Projet Michel Fliess
email: michel.fliess@polytechnique.edu

Cédric Join
email: cedric.join@cran.uhp-nancy.fr

Hebertt Sira-Ramírez

Dr Sira-Ramírez

Non-linear estimation is easy

Keywords: Non-linear systems, observability, parametric identifiability, closed-loop state estimation, closed-loop parametric identification, closed-loop fault diagnosis, closed-loop fault tolerant control, closed-loop perturbation attenuation, numerical differentiation, differential algebra

Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint.

1 Introduction

General overview

Since fifteen years non-linear flatness-based control [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF][START_REF] Sastry | Nonlinear Systems[END_REF]) has been quite effective in many concrete and industrial applications (see also Lamnabhi-Lagarrigue & Rouchon (2002b); [START_REF] Rudolph | Beiträge zur flacheitsbasierten Folgeregelung linearer und nichtlinearer Syteme endlicher und undendlicher Dimension[END_REF]; [START_REF] Sira-Ramírez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF]). On the other hand, most of the problems pertaining to non-linear state estimation, and to related topics, like

• parametric estimation,

• fault diagnosis and fault tolerant control,

• perturbation attenuation, remain largely open in spite of a huge literature 1 . This paper aims at providing simple and effective design methods for such questions. This is made possible by the following facts:

According to the definition given by Diop & Fliess (1991a,b), a non-linear input-output system is observable if, and only if, any system variable, a state variable for instance, is a differential function of the control and output variables, i.e., a function of those variables and their derivatives up to some finite order. This definition is easily generalized to parametric identifiability and fault isolability. We will say more generally that an unknown quantity may be determined if, and only if, it is expressible as a differential function of the control and output variables.

It follows from this conceptually simple and natural viewpoint that non-linear estimation boils down to numerical differentiation, i.e., to the derivatives estimations of noisy time signals 2 . This classic ill-posed mathematical problem has been already attacked by numerous means 3 . We follow here another thread, which started in Fliess & Sira-Ramírez (2004b) and [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF][START_REF] Chambert-Loir | A Field Guide to Algebra[END_REF]: derivatives estimates are obtained via integrations. This is the explanation of the quite provocative title of this paper4 where non-linear asymptotic estimators are replaced by differentiators, which are easy to implement5 .

Remark 1.1. This approach to non-linear estimation should be regarded as an extension of techniques for linear closed-loop parametric estimation [START_REF] Fliess | An algebraic framework for linear identification[END_REF][START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF]). Those techniques gave as a byproduct linear closed-loop fault diagnosis (Fliess, Join & Sira-Ramírez (2004)), and linear state reconstructors (Fliess & Sira-Ramírez (2004a)), which offer a promising alternative to linear asymptotic observers and to Kalman's filtering.

1.2 Numerical differentiation: a short summary of our approach .

Let us start with the first degree polynomial time function p 1 (t) = a 0 + a 1 t, t ≥ 0, a 0 , a 1 ∈ R. Rewrite thanks to classic operational calculus (see, e.g., [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]) p 1 as P 1 = a0 s + a1 s 2 . Multiply both sides by s 2 :

s 2 P 1 = a 0 s + a 1 (1)
Take the derivative of both sides with respect to s, which corresponds in the time domain to the multiplication by -t:

s 2 dP 1 ds + 2sP 1 = a 0 (2)
The coefficients a 0 , a 1 are obtained via the triangular system of equations ( 1)-(2). We get rid of the time derivatives, i.e., of sP 1 , s 2 P 1 , and s 2 dP1 ds , by multiplying both sides of Equations ( 1)-( 2) by s -n , n ≥ 2. The corresponding iterated time integrals are low pass filters which attenuate the corrupting noises, which are viewed as highly fluctuating phenomena (cf. [START_REF] Fliess | Analyse non standard du bruit[END_REF]. A quite short time window is sufficient for obtaining accurate values of a 0 , a 1 .

The extension to polynomial functions of higher degree is straightforward. For derivatives estimates up to some finite order of a given smooth function f : [0, +∞) → R, take a suitable truncated Taylor expansion around a given time instant t 0 , and apply the previous computations. Resetting and utilizing sliding time windows permit to estimate derivatives of various orders at any sampled time instant.

Remark 1.2. Note that our differentiators are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises. Those two fundamental features remain therefore valid for our non-linear estimation6 . This is a change of paradigms when compared to most of today's approaches7 .

Analysis and organization of our paper

Our paper is organized as follows. Section 2 deals with the differential algebraic setting for nonlinear systems, which was introduced in [START_REF] Fliess | Automatique et corps différentiels[END_REF][START_REF] Fliess | Controller canonical forms for linear and nonlinear dynamics[END_REF]. When compared to those expositions and to other ones like Fliess, Lévine, Martin & Rouchon (1995); [START_REF] Fliess | Correcteurs proportionnelsintégraux généralisés[END_REF]; [START_REF] Rudolph | Beiträge zur flacheitsbasierten Folgeregelung linearer und nichtlinearer Syteme endlicher und undendlicher Dimension[END_REF]; [START_REF] Sira-Ramírez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF], the novelty lies in the two following points:

1. The definitions of observability and parametric identifiability are borrowed from Diop & Fliess (1991a,b).

2. We provide simple and natural definitions related to non-linear diagnosis such as detectability, isolability, parity equations, and residuals, which are straightforward extensions of the module-theoretic approach in Fliess, Join & Sira-Ramírez ( 2004) for linear systems.

The main reason if not the only one for utilizing differential algebra is the absolute necessity of considering derivatives of arbitrary order of the system variables. Note that this could have been also achieved with the differential geometric language of infinite order prolongations (see, e.g., [START_REF] Fliess | Deux applications de la géometrie locale des diffiétés[END_REF][START_REF] Sastry | Nonlinear Systems[END_REF])) 8 . Section 3 details Subsection 1.2 on numerical differentiation.

Illustrations are provided by several academic examples 9 and their numerical simulations 10 which we wrote in a such a style that they are easy to grasp without understanding the algebraic subtleties of Section 2:

1. Section 4 is adapting a paper by [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] on a non-linear observer. We only need for closing the loop derivatives of the output signal. We nevertheless present also a state reconstructor of an important physical variable.

2. Closed-loop parametric identification is achieved in Section 5.

(2001), and the references therein, for other non-statistical approaches. 8 The choice between the algebraic and geometric languages is a delicate matter. The formalism of differential algebra is perhaps suppler and more elegant, whereas infinite prolongations permit to take advantage of the integration of partial differential equations. This last point plays a crucial rôle in the theoretical study of flatness (see, e.g., [START_REF] Chetverikov | A nonlinear Spencer complex for the group of invertible differential operators and its applications[END_REF]; [START_REF] Martin | Feedback linearization and driftless systems[END_REF], 1995); van Nieuwstadt, Rathinam & Murray (1998); [START_REF] Pomet | On dynamic feedback linearization of four-dimensional affine control systems with two inputs[END_REF]; [START_REF] Sastry | Nonlinear Systems[END_REF], and the references therein) but seems to be unimportant here. Differential algebra on the other hand permitted to introduce quasi-static state feedbacks (Delaleau & Pereira da Silva (1998a,b)), which are quite helpful in feedback synthesis (see also Delaleau & Rudolph (1998); Rudolph & Delaleau (1998)). The connection of differential algebra with constructive and computer algebra might be useful in control (see, e.g., [START_REF] Diop | Elimination in control theory[END_REF][START_REF] Diop | Differential algebraic decision methods, and some applications to system theory[END_REF]; [START_REF] Glad | Using differential algebra to determine the structure of control systems[END_REF], and the references therein).

9 These examples happen to be flat, although our estimation techniques are not at all restricted to such systems. We could have examined as well uncontrolled systems and/or non-flat systems. The control of non-flat systems, which is much more delicate (see, e.g., [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF][START_REF] Sira-Ramírez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF], and the references therein), is beyond the scope of this article.

10 Any interested reader may ask C. Join for the corresponding computer programs (Cedric.Join@cran.uhp-nancy.fr).

3. Section 6 deals with closed-loop fault diagnosis and fault tolerant control.

4. Perturbation attenuation is presented in Section 7, via linear and non-linear case-studies.

We end with a brief conclusion. First drafts of various parts of this paper were presented in Fliess & Sira-Ramírez (2004b); Fliess, Join & Sira-Ramírez (2005).

Differential algebra

Commutative algebra, which is mainly concerned with the study of commutative rings and fields, provides the right tools for understanding algebraic equations (see, e.g., [START_REF] Hartshorne | Algebraic Geometry[END_REF]; [START_REF] Eisenbud | Commutative Algebra with a View Toward Algebraic Geometry[END_REF]). Differential algebra, which was mainly founded by [START_REF] Ritt | Differential Algebra[END_REF] and [START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF], extends to differential equations concepts and results from commutative algebra11 .

Basic definitions

A differential ring R, or, more precisely, an ordinary differential ring, (see, e.g., [START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF] and Chambert-Loir (2005)) will be here a commutative ring12 which is equipped with a single derivation d dt : R → R such that, for any a, b ∈ R,

• d dt (a + b) = ȧ + ḃ, • d dt (ab) = ȧb + a ḃ.
where da dt = ȧ, d ν a dt ν = a (ν) , ν ≥ 0. A differential field, or, more precisely, an ordinary differential field, is a differential ring which is a field. A constant of R is an element c ∈ R such that ċ = 0. A (differential) ring (resp. field) of constants is a differential ring (resp. field) which only contains constants. The set of all constant elements of R is a subring (resp. subfield), which is called the subring (resp.subfield) of constants.

A differential ring (resp. field) extension is given by two differential rings (resp. fields) R 1 , R 2 , such that R 1 ⊆ R 2 , and qthe derivation of R 1 is the restriction to R 1 of the derivation of R 2 . Notation Let S be a subset of R 2 . Write R 1 {S} (resp. R 1 S ) the differential subring (resp. subfield) of R 2 generated by R 1 and S. Notation Let k be a differential field and X = {x ι |ι ∈ I} a set of differential indeterminates, i.e., of indeterminates and their derivatives of any order. Write k{X} the differential ring of differential polynomials, i.e., of polynomials belonging to k[x

(νι) ι |ι ∈ I; ν ι ≥ 0]. Any differential poly- nomial is of the form finite c finite (x (µι) ι ) αµ ι , c ∈ k.
Notation If R 1 and R 2 are differential fields, the corresponding field extension is often written R 2 /R 1 .

A differential ideal I of R is an ideal which is also a differential subring. It is said to be prime if, and only if, I is prime in the usual sense.

