S Hénaff

I Taralova
email: ina.taralova@irccyn.ec-nantes.fr

C H Moog

Observability and left invertibility of nonlinear discrete time systems with under-sampling

Keywords: nonlinear systems, discrete time systems, under-sampling, observability, system theory

This paper deals with observability and left-invertibility of under-sampled systems. The concept of under-sampling of a discrete time system is introduced in the context of system theory for the first time in this article. Under-sampled systems are explained under a standard realization that allows to apply system theory. In particular, observability and left invertibility of these systems is investigated. It is shown that rewriting the under-sampled system under a standard form allows to characterize completely the observability and the left invertibility of the original system. One of the potential applications concerns the decrease of the number of output measurements without loss of observability of the original system.

Introduction

The phenomenon of under-sampling appears in numerous engineering applications. Typically, under-sampling of the outputs is encountered in real-time controlled systems, where the sensors are connected by a network, subject to failures: the frames (i.e. the embedded output measurements) can be lost in a completely irregular way. Another example is the case of a sensor with temporal failures (e.g. supply problems), or a bloated network, where information packages can be arbitrarily lost. If the processing time of a system needs to be reduced, then an under-sampling may also be of interest since less data will be processed. An overview of networked control systems can be found in [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], where a stochastic approach has been applied. The lossy channel has been modelled by a stochastic process " theta " modelling the dropout, in order to do a state estimation over a lossy network. In [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] the stability of linear time-invariant sampled data system with uncertain and variable sampling intervals has been analyzed using Lyapunov functions.

Another application where the under-sampling can be useful are chaos-based secure communications. The chaotic carrier is chosen deliberately according to a chaotic sampling of the generator's output, i.e. not all output data are used for the transmission [START_REF] Hénaff | Statistical and spectral analysis of a new coupled maps systems[END_REF]. In all the above examples, the output information arrives at irregular, and unpredictable instants. However, the control (or transmission) system must still be reliable, i.e. able to generate the correct action, by reconstructing the missing samples.

Such notions are popular in signal processing, for example in information compression, although no formal generally accepted concepts have been defined to date. For instance, an under-sampling [START_REF] Malik | x -f choice: Reconstruction of undersampled dynamic mri by data-driven alias rejection applied to contrast-enhanced angiography[END_REF] -or subsampling [START_REF] Politis | Subsampling[END_REF], downsampling [START_REF] Abe | Image restoration from a downsampled image by using the DCT[END_REF] -of a discrete signal generates signal that looses some information according to a rule whereas an over-sampling [START_REF] Labeau | Oversampled filter banks as error correcting codes: theory and impulse noise correction[END_REF] -or upsampling [START_REF]Image upsampling via imposed edge statistics[END_REF], interpolation [START_REF] Knott | Interpolating cubic splines, volume 18 of Progress in computer science and applied logic[END_REF] -adds some points between two instants according to a rule.

In the rest of the paper, we will focus on discrete time systems and the notion of under-sampling, which is sketched in Figure 1(b). More precisely, the figure shows the evolution of the time of a signal, indexed by time n and the one of an under-sampling signal, indexed by instant k. Both concepts of over-sampling and under-sampling are widely used in signal processing. In opposition to the main stream of the current literature, a system theoretic approach is introduced in this paper; it is shown to be instrumental to analyze the above challenging problems and to derive appropriate solutions. The new approach is of paramount importance since it leads to a new formalism that highlightens both analysis and synthesis of such under-sampled systems. The possibility to reconstruct -or not -the missing output samples of the original system is naturally recasted in terms of observability matrices and left invertibility. As for the synthesis, if the under-sampled system is still observable, the number of output measurements can be reduced consequently without loss of observability. The possibility to rewrite into a so-called standard form is naturally stated as a realization problem. To our best knowledge, paralleling the open problem of reconstructed missing signal samples with such system theoretic notions as realization and observability has never been studied before, and therefore shall be introduced for the first time in this paper.

Sampling of continuous time systems is still studied including the case of non-uniform sampling [START_REF] Huke | Embedding theorems for non-uniformly sampled dynamical systems[END_REF][START_REF] Kreisselmeier | On sampling without loss of observability/controllability, Automatic Control[END_REF][START_REF] Ding | Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems[END_REF] whereas under-sampling of discrete time systems has surprisingly never been considered to date. The first pioneering results on this topic are established in this paper. A realization study of under-sampled systems allows to apply systems theory and to announce new results. Amongst, special cases of constant, periodic or random, known or unknown samplings shall be considered. Full straight forward characterisations are derived from the observation in such special cases and surprisingly, they have not been reported in the current literature. Preliminary results of this work are available in [START_REF] Hénaff | Actes de la conférence internationale francophone d'automatique[END_REF].

This paper is organized as follows: the concept of under-sampling is formalized and the observability and left invertibility problems for the non-standard discrete time system under undersampling is stated in section two. Section three recasts characterization of observability in terms of realization. Section four examines the observability of systems for different kinds of under-sampling.

Preliminaries

Consider a classical discrete time system:

x(n + 1) = f (x(n), u(n)) y(n) = h(x(n)) (1)
where f : R N × R → R N and h : R N → R are analytic. x represents the states of the system, u its input and y its output. For simplicity, u and y are considered as being one-component vectors but this article can be generalised to multi-component vectors. Under-sampled systems are considered in this paper so to define it, let define the under-sampling. This notion is usually used in signal processing theory. A signal s e is said to be under-sampled of s if there exists an under-sampling step p ∈ N such that s e (n) = s(pn) [START_REF] Oppenheim | Discrete-time signal processing[END_REF]. This paper consider a more general case where p vary. Definition 2.1 (Under-sampled signal). Let s : N → R N a signal.

s e : N → R N is said to be an under-sampled signal of s if the sequence (s e (n)) n∈N is a subsequence of (s(n)) n∈N .

s and s e denote the same physical signal. Therefore, up to the rest of the paper, an undersampled signal would be denoted by the same variable s. However, to avoid confusion, s(n) would denote the original signal and s[k] would denote the under-sampled signal.

Some definitions should be necessary to study under-sampled systems. Let first introduce the under-sampling. Definition 2.2 (Under-sampling). Let s(.) : N → R N a signal and s[.] : N → R N an under-sampled signal of s(.).

q

: N → N that verifies s[k] = s(q[k]), k ∈ N
is called under-sampling of signal s(.).

Proposition 2.1 (Under-sampling). An under-sampling k → n = q[k], q : N → N is an injective and strictly increasing map.

Proof. (s[k]

) k∈N is a subsequence of (s(n)) n∈N and s[k] = s(q[k]), k ∈ N so q is an injective and strictly increasing map.

These definitions allows to define under-sampled systems. The action of the under-sampling q on (1) results into:

x(n + 1) = f (x(n), u(n)) y[k] = h(x(q[k])) (2)
The particularity of this system is that it runs under two different time scales k and n: the state x evolves with the index n whereas the output y evolves with the index k. Indeed, the output of system (1) does not exist for all instants n but only at the instant q[k]. The limit case is attained when q is identity, there would not have no under-sampling.

Let us introduce the under-sampling step p[k]. This is instrumental for writing the realization of the under-sampled system under a standard form. Definition 2.3 (Under-sampling step). The step p of an under-sampling q is defined by

p[k] = q[k + 1] -q[k], k ∈ N
A visualisation of both under-samplings and under-sampling steps is in Figure 2. Definition 2.4 (Left invertibility). The discrete-time undersampled system (2) is said to be left invertible if the input u(q[k]) can be expressed as a function of the state vector, the output, and its forward iterates:

u[k] = G(x[k], y[k], y[k + 1], . . .) (3)
n k 0 1 2 3 4 5 0 1 2 q[0] q[1] q[2] p[0] = 2 p[1] = 3
ρ := min{j ∈ N | ∂y[k + j] ∂u[k] ≡ 0} (4) Note that ρ[k] is necessarily ≤ p[k].
Example 2.1. Consider the following system with the under-sampling p[k] = 8.

           x 1 (n + 1) = x 2 (n) x 2 (n + 1) = x 3 (n) x 3 (n + 1) = x 4 (n) x 4 (n + 1) = u(n) y[k] = x 1 (8k) = x 1 [k]
Here, ρ = 1 and ρ < p.

