Severus Constantin Olteanu
email: severus.olteanu@gmail.com

Abdel Aitouche
email: abdel.aitouche@hei.fr

Lotfi Belkoura
email: lotfi.belkoura@univ-lille1.fr

Adnan A J Jouni

Embedded P.E.M. fuel cell stack nonlinear observer by means of a Takagi-Sugeno approach

Keywords: nonlinear state estimation, PEM fuel cell, Takagi Sugeno, unmeasurable premise variables, embedded observer, hardware in the loop validation

HAL is

Introduction

The Hydrogen Fuel Cell systems have grown in interest in the last decade, despite of the still high production cost, because of their elevated efficiency, reduced pollution level and the targeted independence from fossil fuels. Amongst different types of Fuel Cells (FC) like solid oxide or alkaline ones [START_REF] Kirubakaran | A review on fuel cell technologies and power electronic interface[END_REF], the proton exchange membrane (P.E.M.) type [START_REF] Barbir | PEM Fuel Cells Theory and Practice[END_REF] stands out, because of its low working temperature, and proves to be best suited for vehicle applications. In electrical vehicles, the energy storage plays one of the most important roles [START_REF] Lukic | Energy Storage Systems for Automotive Applications[END_REF], so the research on Fuel Cells will boost the acceptance of electrical vehicles as well. Therefore it directly competes with batteries which have been developing continuously for many years, yet prove inferior in some aspects to hydrogen technology as shown in [START_REF] Thomas | Fuel Cell and Battery Electric Vehicles Compared[END_REF], both as weight per storage capacity as well as energy density; these being two important factors, adding to the slow recharge rate of a battery.

The Fuel Cells are small scaled devices therefore the development of virtual sensors would reduce the price. Also, a state observer may be used for diagnostics [START_REF] Zhang | Techniques for PEM Fuel Cell Testing and Diagnosis[END_REF]. The majority of the papers that take into account the dynamics and not only the static models of FCs, focus only upon the electrical part of the fuel cell ignoring the auxiliary components [START_REF] Kim | Equivalent circuit modeling of PEM Fuel Cell Degradation Combined with a LFRC[END_REF] or treating just the compressor separately [START_REF] Matraji | Cascade control of the moto-compressor of a PEM fuel cell via second order sliding mode[END_REF]. Nevertheless papers like [START_REF] Pukrushpan | Control of fuel cell power systems: principles, modeling, analysis and feedback design[END_REF], have to be mentioned as a thorough review upon all the components used. Indeed, for the more general case of system diagnosis, we find also many alternative approaches to model based techniques (for which a good review is [START_REF] Petrone | A review on model-based diagnosis methodologies for PEMFCs[END_REF]): experimental (ex: impedance spectroscopy [START_REF] Steiner | Non intrusive diagnosis of polymer electrolyte fuel cells by wavelet packet transform[END_REF], neuro-fuzzy techniques). As there is still no standardization in different existing types of FCs, a functional model would be easier to adapt to any particular case instead of experimental approaches that require extensive training data. Also model based approaches [11], [12] have the potential to give fast response to time variations, therefore being very efficient for online diagnosis [13] as well as control [14]. Of course one has to mention the greatest inconvenient of model based techniques that is the difficulty in parameter estimation. The state observer acts as a virtual sensors and it is designed to estimate cathode and anode pressures and mass flows of oxygen and hydrogen which are generally not measured. The mass flow rates of reactant gases play a pivotal role in the reliable and efficient operation of FCS.

For the design of the nonlinear observer, a Takagi-Sugeno (TS) representation has been chosen [15], [16]. This method can be found in literature, acting upon different types of industrial processes [17]. This approach has an advantage over other nonlinear ones in that there is no need for many assumptions regarding the form of the state space model, it has a structured form, it is easy to implement numerically and also that it allows a parallel to be drawn to linear techniques. The construction of state observers based on TS representation has been in a continuous augmentation in the last period. Although many papers consider the premise variables measurable [19], this case in many practical applications is unfortunately unattainable. Among those who have tackled the issue of unmeasurable premise variables, one can cite [18].

It is useful to adopt the use of simulation software to replace the real system in the first hardware in the loop testing stage. For this, AMESim has been chosen [20]. Also, in the last years, small scaled embedded systems have become more and more accessible.

We can distinguish three classes:

 Microcontroller(based boards) as

Arduino boards;

 FPGA which are good for parallel computing;

 Processor based: as Raspberry PI, Beagle board, that act like small computers.

Each of them has certain advantages and disadvantages. In this article the authors have adopted the use of an Arduino Due board, and the development procedure with the hardware in the loop testing being described in [21].

The paper is organized as follows. Second section develops upon the Fuel Cell model. It is followed by a description of the TS representation and nonlinear observer design in the section III. Afterwards, in the fourth section, the embedded platform is described as well as the hardware in the loop (HIL) validation mechanism. The paper ends with results in section V respectively conclusions in section VI.

Modeling

PEM Fuel Cell

The developed model is for a Polymer Electrolyte Membrane Fuel Cell Stack that uses a Nafion 117 membrane, integrating auxiliary components as in Figure 1. The model focuses on components with medium time dynamics from a vehicle like configuration, meaning the focus is on the gaseous part. In vehicle applications, the auxiliary components play an important role, as the piping occupies significant volume.

The current work considers the case of pure Oxygen as input on the cathode side and takes the humidifying and cooling units as ideal elements. Concerning the pressure difference between the anode and cathode, it is kept null by means of a pressure regulator. A constant pressure difference has been seen to offer good performances in many cases.

The temperature inside the fuel cell is considered homogenous, so spatial variations in general are ignored, yet the temperature is not constant, compared to [22].

With the aid of the observer, only by measuring parameters outside the fuel cell will permit us to deduce what happens inside the Anode respectively cathode. Also the gases will be considered ideal.

State Space representation

The mathematical model of the FC is brought in a state space form considering the compressor's flow and electrical current as inputs to the system. Therefore the accumulated mass of oxygen and hydrogen react or pass freely towards the return manifold that consists of pipes and valves. The valve models for gaseous mediums have a nonlinear behavior with two distinct models depending on the pressure difference that arises: choked or unchoked regime. Therefore as the pressure difference may overpass 2 bar both situations have to be taken into account, this being another contribution of the paper. The TS representation will help with this, so that a switching between the two is not required. The equations for a volume chamber and a valve are given by ( 1)

  1 1 2 2( 1) 1 2 1 1 2 1 2 1 in out in a in in out out out out out crit in in in D T out out a in out crit in crit dm W W dt dp R T W T W dt V p p p if P p p p C A p W R T p if P p P                                                                                         (1)
where m is the mass inside the chamber, p in , T in and W in the input pressure, temperature and flow, p out , T out and W out the output pressure, temperature and flow, V the volume of the chamber, γ the adiabatic coefficient of air, Ra the specific perfect gas constant of air (Oxygen for the current case), C D the flow coefficient of the valve and A T the cross section of the valve. The input flow of the supply manifold will be considered as an input to the system with known temperature and pressure and the return manifold will flow toward exterior air.

Regarding the whole FC model, we will suppose that we measure the pressures and mass at the supply and return manifold, as indeed the measurement of the pressure and mass inside the cathode is expensive and impractical. Of course, in order to avoid measuring mass, one can have a pressure and temperature sensor in order to deduce the mass by means of the perfect gas law a PV mR T 

.

One can represent the equations for the cathode side of the Fuel cell as in (2) (the hydrogen side is identical), defining the dynamics for the mass flows of the Supply Manifold-Cathode-Return manifold, as well as the dynamics of the pressures of the same three elements:

      2
dm W W dt dm W W W dt dm W W W dt dp R T W T W dt V dp R T W T W T W dt V dp R T W T W T W dt V                                       (2)
Where the mass flows of the outputs of the Supply manifold, Return manifold and Cathode respectively the Mass flow that reacts in the cell are shown in [START_REF] Lukic | Energy Storage Systems for Automotive Applications[END_REF]. We can see the influence of the electrical current I, on the reacted mass flow depending on the number of cells(n), Oxygen molar mass and Faraday number.

p p p C A p W R T                                                                                  1 2 2( 1) 1 1 2 2( 1)
                                                                                        2 2 2 O O react n I M F                                    (3)
V p V p V T T T R m R m R m       
Therefore by doing a set of transformations and by considering the current as a system input along with the mass flow of the compressor we arrive at its state space equivalent in (4). The critical pressure is the same as the one presented in equation [START_REF] Kirubakaran | A review on fuel cell technologies and power electronic interface[END_REF].

    ( ) 1
                                            (4) 2 2 2 2 min 0 0 2 2 2 2 1 0 0 0 0 T O O O O a cs a r cs rm a smin sm n M n M n M n M R T R T F F F V F V B R T V                             (5)
The matrix A being the carrier of the nonlinear terms.

C A p m V C A p C A p m m V V A x C A p C A p m m V V p C A V V m p p C A V C A V V m V m                                2 1 2 2( 1) 2 2 (5) (5) (5) 1 (4) (4) (4) 2 (5) 1 (4) 2 (6) (6) 1 (5)
(5)

c rm Dcs Tcs cs Drm Drm sm rm c rm crit crit x p p C A V C A V V m m x x x if P x x x x if P x x x x x                                                                                         1 2 2( 1) 2 1 2 2( 1) (6) (5) 2 (6) 1 (5) 2 1 (6) (6) (6
P if P x                                                                                                                                                  (6)
For the values, please refer to the annex.

Observer Design Takagi-Sugeno Representation

The TS model is represented as a sum of nonlinear systems, where the contribution of each system is corrected by a weighing function w i , as presented in equation [START_REF] Matraji | Cascade control of the moto-compressor of a PEM fuel cell via second order sliding mode[END_REF]. Here, it is considered that B and C do not depend on the states and using the notation 'i' for the number of the decomposed linear system; also z represents the nonlinear terms, defined as premise variables.

i i i x w A x Bu y Cx            (7)
Therefore in this new representation, one can observe that the nonlinearities have moved into the membership functions w i , where the convex sum property is satisfied ∑w i =1.

In order to obtain the state observer, it is first needed to modify the state space form, so that it can be used to write the Takagi-Sugeno (TS) representation. So from equation ( 4), in order to get to the form [START_REF] Matraji | Cascade control of the moto-compressor of a PEM fuel cell via second order sliding mode[END_REF] we separate in the matrix A the z nonlinearities:

                                                           (8)
The variables z are called the premises variables, and they contain the nonlinear information for the system. By means of the sector nonlinearity method, as the minimum and maximum of each z i can be determined, the membership functions can be built.