Field extensions

All fields are assumed to be of characteristic zero. Assume also that the differential field extension K/k is finitely generated, i.e., there exists a finite subset S ⊂ K such that K = k S . An element a of K is said to be differentially algebraic over k if, and only if, it satisfies an algebraic differential equation with coefficients in k: there exists a nonzero polynomial P over k, in several indeterminates, such that P (a, ȧ, . . . , a (ν) ) = 0. It is said to be differentially transcendental over k if, and only if, it is not differentially algebraic. The extension K/k is said to be differentially algebraic if, and only if, any element of K is differentially algebraic over k. An extension which is not differentially algebraic is said to be differentially transcendental.

The following result is playing an important rôle:

Proposition 2.1. The extension K/k is differentially algebraic if, and only if, its transcendence degree is finite.

A set {ξ ι | ι ∈ I} of elements in K is said to be differentially algebraically independent over k if, and only if, the set {ξ

(ν) ι | ι ∈ I, ν ≥ 0} of
derivatives of any order is algebraically independent over k. If a set is not differentially algebraically independent over k, it is differentially algebraically dependent over k. An independent set which is maximal with respect to inclusion is called a differential transcendence basis. The cardinalities, i.e., the numbers of elements, of two such bases are equal. This cardinality is the differential transcendence degree of the extension K/k; it is written diff tr deg (K/k). Note that this degree is 0 if, and only if, K/k is differentially algebraic.

Kähler differentials

Kähler differentials (see, e.g., [START_REF] Hartshorne | Algebraic Geometry[END_REF]; [START_REF] Eisenbud | Commutative Algebra with a View Toward Algebraic Geometry[END_REF]) provide a kind of analogue of infinitesimal calculus in commutative algebra. They have been extended to differential algebra by [START_REF] Johnson | Kähler differentials and differential algebra[END_REF]. Consider again the extension K/k. Denote by

• K[ d dt ] the set of linear differential operators finite a α d α
dt α , a α ∈ K, which is a left and right principal ideal ring (see, e.g., [START_REF] Mcconnell | Noncommutative Noetherian Rings[END_REF]);

• Ω K/k the left K[ d dt ]-module of Kähler differentials of the extension K/k; • d K/k x ∈ Ω K/k the (Kähler) differential of x ∈ K.
Proposition 2.2. The next two properties are equivalent:

1. The set {x ι | ι ∈ I} ⊂ K is differentially algebraically dependent (resp. independent) over k.

The set {d

K/k x ι | ι ∈ I} is K[ d dt ]-linearly dependent (resp. independent).
The next corollary is a direct consequence from Propositions 2.1 and 2.2.

Corollary 2.1. The module Ω K/k satisfies the following properties:

• The rank13 of Ω K/k is equal to the differential transcendence degree of K/k.

• Ω K/k is torsion14 if, and only if, K/k is differentially algebraic.

• dim K (Ω K/k ) = tr deg(L/K).
It is therefore finite if, and only if, L/K is differentially algebraic.

• Ω K/k = {0} if, and only if, L/K is algebraic.

Nonlinear systems

Generalities

Let k be a given differential ground field. A (nonlinear) (input-output) system is a finitely generated differential extension K/k. Set K = k S, W, π where 1. S is a finite set of system variables, which contains the sets u = (u 1 , . . . , u m ) and y = (y 1 , . . . , y p ) of control and output variables, 2. W = {w 1 , . . . , w q } denotes the fault variables, 3. π = (π 1 , . . . , π r ) denotes the perturbation, or disturbance, variables.

They satisfy the following properties:

• The control, fault and perturbation variables do not "interact", i.e., the differential extensions k u /k, k W /k and k π /k are linearly disjoint15 .

• The control (resp. fault) variables are assumed to be independent, i.e., u (resp. W) is a differential transcendence basis of k u /k (resp. k W /k).

• The extension K/k u, W, π is differentially algebraic.

• Assume that the differential ideal (π) ⊂ k{S, π, W} generated by π is prime16 . Write

k{S nom , W nom } = k{S, π, W}/(π)
the quotient differential ring, where the nominal system and fault variables S nom , W nom are the canonical images of S, W.

To those nominal variables corresponds the nominal system 17 K nom /k, where K nom = k S nom , W nom is the quotient field of k{S nom , W nom }, which is an integral domain, i.e., without zero divisors. The extension K nom /k u nom , W nom is differentially algebraic.

• Assume as above that the differential ideal (W nom ) ⊂ k{S nom , W nom } generated by W nom is prime. Write

k{S pure } = k{S nom , W nom }/(W nom )
where the pure system variables S pure are the canonical images of S nom . To those pure variables corresponds the pure system 18 K pure /k, where K pure = k S pure is the quotient field of k{S pure }. The extension K pure /k u pure is differentially algebraic.

Remark 2.1. We make moreover the following natural assumptions:

diff tr deg (k u pure /k) = diff tr deg (k u nom /k) = diff tr deg (k u /k) = m, diff tr deg (k W nom /k) = diff tr deg (k W /k) = q
Remark 2.2. Remember that differential algebra considers algebraic differential equations, i.e., differential equations which only contain polynomial functions of the variables and their derivatives up to some finite order. This is of course not always the case in practice. In the example of Section 4, for instance, appears the transcendental function sin θ l . As already noted in Fliess, Lévine, Martin & Rouchon (1995), we recover algebraic differential equations by introducing tan θ l 2 .

State-variable representation

We know, from proposition 2.1, that the transcendence degree of the extension K/k u, W, π is finite, say n. Let x = (x 1 , . . . , x n ) be a transcendence basis. Any derivative ẋi , i = 1, . . . , n, and any output variable y j , j = 1, . . . , p, are algebraically dependent over k u, W, π on x:

A i ( ẋi , x) = 0 i = 1, . . . , n B j (y j , x) = 0 j = 1, . . . , p (3)
where

A i ∈ k u, W, π [ ẋi , x], B j ∈ k u, W, π [y j , x],
i.e., the coefficients of the polynomials A i , B j depend on the control, fault and perturbation variables and on their derivatives up to some finite order. Eq. ( 3) becomes for the nominal system

A nom i ( ẋnom i , x nom ) = 0 i = 1, . . . , n nom ≤ n B nom j (y nom j , x nom ) = 0 j = 1, . . . , p (4)
where

A nom i ∈ k u nom , W nom [ ẋnom i , x nom ], B nom j ∈ k u nom , W nom [y nom j ,
x nom ], i.e., the coefficients of A nom i and B nom j depend on the nominal control and fault variables and their derivatives and no more on the perturbation variables and their derivatives.

We get for the pure system

A pure i ( ẋpure i , x pure ) = 0 i = 1, . . . , n pure ≤ n nom B pure j (y pure j , x pure ) = 0 j = 1, . . . , p (5)
18 Ignoring as above the fault variables in the nominal system yields the pure system.

where

A pure i ∈ k u pure [ ẋpure i , x pure ], B pure j ∈ k u pure [y pure j ,
x pure ], i.e., the coefficients of A pure i and B pure j depend only on the pure control variables and their derivatives.

Remark 2.3. Two main differences, which are confirmed by concrete examples (see, e.g., [START_REF] Fliess | Questioning the classical state space description via circuit examples[END_REF]; [START_REF] Fliess | Generalized state variable representation for a simplified crane description[END_REF]), can be made with the usual state-variable representation

ẋ = F (x, u) y = H(x)
1. The representations (3), ( 4), (5) are implicit.

2. The derivatives of the control variables in the equations of the dynamics cannot be in general removed (see [START_REF] Delaleau | Lowering the orders of derivatives of control in generalized state space systems[END_REF]).

Variational system 19

Call Ω K/k (resp. Ω K nom /k , Ω K pure /k ) the variational, or linearized, system (resp. nominal system, pure system) of system K/k. Proposition 2.2 yields for pure systems

A    d K pure /k y pure 1 . . . d K pure /k y pure p    = B    d K pure /k u pure 1 . . . d K pure /k u pure m    (6)
where

• A ∈ K[ d dt ] p×p is of full rank, • B ∈ K[ d dt ] p×m .
The pure transfer matrix 20 is the matrix

A -1 B ∈ K(s) p×m , where K(s), s = d dt , is the skew quotient field 21 of K[ d dt ].

Differential flatness 22

The system K/k is said to be (differentially) flat if, and only if, the pure system K pure /k is (differentially) flat [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF] 19 See Fliess, Lévine, Martin & Rouchon (1995) for more details. 20 See [START_REF] Fliess | Une inteprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF] for more details on transfer matrices of timevarying linear systems, and, more generally, Fliess, Join & Sira-Ramírez (2004), [START_REF] Bourlès | Systèmes linéaires : de la modélisation à la commande[END_REF] for the module-theoretic approach to linear systems.

21 See, e.g., [START_REF] Mcconnell | Noncommutative Noetherian Rings[END_REF]. 22 For more details see Fliess, Lévine, Martin & Rouchon (1995); [START_REF] Rudolph | Beiträge zur flacheitsbasierten Folgeregelung linearer und nichtlinearer Syteme endlicher und undendlicher Dimension[END_REF]; [START_REF] Sira-Ramírez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF].

• any pure system variable is algebraic over k z pure 1 , . . . , z pure m .

z pure is a (pure) flat, or linearizing, output. For a flat dynamics, it is known that the number m of its elements is equal to the number of independent control variables.

Observability and identifiability

Take a system K/k with control u and output y.

Observability

According to Diop & Fliess (1991a,b) (see also [START_REF] Diop | From the geometry to the algebra of nonlinear observability[END_REF]), system K/k is said to be observable if, and only if, the extension K pure /k u pure , y pure is algebraic.

Remark 2.4. This new definition23 of observability is "roughly" equivalent (see Diop & Fliess (1991a,b) for details24 ) to its usual differential geometric counterpart due to [START_REF] Hermann | Nonlinear controllability and observability[END_REF] (see also [START_REF] Conte | Nonlinear Control Systems -An Algebraic Setting[END_REF]; [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF]; [START_REF] Isidori | Nonlinear Control Systems[END_REF]; [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF]; Sontag (1998)).

Identifiable parameters 25

Set k = k 0 Θ , where k 0 is a differential field and Θ = {θ 1 , . . . , θ r } a finite set of unknown parameters, which might not be constant. According to Diop & Fliess (1991a,b), a parameter θ ι , ι = 1, . . . , r, is said to be algebraically (resp. rationally) identifiable if, and only if, it is algebraic over (resp. belongs to) k 0 u, y :

• θ ι is rationally identifiable if, and only if, it is equal to a differential rational function over k 0 of the variables u, y, i.e., to a rational function of u, y and their derivatives up to some finite order, with coefficients in k 0 ;

• θ ι is algebraically identifiable if, and only if, it satisfies an algebraic equation with coefficients in k 0 u, y .

