text:
Mika Gabel
email: mikagabe@post.tau.ac.il

Tommy Dreyfus
email: tommyd@post.tau.ac.il

The flow of a proof -Establishing a basis of agreement

Keywords: Proof teaching, flow of proof, Perelman's New Rhetoric, mathematical argumentation

The notion of flow of a proof encapsulates mathematical, didactical and contextual aspects of proof presentation, related to the lecturer's choices regarding the presentation. We explore the relationship between mathematics teaching and rhetoric, suggesting Perelman's New Rhetoric (PNR) as theoretical framework to assess different rhetorical aspects of the flow of a proof. In this paper we relate particularly to the establishment of a shared basis of agreement between the lecturer and the students, and to potential fallacies in this basis. We present examples of analysis of the basis of agreement from a lesson in Number Theory, at the beginning undergraduate level.

Theoretical background

Mathematics and Rhetoric -"Can two walk together, except they be agreed?"

Mathematics "possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature…" (Russell, 1917, p. 60) and is "independent of us and our thoughts" (ibid, p. 69). This perception of mathematics seems to stand in drastic contrast to rhetoric, the ancient art of persuasion, which over the centuries became mostly related to the study of the ostentatious and artificial aspects of discourse. Yet, over the last few decades, scholars have begun to discuss the mathematics-rhetoric separation and its consequences.

A pioneering effort of associating mathematics and rhetoric was made by [START_REF] Davis | Rhetoric and mathematics[END_REF] who argued "that mathematics is not really the antithesis of rhetoric, but rather that rhetoric may sometimes be mathematical, and that mathematics may sometimes be rhetorical" (p. 54). Davis and Hersh challenged the opinion that mathematics establishes truth "by a unique mode of argumentation, which consists of passing from hypothesis to conclusion by…small logical steps…", and claimed that "mathematical proof has its rhetorical moments and its rhetorical elements" (ibid, pp. 59-60). They illustrated this by phrases that a college mathematics lecturer may use while presenting a proof (in addition to the expected logical transformations), such as: "It is easy to show …", "… simple computation, which I leave to the student, will verify that…"; they identified these phrases as rhetorical means in the service of proof. They acknowledged that the use of such phrases may be related to context, but rejected the myth that behind each theorem stands a flawless, logical proof. For them 'proof' is an amalgam of formality, of convincing arguments and of appeals to imagination and intuition.

Another example is the 'rhetoric of the sciences' movement, which studies the stylistic forms used by scientists in scientific texts (mathematics included), to persuade others that their claims are valid. So, as in the other sciences, the rhetoric of mathematics plays an essential role in maintaining its epistemological claims [START_REF] Ernest | Forms of knowledge in mathematics and mathematics education: Philosophical and rhetorical perspectives[END_REF]. Ernest relates to another aspect of rhetoric in mathematics, namely the importance of persuasion for mathematics instruction. [START_REF] Reyes | Stranger relations: The case for rebuilding commonplaces between rhetoric and mathematics[END_REF] asserts that it should be in the interest of rhetorical scholars to explore mathematics discourse, as it is the basis of techno-science. He analyses conceptual mathematical metaphors as an example for a mode of analysis of mathematics whose roots lie in rhetorical studies. Elsewhere, Reyes studies the rhetorical process during the invention of mathematics, and explores the introduction of infinitesimals by Newton and Leibniz as an example of the role of mathematical rhetoric in mathematical invention, in addition to its role in communicating the mathematics.

In conclusion, inquiry into relations between rhetoric and mathematics is growing in extent and richness. An increasing number of scholars explore the possibilities offered by the use of rhetorical concepts and ideas to gain better understanding of mathematics and mathematical education. Instead of viewing mathematics as a 'perfected, austere' product, they re-connect it to its 'human features', that in addition to formal logic utilizes persuasive argumentation and exploits rhetorical means.

Argumentation theory and '

The New Rhetoric' [START_REF] Aberdein | Commentary on Andrzej Kisielewicz's: A new approach to argumentation and reasoning based on mathematical practice[END_REF] rejects the common thesis that mathematical reasoning is fundamentally different from everyday reasoning and that formal logic adequately models the practice of mathematical reasoning. Research in mathematical education uses argumentation theory to address aspects of mathematical argumentation other than formal logic, and for that purpose frequently uses Toulmin's model that permits schematic analysis of formal proofs as well as of arguments classified as deductively invalid. Toulmin's model has been shown to be an efficient framework to discuss local arguments as well as global argumentation structures (e.g. [START_REF] Knipping | Revealing structures of argumentations in classroom proving processes[END_REF] and [START_REF] Inglis | Modelling mathematical argumentation: The importance of qualification[END_REF] claim that implementing Toulmin's full model (including rebuttals and qualifiers) should be used for this purpose. However, Toulmin's model has been criticized for not relating to the effect of the arguments on the audience, and for denigrating rhetoric in argumentation (Olbrechts-Tyteca, cited in [START_REF] Frank | Argumentation studies in the wake of the new rhetoric[END_REF].

In 1958, Perelman and Olbrechts-Tyteca published 'The New Rhetoric' (PNR, translated in 1969), an argumentation theory based on the notion that argumentation aimed at justification of choices, decisions, and actions, is a rational activity complementing formal argument. PNR studies techniques used by an arguer to increase audience adherence to the arguer's theses and conditions that allow argumentation to begin and develop. PNR asserts that reducing an argument to its formal aspects undermines the rhetoric features that support its meaning; it recognizes the distance between dialectic and rhetoric but creates an alignment between them. This complex view at times produced an inherent ambiguity in definitions of some concepts. However, PNR adds meaningful layers of analysis beyond the analysis of argument structure and type achieved by using Toulmin's model. PNR describes the 'threads that make the cloth of the argument': the starting points that establish a shared basis of agreement, the scope and organization of arguments, ways of creating presence to arguments, and different argumentation techniques. The audience plays a pivotal role in PNR since each 'thread', or aspect, is tied to what the arguer believes will deeply persuade the audience. This means that argumentation techniques should be adjusted to the audience's frame of reference, its previous knowledge, experiences, expectations, opinions and norms. So arguers construct arguments that they consider persuading for a particular audiences or convincing by a 'universal audience' (an arguer construct consisting of all reasonable humans) [START_REF] Van Eemeren | The new rhetoric[END_REF].

In our study, we wish to analyze rhetorical aspects of proof presentation, in a scenario of a lecturer presenting a mathematical proof to a class of students. We use PNR as a theoretical framework as it incorporates aspects of rhetoric, argumentation and lecturer-classroom relations. Elsewhere [START_REF] Gabel | Affecting the flow of a proof by creating presencea case study in number theory[END_REF], we demonstrate an analysis of other PNR aspects: scope and organization of the argumentation, and presence of proof elements. In this paper we address a different aspect: establishing a shared basis of agreement with the audience.

PNR's basis of agreement and its adaptation to proof teaching

According to PNR, argumentation can be successful if it advances from premises already accepted by the audience, i.e. the arguer established a shared basis of agreement with the audience. These premises are classified as follows: (1) Premises relating to the real: premises where the arguer claims recognition or acknowledgement of the universal audience. Those include: Facts, truths and presumptions. ( 2) Premises relating to the preferable: premises that have to do with the preferences of a particular audience. Those include: Values, value hierarchies and loci of preferable.

Facts and truths are statements already agreed to by the universal audience; they are considered to require no further justification. Truths stand for connections between facts. Presumptions are opinions or statements about what is the usual course of events which need not be proved, although adherence to them can be reinforced, and it is expected that at some point they will be confirmed. Values relate to the preference of one particular audience as opposed to another. They function as guidelines in making choices of the arguer (even though not all would accept them as good reasons). Values are normally arranged in value hierarchies, which are very important since different audiences may possess the same set of values arranged in different hierarchies. Values and value hierarchies generally remain implicit, but the arguer cannot simply ignore them. Loci of the preferable (aka commonplaces, Topoi) are premises used to justify values or hierarchies and express the preferences of a particular audience (e.g. quantity, quality, essence) [START_REF] Van Eemeren | The new rhetoric[END_REF].

We have adapted PNR's classification of premises to the context of our study (analyzing proof presentation in class) as described in Table 1. We do not include in the table the loci of the preferable since they are highly abstract mental constructs which did not need adaptation.

Values

The preference or adaptability to a particular audience of a certain proving method or technique as opposed to another.

Hierarchies of values

The hierarchies of values will affect audience preferences for choosing notation, proving method or mathematical technique. According to Perelman and Olbrechts-Tyteca, lack of agreement concerning the basis of agreement may occur at one or more of the following three levels: a) Status of premises: e.g. if the arguer advances something as a fact but the audience wants to see it proven or if the arguer assumes a value hierarchy not accepted by the audience; b) Choice of premises: e.g. if the arguer uses facts that the audience does not consider relevant to the argument or would have preferred not to see mentioned; c) Verbal presentation of premises: e.g. if the arguer is presenting certain facts (acknowledged as relevant by the audience) in words which have connotations unacceptable to the audience.

The ability of creating a shared basis of agreement with the audience is crucial to the success of argumentation. Arguers should therefore carefully consider the status they ascribe to premises, the selection of premises, and the wording of explicit premises [START_REF] Van Eemeren | The new rhetoric[END_REF]). Examine, for example, two possible values related to proof teaching: (1) Certainty: every argument in the proof should be proved formally or at least justified;

(2) Pedagogy: parts pf the proof should be left for the students. A lecturer may choose to leave parts of the proof as homework because her/his value hierarchy places (2) over (1). However, if the students have an opposite value hierarchy, this implies that the lecturer had a fallacy in the shared basis of agreement at the level of the status of his value hierarchy, which might consequently weaken students' persuasion.

The studydescription and methods

Our research concerns the notion of 'The flow of a proof' [START_REF] Gabel | Affecting the flow of a proof by creating presencea case study in number theory[END_REF] which encapsulates various aspects of the proof presentation. The flow is an outcome of the choices made by the lecturer regarding presentation of: (i) the logical structure of the proof (arranging the proof of the theorem into claims, which are proved in a specific order); (ii) informal features and considerations of the proof and proving process (e.g. examples, intuitions), and (iii) mathematical and instructional contextual factors. One aim of the research was to analyze global and local aspects the flow of the proof, in particular to examine rhetorical aspects of the proof presentation.

