L Yu

J.-P Barbot
email: barbot@ensea.fr

D Boutat
email: driss.boutat@ensi-bourges.fr

D Benmerzouk
email: dbenmerzouk@yahoo.fr

‡ L Yu

Observability forms for switched systems with Zeno phenomenon or high switching frequency

This paper deals with the observability of a class of switched systems with Zeno phenomenon or high switching frequency. Particularly, three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed. Meanwhile, sufficient and necessary conditions for the existence of a diffeomorphism to transform a class of switched systems into one of such forms are presented. Examples and simulations are given at the end to highlight the theoretical results.

Introduction

Even though Zeno phenomenon is due to the high abstraction when modeling physical systems [START_REF] Zhang | Zeno hybrid systems[END_REF], it is still interesting to analyze the observability and design an observer for system with Zeno behavior, since some physical systems may have behavior close to Zeno. On the other hand, the work concerning about the observability for systems with Zeno behavior can also be adopted to analyze that of systems with high switching frequency.

Since at least fifty years ago, some particular types of Zeno phenomenon have been studied [START_REF] Filippov | Classical solutions of differential equations with multivalued right-hand sides[END_REF][START_REF] Filippov | Differential Equations with Discontinuous Righthand sides[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF]. The first order sliding mode [START_REF] Utkin | Sliding Modes in Optimization and Control[END_REF] and the high order sliding mode [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF][START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] have been largely applied in control and observer design for many years, and correspondingly, by considering solutions of differential equations in the sense of Fillipov [START_REF] Aubin | Differential inclusions, set-valued maps and viability theory[END_REF], the main property of sliding mode is the convergence of the system behavior in finite time under some controllability (or observability) matching conditions [START_REF] Drazenovic | The invariance condition in variable structure systems[END_REF][START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF].

In [START_REF] Ames | Is there life after zeno? taking executions past the breaking (zeno) point[END_REF], the Zeno phenomenon is categorized to "Chattering" and "Genuinely" Zeno. Analytically, the Chattering Zeno corresponds to both of the first order sliding mode and the high order sliding mode on the sliding manifold, while Genuinely Zeno corresponds to the high order sliding mode just before reaching the sliding manifold [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF][START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF]. From this point of view, this paper presents observability conditions for switched system with Zeno phenomenon through both sliding mode and normal form approaches.

For the sake of simplicity, output switches and continuous state jumps are not considered in this paper 1 . Let us consider the following class of switched system S:

ẋ = f q (x) y = h(x) (1)
where x(t) ∈ U ⊂ R n is the state, y(t) ∈ R is the measured output, vector fields f q : U → R n , q ∈ Q = {1, ..., N } and function h : U → R are sufficiently smooth for each q. The discrete state q may be driven by a switching function σ : R → Q.

From geometrical view point, through a diffeomorphism z = φ(x), one can transform the switched system S into some observability forms, i.e. some new system S ′ , with following expression:

ż = g q (z) y = z 1 (2)
where z(t) ∈ V ⊂ R n is the new state, vector fields g q : V → R n with q ∈ Q, and z 1 is the first new state. In this paper, two basic observability forms are proposed [START_REF] Yu | Observability normal forms for a class of switched systems wih zeno phenomena[END_REF]: the first basic observability form is directly inspired by the observer matching condition, and the second basic observability form is by differential geometric results [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF][START_REF] Isidori | Nonlinear Control Systems. Communication and Control Engineering Series[END_REF]. Moreover, Henstock-Kurzweil (HK) integral [START_REF] Henstock | The General Theory of Integration[END_REF] is considered instead of the notion of Lebesgue integral, since HK integral much more general than Riemman and Lebesgue integral. Consequently, the analytical representation of the switching signal would be more suitable for special discontinuous signals, for instance the infinite switching signal. Meanwhile, the two basic forms can be easily extended to a more general form, called the "extended observability form". Specifically, we denote S ′ (1), S ′ (2), S ′ (3) the first basic, the second basic and the extended observability form.

In order to deal with the observability of system S, the following assumption for each subsystem is needed throughout this paper.

Assumption 1

Regularly Weakly Locally Observable for Subsystems: For each q ∈ Q the pair (y, f q ) is Regularly Weakly Locally Observable2 (RWLO), i.e. rank{dL j f h, j = 0, ..., n -1} = n. This paper is organized as follows. In section 2, three observability forms are proposed and their corresponding conditions for the existence of a diffeomorphism which transforms the switched system into one of such forms are given. In section 3, the observability for the considered system is analyzed. Simulations are given in section 4.

Observability forms and conditions for existence of a diffeomorphism

In this section, three observability forms are presented respectively. Although the first two basic observability forms have already been discussed in [START_REF] Yu | Observability normal forms for a class of switched systems wih zeno phenomena[END_REF], a new extended observability form with more general form is given here and moreover, the proofs for the theorems are rearranged. On the other hand, one approach to find the required vector fields in the theorems is given (see Remark 1).

First basic observability form

The first basic observability form is inspired by observer matching condition, and its vector field g q is written as follows:

S ′ (1) : g q (z) = A nn z + B n χ(z) (3)
where A nn is a n × n superior nilpotent Jordan matrix, written as

A nn =       0 1 0 . . . . . . 1 0       n×n
B n = (0, 0, . . . 1) T and χ q (z) ∈ R.

Theorem 1 Under Assumption 1, there exists a diffeomorphism z = φ(x) which transforms the system S into S ′ (1) if and only if

L f∆ κ,ν L i-1 fκ h = 0, ∀κ, ν ∈ Q and ∀i ∈ {1, . . . n -1} (4)
where

f ∆κ,ν (x) = f κ (x) -f ν (x) with κ, ν ∈ Q.

Second basic observability form

The second basic observability form, i.e. S ′ (2), is inspired by the geometrical results [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF][START_REF] Isidori | Nonlinear Control Systems. Communication and Control Engineering Series[END_REF], and its vector field g q is written as follows:

S ′ (2) : g q (z) = α(z) + γ q (z 1 ) (5)
where vector fields α : V → R n , γ q : R → R n .

Theorem 2 There exists a diffeomorphism z = φ(x) which can transform the system S into system S ′ (2), if and only if there exists a family of linearly independent vector fields {τ 1 , ...τ n } such that

1. dh • τ 1 = 1 2. dh • τ i = 0, ∀i ∈ {2, ..., n} 3. [τ i , τ j ] = 0, ∀(i, j) ∈ {1, ..., n} × {1, ..., n} 4. [τ i , f ∆κ,ν ] = 0, ∀i ∈ {2, ..., n}, ∀κ, ν ∈ Q
The dynamical systems characterized by the theorem of Krener and Isidori in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] would fulfill conditions of Theorem 2. It is worth to mention that the class which satisfies conditions of the Theorem 2 is larger than the class which satisfies the theorem of Krener and Isidori, since the fact that we avoid a linear condition for α. In Example 1, an academic system which satisfies conditions of Theorem 2 will be given, but it does not satisfy conditions of the theorem in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF].

Remark 1

The previous theorems does not guarantee the uniqueness of vector fields τ . One method to calculate the vector fields τ is using the following induction method: one can first find the vector field τ 1 according to condition dh • τ 1 = 1, then for τ i , i ∈ {2, ..., n}, it can be found according to conditions that

• a) dL i-1 fq h • τ i = 1; • b) dL j fq h • τ i = 0, ∀j < i; • c) [τ j , τ i ] = 0, ∀j < i.
For the extended observability form presented in the following subsection, the previous method also can be adopted to calculate the vector fields needed in the theorems. Example 1 and 2 are using this method to find such vector fields.

Example 1 Define the dynamics of system S as below

f q (x) = x1+x2 x1-x2 + x1 x1-x2 µ(q, x) -x1+x2 x1-x2 -x2 x1-x2 µ(q, x) (6)
h(x) = x 1 + x 2 where µ(q, x) = x 1 x 2 -(x 1 + x 2 ) 3 + q(x 1 + x 2 ) and q ∈ {1, 2}. f ∆ (x) = f 1 (x) -f 2 (x) = x1+x2 x1-x2 (x 1 , x 2 )
T and dh = dx 1 + dx 2 . Then τ 1 can be calculated

τ 1 = 1 x1-x2 (x 1 ∂ ∂x1 -x 2 ∂ ∂x2 ); On the other hand, dL fq h = (x 2 -3(x 1 + x 2 ) 2 + q)dx 1 + (x 1 -3(x 1 + x 2 ) 2 + q)dx 2 which implies τ 2 = 1 x1-x2 (-∂ ∂x1 + ∂ ∂x2
). Afterwards, it can be easily verified that such vector fields τ 1 , τ 2 coincide with conditions in Theorem 2.