From different types of possible membership functions, triangular membership function were chosen as in [START_REF] Petrone | A review on model-based diagnosis methodologies for PEMFCs[END_REF]; where MF is the membership function associated to the i-th premise variable z i . With these values, one is able to construct the normalized membership functions w i as in [START_REF] Steiner | Non intrusive diagnosis of polymer electrolyte fuel cells by wavelet packet transform[END_REF]. , ,

i Max i i i min i Max i i Max i i Max i z z z z MF z MF z z z z z       (9) 3 / 1 / 2 / 3 2 1 1..8 ( ) ( ) ( ) ; i Max Min Max Min Max Min i i i i i h MF z MF z MF z h w h                                         (10) min , ,min , ,min ( ) ; ( )

Takagi-Sugeno Observer

Once the T-S representation has been obtained, the focus can now be directed towards the observer construction. The observer computation will materialize as an optimization problem that will deal with the resolution of some linear matrix inequalities (LMIs). It is considered that the premise variables z are not measurable. Although many cases in the literature deal with the simplifying supposition of measurable premise variables w i (ẑ)=w i (z), in the current case this assumption cannot be satisfied. As a result, the Luenberger like observer is of the form (11).

 

ˆˆˆ( ) ( )

i i i i y C x x w z A x B u L y y                 (11)
To obtain the observer gains, a numerical optimization problem (12) should be resolved, with the aid of an LMI solver. This solution consists of the following LMIs (12); where P is a symmetric and positive definite matrix, with the same dimension as A; and α is a positive scalar:

1..8 2 0 ( ) ( . ) T T i i i i i i i i i A P C Q PA Q C P P P Q P L size Q size A C                            (12)

Demonstration

In the demonstration that follows, the known inequality (13) is employed:

1 0, , : T T T T X Y X Y Y X X X Y Y          ( 13
)
The condition required for the observer to function correctly is to have an estimation error that converges towards zero x x x    . Using the Lyapunov stability method, it is demanded that a chosen Lyapunov function has a negative derivative. The Lyapunov function is chosen as in (14).

; 0

T T V x Px P P      (14)
The derivative of the estimation error can be rewritten so that the estimated and real membership functions multiplied by the states are isolated in a separate term. This is done in order to reduce complications that would appear in the construction of the LMIs. Therefore (15) is reached.

     ( ) ; ( ) ( )
i i i i not i i i i x w z A x B u L C x w z w z A x B u                           (15)
By replacing (15) into ( 14), we can see in ( 16) that the multiplication with the state impedes obtaining a linear inequality. A usual workaround is to apply Lipschitz constants, but the method could prove restrictive, first of all because these values are hard to find and also high constants may have negative effects on the resolution of the LMIs.

    ( ) ( ) T i i i T i i i i i T T w z A L C P x w z P A L C Px x P V x                           (16)
As such, the solution adopted here is to view Δ as a virtual perturbation, a parameter that converges towards zero. This would allow us to impose more relaxed conditions, by demanding a bounded stability instead of an asymptotic one, with an as small as possible. An article with a similar method is [18], employing the notion of Input to state stability [24]. Practically this translates into a problematic of stability with rejection of the symbolic perturbation Δ, as in (17), having R as a positive matrix (identity or not), and a positive scalar  , that will be searched to be as small as possible.

2 0

T T V x Rx          (17)
Therefore, using (16):

    2 ( ) 0; T T i i i i i i T T T T x w z A L C P P A L C x Px x P x Rx                               (18)
In order to eliminate the nonlinear terms that remain in the previous inequality, we make use of the equation ( 13). What is important to note is that this is true for any positive λ, which for (18) translates into:

2 1 2 T T T T T T T T Px x P x Rx x PPx x Rx                           (19)
Adding this to (18) would give the modified inequality (20):

    1 2 ( ) 0; T T i i i i i i T T T e w z A L C P P A L C I e e PPe                             (20)
An interesting observation is that, because  >0 can be any coefficient, then it can be chosen as

T i i i i i i w z A L C P P A L C I PP              (21)
The general following stage is to use the hypothesis that the inequality is true if all the terms of the sum are negative. Although restrictive, this proves to bring small enough constraints to the stability regions of the solution.

    1 1..8 0;

T i i i i i A L C P P A L C I PP               (22)
In order to eliminate the Bilinear Matrix Inequalities (BMIs) forms, a notation i i Q PL  can be made. Also using the Schur transformation, we end up with simple LMIs. One can improve upon the performances by asking for exponential stability, using relation

T T V e Re V          ( 23
)
As for BMIs there are no standard solutions, a choice of  by trial can be made.

Thus LMIs (12) were obtained.

The Embedded Solution

The Arduino Board

The embedded platform chosen is an Arduino Due board with a 32 bit ARM microcontroller from Atmel with 84Mhz clock speed (Figure 3).

The power of the microcontroller comes from the fact that it is based on an ARM Cortex M3 processor (Atmel SAM3x8E). An Ethernet module is attached for data communication.

Hardware In the Loop testing

The integration of specialized simulation software can be done by means of hardware in the loop (HIL) architecture which permits within a closed loop with a physical embedded system, to test the code and performance of the developed platform. By means of AMESim simulation software, that includes a Fuel Cell model amongst other components, a real time connection can be established between the embedded system and the computer. For the computer, a Windows operating system has been chosen, because although it is not well suited for real time operation is the most widespread and known one in the academic medium.

In order to give real time capabilities to windows, Matlab Real Time Kernel and Simulink are used as an interface. A functional schematic is shown in Figure 4. The tested communication protocols were serial and Ethernet, offering an as general solution as possible. For more details see [21]. The AMESim Fuel cell model built is presented in Figure 5. The first few seconds where the observer appears inactive, it is due just to the initialization time of the devices and communication.

By following the evolution of the estimation error in Figure 7, one can notice that the stability is not asymptotic, but bounded, of course the bounds being sufficient to respond to requirements. Also one can notice that indeed in the beginning, the observer manifests oscillations before stabilizing, but the settling time is still in an acceptable interval.

One can observe in Figure 6, at time around 7 seconds for example, there is a communication loss, yet the observer manages to readapt the estimated values to the real ones. The few seconds delay at the beginning is just due to the initialization procedure of the embedded system. The main problem with this particular embedded system is that the dynamic of the valve is fast and that, for a sample time of more than 5ms, oscillations appear. Considering that we have 6x6 and 6x4 matrices, an 8 bit microcontroller arrives at a minimum sampling time of 10-15ms. Therefore we need to use the Arduino DUE, a 32 bit processor, which has 96kBytes of RAM. Furthermore, the Arduino DUE has a 84 MHz clock, allowing to do the floating point calculations for the oxygen supply in 4ms.

Conclusions

In this paper, we have developed a TS unmeasurable premise variable observer for determining parameters inside a PEM Fuel Cell which is able to estimate the pressures and mass flow rates. We also showed the capacities of Takagi-Sugeno modeling to be used on small scale microcontrollers with good sampling and in real-time which can be embedded on an electrical vehicle supplied by a FC system. This solution is more economical than the usual ECU. The real-time interface between AMESim, Matlab/Simulink and an Arduino connected by Ethernet that we developed represents a proper way to test any FC related embedded system and ensures that the observers behave as intended before using them on a real system. Using this connection allows to test any possible loss in the communication and design of the observer. Despite the memory restriction, the calculations remain fast and the limiting factor for real-time can be improved by parallel calculus as T-S is well adapted for it.

Annex

Fig. 1 .

 1 Fig. 1. Block view of fuel cell stack with auxiliary elements

(

  23), where  represents the exponential decay rate.

Fig. 3 .Fig. 2 .

 32 Fig. 3. The embedded HIL validation schematics

Fig. 7 .Fig. 6 .Fig. 5 .Fig. 4 .

 7654 Fig. 7. Real time observer values for the pressures of the oxygen supply of the PEMFC

Table 1 .

 1 Min Max values

	Symbol	Quantity
	min 1 z 	1	max 1 z 	1500

Table 2 .

 2 Parameter Values

	Symbol	Value		Symbol	Value
		1.4		Tc A	600 mm	2
	a R	259.8	. kg K J	Trm A	600 mm	2
	C	Dsm	0.2		sm V	20 L
	Tsm A	550 mm	2	c V	1 L
	sm V	20 L		rm V	20 L
	C	Dsm	0.2		in sm T	353.15 K
	C	Dc	0.2		in sm P	5 bar
	C	Drm	0.075		out rm T	273.15 K
	Tsm A	550 mm	2	out sm P	1.013 bar

http://www.sic.ici.ro

Studies in Informatics and Control, Vol. XX, No. X, Month Year

Acknowledgements

This work was carried out in the framework of PN-7-022-BE i-MOCCA (Interregional Mobility and Competence Centres in Automation), an European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IV A 2 Seas Programme and the Ministry of Education and Research, France.


S C Olteanu
email: sc.olteanu@ed.univ-lille1.fr

A Aitouche
email: abdel.aitouche@hei.fr

L Belkoura
email: lotfi.belkoura@univ-lille1.fr

Advanced embedded nonlinear observer design and HIL validation using a Takagi-Sugeno approach with unmeasurable premise variables

The article's goals are to illustrate the feasibility of implementing a Takagi Sugeno state observer on an embedded microcontroller based platform and secondly to present a methodology for validating a physical embedded system using a Hardware In The Loop architecture, where a simulation software replaces the process. As an application, a three water tank system was chosen. For the validation part, LMS AMESim software is employed to reproduce the process behaviour. The interface to the embedded platform is assured by Simulink on a Windows operating system, chosen as it is the most commonly used operating system. The lack of real time behaviour of the operating system is compensated by a real time kernel that manages to offer deterministic response times. The Takagi-Sugeno observer in the case of this process has the complex form that considers the premise variables to be unmeasurable. The embedded system consists of two Arduino boards connected in parallel, thus offering distributed resources.

Introduction

A consistent increase has been seen in the last decade in the usage of small scaled embedded systems in control, diagnostics and signal manipulation, as these solution become cheaper and more powerful [START_REF] Chen | Designing a spatially aware and autonomous quadcopter[END_REF]. Also, as the requirements of more complex algorithms augment, the study of the possibilities and the feasibility of implementing complex algorithms on these types of platforms becomes more and more useful. As such, we have implemented a nonlinear state observer using a Takagi-Sugeno (T-S) [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF].