Determinable variables

More generally, a variable Υ ∈ K is said to be rationally (resp. algebraically) determinable if, and only if, Υ pure belongs to (resp. is algebraic over) k u pure , y pure . A system variable χ is then said to be rationally (resp. algebraically) observable if, and only if, χ pure belongs to (resp. is algebraic over) k u pure , y pure .

Remark 2.5. In the case of algebraic determinability, the corresponding algebraic equation might possess several roots which are not easily discriminated (see, e.g., [START_REF] Li | A differential-algebraic approach to speed estimation in an induction motor[END_REF] for a concrete example).

Remark 2.6. See [START_REF] Sedoglavic | A probabilistic algorithm to test local algebraic observability in polynomial time[END_REF] and [START_REF] Ollivier | Algorithmes efficaces pour tester l'identifiabilité locale[END_REF] for efficient algorithms in order to test observability and identifiability. Those algorithms may certainly be extended to determinable variables and to various questions related to fault diagnosis in Section 2.8.

Fundamental properties of fault variables 26

Detectability

The fault variable w ι , ι = 1, . . . , q, is said to be detectable if, and only if, the field extension

K nom /k u nom , W nom ι , where W nom ι = W nom \{w nom ι }, is differentially transcen- dental.
It means that w ι is indeed "influencing" the output. When considering the variational nominal system, formula (6) yields

   d K nom /k y nom 1 . . . d K nom /k y nom p    = T u    d K nom /k u nom 1 . . . d K nom /k u nom m    +T W    d K nom /k w nom 1 . . . d K nom /k w nom q   
where T u ∈ K(s) p×m , T W ∈ K(s) p×q . Call T W the fault transfer matrix. The next result is clear:

Proposition 2.3. The fault variable w ι is detectable if, and only if, the ι th column of the fault transfer matrix T W is non-zero.

Isolability, parity equations and residuals

A subset W ′ = (w ι1 , . . . , w ι q ′ ) of the set W of fault variables is said to be [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF] for introductions to this perhaps less well known subject. The definitions and properties below are clear-cut extensions of their linear counterparts in Fliess, Join & Sira-Ramírez (2004). Some of them might also be seen as a direct consequence of Section 2.7. means that the parity differential equation is of order 0, i.e., it is an algebraic equation with coefficients k u nom , y nom .

• Rationally isolable if, and only if, W ′nom belongs to k u nom , y nom . It means that the parity equation is a linear algebraic equation, i.e., any component of W ′nom may be expressed as a rational function over k in the variables u nom , y nom and their derivatives up to some finite order.

The next property is obvious:

Proposition 2.4. Rational isolability ⇒ algebraic isolability ⇒ differentially algebraic isolability.

When we will say for short that fault variables are isolable, it will mean that they are differentially algebraically isolable.

Proposition 2.5. Assume that the fault variables belonging to W ′ are isolable. Then card(W ′ ) ≤ card(y).

Proof. The differential transcendence degree of the extension k u nom , y nom , W ′nom /k (resp. k u nom , y nom /k) is equal to card(u) + card(W ′ ) (resp. is less than or equal to card(u) + card(y)). The equality of those two degrees implies our result thanks to the Remark 2.1.

3 Derivatives of a noisy signal

Polynomial time signals

Consider the real-valued polynomial function x

N (t) = N ν=0 x (ν) (0) t ν ν! ∈ R[t]
, t ≥ 0, of degree N . Rewrite it in the well known notations of operational calculus:

X N (s) = N ν=0 x (ν) (0) s ν+1
We know utilize d ds , which is sometimes called the algebraic derivative (cf. [START_REF] Mikusinski | Operational Calculus[END_REF]; [START_REF] Mikusinski | Operational Calculus[END_REF]). Multiply both sides by d α ds α s N +1 , α = 0, 1, . . . , N . The quantities x (ν) (0), ν = 0, 1, . . . , N are given by the triangular system of linear equations 27 :

d α s N +1 X N ds α = d α ds α N ν=0 x (ν) (0)s N -ν (7)
The time derivatives, i.e., s µ d ι XN ds ι , µ = 1, . . . , N , 0 ≤ ι ≤ N , are removed by multiplying both sides of Eq. ( 7) by s -N , N > N .

Remark 3.1. Remember (cf. [START_REF] Mikusinski | Operational Calculus[END_REF]; [START_REF] Mikusinski | Operational Calculus[END_REF]; [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]) that d ds corresponds in the time domain to the multiplication by -t.

27 Following [START_REF] Fliess | An algebraic framework for linear identification[END_REF][START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF], those quantities are said to be linearly identifiable.

Analytic time signals

Consider a real-valued analytic time function defined by the convergent power series x(t) = ∞ ν=0 x (ν) (0) t ν ν! , where 0 ≤ t < ρ. Introduce its truncated Taylor expansion

x(t) = N ν=0 x (ν) (0) t ν ν! + O(t N +1 ) (8)
Approximate x(t) in the interval (0, ε), 0 < ε ≤ ρ, by its truncated Taylor expansion x N (t) = N ν=0 x (ν) (0) t ν ν! of order N . Introduce the operational analogue of x(t), i.e.,

X(s) = ν≥0 x (ν) (0)
s ν+1 , which is an operationally convergent series in the sense of [START_REF] Mikusinski | Operational Calculus[END_REF]; [START_REF] Mikusinski | Operational Calculus[END_REF]. Denote by [x (ν) (0)] eN (t), 0 ≤ ν ≤ N , the numerical estimate of x (ν) (0), which is obtained by replacing X N (s) by X(s) in Eq. ( 7). The next result, which is elementary from an analytic standpoint, provides a mathematical justification for the computer implementations: [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]) for fundamental theoretical developments. See also [START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF] for most fruitful comparisons and discussions.

Proposition 3.1. For 0 < t < ε, lim t↓0 [x (ν) (0)] eN (t) = lim N →+∞ [x (ν) (0)] eN (t) = x (ν) (0) (9) Proof. Following (8) replace x N (t) by x(t) = x N (t) + O(t N +1 ). The quantity O(t N +1 ) becomes negligible if t ↓ 0 or N → +∞. Remark 3.2. See

Noisy signals

Assume that our signals are corrupted by additive noises. Those noises are viewed here as highly fluctuating, or oscillatory, phenomena. They may be therefore attenuated by low-pass filters, like iterated time integrals. Remember that those iterated time integrals do occur in Eq. ( 7) after multiplying both sides by s -N , for N > 0 large enough.

Remark 3.3. The estimated value of x(0), which is obtained along those lines, should be viewed as a denoising of the corresponding signal.

Remark 3.4. See [START_REF] Fliess | Analyse non standard du bruit[END_REF] for a precise mathematical foundation, which is based on nonstandard analysis. A highly fluctuating function of zero mean is then defined by the following property: its integral over a finite time interval is infinitesimal, i.e., "very small". Let us emphasize that this approach28 , which has been confirmed by numerous computer simulations and several laboratory experiments in control and in signal processing29 , is independent of any probabilistic setting. No knowledge of the statistical properties of the noises is required.

J m θm (t) = κ θ l (t) -θ m (t) -B θm (t) + K τ u(t) J l θl (t) = -κ θ l (t) -θ m (t) -mgh sin(θ l (t)) y(t) = θ l (t) (10)
where

• θ m and θ l represent respectively the angular deviation of the motor shaft and the angular position of the inverted pendulum,

• J m , J l , h, m, κ, B, K τ and g are physical parameters which are assumed to be constant and known. System (10), which is linearizable by static state feedback, is flat; y = θ l is a flat output.

Control design

Tracking of a given smooth reference trajectory y * (t) = θ * l (t) is achieved via the linearizing feedback controller

u(t) = 1 Kτ Jm κ J l v(t)
+ κÿ e (t) +mgh(ÿ e (t) cos(y e (t)) -( ẏe (t)) 2 sin(y e (t))) +J l ÿe (t) + mgh sin(y e (t))

B κ J l y (3) e (t) + κ ẏe (t) + mgh ẏe (t) cos(y e (t) (11)
where

v(t) = y * (4) (t) -γ 4 (y (3) e (t) -y * (3) (t)) -γ 3 (ÿ e (t) -ÿ * (t)) -γ 2 ( ẏe (t) -ẏ * (t)) -γ 1 (y e (t) -y * (t)) (12)
The subscript "e"denotes the estimated value. The design parameters γ 1 , ..., γ 4 are chosen so that the resulting characteristic polynomial is Hurwitz.

Remark 4.1. Feedback laws like ( 11)-( 12) depend, as usual in flatness-based control (see, e.g., [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF][START_REF] Sastry | Nonlinear Systems[END_REF]; Sira-Ramírez & Agrawal ( 2004)), on the derivatives of the flat output and not on the state variables. We might nevertheless be interested in obtaining an estimate [θ m ] e (t) of the unmeasured state θ m (t):

[θ m ] e (t) = 1 κ J l ÿe (t) + mgh sin(y e (t)) + y e (t) (13)

Numerical simulations

The physical parameters have the same numerical values as in [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF]: J m = 3.7 × 10 -3 kgm 2 , J l = 9.3 × 10 -3 kgm 2 , h = 1.5 × 10 -1 m, m = 0.21 kg, B = 4.6 × 10 -2 m, K τ = 8 × 10 -2 NmV -1 . The numerical simulations are presented in Figures 23456789. Robustness has been tested with an additive white Gaussian noise N(0; 0.01) on the output y. Note that the off-line estimations of ÿ and θ m , where a "small" delay is allowed, are better than the on-line estimation of ÿ.

5 Parametric identification

A rigid body

Consider the fully actuated rigid body, depicted in Figure 10, which is given by the Euler equations

I 1 ẇ1 (t) = (I 2 -I 3 )w 2 (t)w 3 (t) + u 1 (t) I 2 ẇ2 (t) = (I 3 -I 1 )w 3 (t)w 1 (t) + u 2 (t) I 3 ẇ3 (t) = (I 1 -I 2 )w 1 (t)w 2 (t) + u 3 (t) (14)
where w 1 , w 2 , w 3 are the measured angular velocities, u 1 , u 2 , u 3 the applied control input torques, I 1 , I 2 , I 3 the constant moments of inertia, which are poorly known. System ( 14) is stabilized around the origin, for suitably chosen design parameters λ 1ι , λ 0ι , ι = 1, 2, 3, by the feedback Write Eq. ( 14) in the following matrix form:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (s)
u 1 (t) = -(I 2 -I 3 )w 2 (t)w 3 (t) +I 1 -λ 11 w 1 (t) -λ 01 t 0 w 1 (σ)dσ u 2 (t) = -(I 3 -I 1 )w 3 (t)w 1 (t) +I 2 -λ 12 w 2 (t) -λ 02 t 0 w 2 (σ)dσ u 3 (t) = -(I 1 -I 2 )w 1 (t)w 2 (t) +I 3 -λ 13 w 3 (t) -λ 03 t 0 w 3 (σ)dσ (15)
  ẇ1 -w 2 w 3 w 2 w 2 w 1 w 3 ẇ2 -w 1 w 3 -w 1 w 2 w 1 w 2 ẇ3   ×   I 1 I 2 I 3   =   u 1 u 2 u 3   It yields

Numerical simulations

The output measurements are corrupted by an additive Gaussian white noise N (0; 0.005). Figure 11 shows an excellent on-line estimation of the three moments of inertia. Set for the design parameters in the controllers ( 15) and ( 16)

λ 1ι = 2ξ̟, λ 0ι = ̟ 2 , ι = 1, 2, 3
, where ξ = 0.707, ̟ = 0.5. The stabilization with the above estimated values in Figure 12 is quite better than in Figure 13 where the following false values where utilized: I 1 = 0.2, I 2 = 0.1 and I 3 = 0.1.