Meanwhile the diffeomorphism can be given as z = φ(x) = (x 1 + x 2 , x 1 x 2 ) T , through which the original switched system defined by ( 6) can be transformed to the form of S ′ (2), written as

   ż1 = z 2 -z 3 1 + qz 1 ż2 = -z 1 y = z 1 (7)

Extended observability form

Then we can easily extend the two basic observability forms to a more general form, called the "extended observability form", i.e. S ′ (3), and its vector field g q can be written as follows:

S ′ (3) : g q (z) = g 1 q (ξ, η) g 2 q (ξ, η) (8)
where

ξ ∈ V 1 ⊂ R r , η ∈ V 2 ⊂ R n-r and z = (ξ, η) T ; g 1 q (ξ, η) = A rr ξ +B r χ q (ξ, η) with A rr a r × r superior nilpotent Jordan matrix, and B r = (0, ..., 0, 1) T ∈ R r ; g 2 q (ξ, η) = α(ξ, η) + γ q (ξ) with vector fields α : V 1 × V 2 → R n-r , γ q : V 1 → R n-r , χ q : V 1 × V 2 → R.
The parameter r is the relative degree of the original system when considering q as an input. Moreover, if 1 ≤ r < n, χ q has the following form χ q (ξ, η) = α 0 (ξ, η) + γ 0 q (ξ) with α 0 : V 1 × V 2 → R and γ 0 q : V 1 → R; but if r = n, χ q does not have this form and the extended form S ′ (3) degenerates into the first basic form S ′ (1).

Theorem 3 There exists a diffeomorphism φ(x) which can transform the system S into the system S ′ (3), if and only if ∀κ, ν ∈ Q

1. L f∆ κ,ν L i-1
fκ h(x) = 0, where ∀i ∈ {1, ..., r -1}; 2. there exist nr independent vector fields {τ 1 , ..., τ n-r }, such that

(a) dL i-1 fκ h • τ j = 0, i ∈ {1, ..., r}, j ∈ {1, ..., n -r}; (b) [τ i , τ j ] = 0, i, j ∈ {1, ..., n -r}; (c) [τ i , f ∆κ,ν ] = 0, i ∈ {1, ..., n -r}.
where 0 < r ≤ n.

Example 2 The dynamics of system S is now given by:

f q (x) =   x 2 + x 3 -µ(q, x) x2+x3 x2-x3 + x2 x2-x3 µ(q, x) -x2+x3 x2-x3 -x3 x2-x3 µ(q, x)   (9) h(x) = x 1 + x 2 + x 3 where µ(q, x) = x 2 x 3 -(x 2 + x 3 ) 3 + q(x 2 + x 3 ), q ∈ {1, 2} and f ∆ (x) = f 1 (x) - f 2 (x) = x2+x3 x2-x3 -∂ ∂x1 +x 2 ∂ ∂x2 -x 3 ∂ ∂x3 .
Then condition 1) can be easily verified with r = 2. Then, dL r fq h = (x 3 -3(x 2 + x 3 ) 2 + q)dx 2 , (x 2 -3(x 2 + x 3 ) 2 + q)dx 3 , and according to Remark 1, the only restriction for τ 1 becomes to dL r fq h • τ 1 = 1, so we can find

τ 1 = -1 x2-x3 ( ∂ ∂x2 -∂ ∂x3 )
, which verifies conditions 2a), 2b), and 2c).

Consequently, there exists a diffeomorphism φ defined as

z = φ(x) = (x 1 + x 2 + x 3 , x 2 + x 3 , x 2 x 3 )
T and it can transform system S to S ′ (3) written as below

       ż1 = z 2 ż2 = z 3 -z 3 2 + qz 2 ż3 = -z 2 y = z 1 (10)

Proofs for theorems

Proof 1 (1. Proof of Theorem 3) Sufficiency:

Condition 1) implies that the relative degree is r. According to the theory of nonlinear feedback for SISO system in [START_REF] Isidori | Nonlinear Control Systems. Communication and Control Engineering Series[END_REF], the following function

φ 1 : U → V 1 φ 1 = h(x), L fκ h(x), . . . , L r-1 fκ h(x)
T is of rank r such that the first r states in the new coordinate frame ξ = φ 1 (x) ∈ V 1 are with the form of (g q ) 1 (ξ). Condition 2b) implies that there exists a smooth function

φ 2 : U → V 2 φ 2 = (η 1 (x), η 2 (x), . . . , η n-r (x))
T such that

dη i • τ i = 1, ∀i ∈ {1, ..., n -r} dη i • τ j = 0, ∀i, j ∈ {1, .
.., n -r}, and i = j.

Moreover, condition 2a) implies that the one form dφ 1 is independent to the one form dφ 2 , and one can set that φ

= (φ 1 , φ 2 ) T is a diffeomorphism defined from U → V 1 × V 2 such that φ * (τ i ) := dφ • τ i = dφ 1 dφ 2 • τ i = ∂ ∂η i
According to the transformation of Lie bracket by mean of addiffeomorphism, we have

∂ ∂η i φ * (f ∆κ,ν ) = [φ * (τ i ), φ * (f ∆κ,ν )] = 0 ∀i ∈ {1, ..., n -r}
where φ * represents the Jacobian Matrix. This result means that in the new coordinate frame, we obtain

φ * (f ∆κ,ν ) = n-r i=1 ϕ i (ξ) ∂ ∂η i
which means that η is in the form g 2 q (ξ, η), and also χ q has the form χ q (ξ, η) = α 0 (ξ, η) + γ 0 q (ξ). Necessity: If r = n, the form S ′ (3) will degenerate into the first basic form S ′ (1), and suppose there exists a diffeomorphism z = φ(x) which can transforms the system S into the form S ′ (1).

One can easily conclude that the form S ′ (1) satisfies Assumption 1 and (4).

L f∆κ,ν L i-1 fκ h = 0
where fq = A nn z + B n χ q (z), f∆κ,ν = fκfν , and h = z 1 .

Obviously, it has

f q | x = ∂φ ∂x -1 fq | z=φ(x)
where f q | x denotes function f q in x coordinate system. Consequently,

L f∆ κ,ν L i-1 fκ h = L f∆κ,ν L i-1 fκ h = 0
If 1 ≤ r < n then suppose that there exists a diffeomorphism φ(x) which transforms S into S ′ (3). Let's write ξ = φ 1 (x) ∈ V 1 and η = φ 2 (x) ∈ V 2 , where φ = φ1 φ2 . From the first part of dynamics of S ′ (3), it can be concluded that the relative degree is r which implies condition 1). Then denote a set of vector fields

π := {π 1 , ..., π n } the columns of ∂φ ∂x -1
and define another set of vector fields {τ 1 , ..., τ n-r }, where τ i = π r+i . Obviously, conditions 2a) and 2b) are satisfied with vector fields τ i such that [τ i , π j ] = 0, ∀i, j.

Consider the (ξ, η) coordinate frame, there exists a set of functions ϕ κ,ν i (ξ) ∈ R, such that

f ∆κ,ν | (ξ,η) = n-r i=1 ϕ κ,ν i (ξ) ∂ ∂η i + γ 0 ∆κ,ν (ξ) ∂ ∂ξ r ( 11
)
where γ 0 ∆κ,ν (ξ) = γ 0 κ (ξ)γ 0 ν (ξ) with κ, ν ∈ Q. Then, rewrite equation [START_REF] Goebel | Hybrid dynamical systems[END_REF] in original x coordinate frame

f ∆κ,ν | x = n-r i=1 ϕ κ,ν i (ξ)τ i + γ 0 ∆κ,ν (ξ)π r | ξ=φ1(x)
which implies condition 2c).

Proof 2 (2. Proof of Theorem 1 and Theorem 2) When r = n, it is just the case of form S ′ (1) and Theorem 1 can be easily proved by setting r = n in Theorem 3. While if r = 1, condition 1) is not valid any more and it degenerates into the form S ′ (2), meanwhile, if setting z 1 = h(x), associating with condition 2a) in Theorem 3, one will obtain the same conditions in Theorem 2. Consequently, Theorem 2 can also be proved by setting r = 1 and providing just one additional condition z 1 = h(x).

Observability analysis

Before analyzing the observability of the system S, it is important to emphasize that instead of the full knowledge of the discrete state q, only the partial knowledge -the value of the filtered discrete state (often a low pass filter) can be obtained. This is due to the fact that the sensor normally has relative low speed comparing to the switching frequency considered in this paper, since it's Zeno (or high switching frequency). In practice, a low pass filter F is often adopted before the sensor, and the passed signal, denote q F , can be approximated as the time average of discrete state q, while the oscillations written as qq F cannot pass the filter F , see Fig. 1.

Figure 1: Diagram of switched system and its observer; The discrete state q (left plot) and its filtered value q F (right plot).

What if the exact switching signal is not available but its filtered value can be captured? Before answer this question, let us recall some conceptions on integral.

Recall of Henstock-Kurzweil Integral and its implications for Zeno

Henstock-Kurzweil integral (H-K integral) [START_REF] Cichon | The henstock-kurweil-pettis integral and existence theorems for the cauchy problem[END_REF][START_REF] Henstock | The General Theory of Integration[END_REF], also known as the gauge integral, is particularly a generalization of the well-known Riemann integral.

