliable benchmark for evaluating semantic similarity measures. As shown in Figure 3, the correlation increases rapidly when the dimension of the target space is from 1 to 10, and achieves the maximal value of 0.812 when the dimension is 11. After the dimension exceeds 11, the correlation coefficient decreases steadily. These experimental results coincide with the intuition that there is a

Algorithm : Semantic Similarity Based on Laplacian Eigenmaps Input:

(1) a set of words S = {wordi}, (2) the dimension N of target space. Output: the similarity matrix for all words in S.

Procedure:

W are calculated by the formula [START_REF] Belkin | Laplacian Eigenmaps for Dimensionality Reduction and Data Representation[END_REF];

D = diagonal(W ); L = D -W ;
Calculate the first N eigenvectors v1, ..., vN which satisfy the generalized eigenproblem Lv=Dv;

Let U be the matrix with v1, ..., vN as columns Let yi be the ith row of matrix U; for eachword-pair < wi, wj >∈ S × S Sim Matrix (i,j) = Simimproved(yi ,yj) as defined in formula (4) ; end for each Return Sim Matrix; 2 presents a comparison of the proposed method with several other methods, including CODC [START_REF] Chen | A study on word similarity using context vector models[END_REF], SemSim [START_REF] Bollegala | Measuring semantic similarity between words using web search engines[END_REF] , Lin [START_REF] Lin | An information-theoretic definition of similarity[END_REF] and Resink [START_REF] Resnik | Using information content to evaluate semantic similarity[END_REF]. We get a correlation coefficient of 0.812 on Miller-Charlers dataset when the dimension of target space is 11. As shown in Table 2, the correlation coefficient of the proposed method is slightly lower than those obtained using the methods of SimSem and Lin, and higher than those obtained using the two other methods.

Conclusion and Future Work

In this paper, we proposed a method based on Laplacian eigenmaps to measure the semantic similarity between words. The main contributions of our work are listed as follows. First, our method takes semantic features into consideration in a natural way when measuring similarity between words. These semantic features are organized as attribute-value pairs. Our method is very flexible and easy to extend, because there is no dependence on the structure of semantic features.

Second, the problem of data sparseness was avoided, because the final similarities were calculated in low dimensional space.

Experimental results on the Miller-Charles dataset achieve a correlation coefficient of 0.812, showing that the proposed method outperforms the traditional corpus-based and thesauri-based measures. The future work will concentrate on the following two directions. One is to transform other knowledge sources into

Fig. 1 .

 1 Fig. 1. Sample interpretations for word "car"

Fig. 2 .

 2 Fig. 2. Semantic Similarity Based on Laplacian Eigenmaps

Fig. 3 .

 3 Fig. 3. Semantic Similarity Based on Laplacian Eigenmaps

Table 1 .

 1 Sample attribute-value pairs for word "car"

	Attribute Name	Attribute Value
	Word Sense	motor
	Synonyms	auto

Table 2 .

 2 Similarity measure comparison on

	Miller-Charlers CO DC	Sem Sim Lin	Resnik Proposed
					Method
	Correlation	1	0.693	0.834 0.823	0.775 0.812
	Coefficient

http://www.dict.cn

Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60773059.


Pauline Bernard

Laurent Praly

Vincent Andrieu

Observers for a non-Lipschitz triangular form

Keywords: triangular observable form, high-gain observer, finite-time observers, homogeneous observers, exact differentiators, explicit Lyapunov functions

come

Introduction

A preliminary step is often required in the construction of observers for controlled nonlinear systems. It consists in finding a reversible coordinate transformation, allowing us to rewrite the system dynamics in a target form more favorable for writing and/or analyzing the observer. For example, the dynamics of a controlled single output system of dimension n which is uniformly observable (see [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]Definition I.2.1.2]) and differentially observable of order m (see [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]Definition I.2.4.2]) with m = n can be written with appropriate coordinates in a Lipschitz triangular form appropriate for the design of a high gain observer ( [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF][START_REF] Gauthier | A simple observer for nonlinear systems application to bioreactors[END_REF]). Such a property is no more true when the order m is strictly larger than the dimension n. Indeed in this case, we may still get the usual triangular form but with functions that may not be Lipschitz ( [START_REF] Bernard | On the triangular canonical form for uniformly observable controlled systems[END_REF]). A particular case of this is when there is only one nonlinear function (in the last line for the single output case). This is the so-called phase-variable form. It has been known for a long time, in particular in the context of dirty-derivatives and output differentiation, that a high gain observer can provide an arbitrary small error as long as the nonlinearity is bounded ( [START_REF] Tornambe | Use of asymptotic observers having high gains in the state and parameter estimation[END_REF] among many others). We also know since [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF] that a sliding mode observer can achieve finite-time convergence under the same assumption. In this paper, we want to build observers for the more general triangular canonical form where non-Lipschitz triangular nonlinearities can appear on any line. As far as we know, this form has not received much attention apart from its well-known Email addresses: pauline.bernard@mines-paristech.fr (P. Bernard), laurent.praly@mines-paristech.fr (L. Praly), vincent.andrieu@gmail.com (V. Andrieu).

Lipschitz version and the convergence results holding for the phase-variable form do not extend trivially. This paper follows and completes [START_REF] Bernard | Non lipschitz triangular canonical form for uniformly observable controlled systems[END_REF]. We show here that the classical high gain observer may still be used when the nonlinearities verify some Hölder-type condition. Nevertheless, the asymptotic convergence is lost and only a convergence with an arbitrary small error remains. When the nonlinearities do not verify the required Hölder regularity, it is also possible to use a cascade of high gain observers, but once again, the convergence is only with an arbitrary small error.

Fortunately, moving to a generalization of high gain observer exploiting homogeneity makes it possible to achieve convergence. It is at the beginning of the century that researchers started to consider homogeneous observers with various motivations: exact differentiators ( [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF][START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF][START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]), domination as a tool for designing stabilizing output feedback ( [START_REF] Yang | Homogeneous observers, iterative design, and global stabilization of highorder nonlinear systems by smooth output feedback[END_REF], [START_REF] Qian | A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems[END_REF], [START_REF] Qian | Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems[END_REF], [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF] and references therein (in particular [START_REF] Andrieu | Nonlinear output feedback design via domination and generalized weighted homogeneity[END_REF])), ... The advantage of this type of observers is their ability to face Hölder nonlinearities.

With the tools introduced in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], we have at our disposal a Lyapunov design to obtain an homogeneous observer with degree in ] -1, 0[ for the triangular form mentioned above. By construction, convergence is guaranteed if the nonlinearities verify a Hölder-type condition. We show here that the same Lyapunov design can be extended to the case where the degree of homogeneity is -1. This is interesting since the constraints on the nonlinearities become less and less restrictive when the degree gets closer to -1. It turns out that, in the absence of nonlinearities, the observer we obtain is actually the exact differentiator presented in [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF] and which is defined by an homogeneous differential inclusion. But as opposed to [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF] where convergence is established via a solution-based analysis, in our case, convergence is again guaranteed by construction since the design gives also a homogeneous strict Lyapunov function. Moreover this function enables us to quantify the effect of the observer parameters on the behavior in presence of Hölder nonlinearities and disturbances. Of course, knowing the convergence of the exact differentiator from [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF], we could have deduced the existence of such a Lyapunov function via a converse theorem as in [START_REF] Nakamura | Lyapunov functions for homogeneous differential incolusions[END_REF]. But with only existence, effect quantifications as mentioned above is nearly impossible. Actually many efforts have been made to get expressions for Lyapunov functions but, as opposed to Lyapunov design, Lyapunov analysis is much harder. As far as we know, expressions of Lyapunov functions have been obtained this way only for m ≤ 3. See [START_REF] Ortiz-Ricardez | Smooth lyapunov function and gain design for a second order differentiator[END_REF].

Finally, to face the unfortunate situation where the nonlinearities verify none of the above mentioned Hölder type conditions, we propose a novel observer made of a cascade of homogeneous observers whose maximal total dimension is m(m+1)

2

. We prove that it converges without requiring anything on the nonlinearities (except continuity) in the case where the system trajectories and the input are bounded.

All along our paper, we sometimes use stronger assumptions than necessary in order to simplify the presentation of our results. We signal them to the reader with a ☼ symbol as in "the trajectories are complete ☼ ". We discuss how they can be relaxed later in Section 7, in particular when we restrict our attention to compact sets.

