Tarek Ahmed-Ali
email: tarek.ahmed-ali@ensicaen.fr

Iasson Karafyllis

Fouad Giri
email: fouad.giri@unicaen.fr

Sampled-Data Observers for Delay Systems and Hyperbolic PDE-ODE Loops

Keywords: Nonlinear observers, sampled-data observers, delay systems, inter-sample predictor

This paper studies the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results can be applied to time delay systems of strict-feedback structure, transport Partial Differential Equations (PDEs) with nonlocal terms, and feedback interconnections of Ordinary Differential Equations with a transport PDE. The proposed design approach consists in exploiting any existing observer that features robust exponential convergence of the error when continuous-time output measurements are available. The (continuous-time) observer is then modified, mainly by adding an inter-sample output predictor, to compensate for the effect of data-sampling. Using small-gain analysis, we show that robust exponential stability of the error is preserved, provided the sampling period is not too large. The results are applied to a chemical reactor and to the class of triangular globally Lipschitz delay systems.

Introduction

With the growing penetration of network technology in control systems, the compensation of the effects of time-delay has become a major issue in control theory [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF][START_REF] Karafyllis | Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers[END_REF][START_REF] Karafyllis | Recent Results on Nonlinear Delay Control Systems[END_REF][START_REF] Karafyllis | Predictor Feedback for Delay Systems: Implementations and Approximations[END_REF][START_REF] Michiels | Stability, Control and Computation for Time-Delay Systems. An Eigenvalue Based Approach[END_REF]. A great deal of interest has recently been paid to the problem of designing state observers for linear and nonlinear systems with measurement delays. The dominant design approach consists in starting with the design of an exponentially convergent observer for the delay-free system, which is described by Ordinary Differential Equations (ODEs), and modifying it mainly by adding predictors: static predictors (see [START_REF] Karafyllis | Predictor Feedback for Delay Systems: Implementations and Approximations[END_REF]) or dynamic (chain) predictors (see [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF][START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF][START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF][START_REF] Cacace | An Observer for a Class of Nonlinear Systems with Time-Varying Measurement Delays[END_REF][START_REF] Cacace | A Chain Observer for Nonlinear Systems with Multiple Time-Varying Measurement Delays[END_REF][START_REF] Germani | A New Approach to State Observation of Nonlinear Systems with Delayed Output[END_REF][START_REF] Kazantzis | Nonlinear Observer Design in the Presence of Delayed Output Measurements[END_REF]). In parallel to this research activity, which takes into account the time-delay explicitly in the model, a separate activity, based on Partial Differential Equations (PDEs) has been initiated in [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. This consists in modeling time-delays by means of first-order hyperbolic PDEs, leading to a representation of the delayed system in the form of an ODE-PDE cascade (see also the recent work [START_REF] Ahmed-Ali | PDE Based Observer Design for Nonlinear Systems with Large Output Delay[END_REF], where a PDE-based chain-observer is constructed).

Most existing results on observer design for delayed systems have been established assuming the measurement delay to be of discrete nature. So far, only a few studies have investigated the case of distributed measurement delays.

The PDE-based observer developed in [START_REF] Bekiaris-Liberis | Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays[END_REF] and the recent observer developed in [START_REF] Ammeh | Observer Design for Nonlinear Systems with Output Distributed Delay[END_REF], are notable exceptions.

The nowadays-digital implementation of observers entails sampling in time of all system signals needed by the observer. Consequently, the design of sampled-data observers is a major issue. Sampled-data observers for ODE systems can be classified in four main categories: 1) observers where data-sampling is accounted for through a standard Zero-Order-Hold (ZOH) sampling of the output estimation error; see for example [START_REF] Ahmed-Ali | Using Exponential Time-Varying Gains for Sampled-Data Stabilization and Estimation[END_REF][START_REF] Raff | Observer with Sampleand-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements[END_REF], 2) observers designed on approximate discrete-time models (see [START_REF] Arcak | A Framework for Nonlinear Sampled-Data Observer Design via Approximate Discrete-Time Models and Emulation[END_REF][START_REF] Biyik | A Hybrid Redesign of Newton Observers in the Absence of an Exact Discrete-Time Model[END_REF]), 3) continuous-discrete time observers where correction terms are employed at the sampling times; see for instance [START_REF] Nadri | Observer Design for Uniformly Observable Systems with Sampled Measurements[END_REF], and 4) sampled-data observers, where the time-varying delay effect (caused by output sampling) is compensated by using inter-sample output predictors; see [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF]. The use of inter-sample output predictors was extended to systems with asynchronous measurements (see [START_REF] Ling | Multi-Rate Sampled-Data Observer Design for Nonlinear Systems with Asynchronous and Delayed Measurements[END_REF]) and systems described by parabolic PDEs (see [START_REF] Karafyllis | Sampled-Data Observers for 1-D Parabolic PDEs with Non-Local Outputs[END_REF]).

The combination of time-delay and data-sampling effects necessarily makes the problem of observer design more complex. Indeed, not only data-sampling introduces a timevarying delay but it also entails information lost. The case of discrete measurement delays, in conjunction with data Preprint submitted to Automatica

2
1 October 2020 sampling, has been investigated in [START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF][START_REF] Ahmed-Ali | Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems[END_REF][START_REF] Karafyllis | Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers[END_REF][START_REF] Karafyllis | Predictor Feedback for Delay Systems: Implementations and Approximations[END_REF][START_REF] Raff | Observer with Sampleand-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements[END_REF]. Results on observer-based output feedback stabilization of delay systems with sampled measurements have been recently given in [START_REF] Di Ferdinando | Sampled-Data Emulation of Dynamic Output Feedback Controllers for Nonlinear Time-Delay Systems[END_REF][START_REF] Pepe | On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems[END_REF] (see also [START_REF] Pepe | Stabilization in the Sample-and-Hold Sense of Nonlinear Retarded Systems[END_REF]).

In the present work, we extend for the first time the use of inter-sample predictors to the case of time-delay systems with state and output (discrete and/or distributed) delays. Moreover, we provide observer-based output feedback stabilization results for delay systems with sampled measurements under appropriate assumptions. More specifically, we consider nonlinear time-delay systems of the form:

( , , ) , ( )   are continuous mappings with (0, 0, 0) 0 f  , (0) 0 h  . The input u is assumed to be avai- lable, but the inputs , d  are unknown and represent possi- ble modeling errors and measurement noise, respectively. The signal t x denotes the " r -history" of x , see definition in the notation subsection at the end of this section. The proposed sampled-data observer design approach consists in starting with any existing observer, that features robust exponential convergence when continuous-time output measurements are available (see Definition 2.1 for the precise meaning of the notion of "robust exponential convergence"). The available observer, based on continuous-time measurements, is then modified, mainly by adding an intersample output predictor, to compensate for the effect of data-sampling. Using small gain analysis, we show that the robust exponential stability feature is preserved, provided that the sampling period is sufficiently small (Theorem 2.2). The sampled-data observer can be used in a straightforward way for the design of observer-based output feedback stabilizers (Corollary 2.4) under certain assumptions.

tt x f x u d y h x     ( , , ) , , nk x u d U D y        (1)
The second contribution of the paper is that it provides a framework where sampled-data observer and feedback design for ODE-PDE loops can be converted to a similar problem for a time-delay system. More specifically, we consider feedback interconnections of ODEs with a first-order, hyperbolic PDE with non-local terms of the form [START_REF] Ahmed-Ali | PDE Based Observer Design for Nonlinear Systems with Large Output Delay[END_REF][START_REF] Bastin | Stability of Linear Density-Flow Hyperbolic Systems Under PI Boundary Control[END_REF][START_REF] Bernard | Adaptive Output-Feedback Stabilization of Non-Local Hyperbolic PDEs[END_REF][START_REF] Santos | Boundary Control with Integral Action for Hyperbolic Systems of Conservation Laws: Stability and Experiments[END_REF][START_REF] Ferrante | Boundary Observer Design for Coupled ODEs-Hyperbolic PDEs Systems[END_REF][START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF][START_REF] Tang | Stability Analysis of Coupled Linear ODE-Hyperbolic PDE Systems with Two Time Scales[END_REF]. For the class of systems ( 2), ( 3), ( 4), [START_REF] Ahmed-Ali | Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems[END_REF], we provide sampled-data observers and observer-based, sampled-data, output feedback stabilizers (Theorem 3.2, Corollary 3.4).

It should be noted that in all cases the results are global. Moreover, we are in a position to consider uncertain sampling schedules (i.e., the sampling times are not a priori known) and guarantee robustness with respect to measurement noise. Finally, in the absence of measurement noise and unknown disturbances, exponential convergence of the observer error is achieved. The fact that the proposed sampled-data observer design approach with an inter-sample output predictor can indifferently be applied to time-delay systems, to transport PDEs, and to interconnections of ODEs with a transport PDE, provides the approach with a strong unifying feature.