Definition 1 (Riemann (respectively H-K) integrable) A number I is called the Riemann integral (respectively the H-K integral) of

f : [a, b] → R if for each constant ε > 0, there exists a constant δ (respectively a function δ : [a, b] → R + ),
such that whenever n is a positive integer and t 0 , t 1 , ..., t n and s 1 , s 2 , ..., s n are some numbers satisfying

a = t 0 ≤ s 1 ≤ t 1 ≤ s 2 ≤ t 2 ≤ ... ≤ s n ≤ t n = b and t i -t i-1 < δ (respectively t i -t i-1 < δ(s i )) for all i, then |I - n i=1 f (s i )(t i -t i-1 )| < ε.
Apparently, the only difference between Riemann integral and H-K integral is that the gauge δ in H-K integral is a function of s i instead of constant value in Riemann integral. Taking a more general form of the gauge δ (instead of δ constant) yields a rich class of possible integrands and permits to obtain that

Riemann integrable functions ⊂ Lebesgue integrable functions ⊂ H-K integrable functions

To make the problem much simpler, without loss of generality, we assume that system S is affine with respect to the discrete state q, hence γ q (y) = γ(y)q. Meanwhile define ṗ γ(y)(qq F ) (

Then, according to the characteristics of Zeno phenomenon, such as the Chattering Zeno (i.e. after some time the dwell time is exactly equal to zero) or the Genuinely Zeno (i.e. the dwell time is never equal to zero) [START_REF] Ames | Is there life after zeno? taking executions past the breaking (zeno) point[END_REF], we can make the following assumption:

Assumption 2 The switching terms γ(y)q and γ(y)q F are both H-K integrable over [t 0 , t 1 ] ⊂ R + , where [t 0 , t 1 ] is the interval with Zeno phenomenon or high switching frequency.

Assumption 2 implies that γ(y)(qq F ) is H-K integrable, hence, ∀ ε > 0, it has

p = sup t∈[t0,t1] t t0 γ(y)(q -q F )dτ < ε (13)
where t 1 > t 0 ≥ 0, t ∈ [t 0 , t 1 ] and the integration is in the sense of H-K.

In [START_REF] Cortes | Discontinuous dynamical systems[END_REF], the authors have introduced Caratheodory solutions and Fillipov solutions for discontinuous systems. Analogically, the solution for the system S can be defined in the same way but in the sense of H-K integral, for example, we can define its Caratheodory solutions as

x(t) = x(t 0 ) + t t0 f q (x(s))ds, t > t 0
where the integral is H-K integral for almost everywhere. Since the fact that H-K integral is much more general than Lebesgue (or Riemann) integral, properties of function for Lebesgue integral are also satisfied for H-K integral. And this is the same for Fillipov solutions.

Remark 2 From Assumption 2, one can also obtain that γ(y) is H-K integrable. But it is not sufficient only assuming that γ(y) is H-K integrable. To highlight this point, one example is given below.

Example 3 Define γ : [0, b[→ R, with γ(y) = 1
y sin 1 y 2 for y = 0 and γ(0) = 0. Hence γ is H-K integrable [START_REF] Henstock | The General Theory of Integration[END_REF]. Now considering the following dynamic:

ṗ = γ(y)(q -q F ) = 1 y sin 1 y 2 (q -q F ) (14)
which has a Caratheodory solution in the H-K meaning for sufficiently smooth (qq F ), but does not have Caratheodory solution if the considered integral are restricted to Lebesgue. Nevertheless, it is important to mention that for a particular (qq F ) , as for example q -F = sign(sin 1 y 2 ) equation ( 14) has also no Caratheodory solution in the H-K meaning.

Observability analysis

As discussed above, H-K integral includes Riemann and Lebesgue integral. Therefor, we adopt this integral definition here in order to treat the more general case, even if Lebesgue integral is enough at most of time.

Then define the observability as follows:

Definition 2 Constructively Regularly Weakly Locally Observability: Switched system with Zeno phenomenon or high switching frequency is said to be Constructively Regularly Weakly Locally Observable (CRWLO), if there exists at least one observer for the considered system such that the error between the observation and the original state is locally asymptotically stable.

Observability for the first basic form

Theorem 4 Under Assumption 1, the form S ′ (1) with Zeno phenomenon or high switching frequency, is CRWLO.

Theorem 4 is evident, since the observability of the continuous states z are independent to the discrete state q. Moreover, RWLO can be guaranteed even without any knowledge on q for this form.

Observability for the second basic form

In order to analyze the observability for the second basic form, we need two more assumptions:

Assumption 3 The pair (y, α(z)) of form S ′ (2) is CRWLO.
Assumption 1 and Assumption 3 imply that there exists an observer ẑ for the pair (y, α(z)), given by ż = β(ẑ, y, ŷ) (15) ŷ = ẑ1 [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] and the continuous state observation error e = zẑ is exponentially stable, so there exists a Lyapunov function V (e) such that V (e) = ∂V ∂e (α(z)β(ẑ, y, ŷ)) < -KV (e) [START_REF] Liberzon | Switching in Systems and Control[END_REF] where K > 0. Then, with the same Lyapunov function V and the observer ẑ for the form S ′ (2) designed as follows ż = β(ẑ, y, ŷ) + γ(y)q F (18)

Theorem 5 Under Assumptions 1, 2, 3 and with the knowledge of q F , the form S ′ (2) with Zeno phenomenon or high switching frequency, is CRWLO.

Proof 3 According to [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], the corresponding observation error e of continuous state will change to ė = α(z)β(ẑ, y, ŷ) + γ(y)(qq F ) Let ǫ = ep, where p is defined in [START_REF] Henstock | The General Theory of Integration[END_REF], then dynamic of ǫ can be written

ǫ = α(z) -β(ẑ, y, ŷ)
Adopt the same Lyapunov function V of observer ( 15)-( 16) on ǫ, and compute its derivation

V = ∂V ∂ǫ (α(z) -β(ẑ, y, ŷ))
Applying Taylor expansion on vector ∂V ∂ǫ around -p, one can easily obtain

V = ∂V ∂e | e (α(z) -β(ẑ, y, ŷ)) - ∂ 2 V ∂e 2 | e (O(p) ⊗ (α(z) -β(ẑ, y, ŷ))) + O( p 2 )
where lim p →0 O( p ) = 0. Then thanks to [START_REF] Hermann | Nonlinear controllability and observability[END_REF] and [START_REF] Liberzon | Switching in Systems and Control[END_REF], it has

V < -KV (e) + ∂ 2 V ∂e 2 | e (O(p) ⊗ (α(z) -β(ẑ, y, ŷ))) + |O( p 2 )| < -KV (e) + |O(ε)| Consequently, it is possible to set ε ≪ V d which guarantees that there exists a small M > 0 such that |O(ε)| < M V d . Then ∀V d > 0, and ∀e ∈ {e : V (e) ≥ V d }, it has V < -K ′ V (e)
where K ′ = K -M > 0.

Observability for the extended form

The extended form is just the composition of the two basic forms, for which the observability have been discussed in the previous subsections. Consequently, one can easily conclude the following proposition.

Proposition 6 Under Assumptions 1, 2, 3 and with the knowledge of q F , the form S ′ (3) with Zeno phenomenon or high switching frequency, is CRWLO.

Observability for system S

The following Corollary can be directly concluded from Theorem 1, 2, 3, 4, 5 and Proposition 6.

Corollary 7 Under Assumptions 1, 2, 3 and with the knowledge of q F , system S with Zeno phenomenon or high switching frequency is CRWLO, if conditions in Theorem 1 (or Theorem 2, 3) are satisfied.

Simulations

The objective of this section is to highlight the efficiencies of the observability for proposed forms with Zeno phenomenon or high switching frequency. Without loss of generality, we just use the same example expressed in Example 1 and 2, while we build a new example for the first basic form. To simulate the Zeno phenomenon, the discrete state q is driven by a function defined as σ(t) = 3+sign(w(t)) 2 with w(t) the Gaussian random process. Afterwards, to simulate the filtered discrete signal q F , a low pass filter with cutoff frequency f c = 100Hz is used, where we must guarantee that f c is bigger than the bandwidth of system. At the end, to built the observer, ALIEN method (an numerical algebraic differential method) [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], high gain method [START_REF] Besançon | High-gain observer based state and parameter estimation in nonlinear systems[END_REF] and sliding mode method [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] are respectively adopted. All ODEs in simulations are implemented with solver "db45".

Simulation 1 (The first basic observability form) To verify the observability of form S ′ (1), we design a simple system as

   ż1 = z 2 ż2 = -2z 1 + z 1 q y = z 1 (19)
Easily, one can apply an observer based on the second order sliding mode method to estimate z 2 . And the simulation results are shown in Fig. 2(A).

Simulation 2 (The second basic observability form) Using high gain method to build an observer for system [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF], we can easily write the observer as below ż = α(ẑ) + γ qF (y) -K (ẑ 1y) where K = (100, 100) T . The observation for z 2 is shown in Fig. 2(B). 19) and its observation using sliding mode with ẑ2 (0) = 0; B: ẑ2 of system [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF] and its observation using high gain method with ẑ2 (0) = 0; C: ẑ2 of system [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] and its observation using ALIEN method with ẑ2 (0) = 0; D: ẑ3 of system [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] and its observation using high gain method with ẑ3 (0) = 0.

Simulation 3 (The extended observability form) Considering system [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF], we use ALIEN method to observe z 2 and high gain method to observe z 3 with K = (100, 100) T . The results3 are shown in Fig. 2(C,D).