It should be noted that if the undersampling p[k] is not constant, than the relative degree may not be constant either, and depends on the time instant [k].

Problem statement 2.1 (Observability). Given an under-sampled system (2), find if possible a function F and two integers ν 1 and ν 2 such that, locally,

x[k] = F (y[k], y[k + 1], . . . , y[k + ν 1 ], u(q[k]), u(q[k + 1]), . . . , u(q[k + ν 2 ]))
This is an observability problem for the non standard system (2).

Although there exists a tremendous literature on observation and observability of non-linear systems, no obvious solution of the above problem can be derived. Before defining observability, let firsty define distinguibility. Definition 2.6 (Distinguishability of under-sampled systems). Let (x 0 , x0 ) ∈ R N . Let y and ỹ denote respectively the output of the under-sampled system (2) (for the same under-sampling q[k]) with initial conditions x 0 and x0 . The states x 0 and x0 are said to be distinguishable if there exists an integer k and an input u

[k] such that ỹ[k] = y[k].
Definition 2.7 (Observability of under-sampled systems). An under-sampled system (2) is said to be locally observable in x 0 if there exists a neighborhood F of x 0 such that for all states x ∈ F \{x 0 }, x and x 0 are distinguishable.

Therefore, if there exists an integer k ∈ N such that ỹ[k] = y[k] for q[k] ≥ n 0 where y denotes the output of the system with initial condition x 0 and ỹ denotes the system with an initial condition x.

Computable criteria for observability of under-sampled systems are reviewed in Section 4. According to this definition, local observability of systems without under-sampling have been characterized. System (1) is locally observable if the observability matrix O 1 below has rank N .

O 1 =       ∂y(n) ∂x 1 (n) ∂y(n) ∂x 2 (n) • • • ∂y(n) ∂x N (n) ∂y(n+1) ∂x 1 (n) ∂y(n+1) ∂x 2 (n) • • • ∂y(n+1) ∂x N (n) . . . . . . . . . ∂y(n+N -1) ∂x 1 (n) ∂y(n+N -1) ∂x 2 (n) • • • ∂y(n+N -1) ∂x N (n)       rank(O 1 ) = N (5)
The under-sampling can lose the observability. The following example show this:

Example 2.2. Consider the following system in the same form as [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF].

       x 1 (n + 1) = x 2 (n) x 2 (n + 1) = x 3 (n) x 3 (n + 1) = x 1 (n) y[k] = x 1 [k] y(3k) = x 1 (0), y(3k + 1) = x 2 (
0) and y(3k + 2) = x 3 (0). So this system is completely observable. Now apply an under-sampling every three iterations:

q[k] = 3k. ∀k ∈ N, y[k] = x 1 [k];
x 2 and x 3 are independent and never modify the output so both components x 2 and x 3 are not observable. So an under-sampling can make a system unobservable whereas the original system can be observable. Now consider an other sampling q(k) = 2k. The output is defined by y(3k) = x 1 (0), y(3k + 1) = x 3 (0) and y(3k + 2) = x 2 (0). This system is completely observable so an under-sampling does not necessary make a system loose its observability.

Moreover, decreasing the number of output samples can preserve the observability of the original system, therefore lower number of measurements (i.e. less memory processing time) could be sufficient to observe the system states.

Realization of an under-sampled system under standard form

Control theory does not give any result to analyze non-uniformly sampled systems. The goal of this section is to find out the conditions under which an under-sampled system can be written under a standard form, more precisely, this is a realization problem using only one single time scale k. Two cases of realization shall be considered hereafter: a standard and a generalized one.

Standard realization

Problem statement 3.1 (Realization). Given (2), find a realization (6) whenever it exists.

z[k + 1] = g(z[k], ū[k]) y[k] = h (z[k]) ( 6
)
where ū is a new input function depending on u, in the general case ū

[k] = [u(q[k]), u(q[k] + 1), u(q[k] + 2), ..., u(q[k + 1] -1)
. z is a priori different from x and their dimensions are not necessary equal. A solution to problem 3.1 does not always exist if ū = u as shown by the following example: Example 3.1. Consider the following system with the under-sampling q

[k] = 2k, p[k] = 2: x(n + 1) = u(n) y[k] = x(2k) = x(q[k]) = x[k]
The corresponding input-output equation is

y[k] = x(2k) = u(2k -1) or equivalently y[k + 1] = x(4k) = u(4k -1)
Im(q) = 2N so ∀k ∈ N, 2k -1 / ∈ Im(q) so there does not exist any k such that u(2k

-1) = u(q[k ]) = u[k ].
Finally, problem 3.1 is not solvable for this system.

The following example show a system that can be written under a realization [START_REF] Abe | Image restoration from a downsampled image by using the DCT[END_REF].

Example 3.2. Consider the following system with the under-sampling q(k) = 2k, p[k] = 2:

   x 1 (n + 1) = x 2 (n) x 2 (n + 1) = u(n) y[k] = x 1 (2k)
It can be reformulated under form [START_REF] Abe | Image restoration from a downsampled image by using the DCT[END_REF] as

z[k + 1] = u[k] y[k] = z[k] Consider an application h k = h O f q [k] that
∀n ∈ N \ Im(q), ∀k ∈ N * | ∂h k [k] ∂u(n) = 0
where N * = N \ {0}.

Proof. Consider f i defined by induction:

f 1 (x(n), u(n)) = f (x(n), u(n)) and f 2 (x(n), u(n), u(n + 1)) = f 1 (x(n + 1), u(n + 1)) = f 1 (f 1 (x(n), u(n)), u(n + 1)) f 3 (x(n), u(n), u(n + 1), u(n + 2)) = f 1 (f 2 (x(n), u(n), u(n + 1)), u(n + 2)) f i+1 (x(n), u(n), . . . , u(n + i)) = f 1 (f i (x(n), u(n), . . . , u(n + i -1)), u(n + i))
Consider system (2) which satisfies condition of theorem 3.1. The following system describes the same output y.

z(n + 1) = f (z(n), ū(n)) y[k] = h(z(q[k])) with ū(n) = u(n) if n ∈ Im(q) 0 else .
so the following realization is equivalent to the previous one

z[k + 1] = f p[k] (z[k], u[k], 0, 0, . . . , 0) y[k] = h(z[k])
This is a realization on the form of ( 6).

From the theorem 3.1 it comes that Corollary 3.1. System (2) is left invertible in the sense of the definition 2.4 if and only if

(i) ∂h k [k] ∂u(n) = 0 ∀ n ∈ Im(q) (ii)∃k, ∂h k [k] ∂u(q[k]) = 0
Example 3.3. Consider the system defined in example 3.1 with q(k) = 2k.

∂y(2k) ∂u(2k -1) = 1, k ∈ N * ∀k ∈ N * , 2k -1 /
∈ Im(q) so problem 3.1 is not solvable.

Example 3.4. Consider the system defined in example 3.2 with q(k) = 2k.

y[k] = u(2k -2), k ∈ N * Im(q) = 2N so ∀k ∈ N, 2k -2 ∈ Im(q) so condition of Theorem 3.1 is satisfied. Corollary 3.2. Problem 3.1 is solvable if (2) is an autonomous system.
Proof. If system ( 2) is an autonomous one,

∂y[k] ∂u(n) = 0, (k, n) ∈ N 2
So conditions of Theorem 3.1 are verified. An equivalent realization is then

x[k + 1] = f p[k] (x[k]) y[k] = h(x[k])
Obviously, the condition is not necessary as shown by example (3.2).