Notations

(1) We define the signed power function as

a b = sign(a) |a| b ,
where b is a nonnegative real number. In the particular case where b = 0, a 0 is actually any number in the set

S(a) =      {1} if a > 0 , [-1, 1] if a = 0 , {-1} if a < 0 . (1)
Namely, writing c = a 0 means c ∈ S(a). Note that the set valued map a → S(a) is upper semicontinuous with nonempty, compact and convex values.

(2) For (z 1 , . . . , z i ) and (ẑ 1 , . . . , ẑi ) (resp. (ẑ i1 , . . . , ẑii ))

in R i , we denote

z i = (z 1 , . . . , z i ) ẑi = (ẑ 1 , . . . , ẑi ) (resp. ẑi = (ẑ i1 , . . . , ẑii )) e ij = ẑij -z j , e j = ẑj -z j , e i = ẑi -z i .
2 Continuous triangular form Consider a nonlinear system of the form

                       ż1 = z 2 + Φ 1 (u, z 1 ) + w 1 . . . żi = z i+1 + Φ i (u, z 1 , . . . , z i ) + w i . . . żm = Φ m (u, z) + w m y = z 1 + v , ( 2
)
where z is the state in R m , y is a measured output in R, Φ is a continuous function which is not assumed to be locally Lipschitz. w can model either a known or an unknown disturbance on the dynamics and v is an unknown disturbance on the measurement. Given locally bounded measurable time functions t → u(t) and t → w(t), we denote Z(z, t; u, w) a solution of (2) going through z at time 0 which, to simplify the presentation, is assumed to be defined for all t ≥ 0 (i.e. the trajectories are complete ☼ ). We are interested in estimating Z(z, t; u, w) knowing y and u.

As mentioned in the introduction, this kind of triangular continuous form, as we call (2), appears when we consider systems which are uniformly observable and differentially observable but with an order larger than the system's dimension. An example is given in Section 8.

The only existing observer we are aware of able to cope with Φ no more than continuous is the one presented in [START_REF] Barbot | Sliding mode observer for triangular input form[END_REF]. Its dynamics are described by a differential inclusion (see Appendix A) :

ż ∈ F (ẑ, y, u)
where (ẑ, y, u) → F (ẑ, y, u) is a set valued map. In the disturbance free context (i.e. v = w i = 0), it can be shown that any absolutely continuous solution gives in finite time an estimate of z under the only assumption of boundedness of the input and of the state trajectory. But the set valued map F above does not satisfy the usual basic assumptions (upper semi-continuous with compact and convex values) (see [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF][START_REF] Smirnov | Introduction to the theory of differential inclusions[END_REF]). It follows that we are not guaranteed of the existence of absolutely continuous solutions nor of possible sequential compactness of such solutions and therefore of possibilities of approximations of F . That is why, in this paper, we look for other candidate observers for the triangular form [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF].

In doing so, we might have to restrict the possible nonlinearities allowed to obtain the existence of an observer.

The restriction we will impose can be described as follows. For a positive real number a, and a vector α in [0, 1]

m(m+1) 2
, we will say that the function Φ verifies the property P(α, a) if :

Property P(α, a)
☼ : For all i in {1, . . . , m}, for all z ia and z ib in R m and u in U , we have1 :

|Φ i (u, z ia ) -Φ i (u, z ib )| ≤ a i j=1 |z ja -z jb | αij . (3)
This property captures many possible contexts. In the case in which α ij > 0, it implies that the function Φ is Hölder with power α ij . When the α ij = 0, it simply implies that the function Φ is bounded. In the following, our aim is to design an observer depending on the values of α.

It is possible to employ the degree of freedom given in (2) by the time functions w to deal with the case in which the given function Φ(u, z) doesn't satisfy P(a, α).

In this case, an approximation procedure can be carried out to get a function Φ satisfying P(a, α) and selecting w = Φ(u, z) -Φ(u, z) which is an unknown disturbance.

The quality of the estimates obtained from the observer will then depend on the quality of the approximation (ie the norm of w). This is what is done for example in [START_REF] Moreno | Approximate high-gain observers for uniformly observable nonlinear systems[END_REF] when dealing with locally Lipschitz approximations. We will further discuss in Section 7 how to relax assumption P(a, α).

In Section 3, we start by showing the convergence with an arbitrary small error of the classical high gain observer when the nonlinearity Φ verifies the property P for certain values of α ij . We deduce in Section 4 the convergence with an arbitrary small error for a cascaded high gain observer when the input and the state trajectories are bounded. On an other hand, in Section 5, we show that replacing the high gain structure by an homogeneous structure enables to obtain convergence under a slightly more restrictive Hölder restriction. Then, a cascaded homogeneous observer is presented in Section 6, which ensures asymptotic convergence when the input and the state trajectories are bounded. As already mentioned, in Section 7, we indicate how the assumptions, marked with ☼ in the text, can be relaxed. Finally, we illustrate our observers with an example in Section 8.

High gain observer

We consider in this section a classical high gain observer:

           ż1 = ẑ2 + Φ 1 (u, ẑ1 ) + ŵ1 -L k 1 (ẑ 1 -y) ż2 = ẑ3 + Φ 2 (u, ẑ1 , ẑ2 ) + ŵ2 -L 2 k 2 (ẑ 1 -y) . . . żm = Φ m (u, ẑ) + ŵm -L m k m (ẑ 1 -y) (4)
where L and the k i 's are gains to be tuned, y is the measurement. The ŵi are approximations of the w i . In particular, when w i represents unknown disturbances, the corresponding ŵi is simply taken equal to 0. In the following, we denote ∆w = ŵ -w .

When Φ satisfies the property P(α, a) with α ij = 1 for all 1 ≤ j ≤ i ≤ m, we recognize the usual triangular Lipschitz property for which the nominal high-gain observer gives an input to state stability (ISS) property with respect to the measurement disturbance v and dynamics disturbance w. It is well known that the ISS gain between the disturbance and the estimation error depends on the high-gain parameter L. Specifically, we have the following well known result. See for instance [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] for a proof.

Proposition 1 (Nominal high-gain) There exist real numbers k 1 , . . . , k m , L * , λ, β and γ such that, a) for all functions Φ satisfying ☼ for all i and for all z ia and z ib in R m

|Φ i (u, z ia ) -Φ i (u, z ib )| ≤ a i j=1 |z ja -z jb | + b i (5)
b) for all L ≥ max{aL * , 1}, c) for all locally bounded time function

(u, v, w, ŵ), all (z, ẑ) in R m × R m ,
any solution Ẑ(ẑ, z, t; u, v, w, ŵ) of ( 4) verifies, for all t 0 and t such that t ≥ t 0 ≥ 0, and for all i in {1, ..., m},

Ẑi (t) -Z i (t)) (6)
≤ max L i-1 β Ẑi (t 0 ) -Z i (t 0 )) e -λL(t-t0) , γ sup 1≤j≤m s∈[t0,t] L i-1 |v(s)|, |∆w j (s)| + b j L j-i+1 .
where we have used the abbreviation Z(t) = Z(z, t; u, w) and Ẑ(t) = Ẑ(z, ẑ, t; u, v, w, ŵ).

Since the nominal high-gain observer gives asymptotic convergence for Lipschitz nonlinearities, we may wonder what type of property is preserved when the nonlinearities are only Hölder. In the following proposition, we show that the usual high-gain observer can provide an arbitrary small error on the estimate providing the Hölder orders α ij satisfy the restrictions given in Table 1 or Equation [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF].

Proposition 2 Assume the function Φ verifies P(α, a) for some (α, a) in [0, 1]

m(m+1) 2 × R + satisfying, for 1 ≤ j ≤ i m-i-1 m-i < α ij ≤ 1 for i = 1 . . . , m -1 , 0 ≤ α mj ≤ 1 (7)
Then, there exist real numbers k 1 , . . . , k m , such that, for all > 0 we can find positive real numbers λ, β, γ, and

. . . m-2 m-1 m i 1 m-2 m-1 2 m-3 m-2 m-3 m-2 . . . α ij > . . . . . . . . . m-2 1 2 1 2
. . . L * such that, for all L ≥ L * , for all locally bounded time function (u, v, w, ŵ) and all (z, ẑ) in R m × R m , any solution Ẑ(ẑ, z, t; u, v, w, ŵ) of (4) verifies, for all t 0 and t such that t ≥ t 0 ≥ 0, and for all i in {1, ..., m},

Ẑi (t) -Z i (t)) ≤ max , L i-1 β Ẑi (t 0 ) -Z i (t 0 )) e -λL(t-t0) , γ sup 1≤j≤m s∈[t0,t] L i-1 |v(s)|, |∆w j (s)| L j-i+1
where we have used the abbreviation Z(t) = Z(z, t; u, w) and Ẑ(t) = Ẑ(z, ẑ, t; u, v, w, ŵ).