The research took place in a Number Theory course, given by the same lecturer to prospective mathematics teachers in two consecutive years. Each year, three lessons including the same suitable proofs (length, richness) were observed and audio-recorded. The three proofs were unrelated to each other. After each lesson students answered a questionnaire relating to cognitive and affective aspects; also, a reflective interview with the lecturer and individual interviews with students were conducted. The post-lesson interviews conducted with the lecturer in Year 1 were analyzed and interpreted, and a list of the lecturer's general considerations regarding proof teaching was produced. In this paper we relate to the second lesson in each year, in which the following theorem related to linear Diophantine equations ax+by=d , , x y Z  was formulated and proved:

Theorem: The greatest common divisor (gcd) of two integers , ab, at least one of which is not 0, equals the smallest natural number of the form ma nb  , where , mn are integers:

gcd( , ) min{ 0 : , } a b ma nb m n Z     .
The full proof of this theorem requires the use of previously proven results. In the next section we present a partial analysis of the shared basis of agreement, demonstrating the different types of lecturer premises (in the PNR sense) reflected in the proof presentation, consider potential fallacies in these premises and demonstrate the lecturer's attempt to fix these fallacies.

Examples of analysis of the basis of agreement

All post lesson interviews conducted with the lecturer in Year 1 were analyzed and interpreted as two sets of lecturer considerations [START_REF] Gabel | Affecting the flow of a proof by creating presencea case study in number theory[END_REF]. One of the sets contains general considerations for proof teaching. In the current paper we relate to three of these general considerations: (a) A proof should be mathematically complete and exact; (b) Some of the proof elements should be left for students to prove by themselves; and (c) Proof structure should be clear. One aspect of the clarity of proof structure was exhibited when the lecturer referred to the myth about Ariadne's thread: "I use… Ariadne's thread many times since mathematical proofs are built in such a way that you need to find the tip of the thread and just follow it…" We relate to these lecturer considerations as values that affect the lecturer's choice and status of premises.

Our examples stem from the last part of the proof as presented by the lecturer, and we will address lecturer premises as reflected in his arguments. In Year 1, just after proving that d is a divisor of a , leaving the (almost identical) proof that d is a divisor of b to the students, the lecturer said:

Lecturer:

The same way we showed that d is a divisor of a it follows that d is a divisor of b , so d is a common divisor of , ab. Now, it can't be smaller than the gcd, yes?

Because once I write the equation ax by d  then…like we said in the beginning of the lesson, we said that this d must be divisible by gcd ( , ) ab … so it can't be smaller then gcd ( , ) ab and that means it is equal to gcd( , ) ab .

Figure 1: Toulmin's scheme representation of 1st explanation

The argumentation in this excerpt is represented by the Toulmin's scheme in Figure 1. We suggest a possible interpretation of the lecturer's explicit and implicit premises reflected in this explanation.

For the lecturer this argumentation requires very little justification (if any) and he presents it as a chain of facts (D1, C1, D2 and possibly C2) that does not need to be discussed, and whose connection results in the conclusion (C4) in a self-evident way. ; he believes that this presumption does not need to be made explicit and that he and the students share this presumption. As for the values reflected in this explanation and their hierarchy, since the lecturer chose to leave some of the proof (that d|b) to the students, in this case the pedagogical value of leaving some of the proof elements for students was placed above the value regrading proof completeness. In addition, we recognize another implicit value: for the lecturer the 'tip of Ariadne's thread' here is to realize that d is a common divisor of a,b from which the rest of the proof just unfolds.

However, the students had difficulties following this first explanation and asked the lecturer to repeat it. A possible reason for this difficulty is that the premises that the lecturer considered as facts were not considered as facts by the students and required further justification. For example, the students probably needed an explicit justification for the argument "if d is a common divisor of a,b ". Moreover, the lecturer's implicit presumption regarding the natural proving technique (the two inequalities) is not necessarily clear and natural to the students. In PNR language, there was a lack of agreement about the status and choice of the lecturer's premises, which caused a fallacy in establishing a shared basis of agreement. So, following the students' request, the lecturer instantly explained again the argument in Year 1 lesson as follows:

Lecturer:

We said that d , as a minimal element of this set [{ 0 : , }

ma nb m n Z    ], is
of the form a  integer+ b integer. Now the first thing I have shown today is that in such a situation, actually this is a result of theorem 1 that we have used before,… it follows that d must be divisible by gcd( , ) ab , yes? Once I can write a number as a linear combination of two numbers , ab, with integer multipliers , mn, this d must be divisible by gcd ( , ) ab . On one hand it must be divisible by gcd( , ) ab ;

on the other hand…it is a divisor of , ab. It can't be smaller then gcd ( , ) ab so it can only be equal to it. Because gcd( , ) ab is the greatest common divisor, yes?

And that finishes the proof… d is a common divisor of , ab that also has to be divisible by gcd( , ) ab so we conclude that The argumentation in this excerpt is represented by Toulmin's scheme in Figure 2. In the second explanation the lecturer added some justification (W1, B1, W3) to the conclusions C1, C2 and C3; we interpret they were not presented as facts but rather as truths, i.e. the lecturer changed the status of the premises to establish a stronger basis of agreement with the students. However, his presumption still remained implicita point which we will revisit shortly.

Before the lesson in Year 2 the lecturer was informed by the researcher about some student difficulties that were found in the post lesson students' questionnaires of Year 1; in particular, the questionnaires reflected that the last part of the proof, where combining the inequalities gcd( , ) Lecturer:

No, we should prove two inequalities, right? Or refute two inequalities, right? I remind you, we already used it: when we wanted to show that two numbers are equal then we need to show that it is impossible that a is smaller than b … it is impossible that a is bigger than b , or in other words … to show that a is not smaller than b is actually showing that ab  , and instead of showing that a is not bigger than b we'll show that ab  . If I want to show that ab  , I need to show that ab  , i.e. not smaller than b , and that ab  , meaning that a is not bigger than b . Once I will show these two inequalities I am done, I've shown that ab  . Here, the lecturer consciously makes his presumption explicit to the students, justifies the choice of this presumption and makes it relevant. By explicating his presumption the lecturer also caused a change of value hierarchies: he enhanced the clarity of the proof structure, making it more explicit before going into the details of the proof. Indeed, the lecturer also explicitly declared:

Lecturer:

It remains to prove the other inequality: gcd( , ) a b d  . In fact, I will show you that this… minimum of the set, d, is a divisor of a,b… and this will end the story… So before formally proving that d is a divisor of a,b, the lecturer spread out his proving plan, identifying "the tip of Ariadne's thread" and explained exactly why this "will end the story". We conclude this short example by stressing that while Toulmin's model enables the presentation and analysis of the argumentation structure, PNR complements it by relating to other argumentation qualities, such as the adaptability to the intended audience. The fallacies that have been mentioned in the example were related to the status and choice of premises.

Concluding remarks

Weber and Mejia-Ramos (2014) demonstrated that mathematics students and mathematicians have different perceptions regrading students' responsibilities when reading a mathematical proof: the students believe that reading a good proof is quite a passive process, one in which they are not expected to construct justifications, diagrams or sub-proofs by themselves, and they may simply follow each and every step. Mathematicians believe the opposite. This tension between the students' and their teachers' beliefs supports our interpretation regarding the different value hierarchies that the lecturer and the students have. But beyond the identification of the difference, we argue that PNR has the potential to explain the consequences of that difference on the effectiveness of argumentation; in other words, PNR provides a suitable framework to identify ways to increase argumentation effectiveness, for instance by referring to the shared basis of agreement.

Moreover, PNR relates to other rhetorical and dialectical aspects of argumentation. Some of these aspects (scope and organization and presence) have been studied in [START_REF] Gabel | Affecting the flow of a proof by creating presencea case study in number theory[END_REF]; others, namely argumentation techniques and the manner by which PNR complements the use of Toulmin's scheme, need further study. One advantage of PNR is that because of its theoretical scope, all these aspects can be studied within a single unifying theoretical framework.

Although Perelman perceived PNR as a complement of formal logic and focused on disputes in which values play a part [START_REF] Van Eemeren | The new rhetoric[END_REF], we argue that PNR can be adapted to be a productive theoretical framework in the context of proof teaching, particularly the flow of a proof: firstly, Perelman was much inspired by formal logic (mainly the work of Frege), and secondly, the

  C2: d cannot be smaller than gcd(a,b) D2: ax+by=d x,y integers D1: d|a and d|b W: We showed before that d is divisible by gcd(a,b) C4: d = gcd(a,b) C1: d is a common divisor of a

  Figure 2: Toulmin's scheme representation of 2nd explanation

  to the students. For lack of space we will concentrate on demonstrating the change in the lecturer's presumptions and his value hierarchy between Years 1 and 2. The lecturer took the reported students' difficulty into account and during the lesson in Year 2, just before the last part of the proof he explained:Lecturer:Here we are doing something similar to what we already had in the past, when we wanted to prove that two numbers are equal…Student:We assume that they are unequal… C3: d cannot be bigger than gcd(a,b) W1: Theorem 1 (previous lessons) W3: gcd(a,b) is the greatest of all common divisors of a,b C4: d = gcd(a,b) C1: gcd(a,b)|d C2: d cannot be smaller than gcd(a,b) B1: d=ma+nb, m,n integers D1: d is a linear combination of a,b D2: d|a and d|b

Table 1 : Adapting PNR types of premises to a proof teaching context

 1

Proceedings of CERME10

context of argumentation in the mathematics classroom resembles PNR's context of persuading an audience. Thus PNR is a comprehensive argumentation theory that can broaden and enrich researchers' perspectives regarding different aspects of mathematics classroom argumentation.


Radhia Ettouil
email: radhiaettouil@gmail.com

Karim Chabir

Dominique Sauter
email: dominique.sauter@univ-lorraine.fr

Mohamed Naceur Abdelkrim

Mohamed Naceur

Abdelkrim Resilient

Resilient synergetic control for fault tolerant control system

Keywords: Synergetic control, adaptive observer, resilient control, fault estimation, pitch system

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Resilient Synergetic Control for fault tolerant control system

Radhia. Ettouil * Karim. Chabir * Dominique. Sauter * * Mohamed Naceur. Abdelkrim *

INTRODUCTION

Wind turbines constitute an interesting part of the renewable energy production. Reliability, safety and fault tolerant wind turbine control are highly required. To increase system reliability, the design of fault tolerant control (FTC) becomes necessary. These systems are control systems that assure tolerating potential faults in the system in order to improve its reliability and safety while maintaining a desirable performance. In other words, they are closed-loop control systems which compensate for fault and provide a desirable performance. Recently, FTC techniques have attracted the attention of several researchers- [START_REF] Gao | A survey of fault diagnosis and fault-tolerant techniquespart i: Fault diagnosis with model-based and signal-based approaches[END_REF] and [START_REF] Cecati | A survey of fault diagnosis and fault-tolerant techniquespart ii: fault diagnosis with knowledge-based and hybrid/active approaches[END_REF].