Example 3.5. Consider the under-sampling (q(0); q(1); q(2)) = (0; 3; 5), then (p(0); p(1)) = (3; 2). The under-sampled system can be reformulated by

       x[1] = f of of (x[0]) x[2] = f of (x[1]) • • • y[k] = h(x[k])

Generalized realization

Not all systems can be written under realization [START_REF] Abe | Image restoration from a downsampled image by using the DCT[END_REF]. This section considers an other equivalent realization which is derived from some redefinition of the input.

Problem statement 3.2 (Realization). Find g and a prolongation ū such that a realization (7) is equivalent to a given (2) whenever it exists.

z[k + 1] = g(z[k], ū[k]) y[k] = h (z[k]) (7)
Theorem 3.2. System (2) has always a realisation [START_REF] Labeau | Oversampled filter banks as error correcting codes: theory and impulse noise correction[END_REF].

Proof. Consider a p[k]-component vector ū defined by

ū[k] = (u(q[k]); u(q[k] + 1); u(q[k] + 2); . . . ; u(q[k + 1] -1))
Let f i defined by induction:

f 1 (x(n), u(n)) = f (x(n), u(n))
and

f 2 (x(n), u(n), u(n + 1)) = f 1 (x(n + 1), u(n + 1)) = f 1 (f 1 (x(n), u(n)), u(n + 1)) f 3 (x(n), u(n), u(n + 1), u(n + 2)) = f 1 (f 2 (x(n), u(n), u(n + 1)), u(n + 2)) f i+1 (x(n), u(n), . . . , u(n + i)) = f 1 (f i (x(n), u(n), . . . , u(n + i -1)), u(n + i))
Let g such that

g(x[k + 1]) = f p[k] (x[k], u(q[k]), u(q[k] + 1), . . . , u(q[k + 1] -1))
Then, realizations ( 2) and ( 7) are equivalent.

It can be noticed that if p only depends on the value of the state x, then the system is stationary.

Example 3.6. Consider the system defined in example 3.1 with q(k) = 2k. Let ū[k] defined by ∀k ∈ N * , ū[k] = (u(2k); u(2k + 1)). Then, the system is equivalent to the following one:

z[k + 1] = ū2 [k] = u(2k + 1) y[k] = z[k]
The system (2) has been rewritten under form (6) by using the standard output function y[k] = h(x[k]) to allow to appear the same indexes in both state and output equations but not all systems can be written under this form. This has been solved by considering a more general equation ( 7) that characterises all under-sampled systems.

Observability and left-invertibility of under-sampled systems

After finding out an equivalent standard realization of any under-sampled systems, this section deals with observability and left-invertibility study whenever the conditions in Theorem (3.1) are fulfilled.

x

[k + 1] = f p[k] (x[k], u[k]) y[k] = h(x[k]) (8)
To our best knowledge, the observability and left-invertibility of such systems has never been investigated. This section considers two observability problems: the under-sampling is firstly considered as being known by the observer before considered as being unknown.

Without loss of generality, assume the system (1) to be left-invertible.

Theorem 4.1 (Existence of a left-inverse). Consider a non autonomous under-sampled system (8).

The following three statements are equivalent:

• there exists a standard realization of the under-sampled system (8),

• the under-sampled system ( 8) is left-invertible,

• the under-sampling step p is a multiple of the relative degree ρ.

Proof. If system (2) is left-invertible in the sense of definition 2.4, the input can be explicitely recovered and there exists a realization, and vice-versa. This proves the equivalence of the two first items in Theorem 4.1.

If the under-sampling step p is a multiple of the relative degree, then q

[k + 1] -q[k] = m * ρ for some m ∈ N * . Thus ∂h m * ρ [k] ∂u(q[k]) = 0
and according to Corollary 3.1 the under-sampled system ( 8) is left invertible. Conversely, it is obvious from Definition 2.5 that if the relative degree and the under sampling step are not multiple, then the under sampled system can not be left-invertible.

Note that for the original system (1), a finite relative degree is necessary and sufficient for left-invertibility. This is absolutely not the case for the under-sampled system, where the relative degree varies as a function of the undersampling which can be unknown (aperiodic, chaotic etc). Here ρ = p since ρ < p. The input can be recovered after one iteration of the under-sampled system.

In the rest of this paper, it is assumed that the under-sampled system (8) is either autonomous or fulfils the conditions in Theorem 4.1.

Problem statement 4.1 (Observability of known under-sampling). Is a given under-sampled system (9) observable for an undistinguished known under-sampling?

x[k + 1] = f p[k] (x[k], u[k]) y[k] = (h(x[k]); p[k]) (9)
To do so, several under-sampling models are considered to make the under-sampling more and more complex. The first model is a classical one (p = 1 or q = id) as system [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]. The second one is the periodic under-sampling (with constant p or linear q). Then, a regular under-sampling characterise a periodic step p. Finally, an undistinguished known under-sampling, a priori aperiodic, is considered.

The final aim of the work is to find out some observers for all of these under-sampling models. But before designing some observers, the conservation of the observability should be verified.

System without under-sampling (p = 1)

Consider the following system

x(k + 1) = g(x(k), u(k)) = f (x(k), u(k)) y(k) = h(x(k))
This kind of system is standard so the observability and the design of observers can be performed classically.

Observability criterion 4.1. An under-sampled system (9) without under-sampling q(k) = k or p = 1 is observable if and only if the rank of the observability matrix is N :

rank      ∂h ∂x 1 ∂h ∂x 2 • • • ∂h ∂x N ∂hof ∂x 1 ∂hof ∂x 2 • • • ∂hof ∂x N . . . . . . . . . ∂hof N -1 ∂x 1 ∂hof N -1 ∂x 2 • • • ∂hof N -1 ∂x N      = N Remark 4.
1. Consider a linear system (10) without under-sampling.

x[k + 1] = A p[k] x[k] + Bu[k] y[k] = (Cx[k]; p[k]) (10)
The criterion becomes rank

     C CA . . . CA N -1      = N

Periodic under-sampling

A periodic under-sampled system is written as:

x[k + 1] = g(x[k], u[k]) = f p (x[k], u[k]) y[k] = h(x[k])
The step p is constant in this model. g is then f composed p times: g = f p so this problem is the same as for the system without any under-sampling but by changing f by f p . Observability criterion 4.2. An under-sampled system (9) with a periodic under-sampling q(k) = kp is observable if the rank of the observability matrix is N :

rank      ∂h ∂x 1 ∂h ∂x 2 • • • ∂h ∂x N ∂hof p ∂x 1 ∂hof p ∂x 2 • • • ∂hof p ∂x N . . . . . . . . . ∂hof p(N -1) ∂x 1 ∂hof p(N -1) ∂x 2 • • • ∂hof p(N -1) ∂x N      = N Remark 4.2.
For linear systems [START_REF] Huke | Embedding theorems for non-uniformly sampled dynamical systems[END_REF], the criterion becomes rank

     C CA p . . . CA p(N -1)      = N

Alternate under-sampling

The alternate under-sampling is defined as being an under-sampled system with a step that alternates from two different steps p 1 and p 2 . The system can be written under the form:

x[k + 1] = g(x[k], u[k]) y[k] = h(x[k]) with g(x[k]) = f p 1 (x[k], u[k]), if k is even g(x[k]) = f p 2 (x[k], u[k]), if k is odd
The above equations are equivalent to [START_REF]Image upsampling via imposed edge statistics[END_REF] with p(k) = p 1 when k is even and p(k) = p 2 when k is odd.

Contrary to both afore under-sampled models, the state equation changes in the time so the study of observability must be different than in the usual case. Vidal, Chiuso et Soatto have studied observability of a particular trajectory of jump linear systems [START_REF] Vidal | Observability and identifiability of jump linear systems[END_REF], which are linear systems whose state function change in the time.