Comparing this inequality with (6), we have now the arbitrarily small non zero ε in the right hand side but this is obtained under the Hölder condition instead of the Lipschitz one.

PROOF. With Young's inequality, we obtain from (3) that, for all σ ij in R + and all ẑ and z in R m

|Φ i (u, ẑi ) -Φ i (u, z i )| ≤ i j=1 a ij |ẑ j -z j | + b ij , (8)
with a ij and b ij defined as

         a ij = 0 , b ij = a , if α ij = 0 a ij = a 1 α ij α ij σ 1 α ij ij , b ij = 1-αij σ 1 1-α ij ij if 0 < α ij < 1 a ij = a , b ij = 0 if α ij = 1 (9) With (8), the assumptions of Proposition 1 are satisfied with b i = i j=1 b ij . It gives k 1 , . . . , k m , L * , λ, β and γ and, if L > max i≥j {a ij L * , 1}
, the solution satisfies the ISS inequality [START_REF] Bernard | On the triangular canonical form for uniformly observable controlled systems[END_REF]. The result will follow if there exist L and σ ij such that

L > max i≥j {a ij L * , 1} , max i,j j =1 γb j L i-j-1 ≤ . (10)
At this point, we have to work with the expressions of a ij and b j given in [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. From [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF], α ij can be zero only if i = m. And, when α m = 0, we get

γb m L i-m-1 = γaL i-m-1 ≤
γa L Say that we pick σ m = 1 in this case. For all the other cases, we choose

σ j = 2jγ (1 -α j )L (m-j-1)
1-α j , to obtain from ( 9)

γb j L i-j-1 ≤ 1 j 1 2L m-i
So, with this selection of the σ j , the right inequality in [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF] is satisfied for L sufficiently large. Then, according to [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], the a ij are independent of L or proportional to

L (m-i-1) 1-α ij α ij . But with (7) we have 0 < (m -i -1) 1 -α ij α ij < 1 .
This implies that aij L tends to 0 as L tends to +∞. We conclude that (10) holds if we pick L sufficiently large.

It is interesting to remark the weakness of the assumptions imposed on the last two components of the function Φ. Indeed, [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF] only imposes that Φ m-1 be Hölder without any restriction on the order, and that Φ m be bounded ☼ .

Cascaded high gain observer

According to Proposition 2, the classical high gain observer can provide an arbitrary small error when the last nonlinearity is only bounded and when there is no disturbance. We exploit here this observation by proposing the following cascaded high gain observer to deal with the case where the functions Φ i do not satisfy ( 7

. żm1 = ẑm2 + Φ 1 (u, ẑ(m-1)1 ) + ŵ1 -L m k m1 (ẑ m1 -z 1 ) żm2 = ẑm3 + Φ 2 (u, ẑ(m-1)1 , ẑ(m-1)2 ) + ŵ2 -L 2 m k m2 (ẑ m1 -z 1 ) . . . żmm = ŵm -L m m k mm (ẑ m1 -z 1 ) (11)
with the gain k ij chosen as in a classical high gain ob-server of dimension i, ŵi are estimations of w i and L i are the high gains parameters to be chosen.

Assuming the input function and the system solution are bounded, it is shown in the following that estimation with an arbitrary small error can be achieved by the cascaded high-gain observer.

Proposition 3 Assume Φ is continuous. For any positive real numbers z and u, for any strictly positive real number , there exist a choice of (L 1 , ..., L m ), a class KL function β and two class K ∞ functions γ 1 and γ 2 such that, for all locally bounded time function (u, v, w, ŵ), for all (z, ẑ) in R m × R m and for all t such that |Z(z, s; u, w)| ≤ z and |u(s)| ≤ u for all 0 ≤ s ≤ t, any solution Ẑ1 (ẑ, z, t; u, v, w, ŵ), ..., Ẑm (ẑ, z, t; u, v, w, ŵ) of ( 11) verifies, for all i in {1, . . . , m},

| Ẑi (t) -Z i (t)| ≤ max ε , β   i j=1 |ẑ j -z j |, t   , sup s∈[0,t] γ 1 (|v(s)|), γ 2 (|∆w(s)|)
where Ẑi is the state of the ith block (see Notation 2) and we have used the abbreviation

Ẑi (t) = Ẑi (ẑ, z, t; u, v, w, ŵ) and Z i (t) = Z i (z, t; u, w).
PROOF. This result is nothing but a straightforward consequence of the fact that a cascade of ISS systems is ISS.

Specifically the error system attached to the high gain observer in block i has state e i (see Notation 2) and input v and δ ij defined as

δ ij = Φ j (u, ẑ(i-1) ) -Φ j (u, z (i-1) ) + [ ŵj -w j ] δ ii = -z i+1 -Φ i (u, z i ) + ŵi -w i
with z m+1 = 0. With Proposition 1, we have the existence of k i1 , . . . , k ii , λ i , β i and γ i such that we have, for all L i ≥ 1, all t ≥ t i ≥ 0, all j in {1, . . . , i} and with e ij (t) denoting the jth error in the ith block evaluated along the solution at time t,

|e ij (t)| ≤ max L j-1 i β i |e i (t i )| e -λiLi(t-ti) , γ i sup 1≤ ≤j s∈[ti,t] L j-1 i |v(s)|, |δ i (s)| L -j+1 i .
But the continuity of the Φ j implies the existence of a function 2 ρ of class K such that, for all j in {1, . . . , m} and for all (

z (i-1) , ẑ(i-1) , u) in R i-1 × R i-1 × U satisfy- ing |z (i-1) | ≤ z and |u| ≤ u, |Φ j (u, ẑ(i-1) ) -Φ j (u, z (i-1) )| ≤ ρ |e (i-1) | .
This implies

|δ i (s)| ≤ ρ(|e i-1 (s)|) + |∆w (s)| , = 1, . . . , j -1 , |δ ii (s)| ≤ z i+1 + Φ i + |∆w i (s)| ,
where

Φ i = max |u|≤u,|zi|≤z |Φ i (u, z i )|.
Hence, we have the existence of c i independent of L i such that

|e i (t)| ≤ c i max L i-1 i |e i (t i )| e -λiLi(t-ti) , sup s∈[ti,t] L i-1 i |v(s)| , sup s∈[ti,t] ρ(|e i-1 (s)|) L 2-i i , sup 1≤ ≤i s∈[ti,t] |∆w (s)| L -i+1 i , z i+1 + Φ i L i .
This

{γ vi (|v(s)|), γ wi (|∆w(s)|)} .
where i is a positive real number defined by the sequences

1 = c 1 z 2 + Φ 1 L 1 , i = c i max z i+1 + Φ i L i , ρ( i-1 ) L 2-i i .
Then by picking L i ≥ L * i where L * i is defined recursively as :

m = , i = min , ρ -1 i+1 ci+1L i-2 i+1 i+1 L * m = c m Φ m ε m , L * i = c i [z i+1 + Φ i ] ε i
we obtain i ≤ for all i, hence the result.

This observer has the advantage of working without any assumption on the nonlinearities besides their continuity. Note however that it requires the knowledge of a bound on the system solution and on the input. Also we may not need to build m blocks, since according to Proposition 2, we need to create a new block only for the indexes i where Φ i does not verify Property P(α, a) for any a ≥ 0 and with α satisfying [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF]. Unfortunately, as it appears from the proof of Proposition 3, the choice of (L 1 , ..., L m ) can be complicated. Besides, only a convergence with an arbitrary small error is obtained. It may thus be necessary to take very high gains which is problematic in terms of peaking and most importantly in presence of noise (see Section 8). In the following two sections, we move our attention to homogeneous observers, and show that they enable to obtain convergence.