Conclusion

This paper mainly discusses the observability, defined as Constructive Regularly Weakly Locally Observability (CRWLO), for a class of switched system with Zeno phenomenon or high switching frequency. Particularly, three observability forms are proposed, and sufficient and necessary conditions for the existence of diffeomorphism that can transform systems into each form are presented. On the other hand, CRWLO for each forms are proved under some reasonable assumptions. Moreover, open problems such as the case of multi-output or switching with state jumps also can be investigated in the same method proposed in this paper.

Figure 2 :

 2 Figure2: A: z 2 of system[START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF] and its observation using sliding mode with ẑ2 (0) = 0; B: ẑ2 of system[START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF] and its observation using high gain method with ẑ2 (0) = 0; C: ẑ2 of system[START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] and its observation using ALIEN method with ẑ2 (0) = 0; D: ẑ3 of system[START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] and its observation using high gain method with ẑ3 (0) = 0.

The bouncing ball can be treated as a particular switched system with state jump at the switching time.

The classical locally weakly observability proposed in[START_REF] Hermann | Nonlinear controllability and observability[END_REF] considers the infinite derivatives, while we add regularity with considering just the first n -1 derivatives in the rank condition in this paper.

The time delay for observation using ALIEN method has been compensated in all simulations, and the technology for compensation is presented in[START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF].


L Yu

J.-P Barbot

D Boutat

D Benmerzouk

Observability Normal Forms for a class of switched systems with zeno phenomena

Keywords: Zeno phenomena, switched system, observability

This paper deals with necessary and sufficient conditions to transform a class of switched systems to a particular form dedicated to observer design with and without zeno phenomena. Meanwhile, sufficient observability conditions for switched system with or without zeno phenomena are given. In the last section, some observer structures are proposed upon two academical examples.

I. INTRODUCTION

Z Eno phenomenon is well known in physical domain, for example the bouncing ball case leading to many theoretical developments and simulations [START_REF] Ames | Is there life after Zeno? Taking executions past the breaking (zeno) point[END_REF], [START_REF] Heymann | Analysis of Zeno behaviours in hybrid systems[END_REF]. In systems theory, the first order sliding mode [START_REF] Utkin | Sliding Modes in Optimization and Control[END_REF] and high order sliding mode [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF] [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] have been powerfully used in control design, as well as in observer design for many years. In control design problems, the main property of sliding mode is convergence of the system behavior in finite time under some matching condition [START_REF] Drazenovic | The invariance condition in variable structure systems[END_REF] (respectively some observability condition [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF] in observer design problems). Moreover, this property corresponds to a particular type of Zeno phenomena (i.e. chattering zeno [START_REF] Ames | Is there life after Zeno? Taking executions past the breaking (zeno) point[END_REF] for the first order sliding mode). Based on this correspondence, this paper presents observability conditions for hybrid system with zeno phenomena by considering sliding mode and normal form approaches. These lead to two normal forms: one directly inspired on the observer matching condition and the other one on the Filippov theorem [START_REF] Filippov | Differential Euations with Discontinuous Righthand sides[END_REF] and geometrical results [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF]. For a sake of simplicity, we don't consider a switch on the output function and reset function at switching time.

Obviously, both observability forms presented in this paper can make sense only if there exist at least one observer for estimating the states when there exists zeno phenomena. Consequently, it is necessary to give some samples of building observers for each observability form presented in this paper. This paper is organized as follows: Section II gives the two proposed observability forms. Then conditions for the existence of diffeomorphism which transforms the system into one of the two normal forms are proposed in section III. Afterwards, sufficient observability conditions are presented L.Yu is with Signal Processing Laboratory, Electronic and Information School, Wuhan University, China, and ECS ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, France. lei.yu@ensea.fr in section IV. In section V, conditions for the existence of diffeomorphism which transforms the system into one of the two normal forms for the linear hybrid system are proposed. In the last part, some observers are built for the two normal forms at the end of this paper.

II. PROBLEM STATEMENT

Let us consider the following class of switched system:

ẋ = f q (x), q ∈ {1, . . . , N } (1) y = h(x) (2)
where x(t) ∈ U ⊂ n is the state, y(t) ∈ is the measured output and functions f q : U → n , q ∈ Q and h : U → are smooth for each q. q is the discrete state which may be driven by a switching function σ, where σ : → Q = {1, . . . , N }. In order to deal with observability of systems (1), we will assume within this paper the following: Assumption 1: For each q ∈ Q the pair (y, f q ) is regularly weakly locally observable 1 . Thus, rank{dL j f q h, j = 0 : n -1} = n.

Hereafter we give two normal forms, each of them corresponds to a particular hybrid observability form and different assumptions on the discrete state are requested in order to consider state observation with zeno phenomena.

A. First observability form

The first proposed normal form gives the following form:

       ξ1 ξ2 ξ3 . . . ξn        =        0 ξ 1 ξ 2 . . . ξ n-1        +        fq 0 0 . . . 0        (3)
y = h(ξ) = ξ n (4)
where q ∈ Q is the discrete state. Remark 1: 1-) It is clear from the form (3)-(4) that the observation of the state ξ is independent with respect to the discrete state q. 2-) In the form (3)-(4) the function fq (ξ) may be considered as a perturbation, consequently it is not surprising that the necessary and sufficient conditions to transform the original system in this form are similar to the observability matching condition [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF].

B. Second observability form

The second case considered in the next section is to exhibit sufficient and necessary conditions for the existence of diffeomorphism z = φ(x) such that the system (1)-( 2) is transformed into the following form:

ż = α 0 (z) + γ q (y), q ∈ Q (5) y = z n (6)
Remark 2: The local weakly observability property of system (5-6) doesn't depend on the term of γ q (y). Thus, it is equivalent to the observability of the pair (α 0 (z), z n ).

III. MAIN RESULTS

In this section, the structural conditions is analyzed in order to transform by diffeomorphism system (1)-( 2) into one of both cases previously introduced. Here the difference between two functions is considered:

f ∆ κ,ν (x) = f κ (x) -f ν (x), κ, ν ∈ Q
And this function will be used in the analysis of both case.

A. Condition for the existence of a diffeomorphism in the first case

Hereafter, sufficient and necessary condition are given for the existence of diffeomorphism ξ = ψ(x) such that system (1)-( 2) is transformed into the form (3)-(4). Moreover, the proof of the next proposition is a constructive one.

Proposition 1: Under assumption 1, there exist a diffeomorphism ξ = ψ(x) which transforms the system (1)-(2) into the form (3)-(4) if and only if the following conditions are fulfilled:

L f ∆κ,ν L j-1 f κ h = 0 ∀κ, ν ∈ Q and ∀j ∈ {1, . . . n -1} (7)
Proof: Necessity: the form (3)-(4) satisfy assumption 1 and condition [START_REF] Busawon | An observer design for uniformly observable nonlinear systems[END_REF]. Sufficiency: setting z n = ψ n = h, condition [START_REF] Busawon | An observer design for uniformly observable nonlinear systems[END_REF] gives (L fκ h = L fν h) and assumption 1 implies that dh, dL fκ h are linearly independent, consequently it is possible to define

z n-1 = L f κ h = ψ n-1 . Finally by induction the diffeomorphism is defined as follow z n-i = L i f κ h for i ∈ {1, . . . n -1}. Remark 3: If f ∆ κ,ν
is not trivial then assumption 1 and condition [START_REF] Busawon | An observer design for uniformly observable nonlinear systems[END_REF]

imply that L f ∆κ,ν L n-1 f κ h = 0.
Note that, this condition is closed to the notion of relative index [START_REF] Isidori | Nonlinear Control Systems, Communication and Control Engineering Series[END_REF] and observability matching condition [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF]. Moreover, under this condition and for a switch σ without Zeno we can estimate the discrete state q. (For more details see [6][15]).

Remark 4: From the observability concept introduced by M. Fliess and S. Diop [START_REF] Diop | On nonlinear observability[END_REF] in the theoretical frames of differential algebra, proposition 1 is equivalent to the existence of a common function F (F is independent to q.), for all subsystems, which verifies the following algebraic equation:

x = F (y, ..., y (n-1) )
where y (j) denotes the j th derivative of y.

B. Condition for the existence of a diffeomorphism in the second case

Hereafter, we will exhibit sufficient and necessary conditions for the existence of the diffeomorphism for the second case φ, which transforms the system (1)-(2) into the system (5)- [START_REF] Boutat | About the observability of piecewise dyamical system[END_REF].