Because it is preferred to test an embedded system prior to its implementation on a real world process in order to avoid unforeseen errors in modeling or communication, it is a useful practice to test the system in a hardware in the loop configuration, where the process itself is replaced by a simulation software, as the AMESim software [START_REF] Bourdon | Integration of Physical AMESim® Engine Model in Hardware in the Loop Environment, Dedicated to Engine Control Unit Testing[END_REF], [START_REF] Feingesicht | HIL validation of an embedded system acting as a nonlinear Takagi-Sugeno state observer on an Arduino board[END_REF]. For the embedded part, based on versatility and size/price ratio, a microcontroller platform (on an Arduino board) is employed.

The T-S methodology, that can be seen as a polytopic linear differential inclusion, has seen a growing interest in the last decade as it has a simple implementation mechanism, as well as a simple structure. A common solution for obtaining a TS model from a general nonlinear one, is to apply a sector nonlinearity transformation [START_REF] Lendek | Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models[END_REF] to the state space form, thus giving an exact approximation. This model consists of a blend of linear systems that are coupled together by means of weighting functions.

As an application, a nonlinear analysis of a three tank system has been selected as it represents a basic example that brings some interesting complexity when looked upon from a nonlinear perspective. After the construction of the state space model, the T-S transformation is applied, and for the newly built representation, the observer is developed. A similar approach, also with unmeasurable premise variables can be seen in [START_REF] Ghorbel | Robust Tracking Control for Takagi-Sugeno Fuzzy Systems With Unmeasurable Premise Variables: Application to Tank System[END_REF]. Others have used the application in nonlinear studies, even fault diagnosis like in [START_REF] Rincon-Pasaye | Fault diagnosis in nonlinear systems: An application to a three-tank system[END_REF].

The article begins in section 2 with a general description of the T-S representation and the construction of the observer, followed in section 3 by the embedded system description as well as the HIL validation technique. Section 4 deals with the application process, meaning the three tank system's modelling and its TS representation. Section 5 presents the implementation of the observer for the system both in simulation and on the physical process. The paper ends with a conclusions section.

Takagi-Sugeno Observer

Takagi-Sugeno Representation

A general TS model presents itself as a sum of linear systems as in [START_REF] Chen | Designing a spatially aware and autonomous quadcopter[END_REF], where each system contributes in a percentage given by the weighting function w i . Here we consider B and C as non dependent on states and we consider i as the number of a fuzzy rule (this representation may also be seen as a fuzzy system).

i i i x w A x Bu y Cx z               (1)
It becomes obvious that this way of writing the system moves the nonlinearities from inside the system matrices into the weighting functions, adding the fact that they respect a convex sum property:

1 i i w   (2)
To arrive from the general nonlinear form to a TS representation the most common procedure is to apply sector nonlinearity technique. But first of all, the system is rewritten in form (3):

(z( ))

; x A x x B u y C x           (3)
The z variables, that are dependent on states, have the name of premise variables, and they contain the nonlinear information for the system. Now the sector nonlinearity method [START_REF] Herrera-Lopez | Observer for a Switching Bioprocess: Sector Nonlinearity Approach[END_REF] can be applied by determining the minimum and maximum of each z i . If we consider a general triangular membership function one arrives at (4) where we design MFs as membership functions:

, , ,min ,min , ,min / 1
i z z MF z z z z z MF z z z w MF z MF z                      (4)

Takagi-Sugeno Observer

i i i i i i i i y x w z A x B u L y y w z C x              (5)
As it can be seen, the weighting functions depend on estimated premise variables z, therefore the computations are more complex. In the literature, there are many cases where premise variables are directly measurable as in [START_REF] Lendek | Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models[END_REF], a supposition that indeed simplifies the stability study ( w i (ẑ)=w i (z) ). This is not the case for the current work.

In order to construct the observer (6), we need to calculate the observer gains. This will be done by employing Matlab's implicit LMI solver or YALMIP toolbox to resolve the LMIs presented in [START_REF] Ghorbel | Robust Tracking Control for Takagi-Sugeno Fuzzy Systems With Unmeasurable Premise Variables: Application to Tank System[END_REF].

T T i i i i T i i i i i A P C Q PA Q C I P P P size P size A P P Q P L size Q size A C scalar                                        (6)

Proof:

As the goal of any observer is to reach stability in 0 of the estimation error ê x x   , the methodology adopted is Lyapunov stability analysis. This dictates to impose condition (8) upon a well chosen Lyapunov function. We have chosen a quadratic one as in [START_REF] Rincon-Pasaye | Fault diagnosis in nonlinear systems: An application to a three-tank system[END_REF].

; 0

T T V e Pe P P    (7) Its derivative being: 0 T T Pe e Pe V e       (8)
By developing the expression of the estimation error's derivative as in [START_REF] Yacine | Vehicle nonlinear observer for state and tire-road friction estimation[END_REF], one can rearrange it to get expression (10).

       ˆ( ) ( ) ( ) ( ) ( ) ; i i i i i i i i i i i i i i e w z Ax Bu w z w z Ax Bu w z Ax Bu L y y              (9)      ( ) ; ( ) ( ) i i i i i not i i i i i e w z A e B u L C e w z w z A x B u                        (10)
We have separated the Δ term as it contains a difference between measured weighting function and estimated one, this expression posing a problem in the resolution of the stability requirement. A solution seen generally is to use Lipchitz constants, yet this solution is quite restrictive for systems where the constant is big. Therefore we adopted another method, which is to consider Δ as a perturbation, a value that converges to 0 when t→inf. Therefore the stability problem passes from an asymptotic one to a bounded stability, where the bound is as small as possible. Similar approach has been treated in [START_REF] Yacine | Vehicle nonlinear observer for state and tire-road friction estimation[END_REF].

From equation ( 8), making use of (10), one obtains: Moreover, imposing stability with rejection of the virtual perturbation Δ, (12) is reached.

    ˆ( ) ( ) T T T T i i i i i i i i e w z A L C P w z P A L C e Pe e P V                  (11)
2 0 T T V e Re        (12)
Where R>0, usually taken as the identity matrix and  is an small as possible positive coefficient. This leading to:

    2 ( ) 0; T T T T T T i i i i i i e w z A L C P P A L C e Pe e P e Re                         (13)
These inequalities still contain variable terms multiplied with other variable terms, so it still has nonlinearities in the last part of the sum. In order to eliminate this, the following known inequality is employed, which is available for all positive λ: 1 0, , :

T T T T X Y X Y Y X X X Y Y          (
    1 2 ( ) 0; T T T T T i i i i i i e w z A L C P P A L C I e e PPe                            (16)
As  >0 can be any scalar, then one can choose 2    , therefore considering also the convex sum property, the inequality becomes:

    1 ( ) 0; T i i i i i i w z A L C P P A L C I PP              (17)
To finalize and to obtain a set of LMI, we use a restrictive hypothesis that each term in the sum is negative, and as w i are all positive, then (18) is reached.

   

1 For 1: 8 0;

T i i i i i A L C P P A L C I PP               (18) Using the notation i i Q PL 
and making use of the Schur transformation, we thus avoid a Bilinear Matrix problem (BMI).

Also, optionally we can improve the performance by imposing exponential stability, by replacing (12) with ( 19), where  is the exponential decay rate.

2 2 0 T T V e Re V          (19)
Of course this would also generate a BMI problem, but this can be easily resolved by choosing manually by trial, a suitable  . So from (18). Adding the exponential stability, we have proven the LMIs (6).

The Embedded system and HIL validation platform

The embedded platform is based on two Arduino Mega 2560, as the number of input/outputs of the ATMEGA 2560 is bigger than the standard UNO version. The two microcontrollers are connected in between themselves as in figure 1. The connection can be done both by serial transfer or parallel one (by digital pins). In the application chosen in the current article both configurations give acceptable results, yet it has to be taken into account that floating point data is transferred (stored on 16 bits), so the serial communication has a slower transfer time.

The implementation of the observer on the microcontroller and possible unpredicted errors can be tested by a hardware in the loop (HIL) architecture, where the microcontrollers based platform is connected to a computer that hosts the simulation software (AMESim). The communication protocol between the computer hosting the software and the embedded system has been chosen to be a general one, adaptable for any situation, therefore an Ethernet UDP connection fits the requirement. For this, a 5 ms sampling time can be reached at the 16Mhz of the microcontroller. An important inconvenient of microcontroller systems is that the floating point operations are done at software level, increasing therefore the processing power.

Figure 1. Hardware implementation

The basic functional schematic at the base of the implementation of the embedded HIL validation structure is shown in Figure 2. Although the operating system is Windows, which by default does not assure real time operation, by using the Matlab Real Time Kernel, only the processing threads of interest to the task at hand are kept at a top priority level being overpassed just by core root threads like system inputs and operating system messaging interrupts.

As AMESim needs Simulink for operating a real time data exchange, the creation of a cosimulation AMESim/Simulink is required. As the co-simulation is built, the Real Time Windows Target toolbox in Simulink manages the compilation of the program that will be executed in the Real Time Windows Kernel of Matlab.

Figure 2. Functional block of the real time system

The coding done in the Arduino IDE is embedded C language, more robust for larger problems than assembly language. A variable discretization is used for the observer, meaning it is recalculated each sampling time inside each microcontroller, this helping in talking the problem of lost samples

The time lost because of the computations done in floating point (which is done at software level) and also due to the fact that TS model is represented as a sum, gave us the possibility to implement a distributed solution, where 2 processors divides the workload at each sampling time. The logical diagram is presented in Figure 3.

The application

System description and modeling

In order to show the functioning principles, an application example has been chosen, specifically a system of three interconnected water tanks. The system is analyzed from a nonlinear perspective to show the potential of using T-S techniques on an embedded platform. As it can be seen in Figure 4, the height of the first tank and the third tank has been considered measurable while the second tank's height is the estimated parameter. In this system, we have an input pump supplying the first tank Q p1 as well as another pump for the third tank Q p2 . The last tank presents a valve evacuating water outwards, where atmospheric pressure P atm is encountered. Also, a valve between it and the middle tank is present. The second tank has only the two interconnection valves while the first tank presents only one valve supplying the second tank. A supposition is made, that the flow of the third tank pump is smaller than the first one, thus the levels in tanks will always remain in a descending step, the first the highest level, then the second and the smallest water level in the third at all times. ; ;

P P P P P d V Q Q dt d V Q Q dt d V Q Q Q dt V S h V S h V S h
                                          (20)
We will use the notations h i as heights of liquid in each tank and Cp the flow coefficient of each valve, all valves being considered identical. By neglecting the atmospheric pressure, equations (20) are found. Developing equation (20), the system equations become (21).