The present paper provides illustrative applications of the theoretical results. In Section 4, we consider the ODE-PDE loop that is used for the mathematical description of a chemical reactor with an exothermic chemical reaction taking place in it and a cooling jacket. A sampled-data observer is designed for this system when we only measure the temperature of the cooling medium at the exit of the jacket. The observer guarantees exponential convergence of the observer error in the noise-free case, no matter what the transport speed inside the reactor is. In Section 5, we deal with uncertain, triangular, globally Lipschitz delay systems of the form

1 1 1, 2 1 1 1 1, 1, 1 1, , ( ) ( , ( )) ( ) ( )
im i f C r       ( 1,..., in  ) with (0) 0 i f  ( 1,..., )
in  are globally Lipschitz functionals. The use of inter-sample predictors in the design of sampled-data observers for [START_REF] Ammeh | Observer Design for Nonlinear Systems with Output Distributed Delay[END_REF] guaranteees exponential convergence of the observer error, no matter how large the maximum delay 0 r  is (Theorem 5.1). The design is based on the high-gain observer design for ODEs, proposed in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF].

Notation. Throughout this paper, we adopt the following notation.

 [0, )     . Let : [0,1] u      be given.
     with (0) 0   . We say that a function K   is of class K  if lim ( )

Main Results for Delay Systems

In the present work we study systems of the form (1) under the following assumptions: (H1) The mappings

0 : ([ ,0]; ) nn f C r U D       , 0 : ([ , 0]; ) nk h C r    where m U  , q D
 are convex sets with 0 U  , 0 D  , are continuous and satisfy the following properties, (i) (0, 0, 0) 0 f  , (0) 0 h  , (ii) for every bounded 0 ([ ,0]; )

n C r U D       the image set n f   ) ( is bounded, (iii) for every bounded 0 ([ ,0]; ) n Cr     the image set () k h    is bounded,
and (iv) for every bounded 0 ([ ,0]; )

n S C r U D      , there exists a constant 0 S L  such that     2 (0) (0) ( , , ) ( , , ) ( , , ) , ( , ,
)

T S x x f x u d f x u d L x x x u d S x u d S         (H2) System (
n x x C r    , ( , ) u d U D  :   ( , , 0) ( , , ) R x u R x u d L x x d      (8)
Assumption (H3) requires that the derivative of the output of system (1) exists and is expressed by the globally Lipschitz (with respect to x ) mapping R . Not every nonlinear time-delay system satisfies (H3). Nevertheless, the class of systems satisfying Assumption (H3) is very wide, including systems of practical interest (see [START_REF] Ammeh | Observer Design for Nonlinear Systems with Output Distributed Delay[END_REF]), like globally Lipschitz delay systems with linear delay-free outputs.

The notion of the Robust Exponential Observer (REO) for system (1) is crucial to the development of the results of the paper and it is given in the following definition. Definition 2.1 (Robust Exponential Observer): Consider the following system ( , , ),

l t z F z y u z    ˆ( ), n tt x z x     ( 9
)
where 0 : ([ , 0]; ) 9) is called a Robust Exponential Observer (REO) for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF]

l k l F C r U       , 00 : ([ ,0]; ) ([ ,0]; ) ln C r C r       are continuous map- pings with (0, 0, 0) 0 F  , (0) 0  . Suppose that the map- ping F is such that, for every bounded 0 ([ ,0]; ) lk C r U       the image set () l F    is bounded and there exists a constant 0 L   such that     2 (0) (0) ( , , ) ( , , ) ( , , ) , ( , , )
T z z F z y u F z y u L z z z y u z y u            System (
           (11)
At this point, it should be noticed that the way the inputs d and  enter the Input-to-Output (IOS) Stability estimate [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF] is different. While the input d comes in estimate [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF] through a (possibly) nonlinear gain function, the input  appears in estimate [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF] with a linear gain and with a fading memory effect (see [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF]). This difference is important and allows, in what follows, the construction of sampled-data observers. Besides the fact that Definition 2.1 introduces the notion of REO for systems with state delays, there are important differences between this notion of a REO in Definition 2.1 and similar notions in the literature (for systems described by ODEs; see [START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF][START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF]): 1) In Definition 2.1, the effect of disturbances is explicitly taken into account (see the term 11)), while in other similar notions in the literature no disturbances are assumed to act on the system.

2)

In Definition 2.1, the IOS estimate ( 11) is assumed to hold uniformly for inputs ( ; )

loc u L U  



, while in other notions in the literature either there is no control input u or the sup-norm of u appears in the corresponding observer error estimate. This difference is important when the observer is to be used in conjunction with a state feedback control law.

The existence of a REO for a control system is a strict requirement which does not hold for all nonlinear systems. A discussion of classes of finite-dimensional systems (which may be considered as a special case of delay systems) which admit a REO can be found in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] (linear detectable systems, globally Lipschitz systems). It should be noticed that Definition 2.1 allows (in general) arbitrarily large measurement delays (included in h ) because the REO may include continuous-time predictors (see [START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF]).

We are now in a position to state our main result. Theorem 2.2 (Sampled-Data Observer Design): Consider system (1) under (H1), (H2), (H3) and suppose that system [START_REF] Bekiaris-Liberis | Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays[END_REF] is a REO for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF]. Moreover, suppose that for every bounded 0 ([ , 0]; )

n S C r U     , there exists a con- stant 0 S L  such that ( ( ), , 0) ( ( ), , 0) S R z u R z u L z z      , ( , ) , ( , ) z u S z u S     ( 12
)
Let 0   and

(0, ]
  be constants that satisfy

  11 ln 1 ( ) L       (13)
where 0 L  is the constant appering in [START_REF] Bastin | Stability of Linear Density-Flow Hyperbolic Systems Under PI Boundary Control[END_REF] and ,0  are the constants appering in [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF]. Then for every sampling sequence

  0 i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every 00 00 ( , ) ([ ,0]; ) ([ ,0]; ) nl x z C r C r       and ( ; ) loc u L U    , ( ; ) loc d L D    , ( ; ) k loc L    
  , the solution ( ( ), ( ), ( ))

x t z t w t of (1) with

    1 0 0 0 ˆ(1 ) exp( ) sup ( ( ) ) tt st x x B t a x z g d s            1 0 (1 ) exp( ) sup ( ) exp( ( )) st B s t s           ( 16
)
where

1 ( ) : (1 ) ( ( ) ( )) g s B g s s      and   : exp( ) 1 / 1 BL       .

Remark 2.3: (a)

The observer ( 14), ( 15) is the REO ( 9) with the unavailable output signal replaced by the signal produced by the inter-sample output predictor 16) guarantees the IOS property for the output map ˆtt Y x x  from the inputs , d  , i.e. from the in- puts expressing the effect of modeling errors and measurement noise, respectively. However, a comparison of ( 11) and [START_REF] Di Ferdinando | Sampled-Data Emulation of Dynamic Output Feedback Controllers for Nonlinear Time-Delay Systems[END_REF] shows that the input gains are higher for the sampled-data observer ( 14), [START_REF] Cacace | A Chain Observer for Nonlinear Systems with Multiple Time-Varying Measurement Delays[END_REF] than the continuous-time REO [START_REF] Bekiaris-Liberis | Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays[END_REF]. It is clear that sampling makes the observer more sensitive to modeling errors and measurement noise. (c) Despite the hybrid nature of the observer ( 14)-( 15), the trajectory of the estimated state features continuity. The proof of Theorem 2.2 is based on a small-gain argument (see Section 5). It is therefore expected that the observer error estimate ( 16) and the upper bound for the diameter of the sampling sequence 0   given by ( 13) are conservative.

1 ( ) ( ( ), ( ), 0) , [ , ) ( ) ( ) ( ) i t i i ii w t R z u t t w h x           (b) Notice that (
However, formulas (13), ( 16) are useful because they indicate which parameters affect the performance of the observer and (qualitatively) how the upper bound for the diameter of the sampling sequence depends on the parameters of the system.

(d) Since the mapping

  11 (0, ] ln 1 ( ) L          is decreasing with     1 1 1 0 lim ln 1 ( ) ( ) LL           
, it is clear from (13) that: (i) Theorem 2.2 requires sampling sequences with diameter 0   being less than 1/ L  , and (ii) the smaller the diameter 0   of the sampling sequence is, the larger the constant 0   is, in the absence of modeling errors and noise (recall ( 16)).