Theorem 1: Under assumption 1, sufficient and necessary conditions for the existence of diffeomorphism φ(x) such that the system (1)-( 2) is transformed into system ( 5)-( 6) are: There exist a family of independent vector fields {τ 1 , ...τ n } such that:

a)- dh • τ n = 1 b)- dh • τ i = 0, ∀i ∈ {1, ..., n -1} c)- [τ i , τ j ] = 0, ∀(i, j) ∈ {1, ..., n} × {1, ..., n} d)- [τ i , f ∆κ,ν ] = 0, ∀i ∈ {1, ..., n -1}, ∀κ, ν ∈ Q Proof:
1) Sufficiency: As the family {τ i } 1≤i≤n is a basis of the tangent fibre bundle T U of U and thanks to condition c) there exists a change of coordinates z = φ(x) such that the differential function of φ satisfies:

φ * (τ i ) = ∂ ∂z i for all i ∈ {1, . . . , n}. (8)
Now we will calculate f ∆κ,ν in the z coordinates, from d) and the linearity of φ * we have for i ∈ {1, . . . , n -1}:

∂ ∂z i (φ * (f ∆κ,ν )) = [φ * (τ i ), φ * (f ∆ )] = 0, (9)
which implies that:

φ * (f ∆ κ,ν ) = n i=1 ϕ i (z n ) ∂ ∂z i
Conditions a) and b) mean that z n = y. Therefore there exist functions ϕ κ,ν i (y) 1≤i≤n on U such that:

f ∆ κ,ν = n i=1 ϕ κ,ν i (y)τ i . ( 10
)
2) Conditions a), b), c) and d) are necessary. In fact, assume that:

f ∆κ,ν (x) = n i=1 ϕ κ,ν i (y) ∂ ∂x i and y = x n . ( 11
)
then the seek base is τ i = ∂ ∂x i for i ∈ {1, . . . , n}. Remark 5: A class of dynamical systems which fulfill the conditions of theorem 1 are the dynamical systems, linearizable by the same diffeomorphism and output injection, which are characterized by the theorem of Krener and Isidori [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF].

Here after we will give an example of dynamical system which satisfies the theorem 1, however, it doesn't satisfy the theorem of Krener and Isidori [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF]. This means that the class which satisfies the theorem 1 is larger than the class which satisfies the theorem of Krener and Isidori, this is obviously due to the fact that we avoid a linear condition for α 0 .

Example 1: Consider this dynamical example,

ẋ1 ẋ2 = -x 1 (x 2 -x 1 ) x 1 (x 1 -x 2 -1) + (x 2 -x 1 ) 2 0 y = x 2 -x 1 (12) ẋ1 ẋ2 = -x 1 (x 2 -x 1 ) x 1 (x 1 -x 2 -1) + 2(x 2 -x 1 ) 2 0 y = x 2 -x 1
Now after assemble calculations, the vector fields τ are given by τ 1 = 1 1 and τ 2 = 0 1 .

From system [START_REF] Fogio | The NV-Integrals[END_REF], the difference between the associated two dynamical functions can be obtained as

f ∆ (x 1 , x 2 ) = -(x 2 -x 1 ) 2 0 .
2 By some calculation, the condition a)

and b) can be easily verified. And the condition c) is obvious since the vector fields τ 1 , τ 2 are with constant values. For condition d),

[τ 1 , f ∆ ] = L τ1 f ∆ = 0.
Through the following diffeomorphism

ξ 1 ξ 2 = φ(x) = x 1 x 2 -x 1
The switched system (12) can be transformed as

ξ1 ξ2 = -ξ 1 ξ 2 -ξ 1 + ξ 2 2 -ξ 2 2 y = ξ 2 ξ1 ξ2 = -ξ 1 ξ 2 -ξ 1 + 2ξ 2 2 -2ξ 2 2
y = ξ 2 Remark 6: A necessary condition for the existence of diffeomorphism φ(x) such that the system (1)-( 2) is transformed into system (5)-( 6) is:

Rank ∂f∆ ∂x ∂h ∂x = Rank( ∂h ∂x ) = 1, on U .
Remark 7: It is important to mention that the previous theorem gives a necessary and sufficient conditions for transforming the system (1)-(2) into the system (5)-( 6) but it is necessary to introduce also some extra conditions on the knowledge of the state q (or equivalently σ) in order to be able to design an observer.

IV. SUFFICIENT OBSERVABILITY CONDITIONS

A. Case One

For system of the form (3)-(4) algebraic estimator [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Mboup | Parameter estimation via differential algebra and operational culculus[END_REF] or step by step sliding mode observer [START_REF] Barbot | Sliding mode observer for triangular input form[END_REF], [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] work well in the continuous state estimation, because in the first equation of (3), ξ1 is never considered. More precisely, the output derivative is considered only until n-1 in algebraic solution and the last step is a sliding mode observer of one in the step by step sliding mode observer, i.e. it is requested that ż1 is bounded and the bound is known.

From these and proposition 1, it is possible to set the following corollary:

Corollary 1: Under assumption 1, a sufficient condition for the observability of the continuous state of system ( 5)-( 6) with zeno phenomena is:

L f∆ κ,ν L j-1 fκ h = 0 ∀κ, ν ∈ Q and ∀j = {1, . . . , n -1} B. Case two
In the form ( 5)-( 6), the discrete state q is not considered as a perturbation, consequently, the following assumption is requested:

Assumption 2: The discrete state q is known.

Obviously, this assumption is practically impossible to be guarantied in the case of zeno phenomena, such as for the Chattering Zeno (i.e. after some time the dwell time is exactly equal to zero) or for the Genuinely Zeno (i.e. the dwell time is never equal to zero) [START_REF] Ames | Is there life after Zeno? Taking executions past the breaking (zeno) point[END_REF]. So assumption 2 is replaced by the following one:

Assumption 3: a-The discrete state q is Henstock-Kurzweil-Pettits integrable 3 [START_REF] Henstock | The General Theory of Integration[END_REF][8] [START_REF] Talvila | Estimates of the remainder in Taylor's theorem using the Henstock[END_REF] or NV-Integrable [START_REF] Fogio | The NV-Integrals[END_REF] and its averages is measured via a low pass filter of sufficiently large bandwidth on time interval [0, α] b-The system (1)-( 2) is affine with respect to the discrete state q. The second condition of the assumption above and the conditions of theorem 1 imply that there exists a diffeomorphism such that system (1)-( 2) can be transformed into:

ż = α 0 (z) + γ(y)q (13) y = h(z) = z n (14)
So practically, it is only possible to obtain the filtered discrete state q f (instead of the real discrete state q). Now, let us consider the system (5)-( 6) reduced to the common dynamics:

ż = α 0 (z) (15) y = h(z) = z n ( 16
)
and assuming: Assumption 4: For the system ( 15)-( 16), there exist an observer such that the continuous state observation error (i.e. the difference between the continuous state ant its estimate) is exponentially stable.

Remark 8: Under some specific assumptions as Lipschitz condition, persistent excitation,.. it is possible to use classical high gain observer [START_REF] Besanc ¸on | High-Gain Observer based State and Parameter Estimation in Nonlinear System[END_REF] or adaptive observer [START_REF] Busawon | An observer design for uniformly observable nonlinear systems[END_REF] For these observer classes, the exponential stability of the continuous state observation error is guarantied.

From the previous assumption and theorem 1, it is possible to set the following proposition:

Proposition 2: • -A) Assumptions 1, 2, 4 and conditions of the theorem 1 are sufficient conditions for the continuous state 3 see also Denjoy-Khinchine integrable. observability 4 of system (1)-( 2) with zeno phenomena. • -B) Assumptions 1, 3, 4 and conditions of theorem 1 are sufficient conditions for the practical 5 continuous state observability of the system (1)-( 2) with zeno phenomena.

Proof: Proof of the part A of the proposition: there exists an observer of the form: ż = β(ẑ, y, ŷ) (17) ŷ = ẑn [START_REF] Henstock | The General Theory of Integration[END_REF] which ensures the exponential stability of the continuous state observation error e = z -ẑ. Consequently, there exists a Lyapunov function V (e) with respect to ( 15),( 16) and ( 17),( 18) such that:

V = ∂V ∂e (α 0 (z) -β(ẑ, y, ŷ)) < -KV ( 19
)
with K > 0.

Modifying ( 17),( 18) as follow:

ż = β(ẑ, y, ŷ) + γ q (y) (20) ŷ = z n ( 21
)
The state observation error for system ( 5)-( 6) and observer ( 20)-( 21) is exponentially stable, because the previous Lyapunov function in this case gives:

V = ∂V ∂e (α 0 (z) -β(ẑ, y, ŷ) + γ q (y) -γ q (y)) < -KV
Proof of the part B: the observation error becomes:

ė = α 0 (z) -β(ẑ, y, ŷ) + γ(y)(q -q f ) (22)
From condition a) of the assumption 3, it can be obtained that ṗ = γ(y

)(q -q f ) (23)
which is Cauchy problem in the sense of Henstock-Kurzweil-Pettis integrals. Using the same method as Filippov in [START_REF] Filippov | Differential Euations with Discontinuous Righthand sides[END_REF] page 17, it is possible to set = e -p and it can be obtained that

˙ = α 0 (z) -β(ẑ, y, ŷ) (24)
which admit a local solution in the framework of Carathodory for t ∈ [0, Γ] ⊂ [0, α]. Moreover, the assumption 4 and the observer ( 17)-( 18) ensure that there exist a Lyapunov function V (e) for (15)( 16), which verifies [START_REF] Hermann | Nonlinear controllability and observability[END_REF]. Consequently, derivation of V ( ) with respect to [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], gives

V ( ) = ∂V ∂ (α 0 (z) -β(ẑ, y, ŷ)) (25)
As = e -p, it implies that ∂V ∂ | = ∂V ∂e | and from the assumption 4, it is possible to rewrite the equation [START_REF] Levant | Robust Extract Differentiation via Sliding mode Technique[END_REF] as follows 4 The observation error can be assign in any measurable vicinity of zero 5 The observation error must be as small as it can be but not zero.