   

 

dh Q C h h g h h dt S dh C h h g h h C h h g h h dt S dh Q C h h g h h C g h dt S

                             (21)
Now, the equivalent state-space form can be built, considering S 1 = S 2 = S 3 and CP 12 = CP 23 =CP 3 . As it can be seen, the inputs are the pump flows, and the states are the three heights and finally the output is the first and last tanks' height: ( )

P P Q u Q h x h h h x y h x h x                                                 (22) 1 2 1 2 1 2 1 2 2 3 2 3 2 3 2 3 3 1 3 1 0 sgn( ) sgn( ) sgn( ) 0 0 1 sgn( ) 0 1 0 0 0 0 1 P h h h h S C g x h h h h h h h h u S h h h h h S x y x x                                                                     (23)
As it can be seen, the nonlinearities are present as square roots, that have no differentiability in 0, yet for the regime in which we work, we will not reach this point.

Takagi-Sugeno transformation

In order to obtain the state observer, it is first needed to modify the state space form, so that it can be used to write the Takagi-Sugeno (TS) representation. So equation ( 23) is written as (24):

0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 P z z C g x z z z z x u S S z z z y x x                                                 (24)
Where:

1 2 1 2 1 2 2 3 2 3 3 3 1 sgn( ) 1 sgn( ) 1 x x x x z z z x x x x z x                                     (25)
For the sector nonlinearity transformation we have to establish the conditions for the heights, where some additional constraints are added so that singular values are eliminated:

h m h m h m            (26)
The three valves have similar characteristics, and the flow on the left pump is bigger, therefore the only possibility when the water levels would be equal is when the pump starts or when the tanks are empty, in which case the observer is not useful. Therefore, it is impossible that the difference between levels of the 3 tanks will change their sign. Even if this had happened, one could resolve this problem, by separating the system into multiple systems that switch according to the value of the difference between heights. Once all the observers are built, the one that will be active is chosen in a switching manner. Of course the main regime is the case when all the square roots are different from 0.

The parameters for the system are: (28)

Observer Implementation

The AMESim System

For the validation part, one has to first construct the AMESim model, by adding all the elements, configuring their parameters, and then the co-simulation block with Simulink needs to be added in Amesim as well as in figure 6. For the Simulink part (Figure 5), Matlab 2012a have been used. The Amesim version is R12. As the connection between Amesim and Simulink is done, by using Matlab real time workshop we add the input/output blocks that will connect via Ethernet to the microcontroller. The real time compilation is then done, and the whole system is run from Simulink.

The real time simulation

By means of the experimental setup described previously, a performance analysis is done in real time, and the parameters are configured for optimal results. As it can be observed in figures 7 and 8, the estimation of the water level in the second tank stabilizes in about 10 seconds, which is an adequate time response. Also the overshoot of the estimation is in an acceptable interval. It is important to notice that perturbation added to the communication in a forcedly manner, can be seen to affect the system like in figure 7 at second 0.7 (in the zoomed pane), but performance remains good overall, thus proving its robustness to data loss.

Validation on a real system

Once the tested had been finished the embedded observer has been installed on a real 3 tank platform AMIRA DTS200 (Figure 9). After the implementation, also adding some filters on the inputs and outputs, the estimation error stabilizes in 8 seconds as can be seen in the following figure 10:

Conclusions and Perspectives

An embedded nonlinear observer was developed using the Takagi Sugeno approach and applied to a three tank system with a prior real time HIL validation. As previously seen, the observer's performances on the HIL testing bench reproduce the behaviour of the real system thus reducing implementation time and also avoiding potential dangerous faults on the real system. The testing also takes into consideration communication losses and memory and computation timing errors. Of course the testing itself can be reproduced for any process that is supported by the simulation platform. The developed observer was applied to a real three tank system to confirm the results. Also another objective of the paper was to show the capability of a T-S system to be represented on small scaled microcontroller platforms. By doing the parallel calculus in between two microcontrollers, the complexity of the system can be augmented even further.

For

Figure 3 .

 3 Figure 3. Logical diagram of the Arduino code

Figure 4 .

 4 Figure 4. Three tanks system representation

Figure 5 .Figure 6 .

 56 Figure 5. Simulink diagram including co-simulation block

Figure 7 .Figure 8 .

 78 Figure 7. Estimation error dynamics of the 3 heights (tank 1, 2 and 3)

Figure 9 .

 9 Figure 9. AMIRA DTS 200

Figure 10 .

 10 Figure 10. AMIRA DTS 200 estimation error dynamics in time By introducing short perturbations, into the system, by means of purge valves openings, the observer still manages to regain stability (as in figure 11):

Figure 11 .

 11 Figure 11. AMIRA DTS 200 Where the numbered areas are: 1) Leakage in tank 1 (fault 1) 2) Leakage in tank 2 (fault 2) 3) Leakage in tank 3 (fault 3) 4) Pump of tank 1 does not work (no input flow) (fault 4) 5) Pump of tank 3 does not work (no input flow) (fault 5) 6) Valve between tank 2 and tank 3 is blocked (fault 6) 7) Valve between tank 3 and atmosphere is blocked (fault 7) 8) Valve between tank 1 and tank 2 is blocked (fault 8).

Acknowledgments

This work was carried out in the framework of PN-7-022-BE i-MOCCA (Interregional Mobility and Competence Centres in Automation), an European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IV A 2 Seas Programme and the Ministry of Education and Research, France.


Robust state estimation of P.E.M. fuel cell systems

Keywords: PEMFC, State Estimation, Non-Linear Observer, Uncertainties

. The model presents some simplifications considering only pure Oxygen as input and also an ideal humidifying and cooling unit. The numerical problem of the disequilibrium between small mass values and big pressure values is resolved by constructing a symmetric system.

INTRODUCTION

In the last decade a growing interest can be seen in the Fuel Cell technology, presenting itself as one of the most efficient alternative power generation technologies. Fuel Cell Stack(FCS) development has generated much interest in both research and industrial areas such that, although still expensive for mass production, the expectations for the future show a change in this aspect. Fuel Cells (FCs), act as ideal power sources, being based on an inverse electrolysis, converting chemical energy into electrical power. They present: lower/zero emission, silent functioning, high efficiency, fast refill time in comparison to batteries. The type of FC analyzed here, Proton Exchange Membrane Fuel Cell(PEMFC) was invented at General Electric in 1960s is still the most popular membrane in use today [START_REF] Larminie | Fuel cell system explained[END_REF]. A FC produces electricity, water, and heat, thereby eliminating pollution at the energy conversion. Although the research in the field of modeling and control of fuel cell systems has known a great boast, there is still a lack of FCS models appropriate for control and diagnosis research, especially in state space form, many diagnosis solutions make use of spectral analysis. Two types of models are known: steady state FCS models and dynamic FCS models. The former ones serve as a mean to design FC components and to choose the operating points. On another hand, even if they are not suitable for control and diagnosis studies, they are handy in establishing the effects of different parameters as pressure, temperature on the fuel cell voltage. Whereas, the latter are described by multiple variables and a strong coupling with profound dynamics. Some dedicated articles have been presented: -Turner et al [START_REF] Turner | Modeling a PEM fuel cell for use in a hybrid electric vehicle[END_REF] included the transient effect of fuel cell stack temperature in his dynamic model. -Pukrushpan et al [START_REF] Pukrushpan | Control of Fuel Cell Power Systems, AIC (Advances in Industrial Control)[END_REF] presented a nonlinear fuel cell system dynamic model suitable for control study. The flow characteristics and inertia dynamics of the compressor, the manifold dynamics, and consequently, the reactant partial pressures were included in the transient phenomena captured of the model he proposed.

In general, many real life system present some parameters that are hard to measure directly by means of sensors so these states are required to be estimated; For example in a Fuel Cell Stack there are many small cells grouped together in a stack and to measure the parameters of each incorporated cell is unsatisfactory. Also, observers can be built to estimate those variables that can generate residuals used to detect some eventual faults in the system. The observers need to be tolerant to perturbations. In literature, there are many methods for general state estimation for non-linear systems utilizing different methods some based on a Luenberger like observer and other; the effective theories for adaptation of a non-linear structure to a linear one show also a broad variety like Sliding mode observers(reference), observers based on Fuzzy Logic as Takagi-Sugeno approach where the system model is transformed into a sum of linear models, each of them scaled according to their weighting function( i w ) as in [START_REF] Olteanu | PEM fuel cell modeling and simulation via the Takagi-Sugeno fuzzy model[END_REF] and those that use the Mean value theorem approach. In many of the existing observers, the computations are manged using Linear Matrix Inequalities (L.M.I.s) to satisfy the stability of the estimation error's dynamics.

The paper is organized as follows: Section 2 will cover the PEMFCS modeling, after which Section 3 will treat the theory behind the observer, based on the Mean Value Theorem. After that in Section 4 the method will be applied to the PEMFCS. Finally conclusions and perspectives are drawn.

MODEL DEVELOPMENT

Modeling of a fuel cell system concerns multi-physical domains using electrochemical, thermo dynamical, electrical and fluid mechanical principles. In this paper, we focus on the study of the most generally used PEM fuel cell system which operates at low temperatures between 50 and 100 degrees Celsius. The fuel cell is a volumetric capacitor including two electrodes named as cathode (ca) and anode (an) which sandwich an electrolyte inside the proton exchange membrane with bipolar plates called Membrane Electrode Assemblies (MEA).The purpose of this work is to focus on the auxiliary elements, and less on the electrical part of the model(the dynamics being at a different scale): Air compressor (with an electrical DC motor); Hydrogen tank (with a control valve attached); Supply manifold (including the volume of pipes from the compressor plus the volume of the cooler and the volume of the humidifier); Return Manifold(the pipe volume at the Fuel Cell's exhaust); Cooling Unit (represents a circuit of de-ionized cooled water system). The application is concentrated on vehicle applicable Fuel cells (this would also take into consideration the adaptability towards embedded algorithms). Graphical description in figure below. The used model is based on [START_REF] Pukrushpan | Control of Fuel Cell Power Systems, AIC (Advances in Industrial Control)[END_REF]. More details can be found in [START_REF] Olteanu | PEM fuel cell modeling and simulation via the Takagi-Sugeno fuzzy model[END_REF]. The obtained system model can then be used in the end both for fault detection and control [START_REF] Aitouche | Fault Detection and Isolation of PEM Fuel Cell System based on Nonlinear Analytical Redundancy. An Application via Parity Space Approach[END_REF]. The State space representation is the following: ( ) ( ) ; ;

x x c x F x x G x u F y C x            ; ; ; ; n p m x R y R u R    2 2 O H sm sm rm m m x m p p                  ; , st T rm I u A        ; sm rm p y p       
; y=direct measurable values;

4 7 0 ; 1 1 0 cst cp c cst cp f W F f                                 0 0 0 1 0 ; 0 0 0 0 1 C          2 , , 5 3 , 1
f k k f f F x k f k z f f z f z f f f                            2 2 10 3 0 4 0 2 ( ) 0 0 0 0 0 O H x cst n M F n M F G x f z                                           
;

The model has 5 states, that includes the return manifold pressure, the supply manifold pressure and gas masses of Oxygen, Hydrogen and mass of Oxygen in the supply manifold; the pressures are considered measured. As inputs we consider the stack current and the return manifold evacuation nozzle's surface. With 'z', we specify the nonlinear terms (their values are presented in the Appendix).