Homogeneous observer

Homogeneous observers are extensions of high gain observers able to cope with some non Lipschitz functions.

As mentioned in the introduction, they already have an old history (see [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF], [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF], [START_REF] Yang | Homogeneous observers, iterative design, and global stabilization of highorder nonlinear systems by smooth output feedback[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Qian | A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems[END_REF], [START_REF] Qian | Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems[END_REF], [START_REF] Andrieu | Nonlinear output feedback design via domination and generalized weighted homogeneity[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF] ). In our context they take the form :

ż1 = ẑ2 + Φ 1 (u, ẑ1 , t) + ŵ1 -L k 1 ẑ1 -y r 2 r 1 ż2 = ẑ3 + Φ 2 (u, ẑ1 , ẑ2 , t) + ŵ2 -L 2 k 2 ẑ1 -y r 3 r 1 . . . żm = Φ m (u, ẑ, t) + ŵm -L m k m ẑ1 -y r m+1 r 1
(12) where r is a vector in R m+1 , called weight vector, the components of which, called weights, are defined by

r i = 1 -d 0 (m -i) , (13)
and where L and the k i 's are gains to be tuned, d 0 is a parameter to be chosen in [-1, 0]. We refer to Notation (1) for the case d 0 = -1, for which the dynamics ( 12) must be understood as a differential inclusion. When d 0 = 0, we recover the high-gain observer studied in Section 3. As mentioned in Proposition 2, the usual high-gain observer can provide an estimation with an arbitrary small error provided the nonlinearity satisfies the property P(α, a) with the α ij verifying [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF]. In the following proposition we claim that asymptotic estimation may be obtained with homogeneous correction terms and when considering nonlinearities which satisfies P(α, a) with the α ij verifying

α ij = 1 -d 0 (m -i -1) 1 -d 0 (m -j) = r i+1 r j , 1 ≤ j ≤ i ≤ m .
(14) Those conditions in the extreme case where d 0 = -1 are summed up in Table 2. On top of that, finite time estimation may be obtained. Proposition 4 Assume that there exist d 0 in [-1, 0] and a in R + such that Φ satisfies P(α, a) with α verifying (14) ☼ . There exist (k 1 , . . . , k m ), such that for all wm > 0 there exist L * ≥ 1 and a positive constant γ such that, for all L ≥ L * there exists a class KL function β such that for all locally bounded time function (u, v, w, ŵ), and all (z, ẑ) in R m × R m system (12) admits absolutely continuous solutions Ẑ(ẑ, z, t; u, v, w, ŵ) defined on R + and for any such solution the following implications hold for all t 0 and t such that t ≥ t 0 ≥ 0, and for all i in {1, ..., m} :

If d 0 > -1 : j 1 2 . . . m-2 m-1 m i 1 m-1 m 2 m-2 m m-2 m-1 . . . α ij = . . . . . . . . . m-2 2 m 2 m-1 . . . 2 3 m-1 1 m 1 m-1 . . . . . .
d 0 = -1 | Ẑi (t) -Z i (t)| ≤ max β(| Ẑ(t 0 ) -Z(t 0 )|, t -t 0 ) , (15) γ sup 1≤j≤i s∈[t0,t] L i-1 |v(s)| r i r 1 , |∆w j (s)| r i r j+1 L µij where µ ij = (j-i+1) r1
rj+1 , and we have used the abbreviation Z(t) = Z(z, t; u, w) and Ẑ(t) = Ẑ(z, ẑ, t; u, v, w, ŵ). Moreover, when d 0 < 0 and v(t) = w j (t) = 0 for all t and j = 1, . . . , m, there exists T such that Ẑ(ẑ, z, t) = Z(z, t) for all t ≥ T .

If d 0 = -1 and |∆w m (t)| ≤ wm : | Ẑi (t) -Z i (t)| ≤ max β(| Ẑ(t 0 ) -Z(t 0 )|, t -t 0 ) , (16)
γ sup 1≤j≤i-1 s∈[t0,t] L i-1 |v(s)| r i r 1 , |∆w j (s)| r i r j+1 L µij
where µ ij , Z(t) and Ẑ(t) are defined above. Moreover, when v(t) = w j (t) = 0 for all t and j = 1, . . . , m, there exists T such that Ẑ(t) = Z(t) for all t ≥ T .

Note that j is in {1, . . . , i} in [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] whereas it is in {1, . . . , i -1} in [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF].

The proof of Proposition 4 for the case d 0 ∈] -1, 0] and without disturbances is given for example in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]. Actually [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF] gives a Lyapunov design of the observer (12) with a recursive construction of both Lyapunov function and observer. Here we are concerned with the case d 0 = -1. In this limit case, the observer ( 12) is a differential inclusion corresponding to the exact differentiator studied in [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF], where convergence is established in the particular case in which Φ i = 0 for j = 1, . . . , m -1 and Φ m is bounded. We prove in Proposition 6 that the Lyapunov design of [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF] can be extended to this case. This allows us to show that the observer (12) still converges if, for each i, Φ i is Hölder with order α ij equal to the values given in Table 2, where i is the index of Φ i and j is the index of e j . We also recover the same bound in presence of a noise v as the one given in [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF].

Actually some effort has been devoted to Lyapunov analysis for establishing the convergence of the observer proposed in [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF]. But, as far as we are aware of, this more difficult route has been successful for m ≤ 3 only. See [START_REF] Ortiz-Ricardez | Smooth lyapunov function and gain design for a second order differentiator[END_REF].

Finally, it is interesting to remark that in the case d 0 = -1 the ISS property between the disturbance w m and the estimation error is with restrictions as defined in [START_REF] Teel | Nonlinear small gain theorem for the analysis of control systems with saturations[END_REF]Definition 3.1]. If |∆w m (t)| ≤ wm and L is chosen sufficiently large, then asymptotic convergence is obtained. However, nothing can be said when |∆w m | > wm . Moreover, it may be possible for a bounded large disturbance to induce a norm of the estimation error which goes to infinity. We believe that this problem could be solved employing homogeneous in the bi-limit observer as in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]. It is shown to be doable in dimension 2 in [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF].

PROOF. The set-valued function e 1 → e 1 0 defined in Notations 1 is upper semi-continuous and has convex and compact values. Thus, according to [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], there exist absolutely continuous solutions to [START_REF] Gauthier | Deterministic observation theory and applications[END_REF].

Let L = diag(1, L, ..., L m-1 ). The error e = ẑ -z produced by the observer [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]

satisfies ė ∈ LS m e + δ + LLK(e 1 + v) (17)
where S m is the shifting matrix of order m,

δ = Φ(u, ẑ) + ŵ -Φ(u, z) -w ,
and K is the homogeneous correction term the components of which are defined as

(K(e 1 )) i = k i e 1 r i+1 r 1
where (k 1 , . . . , k m ) are positive real number and r i is defined in [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF].

Let also V : R m → R + be the function defined as

V (ē) = m-1 i=1 i ēi ēi+1 r i r i+1 x d V -r i r i -ēi+1 d V -r i r i+1 dx + |ē m | d V d V , (18)
where d V and i are positive real numbers such that d V > 2m -1. Note that V is a homogeneous function with weight vector r. It is nothing but the one proposed in [2, Theorem 3.1] for designing an observer homogeneous in the bi-limit with d 0 in ] -1, 0]. There it is shown that, by appropriately selecting the parameters i and k i , V is a strict C 1 Lyapunov function homogeneous of degree d V for the L-independent auxiliary system with state ē :

ė ∈ S m ē + K(ē 1 ) . ( 19
)
With this result in hand a robustness analysis can be carried out on a system of the form [START_REF] Moreno | Approximate high-gain observers for uniformly observable nonlinear systems[END_REF]. In fact, the same approach can be followed for the case d 0 = -1 and the following technical result is proved in Appendix B.