V ( ) = ∂V ∂e | e (α 0 (z) -β(ẑ, y, ŷ)) - 1 2
∂ 2 V ∂e 2 | e [O(p) ⊗ (α 0 (z) -β(ẑ, y, ŷ))]
with lim p→0 O(p) = 0. And [START_REF] Hermann | Nonlinear controllability and observability[END_REF] gives

V ( ) ≤ -KV ( ) + 1 2 ∂ 2 V ∂e 2 | e [O(p) ⊗ (α 0 (z) -β(ẑ, y, ŷ))] (26)
As for all > 0, there exists a filter and t 1 ≥ 0 such that ∀t > t 1 , we have p = sup 0≤t≤Γ

t t 1 γ(y)(q -q f ) < ε. Consequently, it is possible, for each V d > 0 to set ε << V d 2
and the inequality (26) becomes

V ( ) ≤ -KV + |O(ε)| So, one can conclude that for e / ∈ E V d := {e/V (e) < V d }, such that V ( ) ≤ - K 2 V ( ) Because, ε << V d 2 guaranties that |O(ε)| < V d 2 .
V. LINEAR VIEWPOINT For linear case, system (1)-( 2) would be written as

ẋ = A q x, q ∈ {1, . . . , N } (27) y = Cx ( 28
)
where A q is a n × n matrix and C is an n × 1 vector.

A. Condition for the first case

The sufficient and necessary condition for the existence of linear diffeomorphism ξ = Φx such that system (27)(28) can be transformed into the form (3)-( 4) is given by, Proposition 3: Under assumption 1, there exists a linear diffeomorphism ξ = Φx which transforms the system ( 27)-( 28) into the form (3)-( 4) if and only if the following condition is fulfilled:

• CA j κ = CA j ν , where j = 1, ..., n -1, κ, ν ∈ Q Proof: Necessity: System (3)-(4) shows that for each i ∈ {1, ..., n -1}, it is satisfied y (i) κ = y (i) ν ( 29
)
this means that the i-th derivative is independent from the discrete state q, i.e. the necessity of the condition is verified. Sufficiency: Construct a diffeomorphism as

ξ k = CA n-k q
where k ∈ {1, ..., n -1} and q = κ or q = ν. Then, we can easily verify the sufficiency of condition.

B. Condition for the second case

The sufficient and necessary condition for the existence of linear diffeomorphism z = Ψx such that system ( 27)-( 28) can be transformed into the form ( 5)-( 6) is given as, Proposition 4: Under assumption 1, a sufficient and necessary condition for the existence of linear diffeomorphism z = Ψx such that the system ( 27)-( 28) is transformed into system ( 5)-( 6) is that there exist a family of independent vectors {τ 1 , ..., τ n } such that:

a) C • τ n = 1; b) C • τ i = 0, with i ∈ 1, ..., n -1; c) (A κ -A ν )
•τ i = 0, with i ∈ 1, ..., n -1, and ∀κ, ν ∈ Q. Proof: a) Sufficiency: Because the vectors {τ 1 , ..., τ n } are independent, there exists a diffeomorphism which is z = Ψx, where Ψ -1 = col{τ 1 , ..., τ n }. So, consider the following system

ẋ = A κ + A ν 2 x + H q A κ -A ν 2 x
where H q = -1 f or q = κ 1 f or q = ν .

And rewrite the first derivative of z with the system above

ż = Ψ ẋ = Ψ A κ + A ν 2 Ψ -1 z + H q Ψ A κ -A ν 2 Ψ -1 z
From condition (a) -(b), we obtain that y = Cx = z n . And from condition (c), H q Ψ Aκ-Aν 2 Ψ -1 z = B q z n = B q y. b) Necessity: Suppose there exists a diffeomorphism z = Ψx, then

ż = Ψ ẋ = Ψ A κ + A ν 2 Ψ -1 z + H q Ψ A κ -A ν 2 Ψ -1 z
In fact col{τ 1 , ..., τ n } = Ψ -1 , then the condition (a) -(c) can be easily verified.

VI. SIMULATIONS

The objective of this section is to highlight the efficiencies of these observability conditions. Moreover, some observers for each system with zeno phenomena are given. To make the simulations more general, three observer structures are adapted, ALIEN observer [START_REF] Mboup | Parameter estimation via differential algebra and operational culculus[END_REF][3] [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF][9], the second order sliding mode observer [START_REF] Utkin | Sliding Modes in Optimization and Control[END_REF][15] [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], and the high-gain observer [START_REF] Besanc ¸on | High-Gain Observer based State and Parameter Estimation in Nonlinear System[END_REF].

A. Example for the first observability form

Let us consider the very simple academical system in the form (3)-( 4)

ż1 = f q (z 1 , z 2 ), ż2 = z 1 (30) y = z 2 (31)
with f q = -z 1 -2z 2 + q and q = sign(w) where w = N is a white noise. This form of q theoretically leads to the Zeno phenomenon. For this system, we apply the ALIEN observer and the second order sliding mode observer to estimate z 1 , one state of system (30)-(31). ALIEN algorithm is numerical differential algebraic approach, proposed by M. Fliess et al. in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], and roughly speaking, the main fundamental principle of this method is the derivative of Laplace transform with respect to s starting from Taylor expansion in the time domain of the original signal. In this paper, we use the individual estimation algorithm [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]. Meanwhile, the second order sliding mode observers are built for the continuous states [START_REF] Utkin | Sliding Modes in Optimization and Control[END_REF] of observer and the right are the difference between observer and original value of the state. Obviously, the state z 1 can be observed from these observer structures, and from the error plot, defined as e erro = |ẑ 1 -z 1 |, the performance of the observer structures can be considered satisfied.

B. Example for second observability form

Let us consider another very simple academical system in the form ( 5)-( 6) with q = 1.5 + 0.5sign(w) and w = N + e -t , where N is a white noise. This form of q theoretically leads to infinite commutation in finite time.

ż1 = -z 2 , ż2 = z 1 + (-z 3 2 + q) (32) y = z 2 (
In this case, due to assumption 4, high-gain observer is adapted.

For this system (32)-( 33), the high-gain observer is designed as follows

ż = A 0 ẑ + φ(ẑ) -Λ(λ) -1 K 0 (C 0 ẑ -y) ( 34
)
where A 0 = (0, -1; 1, 0), C 0 = (0, 1) and φ((ẑ)) = (0, -ẑ 3 2 + q f ) T . Here, we choose λ = 10, K 0 = (0, 100) T and Λ(λ) -1 = diag (1, λ) Above all, we can make a conclusion that it is possible to observe a hybrid system under specific conditions even if zeno phenomenon occurs. Obviously this work considers a very primary case and many other forms and cases should be considered. One of the most difficult problem for us is to observe hybrid systems with jumps under Zeno phenomenon (i.e. the case of bouncing ball).

Fig. 1 .Fig. 2 .

 12 Fig. 1. (Left)Result of ALIEN Observer without noise; (Right)Corresponding Error.

  [START_REF] Floquet | Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. Fig 1, 3 are the results of ideal condition without any noise in the measurement, and Fig 2, 4 are the results of observers with noisy measurements, which makes the simulation a little more realistic, and the power of the noise are assigned to E noise = 0.01. The left figures of Fig 1,2,3,4 are the results

Fig. 3 .

 3 Fig. 3. (Left)Result of second order sliding mode observer without noise; (Right)Corresponding Error.

Fig. 4 .

 4 Fig. 4. (Left)Result of second order sliding mode observer with noise; (Right)Corresponding Error.

.

  The results are shown in Fig 5 without noise in the measurement, and Fig 6 with noise in the measurement. Obviously in both cases the observers work well.

Fig. 5 .

 5 Fig. 5. (Left)Result of High-Gain Observer without noise; (Right)Corresponding Error.

Fig. 6 .

 6 Fig. 6. (Left)Result of High-Gain Observer with noise; (Right)Corresponding Error.

At the classical locally weakly observability[START_REF] Hermann | Nonlinear controllability and observability[END_REF], it is added the regularity of the first n -1 derivatives.

When the switched system is only composed of two subsystems, then f ∆ 1,2 can be abbreviate to f ∆ .


Romain Cauchois
email: romain.cauchois@gemalto.com

Mohamed Saadaoui

Jacques Legeleux

Thierry Malia

Béatrice Dubois-Bonvalot

Karim Inal

Jean-Christophe Fidalgo

Wire-Bonding on Inkjet-Printed Silver Pads Reinforced by Electroless Plating for Chip on Flexible Board Packages

The nanoporous nature of the inkjet printed silver nanoparticles entail low hardness and surface effective contact area for being compatible with pads that are suitable for wire-bonding in electronic packaging. Electroless nickel plating is a selective metal deposition technique which can brings the required thickness and hardness for further pads processing. Here, a 1.7 µm thick nickel layer is deposited on top of 600 nm thick printed and sintered silver nanoparticles using Kapton polyimide as substrate. Prior to plating, a special attention was put on tuning microstructures of printed silver pads by sintering nanoparticles at various temperature ramps (0.1, 10 and 50°C/s) up to 200°C. Results show that fast sintering exhibits the lowest electrical resistivity which is suitable in printed interconnects. However, wire-bonding on nickel pads is best achieved when low sintering ramp is used. This slow sintering presents the highest adhesion strength at the nickel/silver interface since the pores dimensions were restricted to the nanoscale. The validation of the optimized bonding process came from the low electrical contact resistance between plated nickel and the bonded gold wire, and from the wire-pull test which is in accordance with the MIL-STD 883 standard. The actual results show that a compromise has to be found when emphasis is on patterning low resistive interconnects or stiff pads for wire-bonding applications. This compromise is tailored by the sintering engineering.