One of the categories of faults is that leading to a large variation in system parameters. As noted in [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF]), the occurrence of these faults causes a drastic change in system dynamics and can lead to an abnormal operation of the system. Significant research and numerous strategies have been developed in order to design a specific FTC for that type of faults. A fault reconfiguration approach is proposed in [START_REF] Steffen | Control reconfiguration of dynamical systems: linear approaches and structural tests[END_REF]), based on the change of the control structure in response to the fault. Furthermore, data-driven FTC techniques focus on the problem of the output feedback FTC for a system affected by a fault that changes system dynamics [START_REF] Kulcsár | Modelfree fault tolerant control approach for linear parameter varying system[END_REF], [START_REF] Dong | Data driven fault tolerant control: a subspace approach[END_REF], [START_REF] Yin | Real-time implementation of fault-tolerant control systems with performance optimization[END_REF], [START_REF] Ding | Datadriven realizations of kernel and image representations and their application to fault detection and control system design[END_REF] and [START_REF] Wang | Data-driven outputfeedback fault-tolerant control for unknown dynamic systems with faults changing system dynamics[END_REF]). A data-driven output feedback FTC problem considering optimal performance is studied in [15]. Moreover, a projection-based FTC design for pitch systems is presented in [START_REF] Ding | Data-driven design of fault-tolerant control systems[END_REF]). Using a new form of extended state observer, a reconfigurable control based on LPV system modeling is proposed in [START_REF] Jain | A novel approach to real-time fault accommodation in nrel's 5-mw wind turbine systems[END_REF]).

The sliding mode approach is also used for the FTC issue to compensate for the mentioned faults. The sliding mode observer [START_REF] Edwards | Sliding mode observers for fault detection and isolation[END_REF]and [START_REF] Patel | A sliding mode observer for tyre friction estimation during braking[END_REF]) is used to achieve fault estimation that has been applied to substitute the fault detection and isolation (FDI) approach. Also, a Takagi-Sugeno Sliding Mode Observer (T-S SMO) is proposed in [START_REF] Jiang | Fault estimation in nonlinear uncertain systems using robust/sliding-mode observers[END_REF]) in order to estimate system actuator parameter faults. A novel FTC allocation schema with output integral sliding mode is proposed in [START_REF] Hamayun | A fault tolerant control allocation scheme with output integral sliding modes[END_REF]), when only the system outputs are measured. Recently, an adaptive step by step sliding mode observer has been used in fault tolerant control based on state system estimation and fault indicator [START_REF] Lan | Fault-tolerant wind turbine pitch control using adaptive sliding mode estimation[END_REF]).

In this work, we present a novel technique for compensating for the actuator fault. The new strategy consists in designing a resilient controller using the synergetic theory. The synergetic control allows to drive the system trajectories to operate on the designed manifolds. This approach has been developed by Kolesnikov and Coworkers (Kolesnikov et al. [2000]). The connection and comparison between the synergetic performance and those of sliding mode control have been detailed in [START_REF] Nusawardhana | Nonlinear synergetic optimal controllers[END_REF]. Recently, the synergetic control has been applied in many fields [START_REF] Ju | Nonlinear synergetic governor controllers for steam turbine generators to enhance power system stability[END_REF], [START_REF] Shokouhandeh | An enhanced and auto-tuned power system stabilizer based on optimized interval type-2 fuzzy pid scheme[END_REF] and [START_REF] Bouchama | Reaching phase free adaptive fuzzy synergetic power system stabilizer[END_REF]). The main advantages of this approach are global stability, order reduction and insensitivity to parameter variation. Here, this theory will be used to design a resilient control able to compensate for the dynamic faults and to recover the initial performance.

The remainder of the paper is organized as follows. Section 2 presents the problem formulation. Section 3 introduces briefly the synergetic control and the adaptive observer and presents the proposed resilient synergetic control. A numerical example is simulated in section 4. Finally, some conclusions are drawn in section 5.

PROBLEM FORMULATION

This work focuses on the low pressure fault that affects pitch system of wind turbines. The faulty system is modeled by: ẋ

= Ax + Bu + ∆Af x y = Cx (1)
Where,

A = 0 1 -ω 2 n -2ξ n ω n , B = 0 ω 2 n , C = [ 0 1 ] ∆A = 0 0 ω 2 f -ω 2 n 2ξ f ω f -2ξ n ω n
where f denotes the indicator fault, f ∈ [ 0 1 ]. ω n and ξ n denote respectively the natural frequency and the damping factor, while ω f and ξ f denote their values at low pressure. This work aims to design a resilient synergetic control as an FTC technique to recover the desired performance when an actuator fault occurs. Fig. 1 shows the general structure of the FTC system: Our goal is to use the synergetic theory to design a resilient control in order to compensate for the effect of the fault and ensure stability for a closed loop system.

Synergetic Theory

The design of the synergetic controller is similar to that of the sliding mode controller. The main steps of the procedure can be itemized as follows:

• Design of manifold The manifold s(x) = 0 can be a linear or nonlinear function of the state variables.

It is selected according to the desired performance.

Consider the following dynamical system of a single pitch system in fault-free case: ẋ = Ax + Bu (2) The manifold has the form:

M = s(x) = 0 (3)
It is constructed so that det ∂M ∂x B = 0.

• Design of synergetic control law The synergetic controller is synthesized by solving the first order differential equation:

T Ṁ + M = 0 (4) with T = T T > 0 is a diagonal positive definite matrix, presenting the convergence parameter. Solving the above differential equation for u yields a controller that forces the variables state to lie on the manifold s(x). Note that:

dM dt = ∂M ∂x ∂x ∂t = ∂ ∂x s(x) T ẋ = s x (x) ẋ (5)
Then, solving the above differential equation gives:

T Ṁ + M = T s x (x) ẋ + M = T s x (x)(Ax + Bu syn ) + M = 0 (6) Thus, u syn = -(T s x (x)B) -1 (T s x (x)Ax + M ) = u eq + u s (7)
where (s x (x)B) is invertible. u eq = -(T s x B) -1 T s x Ax denotes the equivalent control law that drives the state trajectories to converge toward the manifold and u s = -(T s x B) -1 s(x) is the smooth component that maintains the system dynamics along the manifolds.

The system dynamics along the manifold M = 0 is given by two equations as follows:

M = 0 ẋ = Ax + Bu syn = (I -B(T s x (x)B) -1 T s x (x))Ax- B(T s x (x)B) -1 M = 0 (8)
These equations prove the advantage of the order reduction of the synergetic control. It provides a reduced-order, n -m-dimensional dynamical system.

Theorem 1. Consider linear system (2) and diagonal positive definite matrix T . Using the control law that satisfies (4), the state trajectories will converge to the manifold and the convergence speed depends on the designated parameterT .

Proof 1. Consider the Lyapunov function defined as

V (s(x)) = s T (x)s(x). Derivation of V (s(x)) gives: V (s(x)) = 2s T (x) ṡ(x) (9)
From (4) we have ṡ

(x) = -T -1 s(x) (10) Hence, V (s(x)) = -2s T (x)T -1 s(x) < 0 , f or T > 0 (11)
Thus, the stability condition is guaranteed for such synergetic control law and the rate of convergence of the system state to the manifold depends on T . It is clear from (7) that the synergetic control depends on the dynamic matrix A . Therefore, if the synergetic control is used when a fault that can change the system dynamics occurs, the system adapts easily to this change and compensates for the effect of the fault. This feature can be considered as an advantage used for designing resilient control when the fault caused by the system dynamics changes.

In what follows we will estimate the fault indicator using the adaptive observer.

Fault Estimation

For designing the resilient synergetic control, we need the fault indicator estimation. To this end, we will use an adaptive observer to get both the state and the fault estimations. Consider the pitch system in faulty case with a disturbance signals d described by

ẋ = Ax + Bu + ∆Af x + Ed y = Cx (12)
where f the fault indicator is a bounded unknown scalar. We assume that (A + ∆Af, C) is observable for all f . E is the distribution matrix of the disturbance. The adaptive observer is given by

ẋ = Ax + Bu + ∆A f x + K (y -ŷ) ˙f = xT ∆A T N (y -ŷ) ŷ = C x (13)
Let e x = x-x and e f = f -f be the error state estimations and the error fault estimation, respectively. It follows from ( 12) and ( 13) that the estimation error system is ėx

= (A -KC + ∆Af )e x + ∆Axe f + Ed ˙f = xT ∆A T N Ce x ( 14
)
Now it is necessary to verify the stability condition of the observer. The linear matrix inequality (LMI) approach is used here to show the existence of the mentioned observer and to guarantee the estimations performance.

Theorem 2. If there exist symmetric positive definite matrix P and a full matrix X which satisfy the following condition that holds for all slow variations of f .

P ∆A = C T N T ∆A ( 15
) P A -XC + P ∆Af + * P E E T P -µ < 0 ( 16
)
There exists an adaptive observer whose estimation errors converge to the origin via the calculation of the observer gains. Note that X = P K and thus the observer gain K is obtained as follows:

K = P -1 X (17)
To minimize the estimation error [ e x e f ] the scalar µ has to be minimized. Proof 2. Let the Lyapunov function be as follows:

V (t) = e T x P e x + e T f e f > 0 (18)
Its derivative with respect to time can be given by V = 2e

For all slowly varying f , we have ḟ = 0, thus

2e T f ėf = -2e T f xT ∆A T N Ce x (21)
From ( 15), we can deduce

P ∆Ax = xT A T f N C T ( 22
)
We define a matrix R and scalar µ such as:

R = P (A -KC + ∆Af ) + (A -KC + ∆Af ) T P + 1 µ P EE T P (23)
Suppose that the scalar µ and the matrix P justify the following inequality:

2e

It follows that V ≤ 0. This proves that the estimation error converges to a small set according to Lyapunov stability approach. Using the the Schur complement in the case of R < 0 can be given otherwise.

P (A -KC + ∆Af ) + * P E E T P -µ < 0 ( 26
)
where * denotes the symmetric component. Substituting P K by X and thus, the proof of the theorem is finished.

There are some difficulties in solving ( 15) and ( 16) simultaneously to extract P and X. So, we can transform (15) into the following optimization problem: Define the scalar η, such as

η = inf η : ηI P ∆A -C T N T ∆A ηI < 0 (27)
where * means a symmetric component. Now it remains to design the resilient synergetic controller.

Resilient Synergetic Control

The main purpose of resilient control is to compensate for the dynamics change induced by low pressure faults in the pitch systems, so that the stability and overall closed-loop system performance can be maintained. To this purpose, we exploit the feature of the synergetic control law that depends on the system dynamics matrix A and we integrate the fault value in the system dynamics matrix. Consider the fault-free dynamics model

ẋ = Ax + Bu nom + Ed y = Cx (28)
After obtaining the fault indicator estimation f , this estimation can be used to define a modified faulty model as follows

ẋ = Ax + Bu r + ∆A f x + Ed y = Cx (29)
where u r is the control in faulty case and u nom is the nominal control. Combining ( 28) and ( 29), the pitch system switches between two models according to fault occurrence. Thus, the pitch system can be modelled as follows: ẋ = A s x + Bu s + Ed y = Cx (30) with,

A s = A, u s = u nom if f < f th A s = A + ∆A f , u s = u r if f ≥ f th (31)
where f th is the threshold value of the fault indicator. Many methods of selecting the threshold have been developed [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]). u nom is the baseline controller used to achieve pitch angle control in nominal case and u r is the control used to compensate for the low pressure fault. Both controllers are designed using the synergetic control approach.