For the design of observed-based output feedback, we need a stabilizability assumption. (H4) The equalities (0, 0, 0) 0 f  and m U  hold. More- over there exist a function K

  , constants ,0 M   and a functional 0 : ([ , 0]; ) nm k C r     with (0) 0 k  and a constant 0 L  such that the inequalities ( ) ( ) ( , ,0) ( , ,0) k x k x f x u f x u L x x L u u          ( , , ) ( , ,0) f x u d f x u d    hold for 0 , ([ ,0]; ) n x x C r    , , m uu , dD  and such that for all 0 0 ([ ,0]; ) n x C r    , the solution () xt of ( ) ( , ( ), 0) t x t f x u t    () t u t k x  ( 17
)
with initial condition 0 x exists for all 0 t  and satisfies the following estimate 0 exp( )

t x M t x  , 0 t  (18)
When Assumption (H4) holds then we obtain the following stabilization result. 1) under (H1), (H2), (H3), (H4) and suppose that system ( 9) is a REO for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF]. Moreover, suppose that for every bounded 0 ([ , 0]; ) 12) holds. Then there exist con- stants , , 0     and functions ˆ,

n S C r U     , there exists a constant 0 S L  such that (
g a K  such that for every sampling sequence   0 i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every 0 0 00 ( , ) ([ ,0]; ) ([ ,0]; ) n l x z C r C r       , ( ; ) k loc L       , ( ; ) loc d L D  
, the solution () xt of ( 1) with ( 14), (15) and

  1 ( ) , [ , ) i ii u t k x t     (19) initial condition 00 ( , ) xz , corresponding to inputs , d  ,
exists for all 0 t  and satisfies the following estimate

  00 ˆêxp( ) tt x x t a x z          00 ˆŝup ( ) sup ( ( ) ) s t s t s g d s       ( 20
)
Moreover, if aK   (the function involved in [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF]) is linear then â is linear too. Example 2.5: To briefly illustrate the results of this section, let us consider the following quite simple example:

( ) ( ) ( ) ( ) ( ), ( ) ( ) t tr x

t x t y t u t d t y t b x s ds

         (21) with 0, b     , 0 b  . Clearly,
( ) ( ) ( ) ( ) , 0 ( ) ( ) ( ) , [ , ) ii z t z t w t u t t w t bz t bz t r t             ˆ,0 tt x z t  (22) ( ) ( ) ( ) i i ii r w b x s ds         (23)
By virtue of Theorem 2.2, the sampled-data observer ( 22), [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF]

( ) ( ) r u k x b x s ds      (24)
It is readily seen that the feedback law ( 24) meets all requirements of Corollary 2.4. It turns out that the observer-based, sampled-data feedback

) ( ~x k u 
, with x ˆ being online generated by the sampled-data observer ( 22), [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF], is an exponentially globally stabilizing regulator.

Hyperbolic PDE-ODE Interconnections

The General Case

When a plant is interconnected with a transport process then we can obtain a system of the form (1) with distributed delays. This is the reason that in this section we consider initial-boundary value problems of the form (2), ( 3), ( 4) with initial condition

0 [0] vv  , 0 (0) xx  (25)
where 0  c is a constant, ()

n xt  , ( , ) v t z  are the states,   0 ; m uC     is an external input,   0 [0,1] aC  , 1 ([0,1]) i bC  ( 1 1,..., iN  ), 0 ([0,1] [0,1]) i C   , 1 ([0,1]) i C   ( 2 1,..., iN  ), 0 n x  ,   1 0 [0,1] vC  sa- tisfy the compatibility conditions 0 (0) 0 v  and 1 2 1 0 0 0 0 1 1 0 (0) (0, ) (0) ( ) (0) ( ) ( ) N N i i i i i i cv g x b v z s v s ds          (26)
and the following assumption holds for the mappings

  0 : [0,1] n m n FC      , :[0,1] n g    : (A1)   0 : [0,1] n m n FC      , 1 ([0,1] ; ) n gC    are continuous mappings with (0, 0, 0) 0 F  , ( , 0) 0 gz  for all [0,1] z 
, for which there exists a constant 0  L such that the inequalities ( , , ) ( , , )

F x v u F y w u L v w L x y       ,   01 max ( , ) ( , ) z g z x g z y L x y     hold for all 0 , ([0,1]) v w C  , , n xy  , m u  .
Under (A1), Theorem 2.2 on page 22 in [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF] shows that for every 3), (4), [START_REF] Karafyllis | Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers[END_REF]. The solution satisfies the formula 27), ( 28) and (5), system (2) with () ( , )

  1 0 [0,1] vC  , 0 n x  ,   0 ; m uC     with 0 (0) 0 v  satisfying (26), there exist unique mappings 1 ( [0,1]) vC     and 1 ( ; ) n xC     satisfying (2), (
  1 1 0 max(0, ) 1 1 1 max(0, ) 1 1 1 1 max(0, ) 1 1 ( , ) max(0, ) exp ( ) exp ( ) ( , ( ( ))) exp ( ) ( ) ( ( ), ) exp ( )
z z ct z z z ct l z z N i i i z ct l z l v t z v z ct c a w dw c c a w dw g l x t c l z dl c c a w dw b l v t c l z z dl c c a w dw                                                     2 1 1 1 max(0, ) 0 ( ) ( ) ( ( ), ) z N i i i z ct l s v t c l z s dsdl            for ( , ) [0,1] tz     (
NN i i i i ii g z x g z x b z z           (31)
( ) ( ) ( ) ( ) NN T j j j i i j i i ii y t q x t b t t         , for tr  , 1,..., jk  (34
) At this point, we need the following technical assumption: (A2) There exists a constant 0 G  such that for every

  1 0 [0,1] vC  , 0 n x  with 0 (0) 0 v  satisfyng (26), there exist   0 [ , 0]; n x C r    ,   1 0 [ , 0]; N Cr     ,   2 0 [ , 0]; N Cr     with 0 (0) xx  , 0 (0) ( ) ii vz   Preprint submitted to Automatica 7 1 October 2020 1 ( 1,..., ) iN  , 1 0 0 (0) ( ) ( ) ii z v z dz    ( 2 1,..., iN 
) and 29), ( 32), [START_REF] Michiels | Stability, Control and Computation for Time-Delay Systems. An Eigenvalue Based Approach[END_REF], [START_REF] Nadri | Observer Design for Uniformly Observable Systems with Sampled Measurements[END_REF] are valid for all ( , ) [0,1] tz     . Moreover, in this case and if the output is sampled, then Theorem 2.2 can be used for sampled-data observer design. We define

  0 0 0 x G v v x         that satisfy   0 0 () ( , (

 

,,

T T T T n xx     , with 12 n n N N    (36)   1 2 2,1 2,
( ) : ( ),..., ( )

T N f x f x f x  ,   2 3 3,1 3, ( ) : ( ),..., ( ) T N f x f x f x    1 2 3
( , , ) : ( ( , , )) , ( ( )) , ( ( ))

T T T T f x u d f x u d f x f x  ( 37
)
for

  0 [ , 0]; n x C r    , m u  and q d
 by means of the following equations:

1 ( , , ) : ( (0), , )

f x u d F x v u d  , with   0 ( ) ( , ( )) () rz C z g cp z x p v z dp C cp z      , [0,1] z  (38) 0 2, ( ) ( , ( )) (
) : ( , (0)) ( ) (0, ( )) ( ) i i i i i i i i rz C z g z cp x p f x g z x C z g x rz dp C z cp         , for 1 1,..., iN  (39)   1 3, 0 10 0 ( ) : ( ) ( , (0)) ( ) (0, ( )) ( ) ( ) ( , ( )) () ii i rz f x z g z x C z g x rz dz z C z g z cp x p dpdz C z cp           for 2 1,..., iN  (40) 1 2 , , 1 1 ( )
: (0) (0) (0) N N T j j j i i j i i i i h x q x b          , 1,..., j k  (41) 1 2 1 1 ( , , , ) : ( ) ( , ) ( , ) ( ) ( ) N N j i j j i j j j g g z x a z g z x c z x b z z z                (42) ( ) : ( ) ( ) ( ) j j j b z a z b z cb z   , 1 1,..., iN  (43) ( ) : ( ) ( ) ( ) j j j z a z z c z      , 2 1,..., iN  (44)
The proof of the following lemma is trivial and is omitted. Lemma 3.1: Suppose that Assumptions (A1), (A2) hold for system (2), ( 3), [START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF]. Moreover, suppose that: (A3) There exists a constant 0 L  such that the inequality

01 max ( , ) ( , ) z g g z x z y L x y z z                 holds for all n y x   , .
Consider system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF] with

{0} D 
, where , fh are defined by [START_REF] Raff | Observer with Sampleand-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements[END_REF], [START_REF] Tang | Stability Analysis of Coupled Linear ODE-Hyperbolic PDE Systems with Two Time Scales[END_REF], ( 39), ( 40), (41). Then Assumptions (H1), (H2), (H3) for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF]

n m k R C r D       defined for all 0 ( , , ) ([ ,0]; ) nm x u d C r D      by 12 12 1 1 1, 2, 1 , 3, 11 1 , 2, , 3, 11 ( , , ) ( ) ( ) ( , , ) ( , , ) ( ) ( )
NN T i i i i ii NN T k k i i k i i ii q f x u d b f x f x R x u d q f x u d b f x f x                  (45)
Based on Theorem 2.2 and Lemma 3.1, we are in a position to show the following result. Theorem 3.2 (Sampled-Data Observer for Hyperbolic PDE-ODE Loops): Suppose that Assumptions (A1), (A2), (A3) hold for system (2), ( 3), [START_REF] Ahmed-Ali | Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements[END_REF]. Consider system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF] with