The study of local observability of a system leads to consider a linearized system. By using the study of the observability of jump linear systems, the observability is determined by the rank of the observability matrix whose dimension is (2(σ + 1) × N ), where σ denotes the number of switches between the under-sampling steps p 1 and p 2 .

Observability criterion 4.3. An under-sampled system [START_REF] Knott | Interpolating cubic splines, volume 18 of Progress in computer science and applied logic[END_REF] without alternate under-sampling p 1 and p 2 , as defined before, is locally observable if the rank of the observability matrix is N :

rank              ∂h ∂x 1 • • • ∂h ∂x N ∂hof p 1 ∂x 1 • • • ∂hof p 1 ∂x N ∂hof p 1 +p 2 ∂x 1 • • • ∂hof p 1 +p 2 ∂x N ∂hof 2p 1 +p 2 ∂x 1 • • • ∂hof 2p 1 +p 2 ∂x N . . . . . . ∂hof σp 1 +σp 2 ∂x 1 • • • ∂hof σp 1 +σp 2 ∂x N ∂hof (σ+1)p 1 +σp 2 ∂x 1 • • • ∂hof (σ+1)p 1 +σp 2 ∂x N              = N Remark 4.3.
For linear systems [START_REF] Huke | Embedding theorems for non-uniformly sampled dynamical systems[END_REF], the criterium becomes rank

     C CA p 1 . . . CA (σ+1)p 1 +σp 2      = N

General known under-sampling

In this section, the under-sampled step p evolves but is known for the observer.

x

[k + 1] = g(x[k], u[k]) = f p[k] (x[k], u[k]) y[k] = (h(x[k]); p[k]) (11)
Following the previous section, the criteria of observability can be generalized.

Proposition 4.1 (Observability). An under-sampled system (11) is locally observable if there exists an integer τ such that the rank of the observability matrix is N .

rank         ∂h ∂x 1 ∂h ∂x 2 • • • ∂h ∂x N ∂hof q[1] ∂x 1 ∂hof q[1] ∂x 2 • • • ∂hof q[1]
∂x N ∂hof q [2] ∂x 1

∂hof q [2] ∂x 2

• • • ∂hof q [2] ∂x N . . . . . . . . .

∂hof q[τ ] ∂x 1 ∂hof q[τ ] ∂x 2 • • • ∂hof q[τ ] ∂x N         = N (12)
Remark 4.4. For linear systems [START_REF] Huke | Embedding theorems for non-uniformly sampled dynamical systems[END_REF], the criterion becomes rank [1] . . .

     C CA q
CA q[τ ]      = N
The use of the integer τ is necessary for determining the observability because an under-sampled system defined by analytic functions f and h can be unobservable for an iteration τ -1 > N and observable for an iteration τ . The following example illustrates this for a second order system defined by linear functions f and h. Example 4.2. Consider the following system:

   x 1 (n + 1) = √ 2 2 (x 1 (n) + x 2 (n)) x 2 (n + 1) = √ 2 2 (-x 1 (n) + x 2 (n)) y[k] = x 1 (q[k])
Consider the under-sampling defined by: (q[0]; q [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]; q [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]; q[3]) = (0; 4; 8; 9).

x 1 (n + 4) = -x 1 (n) x 2 (n + 4) = -x 2 (n)
So the observability matrix is:

    1 0 -1 0 1 0 √ 2 2 √ 2 2    
Unlike the classical case, this second order system requires four iterations in order to be observable.

Proposition 4.2. The ranks of the observability matrix O 1 of the system (1) and the observability matrix of O 2 of the under-sampled system (2) satisfy :

rank(O 2 ) ≤ rank(O 1 )
Proof. Let O 1 be the observability matrix of system (1) without under-sampling and O 2 the observability matrix of the system with an under-sampling [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]. The observability matrix of (1) is

O 1 =       ∂y(n) ∂x 1 (n) ∂y(n) ∂x 2 (n) • • • ∂y(n) ∂x N (n) ∂y(n+1) ∂x 1 (n) ∂y(n+1) ∂x 2 (n) • • • ∂y(n+1) ∂x N (n) . . . . . . . . . ∂y(n+σ-1) ∂x 1 (n) ∂y(n+σ-1) ∂x 2 (n) • • • ∂y(n+σ-1) ∂x N (n)      
The under-sampled system (2) is defined as being (1) without some measurements. So the observability matrix O 2 is O 1 where some rows have been dropped out, thus rank(O 2 ) ≤ rank(O 1 ).

Unknown under-sampling

Unlike the previous models, the under-sampling step p is now unknown for the observer. This is the most intricate case with respect to the characterization of observability. This problem is qualitatively different from the previous one where the under-sampling step was known. A n = cos(πnω) -sin(πnω) sin(πnω) cos(πnω)

The computation of the output iterates yields:

⇒            y[0] = x 1 [0] y[1] = cos(πq[0]ω)x 1 [0] -sin(πq[0]ω)x 2 [0] y[2] = cos(πq[1]ω)x 1 [0] -sin(πq[1]ω)x 2 [0] y[3] = cos(πq[2]ω)x 1 [0] -sin(πq[2]ω)x 2 [0] . . .
The state x 1 [0] is easily obtained from the above. However, the state x 2 [0] can not be computed whenever q[i] is unknown. The number of unknowns is always larger then the number of considered equations.

To circumvent this drawback, the state vector is extended with a new exogenous state variable: the under-sampling step. Namely the following problem is considered now. Double state-step observability. Consider the following system.

x

[k + 1] = f p[k] (x[k], u[k]) y[k] = h(x[k]) (13)
where f : R n → R n , h : R n → R and p is an unknown variable.

Problem statement 4.2 (Step and state observability of unknown under-sampling). Given the dynamical system (13) with an unknown under-sampling. Find if possible a function F and a σ such that

(x[0]; p[1]; p[2]; . . . ; p[σ -1]) = F (y[0]; y[1]; . . . ; y[σ])
So p, as well as the state x, must be observed. As a consequence, p becomes a component of the new state to observe. This will be explicitely stated in Proposition 4.4 below. Definition 4.1 (local state and step observability). A system is said to be locally state and step observable in x if ∃V(x), ∀x ∈ V(x) \ {x}, ∀p : N → N * , ∀p : N → N * ,, (x, p) and (x, p) are distinguishable for an under-sampled system [START_REF] Hénaff | Actes de la conférence internationale francophone d'automatique[END_REF].

Example 4.4. Consider the following system:

   x 1 (n + 1) = x 2 (n) x 2 (n + 1) = αx 2 (n) y[k] = h(x(q[k]))
where 0 < α < 1 and x 2 (0) > 0.

The outputs are equal to:

   y[0] = x 1 [0] y[1] = α q[0]-1 x 2 [0] y[2] = α q[1]-1 x 2 [0]
Global observability This system is not globally state and step observable. Let x2 (0) being an other initial condition such that x 2 (0) = αx 2 (0), or equivalently, x 2 (0) = x2 (1).

Then let p being the under-sampling defined such that q

[k] = q[k] + 1∀k ∈ N. It comes ỹ[1] = α q[0]-1 x2 [0] = y[1]. And ỹ[k] = y[k]∀k ∈ N.
So the system is globally state and step unobservable.

Local observability

More generally, a given initial condition x(0) is undistinguishable with x(0) if x 1 (0) = x1 (0) and x2 (0) = α k x(0), k ∈ Z. So the system is locally observable. Finally, the system (4.4) is not double state-step observable but is locally double state-step observable.

Linear systems

This section looks for conditions of unobservability of under-sampled linear system. Consider the under-sampled system (14) reformulated under the standard form [START_REF] Vidal | Observability and identifiability of jump linear systems[END_REF].

x(n) = A(x(n)) y[k] = Cx(q[k]) (14)
x

[k] = A q[k] (x[k]) y[k] = Cx[k] (15)
Proposition 4.3. If a linear system ( 14) is defined such that there exists an integer s such that A s is similar to the right hand side of (16) and such that A and C verify (4.5), then the under-sampled system (14) is unobservable for q[k] = sk.