Our objective now is to construct a control strategy that not only drives the closed-loop system trajectory approaches to get the manifold s(x) = 0 but also adapts to the system dynamics change. Firstly we design the manifold using the procedure of [START_REF] Slotine | Applied nonlinear control[END_REF]) as follows:

s(x) = ( d dt + λ)(x 1 -x 1ref ) ( 32
)
With λ is selected by the constructer so that det ∂M ∂x B = 0 and x 1ref is a constant value representing the reference value of the first variable state x 1 . This manifold leads the trajectory of the variable state x 1 to converge to the reference value x 1ref . The resilient synergetic control is obtained by solving the first-order differential equation T ṡ(x) + s(x) = 0 that is controlled by appropriately choosing the convergence parameter T . To proceed, note that ṡ

(x) = λ ẋ1 + ẋ2 = S ẋ (33)
where S = [ λ 1 ], This variable is designed so that (SB) is invertible. The resilient control is derived from the differential equation using the faulty model ( 29) when f ≥ f th as follows:

T ṡ(x) + s(x) = T S ẋ + s(x) = T S((A + ∆A f )x + Bu r + Ed) +s(x) = 0 (34)
Thus, we obtain the synergetic control law

u r = -(T SB) -1 T S(A + ∆A f )x + T SEd + s(x) (35)
Hence, the obtained synergetic control comprises two terms given by

u r = -(T SB) -1 (T SAx + T SEd + s(x)) -(T SB) -1 ∆A f x = u nom + u resilient (36)
with,

u resilient = -(T SB) -1 ∆A f x (37)
It can be seen that the resilient control depends on the additive term that represents the fault which allows to compensate for the effect of the considered fault. In fact, the designed resilient controller drives system dynamics A s that includes dynamics variation ∆A to converge towards the desired performances.

The stability of the obtained controller as mentioned in the previous subsection has the simplest form T > 0.

Theorem 3. The obtained controller (36) can compensate for the fault f and recover the nominal pitch dynamics. Proof 3. The dynamics of the pitch system along the manifold in presence of fault is obtained by substituting (36) in ( 29) yielding

ẋ = Ax -B((T SB) -1 (T SAx + T S∆A f x +T SEd + s(x))) + ∆A f x + Ed = -(T S) -1 s(x) (38)
It is clear that the pitch system dynamics along the manifold is independent of the fault, which proves that the resilient controller can compensate for the fault.

NUMERICAL EXAMPLE

In this section, the pitch system is used to prove the effectiveness of the proposed resilient synergetic control.

In the nominal case, we have ω n = 11.11 rad/s and ξ n = 0.6 rad/s and under actuator fault, we have ω f = 3.42 rad/s and ξ f = 0.9 rad/s. We suppose that there exists a measurement noise modeled by a zero mean white Gaussian noise. In this work we aim to drive the pitch angle β to converge to the reference pitch angle β ref = 6 . The synergetic control is characterized by T = 1 and λ = 6. The pitch angle β is shown in fig. 2. Then, we aim to study the effect of the variation of the synergetic parameter T on the variable state evolutions (Fig. 3). The effect of variation of the design parameter T on the pitch system dynamics is illustrated in Fig. 3. We can see from this figure that the increase of T gives a lower rate of state variable convergence toward the desired value. So, the convergence of the system trajectories toward the manifold is inversely proportional to the value of T . This is explained by the fact that the increase of T is correlated with the decrease of the synergetic control law gains. Parameters T and λ can affect stability and performance of the pitch system, thus they must be well chosen.

Resilient synergetic control

In the time period t ∈ [200; 300] s , the pitch system is affected by a drop in pressure. The fault estimation is carried out by an adaptive observer (see Fig. 4). The simulation results are shown in the following figures. As Remark: We observe that the value of the synergetic parameter λ affects the fault estimation. When we increase the value of λ from 2 to 6, the fault estimation becomes more accurate and approaches the real fault. Therefore, an appropriate value of gives an adequate fault estimation. [200; 300] s. In order to demonstrate the resilient control performance, the pitch angle, before and after using the synergetic resilient control, is plotted in Fig. 7 and Fig. 8. It can be seen from Fig. 7 that the proposed resilient synergetic control succeeded in compensating for the considered fault and recovering the pitch angle to the desired value. The resilient control increased from 6 to 12 when the fault occurs which maintains the pitch system in the desired performance.

CONCLUSIONS

In this work, a fault tolerant control (FTC) system was proposed for system with fault changing system dynamics to recover the nominal operation and the desired performance. The FTC system included an adaptive observer to detect and estimate the fault and a synergetic resilient control to compensate for parameter faults. The proposed synergetic resilient control based on the synergetic theory taking into consideration the feature of dependence of the system dynamics. Finally, a numerical example proved the effectiveness of the proposed synergetic resilient control

Fig. 1 .

 1 Fig. 1. General structure of the FTC system

  Fig. 2. Pitch angleAs expected, the closed loop trajectory generated by the synergetic control law converges to the manifold. In fact, the pitch angle converges to the reference value 6.

Fig. 3 .

 3 Fig. 3. The effect of T on the evolution of the pitch angle

  Fig. 4. Fault estimation can be seen from Fig 4, the adaptive observer allows satisfactory fault estimation in spite of a light difference between real and estimate fault due to the noise applied to the pitch system.Remark: We observe that the value of the synergetic parameter λ affects the fault estimation. When we increase the value of λ from 2 to 6, the fault estimation becomes more accurate and approaches the real fault. Therefore, an appropriate value of gives an adequate fault estimation.

  Fig. 5. Fault estimation according to the value of λ

Fig

  Fig. 7. Pitch angle


Ricardo Schacht Rodriguez
email: ricardo.schacht-rodriguez@

Jean-Christophe Ponsart
email: jean-christophe.ponsart@univ-lorraine.fr

Carlos Daniel

Carlos Manuel Astorga-Zaragoza

Carlos Manuel

Prognosis & health management for the prediction of UAV flight endurance

Keywords: Prediction methods, Extended Kalman Filters, Mobile robots, Threshold voltage, Energy dependence

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Nowadays, the use of Lithium Polymer (Li-Po) battery as powertrain for multirotor UAVs with capabilities of vertical take-off and landing (VTOL) has become a capable and economical power source, due to its high energy and power densities, discharge rate (C-rate), and zero emission of pollutants [START_REF] Donateo | A new approach to calculating endurance in electric flight and comparing fuel cells and batteries[END_REF]). By considering a new and fully charged battery, the flight endurance is between 10 to 30 minutes [START_REF] Roberts | Energy management for indoor hovering robots[END_REF]; [START_REF] Morbidi | Minimum-Energy Path Generation for a Quadrotor UAV[END_REF]; [START_REF] Chang | LiPo battery energy studies for improved flight performance of unmanned aerial systems[END_REF], where 85% of the energy is consumed by the propulsion system. The remaining consumption is caused by the sensors, (such as GPS, accelerometers, compass, gyroscope [START_REF] Aleksandrov | Energy consumption of mini UAV helicopters with different number of rotors[END_REF])), the flight control computer, and for some payload (gimbal system, cameras, and other kind of sensors). However, the flight endurance is gradually reduced over timer due to aging of the battery. The effect of aging is characterized by a capacity loss and immediately power fade. The aging is caused by several factors such as highrate cycling, overdischarge and overcharge, and drastically changes of operation temperature [START_REF] Watrin | Review of adaptive systems for lithium batteries State-of-Charge and State-of-Health estimation[END_REF]). To avoid damages and reduce the aging rate during the This work was supported by CONACYT (Consejo Nacional de Ciencia y Tecnología) charge/discharge cycles of the battery, it is necessary to monitor the State of Charge (SoC) and to establish safety limits around the End of Discharge (EoD) (also called cutoff voltage). At a given time, the SoC is the proportion of the available charge, compared to the total charge available when the battery is fully charged, it is analog to a car gas fuel gauge. On the other hand, the EoD indicates that the battery is fully discharged. In the literature of Electrical Cars, such concepts are the main criterion in the design of Battery Manage Systems (BMS) [START_REF] Saha | Battery health management system for electric UAVs[END_REF], where the main objective is to define adequate operation regimes to extend the life of the battery and maximize the energy supplied. In the case of UAVs, the monitoring of SoC battery during the development of a mission will allow to establish a voltage safety limit before reaching the EoD to ensure the fulfillment of the mission, safeguard the integrity of the battery, and have an adequate use of the energy supplied to maximize the flight endurance. Several researches have addressed the problem of flight endurance for different types of UAV considering mainly three types of powertrains such as batteries, fuel cells, and solar cells [START_REF] Meyer | Design considerations for a low altitude long endurance solar powered unmanned aerial vehicle[END_REF]; [START_REF] Saha | Battery health management system for electric UAVs[END_REF][START_REF] Swider-Lyons | Hydrogen fuel cell propulsion for long endurance small uavs[END_REF][START_REF] Traub | Range and endurance estimates for batterypowered aircraft[END_REF]). The main objectives of such researches were the development of methodologies and strategies to increase the flight endurance taking into account the geometry of the UAV, disturbance as the wind, the type of trajectory, the maximum altitude and the power requirement to perform a mission. In the specific case of multirotor UAVs powered by Li-Po batteries, the efforts have been focused to generate strategies to ensure the fulfillment of a mission and increase flight time, e.g. [START_REF] Souza Cândido | Control and energy management for quadrotor[END_REF]) are presented an integrated planning and control approach to increase the probability of completion of fly-by missions by batterypowered UAV quadrotor. The authors in (Abdilla et al. (2015a)) proposed a technique to extend the endurance of battery-powered rotorcraft by sub-diving the monolithic battery into multiple smaller capacity batteries which are sequentially discharged and released. Also, in [START_REF] Abdilla | Power and endurance modelling of battery-powered rotorcraft[END_REF]) the authors characterized the power consumption of multirotors and derived an endurance estimation model. [START_REF] Morbidi | Minimum-Energy Path Generation for a Quadrotor UAV[END_REF]) presented two optimal control problems with respect to the angular accelerations of four electrical motors of an UAV quadrotor to improve its flight endurance. More recently [START_REF] Dietrich | An empirical study on generic multicopter energy consumption profiles[END_REF]) are presented experimental results of an analysis of the energy consumption and derivation and validation of empirical formulas to estimate the power consumption and the remaining flight time. The works mentioned above were mainly engaged in developing strategies to improve the consumption of the energy supply, which allows the battery to extend the power consumption for UAV applications, to extend the flight endurance and to ensure the fulfillment of a mission. The main contribution of this work is to present a methodology to predict the flight endurance in the context of Prognosis & Health Management through a Model-based Prognosis algorithm based on the mathematical model of the propulsion system of an UAV multirotor. The prediction of flight endurance is developed in three sequential steps: 1) the estimation of the battery SoC through an Extended Kalman Filter, followed by 2) the propagation of the estimated SoC in the future until reaching a safety threshold value before the EoD, and the 3) update of the prediction. The proposed methodology is applied to an UAV hexarotor to demonstrate: i) the fulfillment of the UAV mission, ii) avoiding overdischarges in the battery, and iii) to be able to take decisions over the mission during the flight endurance prediction. Such considerations must be taken into account to perform routines of mission planning or trajectory reconfiguration. The remaining of the paper is organized as follows: The mathematical model of the propulsion system of an UAV multirotor is presented in the next section. The main concepts around Prognosis & Health Management and the Model-based Prognosis algorithm to predict the flight endurance are explained in Section 3. The development of the flight endurance is detailed in Section 4. The simulation results and the discussion are described in Section 5. Finally, the conclusion and perspectives of the future work are defined in the last Section.