Introduction

Metallic nanoparticles have been widely studied over the past few years notably for their applications in microelectronics [START_REF] Lu | Low-temperature and pressureless sintering technology for high-performance and hightemperature interconnection of semiconductor devices[END_REF][START_REF] Yamaguchi | Sintering mechanism of composite nanoparticle and its application to bonding in electronics[END_REF][START_REF] Moon | Thermal behavior of silver nanoparticles for low-temperature interconnect applications[END_REF]. In addition to its use as a filler in conductive (ACA/ACF) and non-conductive (NCF) films dedicated to fine-pitch flip-chip applications [START_REF] Li | Novel Monolayer-enhanced Non-Conductive Film (NCF) for Ultra-Fine Pitch High Performance Interconnect in Lead-free Electronics[END_REF], nanoparticles can be used to perform circuitry on rigid or flexible substrates after being dispersed in a solvent based vehicle [START_REF] Kim | Direct writing of silver conductive patterns: Improvement of film morphology and conductance by controlling solvent compositions[END_REF]. Prevalence of their high surface area to volume ratio and its benefit on sintering activation by reducing melting temperature has been reported in previous works [START_REF] Buffat | Size effect on the melting temperature of gold particles[END_REF][START_REF] Safaei | Modelling the size effect on the melting temperature of nanoparticles, nanowires and nanofilms[END_REF]. Those colloidal suspensions exhibit a high processability which enables the use of high resolution and low cost techniques especially inkjet printing. This technique allows the reduction of process steps by a factor of three compared with common photolithography. The later is basically divided in six steps consisting in coating the base layer with several materials, before masking to transfer the circuitry patterns from a dedicated mask. After that, etching and stripping steps are performed. On comparison, inkjet is limited to the pattern direct printing on the base layer and to the curing of the deposited ink in order to remove solvents, to initiate the film cohesion and to improve electrical contact between nanoparticles. This curing step can be carried out using either a conventional oven or other selective techniques such as laser, microwave and joule heating [START_REF] Ko | All-inkjet-printed flexible electronics fabrication on a polymer substrate by low-temperature high-resolution selective laser sintering of metal nanoparticles[END_REF][START_REF] Perelaer | Ink-jet Printing and Microwave Sintering of Conductive Silver[END_REF][START_REF] Allen | Electrical sintering of nanoparticle structures[END_REF]. This additive approach is thus time and cost-saving for the fabrication of 2D interconnects and passive electronics. Nonetheless, films produced by this technique are usually nanoporous which simultaneously affects the electrical and the mechanical properties of printed structures [START_REF] Xiang | Mechanical properties of porous and fully dense low-k dielectric thin films measured by means of nanoindentation and the plane-strain bulge test technique[END_REF].

In this paper, we report on new developments of wirebondable pads for flexible chip-on-board and lead-frame applications using thin films made of sintered silver nanoparticles followed by nickel electroless plating. In fact, low electrical resistivity of nanoporous silver is obtained through the optimization of sintering temperature ramps. Mechanical characterization shows that the Young's modulus and hardness still has to be improved for gold wire bonding process. This is achieved by surface finishing using nickel plating that shows a dependence to the initial microstructure of printed silver pads especially by affecting adhesion strength at the silver/nickel interface. The present technology is demonstrated through the implementation of IC chips in flexible boards.

Tuning of inkjet printing parameters

Patterns have been designed and printed using 40 nm silver nanoparticles ink on A4 Kapton sheet. Test structures include Kelvins, wire-bonding chain pads, a lead-frame for IC chips rerouting and squares for Young's modulus measurements and pull-off adhesion characterization. An in-line inkjet printer prototype developed in the lab (JetPac) was employed to pattern the test vehicle. This fully automated prototype allows sequential pre-treatment and post-processing of printed substrate. The inkjet system is based on drop-on-demand technique which operates by propelling 30 pL droplets through each of the 256 nozzles. Ejections are driven by a bipolar pulse waveform sent to a piezoelectric print head which control the volume and the ejection velocity of droplets. An automaton ensures the coordination between the conveyor speed and the piezoelectric operating frequency for achieving patterns with line width inferior to 100 µm. A commercially available 125 µm thick Kapton HN polyimide substrate from DuPont has been selected for its flexibility, chemical resistance and low shrinkage behaviour at curing temperature up to 350°C. Inkjet process parameters, such as meniscus and droplets ejection velocity, have been finely tuned to achieve high resolution patterns and defect free interconnects. Time of flight adjustment also ensures the quality of the pattern by preventing formation of satellite drops and by reducing geometrical discrepancies.

Microstructural evolution with sintering

Once silver lines are properly printed on the substrate, their electrical conduction is activated by the evaporation of solvents and by the sintering of metallic nanoparticles. Both of these steps are activated by thermal energy. The sintering step is achieved by providing thermal energy to nanoparticles in order to initiate diffusion regimes that will consequently cast bonds between nanoparticles. This phenomenon, called the Gibbs-Thomson effect, results from significant contributions of the surface energy to the free energy of the system. Due to their high value of surface area to volume ratio, diffusion regimes in silver nanoparticles are activated at lower temperature than the bulk melting point (961°C) [START_REF] Buffat | Size effect on the melting temperature of gold particles[END_REF][START_REF] Allen | Small particle melting of pure metals[END_REF]. Nonetheless, densifying and non-densifying sintering regimes have been previously identified as consuming the given energy. Control of the time-temperature couple showed that these various regimes diverge as much by their physical mechanisms [START_REF] German | Sintering Theory and Practice[END_REF][START_REF] Swinkels | A second report on sintering diagrams[END_REF] as by their effects on pore density reduction and thin-film shrinkage.

The electrical properties of the printed thin-film are closely linked to the time and temperature couple during sintering. Since the mean free path of electrons in silver (=53nm) is of the same order of the nanoparticles average radius (<r 0 >=40nm), electrical resistance between nanoparticles is a combination of ohmic and ballistic transport. This physical phenomenon has been modelled by Wexler function (see equ. 1) which states that the higher the contact area between nanoparticles is, the lower the electrical resistivity will be [START_REF] Wexler | The size effect and the non-local Boltzmann transport equation in orifice and disk geometry[END_REF]:

2 3 4 ) ( a a K R W       (1)
where K, (K), a are respectively the Knudsen number (), a factor of the order of unity, the bulk resistivity and the contact radius. Sintering of nanoparticles is thus a critical step toward, promoting grain growth and controlling pores density. By optimizing thermal annealing, different electrical and mechanical properties can be achieved.

Previous work included the optimisation of thermal sintering process conditions for performing low-resistive interconnects on silicon nitride substrate [START_REF] Cauchois | Thermal optimization of silver nanoparticles sintering for low resistive printed applications[END_REF]. The use of organic substrates indeed greatly limits the sintering temperature because of the low glass transition temperature and the high coefficient of thermal expansion. For this purpose, the working temperature on Kapton is then fixed at 200°C.

When sintering temperature is fixed, microstructure of printed features can be tailored with the thermal ramping. Annealing experiments have been performed in nitrogen atmosphere using a Rapid Thermal Anneal (RTA) oven between room temperature and 200°C, at 0.1°C/s, 10°C/s and 50°C/s. Microstructural response to these sintering conditions is shown in Figure 2 through SEM observations. According to Figure 2, increasing the heating rate tends to enhance both the grain growth and the Ostwald ripening of porosities. Image analysis was performed to derive microstructural parameters such as the number of porosities by unit of area (density), the average Feret's diameter of porosities which relates with their size distribution (see table below). The mean grain diameter is growing from 24 nm to 57 nm, according the ramp. This grain growth of silver nanoparticles is intended to reduce interconnects electrical resistivity, regarding Equation 1. However, grain growth comes along with a merge of porosities which limits the film densification and can worsen its conductivity (see Table 2).

Microstructural evolution of printed films was linked to their electrical performances through the measurement of Kelvin test structures. Results are reported jointly with the mean grain size in Table 2. Structures treated at 0.1°C/s show no densification compared with the initial state. Indeed, the mean grain size is similar to the initial nanoparticles size, as if microstructure has been frozen throughout the solvent evaporation step in spite of sintering bonds development. Advanced sintered state is attested by a resistivity of 3.8 µOhm-cm. Interconnects having undergone a 10 and 50°C/s ramp clearly exhibit grain growth and porosities merging which affect electrical conduction as seen in Table 2. The poor densification and high pore size is attributed to the low temperature treatment and to the increasing thermal shock caused on polyimide substrate during RTA treatment. Despite good electrical performances, the aim of the present work is more focused on optimizing the microstructures of printed silver nanoparticles for further wire-bonding experiments.