P -1 A s P = A 1 0 0 A 2 (16)
CA s = CP -1 A 1 0 0 0 P (17)

Proof. Consider a linear system [START_REF] Oppenheim | Discrete-time signal processing[END_REF] which satisfies [START_REF] Vesentini | An introduction to topological dynamics in dimension one[END_REF]. It comes that for all t ∈ N * , P -1 A ts P = A t 1 0 0 A t 2 Furthermore, if it satisfies (4.5), it comes CA ts = CP -1 A t 1 0 0 0 P Let D defined by

D = P -1      0 . . . 0 1     
It comes that CA ts D = 0, t ∈ N * . Then, there exists a periodic under-sampling q[k] = sk such that the under-sampled system is unobservable.

The following example illustrates proposition .

Conclusion

The open problems of observability and left invertibility of under-sampled systems have been defined and characterized. The notion of under-sampling has been introduced and formalized. This formalization has then permitted to consider the realization problem in order to equivalently rewrite an under-sampled system in a standard form from system theory point of view.

Under-sampled systems have been separated into two families: systems where the sampling is known and systems where the sampling is unknown. The study of observability in the context of known under-sampling has led to an extension of the classical observability criterion. Some examples have illustrated the case of unknown under-sampled systems. A potential application of this study is the decrease of required number of output measurements, without loss of observability of the original system.

Figure 1 Figure 1 :

 11 Figure 1: over and under-sampling time representation

Figure 2 :

 2 Figure 2: under-sampling and under-sampling steps

  links y to the initial state x(0) and the inputs u: y[k] = h k (x(0); u(0); u(1); u(2); . . . ; u(q[k] -1)) Theorem 3.1. System (2) can be written under a standard form (6) with ū[k] = u[k] if and only if

Example 4 . 1 .

 41 Consider the system (4.1), where the under-sampling is a multiple m = 8 of the relative degree: q(k) = 8k, p[k] = 8, ρ = 1.

Example 4 . 3 .

 43 Let the second order system f : x → Ax et h : x → Cx where A = cos θ -sin θ sin θ cos θ and C = 1 0 .

Example 4 . 5 .

 45 Consider a system defined by matrices A and C:A = cos θ -sin θ sin θ cos θ C = 1 0A n is calculable and it comesA n = cos(πnω) -sin(πnω) sin(πnω) cos(πnω) | θ = πω So, if ω ∈ Q, let ω = a/b, (a, b) ∈ N 2, there exists an integer n, n = 2b, such that is unobservable for sampling steps p[k] ∈ 2bN.This section we introduce an 'intelligent step sensor' to measure the unknown step by extending the state vector x[k] with an additional state variable z[k].

Proposition 4 . 4 (

 44 Intelligent Step Sensor). The dynamics of z[k] is governed by the function g(z[k]), which plays the role of a step sensor: it is exogenous to the system and accounts explicitly for the step p[k].    x[k + 1] = f p[k] (x[k]) z[k + 1] = α p[k] y[k] = (h(x[k]), z[k]) (18)where h : R n × N * → R × N * , α > 1, z is the extended state variable, and p[k] is the undersampling step. In this way the under-sampling step is now known. It is included in the output vector y[k] and the problem falls in the case of undistinguished known under-sampling which has already been solved in section 4.4.


A Ludwig

J Mogerisch

M Kolbe

G Zimmermanm

L Sturz

N Bergeon

B Billia

G Faivre

S Akamatsu

S Bottin-Rousseau

D Voss

Advanced Solidification Studies onTransparent Alloy Systems: A new European SolidificationInsert for MSG on Board the ISS

Investigations on solidifying transparent model alloys have served frequently to gain knowledge on physical phenomena occurring during solidification of metallic alloys. However, quantitative results were only possible to obtain in thin samples where convection can successfully be suppressed. Quantitative studies on three dimensional phenomena not being affected by natural convection are thus only possible under microgravity conditions. Therefore, the European Space Agency (ESA) is planning to launch a new insert for the Material Science Glovebox (MSG) on board of the International Space Station (ISS) for studies on solidification phenomena in thick samples. Four different classes of transparent model alloys will be used to address the following scientific topics: (i) Columnar to Equiaxed Transition in Solidification Processing, (ii)Novel Peritectic Structures and In-Situ Composites; (iii) Solidification along an Eutectic Path in Binary Alloys; and (iv) Solidification along an Eutectic Path in Ternary Alloys. Here, we give details on the scientific objectives and the operational features ESA's new solidification device will offer.

Introduction:

Studies on solidification of transparent model alloys have since long leadto breakthrough findings which permitted deeper understanding of physical phenomena occurring during solidification of daily life alloy systems. With that knowledge it was possible to setup advanced computer codes which nowadays help the metallurgical industry to optimize product quality and reduce production time. The aim of the planned experiments is to study the morphological instabilities of directionally solidified transparent alloys under purely diffusive conditions. Observations will be performed in real time, and the dynamics of the solidification structures will be followed with a micron-scale resolution, over a large (centimetric) space scale, and over long (up to several 10 hours) periods of time. Such observations would be strongly sensitive to convective motions in the liquid, which, in ordinary conditions on earth, entail a detrimental redistribution of the solute on a scale comparable to the container size. Such convective motions are suppressed in microgravity.

In the present publication, the research objectives ofthree from the aforementioned four research teams are described in so far as they are concerned with in-situ observation of transparent model alloys. In addition, we present the main features of the new piece of hardware, which is termedthe "TRANSPARENT ALLOYS".

Projects Objectives:

In the following the objectives and specific goals are given:

CETSOL (Columnar to Equiaxed Transition in Solidification Processing)

The grain structure in many castings is often a competition between columnar and equiaxed dendritic growth. The investigation of the transition between these grain structures is the objective of the CETSOL research program, both experimentally and numerically. From experimental point-of-view transparent organic materials offer in-situ and real-time observation with microscopic optics. The underlying phenomena and physics like columnar dendritic growth and grain selection, nucleation and equiaxed dendritic growth, as well as flow phenomena in the melt interacting with solidification can be assessed [Trivedi 2005]. A significant contribution to the improvement of integrated modeling of grain (crystal) structure in industrially important castings is expected. In particular, this is aimed to give scientists and industries confidence on the reliability of the relationships and numerical tools introduced in the integrated numerical models of casting they are using for in-house optimizing of processes.

The major aim of solidification experiments of the CETSOL team in the TRANSPARENT AL-LOYS instrument using transparent model alloys will be to identify growth regimes (columnar or equiaxed or mixed) and physical mechanisms in dependence of the experimental parameters (solidification velocity, temperature gradient) for diffusive heat and mass transport and without gravity effects. The critical parameters for the columnar to equiaxed transition will then be determined and compared to numerical predictions. Convective transport in the melt and sedimentation of equiaxed grains or inoculation particles will be avoided in the low-gravity regime and enable modeling approaches a basis for sound comparison.

The CETSOL experiments focus on different aspects of the CET using the two binary alloys Neopentylglycol-(D)Camphor and the Succinonitrile-(D)Camphor. For the first system heterogeneous nucleation and subsequent equiaxed growth in the bulk melt ahead of the columnar front was already observed [Sturz 2011]. The TRANSPARENT facility shall enable a systematic investigation of the critical parameters under low-gravity conditions and for different alloy compositions (20-40 wt% (D)Camphor). Pulling rate jumps in the Bridgman-type solidification set-up will be carried out at constant thermal gradient. The sequence shall be repeated at different thermal gradients and for different pulling rates to identify the influence of these solidification parameters on CET. Fig. 1 shows an example for columnar and equiaxed dendritic growth obtained in the TRANSPARENT facility.