MATHEMATICAL MODEL OF PROPULSION SYSTEM OF UAV MULTIROTOR

The propulsion system of a UAV multirotor is composed by a set of BrushLess DC Motors (BLDCM) powered by a Lithium Polymer battery as it can be seen in Fig. 1, where i denotes the i-th ESC-BLDCM. The connection between each motor and the battery is realized through an Electronic Speed Control (ESC). In Fig. 2 the connection between the Li-Po battery -ESC -BLDCM is detailed for one motor. The ESC adjusts the angular velocity of the BLDCM through a control signal which is a PWM signal.

In this study the dynamic around the ESC is neglected and it is assumed that the voltage supply by the battery and the generated current by the motors are averaged with respect to the duty cycle value produced by the control signal.

Lithium

SoC (t) = SoC (t o ) - 1 3600 • C T t to I batt (t)dt, (1)
where t o represents the initial time, C T is the total capacity of the battery expressed in (Ah). The mathematical model, describing the dynamic behavior of the voltage in a Li-Po battery or a single cell, is based on an Equivalent Circuit Representation (ECR) according to Fig. 3. The ECR was presented in [START_REF] Chen | Accurate electrical battery model capable of predicting runtime and IV performance[END_REF]). In The mathematical model of the battery is given by:

VSoC (t) = - I batt (t) 3600 • C T Vd (t) = - V d (t) R d • C d + I batt (t) C d V Batt (t) = V OCV (V SoC (t)) -V d (t) -R int I batt (t), (2)
where, the V OCV (V SoC (t)) is defined experimentally as:

V OCV (V SoC (t)) = n i=0 λ i V SoC (t) i + ln(V SoC )V SoC (t) (3)
It can be noted that, the nonlinear function ( 3 2013)). In Fig. 6, the aging effect in the voltage of a battery cell is shown. The battery cell was submitted to 150 charge/discharge cycles (N cycle ) [START_REF] Saha | Battery data set[END_REF]). The EoD was established at 2.6 V, and as it can be observed, the time to reach the EoD is reduced according to the number of charge/discharge cycles denoted by (N cycle ). The evolution of the numerical value of C T , is plotted in Fig. 7. It can be noted that this evolution depends on the number of charge/discharge cycles (N cycle ), and the SoH can be defined as

SoH(N cycle ) = C a (N cycle ) C int C T = C 0 • SoH(N cycle ), ( 4
)
where SoH(N cycle ) is the SoH as an effect of capacity loss, C a (N cycle ) is the value of the capacity in a specific charge/discharge cycle, and C 0 is the value of the capacity when the battery is new. Similar to SoC, the SoH is in a range of SoH ∈ [ 0 1 ], i.e the SoH is 1 when the battery is new, and according to the use, damage, and other factors the SoH will be reduced. It can be noted that the decrease of the capacity affects directly the SoC and consequently the battery voltage (as it can be seen in equation ( 3)). This relationship is plotted in Fig. 8, where the dependence between the SoC and the battery voltage is shown. In that sense, the time when the EoD is reached can be determinate when the SoC reaches 0. It should be mentioned that one characteristic in the dynamic associated to the battery voltage during a discharge is the exponential fall of the voltage before reaching the EoD. The voltage fall can be seen in Fig. 8, which occurs after the SoC reach the 0.1 %.

BrushLess DC Motor

The mathematical model of the BrushLess DC Motor (BLDCM) is divided into the electrical and the mechanical sub-models described by the following equations [START_REF] Moseler | Application of model-based fault detection to a brushless DC motor[END_REF]):

vbatt (t) = R ībatt (t) + K E ω(t), ω(t) = K E J T P ībatt (t) - d J T P ω 2 (t) + D f J T P ω(t) - T f ric J T P , (5)
where

R = 2 3 ( 3 i=1 R i )
is the equivalent electric resistance of each coil, K E is the back electromotive force, ω(t) is the angular velocity, T f ric is the motor friction torque, D f is the viscous damping coefficient of the motor, d is the drag constant, associated to the geometry of the propeller, and J T P is the inertia of the motor. The average voltage vbatt (t) and current ībatt (t) are the voltage and current generated by the ESC and they are computed as follows:

vbatt (t) = V batt (t) • D cycle I batt (t) = ībatt (t) • D cycle , (6)
where V batt (t) is the battery voltage, I batt (t) is the current generated by the motors, and D cycle is the duty cycle of Pulse-Width Modulation (PWM) signal, which corresponds to control signal of the motor speed. D cycle can be defined as a function of the square of the angular velocity of the motors and it is determined through experimental correlations, such as D cycle = f (ω(t) 2 ).

PROGNOSIS & HEALTH MANAGEMENT

According to [START_REF] Goebel | Stateof-Charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model[END_REF]; [START_REF] Kim | Prognostics and Health Management of Engineering Systems[END_REF]) the Prognosis & Health Management (PHM) is a novel engineering approach which enables the health assessment of a system according to its actual operating conditions, as well as the prediction of its future condition based on up to date information. One of the main objectives of the PHM is to predict how the system will behave in the future in order to know if more stress or changes in the nominal operation on the system will likely cause an acceleration towards a certain undesirable event or failure condition, and the time when such event will occurs. In order to illustrate how Fig. 9. Evolution of the system behavior.

the prognosis is performed, let us consider Fig. 9. The solid dark blue line represents the past evolution of the system behavior until present time t a , and the brown area is the Safety Zone (SZ) or the nominal operational limit.

According to different factors such as the inputs of the system, aging, or even external disturbances, the system behavior could be progress until reach values outside of its nominal operation. In that sense, if such factors are known, the future trajectory of the system behavior can be predicted until reaching the SZ (segmented blue lines). In the prognosis literature, the End of Life (EoL) is reached, when the system behavior reaches or exceed the upper limit of the SZ at some time t fi , and the Remaining Useful Time (RUL) until reach the EoL is computed as

RU L i = t fi (EoL i ) -t a .
(7) In this work, the system under study is the propulsion system of an UAV multirotor. In that sense, several characteristics of the system behavior could be predicted, such as: i) the battery SoC in a given time. ii) the fulfillment of a mission before to reach the EoD, iii) the flight endurance of the UAV and the Remaining Mission Time. In this work the RUL is considered as the RMT of the UAV.

Model-based Prognosis

Depending on the accuracy level of the prediction and the complexity of the system behavior, different approaches have been used to perform prognosis [START_REF] Goebel | Stateof-Charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model[END_REF]; [START_REF] Coble | Mergin data sources to predict remaining useful life -an automated method to identify prognostic parameters[END_REF]). The Model-based Prognosis or Conditionbased Prediction assess the behavior of individual components of a system as well as the entire system based on a mathematical model that describe the evolution of some system variables (i.e temperature, voltage, position, speed) over time. If such mathematical model exists, the future behavior of the system can be determinate by propagating the mathematical model in the future time. However, to determinate a mathematical model of the interested system in order to perform a prediction of its future behavior is not a trivial task. First, it is necessary to define which system variables are necessary to evaluate the entire system behavior, and determine if such variables are directly measured or not. If such variables are measured, a mathematical model can be defined and used to perform the prediction in the future time. Otherwise, if such variables are unmeasured, they must be estimated. In that sense, this work presents in Fig. 10 a Modelbased Prognosis methodology based on an estimation of the battery SoC in order to predict the flight endurance of the UAV. The prediction in three sequential steps: 1) estimation of unmeasured states of the system, 2) propagation and prediction of the estimated state, and 3) update of the prediction.

State Estimation.

In several dynamic systems, not all state variables can be measured. For such cases, an observer needs to be designed in order to estimate the unmeasured states considering the system model and the available input-output signals. In this work an Extended Kalman Filter (EKF) is used to estimate the battery SoC. The EKF has demonstrated to be an adequately tool to estimate the battery states as well as parameters in battery systems [START_REF] Plett | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation[END_REF]; He et al. ( 2011)). In this work the EKF is developed considering a simplified mathematical model where the unique nonlinear term is the Open Circuit Voltage. The EKF addressed the general problem of state estimation of a nonlinear system, expressed in discrete time as:

x k+1 = f (x k , u k ) + w k y k = g(x k , u k ) + v k , (8)
where f (x k , u k ) and g(x k , u k ) are the nonlinear state transition function and nonlinear measurement function, respectively, and they are assumed to be differentiable at each operating point. The random variables w(t) and v(t) are the process and measurement noise. They are assumed to be independent, white, and with normal probability distributions:

p(w) ∼ N (0, Q) p(v) ∼ N (0, R). ( 9
)
The matrices Q and R are the process noise covariance and measurement noise covariance, respectively. At each sample time, f (x k , u k ) and g(x k , u k ) are linearized through a first-order Taylor-series developed as

A = ∂f (x k , u k ) ∂x k-1 x k-1 =x - k-1 (10) C = ∂g(x k , u k ) ∂x k x k =x - k (11)
The completed equations of the EKF are resumed in Algorithm (1).

Algorithm 1 Extended Kalman Filter algorithm [START_REF] Plett | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation[END_REF].

Initial values of x-

k-1 and P k-1 To compute:

A k = ∂f (x k , u k ) ∂x k-1 x k-1 =x - k-1 C k = ∂g(x k , u k ) ∂x k x k =x - k State estimate update: x- k = f (x k-1 , u k-1
), Error covariance matrix:

P - k = A k P k A T k + Q, Kalman gain: K k = P - k C T k (C k P - k C T k + R) -1 , State estimate measurement update: xk = x- k + K k (y k -g(x - k , u k )
), Error covariance matrix update:

P k = (I -K k C k )P - k .
Propagation and Prediction of estimated states.