Interconnects reinforcement by electroless nickel plating

Nanoporous films usually exhibit different mechanical properties compared to the bulk, especially when the elastic modulus and hardness are considered [START_REF] Greer | Mechanical characterization of solution-derived nanoparticle silver ink thin films[END_REF]. Basically, those two parameters have low values that are greatly affected by the porosity as well as by the grain size. For successful wire-bonding operation, a thick and harder pad metallization is required in order to absorb the stress resulting from wire loading.

The direct bondability of gold wire on printed silver pads has been performed by connecting the lead-frame to the IC chip. It has been observed that whatever the microstructure of silver, the bonding step failed. Pad damages including silver splash and lifting are observed (see Figure 3) showing that both cohesive and adhesive fracture modes occurred. Those failure observations were attributed to the low Young's modulus and hardness of sintered silver pads, compared to the gold mechanical properties.

This key issue has been addressed using selective electroless nickel plating for pads reinforcement for gold wire bonding. This process has been widely adopted in microelectronics industry for the past few years as a consequence to lead-free solders specifications and for under-pads damage protection. Electroless nickel is an auto-catalytic process that is deposited from a hypophosphated nickel bath. For electroless nickel plating on porous printed silver pads, a high pH nickel bath is used at 65°C since acidic plating bath tends to chemically etch the silver pads. The growth of electroless nickel has been performed during 5 min in order to achieve a 1.7 µm-thick nickel layer. This thickness is sufficient to improve the wear mechanical resistance and the film hardness and tends to clog the film porosities (See Figure 4 and Figure 5). of inkjet-printed silver nanoparticles has been determined by nanoindentation system in continuous stiffness measurement mode at 45 Hz using Berkovich tip. In order to take into account the high disparity of porosities on silver surface, 64 indentations has been performed on each sample and the resulting average is presented in Figure 6. The Bec model has been used to extract the film Young's modulus from the reduced one corresponding to the film/substrate composite measured by the equipment [START_REF] Bec | A simple guide to determine elastic properties of films on substrate from nanoindentation experiments[END_REF]:

Ni 2+ + H 2 PO 2 -+ H 2 O  Ni + H 2 PO 3 -+ 2H +
              s C f C C C i i r E A E A t A t A E E 2 2 ) 1 ( 2 1 1 2 2     ( 2
)
where E and  are the Young's modulus and Poisson's ratio (r, i, f and s subscripts stand respectively for reduced, indenter, film and substrate properties), A c is the projected contact area and t is the film thickness. Extracted Young's moduli are consistent with microstructures observed before. Indeed, the Young's Modulus is greatly influenced by the presence of pores in the printed structures. In the case of high porosities size, the modulus shrinks to half the bulk Young's modulus. On the other hand, nickel plating increases the overall modulus to values higher than 100 GPa in all cases, due to anisotropic nickel growth. Results from Figure 7 show that the Young's modulus is all the more great that the porosity content is high. Nonetheless, in all cases, Ni-Au plating enabled wedge-bonding of 25µm diameter gold wire with a bonding force comprised between 18 and 20g (see Figure 8).

Packaging performance

Once the feasibility is demonstrated, the process has to be validated by proving its capability in microelectronics systems. A lead-frame has thus been purposely designed and inkjet-printed on a polyimide substrate to host an IC chip in order to test performance of package from adhesion of plated interconnects to wirebonds robustness, including contact resistance measurements between plated pads and gold wire. The testing package is shown on Figure 9. Thermosonic bonding process involves ultrasonic waves together with thermal heating of the wire. The pads thus have to be stiff enough and sufficiently bounded to the substrate to resist such treatment. Pull-off test consists in sticking pre-treated aluminium studs on both side of the film/substrate system [START_REF] Turunen | Pull-off test in the assessment of adhesion at printed wiring board metallisation/epoxy interface[END_REF]. The microporous surface of the studs enables mechanical interlocking of the adhesive after crosslinking. Both studs are then separated applying a normal load until break is observed. Experiments were conducted on silver films with and without electroless nickel plating to illustrate the influence of the process on the adhesion quality (see Figure 10). Experimental results indicate a marked degradation of adhesion properties of the film to the substrate after the plating step. It is believed that the acidic solutions used during electroless nickel plating are the cause of this degradation since cyanide ions in solution corrode silver atoms and thus weaken physicochemical bonds between film and substrate. Robustness of wire bonds is evaluated performing wire-pull tests along the vertical axis. Due to the central position of the hook on the loop, this test sollicitates the wedge bonds both in normal and shear load. During the tests, 90% of the failure modes involved neck break of the wedge bond located at the die side. The maximum pull load remains constant at 3.6g whatever the previous thermal treatment. This shows the good surface finishing of the electroless nickel plating on silver which enables repeatability of the bonding process in an industrial environment without changing bonding parameters.

The good adhesion of gold wires to reinforced pads ensures sufficient electrical conduction for signal transmission while the device is under operation. Nonetheless, the interface between gold and nickel presents a given contact resistance that has to be measured to prove applicability of the whole process. A chain of wire bonds has been processed on Kelvin type structures printed and consequently plated with Ni/Au. Four point probing was used to eliminate the contact resistance of the probe tip during measurements. While current is injected and collected at both ends of the chain, each connection is probed in order to be averaged.

Contact resistance measurements show no tendency with variations comprised between 10mOhm and 45 mOhm. The final validation came from the functionality test of the assembly where the IC-chip interconnected to the printed lead-frame using the above-mentioned technology has been reported in a ceramic package. The reporting step has been performed to be compatible with conventional electrical testing for microelectronic devices. Smartcard ISO pads on a PCB permitted Answer to Request query of the device, a test which happened to be conclusive with full functionality. This final test validated the pertinence of using inkjet printing jointly with electroless plating finishing for packaging applications.

Conclusions

In this paper, we reported the latest developments on inkjet-printed interconnects compatibility with standard wired connections. Wedge bonding of 25 μm diameter gold wire has been performed on finished surfaces using electroless nickel immersion gold plating process. Electrical and microstructural characterization of the base silver layer has been performed so that the latching of the nickel growth could be tailored. The challenge, here, was to successfully perform wired connections on printed and malleable bond-pads whose thickness can fall down to 300 nm. The improved mechanical behaviour measured by nanoindentation has been correlated to the good performances of the wedge bonding, both electrically and mechanically in tensile stress. Robustness of the wired connections seemed controlled more by the surface state of nickel layer than to the initial microstructure of silver layer. A functionality test finished to validate the process exposed in this paper for chip-on-flexible devices.

The prospective advantage of direct printing for interconnects and bond-pads technologies combined with electroless plating is thus a good alternative to conventional metallization techniques and offers new opportunities for low cost and flexible electronic packages.

Figure 1 :

 1 Figure 1: Inkjet-printed test structures on Kapton substrate.

Figure 2 :

 2 Figure 2: Surface microstructures of silver interconnect after sintering at 0.1, 10 and 50°C/s (resp. from left to right) via SEM observations.

Figure 3 :

 3 Figure 3: SEM image of the gold ball bonding failure on a printed silver pad, showing layer decohesion and lift.

Figure 4 :

 4 Figure 4: Surface microstructures of electroless nickel plating, showing characteristic nickel nodules, on silver sintered at 0.1, 10 and 50°C/s (resp. from left to right) via SEM observations.

Figure 5 :

 5 Figure 5: SEM view of tilted cross-section of the pad before and after nickel plating process

Figure 6 :

 6 Figure 6: Load vs. displacement curves of inkjet-printed silver film before and after electroless nickel process.

Figure 7 :

 7 Figure 7: Variation of Young's modulus (histogram) and of initial pore densities (curve) according the thermal treatment before and after electroless nickel plating process.

Figure 8 :

 8 Figure 8: Gold wire-bonds on silver printed structures plated with electroless nickel process on a polyimide substrate.

Figure 9 :

 9 Figure 9: Dummy flexible package with wire-bonds from an IC chip to plated lead frame (front right and bottom) and between plated pads (front left).

Figure 10 :

 10 Figure 10: Results of pull-off test performed on printed silver films (black square) and electroless nickel plated films (red circle).

Figure 11 :

 11 Figure 11: Apparatus used for functionality tests where the chip-on-flex has been reported in a ceramic package (inset view) to be plugged-in an ISO test board.

Table 1 :

 1 Microstructural properties derived from image analysis.

		0.1°C/s 10°C/s 50°C/s
	Porosities density (µm -²)	139	78	36
	Avg. Feret's diameter (nm)	47	65	88

Table 2 :

 2 Microstructural and electrical evolution of printed silver nanoparticles under different thermal ramping (200°C, 15min)

		0.1°C/s	10°C/s	50°C/s
	Resistivity (µOhm-cm)	5.01	4.52	3.77
	Mean grain diameter (nm)	24	42	57

Acknowledgments

Presented printings were conducted on the JetPac equipment, an inkjet printer prototype developed in the lab with funding from EU (ERDF) and local authorities' research initiatives. This work was supported by the EURIPIDES office through COSY-3D project.


Séverine Sabia

Alice Guéguen

Claudine Berr

Lisa Berkman

Joël Ankri

Marcel Goldberg

Marie Zins

Archana Singh-Manoux

High alcohol consumption in

Figure S1 .

 S1 Figure S1. Analyses restricted to participants without depressive symptoms (CESD<16; N=2992) † † Adjusted for age, screening centre, marital status, and smoking history. B. Non-depressed participants (CESD<16)