6 Fault diagnosis and accommodation 6.1 A two tank system 32

Consider the cascade arrangement of two identical tank systems, shown in Figure 14, which is a popular example Its mathematical description is given by

ẋ1 (t) = - c A x 1 (t) + 1 A u(t) (1 -w(t)) +̟(t) (17) ẋ2 (t) = c A x 1 (t) - c A x 2 (t) y(t) = x 2 (t)
where:

• The constant c and the area A of the tank's bottom are known parameters. • The actuator failure w(t), 0 ≤ w(t) ≤ 1, is constant but unknown. It starts at some unknown time t I >> 0 which is not "small".

• Only the output y = x 2 is available for measurement.

The corresponding pure system, where we are ignoring the fault and perturbation variables (cf. Section 2.4.1),

ẋpure 1 = -c A x pure 1 + 1 A u pure ẋpure 2 = c A x pure 1 -c A x pure 2 y pure = x pure 2 is flat. Its flat output is y pure = x pure 2
. The state variable x pure 1 and control variable u pure are given by

x pure 1 = A c ẏpure + √ y pure 2 (18) u pure = 2A A c ẏpure + √ y pure A c ÿpure + ẏpure 2 √ y pure +c A c ẏpure + √ y pure (19)

Fault tolerant tracking controller

It is desired that the output y tracks a given smooth reference trajectory y * (t). Rewrite Formulae ( 18)-( 19) by taking into account the perturbation variable ̟(t) and the actuator failure w(t):

x 1 (t) = A c ẏ(t) + y(t) 2 (20) u(t) = 1 1 -w(t) -A̟ +2A A c ẏ(t) + y(t) × ( A c ÿ(t) + ẏ(t) 2 y(t) +c A c ẏ(t) + y(t)
With reliable on-line estimates ŵ(t) and ̟(t) of the failure signal w(t) and of the perturbation ̟(t), we design a failure accommodating linearizing feedback controller. It incorporates a classical robustifying integral action:

u(t) = 1 1 -ŵ(t) -A ̟(t) +2A A c ẏe (t) + y e (t) A c v(t) + ẏe (t) 2 y e (t) +c A c ẏe (t) + y e (t) v(t) = ÿ * (t) -G ⋆ (y e (t) -y * (t))
This is a generalized proportional integral (GPI) controller (cf. [START_REF] Fliess | Correcteurs proportionnelsintégraux généralisés[END_REF]) where

• ⋆ denotes the convolution product,

• the transfer function of G is

λ 2 s 2 + λ 1 s + λ 0 s(s + λ 3 )
where λ 0 , λ 1 , λ 2 , λ 3 ∈ R,

• y e (t) is the on-line denoised estimate of y(t) (cf. Remark 3.3),

• ẏe (t) is the on-line estimated value of ẏ(t).

Perturbation and fault estimation

The estimation of the constant perturbation ̟ is readily accomplished from Eq. ( 17) before the occurrence of the failure w, which starts at time t I >> 0:

ẋ1 (t) = - c A x 1 (t) + 1 A u(t) + ̟ if 0 < t < t I
Multiplying both sides by t and integrating by parts yields33

̟ = arbitrary 0 < t < ǫ 2 tx1(t)-t 0 x1(σ)-σ( c A √ x1(σ)- 1 A u(σ)) dσ t 2 ǫ < t < t I
where ǫ > 0 is "very small". The estimated value x1 (t) of x 1 (t), which is obtained from Formula (20), needs as in Section 6.2 the on-line estimation y e (t) and ẏe (t).

The estimated value ŵ of w, which is detectable and algebraically isolable (cf. Section 2.8.2), follows from

ŵ = 1 - 1 u(t) 2A A c ẏe (t) + y e (t) × A c ÿe (t) + ẏe (t)
2 y e (t) +c A c ẏe (t) + y e (t) -A ̟

Numerical simulations

Figure 15 shows the closed-loop performance of our trajectory tracking controller. The simulation scenario is the following:

• The actuator fault w = 0.7 occurs at time t I = 1.5s.

• We estimate before the unknown constant perturbation ̟ = 0.2 and use it for estimating w.

• The fault tolerant control becomes effective at time t = 2.5s.

Robustness is checked via an additive Gaussian white noise N (0; 0.01). Comparison between Figures 16 and 15 confirms the efficiency of our fault accommodation. Suppose we are given a linear perturbed second order system

ÿ(t) + y(t) = u(t) -z(t) + C1(t -t I ) (21)
where

• z(t) is an unknown perturbation input,

• 1(t) is the Heaviside step function, i.e., • s 4 + c 3 s 3 + c 2 s 2 + c 1 s + c 0 is the characteristic polynomial of the unperturbed closed-loop system. The coefficients c 0 , c 1 , c 2 , c 3 are chosen so that the imaginary parts of its roots are strictly negative.

1(t) = 0 if t < 0 1 if t ≥ 0 • C is
Like usual proportional-integral-derivative (PID) regulators, this controller is robust with respect to un-modeled piecewise constant errors The computer simulations were performed with z(t) = 10t 3 sin(2t) 1 + t 2 + t 3

The unknown constant perturbation suddenly appears at time t I = 4 with a permanent value C = 1.25. The coefficients of the characteristic polynomial were forced to be those of the desired polynomial P d (s) = (s 2 +2ζω n s+ω 2 n ) 2 , with ζ = 0.81, ω n = 4. We have set y ⋆ (t) = sin ωt, ω = 2.5[rad/s]. Figure 17 (resp. 18) shows the reference signal y ⋆ (t) and the output signal y(t) without estimating z e (t) (resp. with the estimate z e (t)). We added in the simulations of Figure 18 a Gaussian white noise N (0; 0.025) to the measurement y(t). The results are quite remarkable.

Remark 7.2. The same technique yields an efficient solution to fault tolerant linear control, which completes Fliess, Join & Sira-Ramírez (2004). Just think at z(t) as a fault variable.

Non-linear extension

Replace the term y(t We have proposed a new approach to non-linear estimation, which is not of asymptotic nature and does not necessitate any statistical knowledge of the corrupting noises 34 . Promising results have already been obtained, which will be supplemented in a near future by other theoretical advances (see, e.g., [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF] on observers with unknown inputs) and several concrete case-studies (see already [START_REF] García-Rodríguez | Trajectory tracking via algebraic methods for state estimation[END_REF]; [START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF]). Further numerical improvements 34 Let us refer to a recent book by [START_REF] Smolin | The Trouble with Physics -The Rise of String Theory, the Fall of Science, and What Comes Next[END_REF], which contains an exciting description of the competition between various theories in today's physics. Similar studies do not seem to exist in control.

)

,

  ): the algebraic closure Kpure of K pure is equal to the algebraic closure of a purely differentially transcendental extension of k. It means in other words that there exists a finite subset z pure = {z pure 1

  3. Differential algebra has already been employed but in a different context by several authors (see, e.g., Martinez-Guerra & Diop (2004); Martìnez-Guerra, González-Galan, Luviano-Juárez & Cruz-Victoria (2007); Staroswiecki & Comtet-Varga (2001); Zhang, Basseville & Benveniste (1998)).

  Figure 1: A single link flexible joint manipulator

Figure 2 :

 2 Figure 2: Output (-) and reference trajectory (--)

Figure 5 :

 5 Figure 5: y: (--); on-line noise attenuation y e (-)

Figure 6

 6 Figure 6: ÿ (--); on-line estimation ÿe (-) ,

Figure 7 :

 7 Figure 7: θ m (--); on-line estimation [θ m ] e (-)

Figure 8

 8 Figure 8: ÿ (--); off-line estimation ÿe (-)

Figure 9 :

 9 Figure 9: θ m (--); off-line estimation [θ m ] e (-)

Figure 3 :

 3 Figure 3: Control

Figure 4 :

 4 Figure 4: Output noise

  Figure 10: Rigid body

  estimates [I 1 ] e , [I 2 ] e , [I 3 ] e of I 1 , I 2 , I 3 when we replace w 1 , w 2 , w 3 , ẇ1 , ẇ2 , ẇ3 by their estimates 31 . The control law (15) becomes u1(t) = -([I2]e -[I3]e)[w2]e(t)[w3]e(t) +[I1]e -λ11[w1]e(t) -λ01 t 0 [w1]e(σ)dσ u2(t) = -([I3]e -[I1]e)[w3]e(t)[w1]e(t) +[I2]e -λ12[w2]e(t) -λ02 t 0 [w2]e(σ)dσ u3(t) = -([I1]e -[I2]e)[w1]e(t)[w2]e(t) +[I3]e -λ13[w3]e(t)

Figure 13 :

 13 Figure 13: Feedback stabilization without parametric estimation

Figure 11 :

 11 Figure 11: Zoom on the parametric estimation (-) and real values (--)

Figure 12 :

 12 Figure 12: Feedback stabilization with parametric estimation

Figure 15 :Figure 16 :

 1516 Figure 15: y ⋆ (t) (--) and y(t) (-) with fault accommodation

Figure 17

 17 Figure 17: y ⋆ (t) (--) and y(t) (-) without perturbation attenuation

Figure 19

 19 Figure 19: y ⋆ (t) (--) and y(t) (-) without perturbation attenuation

Figure 20 :

 20 Figure 20: y ⋆ (t) (--) and y(t) (-) with perturbation attenuation.

•

  Differentially algebraically isolable if, and only if, the extension k u nom , y nom , W ′nom /k u nom , y nom is differentially algebraic. It means that any component of W ′nom satisfies a parity differential equation, i.e., an algebraic differential equations where the coefficients belong to k u nom , y nom .• Algebraically isolable if, and only if, the extension k u nom , y nom , W ′nom /k u nom , y nom is algebraic. It

	26 See, e.g., Chen & Patton (1999); Blanke, Kinnaert, Lunze &
	Staroswiecki (2003); Gertler (1998); Vachtsevanos, Lewis, Roemer,

  an unknown constant and thus C1(t -t I ) is a constant bias, of unknown amplitude, starting at time t I ≥ 0.