Lemma 1 For all d 0 in [-1, 0], the function V defined in ( 18) is positive definite and there exist positive real numbers k 1 , . . . k m , 1 , . . . m , λ, c δ and c v such that for all ē in R m , δ in R m and v in R the following implication holds :

if | δi | ≤ c δ V (ē) r i+1 d V , ∀i , and |v| ≤ c v V (ē) r 1 d V then 3 max ∂V ∂ē (ē)(S m (ē) + δ + K(ē 1 + v)) ≤ -λV (ē) d V +d 0 d V .
This Lemma says V is a ISS Lyapunov function for the auxiliary system [START_REF] Ortiz-Ricardez | Smooth lyapunov function and gain design for a second order differentiator[END_REF]. See [24, Proof of Lemma 2.14] for instance. Consider now the scaled error coordinates ε = L -1 (ẑ -z). Straightforward computations from [START_REF] Moreno | Approximate high-gain observers for uniformly observable nonlinear systems[END_REF] give the error system

1 L ε ∈ S m ε + D L + K(ε 1 + v)
with D L = L -1 δ. Since Φ satisfies P(α, a), with ( 14) and

ri+1 rj ≤ 1, we obtain, for all L ≥ 1 |D L,i | ≤ a L i j=1 L (j-1) r i+1 r j -i+1 |ε j | r i+1 r j + |∆w i | L i , ≤ a L i j=1 |ε j | r i+1 r j + |∆w i | L i , ≤ c L V (ε) r i+1 d V + |∆w i | L i ,
where c is a positive real number obtained from Lemma 3 in Appendix D. With Lemma 1, where δi plays the role of D L,i , v the role of v and ē the role of ε, we obtain that, by picking L * sufficiently large such that c L * ≤ c δ 2 , we have, for all L > L * , if

   |∆w i | L i ≤ c δ 4 V (ε) r i+1 d V , ∀i |v| ≤ c v V (ε) r 1 d V (20)
⇒ 1 L max ∂V ∂e (ε) ε ≤ -λV (ε) d V +d 0 d V .
Now, when evaluated along a solution, ε gives rise to an absolutely continuous function t → ε(t). Similarly the function defined by t → ν(t) = V (ε(t)) is absolutely continuous. It follows that its time derivative is defined for almost all t and, according to [22, p174], (20) implies, for almost all t,

if    |∆w i | L i ≤ c δ 4 ν(t) r i+1 d V , ∀i |v| ≤ c v ν(t) r 1 d V (21) ⇒ 1 L ν(t) ≤ -λν(t) d V +d 0 d V
.

Here two cases have to be distinguished.

(1) If d 0 is in ] -1, 0], with Lemma 5 in Appendix D (see also [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]), we get the existence of a class KL function β V such that 4

V (ε(t)) ≤ max i∈[1,m] β V (V (ε(0)), λLt), sup s∈[0,t]    4|∆w i (s)| L i c δ d V r i+1 , |v(s)| d V r 1 c v    .
The result holds since with Lemma 3 there exist a positive real number c 1 such that

e i L i-1 ≤ c 1 V ( ) r i d V .
Moreover, when v(t) = ∆w j (t) = b j = 0 for j = 1, . . . , m, (21) implies finite time convergence in the case in which d 0 < 0.

(2) If d 0 = -1, then r m+1 = 0. We choose L * sufficiently large to satisfy wm (L * ) m ≤ c δ [START_REF] Barbot | Sliding mode observer for triangular input form[END_REF] .

We obtain that the first condition in ( 21) is satisfied for i = m when L ≥ L * . With Lemma 5 in Appendix D (see also [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]), the implication (21) implies the existence of a class KL function

β V such that 4 V (ε(t)) ≤ max i∈[1,m-1] β V (V (ε(0)), λLt), sup s∈[0,t]    4|∆w i (s)| L i c δ d V r i+1 , |v(s)| d V r 1 c v    .
And the result holds as in the previous case.

Cascade of homogeneous observers

When we cannot find d 0 in [-1, 0] and a such that the nonlinearities satisfy P(α, a), with α defined in [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF], we may lose the convergence of observer [START_REF] Gauthier | Deterministic observation theory and applications[END_REF], or the possibility of making the final error arbitrarily small. In such a bad case, we can still take advantage of the fact that, for 4 according to Lemma 5, βV (s, t) = max{0, s

-d 0 d V -t} d V -d 0
α verifying ( 14) with d 0 = -1, P(α, a) does not impose any restriction besides boundedness of the last functions Φ m (see Table 2).

From the remark that observer [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] (1) can be used for the system

ż1 = z 2 + ψ 1 (t) . . . żk-1 = z k + ψ k-1 (t) żk = ϕ k (t)
provided the functions ψ i are known and the function ϕ k is unknown but bounded, with known bound.

(2) gives estimates of the z i 's in finite time, we see that it can be used as a preliminary step to deal with the system

ż1 = z 2 + ψ 1 (t) . . . żk-1 = z k + ψ k-1 (t) żk = z k+1 + Φ k (u, z 1 , . . . , z k ) żk+1 = ϕ k+1 (u, z 1 , . . . , z k+1 )
Indeed, thanks to the above observer we know in finite time the values of z 1 , . . . , z k , so that the function Φ k (u, z 1 , . . . , z k ) becomes a known signal ψ k (t).

From this, we can propose the following observer made of a cascade of homogeneous observers :

ż11 ∈ ŵ1 -L 1 k 11 S(
+ ŵ1 -L m k m1 ẑm1 -y m-1 m . . . żm(m-1) = ẑmm + Φ m-1 (u, ẑ(m-1)1 , . . . , ẑ(m-1)(m-1) ) + ŵm-1 -L m-1 m k m(m-1) ẑm1 -y 1 m żmm ∈ ŵm -L m m k mm S(ẑ m1 -y) (22)
where the k ij and L i are positive real numbers to be tuned.

As a direct consequence of Proposition 4 and following the same steps as in the proof of Proposition 3, we have Proposition 5 Assume Φ is continuous. For any positive real numbers z, u w, we can find positive real numbers k ij and L i , two class K functions γ 1 and γ 2 and a class KL function β such that, for all locally bounded time function (u, v, w, ŵ), and all (z, ẑ) in R m × R m , the observer (22) admits absolutely continuous solutions Ẑ1 (ẑ, z, t; u, v, w, ŵ), ..., Ẑm (ẑ, z, t; u, v, w, ŵ) which are defined on R + and for any such solution we have for all i in {1, ..., m} and for all t such that |Z(z, s; u, w)| ≤ z, |u(s)| ≤ u and |∆w(s)| ≤ w for all 0 ≤ s ≤ t:

| Ẑi (t) -Z i (t)| ≤ max β(|z -ẑ|, t), sup 1≤j≤i-1 s∈[t0,t] {γ 1 (|v(s)|), γ 2 (|∆w j (s)|)} .
where Ẑi is the state of the ith block (see Notation 2) and we have used the abbreviation Ẑi (t) = Ẑi (ẑ, z, t; u, v, w, ŵ) and Z i (t) = Z i (z, t; u, w). Moreover, when v(t) = ∆w j = 0, there exists T such that Ẑi (ẑ, z, t) = Z i (z, t) for all t ≥ T . This observer is an extension of the cascaded high gain observer [START_REF] Gauthier | A simple observer for nonlinear systems application to bioreactors[END_REF] presented in Section 4. The use of homogeneity enables here to obtain convergence without demanding anything but the knowledge of a bound on the input and on the system solution. A drawback of a cascade of observers is that it gives an observer with dimension m(m+1) 2 in general. However, as seen in Section 4, it may be possible to reduce this dimension since, for each new block, one may increase the dimension by more than one, when the corresponding added functions Φ i satisfy P(α, a) m for some α verifying [START_REF] Levant | Higher-order sliding modes and arbitrary-order exact robust differentiation[END_REF] with d 0 = -1 and for some a.

Finally, note that the result of Proposition 5 does not mean that the observer is ISS with respect to ∆w. Indeed, ∆w must be bounded to obtain this ISS-like inequality : the system is ISS with restrictions. Again, we believe that this problem could be solved employing homogeneous in the bi-limit observer as in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF].

7 Relaxing the assumptions marked with ☼ First, if System (2) is not complete, every ISS inequalities still holds for any solution Z(z, t; u, w) but only on [0, T (z)[ where T (z) is its maximal time of existence.

The global aspect of boundedness, Höder, P(α, a), . . . , can be relaxed as follows. Let U be bounded and let M be a given compact set. We define Φ, to be used instead of Φ in the observers, as Φi (u, z 1 , ..., z i ) = sat(Φ i (u, z 1 , ..., z i ), Φi ) [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] where Φi = max u∈U,z∈M (Φ i (u, z 1 , ..., z i )) and the saturation function is defined on R by sat(x, M ) = max(min(x, M ), -M ) .