The other parameters are constant values also presented in the Appendix.

Fig. 1 The PEMFC system with auxiliary elements

OBSERVER DEVELOPMENT 3.1 General Observer

A general Luenberger observer is adopted in this paper, observer for which it has been utilized the Mean Value Theorem / Bounded Jacobian approach to integrate the nonlinearities.

The system has to be of the special form, showed in parallel to the associated observer:

( ) ( , ); ;

x A x x g y u y C x              ˆˆˆ( ) ( , ) ( ); ˆˆ; x A x x g y u L y y y C x                 (1) ˆ=estimated states; Observer's Gain Matrix; y=Real Measured Output; =Estimated Output; x L y 
To calculate the gain matrix L we search for a P symmetric, K, so that the following LMISs present a solution for all i ,j <=number of system states:

max max min min ( ) ( ) 0; ( ) ( ) 0; 0; 0; T T T ij ij T T T ij ij P A H A H P C K KC P A H A H P C K KC P K                      (2)
i i ij ij j j T T ij n n ij ij n n ij T n Where h h x x

H Z e i e j h H Z e i e j h e i Z n n n

                          
Proof: We start from the estimation error dynamics, that becomes:

( ) ( ); x A L C x x           Where ˆ; x x x     ˆ( ) ( ) ( ); x x x      
(3) The Lyapunov function candidate may be defined as ; 0;

T n n V x P x P P        
 where "n" is the number of system states.

[( ) ( )] ;

T T T T V x A L C P P A L C x x P P x                         (4)
We want 0; V   yet we know from the Mean value theorem [START_REF] Phanomchoeng | Nonlinear Observer for Bounded Jacobian Systems with Applications to Automotive Slip Angle Estimation, Automatic Control[END_REF] that :

      , , max max min min , , ˆn n n n ij ij ij ij i j i j x x H H x x                                     (5)
Where

;

n n T T ij ij i j T T n n n n n n ij ij ij ij ij ij i j i j i j V x A L C P P A L C P H P H H P H P x                                                       (6)
After some other computations, because max min

1 ij ij     ;
we can write sums for all the terms in [START_REF] Olteanu | PEM fuel cell modeling and simulation via the Takagi-Sugeno fuzzy model[END_REF].

Afterwards for eliminating the max min , ij ij   terms that are unknown we make the restrictive hypothesis that if all the terms in a sum are less than 0 then their sum is les than 0. We consider

1 ; K P L L P K      
So we will reach the LMIs mentioned in (2).

Robustness

We can also take into consideration some additive perturbations both of the system dynamics and the output. This has been achieved in a similar manner with [START_REF] Zemouche | A New Observer design method for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time-Delay. Extension to H∞ Performance Analysis[END_REF], by considering additive perturbations. This is resolved using H infinity performance. The gain matrix can be reached by finding a matrix P>0, K>0 and a lambda>0 scalar, so that the following LMIs have a solution:

        max max 1 2 1 2 2 min min 1 2 1 2 2 0 1 0 1 0 T T T ij ij T T T T ij ij T H A P C K P H A K C I P W K W P W K W I H A P C K P H A K C I P W K W P W K W I P                                                                                         (7)
Proof: We start from the general system format presented below:

1 2 ( ) ( , ) ( );
( );

x A x x g y u W w t y C x W w t                  (8) 1 2 1 2 [ 0]; [0 ]; ( ) [ ] T W E W D w t w w   
to express it in the general state space format.

By using Lyapunov properties, the following conditions have to apply in order to reduce the effects of the perturbations upon the system: lim ( ) 0 for w(t)=0;

t x t    and 2 2 2 ( ) ( ) for w(t) 0 and x(0)=0 L L x t w t      

;

So we have to find a positive scalar Lambda where 2 0

T T V x x w w         (9)
The observer form remains (1), but the estimation error dynamics becomes:

1 2 ( ) ( ) ( ) ; x A LC x x W L W w              ˆ; x x x       ˆ; x x x    ˆ( ) ( ) ( ); x x x       (10)
The Lyapunov function candidate may be defined as ; 0; ;

T n n V x P x P P         
Considering also (11) we obtain:

1 2 1 2 [( ) ( )] ( ) ( ) ; T T T T T T T V x A L C P P A L C x x P P x w W L W P x x P W L W w                                        (11) From (11) and (9)=> 2 2 1 2 1 2 [( ) ( ) ] ( ) ( ) ; T T T T T T T T T T V x x w w x A L C P P A L C I x x P P x w W L W P x x P W L W w                                                    (12)
Writing it as a Matrix we will have:

    1 2 2 1 2 [( ) ( ) ] ( ) ; ( ) T T T T T T A L C P P A L C I P W L W x x w x P P x w W L W P I                                                   (13)
Yet from the mean value theorem [START_REF] Phanomchoeng | Nonlinear Observer for Bounded Jacobian Systems with Applications to Automotive Slip Angle Estimation, Automatic Control[END_REF], using [START_REF] Aitouche | Survey and Analysis of Diagnosis of Fuel Cell Stack PEM Systems[END_REF] and knowing that max min

1 ij ij     we conclude :       , , max max max min min min 1 2 , 1 , 1 2 1 2 [ ( ) ( ) ( ) ( ) ] ( ) 0;
( )

n n n n T T ij ij ij ij ij ij i j i j T H A L C P P H A L C I H A L C P P H A L C I P W L W W L W P I                                                                      (14)
Using the Schur transformation of Matricial inequalities and rearranging the terms it resultes that

    , m a x m a x m ax 1 2 1 2 2 , 1 m in m in m in 1 2 1 2 2 , 1 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) n n T T ij ij ij i j T T ij ij ij i j H A L C P P H A L C I P W L W I W L W P H A L C P P H A L C I P W L W I W L W P                                                                                        , 0; n n   (15)
Further, we can make an assumption, that is indeed restrictive concerning the solution, method already implemented in [START_REF] Phanomchoeng | Nonlinear Observer for Bounded Jacobian Systems with Applications to Automotive Slip Angle Estimation, Automatic Control[END_REF]: Statement: If each term in the upper sums is negative then the whole negativity rests negative.Than we can transform the inequality in multiple inequalities(also taking into consideration ).For all i,j, using the inverse Schur format, and considering

1 ; K P L L P K      
we arrive at the wanted LMIs.

APPLICATION OF THE OBSERVER ON A PEMFC MODEL 4.1 General System Format

By applying the theory to our system we change the system format so that we have the general form required (1).

                 ; , st T rm I u A        ; sm rm p y p        ; y=direct measurable values;   2 , , 5 3 , 1 , 8 7 6 1
f k k f f k f k A f f f                          g(y,u)= 2 2 10 3 4 2 0 0 O st H st cst Tm n M I F n M I F f z A                                              ; 4 7 0 ; 1 1 0 cst cp cst cp f W Fc f                                  
                                    ; 0 0 0 1 0 ; 0 0 0 0 1 C       
The general observer schema will be in consequence:

Fig. 2 General Observer Schema adapted to our FC System

Observability

The condition of observability has to be satisfied by the virtual linear system determined by A and C and all max,min , ( )

i j
A H  . C = [0 0 0 1 0; 0 0 0 0 1]=> Observability_Matrix will not have a full rank, so we apply a procedure to force the observability. Also if we look at the eigenvalues of A(that will affect the consequent numerical computations), the results are scattered at different scales.

For this we add some terms to matrix A and then subtract the same terms from matrix ( ) x  ; A [START_REF] Phanomchoeng | Nonlinear Observer for Bounded Jacobian Systems with Applications to Automotive Slip Angle Estimation, Automatic Control[END_REF][START_REF] Phanomchoeng | Nonlinear Observer for Bounded Jacobian Systems with Applications to Automotive Slip Angle Estimation, Automatic Control[END_REF] and A [START_REF] Aitouche | Fault Detection and Isolation of PEM Fuel Cell System based on Nonlinear Analytical Redundancy. An Application via Parity Space Approach[END_REF][START_REF] Aitouche | Fault Detection and Isolation of PEM Fuel Cell System based on Nonlinear Analytical Redundancy. An Application via Parity Space Approach[END_REF] are chosen specifically to increase the A's Eigenvalues. While the others, to increase the rank of the Observability matrix of the pairs (A,C). Also the values chosen are carefully picked so that we can do a numerical modification to the LMIs so that the calculations will not have ill posed matrices. A(4,2)=3.21*10^8; A(4,3)=5.4321*10^7; A(3,3)=13.3; A(4,3)=69; And the eigenvalues of the new A are also normalized by the upper procedure.