{0} D 
, where , fh are defined by ( 37), ( 38), ( 39), ( 40), (41). Suppose that system ( 9) is a REO for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF] [START_REF] Biyik | A Hybrid Redesign of Newton Observers in the Absence of an Exact Discrete-Time Model[END_REF] holds with R being defined by (45). Then there exist constants , , 0 P   and aK   such that for

every sequence   0 i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every   1 0 [0,1] vC  , 0 n
x  with 0 (0) 0 v  satisfying (26),

0 0 ([ , 0]; ) l z C r    ,   0 ; m uC     , ( ; ) k loc L       , the solution ( ( ), [ ], ( ), ( ))
x t v t z t w t of ( 2), ( 3), (4) together with ( 14) and

1 2 1 , , 1 1 0 ( ) ( ) ( ) ( , ) ( ) ( , ) N N T j i i j i j l i l j l l i l l w q x b v z s v s ds                  , 1,..., jk  (46) ( ) ( ( ), ( )) ˆ( , ) ( ( )) t t rz C z g z c t p x p v t z dp C z c t p      , for 0 t  , [0,1] z  ( 47
)
initial condition 0 0 0 ( , , )

x v z , corresponding to inputs , u  , exists for all 0 t  and satisfies the following estimate:

  0 0 0 0 ˆ( ) ( ) [ ] [ ] exp( ) x t Hx t v t v t t a v v x z               0 sup ( ) exp( ( )) st P s t s      ( 48
)
where nn H   is the matrix for which the relation  

,, [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF]) is linear then a is linear. Theorem 3.2 can be used for the construction of sampleddata observers for PDE-ODE loops. It transforms the sampled-data observer design problem for a PDE-ODE loop to the construction of a REO for a delay system.

T T T T x H x   (49) holds for n x  , 1 N   , 2 N   . Moreover, if aK   (involved in

Preprint submitted to

Indeed, for every solution of system (2), ( 3), ( 4) there exists a solution of system (50) for which ( 29), ( 32), (33) hold. However, not every solution of system (50) provides a solution of system (2), ( 3), ( 4) by means of ( 29), ( 32), [START_REF] Michiels | Stability, Control and Computation for Time-Delay Systems. An Eigenvalue Based Approach[END_REF]. For such a thing, the initial condition of the solution ( , , ) 3), ( 4) is immersed into system (50).

t

Stabilization of Hyperbolic PDE-ODE Loops

For stabilization purposes, we assume (A1), (A2), (A3) as well as the following (stabilizability) assumption. (A4) There exists

  0 : [0,1] nm kC     such that (H4) holds for (1) with {0} D 
, where f is defined by (37), [START_REF] Tang | Stability Analysis of Coupled Linear ODE-Hyperbolic PDE Systems with Two Time Scales[END_REF], ( 39), (40), for

0 : ([ , 0]; ) nm k C r     given by   00 ( , , ) , k x k x v   (51)
for every

  1 0 [0,1] vC  , 0 n
x  with 0 (0) 0 v  satisfying [START_REF] Karafyllis | Recent Results on Nonlinear Delay Control Systems[END_REF], where

  0 [ , 0]; n x C r    ,   1 0 [ , 0]; N Cr     ,   2 0 [ , 0]; N Cr     with 0 (0) xx  , 0 (0) ( ) ii vz   1 ( 1,..., ) iN  , 1 0 0 (0) ( ) ( ) ii z v z dz    ( 2 1,..., iN 
) are the functions involved in [START_REF] Pepe | Stabilization in the Sample-and-Hold Sense of Nonlinear Retarded Systems[END_REF].

Under (A4), we obtain the following stabilization result.

Corollary 3.4 (Global Stabilization of Hyperbolic PDE-ODE Loops with Observer-Based Sampled-Data Feedback): Suppose that (A1), (A2), (A3), (A4) hold for system (2), (3), (4). Consider system (1) with {0} D 

, where f , h are defined by [START_REF] Raff | Observer with Sampleand-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements[END_REF], [START_REF] Tang | Stability Analysis of Coupled Linear ODE-Hyperbolic PDE Systems with Two Time Scales[END_REF], (39), ( 40), (41). Suppose that system ( 9) is a REO for system [START_REF] Ahmed-Ali | Cascade High Gain Predictors for a Class of Nonlinear Systems[END_REF]. Moreover, suppose that for every bounded 0 ([ ,0]; ) nm S C r     , there exists a constant 0 S L  such that [START_REF] Biyik | A Hybrid Redesign of Newton Observers in the Absence of an Exact Discrete-Time Model[END_REF] holds with R given by (45). Then there exist constants , , 0 P

  and a function aK  

such that for every sampling sequence   0

i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every 0 0 ([ ,0]; ) l z C r    , ( ; ) k loc L       ,   1 0 [0,1] vC  , 0 n
x  with 0 (0) 0 v  satisfying (26), the solution of ( 2), ( 3), ( 4), ( 14), ( 46), (47) and ˆ(

) ( ( ), [ ]) ii u t k Hx v   , for 1 [ , ) ii t    (52) initial condition 0 0 0 ( , , )
x v z , corresponding to input  , exists for all 0 t  and satisfies the following estimate:

 0 0 0 0 ˆ( ) ( ) [ ] [ ] exp( ) x t Hx t v t v t t a v v x z               0 sup ( ) exp( ( )) st P s t s      ( 53
)
where H is the matrix involved in (49).

Application to Chemical Reactor

model of a chemical reactor with an exothermic chemical reaction taking place in it and cooling jacket with negligible axial heat conduction of the cooling medium can be found in Chapter 2 and Chapter 8 of [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF]. It consists of the following ODE and PDE:

    1 0 ( ) ( ) 1 ( ) ( , ) dx t x t x t v t z dz dt        (54) ( , ) ( , ) ( , )
( ) vv t z c t z v t z x t tz        (55)
with boundary condition (4), where ,0  are real con- stants, () xt is the (dimensionless) reactor outlet tempera- ture, ( , ) v t z is the (dimensionless) temperature of the coo- ling medium at position [0,1] z  in the jacket (0 z  is the entrance of the jacket and 1 z  is its exit) and :

    is a 1 C globally Lipschitz function with (0) 0  . When the reactor inlet temperature is not constant then equation (54) is modified as follows:

    1 0 ( ) ( ) 1 ( ) ( , ) ( ) dx t x t x t v t z dz u t dt         (56)
where () ut  is the input that expresses the variation of the reactor inlet temperature. Moreover, it is indeed the case that the measured temperature is the temperature of the cooling medium at the exit of the jacket, i.e., the measured output is ( ) ( ,1) y t v t  . The reactor model ( 4), ( 55), (56), is a system of the form (2), ( 3), ( 4), ( 5) with ()

az   , 1 n  , 1 1 N  , 1 1 z  , 1 k  , 1 ( ) 0 bz ,     1 0 ( , , ) 1 ( ) F x v u x x v z dz u         , ( , ) g z x x   , 1 0 q  , 1,1 1 b  ( 2 N , , ii  , 1,i  are irrelevant). Assumpti- ons (
A1), (A3) are automatically verified. Assumption (A2) also holds, since for every 4), (55), (56) turns out to be equivalent to the following system with distributed delays:

  1 0 [0,1] vC  , 0 x  with 00 (0) c v x    , 0 (0) 0 v  ,
  0 0 0 x G v v x        holds with     1 1 exp G r
    1 0 ( ) ( ( )) ( 1) ( ) ( ) ( ) exp ( ) ( ) exp
i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every   1 0 [0,1] vC  , 0 x  with 00 (0) c v x    , 0 (0) 0 v  , 02 0 ([ , 0]; ) z C r    ,   0 ; uC     , ( ; ) loc L       , the solution ( ( ), [ ], ( ), ( ))
x t v t z t w t of ( 4), ( 55), ( 56) with

          1 2 2 1 1 1 2 2 2 2 1 1 2 0 2 2 1 1 ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( ( )) ( 1) ( ) ( ) ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) exp
                                        (60) ( ) ( ) ( ,1
)

j i i i wv      (61)   2 ˆ( , ) exp ( ) ( ) t t rz v t z t p x p dp       ( 62
)
for 0 t  , [0,1] z 
, initial condition 0 0 0 ( , , )

x v z , correspon- ding to , u  , exists for all 0 t  and satisfies

  2 0 0 0 0 ˆ( ) ( ) [ ] [ ] exp( ) x t x t v t v t t M v v x z               0 sup ( ) exp( ( )) st P s t s      (63)
The proof of Theorem 4.1 is constructive and provides formulae for the observer gains [START_REF] Biyik | A Hybrid Redesign of Newton Observers in the Absence of an Exact Discrete-Time Model[END_REF] , kk (see formulae ( 103), (104), ( 105)), although other choices for the observer gains are possible. It should be noticed that there is no restriction in the speed 0 c  of the transport process (or equivalently, in the delay 0 r  ; see (57)). We next evaluate numerically the sampled-data observer (60), (61), (62), together with the inter-sample predictor, considering the following parameter values:

1/ 2

  0.05   ,         1 1
exp 0.1( 1) exp 0.1 ( ) 10 1 exp 0.1 1 exp 0.1( 1)

x x x                    for 1 x  ,     10 exp 0.1 () 1 exp 0.1 x      for 1 x  , 1 10 k  , 2 30 k  , 0.1 r  , ( ) sin(10 ) u t t  , ( ) 0 t   , 0.4   , i i   , for 0,1, 2,... i    0 ( ) 0.5 1 exp( ) v z rz    , for [0,1] z 
We also consider comparing of the novel sampled-data observer, defined by (60), ( 61) and (62), with the ZOH observer where the unknown output signal is simply replaced by its most recent measurement. Specifically, the ZOH observer consists of (61), (62) and

        1 2 2 1 1 1 2 2 2 2 1 1 2 0 1 ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( ( )) ( 1) ( ) ( ) ( ) ( ) ( ) exp ( ) ( ) 0
                              (64)
Preprint submitted to Automatica

10 1 October 2020
The study is completed by comparing the observer (60), ( 61), (62) with the continuous-time observer that uses the continuous output signal () yt , i.e., (62) and . The results are shown in Fig. 1. We found that the state estimates provided by ( 60), ( 61), ( 62) with the inter-sample predictor were indistinguishable from those of the continuous-time observer (65), (62). Exponential convergence of the error to zero for observer (60), ( 61), (62), is apparent in Fig. 1. On the other hand, the error for the ZOH observer (64), ( 61), (62) does not converge to zero but presents an oscillation with amplitude 0.02. Clearly, the inter-sample predictor compensates well for the effects of sampling. These results were confirmed for a wide range of parameter values that were tested. Pushing further the study, we seek global stabilization of the reactor by observer-based sampled-data feedback. It is readily checked that ( 4), ( 55), ( 56) satisfies (A4), with 1 0 ( , ) :

        1 2 2 1 1 1 2 2 2 2 1 1 2 0 ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( ( )) ( 1) ( ) ( ) ( ) ( ) ( )
                           (
( )

k x v Qx v z dz      , for x ,   0 [0,1] vC  (66)
where 0 Q  is a sufficiently large constant. It follows from Theorem 2.1 and Corollary 3.4 that there exist constants , , 0 P   and a function 0 ( ; ) aC     with (0) 0 a  such that for every sequence   0

i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i  , for every   1 0 [0,1] vC  , 0 x  with 00 (0) c v x    , 0 (0) 0 v  , 02 0 ([ , 0]; ) z C r    , ( ; ) loc L       ,

the solution ( ( ), [ ], ( ), ( ))

x t v t z t w t of ( 4), ( 55), (56) together with (61), (60), (62) and

1 2 0 ˆ( ) ( ) ( , ) ii u t Qx v z dz        , for 1 [ , ) ii t    ( 67
)
with initial condition 0 0 0 ( , , )

x v z , corresponding to input  , exists for all 0 t  and satisfies the estimate

  0 0 0 0 ˆ( ) ( ) [ ] [ ] exp( ) x t x t v t v t t a v v x z               0 sup ( ) exp( ( )) st P s t s      (68)

Triangular Globally Lipschitz Delay Systems

In this section we consider systems of the form ( 6), where we assume that there exists a constant 0 ~ L such that the following inequalities hold for 1,..., in  : 6), (69), we get that for all 0 0 ([ , 0]; )  ,

n x C r    and ( ; ) m loc uL      , ( ; ) n loc dL     
1 2 3 , , , 1 Q Q Q   such that for every sampling sequence   0 i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i 
, for every

0 0 00 ( , ) ([ ,0]; ) ([ ,0]; ) n n x z C r C r       , ( ; ) m loc uL      , ( ; ) n loc dL      , ( ; ) loc L   
   , the solution ( ( ), ( ), ( ))

x t z t w t of ( 6) with 1 October 2020

1 1, 2 1 1 1 1, 1, 1 1, , 1 1, 2 1 ( )
                   ˆtt xz  (73) 1 ( ) ( ) ( ) i i i wx      ( 74
)
initial condition 00 ( , ) xz , corresponding to inputs ,, ud , exists for all 0 t  and satisfies the following estimate

  1 0 0 ˆexp( ) tt x x t Q x z          23 00
sup ( ) exp( ( )) sup ( )

s t s t Q s t s Q d s          (75)
Clearly, the construction of the observer (73), (74) consists in combining a high-gain observer with an inter-sample predictor. The proof of Theorem 5.1, presented in Section 6, makes use of Lyapunov analysis together and small-gain arguments. The parameter 1

  depends on the maximum delay 0 r  (see inequality (122) in the proof of Theorem 5.1, which gives a formula for 1   ). Thus, the design of a sampled-data observer for system ( 6) is straightforward:

1) Select 1 ( ,..., ) Tn n K K K    so that the matrix () T A Kc  is Hurwitz. 2) Select 1
  sufficiently large (so that inequality (122) in the proof of Theorem 5.1 holds). The proof of Theorem 5.1 in conjunction with Theorem 2.2 allow us to prescribe the convergence rate 0   . Recent works have studied this particular problem in the finite-dimensional case (see for instance [START_REF] Cacace | An Exponential Observer with Delay-Dependent Gain for a Class of Nonlinear Systems with Time-Varying Delay[END_REF]). Indeed, the proof of Theorem 5.1 shows that [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF]

  .
To numerically evaluate the relevance of the proposed sampled-data design, we consider the system

1 2 2 1 1 ( ) ( ) ; ( ) 2 ( ) ( ) ( ) ( ) x t x t x t x t r u t y t x t      (76) with 0.1 r  , ( ) 0.5sin(2 ) u t t  , 1 ( ) 0.5 xt for [ 0.1,0] t 
, 2 (0) 0.5 x  We next compare the sampled-data observer with inter-sample output predictor

    1 2 1 2 1 1 21 ( ) ( ) 3 ( ) ( ) ( ) 2 ( ) ( ) 9 ( ) ( ) ( ) ( ) , [ , ), 0,1, 2,... ( ) ( ) , 0,1, 2,
             (77)
with the ZOH sampled-data observer that uses the most recent measurement, i.e., the observer

    1 2 1 2 1 1 1 ( ) ( ) 3 ( ) ( ) ( ) 2 ( ) ( ) 9 ( ) ( ) ( ) 0 , [ , ), 0,1, 2,... ( ) ( ) , 0,1, 2,
            (78)
Both sampled-data observers will be compared to the continuous-time observer

    1 2 1 2 1 1 ( ) ( ) 3 ( ) ( ) ( ) 2 ( ) ( ) 9 ( ) ( ) ˆ( ) ( ) z t z t z t y t z t z t r u t z t y t x t z t          (79)
All observers are initiated with 1 ( ) 0 zt for [ 0.1,0] t  , 2 (0) 0 z  and operated with the sampling schedule was /10

i i   for 0, 1, 2,... i 
. The results of comparison are shown in Fig. 2. Clearly, the ZOH sampled-data observer fails to estimate the state, unlike the observer involving the inter-sample predictor. The latter features a behavior that is comparable to the continuous-time system.

Proofs

In this section we provide the proofs of all main results. Proof of Theorem 2.2: Let 0 i  be an integer for which i z  exists. We show first that t z exists for all 1 [ , ] ii t    .

Due to the regularity assumptions of Definition 2.1 and due to [START_REF] Biyik | A Hybrid Redesign of Newton Observers in the Absence of an Exact Discrete-Time Model[END_REF] there exists max 0 t  such that the solution of ( ) ( , ( ), ( ))

t z t F z w t u t  ( ) ( ( ), ( ), 0) t w t R z u t  (80) is defined on max [ , ) i tt   , where max i t   is the maximal existence time of the solution of (80). If max 1 i t   
then there is nothing to show. We next focus on the case max

1 i t    . Define: ( ) : ( ) ( ) t v t w t h x  (81)
The component of the solution t z of (80) is a solution of ( ) ( , ( ) ( ), ( ))

tt z t F z h x v t u t  (82) for max [0, ) tt 
. Therefore, by virtue of (11) and since   , the following estimate holds for all max [0, ) tt  :

  00 ˆexp( ) tt x x t a x z          00 sup ( ) exp( ( )) sup ( ( ) ) s t s t v s t s g d s          (83)
Using ( 7), ( 8), ( 14) and ( 15

( ) ( ( )) t ss qt v t q t L x x ds          () ( ( )) sup ( ) q t s t t q t d s    (84)
where

  ( ) max : ii q t t   . (85)
Using (84) and the fact that ()

t q t   (a consequence
of definition (85) and the fact that

1 0 ii        for 0,1,... i  ), we obtain for all max [0, ) tt  :       0 () ( ) exp( ) exp( ) sup ( ) ( ( )) exp( ( )) exp( ) exp( ) 1 ŝup exp( ) st ss q t s t v t t t d s q t q t L x x s                 which implies the following estimate for all max [0, ) tt  :     0 0 ( ) exp( ) exp( ) sup ( ) exp( ) exp( ) 1 ŝup exp( ) st ss st v t t s s L x x s                 0 exp( ) sup ( ) st t d s      (86)
Combining estimates (83), (86) we get for max [0, ) tt  :  is not the maximal existence time of the solu- tion of (80), a contradiction. Therefore, the case max    , it follows that t z exists for all 0 t  . Moreover, [START_REF] Di Ferdinando | Sampled-Data Emulation of Dynamic Output Feedback Controllers for Nonlinear Time-Delay Systems[END_REF] holds for all 0 t  . Proof of Corollary 2.4: Assumptions (H1), (H2), (H3), (H4) guarantee that ( 1) with ( 14), ( 15) and ( 19) is forward complete, i.e. its unique solution exists for 0 t  , for arbitrary initial conditions, inputs and arbitrary sequences