As mentioned previously, if a mathematical model that describes the evolution of system behavior as a function of the time exists, the prediction of future behavior can be determined by propagating the model in the future time. However if the system involves unmeasured variables, they must be estimated first, and the prediction of its future behavior is based on the propagation of the estimated variables in the future. In order to illustrate how the propagation and prediction are performed, let us consider Fig. 9 where the trend of the past evolution of system behavior at time t a is known. It is possible to define a function that approximate the trend with the time, and propagate the function in the future until reaches the SZ. This function is called prediction function, and the accuracy of the prediction will depend on the amount of past data measure. On the other hand, the prediction function can be updated by using new data of the system. Due to the fact that there are no abrupt changes or discontinuities in the evolution of estimation SoC, the prediction function is defined as a polynomial function of the time, such as:

η (t, α) = m j=0 α j• t j = α 0 + α 1 • t + • • • + α m • t m , (12)
where η(t) represents the predicted variable, t is the time, α i are identified parameters associated to measured data trend and m is the order of the polynomial function. The method of Weighted Least Squares (WLS) is considered to estimate the parameters (α i ) of ( 12). The WLS is defined as:

H = [ 1 t • • • t m ] , α = [H T R -1 H] -1 H T R -1 Y, (13)
where H contains the information of the relationship between the time and the polynomial order, α is the vector with the estimated parameters α i , R is the measurement covariance matrix, and Y is the vector of the measured data (variable to be identified). The approximation is evaluated the coefficient of determination R 2 , which is limited as R 2 ∈ [ 0.9 1 ] in this study, and the polynomial order depends on the accuracy between the data and the identified function. The iterative sequence of propagation and prediction is given in the Algorithm 2.

Algorithm 2 Propagation and prediction algorithm.

(1)

Delay (t p ) (2) Recollect data until Ŝ oC(t a ) (3) Define H and m for H = [ 1 t • • • t m ] and η (t, α) = m j=0 α j• t j = α 0 + α 1 • t + • • • + α m • t m (4) Estimate α with Y = Ŝ oC(t 0 : t a ) α = [H T R -1 H] -1 H T R -1 Y (5) Verify R 2 to Y = Ŝ oC(t 0 : t a ) with η (t 0 : t a , α) (6) Propagate η (t 0 : t a , α) until t(SV T ) to determinate the Flight Endurance (FE) F E = t(SV T ) (7) Compute Remaining mission Time (RMT) RM T = t(SV T ) -t a

PREDICTION OF FLIGHT ENDURANCE OF UAV MULTIROTOR

The methodology explained previously will be applied to predict the flight endurance of an UAV hexarotor. During the development of a mission of some UAV, the End of Discharge (EoD) must be taking into account to ensure the safety of the UAV, the fulfillment of the mission, and the maximization of flight endurance. In that sense, the total flight endurance is considered as the time between the beginning of the mission until reaches a Safety Voltage Threshold before EoD value. Several factors are implied in the flight endurance of the UAV, such as the total current demanded by the number of the BLDCM (I batt (t)), and C-Rate, the rate discharge (C), the initial charge SoC(t 0 ), and the State of Health SoH. Then, assuming a new and fully charged battery, the EoD is directly associated with the battery voltage. The time when EoD is reached can be computed considering the demanded current (I batt (t)) through relation of C-Rate discharge. However according to the number of charge/discharge, the capacity of the battery decreases and the discharge rate is modified as the aging of the battery increases. In that sense, the computing of EoD time should consider the effect of the aging to obtain the total flight endurance according to the battery SoH. As it can be observed from (2), the variation of C T affects directly the SoC and consequently the OCV(3). Then, considering the relationship between the battery voltage and the SoC shown in Fig. 8, the EoD can be directly associated to the SoC. Taking into account that the total current demanded by the set of BLDCM is known or measure, the SoC of the battery during the mission can be computed using equation (1) or estimated through EKF, and its future trajectory can be propagated and predicted in the future until reaching the EoD, leading to prediction of the flight endurance. The development of each step in the PHM architecture shown in Fig. 10 to predict the flight endurance of the UAV is explained in the next section.

SoC estimation

The prediction of the flight endurance is based on the computation or estimation the battery SoC during the UAV mission. In that sense, two cases are considered in this work: 1) the direct computation of the SoC using the Amper counting method, and 2) the estimation of the SoC using a EKF. In the first case, the SoC is computed directly using equation ( 1). However, this method accumulates errors in measurements and may lead to large SoC errors in real-world applications [START_REF] Gholizadeh | Estimation of state of charge, unknown nonlinearities, and state of health of a lithiumion battery based on a comprehensive unobservable model[END_REF]).

In that sense, the estimation of SoC is considered based on the design of an Extended Kalman Filter applied on system (2).

EKF of Li-Po battery.

The estimation of SoC is developed according to Algorithm 1. System (2) is discretized through forward Euler method, and rewriting in state space:

x k+1 = 1 0 0 1 - T s R d C d x k +    - T s C T T s C d    u k + w k y k = ∂V OCV (V SoC ) ∂V - SoC -1 x k -R int u k + v k , (14)
where the discrete state space vector is

x k = [ V SoC V d ]
T , the input u k = I batt , the output y k = V batt , and T s is the sampling time. The nonlinear term is the OCV shown in equation ( 1), which is in the matrix C. The variation in the measurement and process noise are established at ±0.1 V , and ±0.01 V and the process and measurement covariance matrices are defined using experimental data from a real battery [START_REF] Schacht-Rodriguez | SoC estimation using an Extended Kalman filter for UAV applications[END_REF]) as:

Q = 1 × 10 -2 0 0 1 × 10 -2 R = [ 0.1 ] . (15)
The propagation and prediction of the estimated SoC is developed according to Algorithm (2). It is important to mention that the methodology shown in Fig. 10, considers two types of sampling time during its implementation, i.e. whiles the state estimation stage is subject to the sampling time of the system in order of milliseconds, the propagation and evaluation could be performed in a sampling time in the order of seconds, or even minutes.

RESULTS AND DISCUSSION

The methodology introduced in this paper to predict the flight endurance was tested in simulation considering the propulsion system of an UAV hexarotor. A Li-Po battery of four cells was considered as powertrain and the parameters were obtained using experimental data considering the methodology presented in [START_REF] Schacht-Rodriguez | SoC estimation using an Extended Kalman filter for UAV applications[END_REF]). Considering the relationship between the battery voltage and the SoC, the SVT is defined according to Fig. 11. As it can be observed, before the voltage reaches the EoD, there is an abrupt exponential fall of voltage.

Taking into account this SVT is defined before the voltage reaches the EoD. The UAV was sub- The propagation and prediction was developed over sampling time of 1 minute, and the SoC estimation each 10 milliseconds to collect enough data to estimate the parameters of the prediction function ( 12). Two 1 st and 2 nd order polynomials were considered for the prediction function. The predictions of the total flight endurance for both prediction functions are plotted in Fig. 15. The blue marks correspond to the predicted flight endurance with the 1 st order polynomial, and the red marks with the 2 nd order polynomial. As it can be observed, the first predictions (1 -4 min) with the 1 st order polynomial are closer to the real flight endurance, whilst an approximation of the real flight endurance is displayed from 6 minute onwards with the 2 nd order polynomial. However the closest prediction of real flight endurance is obtained at time 8 min with the 2 nd order polynomial. In Fig. 16 and Fig. 17 SoC was propagate in the future time before reaching the EoD. The propagation and prediction algorithm presented can be applied on other type of electrically powered UAV. On the other hand, the prognosis methodology also enabled to define safe operating limits in the battery to avoid damages and guarantee the fulfillment of the UAV mission. The simulation results exhibited that it is possible to predict the total flight endurance and the remaining mission time before the end of the mission considering a Decision Threshold between the beginning of the mission and the time when the prediction is closest with the real value. The Decision Threshold is necessary in order to define a scheme of Decision Making for mission planning or trajectory planning. In that sense, the prediction of the flight endurance will be the core of a path planning and re-planning strategy to take actions over the evolution of a mission as future work.

  Prognosis & Health Management for the prediction of UAV flight endurance R. Schacht-Rodríguez * , * * J.-C. Ponsart * * C. D. García-Beltrán * C. M. Astorga-Zaragoza * 1. INTRODUCTION

Fig. 1 .

 1 Fig. 1. Propulsion system of UAV multirotor.

Fig. 2 .

 2 Fig. 2. Connexion of Li-Po battery -ESC -BLDCM.

Fig. 3 .

 3 Fig. 3. Electrical equivalent a Lithium battery.

Fig. 3 ,

 3 (on the left-side of the circuit), the voltage V SoC (t) models the state of charge SoC(t) of the battery from the capacity C T . The voltage V OCV (V SoC ) is the Open Circuit Voltage (OCV), i.e. it is the effective voltage in the terminals of battery, and it is modeled as a function of the state of charge of the battery. The voltage V Rint (t) characterizes the ohmic over-potential due to the internal resistance of the battery R int . V d (t) represents the transitory response of the voltage when a current is demanding to the battery. It is characterized by a first order response and its time constant depends on the value of parameters R d and C d .

  Fig. 4. Demanded Current and battery voltage at 25 • C.

Fig. 5 .

 5 Fig. 5. Characteristics of the battery voltage.is the cut-off voltage and it indicates that the battery is fully discharged. If this value is reached and the discharge continues, an irreversible damage is generated to the battery. One way to quantify the aging is to consider the State of Health (SoH). The SoH represents the actual conditions of a battery, in comparison with its nominal conditions. The main phenomena that characterizes the battery SoH is the capacity loss or capacity fade, which is caused by the reduction of the capacity value C T according to the number of charge/discharge cycles (Cordoba-Arenas et al. (2013)). In Fig.6, the aging effect in the voltage of a battery cell is shown. The battery cell was submitted to 150 charge/discharge cycles (N cycle )[START_REF] Saha | Battery data set[END_REF]). The EoD was established at 2.6 V, and as it can be observed, the time to reach the EoD is reduced according to the number of charge/discharge cycles denoted by (N cycle ). The evolution of the numerical value of C T , is plotted in Fig.7. It can be noted that this evolution depends on the number of charge/discharge cycles (N cycle ), and the SoH can be defined as

Fig. 6 .

 6 Fig. 6. Variation of voltage of a battery cell subject to 160charge/discharge cycles[START_REF] Saha | Battery data set[END_REF]).

Fig

  Fig. loss and battery SoH[START_REF] Saha | Battery data set[END_REF]).

Fig. 8 .

 8 Fig. 8. Relationship between battery voltage and SoC.

Fig. 10 .

 10 Fig. 10. Model-based Prognosis methodology.

Fig. 11 .