Bounding the Jacobian Matrix

Now we calculate the jacobian, and so we reach the following matrix:

cst cst cst cst cst cst J x z f f z z f z f z f z f z                                                    (15)
We now find the minimum and maximum of each term. This has been done by means of simulating at different initial conditions. The following possibilities were tested:

          1 0 .0 0 5 0 .0 6 2 1 0 ^4 2 * 10 ^3 3 0 .01 0 .1 2 4 5 * 1 0 ^4 6 * 1 0 ^5 5 5 * 1 0 ^4 5 * 1 0 ^5 x x x x x             =>       4 ,1 8 4 , 2 4 ,3 4 , 4 3
H                       
e 009 e

H                         
e 00 e 00

(16)

Building a Symmetric System

For the Luenberger Observer's Gains, we need to perform a system of 8 inequalities. The problem is that the matrices are ill posed (amplitudes of different scales).To solve this we apply a mathematical numerical solution: We multiply the inequality left and right with a matrix on each side. The matrices will be diagonal, and the values of each of them has to be chosen, following all the possibilities of

min i A H  respectively max i A H 
. In the end we choose(by means of experiment):

7 7 1 7
1 10 0 0 0 0 0 10 0 0 0 ; 0 0 10 0 0 0 0 0 10 0 0 0 0 0 1

P                    7 7 7
10 0 0 0 0 0 10 0 0 0 ; 0 0 10 0 0 0 0 0 10 0 0 0 0 0 1

P                     (17)
The symmetry is to be taken into consideration

; t P P 
We apply the similarity transformation, which implies that if we consider a new state as being 1 ; x P x     then the system becomes :

1 1 1 1 1 1 1 ( ) ( , ) ;
; P x P A P P x P P x P g y u P Fc

y C P P x                                => 1 ( ) ( , ) ; ; c x A x P x g y u F y C x                         (18)
If we make the notations:

1 ; x P x     y y   ; ; C C P    1 ; g P g     1 ( ) ( ) ; A P A P      1 ( ) ; c c F P F    
Important to be noted the behavior of ( ) x  , dependent on x, and not the new state, because the transformation of x is not possible being situated in nonlinear equations. The now observer becomes:

1 ˆˆ( ) ( , ) ( ) ; ˆˆ; c x A x P x g y u L y y F y C x                              (19)
The estimation error dynamics is

( ) ( ); x A L C x x               ˆ; x x x       1 ˆ( ) ( ( ) ( )); x P x x        
Then the Lyapunov function candidate may be defined as 5 5 ; 0;

T V x R x R R            => [( ) ( )] 0; T T T T V x A L C R R A L C x x R R x                                 
(20) Considering as previous the Mean Value Theorem [START_REF] Aitouche | Survey and Analysis of Diagnosis of Fuel Cell Stack PEM Systems[END_REF] and also taking into account that Jacobian( 1 ( )

P x    ) = 1 ( ( ) ( )) P Jacobian x x     
; by oversimplifying the notations we will have

  1 1 ˆˆ( ) ( ( ) ( )) ( min, max( ) ( ); x P x x P Sum H x x x              But yet we need ˆ; x x x       =>   1 1 ˆ( ) ( min, max( )) ( ); x P Sum H x P P x x          =>     1 ( ) (
T T T ij ij T T T ij ij R A H A H R C L R R LC R A H A H R C L R R LC R                                 (21)

Applying robustness to the observer

Starting from the attached theory previously presented, we will have the system:

1 1 ( ) ( , ) w(t); w(t); x A x x g y u Fc W y C x W                 (22) where 2 0 0 0 0.05 0 ; 0 0 0 0 0.05 W        1 0 0 0 0 0 ; 0 0 0 0 0 W       
If we make the notations:

1 ; x P x     y y   ; ; C C P    1 ; g P g     1 ( ) ( ) ; A P A P      1 ( ) ; c c F P F     1 1 1 ( ) ; W P W     2 2 ( ) ; W W  
we will therefore obtain the LMIs (22):

        m a x m a x 1 2 1 2 2 m in m in 1 2 1 2 2 0 1 0 1 0 T T T ij ij T T T T ij ij T H A R C K R H A K C I R W K W R W K W I H A R C K R H A K C I R W K W R W K W I R                                                                                                                   (23)

Results

The observer has been implemented in Matlab/Simulink environment. The inputs of the system are: Electrical current steps and step like variations of the Surface nozzle of the return manifold. The evolution of the pressure in the supply manifold in Fig. 3 shows the system's robustness(in the green rectangle it is shown the difference of the estimation error between the robust and non-robust case).

CONCLUSIONS AND PERSPECTIVES

Not many state space models exist in the scientific literature for the auxiliary elements. Many existing models are experimental or mathematical-physical but are not concentrated on the dynamics and more on the static equations implied by the Fuel Cell. Also this model possesses two interesting aspects: the high number of states and nonlinearities and also the high difference in magnitude of the parameters implied in the model (Pressures vary around 10^5 and Masses vary around 10^-3); these bring a generality to the estimation method so that it can be extended to a model of a higher degree of complexity. So this work brings an important contribution to F.C. observer development applicable even for a more complex and complete F.C. stack model.

As further development of this work, it is envisioned a validation of the system model and observer implementation using AMESim and after that, using a real F.C. Stack situated in an University in Lille.

Other methods for estimation better suited for the Fuel Cell are to be analyzed.

Fig. 3

 3 Fig.3 Evolution of the Pressure in the Supply Manifold

Fig. 4a

 4a Fig.4a Mass of Oxygen, real and estimated evolution Fig.4b Mass of Oxygen, estimation error in the presence of initial state difference and a perturbation evolution

Fig. 5

 5 Fig.5 Hydrogen mass evolution Fig.6 Return Manifold Pressure evolution (real/estimated) (real/estimated)


M Feingesicht
email: maxime.feingesicht@hei.fr

S C Olteanu
email: sc.olteanu@ed.univ-lille1.fr

A Aitouche
email: abdel.aitouche@hei.fr

L Belkoura
email: lotfi.belkoura@univ-lille1.fr

HIL VALIDATION OF AN EMBEDDED SYSTEM ACTING AS A NONLINEAR TAKAGI-SUGENO STATE OBSERVER ON AN ARDUINO BOARD

The article presents a method for validating a physical Embedded state observer by using a Hardware In The Loop architecture. As an application, a two water tank system was chosen. For the validation of the observer, a software has been employed, specifically LMS AMESim, interfaced by Simulink on a Windows operating system. Even though Windows is not well suited for real time operations, it is the most common operating system and it has proven to be robust enough to sustain our tests. The article also shows the capability of small scaled microcontroller based systems to handle complex algorithms. It also promotes the simplicity of implementing Takagi-Sugeno algorithms in small scaled numerical systems. Although tests were made on a Arduino UNO version, by adding the benefits of the powerful, in development, ArduinoTRE Board, the power bonus of Texas Instruments processor allows more complexity as well as signal filtering and faster and more secure communication protocols.

INTRODUCTION

It is often advised to test an embedded system before connecting it to the real process, this way avoiding a potential instability or lack of robustness towards uncertainties of the model or communication. Therefore a high precision simulation by use of a special software becomes an essential step in any development process. For this, AMESim comes to play [START_REF] Bourdon | Integration of Physical AMESim® Engine Model in Hardware in the Loop Environment, Dedicated to Engine Control Unit Testing[END_REF]. As an embedded system, an Arduino platform is employed, because of its simplicity, size and price ratio. In order to compensate for the complexity of a full scale implementation of a standalone system, the Arduino TRE board [START_REF]ArduinoTRE documentation[END_REF] is envisaged. It fully responds to the demand; as the Texas Instruments Sitara processor completes the Atmel microcontroller.

Another objective of the current work being to show the ease in implementation of the Takagi-Sugeno(TS) approach to deal with nonlinear systems [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. This methodology has grown in interest in the last decade because of its structured character. The T-S model is easily obtained considering a sector nonlinearity technique [START_REF] Herrera-Lopez | Takagi-Sugeno Fuzzy Observer for a Switching Bioprocess: Sector Nonlinearity Approach[END_REF]. Once constructed, the model presents itself as a blending of linear systems, that by means of their interrelations give a perfect estimation of the initial system.

A nonlinear analysis of a two tank system has been chosen as it represents a classical example, that also adds some interesting complexity when analyzed from a nonlinear perspective. This process has been approached by means of a Takagi-Sugeno representation in [START_REF] Ghorbel | Robust Tracking Control for Takagi-Sugeno Fuzzy Systems With Unmeasurable Premise Variables: Application to Tank System[END_REF] where premise variables are unmeasurable. In the current work we have chosen a more academic approach, considering the premise variables measurable, as the restrictive suppositions made in [START_REF] Ghorbel | Robust Tracking Control for Takagi-Sugeno Fuzzy Systems With Unmeasurable Premise Variables: Application to Tank System[END_REF] would still limit the domains of stability.

In the following, a general modeling of the implemented system is presented, followed by the Takagi-Sugeno transformation and the observer construction. Finally, the embedded system implementation is described, ending with conclusions and perspectives.

THE EMBEDDED SYSTEM

The hardware in the loop architecture permits the integration of a specialized software within a closed loop with a physical device in order to test the code and performance of the developed embedded system.

Figure 1 -Functional block of the real time system

As such the communication between the computer that hosts the software and the physical device has been chosen 978-1-4799-6842-8

Proceedings of the 6th European Embedded Design in Education and Research, 2014 as general as possible so that it's easily adapted to any potential microcontroller. Therefore an ethernet UDP connection has been established. Also a serial connection was tested and proven to be working as required, with a minimal sampling time of 5ms.

The interactions schematic behind the implementation of the embedded validation structure is shown as functional blocks in Figure 1. Although all the processes including the Matlab real time kernel, are managed by Windows, it is capable to give and maintain higher priority only to inputs and essential interrupts.

As AMESim needs Simulink for operating a real time connection, the creation of a co-simulation Amesim/Simulink is performed. Finally the Real Time Windows Target toolbox in Simulink will manage the compilation of the program that will run in the Real Time Windows Kernel of Matlab.

The Arduino code has been written directly in the Arduino's associated editor/compiler. The fact that the processor is single core does not restrict the system from obtaining a fast enough sampling time.

Figure 2 -Logical diagram of the Arduino code

When the observer has been programmed, a variable discretization was employed, therefore lost samples do not affect the overall performance.

The logical diagram behind the Arduino code is presented in Figure2. Now, in order to show the functioning principles, an application example has been chosen: System of two interconnected water tanks. The system is analyzed from a nonlinear perspective to show the potential of using T-S techniques on an embedded platform.

THE SYSTEM MODEL

For the two tank system(Figure 3), the height of the first tank has been considered measurable while the second tank's height is the estimated parameter.

Figure 3 -Two tanks system representation

In this system, we have an input pump supplying the first tank. This tank presents a valve evacuating water outwards, where atmospheric pressure P atm is encountered and another valve supplying the second tank. The second tank has only one valve, which goes towards the exterior. We will note Q p the input flow from the pump, h i heights of liquid in each tank and Cp the flow coefficient of each valve, all valves being considered identical. By neglecting the atmospheric pressure, the following equations are found:

  1 1 2 1 2 2 1 2 1 2 1 1 sgn( ) sgn( ) 0 1 0 P x x x x x C g x u S S x x x x x y x x                                     (1)
Where the input is the pump flow, and the states are the two heights and finally the output is the first tank's height: ( )

P u Q h x h y h x h x                   (2)
As it can be seen, the nonlinearities are present as square roots, that present no differentiability in 0.

OBSERVER DESIGN

Takagi-Sugeno Representation

The TS model is of the form (3), considering that B and C don't depend on the states and considering i the number of the fuzzy rule: 8 1 (z) nonlinear terms(premise variables)

i i i x w A x Bu y Cx z               (3)
As it can be seen, this new form represents a sum of linear coupled systems for which the nonlinearities have moved into the membership functions w i , where:

1 i i w   (4)
In order to advance towards the TS form, a transformation is applied to the system (1) to obtain the affine in control system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF].