	Remark 7.1. The difference C1(t -t I ) -z(t) is a ratio-nally determinable variable according to Section 2.7.3.
	The estimate z e (t) of z(t) is given up to a piecewise con-
	stant error by
	z e (t) = -ÿ e (t) -y e (t) + u(t)
	where y e (t) and ÿe (t) are the on-line estimated values
	of y(t) and ÿ(t). We design a generalized-proportional-
	integral (GPI) regulator, in order to track asymptotically
	a given output reference trajectory y ⋆ (t), i.e.,
	c2s 2 +c1s+c0
	s(s+c3)

u(t) = y e (t) + z e (t) + ÿ⋆ (t) + G ⋆ (y e (t) -y * (t)) (

22

) where • G is defined via its rational transfer function

  (t) = -ÿ e (t) -y e ẏe (t) + u(t)where y e (t), ẏe (t) and ÿe (t) are the estimates of y(t), ẏ(t) and ÿ(t). The feedback law (22) becomesu = y e (t) ẏe (t) + z e (t) + ÿ⋆ (t) + G ⋆ (y e (t) -y * (t)) (24) -x 1 (t)x 2 (t) + u(t) -z(t) + C1(t -t I ) y(t) = x 1 (t)Applying the feedback law (24) amounts possessing good estimates of the two state variables.Figures 19 and 20 depict the computer simulations with the same numerical conditions as before. The results are again excellent.

	Remark 7.3. Rewrite system (23) via the following state
	variable representation
	 	ẋ1 (t) = x 2 (t)
		ẋ2 (t) =
	

in system (21) by the product y(t) ẏ(t):

ÿ(t) + y(t) ẏ(t) = u(t) -z(t) + C1(t -t I ) (23)

The perturbations z(t) and C1(t -t I ) are the same as above. The estimate z e (t) of z(t) up to a piecewise constant is given by z e

There are of course situations, for instance with a very strong corrupting noise, where the present state of our techniques may be insufficient. See also Remark 2.5.

Other authors like[START_REF] Slotine | Applied Nonlinear Control[END_REF] had already noticed that "good" numerical differentiators would greatly simplify control synthesis.

6 They are also valid for the linear estimation questions listed in Remark 1.

1.7 See, e.g.,[START_REF] Schweppe | Uncertain Dynamic Systems[END_REF];Jaulin, Kiefer, Didrit & Walter

Algebraic equations are differential equations of order 0.

See, e.g.,[START_REF] Atiyah | Introduction to Commutative Algebra[END_REF];[START_REF] Chambert-Loir | A Field Guide to Algebra[END_REF] for basic notions in commutative algebra.

See, e.g., McConnell & Robson (2000).

See, e.g., McConnell & Robson (2000).

See, e.g., Eisenbud (1995).

Any reader with a good algebraic background will notice a connection with the notion of differential specialization (see, e.g.,[START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF]).

Let us explain those algebraic manipulations in plain words. Ignoring the perturbation variables in the original system yields the nominal system.

See Fliess & Rudolph (1997) for a definition via infinite prolongations.

 24 The differential algebraic and the differential geometric languages are not equivalent. We cannot therefore hope for a "one-to-one bijection" between definitions and results which are expressed in those two

settings.25 Differential algebra has already been employed for parametric identifiability and identification but in a different context by several authors (see, e.g.,[START_REF] Ljung | On global identifiability for arbitrary model parametrizations[END_REF];[START_REF] Ollivier | Le problème de l'identifiabilité structurelle globale : approche théorique, méthodes effectives et bornes de complexité[END_REF] Saccomani, Audoly & D'Angio (2003)).

This approach applies as well to multiplicative noises (see[START_REF] Fliess | Analyse non standard du bruit[END_REF]). The assumption on the noises being only additive is therefore unnecessary.

 29 For numerical simulations in signal processing, see[START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF][START_REF] Chambert-Loir | A Field Guide to Algebra[END_REF]; Fliess, Join,[START_REF] Fliess | Estimation des dérivées d'un signal multidimensionnel avec applications aux images et aux vidéos[END_REF]. Some of them are dealing with multiplicative noises.

See Sira-Ramírez & Fliess (2006) and[START_REF] Reger | Robust algebraic state estimation of chaotic systems[END_REF] for other interesting examples of state reconstructors which are applied to chaotically encrypted messages.

We are adapting here linear techniques stemming from[START_REF] Fliess | An algebraic framework for linear identification[END_REF][START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF].


2,77°C

•Fig. 3

 3 Fig. 1 Circuit électronique du capteur associé au circuit du découplage actif optique.

Fig. 5 RSBFig. 6

 56 Fig. 5 Images acquises par la bobine volumique de l'IRM 3,0 T pour imager un fantôme creux lors de la présence de différents types de capteurs endoluminaux.

V. Detti, et al., MRM, vol. 66, p. 448-455, 2011.

R. Aydé, et al., ISMRM-ESMRMB, Milan, Italy, 2014.

Bibliographie


I Saniour

R Aydé

A L Perrier

G Gaborit

L Duvillaret

R Sablong

O Beuf

Active Optical-Based Detuning Circuit for Receiver Endoluminal Coil

Keywords: magnetic resonance imaging, optical detuning, endoluminal coil, fibered sensor, galvanic link, photodiodes

In this paper we demonstrate the effectiveness of an active optical detuning circuit for magnetic resonance imaging (MRI) endoluminal receiver coil. Three endoluminal coils prototypes were built: a coil without any detuning circuit, a coil with a galvanic (classic) detuning circuit using a PIN diode, and a coil with an optical detuning circuit using two photodiodes in parallel with a PIN diode. These coils were built and characterized on a laboratory experimental bench. Then, an in vitro experiment was performed with a 3.0 T MR system to evaluate the impact of the endoluminal receiver coils in detuned phase on the image uniformity distribution measured using the body coil. Next, the endoluminal coil was used as a receiver coil to compare the signal-to-noise ratio (SNR) distribution based on iso-contour maps. On experimental bench, the results show an increase delay of the switching times (tuned-detuned or detuned-tuned) for optical-detuned coils of about 10 µs due to the electro-optical circuits, delay still compatible with requirements. When the body coil is used as a transceiver, the SNR uniformity is similar whether the galvanic or the optical detuning circuit is used. Finally, the SNR iso-contours of the different endoluminal coils prototypes are comparable.

Introduction

Magnetic resonance imaging (MRI) is widely used to diagnose and analyze inflammatory bowel diseases [START_REF] Geenen R W F | Current status of MRI in patients with inflammatory bowel disease colitis[END_REF] and to determine rectal cancer stages (Beets et al 2013, Dewhurst et al 2013, Torkzad et al 2010). However, accurate exploration of bowel diseases and detailed information about the colon wall layers, mandatory for proper management and effective treatment, are still not available. Indeed, despite the huge progresses in radiofrequency (RF) coils, based on the phased array technology with increase of channels [START_REF] Giusti | Preoperative rectal cancer staging with phased-array MR[END_REF], the external MRI coils have remained unable to provide adequate SNR to reach required high spatial resolution images. By comparison several authors have demonstrated that internal coils significantly increase the local SNR and makes possible either to perform fast in vivo MR spectroscopy of heart tissues or to obtain high spatial resolution MR images of colon walls in preclinical context (Beuf et al 2004[START_REF] Dorez | Endoluminal high-resolution MR imaging protocol for colon walls analysis in a mouse model of colitis[END_REF][START_REF] Kantor | In vivo 31 P nuclear magnetic resonance measurements in canine heart using a catheter-coil[END_REF][START_REF] Pilleul | High-resolution MR imaging appearance of colonic tissue in rabbits using an endoluminal coil[END_REF][START_REF] Ramgolam | Optical spectroscopy combined with high-resolution magnetic resonance imaging for digestive wall assessment: endoluminal bimodal probe conception and characterization in vitro on organic sample and in vivo on a rabbit[END_REF].

However, for clinical use, conducting coaxial cables linking the coil (deeply introduced into the patient) to the connecting plug of the MR system may induce safety issues. In fact, the electrical field → E , accompanying the RF magnetic field → B 1 , induces currents in the conducting cables at the same frequency that can lead to a strong increase of the local specific absorption rate (SAR) by the tissues [START_REF] Detti | Assessment of radiofrequency self-heating around a metallic wire with MR T1-based thermometry[END_REF][START_REF] Yeung | RF safety of wires in interventional MRI: using a safety index[END_REF]. Different solutions were proposed to diminish the SAR issue, i.e., coaxial chokes [START_REF] Ladd | Reduction of resonant RF heating in intravascular catheters using coaxial chokes[END_REF], transformer coupled transmission lines [START_REF] Weiss | Transmission line for improved RF safety of interventional devices[END_REF], baluns matching circuits, and traps [START_REF] Peterson | Common mode signal rejection methods for MRI: reduction of cable shield currents for high static magnetic field systems Concepts[END_REF]. These methods do not ensure a total safety for the patient. The use of optical fibers instead of coaxial cables appears as an attractive alternative to solve definitely safety issues. Different solutions have already been explored to ensure an optical connection between MRI and some coils [START_REF] Fandrey | A novel active MR probe using a miniaturized optical link for a 1.5-T MRI scanner[END_REF][START_REF] Koste | Proc. ISMRM, 13th Scientific Meeting and exhibition[END_REF][START_REF] Memis | Miniaturized fiberoptic transmission system for MRI signals[END_REF][START_REF] Possanzini | dStream Architecture[END_REF][START_REF] Yuan | A direct modulated optical link for MRI RF receive coil interconnection[END_REF]. For example in reference [START_REF] Possanzini | dStream Architecture[END_REF], the dStream TM architecture of Philips has introduced a hybrid technique to amplify, filter, and sample the detected nuclear magnetic resonance (NMR) signal to a numerical one to be then converted into an optical signal and transmitted using an optical fiber. In this latter case, the active detuning is classically performed using coaxial cables. Regarding endoluminal coils, whereas the major issue is the optical conversion and transmission of the NMR signal, the optical tuning/detuning is also an important issue to have a fully optical receiver coil. The receiver coil has to be detuned during RF transmit phase to avoid the strong coupling between coils and → B 1 concentrations close to the receiver coil, which affects the image uniformity and involve again the SAR issues due to E-field concentrations. Thus in the RF-transmission phase, the endoluminal coil should be detuned from the resonance frequency to ensure isolation between transmit and receive coils. PIN diodes are usually used for passive or/and active detuning of the receiver coils. In passive detuning, a crossed pair of fast switching PIN diodes is added to the resonant circuit. When the RF field is transmitted, the induced current alternatively switches one of the diodes within the circuit. Although it is a wireless method, passive detuning is not sufficiently effective because, depending on RF pulses used, the induced current is not always sufficient to activate the PIN diode which can degrade again image uniformity. Moreover, the high conduction losses in fast switching diodes may induce for endoluminal coil considerable heat inside patient's body [START_REF] Burl | Transmit mode coil detuning for MRI systems[END_REF]. On the other hand, active detuning can be done by using a PIN diode added to the resonant circuit. The MR system transmits a direct current (DC) to activate the PIN diode when the → B 1 field is emitted. Hence the low impedance of the PIN diode changes the self-resonant frequency of the coil. This method is faster, more reliable and safer than the passive detuning methods. Conventionally, the DC bias current is provided by the MR system and transmitted using a coaxial cable. For some specific applications, optical active detuning was already carried out. For instance, a safe, reliable and precise catheter tracking was demonstrated for intervention MRI procedures [START_REF] Weiss | In vivo safe catheter visualization and slice tracking using an optically detunable resonant marker[END_REF][START_REF] Wong | An optical system for wireless detuning of parallel resonant circuits[END_REF]. But those studies were not used for imaging and did not require an optical detuning system with high switching performance. Korn et al have used optical detuning for coils dedicated to compensate the physiological motion [START_REF] Korn | Optically detunable, inductively coupled coil for self-gating in small animal magnetic resonance imaging[END_REF]. The switching time was not considered as an important parameter. Moreover, the low induced photocurrent (110 µA) in this detuning system would not be sufficient to detune a larger coil. Furthermore, Fandrey et al have used a passive technique to detune of the coil and an optical modulator to transmit the RF detected signal to avoid any additional galvanic connection [START_REF] Fandrey | A novel active MR probe using a miniaturized optical link for a 1.5-T MRI scanner[END_REF]. In contrast, Memis et al have worked with an active optical detuning circuit where a phototransistor associated with a battery instead of a PIN diode [START_REF] Memis | Miniaturized fiberoptic transmission system for MRI signals[END_REF]. This proof of concept was however not suited to MRI applications due to its expected non-magnetic compatibility. Finally, as already mentioned, dStream TM architecture of Philips uses a galvanic active detuning method.