 11 Fig. 11. Safety Voltage Threshold of the Li-Po battery.jected to a mission which consisted in following circular trajectory with an area of 785400 m 2 around x -y axis and an altitude of 20 m (Fig.12). Such trajectory was developed considering the mathematical model of UAV presented in (Arellano-Muro et al. (2013)). Considering that the battery is fully charged and new, the initial SoC and SoH were established in 1 respectively. The current

  Fig. 13. (a) Demanded current I batt and (b) comparison of V batt and Vbatt .

Fig. 14 .

 14 Fig. 14. Comparison between SoC CC and Ŝ oC.

  Fig. 15. Flight Endurance prediction.the prediction for the 1 st and 2 nd order polynomial at time 1, 6 and 8 min, and their comparison with the estimated SoC are shown. A Decision Threshold (DT) at the 0.6% (at time 10 min of the total flight endurance) of estimated SoC was defined to determinate an operation range where the predictions will be useful to take decisions on the progress of the mission. The computation of Remaining Mission Time (RMS) was made from the first prediction, and the results are shown in Fig.18. As it can be seen, it is possible to predict the total flight endurance and the RMS from the beginning of the mission considering the defined DT.

Fig. 16 .

 16 Fig. 16. Prediction of FE with 1 st order polynomial.

Fig. 18 .

 18 Fig. 18. Evolution of Remaining Mission Time.


Marcin Pazera
email: m.pazera@issi.uz.zgora.pl

Marcin Witczak
email: m.witczak@issi.uz.zgora.pl

Christophe Aubrun
email: christophe.aubrun@univ-lorraine.fr

Jozef Korbicz
email: j.korbicz@issi.uz.zgora.pl

Józef Korbicz

Simultaneous estimation of multiple sensor and process faults for non-linear discrete-time systems

Keywords: Fault diagnosis, multiple fault estimation, sensor fault, process fault, robust estimator

INTRODUCTION

The growing requirements for reliability and safety of contemporary complex processes and systems are the inspiration for researchers to develop newer and newer approaches concerning the Fault Diagnosis (FD) [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF][START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic Systems[END_REF][START_REF] Isermann | Fault-diagnosis applications: modelbased condition monitoring: actuators, drives, machinery, plants, sensors, and fault-tolerant systems[END_REF][START_REF] Korbicz | Fault diagnosis. models, artificial intelligence, applications[END_REF][START_REF] Patton | Issues of fault diagnosis for dynamic systems[END_REF]. Significant attention for the latest few decades is dedicated to robust Fault-Tolerant Control (FTC) approaches which can be divided into active and passive ones. Contrarily to the passive FTC, which is designed in such a way to be robust against some predefined faults treated in the same fashion as disturbances, the active one is especially attractive. It allows changing the control action according to occurring faults to compensate their impact effect on the controlled system and as a consequence to guide the system correctly and safely even in a faulty case. In order to achieve such a goal, it is necessary to obtain an information about faults in every instances of time. In the literature, there are plenty methods of identification and estimation to acquire the knowledge about faults. Among them in [START_REF] Foo | A sensor fault detection and isolation method in interior permanent-magnet synchronous motor drives based on an extended kalman filter[END_REF][START_REF] Keller | Two-stage kalman estimator with unknown exogenous inputs[END_REF] a Kalman filterbased schemes are considered. A sliding mode observer and an high-gain observer are presented in [START_REF] Brahim | An h ∞ sliding mode observer for Takagi-Sugeno nonlinear systems with simultaneous actuator and sensor faults[END_REF][START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]. In the works [START_REF] Zhang | Fault diagnosis of a class of nonlinear uncertain systems with lipschitz nonlinearities using adaptive estimation[END_REF][START_REF] López-Estrada | Robust sensor fault estimation for descriptor-lpv systems with unmeasurable gain scheduling functions: Application to an anaerobic bioreactor[END_REF] adaptive estimators are proposed. Moreover, a minimum-variance estimator is considered in [START_REF] Gillijns | Unbiased minimumvariance input and state estimation for linear discretetime systems[END_REF] while in [START_REF] Chen | Robust fault estimation using an lpv reference model: Addsafe benchmark case study[END_REF][START_REF] Nobrega | Robust fault estimation of uncertain systems using an lmi-based approach[END_REF][START_REF] Zhang | Robust H ∞ adaptive descriptor observer design for fault estimation of uncertain nonlinear systems[END_REF] an H ∞ approach is used.

Having in mind the fact that modern control systems can be split into actuators, sensors and process dynamics [START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic Systems[END_REF][START_REF] Ducard | Fault-tolerant flight control and guidance systems: Practical methods for small unmanned aerial vehicles[END_REF], the three classes of faults may be taken into consideration: actuator faults, sensor faults and process (or component) faults. However, most of the listed strategies are dedicated to the one of them only. In the other words, they can be used to either actuator or sensor fault estimation, exclusively.

The main objective of the paper is to deal with the problem of simultaneous estimation of the process and sensor faults. The proposed strategy is robust against the process and measurement uncertainties and it is based on the H ∞ approach. The proposed estimator is able to estimate the process and sensor faults and as a consequence it gives a state estimate free from sensor faults. Indeed, it is very important to take into account simultaneous estimation of the process and sensor faults because in the real environmental conditions both of them might appear at the same time.

The paper is organised as follows: Firstly, section 2 presents preliminaries regarding the class of considered systems. Subsequently, section 3 shows the robust design procedure for the estimator able to estimate the state as well as sensor and process faults. Section 4 concerns an illustrative example with application to the twin-rotor aerodynamical MIMO system and finally, Section 5 concludes the paper.

PRELIMINARIES

Let us consider a discrete-time system:

x k+1 = Ax k + Bu k + n i=1 D i f i,k x i,k + g (x k ) + W 1 w k , (1)
y k = Cx k + f s,k + W 2 w k ,
(2) where x k ∈ X ⊂ R n is the state vector, u k ∈ R r stands for the control input, y k ∈ R m is the output and g (x k ) describes the non-linear behaviour of the system. It is assumed that g (x k ) is differentable and has bounded derivatives. Moreover f s,k ∈ F s ⊂ R m is the sensor fault vector and w k express the exogenous disturbance vector obeying

l 2 = {w ∈ R n | w l2 < +∞}, (3)
w l2 = ∞ k=0 w k 2 1 2 . ( 4
)
Note that, it can be easily shown that the disturbance vector can be spilt in such a way

w k = w T 1,k , w T 2,k
T which represent the process and measurement uncertainties, respectively. Furthermore, W 1 and W 2 denote the noise distribution matrices. Moreover, D stands for the process fault distribution matrix, in other words, it describes the way in which the fault influences the system. It can be observed that the problem of estimating f i,k directly is non-linear because of multiplication, even for the linear system. To tackle this problem, throughout the paper, the system (1)-( 2) is transformed into the following form:

x k+1 = Ax k + Bu k + Dz k + g (x k ) + W 1 w k , (5)
y k = Cx k + f s,k + W 2 w k , (6)
where the vector z k ∈ R n is composed with a multiplication of the fault f k and state x k in such a way as

z k =     f 1,k • x 1,k f 2,k • x 2,k . . . f n,k • x n,k     . (7)
For further deliberations, let us call the vector z k a process fault. The idea is to estimate z k instead of estimating f i,k directly.

JOINT SENSOR AND PROCESS FAULTS ESTIMATOR DESIGN

The main goal of the section is to design the estimator, which will be able to estimate the state as well as the process and all the sensor faults, simultaneously. To cope with such requirements, the estimator of the following form is proposed:

xk+1 = Ax k + Bu k + Dẑ k + g (x k ) + K x y k -C xk -f s,k , (8) ẑk+1
= ẑk + K z y k -C xk -f s,k , (9)
f s,k+1 = f s,k + K s y k -C xk -f s,k , (10)
where K x , K z and K s are the estimator gain matrices for the state, process fault and sensor fault, respectively. Having regard ( 5)-( 6) as well as ( 8)-( 10) the state estimation error is given

e k+1 = x k+1 -xk+1 = Ax k + Bu k + Dz k + g (x k ) + W 1 w k -Ax k -Bu k -Dẑ k -g (x k ) -K x Cx k -K x f s,k -K x W 2 w k + K x C xk + K x f s,k = [A -K x C] e k + g (x k ) -g (x k ) -De z,k -K x e s,k -K x W 2 w k , (11)
with e z,k = z k -ẑk and e s,k = f s,k -f s,k . In a similar way, the process fault estimation error is defined

e z,k+1 = z k+1 -ẑk+1 = z k+1 + z k -z k -ẑk -K z Cx k -K z f s,k -K z W 2 w k + K z C xk + K z f s,k = ε z,k + e z,k -K z Ce k -K z e s,k -K z W 2 w k , (12)
while the sensor fault estimation error is governed by

e s,k+1 = f s,k+1 -f s,k+1 = f s,k+1 + f s,k -f s,k -f s,k -K s Cx k -K s f s,k -K s W 2 w k + K s C xk + K s f s,k = ε s,k + [I -K s C] e s,k -K s Ce k -K s W 2 w k .
(13) Subsequently, using Differential Mean Value Theorem (DMVT) [START_REF] Zemouche | Observer design for Lipschitz non-linear systems: the discrete time case[END_REF], it can be shown that

g (x k , u k ) -g (x k , u k ) = M k (x k -xk ) , (14) with
M k =       ∂g 1 ∂x (c 1 , u k ) . . . ∂g n ∂x (c n , u k )       , (15)
where c 1 , . . . ,

c n ∈ Co(x k , xk ), c i = x k , c i = xk , i = 1, . . . , n.
Having regard the fact that all states are bounded in real system, x k ∈ X let

x i,j ≤ ∂g i (x) ∂x j ≤ xi,j , i = 1, . . . , n, j = 1, . . . , n, (16)
it is clear that there exist

M k ∈ M such that M = M k ∈ R n×n |x i,j ≤ m k,i,j ≤ xi,j , i, j = 1, . . . , n, . (17)
Indeed, now the state estimation error can be rewritten without the non-linear part into the following form

e k+1 = x k+1 -xk+1 [A + M k -K x C] e k -De z,k -K x e s,k -K x W 2 w k . ( 18
)
By constructing the following super-vectors:

ēk+1 = e k+1 e z,k+1 e s,k+1 , (19)
v k = w k ε z,k ε s,k , (20)
let us to present the state as well as process and sensor faults estimation error in a compact form ēk+1 = X k ēk + Zv k , (21) with:

X k = Āk -K C , (22) Z = W -K V , (23)
where:

Āk = A + M k D 0 0 I 0 0 0 I , C = [C 0 I] , W = W 1 0 0 0 I 0 0 0 I , V = [W 2 0 0] , K = K x K z K s . (24)
Based on the aforementioned estimation error for the state and faults, the following theorem can be defined: Theorem 1. For a prescribed disturbance attenuation level µ > 0 for the state and fault estimation error (21), the estimator ( 8) -( 10) design problem for the system ( 5)-( 6) and estimator ( 8)-( 10) is solvable if there exist matrices P , U and N such that the following inequality is satisfied