For Succinonitrile-(D)Camphor, a thermal gradient decrease will be performed. These experiments are closer to casting conditions and should give information on the fragmentation phenomenon (Fig. 2). The sequence shall be repeated using different cooling rates.The parameters gained by microgravity experiments will be determined and compared to groundbased experiments and numerical predictions.

METCOMP (Metastable Solidification of Composites: Novel Peritectic Structures and In-Situ Composites)

Investigations on peritectic metallic systems show a wide range of possible microstructures.Bands, islands, tree-like microstructures and coupled peritectic growth are appearing when the primary and peritecticphase solidify in a competitive manner. Post-mortem analyses ofquenched peritectic alloys were studied to estimate the process conditions necessary for the occurrence of the different microstructures [Hunz. 1998]. However, the dynamic of morphological transitions can only be studied by in-situ observations. Therefore, corresponding studies on transparent peritectic model systems are highly wanted. In such a system, both the pro-and the peritectic phase must reveal a non-facetted solid/liquid interface, so that their growth morphology is indeed comparable with metals; a condition which could not be full filled till a corresponding system was first described in literature in 1995 [Barrio 1995].

Today a few more model systems have been reported to show a peritectic phase diagram and therequired plastic phases needed for non-facetted solid/liquid interfaces. For the investigations of the METCOMP team, the peritectic system NPG-TRIS was selected as the tem-perature range necessary for in-situobservationsof peritectic solidification phenomena in a micro Bridgman-furnacesetup is still accessible [Moger. 2009a]. The most problematic fact with this system is the instable behavior of TRIS at elevated temperature [Moger. 2009b]. It must strictly be ensured that corresponding NPG-TRIS alloys must never be heated above a given temperature limit. This condition restricts of course the use of larger temperature gradients and in consequence extends the duration of a single experiment to 10 hours or more.

During such long experiments natural convection in the liquid ahead of the solid/liquid interface affects the stable growth conditions and thus alters the growth morphologies. It is thus necessary to reduce natural convection to a minimum. Long time experiments and reduced natural convection makes corresponding studies on board of the International Space Station (ISS)compulsory.

In order to prepare for corresponding experiments on the ISS, laboratory experiments on earth were performed. We have tested a variety of cartridge geometries, temperature gradients and alloy composition and worked out process conditions for the occurrence of a wide range of peritectic microstructures above, close and below the limit of constitutional undercooling [Ludwig 2012]. For instance oscillating behavior wasfound close to the peritectic concentration at pulling rates above the critical velocity [Ludwig 2009]. Here, both phasesgrow in a competitive manner whereby the primary phase solidifies in form of dendrites/cells. Thesecond phase solidifies within the interdendritic liquid and influences the solidification structure of the primary phase in a way that oscillating solidification occurs. At a solidification rate close and below thecritical velocity only a planar solidification front is observed. In a few cases the growth of bands isobserved at the beginning of the solidification. In other cases the formation of coupled peritectic growth and its destabilization was observed (Fig. 3).

SEBA (Solidification along an Eutectic Path in Binary Alloys)

The team will focus on the dynamics of formation of rod-like eutectic structures (a regular dispersion of thinfibers of one phase into a continuous matrix of the other eutectic solid). Such eutectic microstructures resultfrom a coupled growth of the two eutectic phases. Ideally, the coupled-growth dynamics leads to theformation, in steady regime, of periodic patterns with a hexagonal symmetry along the solidification front [Akam. 2002].Specific goals are:

 To study the formation and the relaxation of topological defects in rod-like structures  To study the rod-to-lamellar transition of eutectic growth patterns  To study the forcing effects of the distortions of the thermal gradient These topics can be detailed as follows:

1) Hexagonal eutectic patterns most often exhibit a large density of topological defects (this is due to therotational degeneracy of hexagonal patterns about the solidification axis) [Akam. 2004]. To date, the dynamics ofsuch defects is essentially unknown.

2) A transition from hexagonal (rod-like microstructure) to banded (lamellar microstructure) can occur,which is known to depend primarily on the concentration of the alloy. However, there are somepieces of evidence in the literature that hexagonal and banded patterns can coexist in a givenexperiment [Parisi 2006]. The conditions under which such coexistence is possible still remain to be determined.

3) Experimental systems can deviate slightly from the ideal case, defined as being isotropic (nocrystallographic effect) and planar (axial thermal gradient). In particular, the difference of thermalconductivity of the liquid, the solid and the container walls that are in contact with each other creates a distortion of the thermal field which induces a continual forcing of the structure. This effectremains to be characterized in long-duration experiments.

The team proposes to perform real-time observations using the method described in [Bottin 2007] of a directional-solidification front in bulk samples of transparent eutectic alloys. The growth front is observed obliquely in dark field through the liquid and a glass wall of the containerwith a long-distance microscope. It is shown that a focused image of the whole growth front can beobtained at a certain tilt angle of the microscope. At this tilt angle, eutectic fibers of about 3-5μm indiameter can be clearly seen over the whole growth front in 400μm thick samples [Akam 2007[START_REF] Perrut | [END_REF]. Samples ofsuccinonitrile-dcamphor alloys of neareutectic composition are proposed. For minimizingcrystallographic effects, the samples would comprise a crystal selector. Solidification will be performed atvarious pulling speed values. The misalignment of the thermal gradient will be tunable.

Directional Solidification Device:

ESA's "TRANSPARENT ALLOYS" instrument is a kind of classical Bridgman furnace, which is based on the famous thin-sample directional solidification method. It possesses three specific features. First, the thickness of the cartridge will be considerably larger (in the millimeter range) than that of the widely used thin samples (in the 10µm range). Second, observation will be possible both in side view (optical axis perpendicular to the sample plane) and in oblique view. Third, a grain selector will allow one to study growth of large crystals or eutectic grains of given orientation.

The solidification bench is composed of a "hot zone" and a "cold zone", made of thermally regulated metallic blocks fixed at a defined distance from each other (the so-called adiabatic zone). The temperatures of the cold and hot zones will be tuned in such a way that the solidliquid interface is located near the middle of the adiabatic zone. Solidification (melting) will be controlled by moving the sample along the thermal axis at a tunable velocity toward the cold (hot) zone.

Flat glass-wall cartridges will be used, with inner dimensions of 100mm along the thermal gradient axis), 60mm perpendicular to the optical axis (parallel to the isotherms), and 1 or 6mm in the third direction (thickness). Inside the cartridge special means will be applied to compensate for any volume changes (especially the around 5% density difference between solid and liquid).

The length of the adiabatic zone will be 7mm. The thermal environment will be such that the solidification process will be controlled by the hot-and cold-zone temperatures and by the pulling rate of the sample. The hot zone temperature will be adjustable between 323K and 443K and the zone temperature between 263K and 400K. Having set the temperature of the hot and the cool zone, the temperature gradient will range from about 5 to 80K/cm. As the thermal gradient in the sample alloy cannot be measured directly, a dedicated reference cartridge containing at least one thermocouple within the melt will be used to verify the actually achieved gradients inside the sample alloy in a thermally representative set-up during ground reference experiments.

The cartridge will be translated in longitudinal Z-direction in a controlled way over the whole length of the samples. Translation speed of the cartridge will be 0.01μm/s <v< 100μm/s. The mechanism is programmable in order to perform various speeds in steps of ≤ 0.003μm/s during one experiment. The mechanism will offer long-term stability of 1% over a time period > 12hrs.

The experiment will run following a pre-programmed experiment procedure, but interaction from ground will also be possible. Heater and cooler temperatures, pulling velocity and optical parameters will be adjustable from ground with a reaction time compatible to the experiment objectives. The system will have the possibility to store images on-board and to send selected images to ground during the experiment runs.