The overall objective of our project is to develop an endoluminal receiver coil devoid of any galvanic connection and excluded from any active component. The fully optical receiver coil has then to ensure two functionalities: the active optical detuning of the coil and the optical transmission of the detected NMR. The optical transmission was validated on an optical bench [START_REF] Aydé | Potentialities of an electro-optic crystal fed by nuclear magnetic resonant coil for remote and low-invasive magnetic field characterization[END_REF][START_REF] Aydé | Unbiased Electro-Optic waveguide as a sensitive nuclear magnetic resonance sensor[END_REF]. In the following, the first functionality of the coil is presented. The optical detuning method was designed to ensure a reliable detuning based on the bias signal provided through optical fiber. After conception, the detuning circuit was associated with a single loop endoluminal coil with conventional NMR signal reception.

The coils and detuning circuits were electrically characterized on a measuring bench before to be compared in vitro on a 3.0 T MR system in terms of signal uniformity and SNR.

Methods

Coils Prototypes

A rectangular single loop coil was designed using a 35 µm-thick copper trace (width w = 5.1 mm, and length l = 47 mm on a FR4 substrate (relative permittivity r = 4, thickness t = 0.8 mm). Previous in vivo works [START_REF] Armenean | Optimization of endoluminal loop radiofrequency coils for gastrointestinal wall MR imaging[END_REF], Beuf et al 2004[START_REF] Pilleul | High-resolution MR imaging appearance of colonic tissue in rabbits using an endoluminal coil[END_REF] have demonstrated that such size and geometry of the loop enable the depiction of bowel abnormalities by imaging the colon wall layers with an adequate penetration depth and a high in-plane spatial resolution. Figure 1 shows the photograph and the electric circuit of the different used coils. The endoluminal receiver coils were designed to operate with a discovery MR750 3.0 T MR System (GEHC, Milwaukee, USA). These coils are matched to 50 Ω at a 127.73 MHz frequency by using tuning and matching non-magnetic capacitors (American Technical Ceramics (ATC), New York, USA) on the distal and proximal parts of the coil.

A rectangular single loop without detuning circuit was first built. The adjustment of tuning and matching capacitors gives C t = 34.4 pF and C m = 203.2 pF.

The galvanic-detuned coil (reference coil) can be carried out by adding a DH 80055 PIN diode (Temex Ceramics, Pessac, France) to the rectangular coil circuit in the distal part to ensure the detuning of the coil when activated. The tuning and matching capacitor values of this coil are respectively C t = 36.3 pF and C m = 190 pF.

Similarly, the optical-detuned coil (ODC) was built by adding a PIN diode and two photodiodes to the rectangular coil. A silicon S6931-01 photodiode (Hamamatsu Photonics, Japan) was used, with a 6.6 mm 2 photosensitive area. This photodiode is considered as "low-magnetic" based on MR images depicting very negligible local deformation in shape and intensity. Two photodiodes in series were used instead of one to ensure the requested voltage for a PIN diode to operate in direct mode. Tuning and matching capacitor values were 44.4 pF and 206.2 pF respectively. Two choke inductors of 1.2 µH (SRT Resistor Technology, Germany) were added between the loop and the two photodiodes to block any alternating currents (AC) from interacting with the detuning circuit during NMR reception phase. When switching from detuning to tuning phase, the response of the coil is slow. Indeed parasitic capacitance of illuminated photodiodes enables photoelectrons to accumulate. Once light is off, these residual charges slowly flow out through the PIN diode which is still weakly biased (under threshold voltage) [START_REF] Saleh | Fundamentals of Photonics[END_REF]. Hence, a shunt resistance R 0 of 1.8 kΩ was added in parallel to force a rapid discharge of the photodiodes. The switching times form tuning to detuning and reciprocally depend effectively on the electro-optical circuits and the added resistancevalue. For a high resistance value, the commutation time to tuning mode increases (Figure 2). On the other hand, the resistance should be sufficiently high to have the adequate voltage to activate the PIN diode. Taking these factors into consideration, an appropriate trade off was obtained for a resistance value of 1.8 kΩ.

Optical Detuning Unit

In order to detune the coil, the two photodiodes added to the coil are illuminated by using two 30 mW fibred modulated laser diodes (Laser Components R , Germany, λ = 650 nm). The optical detuning system begins when the MR system provides a bias DC current (150 mA) or a negative voltage to respectively detune the coil by turning on the laser diodes or tune the coil back by turning them off. With the connector used for single channel coil, the DC current is provided on the coaxial RF signal cable. A AC-DC filter was first placed to derive the DC bias signal. This DC bias signal was then used to drive the modulation of laser diodes. Given that these laser diodes must be powered and the electronic circuit could contain magnetic components, it was important to place the laser command circuit (including the laser diodes) distant enough from the MR system. To transmit the DC current, a coaxial cable cannot be used; because it is long enough to act as an antenna and depict signal from RF pulses. For this reason our solution was to convert directly the outgoing DC current from the filter to an optical signal using an electrical-to-optical conversion circuit, including an HFBR transmitter (Avago Technologies, 1404, 820 nm, low optial power around 100 µW). The converted signal is transmitted via a multimode optical fiber (Radiospares, ∅ = 0.2 mm) to the laser command circuit. This latter consists of an optical to electrical converter: an HFBR receiver (Avago Technologies, 2404, 820 nm) followed by a non-inverting amplifier to modulate the states of the laser diodes and then decouple the coil.

Experiments

Coils were characterized using a vector network analyzer (VNA) (Agilent Technologies Inc., Santa Clara, CA, USA). The quality factor of each coil is determined from the reflection coefficient |S 11 | when the coils were loaded with a cylindrical phantom that will be later described in this section. |S 11 | is studied to assess detuning performance of each coil prototype. Furthermore, the switching time from tuned to detuned state of the coil and reciprocally was measured. The switching times were characterized on an experimental bench as presented in figure 3. A RF magnetic field was generated using a circular wide band RF emission coil excited by a high-frequency generator. The electrical power of the generator was 13 dBm at a frequency of 127.73 MHz. The endoluminal receiver coil, resonating at the same frequency, was located in front of the RF emission coil to detect the magnetic field. The detuning of the reference coil was ensured by a low-frequency signal generator providing a square electrical 0 V/5 V signal. This is equivalent to the decoupling signal from a MR System (Figure 3(a)). A filter was used here to visualize the DC and RF signals separately on the oscilloscope. On the other hand, the optical detuning was ensured by the optical detuning system presented in figure 3(b). The signal received by the endoluminal coils was measured using a numerical 500 MHz oscilloscope (Agilent Technologies Inc., Santa Clara, CA, USA).

In vitro experiments were performed on a 3.0 T MRI. Figures 4(a) and 4(b) show the experimental setup. The coils were inserted into a plastic cylindrical phantom of 11 mm inner diameter, 90 mm outer diameter, and 100 mm-long. The phantom was filled with a solution of 1.25 g NiSO 4 × 6H 2 O + 5 g NaCl per liter of distilled water to mimic the loading conditions of an internal coil inserted in a patient. First, the body coil was used as a transceiver to validate the efficiency of the detuning circuit by comparing the images acquired while using the tuned and the detuned coils. Corresponding images of the phantom without any endoluminal coil were also acquired to provide a basic reference. Two sequences were performed: a fast spin echo (FSE) sequence and a gradient echo sequence (GRE). Imaging parameters for the FSE sequence are T R/T E = 3275 ms/13.3 ms, axial images with a F OV = 160 × 160 mm 2 , slice thickness = 2 mm, spacing between slices = 0.5 mm, matrix = 256 × 256, receiver bandwidth = 16.67 kHz or ±8.335 kHz (pixel receiver bandwidth = 65.11 Hz/Px). Image parameters for the GRE sequence are: T R/T E = 400 ms/8.9 ms, f lip angle = 80 • , axial images with a F OV = 160 × 160 mm 2 , slice thickness = 2 mm, spacing between slices = 0.5 mm, matrix = 256 × 256, receiver bandwidth = 11.90 kHz or ±5.95 kHz (pixel receiver bandwidth = 46.48 Hz/Px). The SNR calculation and the registration of the MR images were performed using MATLAB language (Mathworks, Milwaukee, USA). The SNR was determined by the mean of the signal intensity in the entire MR image divided by the mean standard deviation of a region of interest chosen outside the image of the phantom (in the air) which is the noise. For the registration, a self-made MATLAB program was developed to align MR images to a common coordinate system. For the purpose of quantifying the performance of the endoluminal coils in terms of SNR uniformity distribution, the same GRE and FSE sequences were applied, this time, using the endoluminal coil as a receiver. The comparison was done by drawing and identifying some specific SNR iso-contour lines in the images acquired by the different coils. Finally, the local variation of the temperature was measured for both reference coil and ODC used as receivers in the MR experiment. This measurement was performed using the optical fiber temperature system (Opsens, Canada). Fibers tips were mounted to the coil using Teflon rubber, and the whole was immersed in 1.5 % agar-agar gel. For the reference coil, the fibers were placed at the proximal side of the loop, the distal (near the PIN diode), and the middle of the loop. For the ODC, the temperature variation was studied near the PIN diode and the two photodiodes.