It can be shown that, for any compact set M strictly contained in M, there exists ã such that (3) holds for Φ for all (z a , z b ) in R m × M. Then, since Φ = Φ on M, we can modify the assumptions -in Proposition 1, so that (5) holds only on the compact set M; -in Propositions 2 and 4, so that Φ verifies P(α, a) only on the compact set M; -Propositions 3 and 5 remain unchanged. In this case, the results hold for the particular system solutions Z(z, t; u, w) which are in the compact set M for t in [0, T (z)[. Precisely, for these solutions, the bounds on Ẑi (t) -Z i (t) given in these Propositions hold for all t in [0, T (z)[. Note also that if P(α, a) holds on a compact set, then for any α such that αij ≤ α ij for all (i, j), there exists ã such that P(α, ã) also holds on this compact set. It follows that the constraints given by ( 14) or Table 2 in Proposition 4 can be relaxed to

α ij ≥ 1-d0(m-i-1)
1-d0(m-j) , and the less restrictive conditions one may ask for are obtained for d 0 = -1.

Finally, in Propositions 1, 2 and 4, it is possible to consider the case where Φ depends also on time as long as any assumption made on Φ is satisfied uniformly with respect to time.

Example

Consider the system ẋ1 = x 2 , ẋ2 = -x 1 + x 53 x 1 , ẋ3 = -x 1 x 2 + u , y = x 1 (24) with u as input. It would lead us too far from the main subject of this article to study here the solutions behavior of this system. We note however that, when u is zero, they evolve in the 2-dimensional surface {x ∈ R 3 : 3x 2 1 + 3x 2 2 + x6 3 = c 6 } which is diffeomorphic 5 to the sphere S 2 . Thanks to Poincaré-Bendixon theory, we know the solutions are periodic and circling the unstable equilibria (x 1 = x 2 = 0, x 3 = ±c). So we hope for the existence of solutions remaining in the compact set

C r, = x ∈ R 3 : x 2 1 + x 2 2 ≥ , 3x 2 1 + 3x 2 2 + x 6 3 ≤ r
for instance when u is a small periodic time function, except maybe for pairs of input u and initial condition (x 1 , x 2 , x 3 ) for which resonance could occur. Moreover, due to their periodic behavior, such solutions are likely to have their x 3 component recurrently crossing zero.

Uniform and differential observability

On S = x ∈ R 3 : x 2 1 + x 2 2 = 0 , and whatever u is, the knowledge of the function t → y(t) = X 1 (x, t) and therefore of its three first derivatives

ẏ = x 2 ÿ = -x 1 + x 5 3 x 1 ... y = -x 2 -5x 4 3 x 2 1 x 2 + x 5 3 x 2 + 5x 4 3 x 1 u
gives us x 1 , x 2 and x 3 . Thus, System ( 24) is uniformly observable on S. Besides, the function

H 4 (x) =      x 1 x 2 -x 1 + x 5 3 x 1 -x 2 -5x 4 3 x 2 1 x 2 + x 5 3 x 2     
is injective on S and admits the left following inverse, defined on z ∈ R 4 : z 2 1 + z 2 2 = 0 , is:

H -1 4 (z) =        z 1 z 2   (z3+z1)z1+ (z4+z2)+3|(z3+z1) z1 3 2 | 4 5 z2 z2 z 2 1 +z 2 2   1 5       
However, H 4 is not an immersion because of a singularity of its Jacobian at x 3 = 0. So the system is differentially observable of order 4 on S but not strongly. According to [START_REF] Bernard | On the triangular canonical form for uniformly observable controlled systems[END_REF], it admits a triangular canonical form of dimension 4 but with functions Φ maybe non Lipschitz.

8.2 Triangular form and property P(d 0 , c, 0)

The triangular canonical form of dimension 4 mentioned above is

ż1 = z 2 ż2 = z 3 ż3 = z 4 + Φ 3 (u, z 1 , z 2 , z 3 ) ż4 = Φ 4 (u, z) y = z 1 . (25)
where

Φ 3 (u, z 1 , z 2 , z 3 ) = 5u|z 3 + z 1 | 4 5 z 1 1 5
and Φ 4 is a continuous non-Lipschitz function the expressions of which is complex, fortunately with no interest here. The function Φ 3 is not Lipschitz at the points on the hyperplanes z 3 = -z 1 and z 1 = 0 (image by H 3 of points where x 3 = 0 or x 1 = 0) known to be visited possibly recurrently along solutions. This example thus falls precisely into the scope of the paper.

The function Φ 4 is continuous and therefore bounded on any compact set including H 4 ( Cr, ). Besides, for ẑ3 and z 3 in a compact set including H 3 ( Cr, ), there exist c 1 and c 3 such that

|Φ 3 (u, ẑ1 , ẑ2 , ẑ3 ) -Φ 3 (u, z 1 , z 2 , z 3 )| ≤ c 1 u|ẑ 1 -z 1 | 1 5 + c 3 u|ẑ 3 -z 3 | 4 5 .
This implies that Φ 3 is Hölder with order 1 5 .

Hence the nonlinearities Φ 3 and Φ 4 verify the conditions of Table 1. This implies that for L sufficiently large, convergence with an arbitrary small error can be achieved with the high gain observer (4) . However, Φ 3 does not verify the conditions of Table 2. Thus, there is no theoretical guarantee that the homogeneous observer [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] with d 0 = -1 will provide exact convergence.

An observer of dimension 4 ?

We consider the solution to system (24) with initial condition x = (1, 1, 0) and u = 5 sin(10 t). This solution is periodic and regularly crosses the Lipschitzness singularities x 3 = 0 or x 1 = 0, as illustrated in Figure 1. In the following, we use the same noised measurement y, shown on Figure 1, in every simulation with noise. It is a filtered gaussian noise with standard deviation σ = 0.03 and 1st order filtering parameter a = 50.

We first implement a high gain observer of dimension 4, in the absence of noise, initialized at x = (0.1, 0.1, 0), and with the gains

k 1 = 14, k 2 = 99, k 3 = 408, k 4 = 833.
As an illustration of Proposition 2, the convergence with an arbitrary small error is achieved and is illustrated in Table 3. However, we observe that the decrease of the errors, especially for e z,4 , is very slow compared to the increase of the peaking and a very high gain is needed to obtain "acceptable" final errors. In presence of noise, the tradeoff between final error and noise amplification becomes very difficult : with the noised measurement of Figure 1, the smallest final error e z,4 is 200, achieved for L = 2. Of course, there might exist a choice of the gains k i giving better results. But overall a high gain observer may not be a systematic solution in practice for non-Lipschitz triangular systems, especially when the solution regularly crosses the Lipschitz-singularities. 3 Decrease of the final error in the z-coordinates (ez,i = ẑi -zi) with the gain L, with a high gain observer and in the absence of noise.

Let us now implement an homogeneous observer of dimension 4 with an explicit Euler method with fixed measurement and integration steps equaling 10 -5 , and with the Matlab sign function. The degree is d 0 = -1, and the gains are chosen according to [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], i-e k 1 = 5, k 2 = 8.77, k 3 = 4.44, k 4 = 1.1. For a gain L = 3, the convergence is achieved with a final error of 8 . 10 -4 on z 4 , even though the Hölder restriction of Proposition 4 is a priori not satisfied around z 1 = 0. Unfortunately, the final errors are heavily impacted in presence of noise, as illustrated in Table 4. This may also come from a lack of ISS property. Notice that the amplification of the noise by the gain L is not as rapid as expected from the bound in Proposition 4. The final errors remain nonetheless too large, although, once again, we did not optimize our choice of gains k i . 4 Final errors in the z-coordinates given by a homogeneous observer of degree -1 in presence of noise.

Cascaded observers

In the absence of noise, the cascaded observers presented in Sections 4 and 6 give similar results to the corresponding observers in dimension 4, i-e arbitrary small asymptotic error and finite time convergence respectively. However, they seem to provide better accuracies in presence of noise.

In the case of a high gain cascade observer, the errors, although smaller than in the high gain observer of dimension 4, remain too large to consider it a viable solution.

On the other hand, the homogeneous cascade observer : Comparing to Table 4, we see that implementing an intermediate homogeneous observer of dimension 3 enables to obtain much better estimates of the first three states z i , which are then used in the nonlinearity of the second block, thus giving a better estimate of z 4 .