The instrument will be equipped with two digital cameras with image resolution ≥1280x980 pixels and ≥8bit. It will allow to select image recording frequency up to 15Hz. The field of view will be 6mm x 5mm in Y-Z direction for side-view observation and not less than 2.4mm x 2.1 mm in Y-Z direction in oblique view. It will be possible to scan through and focus over the whole depth of the sample (X-direction) with accuracy better than 25% of the depth of field in operation for both camera angles. Scanning through the whole depth of the sample will be completed in ≤ 5 seconds.

The illumination system will support both bright and dark field illumination; will be adjustable about the Y-axis for ensuring bot side-and oblique-view; allows switching the illumination without detectable thermal perturbation; and flashing of the illumination and synchronization with camera image taking will be possible.

Summary:

Microgravity conditions are still of great importance for performing solidification experiments on transparent model systems without the ambiguous effect of natural convection.

With the design and the implementation of ESA's "TRANSPARENT ALLOYS instrument" on board of the ISS, the scientific community will get access to a new Bridgman-type facility which alloy to process thick samples under controlled conditions. During the Breadboard tests it was shown that the possibility of observing the solid/liquid interface not only from a side-view but also from an oblique-view will result in fascinating pictures/videos from three dimensional phenomena happening at the solid/liquid interface. It is thus understandable that four different European research teams are looking forward to use the new facility forstudying different 3D aspects of alloy solidification. The clear observation of secondary branches allows to study their growth dynamics and to reveal particular phenomena such as bending or detachment that may be involved in CET [Jung 2009].

  Currently, four European research teams together with the European Space Agency (ESA) are involved in the planning, design and implementation of a new solidification related insert for the Materials Science Glovebox (MSG) on board of the International Space Station (ISS). These four teams are  CETSOL (Columnar to Equiaxed Transition in Solidification Processing)  METCOMP (Metastable Solidification of Composites: Novel Peritectic Structures and In-Situ Composites)  SEBA (Solidification along an Eutectic Path in Binary Alloys)  SETA (Solidification along an Eutectic Path in Ternary Alloys)

Figure Captions : Figure 1 :

 :1 Figure Captions: Figure 1: Typical situation of equiaxed dendrites made from Neopentylglycol solid-solution, nucleation and growing ahead of the columnar dendritic front.

Figure 2 :

 2 Figure 2: Typical image of dendritic columnar growth (Succinonitrile -8.4 wt% Camphor).The clear observation of secondary branches allows to study their growth dynamics and to reveal particular phenomena such as bending or detachment that may be involved in CET[Jung 2009].

Figure 3 :

 3 Figure 3: Example of isothermal peritectic coupled growth in thin samples (NPG-0.46% mol fraction TRIS). Although the isotherms are horizontal the left side of the picture shows a curved solid/liquid interface which is a result of convection in the cartridge.

Figure 4 :

 4 Figure 4: Images of a eutectic solid/liquid interface resulting from scientific Breadboard tests by SEBA team and confirmed to be ofsufficient quality for analysis.

Acknowledgement:

The authors kindly acknowledge ESA for financing planning, design and implementation of the "TRANSPARENT ALLOYS instrument". In addition, AL and JM are grateful to the Austrian Research promotion Agency (FFG) for financial support. GZ and LS gratefully acknowledge financial support from the German Space Agency DLR. For this research the INSP team received a financial support by the French space agency (CNES).


S Akamatsu
email: akamatsu@insp.jussieu.fr

S Bottin-Rousseau

G Faivre

S Ghosh

M Plapp

Lamellar eutectic growth with anisotropic interphase boundaries

We present a numerical study of the effect of a free-energy anisotropy of the solidsolid interphase boundaries on the formation of tilted lamellar microstructures during directional solidification of nonfaceted binary eutectic alloys. We used two different methods -phase-field (PF) and dynamic boundary-integral (BI) -to simulate the growth of periodic eutectic patterns in two dimensions. For a given Wulff plot of the interphase boundary, which characterizes a eutectic grain with a given relative orientation of the two solid phases, the lamellar tilt angle depends on the angle between the thermal axis z and a reference crystallographic axis. Both PF and BI results confirm the general validity of a recent approximate theory which assumes that, at the trijunctions, the surface tension vector of the interphase boundary is parallel to z. In particular, a crystallographic locking of the lamellae onto a direction close to a deep minimum in the Wulff plot is well reproduced in the simulations.

Introduction

In many nonfaceted eutectic alloys, two-phase microstructures that are delivered by coupledgrowth front patterns during directional solidification (DS) present a marked morphological dependence on the crystal orientation of the two solid phases [1,2,3]. In spite of their pratical importance, such crystallographic effects have been largely left aside in previous theories and models. Let us consider the case of lamellar eutectics. Experimentally, it has been known for a long time that eutectic lamellae can grow tilted with respect to the thermal-gradient axis z, and that the lamellar tilt is eutectic-grain dependent (a eutectic grain is a region of uniform crystal orientation of the solid phases α and β [4]). On the basis of in situ experimental observations [5,6,7], two categories of eutectic grains have been previously distinguished (Fig. 1a), namely, floating (eutectic) grains (Fig. 1b1), within which the growth dynamics is well described by the standard theory of regular eutectics, and locked grains, within which the growth direction of the eutectic lamellae is essentially aligned to a certain crystallographic plane, and is generally inclined (or tilted) with respect to the main growth axis z. Lamellar patterns with a strong tilt angle are usually observed when the two solid phases exhibit an epitaxial orientation relationship and interphase-boundary planes with a low free energy [3]. Recently, an approximate theory has been formulated that relates the value of the lamellar tilt to the surface free-energy anisotropy of the α-β interphase boundaries (interfacial anisotropy) in the solid [6]. It has been proposed in coherence with in situ observations indicating that, even for lamellae that grow at a large tilt angle, the solid-liquid interface shape exhibits a mirror symmetry with respect to the mid-plane of the lamellae, similar to the one observed for well-aligned lamellae. For isotropic solid-liquid interfaces, a symmetric shape is obtained under the condition that the contact angles of the solid-liquid interfaces at the trijunctions are the same on both sides of a lamella (Fig. 1b2) [6,7]. For anisotropic interphase boundaries, the symmetric-pattern (SP) approximation thus entails that the Cahn-Hoffman surface tension vector σ [8] is aligned with z (it is assumed that, in a nonfaceted alloy, the anisotropy of the solid-liquid interfaces has a negligible effect on the lamellar growth dynamics). Here, we present a test of the SP approximation by numerical simulations. We carried out two-dimensional calculations using two different simulation methods, namely, (i) a sharpinterface code, which uses the dynamic boundary-integral (BI) formalism previously developed by Karma and Sarkissian [9], and (ii) a multi-phase-field (PF) model, based on recent grandcanonical formulations of alloy solidification [10,11]. Both methods demonstrate that the tilt angle follows quite closely the SP approximation. In addition, a strong locking of the lamellae onto the direction of the peaked minimum of the anisotropy function is well reproduced in both BI and PF simulations.

The symmetric-pattern approximation

The relative orientation of the α and β crystals (constant within a grain) determines the interphase boundary free energy and its anisotropy. In two dimensions (ideally, thin-sample DS), the grain orientation and the inclination of the interphase boundary are specified each by a single angle with respect to the thermal axis. Let n be the unit normal vector of the interphase boundary, and θ the angle between n and the x axis ( t = -dn/dθ is the tangent vector). For a reference grain orientation, the anisotropic interphase boundary energy is given by

γ αβ (θ) = γαβ a c (θ), ( 1
)
where γαβ is a constant and a c (θ) is a dimensionless function. The Cahn-Hoffman ξ and σ vectors [8] are defined by

ξ = γ αβ n -γ αβ t ; σ = γ αβ t + γ αβ n (2)
where γ αβ = dγ αβ /dθ. The Wulff plot is given by r(θ) = γ αβ (θ)n and the minimum-energy, or Wulff shape, by the vector ξ(θ). For a given eutectic grain, we search the steady-state growth pattern as a function of the orientation angle θ R of the bicrystal with respect to z. The interphase energy becomes

γ αβ (θ) = γαβ a c (θ -θ R ), (3)
where θ R = 0 is chosen such that an interphase orientation of minimal energy is aligned with z.