      00 0 0 ˆexp( ) exp( ) sup ( ) exp( ) exp( ) 1 ŝup exp( ) tt st ss st x x t a x z ss L x x s                     00 exp( ) sup ( ( ) ) sup ( ( ) ) s t s t t d s g d s            (87) Since   : exp( ) 1 / 1 BL       , it
  0 i i    with 0 0   ,   lim i    , 1 0 ii        for 0,1,... i 
, where 0   is sufficiently small. Using a mod- ification of the proof of Theorem 5 in [START_REF] Pepe | On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems[END_REF] (applying a smallgain analysis instead of Halanay's inequality), (H4) guarantees the existence of constants , , , 0 M     and a function gK  such that for every sequence   0 [START_REF] Besancon | Asymptotic State Prediction for Continuous-Time Systems with Delayed Input and Application to Control[END_REF], ( 16), (92)) is linear then â is linear. Proof of Theorem 3.2: Under (A1), (A2), (A3), the solution 1 ( [0,1]) vC     and 1 ( ; )

i i    with 0 0   ,   lim i    , 1 0 i i        for 0,
n xC     of system (2), (3)
, ( 4), [START_REF] Ahmed-Ali | Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems[END_REF] with initial conditions

  1 0 [0,1] vC  , 0 n x  with 0 (0) 0 v  satisfying (26), corresponding   0 ; m uC     , is
expressed by [START_REF] Karafyllis | Sampled-Data Observers for 1-D Parabolic PDEs with Non-Local Outputs[END_REF], where ( ) ( ( ), ( ), ( ))

x t x t t t   is the solution of (1) with {0} D  , f , h being defined by ( 37), ( 38), (39), ( 40), (41), initial condition provided by (A2) and corresponding to the same input u . It follows that Theorem 3.2 is a consequence of Theorem 2.2, Lemma 3.1 and the fact that there exists 0 G  such that the initial condition of system (1) satisfies

  0 0 0 x G v v x     
(recall assumption (A2) and ( 36)). Indeed, using ( 29), ( 31), [START_REF] Pepe | On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems[END_REF] ) and g ) such that:

ˆ[ ] [ ] tt v t v t B x x     , for 0 t  (93)
Inequality ( 48) is obtained by combining ( 16), (93) and using the fact the initial condition of system (1) satisfies the inequality

  0 0 0 x G v v x      .
Proof of Corollary 3.4: Under (A1), (A2), (A3), (A4), the solution of ( 2), ( 3), ( 4), ( 14), ( 46), (47) and ( 52) is expressed by ( 29), ( 32), (33) using the solution ( ( ), ( ), ( ))

x t z t w t of (1), ( 14), ( 15) and ( 19) with {0} D 

, where , fh are defined by [START_REF] Raff | Observer with Sampleand-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements[END_REF], [START_REF] Tang | Stability Analysis of Coupled Linear ODE-Hyperbolic PDE Systems with Two Time Scales[END_REF], (39), (40), (41), with initial condition provided by assumption (A2). Following the proof of Theorem 3.2, we conclude that Corollary 3.4 is a consequence of Corollary 2.4 and the fact that there exists 0 G  such that the initial condition of (1) satisfies

  0 0 0 x G v v x      (recall
                          2, ˆtt xz  (94)
is a REO for system (59). We consider the functional

    2 2 2 ( ) ( ) ( ) exp ( ) t tr V t Q z s x s t s ds            2 2 1 1 2 2 1 1 ( ) ( ) / 2 ( ) ( ) ( ) ( ) / 2 R z t x t z t x t bz t bx t       (95)
where , , 0 R b Q  are constants to be selected. For every ( ; ) loc L       , the time derivative of () Vt along the tra- jectories of (94), (59) with

1 ( ) ( ) ( ) y t x t t   is               22 11 2 2 2 2 2 2 22 1 2 1 2 2 22 ( ) ( ) ( ) ( 1 ) ( ) ( ( )) ( ( )) ( ) ( ) ( ) exp ( ) 2 ( ) ( 1 ) ( ) ( ) ( ) ( ) exp t tr V t k R R b Qb E t b Q E t z t x t E t Q z s x s t s ds R bQ b k k b b E t E t Q z t r x t r r                                                 2 1 2 2 2 1 2 1 1 ( ) ( ) exp( ) ( ) ( ) ( ) ( ) ( ) bE t RE t r z t r x t r k bk E t Rk E t t               1
( ( )) ( ( )) ( ) ( ) z t x t E t b E t        (99)
In the above inequality we have used the triangle inequality and (97), (98). By using the inequalities

2 2 2 1 2 2 1 ( ) ( ) ( ) ( ) / 4 b E t E t E t b E t          2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) E t z t r x t r E t z t r x t r           2 2 1 2 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) E t z t r x t r E t z t r x t r         2 22 2 1 2 2 2 1 ( ) ( ) ( ) ( ) / 4 k bk E t t E t k bk t      2 2 2 1 1 1 1 ( ) ( ) ( ) ( ) / 4 k E t t E t k t  
we obtain from (96), (99) the inequality for 0 t  a.e.:

            exp( ) ( ) ( ) ( ) ( ) exp ( ) ( ) exp t tr V t b Q b r E t b k R R b Qb R r R E t R bQ b k k b b E t E t b R Q r z t r x t r Q z s x s t s ds Et                                                                   1 22 0 ( ) ( ) ( ) t t rl t s z s x s dsdl       2 22 2 1 1 ( ) / 4 k bk Rk t     (100)
Preprint submitted to Automatica 14 1 October 2020

Using the Cauchy-Schwarz inequality (twice), we get for the integral appearing on the right hand side of (100):

           1 2 2 0 1/2 2 1 2 2 0 1/2 2 2 2 1/2 2 2 2 exp ( ) ( ) ( ) exp ( ) ( ) ( ) exp ( ) exp ( ) ( ) ( ) 1
                                                                 
Using the above inequality in conjunction with estimate (100), we obtain the following inequality for 0 t  a.e.:

            () t tr V t b Q b r E t b k R R b Qb R r R E t R bQ b k k b b E t E t b R Q r z t r x t r Q z s x s t s ds Et                                                                      2 2 2 exp( ) ( ) ( ) exp ( ) t tr r z s x s t s ds            2 22 2 1 1 ( ) / 4 k bk Rk t     (101)
Finally, using the inequality

        1/2 2 2
                           
in conjunction with estimate (101), we obtain the following inequality for 0 t  a.e.:

            ( ) ( ) exp ( 2 V t b Q b r E t b k R R b Qb R r R E t R bQ b k k b b E t E t Q b R r z t r x t r r Q z s x s                                                                          ) t tr t s ds      2 22 2 1 1 ( ) / 4 k bk Rk t     (102) By selecting 2 2 (1 exp( )) r R     ,   4 ( 1) 1 exp( ) R b r         (103) ( ) / 2 Q b R   , 21 ( ) ( ) ( 1 ) k R b b R b k b b              (104) 2 2 1 11 ( ) exp( ) 1 2 4 2 b k b b R b r RR             (105)
we obtain from (102) and (95) the following differential inequality for 0 t  a.e.:

  2 22 2 1 1 ( ) ( ) / 2 ( ) / 4 V t V t k bk Rk t       (106)
Applying Lemma 2.12 in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] in (106), we get for 0 t  :

  ( ) exp / 2 (0) V t t V         2 22 2 1 1 0 1 exp ( ) / 2
  2 1 1 ( ) ( ) exp 2 (0) z t x t K t V        2 2 2 1 1 2 0 1 sup ( ) exp( 2 ( )) 2 ( 4 ) st k bk Rk s t s K         (108)
   2 1 0 0 ( ) ( ) / exp z t x t K K t x z        2 2 2 1 1 0 1 sup ( ) exp( ( )) 2 ( 4 ) st k bk Rk s t s K         (109)
Notice that since 21 KK  , inequality (109) holds for all tr  . Consequently, we obtain from (109) for tr  :   so large so that     shows that (112) is a REO for system [START_REF] Ammeh | Observer Design for Nonlinear Systems with Output Distributed Delay[END_REF].