ż11 = ẑ12 -L 1 k 11 ẑ11 -y 2 3 ż12 = ẑ13 -L 2 1 k 12 ẑ11 -y 1
Unfortunately, the presented results are still unsatisfactory in presence of noise, which leaves the question of the construction of robust observers for such systems unanswered.

Conclusion

To summarize the most important ideas, we provide in Table 9 a synthetic comparison of the four observers studied in this paper, in the usual case where the system state and the input are bounded.

We have shown the convergence with an arbitrary small error of the classical high gain observer in presence of nonlinearities verifying some Hölder-like condition. The same result could probably be obtained for the high gainlike observer presented in [START_REF] Besançon | Further results on high gain observers for nonlinear suystems[END_REF]. Also, for the case when this Hölder condition is not verified, we proposed a novel cascaded high gain observer. Under slightly more restrictive assumptions, we proved the convergence of an homogeneous observer and of its cascaded version with the help of an explicit Lyapunov function.

Our numerical experience indicates however that to improve the performances in presence of measurement noise, it is very difficult to tune the gains of both high gain and homogeneous observers, although it is slightly simpler for the latter since smaller gains are sufficient to ensure convergence. Simulations on our example suggest that the situation may be more favorable with the cascaded homogeneous observer. Our ISS bounds in this paper being far too conservative, it is necessary to carry out a finer study if we want to optimally tune the gains of the observers. It may also be appropriate to use online gain adaptation techniques since large gains should be necessary only around the points where the nonlinearities are not Lipschitz. About these two aspects, we refer the reader to the survey in [13, Sections 3.2.2 and 3.2.3] and the references therein.

A Barbot et al's observer

The set valued map proposed in [START_REF] Barbot | Sliding mode observer for triangular input form[END_REF] to obtain an observer for a triangular canonical form where the functions are only locally bounded is defined as follows. Given (ẑ, y, u), (v 1 , . . . , v m ) is in F (ẑ, y, u) if there exists (z 2 , ..., zm ) in R m-1 such that:

v 1 = z2 + g 1 (y) u z2 ∈ sat(ẑ 2 ) -k 1 S(y -ẑ1 ) . . . v i = zi+1 + g i (y, z2 , . . . , zi ) u zi+1 ∈ sat(ẑ i+1 ) -k i S(ẑ i -zi ) . . . v m ∈ ϕ m (y, z2 , . . . , zm ) +g m (y, z2 , . . . , zm ) u -k m S(ẑ m -zm )
where sat is some saturation function.

B Proof of Lemma 1

The proof is based on the following Proposition the proof of which is given in the following section for the case d 0 = -1 and can be found for d 0 in ] -1, 0] in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]. This proposition establishes that for a chain of integrator it is possible to construct homogeneous correction terms which provide an observer and that it is possible to construct a smooth strict homogeneous Lyapunov function.

Proposition 6 For all d 0 in [-1, 0], the function V defined in ( 18) is positive definite and there exists positive real numbers k 1 , . . . k m , 1 , . . . m , λ such that for all e in R m , the following holds :

max ∂V ∂ē (ē) (S m (ē) + K(ē 1 )) ≤ -λV (ē) d V +d 0 d V . (B.1)
Let K(ē 1 , s) be the function defined as

K(ē 1 , s) i = (K(ē 1 )) i , i ∈ [1, m -1] ,
and,

K(ē 1 , s) m = k m s , when d 0 = -1 (K(ē 1 )) m , when d 0 > -1 .
Note that K is a continuous (single) real-valued function which satisfies for all ē1 in R K(ē 1 ) = { K(ē 1 , s) , s ∈ S(ē 1 )} .

Consider also the functions

η(ē, δ, v, s) = ∂V ∂ē (ē)(S m (ē) + δ + K(ē 1 + v, s)) + λ 2 V (ē) d V +d 0 d V , and
γ( δ, v) = m i=1 | δi | d V +d 0 r i+1 + |v| d V +d 0 r 1
.

With (B.1), we invoke Lemma 3 to get the existence of a positive real number c 1 satisfying :

∂V ∂ē (ē)(S m (ē) + ∆ + K(ē 1 + v, s)) ≤ - λ 2 V (ē) d V +d 0 d V + c 1 m i=1 δ d V +d 0 r i+1 i + c 1 |v| d V +d 0 r 1 .
This can be rewritten,

∂V ∂ē (ē)(S m (ē)+ δ+ K(ē 1 +v, s)) ≤ - λ 2(m + 2) V (ē) d V +d 0 d V + m i=1 c 1 | δi | d V +d 0 r i+1 - λ 2(m + 2) V (ē) d V +d 0 d V + c 1 |v| d V +d 0 r 1 - λ 2(m + 2) V (ē) d V +d 0 d V .
Consequently, the result holds with λ =

λ 2(m+2) , c δ = c v = α c1 r 1 d V +d 0 .
C Proof of Proposition 6 when d 0 = -1

In this section, we denote E i = (e i , ..., e m ). Let d V be an integer such that d V > 2m -1 and the functions K i recursively defined by :

K m (e m ) = -e m 0 , K i (e i ) =   -i e i r i+1 r i K i+1 i e i r i+1 r i   . Let V m (e m ) = |em| d V d V
and for all i in {1, . . . , m -1}, let also Vi : R 2 → R and V i : R n-i+1 → R be the functions defined by

Vi (ν, e i+1 ) = ν ei+1 r i r i+1 x d V -r i r i -e i+1 d V -r i r i+1 dx , V i (E i ) = i j=m-1 Vj ( j e j , e j+1 ) + V m (e m ) .
With these definitions, the Lyapunov function V defined in ( 18) is simply V (e) = V 1 (e) and the homogeneous Comparison between observers when the system state and the input are bounded.

vector field K(e 1 ) = K 1 (e 1 ) with

k i = ri+1 r i i r i+1 r i-1 i-1 ... r i+1 r 2 2 r i+1 r 1 1 .
Note that the jth component of K i is homogeneous of degree r j+1 = m -j and, for any e i in R, the set K i (e i ) can be expressed as

K i (e i ) = { Ki (e i , s) , s ∈ S(e i )} , where Ki : R×[-1, 1] → R is a continuous (single valued) function.
The proof of Proposition 6 is made iteratively from i = m toward 1. At each step, we show that V i is positive definite and we look for a positive real number i , such that for all E i in R n-i+1 max s∈S(ei)

∂V i ∂E i (E i )(S m-i+1 E i + Ki (e i , s)) ≤ -c i V i (E i ) d V -1 d V , (C.1)
where c i is a positive real number. The Proposition will be proved once we have shown that the former inequality holds for i = 1.

Step i = m : At this step, E m = e m . Note that we have max

s∈S(em) ∂V m ∂E m (E m ) Km (e m , s) = -|E m | d V -1 , = -c m V m (E m ) d V -1 d V , with c m = d d V -1 d V V
. Hence, equation (C.1) holds for i = m.

Step i = j : Assume V j+1 is positive definite and assume there exists ( j+1 , . . . , m ) such that (C.1) holds for j = i-1. Note that the function

x → x d V -r j r j -e i+1 d V -r j r j+1
is strictly increasing, is zero iff x = e j+1 r j r j+1 , and therefore has the same sign as x -e j+1 r j r j+1 . Thus, for any e j+1 fixed in R, the function ν → V j (ν, e j+1 ) is non negative and is zero only for v = e j+1 r j r j+1 . Thus, Vj is positive and we have

V j (E j ) = 0 ⇔ V j+1 (E j+1 ) = 0 V j ( j e j , e j+1 ) = 0 ⇔ E j+1 = 0 j e j = e j+1 r j
r j+1 = 0 so that V j is positive definite.

On another hand, let Ṽj (ν, E j+1 ) = V j+1 (E j+1 ) + Vj (ν, e j+1 ) and let T 1 be the function defined

T 1 (ν, E j+1 ) = max s∈S(ν)
T1 (ν, E j+1 , s)

with T1 continuous and defined by

T1 (ν, E j+1 , s) = ∂ Ṽj ∂E j+1 (E j+1 )(S m-i-1 E i+1 + Kj+1 ( ν r j+1 r j , s)) + c j+1 2 Ṽj (ν, E j+1 ) d V -1 d V
.