 

1 3 2 3 1 1 0 1 0 P z z C g x x u S z z S y x x                              (5)
Where:

1 1 1 2 2 1 3 1 2 1 2 2 sgn( ) x x z x z z x z x x x x x                                   (6)
The variables z are called premises variables, and they contain the nonlinear information for the system. In the following steps of the procedure, this new representation ( 5) is going to be used. By means of the sector nonlinearity method, the minimum and maximum of each z i can be determined. As such, we establish the following conditions for the heights, where some additional constraints are added so that the singular values are eliminated:

1 1 2 2 3 0.8 1.1 0, 05 1 0, 2 0,8 0, 05 1 1.1 12 z h m z h m z                     (7)

Takagi-Sugeno Observer

In this part, the focus is directed towards the observer construction, that will end with the construction of a set of conditions for their synthesis in terms of linear matrix inequalities (LMIs).

It is consider that the premise variables z are measurable, indeed an acceptable hypothesis, as the focus of the paper falls more on the embedded implementation and not on the T-S theory itself. Also the real time observer form is not affected very much by this assumption, from the point of view of the real time implementation. This means that, as in most of the cases in the literature [START_REF] Lendek | Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models[END_REF], w i (ẑ)=w i (z). Then, the Luenberger like observer is of the form :

  ˆˆ( ) ( ) ( ) i i i i i i i i y x w z A x B u L y y w z C x              (8)
To obtain the observer gains, a simple numerical solution that can be resolved in Matlab, either using the incorporated optimization LMI solver, or YALMIP toolbox. This solutions consists of the following LMIs:

      ( ) 1: 8, 2 0
( ) ( ) P 0 ( ) ( . ) 0 ; T i i i i T i i i PA Q C PA Q C P size P size A P P size Q size A C scalar                          (9)

Proof:

In order to construct the stability it is required to start by demanding stability of the estimation error dynamics. Therefore the derivative of the estimation error is of the form (10) where the error is

ê x x   .         ˆˆ( ) ( ) ( ) ˆ( ) ( ) ( ) ( ) i i i i i i i i i i i i i i i i i e x

x w z Ax Bu w z Ax Bu L y y e w z A x x L y y w z A LC e

                      (10)
For the stability, based on the Lyapunov theory, the following quadratic Lyapunov function is chosen:

; 0

T T V e Pe P P    (11) Its derivative is : T T Pe e Pe V e      (12)         ( ) ( ) ( ) ( ) T T i i i i i i i i T T T i i i i i i i i w z A L C e Pe e P w z A L C e w z e Pe w z e P e V V A L C A L C                           (13)
Finally :

      ( ) T T i i i i i i w z A L C P P A L C e V e             (14)
In order for the estimation error to decrease to 0, it is needed to have :

0 V   (15)
Therefore the following inequality is obtained:

      ( ) 0 T i i i i i i w z A L C P P A L C            (16)
One easy way to be sure that these conditions are satisfied is to resolve the inequality (17), which even though restrictive, it is sufficient, leaving out a small interval of potential solutions:

( ) 1:8, 0 0

i i i i x x       (17)
Which applied to (16) gives:

      ( ) 1: 8, 0 T i i i i i A L C P P A L C       (18)
One can easily deduce the set of LMIs, where new variables are created namely i i Q PL 

. Then, using the symmetry of P :

      ( ) 1: 8, 0 T i i i i i PA Q C PA Q C       (19)
Moreover, if it is wanted for the dynamic of the error of our observer to be exponentially decreasing, one can use the following equation :

2 0 V V     (20)
where  is the exponential decay rate. This modifies equation (20) as to obtain exactly the initial inequality (9), thus ending the proof.

In the case where it is aimed to optimize the value of  , a Bilinear Matrix Inequality (BMI) solver is needed. Here, we will only use LMIs, therefore we suppose that  is known and the LMIs in consequence, can be solved.

OBSERVER IMPLEMENTATION

The AMESim system

Once the model has been constructed in AMESim, by adding all the blocks, creating the connections and configuring the parameters, one has to add the cosimulation block that will create the dynamic link with Simulink. For this, AMESim R12 and Matlab 2012a have been used. On the other hand, Simulink has to have the corresponding co-simulation block as well, as in Figure 4.

The real time simulation

By means of the experimental setup shown in the Figure 6, a performance analysis is done.

Figure 7 -Estimation error dynamics

As it can be observed in Figures 7 and8, the estimation of the water level in the second tank stabilizes in 3 seconds, which is an adequate time response. Also the overshoot of the estimation is in an acceptable interval.

Figure 8 -Real/Estimated water levels time evolution

In the previous figures, the perturbation added to the communication in a forcedly manner, does not affect the observer's behavior.

CONCLUSIONS AND PERSPECTIVES

The observer's performance, despite the nonlinear character of the model, confirm the capability of both the embedded system as well as the Takagi-Sugeno algorithm. Also, the validation method can be applied to many processes, ranging from mechanical to chemical or electrical. The hardware in the loop, real time testing permits a robust validation of the observer/controller or diagnostic system, as it adds unpredictable communication and time response errors.

As perspectives, the authors envision to make use of the power of the Arduino TRE to ensure also a robust data acquisition and also better secured communication, as well as to eliminate the restrictiveness of the measurability condition of the premise variables.

APPENDIX

L L L L L L L L                                                                 (

Figure 4 -Figure 5 -

 45 Figure 4 -Simulink diagram including cosimulation block

Figure 6 -

 6 Figure 6 -Hardware implementation Using the obtained LMIs, one gets the gains (23).

ACKNOWLEDGEMENTS

This work was carried out in the framework of PN-7-022-BE i-MOCCA (Interregional Mobility and Competence Centres in Automation), an European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IV A 2 Seas Programme and the Ministry of Education and Research, France.


Sonia Maalej
email: sonia.maalej@ec-lille.fr

Alexandre Kruszewki
email: alexandre.kruszewski@ec-lille.fr

Lotfi Belkoura

A Kruszewski

Derivative Based Control For LTV System With Unknown Parameters

Keywords: Linear Time Varying systems, Robust control, Uncertainties, Linear Matrix Inequalities, Polytopic systems

This paper deals with the robust stabilization of a class of linear time varying systems. Instead of using a state observer or having dynamic structure, the controller is based on output derivative estimation. This allows the stabilization of linear time varying systems with very large parameter variation and a small number of controller parameters. The proof of stability is based on the polytopic representation of the closed loop and Lyapunov conditions. The result is proposed in a Linear Matrix Inequality (LMI) form. The validity of this approach is illustrated by a second order system case of study.

I. Introduction

Since many years, state feedbacks and dynamic output feedbacks [START_REF] Sato | Gain-scheduled output-feedback controllers depending solely on scheduling parameters via parameter-dependent lyapunov functions[END_REF] are largely used to guarantee stability and performances. Despite the fact that dynamic output feedbacks are more robust than observer based controllers [START_REF] Luenberger | Observing the state of a linear system[END_REF] [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF], they still need a model of the plant without too much uncertainties nor time varying parameters with a large variation set. Getting a precise model is time consuming and the controller may become complex if the time varying parameters are the sources of instability. Considering more universal controllers with simple design procedure seems to be important in that context especially if the controlled plant is just an actuator or an auxiliary subsystem.

Recently, some works have addressed issues related to controllers with an another point of view. For exemple, Cedric et al. [START_REF] Cédric | Étude préliminaire d'une commande sans modèle pour papillon de moteur ..... A modelfree control for an engine throttle: a preliminary study[END_REF] introduced a model f ree controller. The latter is signal based and thus don't require many informations about the system. To implement this controller with the less dependence on a good model, the output derivatives are used by the controller.

In the literature, there exist many techniques allowing the derivative estimation of a signal. The simplest one is to use a filtered differentiator i.e. any stable transfer function with a pure differentiator for the numerator. Other possibilities consist in the use of algebraic differentiators [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF] , [START_REF] Cédric | Étude préliminaire d'une commande sans modèle pour papillon de moteur ..... A modelfree control for an engine throttle: a preliminary study[END_REF], unknown input Luenberger observer or sliding mode observers [START_REF] Slotine | Sliding controller design for non-linear systems[END_REF], [START_REF] Nounou | Robust adaptive sliding-mode control for continuous time-delay systems[END_REF], [START_REF] Yan | Robust sliding mode observer with parameters estimation for a class of nonlinear time-delay systems[END_REF], [START_REF] Feng | Terminal sliding mode observer based parameter estimation method for permanent magnet synchronous motor control system[END_REF].

In this paper, a controller inspired by [START_REF] Cédric | Étude préliminaire d'une commande sans modèle pour papillon de moteur ..... A modelfree control for an engine throttle: a preliminary study[END_REF] based on the simplest differentiator is presented. This choice allows to study the stability of the closed loop for a class of systems. This study is performed through the Lyapunov theory and thanks to a polytopic transformation, the stability conditions will be presented as a set of LMI (Linear Matrix Inequalities). In a first part, the class of considered systems is presented, then, the controller and the used differentiation technique are given. The second part provides a polytopic representation of the closed loop and the global stability conditions. The third one is a study case for second order system providing more specific conditions. Some numerical examples are then presented showing the effectiveness of the controller on a set of system with a large parameter variations. The last part gives some conclusions and perspectives.

II. Problem formulation

A. The class of system

In this paper, we consider the following class of single input single output linear system with time varying parameters :

y (n) (t) = -a 0 (t)y(t) -. . . -a n-1 (t)y (n-1) (t) + αu(t). (1)
where y(t) ∈ R is the output, u(t) ∈ R is the control input and a i are scalar time varying parameters.

The aim of many researches is to provide the most parameter independent controller ensuring the stabilization of this system.

To achieve the proposed goal, the system will be viewed as the following equation from the controller point of view :

y (n) (t) = f (t) + αu(t) (2)
that means that only the order and an approximation of the parameter α are given. In the formulation of the system equation given in [START_REF] Luenberger | Observing the state of a linear system[END_REF], f (t) is the function which contains all the dynamic information, i.e. the global dynamic of the system and the possible disturbances. The function f (t) MUST be control independent.

Since the only information on the system are given by Equation (2), the idea of the controller structure is to nullify the system dynamic f (t) and then replace it with the ideal dynamic for the closed loop. Considering this idea, we get the following controller :

u(t) = - 1 α ( f (t) + K Ŷ (t) -k 0 r(t)). ( 3
)
where f (t) is the estimate of f (t), α is an approximation of α, r(t) is the reference, Ŷ (t) = [z 0 (t) . . . z n-1 (t)] T is a vector composed with z i (t) the estimation of y (i) (t) and K = [k 0 . . . k n-1 ] is a vector with the coefficients of the ideal closed loop dynamic given by the specifications.