In a sharp-interface model (with isotropic solid-liquid interfaces, and no diffusion in the solid), the local equilibrium at the trijunction is given by a Young-Herring equation, that is,

γ αL tαL + γ βL tβL + σ = 0. ( 4
)
As stated above, a strictly symmetric shape is thus possible only if σ (which is not parallel to the interphase boundary) is aligned with z (Fig. 1b2), which writes

γ(θ -θ R ) sin θ + γ (θ -θ R ) cos θ = 0. ( 5
)
For a fixed eutectic-grain orientation θ R , this is a nonlinear equation for the interface orientation, which can easily be solved numerically for any γ(θ) function. We will note θ sp the SPapproximation value of the tilt angle, solution of Eq. ( 5), and θ t the steady-state tilt angle obtained in the numercial simulations.

Results and discussion

We simulated a model binary alloy with a symmetric eutectic plateau (and parallel liquidus and solidus lines), equal α-and β-liquid capillary lengths, at eutectic concentration. Temperature gradient, growth velocity, and lamellar spacing (close to the minimum undercooling spacing) are held constant. In the BI model, the interfacial anisotropy is incorporated in the calculation of the position of the trijunctions, using the Young-Herring condition (Eq. 4). In the PF model, the anisotropy of the interphase boundary is directly incorporated in the free-energy functional [11].

We implemented first a standard anisotropy function with a 4-fold symmetry [a c (θ) = 1-4 cos 4θ]. The results for 4 = 0.04 are shown in figure 2a. Both sets of simulation data follow quite closely the SP approximation, up to differences that do not exceed a degree. The value of θ t passes through a maximum for an orientation that depends on the anisotropy function (as expected, symmetric, non-tilted patterns are found when either a minimum or a maximum of γ is aligned with z). The tilt angle always remains much smaller than θ R , which is characteristic of a weak crystallographic effect. Note that the values of the capillary length, the thermal length, and the lamellar spacing are actually slightly different for BI and PF, which demonstrates that the influence of all of these parameters is weak in realistic conditions.

We then reproduced a strong lamellar-locking effect by using an anisotropy function with a deep minimum, of the form a c (θ) = 1g exp -(θ/w g ) 2 (the smoothed gaussian cusp allows us to circumvent some obvious numerical difficulties [12]). Setting g = 0.2 and w g = 0.1 entails the appearance of two (quasi) facets in the Wulff shape of the interphase boundary (see inset in Fig. 2b). In this case, the ξ(θ) plot has self-intersections. The Wulff shape is given by the inner convex part of that plot. The other parts, often called "ears", consist of three segments delimited by turning points. The interface stiffness τ (θ) = γ αβ (θ) + γ αβ (θ) is negative only on the middle segment (a flat interface within this range is unstable with respect to the formation of a hill-and-valley structure [13]), but all orientations located on the "ears" are missing from the physically observable convex equilibrium shape. The SP approximation then predicts three distinct parts in the θ sp vs θ R curve: (i) an essentially linear, strongly locked branch with a slope close to 1, which runs from θ R = 0 to a first τ = 0 turning point (noted θ l ); (ii) a symmetricpattern branch along which there is no anisotropy effect for θ R ranging from a second turning point (noted θ u ) to π/2; (iii) an intermediate, unstable branch. In the [θ u , θ l ] interval, two kinds of lamellar patterns (one is locked, the other is isotropic) are possible for a given eutectic-grain orientation.

Both BI and PF simulations describe the two separate branches (figure 2b) and demonstrate the existence of a strong locking effect over a large orientation range, as predicted by the SP approximation. The BI simulations clearly reproduce a hysteretic behavior, which is expected for a bistable system. It should be noted, however, that the approach of the limit tilt angle (about 50 degrees) along the locked branch also corresponds to a steep, apparently diverging increase of the angle of the σ vector with respect to the thermal axis (Fig. 2b). This indicates that, at large tilt angles, the SP approximation, while it still correctly predicts the locked-tilt angle, becomes inaccurate with regards to other aspects of the dynamics. In contrast with the BI simulations, no hysteresis is observed in the PF simulations.

We can draw an interesting conclusion, of practical relevance, from these results. It is related to the recent development of an experimental method called rotating directional solidification (RDS), during which solidification is performed in a fixed thermal gradient by slowly rotating a thin alloy sample about an axis perpendicular to the xz plane [6,7]. The crystal orientation of a given eutectic grain (fixed in the sample reference frame) with respect to the main solidification axis is then varied continuously while solidification proceeds. In the case of eutectic growth, a lamellar pattern then adapts its shape in a quasistationary way. It was demonstrated in Ref. [6] that, under the SP approximation, and with suitably chosen RDS settings, the solidification microstructure is then homothetic to the Wulff shape of the interphase boundaries in the sample plane, and that, accordingly, the morphological analysis of a RDS pattern gives direct access to a two-dimensional section of the corresponding Wulff plot. The present study shows that this method is accurate, within a few percents, which is quite satisfactory as compared to other ordinary sources of experimental uncertainties.

Conclusion

We performed time-resolved numerical simulations of tilted lamellar patterns during directional solidification of a model eutectic alloy in the presence of an anisotropy of the interphase boundaries. The simulations confirm the semi-quantitative validity of the SP approximation that assumes that the surface tension vector of the interphase boundary at the trijunctions is parallel to the main growth axis. The most salient result is that a strong locking phenomenon associated with the existence of a sharp minimum in the Wulff plot (this corresponds to a facet in the equilibrium shape of the interphase boundary) is well reproduced by the simulations.

Another interesting conclusion, of practical relevance, is the establishment of the RDS method as a good tool for extracting quantitative information on the Wulff plot of interphase boundaries during a solidification experiment. Moreover, anisotropy functions with more detailed features than the test ones used in the present work can be implemented in the simulation codes, and could serve to reproduce "real" Wulff plots measured experimentally, or calculated by moleculardynamics methods [14]. Further simulations are planned to study the dependence of the lamellar tilt on the lamellar spacing, and the influence of a non-symmetric phase diagram. Simulations in large systems will be needed to study the dynamics of "sawtooth" patterns [5], which have been observed experimentally, and may be associated with a bistable behavior in strongly anisotropic cases. Phase-field simulations of eutectic-growth patterns with anisotropic interphase boundaries will be extended to three-dimensional, bulk-solidification geometries.

Figure 1 .

 1 Figure 1. a) Two eutectic grains, one of the floating (lef-thand side) and the other of the locked (right-hand side) type, during real-time DS of a slightly hypoeutectic CBr 4 -C 2 Cl 6 alloy in a 12-µm thin sample (V = 0.5µms -1 ; bar: 20 µm). Sketches: b1) steady lamellar pattern with isotropic interfaces ; b2) tilted lamellar pattern in a locked eutectic grain under the SP approximation ( σ z).

Figure 2 .

 2 Figure 2. Lamellar tilt angle θ sp predicted by the SP approximation (thick line), and θ t calculated with the BI (filled circles) and the PF codes (diamonds) as a function of the eutecticgrain orientation angle θ R . a) weak 4-fold anisotropy function; b) anisotropy function with a peaked gaussian minimum (see text). Open circles: angle of the surface tension vector σ with z (BI data). Inset: half view of the Wulff plot (thin line) and shape (thick line). Dotted lines: Herring-unstable orientations.

Acknowledgments

We thank Alain Karma for sharing with us his original BI code. This work was financially supported by Centre National d'Etudes Spatiales (CNES), France.