    2 1 0 0 sup ( ) ( ) exp( ) / r s t z s x s s K K x z          2 2 2 1 1 0 1 sup ( ) exp( ) 2 ( 4 ) st k bk Rk ss K       (110) Using the fact that       sup ( ) ( ) exp exp ( ) tt t r s t z s x s s t r z x         , we obtain from (110) for all 0 t  :    2 1 0 0 / exp ( ) tt z x K K t r x z            2 2 2 1

Concluding Remarks

The present work studied the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results are applicable to time delay systems of strict-feedback structure, transport PDEs with nonlocal terms, and feedback interconnections of ODEs with a transport PDE. The study constitutes a unified theoretical framework to deal with ODEs, delay systems and PDEs.

  mean the essential supremum of d on I but the actual supremum of d on I

Corollary 2 . 4 (

 24 Global Stabilization by Means of Observer-Based Sampled-Data Output Feedback): Consider system (

Fig. 1 :u

 1 Fig. 1: The error

Theorem 5 . 1 :

 51 There exist constants ,0

Fig. 2 :

 2 Fig. 2: The error ( ) ( ) x t x t  for observer (77) with the inter-sample predictor (grey line), the ZOH observer (78) (red line) and the continuous observer (79) (blue line).

  1) is Forward Complete, i.e., for every

				 d h x ( ) tt  ( , ( ), ( )) R x u t d t dt 	(7)
	Moreover, there exists a constant	0 L  and a function
	 	K	such that the following inequality holds for all
	,		0	([ ,0]; )
						0 x C 	0	([ ,0]; ) n r  	and for every	( ; ) U   loc u L 	,
						( ; ) D   loc d L 	the solution of (1) with initial condition 0 x
						, corresponding to inputs , ud, exists for all	0 t  .
						Assumption (H1) is a standard assumption for time-delay
						systems that guarantees existence and uniqueness of solu-
						tions for system (1), i.e. guarantees that for every
						0 x C 	0	([ ,0]; ) n r  	and for every	( ; ) U   loc u L 	,
						( ; ) D   loc d L 	there exists max (0, ] t   and a unique
						continuous mapping	max :[ , ) x r t 	n   which is absolutely
						continuous on	[0,	t	max	)	and satisfies	0 xx   ( ) ( )	for
						[ , 0] r  	and (1) for	[0, tt 	max	)	a.e.. This mapping
						max :[ , ) x r t 	n   is the solution of (1) with initial condi-
						tion 0 x , corresponding to inputs , ud. Assumption (H2)
						guarantees that max t   for every	0 x C 	0	([ , 0]; ) n r   ,
						( ; ) U   loc u L 	and	( ; ) D   loc d L 	. Assumption (H2) is
						usual in observer studies. Indeed, if an observer is expected
						to achieve asymptotic estimation of the state then the solu-
						tion of (1) must exist for all times. On the other hand, the
						(possible) state and measurement delays are respectively ac-
						counted for through t x in the quantities ( , , ) t f x u d and
						() t hx .
						The following assumption plays a crucial role.
						(H3)		There	exists	a	continuous	mapping
						0 : ([ ,0]; ) nk R C r U D       with the following pro-
						perty: for every	0 x C 	0	([ ,0]; ) n r   ,	( ; ) U   loc u L 	,
						( ; ) D   loc d L 	the unique solution x of (1) with initial
						condition 0 x , corresponding to inputs , ud, satisfies for
						0 t  a.e. the following equation:
	Preprint submitted to Automatica		4			1 October 2020

, if there exist constants ,0

											 and func-
	tions		, a g K  	such that for all	( ; ) U   loc u L 	,
	00 ( , ) x z		C	0	([ ,0]; ) n r   	C	0	([ ,0]; ) l r   ,	( ; ) D   loc d L 	,
	loc L   	( ; ) k    the solution	(	x	( t	),	z	( t	))	of
							( , , ) t x f x u d 
							( , ( ) , ) tt z F z h x u  
							ˆ() tt xz 	(10)
	with initial condition	00 ( , ) xz , corresponding to inputs
	,, ud , exists for all	0 t  and satisfies the following esti-
	mate for all	0 t  :
						ˆexp( tt x x  	 	 t a x 00 ) z 	
					 00  s t sup ( ) exp( ( )) sup ( ( ) )  s t s t s g d s	

  this system is of the form (1) which meets all requirements of Theorem 2.2.

					Consider
	the continuous-time observer ( ) z t	  	( ) ( ) ( ) z t y t u t  	,
	which is a REO. Indeed, it is straightforward to show that
	(11) holds with	1/  	. Moreover, assumption (H3) holds
	with ( , , ) R x u d bx 	(0)	( ) bx r   and (8) holds with	2 Lb 
	. The sampled-data version of this observer is:
					1

  )

	v t z		t t rz  	Cz C z c t p 	g z c t p x p  	p 	p dp
				for tr  ,	z 	[0,1]	(29)
	with
				( ) : exp C z 	0 z r a s ds ( )    	(30)
						12
			( , , , ) : ( , )		( )	( )
						11


Y Buvat

E Bouyssou

B Morillon

Gaël Gautier

Ageing of glass passivated TRIAC devices under thermal and electrical stress

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

Introduction

Several type of power electronic switches, such as diodes, transistors, thyristors or TRIAC devices (Triode for Alternating Current), exist on the market with their benefits and withdraws. Thyristors and TRIAC devices are bidirectional semiconductor switches and are considered as power devices. They can be found in a variety of applications such as household electrical appliances (e.g. washing machines, cookers, coffee machines...), AC loads drive and more recently in industrial sector [START_REF] Gautier | Proceedings of ICETECT[END_REF] as well as automotive market [START_REF] Shin | ICEE Asia[END_REF] with the electrification of vehicles.

TRIAC devices must withstand high voltage with low leakage current during the OFF state and conduct enough current with minimum loss during the ON state. The die edge termination, also called "periphery" is the device part dedicated to the OFF state. So, its properties are strongly correlated with the blocking performances of the device. TRIAC devices are manufactured through three main technologies which are closely based on the edge termination as illustrated in Figure 1: "planar," "top glass," and "double mesa" [START_REF] Menard | [END_REF]. These technologies are a result of various trade-offs between cost and device performances. The "planar" technology enables to reach good electrical performances, but it remains relatively expensive, due to its large periphery area. However, "top glass" technology and even more "double mesa" technologies are cost effective and have been demonstrated to yield high blocking voltage with a limited periphery area.

TRIAC devices usually work at environment temperatures from 60°C up to 80°C but the junction temperature can reach higher levels when dealing with high power during ON state. In addition, new applications of TRIAC require to withstand higher temperature and excellent reliability performances. So, even if TRIAC devices technologies can be considered as mature, some research is still performed to improve reliability by adding a stop channel [5] or searching for new dielectric passivations [START_REF] Matsumoto | ICEEE, Mexico[END_REF][START_REF] Wang | [END_REF]. Some work has been done also to reach a better cost-effective technology by reducing the periphery area of planar structure [4].

The die termination of mesa glass devices consists in a groove which is passivated with a specific silicate glass containing metallic oxides such as lead oxide; silicate glass is generally effective for achieving good reliability performances in high voltage silicon power devices. Charges existing in the passivation layer are known to influence the electrical and reliability behaviour when subjected to thermal and electrical stress [8]. Also, the functional reliability of the device can be altered by the presence of mobile ion in the passivation [9]. Consequently, the failure of TRIACs should be studied and monitored by the means of accelerated lifetime tests and by observing the time evolution of leakage current. Moreover, the stability of the latter is crucial in order to avoid thermal runaway during the device operation.

Finally, new environmental regulation on electronic component specifies that hazardous materials such as lead are forbidden, which implies that substitute materials must be used. Few alternatives exist to remove lead in passivation layer. The use of other silicate glasses [10][11][12] is considered as a best option, as it is close to the existing solution.

Considering the new TRIAC devices application requirements and the new environmental regulations, a new silicate glass was studied. Hence, the aim of this paper is to characterize the electrical charges in this new silicate glass passivation and evaluate the impact on reliability performances of TRIAC devices.

Experimental procedure

Two kinds of test vehicles were processed: Metal-Glass-Semiconductor (MGS) structures, to characterize the electrical © 2020 published by Elsevier. This manuscript is made available under the CC BY NC user license https://creativecommons.org/licenses/by-nc/4.0/ Version of Record: https://www.sciencedirect.com/science/article/pii/S0026271420304339 behaviour of glass material, and TRIAC devices in order to perform the glass reliability assessment on final products.

MGS structures

Metal-Glass-Semiconductor structures (fig. 2) were processed on 150 mm N type silicon substrates with a resistivity of 33-39 Ohm.cm in order to characterize the glass material. Powder glass mixed with a photoresist is first deposited by spin coating technique, followed by a pyrolysis at 500°C to remove the resist. Finally, a firing step is done in order to melt glass powder and to form an amorph glass layer. Aluminum electrodes are then evaporated directly on glass film before backside metallization. The final thickness of the glass film is 8 µm and the aluminum electrodes area is 25 mm².

TRIAC devices

The use of silicate glass as a passivation is known to eventually influence the level of leakage current due to additional carrier generation/recombination at the silicon-glass interface [12,13]. Consequently, two sets of TRIAC devices were manufactured: one set with the new glass material as unique passivation layer and one set with a combination of a semi-insulating layer and the new glass material. The semiinsulating layer has been deposited using chemical vapor diffusion technique with a gas combination of SiH4/N2O. The thickness of the semi resistive layer is 500 nm. Wang et al. [START_REF] Wang | [END_REF] reported that any undesired charges disturbing the electric field distribution can be compensated by the semi-insulating layer.