Let also T 2 be the continuous real-valued function defined by

T 2 (v, E j+1 ) = - ∂ Ṽj ∂ν (ν, E i+1
)(e j+1 -ν r j+1 r j ) .

Note that T 1 and T 2 are homogeneous with weight r j for ν and r i for e i and degree d V -1. Besides, they verify the following two properties :

-for all E j+1 in R m-j , ν in R , E j+1 , s) = ∂V j+1 ∂E j+1 (E j+1 )(S n-i E j+1 + Kj+1 (e j+1 , s))

+ c j+1 2 V j+1 (E j+1 ) d V -1 d V ≤ - c j+1 2 V j+1 (E j+1 ) d V -1 d V
,

where we have employed (C.1) for i = j -1.

Using Lemmas 4 in Appendix D, there exists j such that T 1 (ν, E j+1 )j T 2 (ν, E j+1 ) ≤ 0 , ∀ (ν, E j+1 ) .

Finally, note that max s∈S(ei)

∂V j ∂E j (E j )(S m-j+1 E j + Kj (e j , s)) = T 1 ( j e j )j T 2 ( j e j , E j+1 ) -

c j+1 2 V j (E j ) d V -1 d V
Hence, (C.1) holds for i = j.

D Technical lemmas

Lemma 2 Let η be a continuous functions defined on R n+1 and f a continuous function defined on R n . Let C be a compact subset of R n . Assume that, for all x in C and s in S(f (x)), η(x, s) < 0 . Then, there exists α > 0 such that for all x in C and s in S(f (x)) η(x, s) < -α . PROOF. Assume that for all k > 0, there exists x k in C and s k in S(f (x k )) ⊂ [-1, 1] such that 0 > η(x k , s k ) ≥ -1 k .

Then, η(x k , s k ) tends to 0 when k tends to infinity. Besides, there exists a subsequence (k m ) such that x km tends to x * in C and s km tends to s * in [-1, 1]. Since η is continuous, it follows that η(x * , s * ) = 0 and we will have a contradiction if s * ∈ S(f (x * )). If f (x * ) is not zero, then by continuity of f , s * is equal to the sign of f (x * ), and otherwise, s * ∈ [-1, 1] = S(f (x * )). Thus, s * ∈ S(f (x * )) in all cases.

Lemma 3 Let η be a function defined on R n homogeneous with degree d and weight vector r = (r 1 , ..., r n ), and V a positive definite proper function defined on R n homogeneous of degree d V with same weight vector r. Define C = V -1 ({1}). If there exists α such that for all x in C η(x) < α , then for all x in R n \ {0}, η(x) < αV (x)

d d V .
PROOF. Let x in R n \ {0}. We have x = x i V (x) where η is a continuous function defined on R n+1 and f a continuous function defined on R n . Consider a continuous function γ homogeneous with same degree and weight vector such that, for all x in R n \ {0} and s in S(f (x))

γ(x) ≥ 0 , γ(x) = 0 ⇒ η(x, s) < 0 .

Then, there exists k 0 > 0 such that, for all x in R n \ {0}, η(x) -k 0 γ(x) < 0 . Therefore, there exists k 0 such that η(x, s) -k 0 γ(x) < 0 for all x in C and all s in S(f (x)). Thus, with Lemma 2 there exists α > 0 such that η(x, s) -k 0 γ(x) ≤ -α so that η(x) -k 0 γ(x) < 0 for any x in C. The result follows applying Lemma 3.

Lemma 5 For a positive bounded continuous function t → c(t) and an absolutely continuous function t → ν(t) satisfying for almost all t such that ν(t) ≥ c(t) then ν(t) ≤ -ν(t) d with d in ]0, 1[. Then, for all t in [0, T [ ν(t) ≤ max 0, max{ν(0) -c(0), 0} 1-d -t PROOF. This is a direct consequence of the fact that we have for almost all t such that max{ν(t) -c, 0}) is C 1 ˙ max{ν(t) -c, 0} ≤ -max{ν(t) -c, 0} d

  ): ż11 = ŵ1 -L 1 k 11 (ẑ 11 -z 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ż21 = ẑ22 + Φ 1 (u, ẑ11 ) + ŵ1 -L 2 k 21 (ẑ 21 -z 1 ) ż22 = ŵ2 -L 2 2 k 22 (ẑ 21 -z 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

  Simply take ρ(s) = max |u|≤u,|z j |≤z,|e|≤s |Φj(u, zj + e) -Φj(u, zj)|.

Table 2 :

 2 Hölder restrictions on Φ for a homogeneous observer with

2 ż22

 2 ẑ 11 -y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ż21 = ẑ22 + Φ 1 (u, ẑ11 ) + ŵ1 -L 2 k 21 ẑ21 -y 1 ∈ ŵ2 -L 2 2 k 22 S(ẑ 21 -y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . żm1 = ẑm2 + Φ 1 (u, ẑ11 )

Fig. 1 .

 1 Fig.1. Trajectory of System[START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], with the noised measurement y.

3 ż13 ∈ -L 3 1 k 4 ż22 = ẑ23 -L 2 2 k 22 ẑ21 -y 1 2 4 ż24 ∈ -L 4 2 k

 314242 13 S(ẑ 11 -y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ż21 = ẑ22 -L 2 k 21 ẑ21 -y 3 ż23 = ẑ24 + sat(g 3 (ẑ 11 , ẑ12 , ẑ13 ))u -L 3 2 k 23 ẑ21 -y 1 24 S(ẑ 21 -y) with the coefficients k 1j chosen, according to [16], as k 11 = 3, k 12 = 2.6, k 13 = 1.1, and k 2j as above, and with the gains L 1 = 2.5 and L 2 = 3, gives the following final errors : e z,11 = 0.05, e z,12 = 0.4, e z,13 = 2.5, e z,24 = 12

T 2 (

 2 (ν, E j+1 ) ≥ 0 have the same sign) -for all (ν, E j+1 ) in R m-j+1 \ {0}, and s in S(ν), we have the implicationT 2 (ν, E j+1 ) = 0 ⇒ T1 (ν, E j+1 , s) < 0since T 2 is zero only when ν

Lemma 4

 4 Let η be a homogeneous function of degree d and weight vector r defined on R n by η(x) = max s∈S(f (x))η(x, s)

  consider the compact set C = V -1 ({1}). Assume that for all k > 0, there existsx k in C and s k in S(f (x k )) such that η(x k , s k ) ≥ k γ(x k ) ≥ 0η is continuous, and thus bounded on the compact set C × [-1, 1]. Therefore, γ(x k ) tends to 0 when k tends to infinity. Besides, there exists a subsequence (k m ) such that x km tends to x * in C and s km tends to s * in [-1, 1]. It follows that γ(x * ) = 0 since γ is continuous. But with the same argument as in the proof of Lemma 2, we have s * ∈ S(f (x * )). It yields that η(x * , s * ) < 0 by assumption and we have a contradiction.

Table 5

 5

		High gain (4)	High gain cascade (11)	Homogeneous (12)	Homogeneous	cascade
					(22)
	Assumption	Hölder with order greater	Continuous	Hölder with order greater	Continuous
	on gi	than in Table 1		than in (14) or Table 2 for
				d0 = -1
	Convergence Arbitrary small error	Arbitrary small error	Asymptotic convergence	Asymptotic convergence
	Advantages	Easy choice of gains	No constraint on gi	Not necessarily large	No constraint on gi, con-
				gains because conver-	vergence, apparently bet-
				gence	ter in terms of noise
	Drawbacks	Large gains necessary to	Same as for high gain,	Implementation of the	Large dimension and a
		obtain small error ⇒ nu-	but also gains difficult to	sign function if d0 = -1	lot of gains to choose
		merical problems (peak-	choose and large dimen-	(chatter etc)
		ing) and sensitivity to	sion
		noise

Actually Φi can depend also on zi+1 to zm as long as (3) holds. It can also depend on time requiring some uniform property (seeSection 7).

Here the max is with respect to s in ē1 + v) 0 appearing in the mth component K(ē1 + v)m of K(ē1 + v).

A diffeomorphism from the unit sphere to the set is x → xρ(x) where ρ is the unique positive solution (hint: x3 ≤ 1) of ρ

x 6 3 + 3ρ 2 (1 -x 2 3 ) -1 = 0