Results

Experimental Bench Measurement

The reflection coefficient modulus |S 11 | as a function of the frequency is measured for each coil prototype in the tuned and detuned modes. The resonance frequency of the three coils is around 127.73 MHz. While the coil detuning, the resonance frequency is shifted except for the non-detuned coil. The quality factor Q of the tuned coils is calculated on tuning mode following the procedure described in reference [START_REF] Beuf | Small-animal MRI: signal-to-noise ratio comparison at 7 and 1.5 T with multiple-animal acquisitions strategies[END_REF]. The coil without detuning circuit has the highest quality factor (Q = 74). The quality factor decreases slightly for both reference coils and ODCs due to the added PIN diode and photodiode components. For the two coil prototypes considered in this work, the quality factors Q are 53 and 55 respectively for the reference coil and ODC. On tuned mode, at 127.73 MHz, |S 11 | = -39,33 dB and |S 11 | = -42,96 dB for coils with optical and galvanic detuning circuits respectively. While on detuned mode, at 127.73 MHz, the reflection coefficient of the coil with optical detuning system is close to 0 dB (-0.70 dB); which is very similar to that of the reference coil (-0.66 dB). This result demonstrates that the current provided by the photodiodes is sufficient to detune the coil. Second, the switching times using the reference coil are fast compared to a typical RF pulse durations (in the order of milliseconds). Precisely, switching to tuning mode takes about 750 ns, and switching back to detuning mode takes about 250 ns. For the ODC, switching time is longer. By adding a resistance, the switching time to tuning state can decrease from approximatively 100 µs to 10 µs (Figure 5). Using the resistance-value mentioned above in the methods section (R 0 =1.8 kΩ), the switching times for ODC are 13.6 µs and 1.7 µs for tuning and detuning respectively.

In vitro MR Experiments

The MR images acquired to evaluate detuning methods using the body coil as a transceiver are shown in figure 6. For each acquisition, twenty-one axial images were acquired over the entire length z of the loop. The displayed axial signal intensity images and their SNR mapping corresponding to the central image out of stack. With the exception of the coil without detuning circuit, the signal magnitude of the MR images are very comparable with uniform signal distribution. Signal intensity and SNR maps corresponding to the non-detuned coil show clearly the non-uniform signal distribution close to the coil caused by the transmit RF → B 1 field. The coil detuning is absolutely

mandatory.

The SNR values are ranging between 30 and 40 for the FSE and GRE sequences. For both sequences used, the SNR of MR images taken by the ODC with low-magnetic photodiodes are uniform and very comparable to the image of the phantom without any inserted coil. To study the SNR uniformity along the coil length z, the mean SNR was measured in two square form regions of interest (ROI) of about 100 mm 2 as shown in figure 7. The positions of these two ROIs are identical for each image slice and they were chosen relatively to the positions of hyper and hypo signals of the image acquired while using a non-detuned coil (Figure 7(e)). The SNR uniformity was characterized by comparing the mean SNR of the two ROIs within each image of the stack. For the non-detuned coil, the figures 7(a) and 7(b) and figures 7(c) and 7(d) show that the SNR difference between the two ROI exceeds 30. For all detuned coils, SNR profiles along z for both ROIs are similar for the FSE and GRE sequences. It should be noted that for the ODC, the choice of the photodiode is important. It should have a small-size, a fast response time, and a low-magnetism. Figure 8 exhibits clearly that using magnetic photodiodes induce MR images distortions especially near the location of photodiodes.

The next experiment was done using the body coil as the RF transmitter and the endoluminal coil as a receiver. Given that the intensity of the image taken by a non- detuned endoluminal coil is strongly altered by the cross-coil coupling, images were not acquired within these conditions. Images in figure 9 show comparable behavior between coils in terms of intensity and signal uniformity. According to figure 10, SNR iso-contours of the images taken with different coil prototypes are overlapping and this for both FSE and GRE sequences.

Finally, the variation of temperature for both the reference coil and ODC was measured during the in vitro experiments. For a galvanic detuning of the coil, the elevation of the temperature is around 2.77 • C near the PIN diode owing to the high detuning current emitted (150 mA). In the case of optical detuning, no change in temperature was discerned at the place of the PIN diode, this is due to the low photocurrent emitted (11 mA) by the photodiodes. And the negligible variation in the temperature (∆T = 0.24 • C) near the illuminated photodiodes proves that the laser diodes (30 mW, 650 µm) could be used without any potential risks to the imaged person. This result emphasizes that the use of optical detuning without RF induced heating effects could be considered as a safe active detuning mean.

Discussion and conclusions

The uniformity of signal intensity of images presented in figure 6 confirms that endoluminal ODC is as efficient as the galvanic-detuned coil (reference coil). The weak point of the ODC can be caused by the relatively low current injected into the PIN diode by the photodiodes. This current is actually around 11 mA which is relatively low compared to the 150 mA injected by the MR system to detune the reference coil. With this low intensity, no elevation of the temperature was recorded using the detuning circuit. But in order to fully ensure patient safety, the developed active detuning system should be associated with an endoluminal coil with a novel optical method of NMR signal transmission [START_REF] Aydé | Potentialities of an electro-optic crystal fed by nuclear magnetic resonant coil for remote and low-invasive magnetic field characterization[END_REF][START_REF] Aydé | Unbiased Electro-Optic waveguide as a sensitive nuclear magnetic resonance sensor[END_REF]. A number of optical systems for NMR signal transmission have already been built [START_REF] Fandrey | A novel active MR probe using a miniaturized optical link for a 1.5-T MRI scanner[END_REF][START_REF] Koste | Proc. ISMRM, 13th Scientific Meeting and exhibition[END_REF][START_REF] Memis | Miniaturized fiberoptic transmission system for MRI signals[END_REF][START_REF] Possanzini | dStream Architecture[END_REF][START_REF] Yuan | A direct modulated optical link for MRI RF receive coil interconnection[END_REF]. The associated detuning methods were either active and linked to MR console by a coaxial cable, or passive without any connection. In both cases, patient safety is very difficult to ensure for narrow endoluminal coil deeply inserted in the body. Other active optical detuning circuits were proposed for different MR applications [START_REF] Korn | Optically detunable, inductively coupled coil for self-gating in small animal magnetic resonance imaging[END_REF][START_REF] Weiss | In vivo safe catheter visualization and slice tracking using an optically detunable resonant marker[END_REF][START_REF] Wong | An optical system for wireless detuning of parallel resonant circuits[END_REF]. In the presented work, a PIN diode having very high impedance on blocking mode was used in parallel to tuning capacitor. The PIN diode was activated by two photodiodes being isolated from RF signals by two choke inductors. Using this detune method, Q factor is not modified compared to that chosen by [START_REF] Weiss | In vivo safe catheter visualization and slice tracking using an optically detunable resonant marker[END_REF] where a relatively low impedance photodiode non-isolated from the resonant circuit was used instead of the PIN diode. In the latter case, it was however not a real issue due to the pursued objectives of safe catheter tracking. Then an analysis of the behavior of the circuit in the reference [START_REF] Korn | Optically detunable, inductively coupled coil for self-gating in small animal magnetic resonance imaging[END_REF] shows that the switching time to tuning mode was not evaluated and probably not optimal. This would increase the switching time and limit temporal response of the receiver. With the use of a resistance R 0 , the switching times of our circuit (around ten of µs at the slowest) are still quite small relative to the regular pulse duration (in the range of ms) for most MR imaging sequences including fast and even ultra-fast imaging sequence such as echo-planar imaging. However, our circuit could not be fast enough for very specific sequence such as ultra-short echo time sequences (e.g. UTE). Moreover, in this work, a higher current was provided by the photodiodes to the PIN diode compared to optical detuning system of reference [START_REF] Korn | Optically detunable, inductively coupled coil for self-gating in small animal magnetic resonance imaging[END_REF] (11 mA > 100 µA) which is leading to more efficient detuning. Finally, unlike these previous works, our system configuration is suitable for all size of coil or loop. In fact, it can be easily adjusted in order to respond to all detuning applications (coils dimensions, tracking or imaging) by increasing the power of the laser or the radiant sensitive area dimension of the photodiode. It should be notice that in vivo studies of optical detuning is not adequate at this stage for several reasons. First, evaluating different detuning performances by comparing subtle changes with variable biological environments is not relevant. Second, keeping in mind that it is a safety issue, it would be unethical to perform this experiment in vivo. In conclusion, a novel compact active optical detuning circuit associated to a deported unit was developed and demonstrated with MRI endoluminal coil. The optical detuning circuit was based on the previous work of [START_REF] Korn | Optically detunable, inductively coupled coil for self-gating in small animal magnetic resonance imaging[END_REF]. The use of optical detuning was proposed to ensure total the patient safety. This strategy could be more widely developed and adapted in the future with multi-channel array coils. Optical-based detuning circuit associated with signal optical transmission via optical fibers could be the near future (Yuan et al 2016).

Figure 1 .

 1 Figure 1. Photographs and electric circuits of the different endoluminal coil prototypes: non-detuned coil, reference coil, and ODC.

Figure 2 .

 2 Figure 2. Switching time of the coil for tuning and detuning as a function of resistorvalue R 0 .

Figure 3 .

 3 Figure 3. Schematic representation of switching time measurements procedure of an endoluminal coil added to a) a reference circuit and b) an ODC.

Figure 4 .

 4 Figure 4. Schematic setups of coil detuning experiments performed on a 3.0 T MR system using an endoluminal coil with a) a galvanic-detuned circuit and b) an opticaldetuned circuit.

Figure 5 .

 5 Figure 5. Response time graph showing the evolution of the voltage across the PIN diode versus time for ODC a) without and b) with resistor R 0 .

Figure 6 .

 6 Figure 6. Axial images and SNR mapping of the phantom acquired using FSE and GRE sequence with the body coil used as a transceiver. The three endoluminal coils were located inside the phantom and they were continuously detuned.

Figure 7 .