C. Derivative estimation

In this part, the problem of estimation of f (t) and the successive derivatives of y(t) are considered. In the literature, many approaches were proposed : one can consider sliding mode estimation, [START_REF] Slotine | Sliding controller design for non-linear systems[END_REF], [START_REF] Nounou | Robust adaptive sliding-mode control for continuous time-delay systems[END_REF], [START_REF] Yan | Robust sliding mode observer with parameters estimation for a class of nonlinear time-delay systems[END_REF], [START_REF] Feng | Terminal sliding mode observer based parameter estimation method for permanent magnet synchronous motor control system[END_REF], Luenberger observer based estimation and algebraic estimation [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF]. In order to get a simple solution, we consider a simple filtered derivative approach :

   z0(s) y(s) = 1 τ s+1 zi(s) y(s) = E(s) i = s τ s+1 i+1 ∀i = 1 . . . n (4)
This estimator is causal and ensures a good estimation if τ is sufficiently smaller than the fastest dynamic of the system. It provides the successive estimations z i (t) of y (i) . The estimation of f (t) is made by inverting the dynamic Equation (2) but instead of using the value of u(t), a filtered version of it must be used. This is mandatory to avoid the algebraic loop between u(t) and f (t).

f (t) = z n (t) -αû(t) (5)
with û(s) u(s) = 1 τ s+1 and z n (t) being the estimation of y (n) (t).

III. Main result

The stability of the closed loop needs to be proven. Since it is impossible to stabilize all the models in the form of (1), the stability for a set of variation of the parameters is studied. The conditions used come from the polytopic framework.

A. Closed loop state space representation

The state space representation of the different dynamic equation of the last section are given here. The model in the state space form is given by choosing the state vector as x m (t) = [y(t) . . . y (n-1) (t)] T :

ẋm (t) = A m (t)x m (t) + B m u(t) y(t) = C m x m (t) (6)
Where :

A m (t) =         0 1 0 . . . 0 0 0 1 0 . . . . . . . . . 0 . . . 0 0 . . . . . . . . . 1 -a 0 (t) -a 1 (t) . . . . . . -a n-1 (t)         , B m =        0 0 . . . 0 α        and C m = 1 0 . . . . . . 0 ,
The estimator can be represented as it follows where ẋe

(t) = [z 1 (t) . . . z n (t) û(t)] T (see Proof 1) :    ẋe (t) = A e x e (t) + B ey y(t) + B eu u(t) Ŷ (t) = C eyx x e (t) + C eyy y(t) û(t) = C eu x e (t) (7)
With :

A e =            -1 τ 0 . . . . . . . . . 0 0 -1 τ 2 -1 τ 0 . . . . . . 0 0 -1 τ 3 -1 τ 2 -1 τ 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . -1 τ n-1 -1 τ n-2 . . . . . . -1 τ 0 0 -1 τ n -1 τ n-1 . . . . . . . . . -1 τ 0 0 0 . . . . . . . . . 0 -1 τ            , B T ey = 1 τ 1 τ 2 . . . 1 τ n 0 , B T eu = 0 (1×n) 1 τ , , C eyx = 1 0 (1×n) A e ((1:n-1)×(1:n))
0 , C T eyy = 0 B ey (1:n-1) and C eu = 0 (1×n) 1 . Then, the estimation of the dynamic f (t) is :

f (t) = C f x x e (t) + C f y y(t),
where :

C f x = -1 τ n -1 τ n-1 . . . -1 τ -α and C f y = 1 τ n .
Finally the closed loop of the system and its controller are :

ẋb (t) = Ã(t)x b (t) + Br(t) y b (t) = Cx b (t) , ( 8
)
with the closed system state x b (t) = [x m (t) x e (t)] T , Ã(t) = A m (t) + Ã11 Ã12 Ã21 Ã22 , B = k 0 α B m B eu
and C = 0 (1×2n-1) 1 0 .

where :

Ã11 = -1 α B m ( 1 τ n + KC eyy )C m Ã12 = -1 α B m (C f x + KC eyx ) Ã21 = B ey C m -1 α B eu ( 1 τ n + KC eyy )C m Ã22 = A e -1 α B eu (C f x + KC eyx ) B.

Polytopic representation and stability analysis

There exist several ways to derive stability conditions. Here we chose to represent the closed loop system given by Equation ( 8) as a polytopic system [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. Then the stability conditions are given in the form of a set of LMI to satisfy. Since the closed loop matrices are assumed to be bounded, they can be described/over-approximated by a convex combination of matrices :

A = { Ã(t) : Ã(t) = 2 n i=0 µ i (t) Ãi ; µ i (t) ∈ ∆ µ }; (9)
∆ µ = {µ i (t) ∈ R 2 n : 2 n i=0 µ i (t) = 1; 0 ≤ µ i (t) ≤ 1}.
Where n is the number of time varying parameters. The vertices Ãi of this convex form can be derived by using the using the non-linear sector approach. The stability theorem of this class of system is given by : Theorem 1: If it exist a matrix P = P T > 0 such that :

ÃT i P + P Ãi < 0, ∀i = 1, . . . , 2 n (10)
then, the closed loop 8 is asymptotically stable. The proof is obvious considering the results of [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF].

IV. Study case : 2 nd order LTV system

The stability of the closed loop for a given system was proven in the last section. Since this study aims to allow the stabilization of the largest set of system with the same controller (with the less information possible), the second order case is considered :

A m (t) = 0 1 -a 0 (t) -a 1 (t) , B m = 0 α and C m = 1 0 ,
Remark 1: For this study case, the closed loop dynamic is chosen to be given by K = [1 2] (i.e. a closed loop differential equation given by ÿ(t) = r(t)y(t) -2 ẏ(t)) and the value of τ is given by τ = 0.01.

A. Linear Time Invariant case analysis

The first result considers the Linear Time Invariant (LTI) case i.e. the parameters a 0 (t) = a 0 and a 1 (t) = a 1 of the system dynamic are constant and unknown. The Fig. 1 shows the stability region according to the values of a 0 , -a 1 and the value of α α .

We note from this figure (Fig. 1) that the controller is robust and guarantees the stability even for a large variation of the parameters.

The Fig. 2 shows the state trajectory for the case [a 0 , a 1 ] = [20, -40] (i.e. the system is unstable). From this figure, we note that this estimator is perferment. In fact, the system output converge to the one imposed by the model function. Also, Fig 3 and

B. Linear Time Varying case analysis

This second result considers the LTV case. We consider that a 0 (t) ∈ [a 0 a 0 ] and a 1 (t) ∈ [a 1 a 1 ]. This implies that the vertices of the polytopic model are given by :

A m1 (t) = 0 1 -a 0 -a 1 , A m2 (t) = 0 1 -a 0 -a 1 A m3 (t) = 0 1 -a 0 -a 1 , A m4 (t) = 0 1 -a 0 -a 1
Remark 2: For this case, the parameters bonded are chosen such that a 0,1 = a 0,1 and a 0,1 = -a 0,1 .

The Fig. 5 shows the maximum value of a 1 allowed for a given value of a 0 . This set decreases with α α making the approximation of the parameter α an important point.

Then, Fig. 6 presents the state trajectory for the case a 0 = 20 sin(t) and a 1 = -40 sin(t). Fig. 6 shows that the system output converge to the one imposed by the model function. Remark 3: In both cases (LTI and LTV case), the stability region of the system is directly linked to the value of τ . These sets grow when τ decreases. In fact, if the signals vary quicker than the time constant τ , the estimation error of the system output derivatives as well as the open loop dynamic f (t) will be increased.

V. conclusions

This paper provides a robust controller based on output derivative estimation. Stability conditions are provided for the case where the open loop system has bounded time varying parameters. A study case shows that this controller can stabilize a large set of LTI/LTV second order system if an approximation of some parameters or their bounds on the parameters are available. Our future work will be devoted to deriving conditions without solving any LMI problems and studying system performance.

VI. Appendix

Proof 1: We distinguish tow cases :

• For n = 1:

For this case, the estimator has y(s) as input and z 1 (s) as output. From (4), the transfer function of this estimator is given by :

z 1 (s) y(s) = s τ s + 1 .
This implies :

y(s)s = τ sz 1 (s) + z 1 (s).
Then, we note x ei (t) = z i (t)dt and by integrating one time the result we get : In this case, the transfer function is given by :

z 1 (t) = 1 τ (y(t) -x e1 (t)) (11)
z n (s) z n-1 (s) = s τ s + 1 . ( 12
)
We improve by recurrency that for k = 2 . . . n -1: Thus:

z n (t
sz n (s) = τ sz n+1 (s) + z n+1 (s).

Then, after one integration:

z n+1 (t) = 1 τ (z n (t) -x en+1 (t))
Then, from (13):

z n+1 (t) = 1 τ ( 1 τ n y(t) - 1 τ n x e1 (t) -. . . - 1 τ n-k x e k (t) -x en+1 ) = 1 τ n+1 y(t) - 1 τ n+1 x e1 (t) -. . . - 1 τ n-k+1 x e k (t) - 1 τ x en+1 .
Then the equation ( 13) is true for all n ∈ N.

  Fig 4 show that the model and the control outputs follow rapidly their estimate values.

Fig 7 and

 7 Fig 8 show the model and control output evolution when noises are added to the system output.

1 Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 1123456 Fig. 1. Stability region of the LTI case (zoom in)

Fig. 7 .Fig. 8 .•

 78 Fig. 7. Evolution of the model function and its estimate for the LTV system with r(t) = 0 : (a)-without noise injection and (b)-with noise injection

  1 (s) = τ sz 2 (s) + z 2 (s).This proves that (13) is true for n = 2. Now let us check if equation (13) is true for n = k, it is olso for n = k + 1.

	Or:
	z 2 (t) =	1 τ	(z 1 (t) -x e2 (t)).
	Using (11):
	z 2 (t) = = 1 τ 2 y(t) -1 τ (-1 τ x e1 (t) + 1 τ 2 x e1 (t) -1 y(t) -x e2 (t)) τ 1 τ x e2 (t)(t).
	z n+1 (s) z n (s)	=	s τ s + 1	.
					) =	1 τ n y(t) -	1 τ n x e1 (t) -. . . -	1 τ n-k x e k (t). (13)
					First of all, let us verify if (13) is true for n = 2.
					From (4) we have :
					z 2 (s) z 1 (s)	=	s τ s + 1	.

sz

This work was supported by the INTERREG IV A 2 Mers Seas Zeen Cross-border Cooperation Program 2007-2013 under SYSIASS 6-20 project.