The reliability test at package level is not the best solution as we want to study the behavior of the silicate glass passivation without any interaction with a packaging effect. Indeed, Lantz et al. [14] reported the potential impact of the encapsulation resin on the reliability performances due to the presence of mobile ions. Therefore, TRIAC dies have been especially soldered on a silicon substrate (fig. 3) in order to perform electrical reliability tests using a semi-automatic probe-station.

Measurement procedures

Capacitance-Voltage measurement

Capacitance-Voltage (CV) measurements were performed using a 4200 Semiconductor Parametric analyzer associated with an external 2657A high voltage Source Measure Unit and a high voltage bias tee. The voltage sweep was performed between -500 V to 500 V at 1 MHz.

Bias Temperature Stress method

Bias Temperature Stress (BTS) was performed at wafer level to identify the presence of mobile ions and evaluate their effect on the blocking performances. This test consists in stressing the Metal-Glass-Semiconductor structure by applying a DC voltage at high temperature using a semi-automatic probe station associated with a temperature regulated chuck and performing Capacitance-Voltage readout measurements at ambient temperature (fig. 4). Several Bias Temperature Stress tests were appended up to 220 h in order to observe the evolution of TRIAC devices key parameter. At the end of the test, 26 Capacitance-Voltage readout measurements were performed.

The aging tests were performed from 100°C up to 150°C, for voltage stress levels ranging between 300 V and 700 V. The equivalent electrical field applied in the new glass in the Metal Glass Semiconductor is calculated between 375 kV/cm and 875 kV/cm respectively.

Die reliability method

The die reliability method employed is a DC High Temperature Reverse Bias test (HTRB) and is quite similar to the Bias Temperature Stress method except that Current-Voltage (IV) characteristics are performed instead of Capacitance-Voltage measurement (fig 4). Moreover, the time evolution of the leakage current is recorded during the stress period. A custom program was made to regulate the temperature and control the test instruments and the semi-automatic probestation.

Like other power devices, TRIAC devices are sensitive when applying DC stress at high temperature. If the leakage current is too high, the TRIAC devices are likely to switch in ON state. That is why the voltage stress applied was fixed at 100V, which correspond to an electrical field of 33 kV/cm simulated by TCAD (Technology Computer Aided Design) tool as illustrated in figure 5. Consequently, the voltage acceleration effects will not be studied on TRIAC devices. The temperature stress was fixed between 120°C and 160°C.

Results and discussion

In this part, the capacitance-voltage behavior will be explained. Then, the results on the voltage stress and the temperature stress effects of the Metal Glass Semiconductor structure and the TRIAC devices will be shown. At last, the addition of a semi-resistive layer between the new glass and the silicon in TRIAC devices will be explained and discussed regarding the reliability performances.

Capacitance-voltage

First, a Capacitance-Voltage measurement (fig. 6) can give us information on the presence of charge in the glass. It is well known that the initial shift of the Capacitance-Voltage curve towards the voltage axis compared with a theoretical curve indicates a presence of charges in the dielectric. In a MGS structure, the energy band of the semiconductor is considered flat for a specific voltage bias. This specific voltage bias is called the flat-band voltage. Charge density at the flatband equilibrium can be extracted from CV measurements by the following formula [15]:

𝑁 𝐹𝐵 = 𝐶 𝑂𝑋 (𝑊 𝑀𝑆 -𝑉 𝐹𝐵 ) 𝑞 × 𝐴
where WMS is the contact potential difference in MGS structure [V], COX is dielectric capacitance [F/cm²], VFB is the flat-band voltage [V], q the elementary charge [C], A the area of the structure [cm²]. The Flat-Band Voltage parameter (VFB) has been determined by a graphical method [START_REF] Piskorski | MIPRO[END_REF], which consists in extrapolating the linear part of the following characteristic when the slope crosses the voltage axis:

[( 𝐶 𝑂𝑋 𝐶 ) 2 -1] (𝑉𝑜𝑙𝑡𝑎𝑔𝑒)
A negative charge density of 9.9 10 +11 /cm² has been calculated which is consistent with the value given by the glass powder supplier which is 1 10 +12 /cm². (2) The voltage acceleration effect in the range between 300V to 700V does not seem to be significant, however, the absolute variation of flat-band voltage is much important as the electrical field is higher. This behavior confirms the choice to ignore voltage acceleration effects on TRIAC devices.

Voltage stress effect in MGS structure

Temperature stress in MGS structure

Figure 8 shows the flat-band voltage parameter evolution over time of Metal Glass Semiconductor structures for different temperature stresses after BTS tests. As a matter of fact, the curves can also be separated in three sections as observed before however, the temperature acceleration effect appears to be more important compared with voltage acceleration effect.

For each curve of figure 8, an arbitrary degradation criterion was defined (red dash line) as the characteristic time for which the flat-band voltage evolution reaches 250 V when the curves increase. The kinetics of the degradation can here be characterized following an Arrhenius law, with an activation energy of 0.9eV (fig. 10). Hence, the evolution of the flat-band voltage parameter described before might be attributed to the presence of mobile ions [START_REF] Shimbo | [END_REF].

Temperature stress in TRIAC devices

Figure 9 shows the leakage current drift evolution of TRIAC devices for different temperature levels during HTRB tests. Three slopes were observed on almost all curves. A low slope appears at the beginning of the curves followed by an important slope and once again, a low slope at the end of the curves.

The leakage current drift turns out to be accelerated as the temperature increases. For each curve, a characteristic time, at 30% of leakage current drift, has been extracted (red dash line). The mechanism is activated by the temperature and follows Arrhenius law with an activation energy of 0.9eV (fig. 10). The voltage stress is 100 V and the red dash line is the degradation criterion

The beginning of the drift after 2 h of stress observed on Metal Glass Semiconductor structures and TRIAC devices seem to be related to the same mechanism. The accumulation of mobile ions near the interface between the new glass and the silicon might be at the origin of the deterioration of the reliability performances as observed elsewhere [START_REF] Shimbo | [END_REF].

Reliability performance improvement with the semi resistive layer

Since the origin of degradation is supposed to be at the glass-silicon interface, a semi-insulating passivation layer was deposited between the new glass passivation and the silicon. Figure 11 shows the leakage current drift evolution over time in TRIAC devices. A DC HTRB test was performed with the same conditions of stress, no leakage current drift has been observed with the presence of the semi-insulating layer. Wang et al. [START_REF] Wang | [END_REF] reported that the surface of silicon directly passivated with the semi-insulating passivation layer induced charges of opposite polarity near the silicon surface due to its high density of states. Consequently, the induced charges either neutralize the external charges or form a space charge region within the passivated layer. The addition of this semi-insulating passivation layer is of great interest as it seems to trap mobile ions from near the glass-silicon interface [START_REF] Wang | [END_REF], [18]. Therefore, very promising reliability performances are obtained with this semiinsulating passivation layer.

Conclusion

Dedicated Metal Glass Semiconductor structures and TRIAC devices containing new silicate glass as a passivation layer were studied by means of electrical and temperature stress. Flat-band voltage drift for MGS structures and leakage current drift for TRIAC devices seems to be related to the accumulation of mobile ions near the interface of the glass and the silicon. A semi-insulating passivation layer between the glass and the silicon improves reliability performances as the traps present in semi-insulating passivation layer prevent the mobile ions to accumulate near the glass-silicon interface. This semi-insulating passivation layer is well suitable for high power and high temperature devices and opens up the field of possibilities for exploring new types of glass.

Fig. 1 .

 1 Fig. 1. Simplified diagrams of planar technology (left), top glass technology (right) and double mesa technology (center)

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Schematic cross section view of an MGS structure

Fig. 5 .Fig. 6 .

 56 Fig. 5. TCAD simulation of generated electrical field along the glass-silicon interface when applying 100 V across a TRIAC device.

Figure 7

 7 Figure 7 exhibits the flat-band voltage of Metal Glass Semiconductor structures over time evolution for different stress voltage after BTS tests. The curves are composed of different sections defined by an increase or a decrease of the flat-band voltage parameter. The first section corresponds to the decrease of the flat-band voltage parameter and can be well observed when the structure is stressed at 100°C. The phenomenon is too fast to be monitored at 125°C and 150°C. Another decrease of the flat-band voltage parameter, identified as the third section, occurred at the end of the test. The second section which corresponds with the increase of the flat-band voltage parameter is observed on the three curves.The voltage acceleration effect in the range between 300V to 700V does not seem to be significant, however, the absolute variation of flat-band voltage is much important as the electrical field is higher. This behavior confirms the choice to ignore voltage acceleration effects on TRIAC devices.

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 7. VFB parameter evolution over time in MGS structures for three different stress voltages with the new silicate glass. The temperature stress is 150°C

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Arrhenius plot of temperature activated mechanism of MGS structure and TRIAC devices