   I -P 0 ĀT k U - CT N T 0 -µ 2 I W U - V T N T U Āk -N C U W -N V P -U -U T    ≺ 0. (25)
Proof. The problem of the designing the H ∞ observer [START_REF] Li | A linear matrix inequality approach to robust H ∞ filtering[END_REF][START_REF] Zemouche | Observers for a class of lipschitz systems with extension to H ∞ performance analysis[END_REF] is to obtain matrices N , U and P such that

lim k→∞ ēk = 0 for v k = 0, (26)
ēk l2 < µ v k l2 for v k = 0, ē0 = 0. (27)
To solve such a problem, it is satisfactory to find a Lyapunov function of the following form

∆V k + ēT k ēk -µ 2 v T k v k < 0, (28)
where:

∆V k = V k+1 -V k , V k =
(∆V k ) = V s,∞ -V s,0 , V s,∞ = V s,0 = 0, then (28) yields: ∞ k=0 (∆V k ) + ∞ k=0 ēT k ēk -µ 2 ∞ k=0 v T k v k < 0 =⇒ V 0 + ∞ k=0 ēT k ēk -µ 2 ∞ k=0 v T k v k < 0 =⇒ ∞ k=0 ēT k ēk -µ 2 ∞ k=0 v T k v k < 0 =⇒ ∞ k=0 ēT k ēk < µ 2 ∞ k=0 v T k v k =⇒ ēk l2 < µ v k l2 ,
which leads to (27). As a consequence by using (21) it is easy to show that

∆V k + ēT k ēk -µ 2 v T k v k = ēT k X T k P X k + I -P ēk + ēT k X T k P Z v k + v T k Z T P X k ēk + v T k Z T P Z -µ 2 I v k < 0, (30)
and by introducing

vk = ēk v k , (31)
it can be shown that (30) can be rewritten into the following form

vT k X T k P X k + I -P X T k P Z Z T P X k Z T P Z -µ 2 I vk ≺ 0, (32)
which is equivalent to

X T k Z T P [X k Z] + I -P 0 0 -µ 2 I ≺ 0. ( 33
)
Now, let us recall the following Lemma: Lemma 1. The following statements are equivalent:

(1) There exists X k ≻ 0 such that

V T X k V -W ≺ 0. ( 34
)
(2) There exist

X k ≻ 0 such that -W V T U T U V X k -U -U T ≺ 0. ( 35
)
Applying lemma 1 to (33) gives

  I -P 0 X T k U T 0 -µ 2 I Z T U T U X k U Z P -U -U T   ≺ 0, ( 36
)
and then substituting:

U X k = U Āk -U K C = U Āk -N C , ( 37
) U Z = U W -U K V = U W -N V , (38)
completes the proof.

Note that, M specified by ( 17) can be equivalently expressed by

M = M (α) : M (α) = N i=1 α i M k , N i=1 α i = 1, α i ≥ 0 , ( 39
)
where N = 2 n 2 . Note that, this is a general description, which does not take into account that some elements of M may be constant. In such cases, N is given by N = 2 (n-c) 2 , where c stands for the number of constant elements of M . Thus, the system can be described in a Linear Parameter Varying (LPV) form. Solving ( 25) is equivalent to (for i = 1, . . . , N )   

I -P 0 ĀT i U T - CT N T 0 -µ 2 I W T U T - V T N T U Āi -N C U W -N V P -U -U T    0.
(40) As a result, the gain matrices are obtained as follows

K = K x K z K s = U -1 N . ( 41
)
The purpose of the subsequent section is to show the correctness of the proposed approach by its application to the twin-rotor MIMO system.

ILLUSTRATIVE EXAMPLE

Results

The performance of the proposed approach can be efficiently shown by its implementation to the non-linear twin-rotor MIMO system. The considered system is depicted in Fig. 1. Such a system has been designed to simulate the flight object in laboratory conditions. The system can be described by a highly non-linear model with cross coupled axes using the following equations:

dω v dt = k b k 2 J mr R b u v - B mr J mr + k 2 b J mr R b ω v - g 4 (ω v ) J mr , ( 42
)
dΩ v dt = l m g 5 (ω v ) + kgΩ h g 5 (ω v ) cos(θ v ) -k ov Ω v J v + g ((K A -K B ) cos(θ v ) -K C sin(θ v )) J v - Ω 2 h K H sin(θ v ) cos(θ v ) J v + k t kak1 Ra u h -B tr + k 2 a Ra ω h -g 1 (ω h ) J v J tr , ( 43
)
dθ v dt = Ω v , (44)
dω h dt = k a k 1 J tr R a u h - B tr J tr + k 2 a J tr R a ω h - g 1 (ω h ) J tr , ( 45
)
dΩ h dt = k oh g 2 (ω h ) cos(θ v ) -k oh Ω h -g 3 (θ h ) + g 6 (θ v ) Φ + k m ω v sin(θ v )Ω v -Φ -2K E cos 2 (θ v ) (Φ) 2 + k m sin(θ v ) k b k2 R b u v -B mr + k 2 b R b ω v -g 4 (ω v ) J mr Φ , ( 46
) dθ h dt = Ω h , (47)
where

Φ = K D cos 2 (θ v ) + K E sin 2 (θ v ) + K F , ω v , Ω v , θ v , ω h
, Ω h and θ h are the rotational velocity of the main rotor, angular velocity around horizontal axes, pitch angle of the beam, rotational velocity of the tail rotor, angular velocity around vertical axes and yaw angle of the beam, respectively. For more information the reader is referred to [START_REF] Rotondo | Quasilpv modeling, identification and control of a twin rotor mimo system[END_REF]. The system state vector is

x = ω T v , Ω T v , θ T v , ω T h , Ω T h , θ T h T and the system input vector is u = u T v , u T h
T where u v and u h are the voltages of the main and tail rotors. The rest of the parameters are inherited from [START_REF] Rotondo | Quasilpv modeling, identification and control of a twin rotor mimo system[END_REF].

The non-linear model, which describes the behaviour of the system has been discretized with a sampling time 5)-( 6). It is worth to emphasize that the angular velocity around both, vertical and horizontal axes, were not measured directly during the experiment, which implies the output equation expressed by

C =    1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1    . (48)
Moreover, let us consider following fault scenarios for the sensor fault

f s,k = f s,1,k , f s,3,k , f s,4,k , f s,6,k with: f s,1,k = 0 f s,3,k = y 2,k -0.2, 60[s] ≤ t ≤ 80[s], 0, otherwise, f s,4,k = y 3,k + 77, 70[s] ≤ t ≤ 90[s], 0, otherwise, f s,6,k = 0 (49)
and for the process fault

f k f 1,k = -0.01, 58[s] ≤ t ≤ 72[s], 0, otherwise, (50)
with the process fault distribution matrix

D =       
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

       . ( 51
)
The fault distribution matrix D indicates that the process fault acts directly onto the rotational velocity of the main rotor ω v . Moreover, the rest of the process faults influencing directly onto variables Ω v , θ v , ω h , Ω h , θ h are equal to 0. The number describing the sensor fault is adequate to the state of the system. By analysing the above scenarios, it can be concluded that the intermittent faults occur in two sensors. At some time instances the sensor faults overlap each other. This means that they give wrong measurements, simultaneously. Moreover, at the same time, as it is for the sensor faults, the process fault occurs. Such a dangerous situation might happen in real industrial system and could cause its incorrect operation. As a consequence, it even might be a reason of a total failure of the system. Summarising, in the above scenarios it is considered that two sensor faults and the process fault are occurring simultaneously.

Let us assume that the initial state for the system and for the estimator are x 0 = [0, 0, 0.001, 0, 0, 0.001] and x0 = [1, 0, 0.01, 1, 0, 0.01], respectively. The sensor faults as well as the process fault are initialised by f s,0 = [0, 0, 0, 0] and f 1,0 = 0, respectively.

By solving a set of LMIs (25) described in previous section, the following gain matrices for the state as well as for the sensor and process faults have been obtained:

K x =       
0.0012 0.0000 0.0000 -0.0000 -0.0001 0.0003 -0.0001 -0.0001 0.0000 -0.0010 0.0000 0.0001 0.0000 -0.0001 0.0003 -0.0004 0.0001 0.0001 -0.0001 0.0002 -0.0000 0.0001 0.0001 -0.0003

       , (52)
K z =       
0.5930 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000

       , (53)
K s =   
0.9907 -0.0001 0.0003 -0.0007 -0.0001 0.9750 -0.0014 0.0035 0.0002 -0.0017 0.9858 0.0052 -0.0002 0.0002 -0.0055 0.9843

   . ( 54
)

Discussion

Figures 2-4 present the exemplary response of the system in the open-loop without the control feedback. In those pictures, a black solid line represent the real state of the system. A red dashed line stands for the estimate and a dash-dotted blue line is the measured output. The state estimate for the ω v has been following the real state during the experiment in spite of the process fault occurrence. Meanwhile, the other two states have been measured incorrectly. This means that the sensor faults occurred. However, despite the sensor faults the state estimates have not been following the measured, faulty output but they have been estimating the real state with a very good accuracy. The real sensor faults (red dashed The paper deals with the simultaneous estimation problem for the state as well as for the sensor and process faults.

The proposed estimator was able to estimate the sensor and process faults, simultaneously. Having a knowledge concerning the sensor faults, provided by the fault estimator, it was possible to estimate the real state of the system, instead of the faulty output, which gives a sensor fault-free results. To cope with the process and measurement uncertainties the proposed strategy is based on H ∞ approach. It provides a convergence analysis for the proposed estimator and guaranties a predefined disturbance attenuation level.

The design procedure brings down to solve a set of LMIs.

The final part of the paper showed an illustrative example with an application to the twin-rotor aero-dynamical system. The achieved results confirm the correctness and performances of the proposed approach. The future work will be focused on the problem of compensating the simultaneous sensor and process faults. The controller design strategy will be proposed to cope with the stated problem for the considered system.

  ēT k P ēk and P ≻ 0. If v k = 0, then the Lyapunov function (28) takes the following simplified form ∆V k + ēT k ēk < 0 (29) and hence ∆V k < 0, which leads to (26). If v k = 0 and taking into account the fact that ∞ k=0

Fig. 1 .

 1 Fig. 1. Twin-rotor aero-dynamical system T s = 0.01[s] which leads to the state-space representation (5)-(6). It is worth to emphasize that the angular velocity around both, vertical and horizontal axes, were not measured directly during the experiment, which implies the output equation expressed by

Fig. 2 .Fig. 4 .Fig. 6 .

 246 Fig. 2. Rotational velocity of the main rotor and its estimate line) for the pitch angle and for the rotational velocity of the tail rotor as well as their estimates (blue solid line) are