 7 Figure 7. The measured signal-to-noise ratio of two regions of interest (ROI) for twenty-one axial images acquired by the body coil while using three different coils with (a-b) a fast spin echo sequence and (c-d) a gradient echo sequence. e) The choice of the positions of the two regions of interest (ROI-1 and ROI-2) is depending on the hyper and hypo signals presented in the MR image while using a non-detuned coil. The lines present a curve fitting to confirm the minimal variation of the SNR along the reference coil and the ODC.

Figure 8 .

 8 Figure 8. Coronal images of the phantom using a) low-magnetic photodiodes, and b) magnetic photodiodes.

Figure 9 .

 9 Figure 9. Axial MR images of the phantom acquired by the endoluminal coils.

Figure 10 .

 10 Figure 10. SNR iso-contours of the axial images acquired by the different coils using a) a FSE sequence and b) a GRE sequence.

Acknowledgment

This work was supported by the Rhône-Alpes region (ADR and CIBLE projects) and the DGA (French Military Programs Management and Procurement Agency -RAPID and SNIFER projects). This work was performed within the framework of LABEX PRIMES (ANR-11-LABX-0063) of Lyon University, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Isabelle Saniour

Gwenaël Gaborit

Lionel Duvillaret

Anne-Laure Perrier

Olivier Beuf

Optical-based probe for real time assessment of RF electrical field during MRI exam

During MRI exam, Specific Absorption Rate (SAR) is essential to be controlled and can be evaluated by measuring either indirectly for instance the rise in temperature or directly the radiofrequency induced electrical E-field. In the current study, we proposed an optical probe based on the Pockels effect for subcentimeter resolution measurements of the E-field without altering the surrounding media. Measurements were performed at 4.7 T and 3.0 T. Results show that the probe has an excellent linear response and allow a real time estimate of the three components of the E-field produced during MRI examination.

Introduction

Patient safety is controlled during an MRI exam by estimating the global specific absorption rate (SAR). Without dedicated devices such radiofrequency (RF) traps, the electrical E-field induces RF currents, resulting heating near metallic conductor. The SAR-parameter is determined either by the E-field or the temperature rise. E-field probes are fast and more sensitive than thermal sensors but hard to implement. Dipole-based probes were largely used to measure the E-field in MRI . However, their metallic structure may perturb the electromagnetic fields and the spatial resolution is limited by the probe's dimensions. Recently, optical-based methods were used to measure the E-field . But, the value of the E-field was not precisely determined, besides only one component of the E-field was estimated. In the following, the characteristics of the optical probe and real time measurements of the three components of the RF E-field are presented.

Methods

Birefringence characterizes crystals that change their reflective indices with an applied E-field. Based on this principle, a sub-cm optical probe using isotropic crystals (ZnTe, BSO…) was designed and built. A probe with isotropic crystal allows the measurement of both x-and y-orientations of the Efield so-called transverse probe. The z-component is measured by a longitudinal probe. A fiber optic was connected to the crystal to drive a laser diode (λ = 1.55 μm, relative intensity noise (RIN) < 150 dB/Hz) having an incident polarization that changes with the applied E-field. The modulation of the polarization was analyzed using a processing unit (eosens, Kapteos, France) and then converted into an electrical signal using a photodiode. A spectrum analyzer was used to display the envelope of the detected electro-optical (EO) signal on a fixed and narrow frequency bandwidth (30 kHz). First, the E-field measurements were performed on 4.7 T preclinical MRI (Bruker). A high-pass 60 mm-diameter linear transceiver birdcage with capacitors placed on the two end sides was used. The probe was fixed on a 3-axis translation bench (Fig 1). The measurements were performed in air in order to verify the sensing abilities of the probe. The probes were calibrated outside the MRI to obtain the proportionality factor between the EO signal and the E-field. A FLASH sequence with 1 ms Hermite shape RF excitation pulse was ran. In order to determine the relation between the emitted magnetic B -field and the detected E-field, the flip angle of the RF pulse was varied and the corresponding E-field was measured. Besides, the two transverse and longitudinal probes were used separately to measure the three components of the E-field along the zaxis. Finally, this measuring protocol was also applied on a 3.0 T clinical MRI (GEHC MR750). MR images of an orange with and without inserted probe were acquired.

Results

Figure 2 exhibits an excellent linearity between E-and B -fields for both probes measuring radial and longitudinal components. The B -field was derived from the shape (Hermite equation) and amplitude (flip angle). The maximum values of the E-field were measured at 1078 V/m and 488 V/m for transverse and longitudinal probes, respectively. The sensitivity was established at E =0.27V/m.Hz for transverse probe and E =0.087V/m.Hz for longitudinal probe. Figure 3 displays the three components of the E-field along z-axis. Two maximum peaks were found at Z=5cm and Z=11cm for all curves. Next, the exact shapes of three different excitation pulses having the same excitation time (1ms) and power (138W) were detected by the optical probe (Fig 4). The difference between the MR images of the orange with and without the inserted probe is negligible and the measured E-field was equal to 198 V/m (Fig 5).

Discussion

The E-field has a linear dependence with respect to the B -field. The highest value of the E-field is obtained when the flip angle is equal to 90˚ which corresponds to the highest value of the emitted B -field. The EO probe is highly sensitive to the variation of the E-field value regarding the RF pulse shape. The y-component of the E-field is the dominant component noting that the RF B excitation pulse is linear and emitted along the x-axis. The detected peaks along z-direction are spatially separated by about 60 mm which is the birdcage length. The high-concentrations of the E-field happen near the capacitor areas. Finally, based on MR images comparisons, the probe has not noticeable impact on the B -field.

Conclusion

The described optical probe device is a suitable to measure E-field within MRI tunnel without altering the B -field. An excellent linearity and a good sensitivity of this optical-based probe were demonstrated. Besides, the probes can be used at different B . The three components of the E-field were measured using a transverse probe (TP) that measures the x and y-components of the E-field and a longitudinal probe (LP) that measures the z-component of the E-field. The probe was placed at X=-2.5cm and Y=0cm and moved with a 1 cm step along the z-axis. For the three curves, two high-concentrations located at Z=5cm and Z=11cm were found and the minimum was detected at Z=8cm. The distance between the two peaks is 60 mm which corresponds to the locations of the capacitors at distal and proximal sides of the birdcage.

  Figures

  Fig 1. Schematic of the experimental setup for E-field measurements. The probe was fixed on a three-axis translation bench to move precisely the EO probe within the birdcage coil. The emitted and reflected lasers were transmitted by the same optical fiber. The signal processing unit contains optoelectronics components to analyze the modulation of the polarization of the laser. A photodiode was used to insure the conversion of the optical signal into an electrical signal. A spectrum analyzer and an oscilloscope were used to visualize the envelope of RF signal in the frequency and time domains.

Fig 2 .

 2 Fig 2. The detected E-field is plotted as a function of the emitted B -field. The B -field was derived from the shape (Hermite equation) and amplitude (flip angle). The factors of proportionality between the EO signal and the E-fields were equal to 417714.9 m and 76682.8 m for transverse and longitudinal probes, respectively.

Fig 3 .

 3 Fig 3.The three components of the E-field were measured using a transverse probe (TP) that measures the x and y-components of the E-field and a longitudinal probe (LP) that measures the z-component of the E-field. The probe was placed at X=-2.5cm and Y=0cm and moved with a 1 cm step along the z-axis. For the three curves, two high-concentrations located at Z=5cm and Z=11cm were found and the minimum was detected at Z=8cm. The distance between the two peaks is 60 mm which corresponds to the locations of the capacitors at distal and proximal sides of the birdcage.

Fig 4 .

 4 Fig 4. The detected shape of three different RF pulses (Bloc pulse, Hermite pulse and Sinc3 pulse) with the optical probe. The measurements were performed at a fixed point. For the three pulses, the excitation time was 1 ms and the excitation power was 138 W. The E-field peaks were 139.91 V/m, 844.04 V/m and 753.21 V/m for the bloc pulse, Hermite pulse and Sinc3, respectively.

Fig 5

 5 Fig 5. a) MR images of an orange with and without the insertion of the optical probe and the difference between the two images. b) The detected EO which is the envelope of the detected RF signal. 127 MHz corresponds to the proton's resonant frequency at 3.0 T. The orange was located in the center of the body coil; the electrical field peak was 198 V/m. The exact shape of the emitted excitation pulse was detected by the optical probe. The maximal E-field peak was detected near the location of the gradients and was equal to 15680 V/m.

Acknowledgements

This work was supported by the Rhône-Alpes region and performed within the framework of LABEX-PRIMES (ANR-11-LABX-0063), and the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency.


Isabelle Saniour

Gwenaël Gaborit

Lionel Duvillaret

Anne-Laure Perrier

Olivier Beuf

Experimental and simulated distribution of the RF electrical field inside a birdcage coil

High-field MR systems bring additional safety issues regarding the patient. One of main concern is the heating effect caused by the high-frequency and high-power signals that can be picked up by the conductors. It is thus required to identify these regions and to quantify the electrical E-field knowing that it is proportional to the square of the temperature elevation. For a direct measurement, an optical probe was used to perform a mapping of the E-field distribution inside a birdcage of a preclinical MRI. Experiments and simulations were found similar with E-field concentrations mainly located close to the capacitors.

Introduction

The spatial resolution of MR images could be improved by developing sophisticated radiofrequency (RF) transceiver coils or by increasing the static magnetic field B -strength . However, the fast and successful progresses of MR scanners could be the reason for additional safety issues regarding the patient . The specific absorption rate (SAR) is proportional to the square of B thus local heating is caused by the transmission of highfrequency and high-power electrical E-field along conductors or passive components e.g. capacitors. This requires a fast and an accurate assessment of the E-field concentrations in order to control the SAR-parameter during MRI exam. Detecting of the distribution of the E-field inside MRI or near metallic implants was already evaluated by dipole-based probes inserted in a phantom and numerical simulations . But, the metallic structure of the dipole probe may alter the electromagnetic fields and thus limits the accuracy of the measurements. In this study, we propose an optical-based probe for E-field measurements and we present a comparison between the simulated E-field distribution and the experimental map of the E-field using an optical-based probe inside a MRI birdcage.

Methods

A finite element method simulation was performed using COMSOL Multiphysics. The birdcage consists of two circular conductive loop (end rings) where the capacitors are placed and nineteen conductive elements referred to as rungs (Fig 1). The capacitors of the coil are defined using lumped elements. The birdcage is linear thus only one lumped port boundary is used for RF excitation (P = 138W). The birdcage was placed inside a cylindrical RF shield. The experimental measurement of the E-field was performed in a 4.7-T preclinical MRI (Bruker) using an optical-based probe. The E-field probe operates based on Pockels effect; it consists of isotropic crystal that changes its optical proprieties according to the variation of the E-field. A laser diode was simultaneously applied to the crystal. The polarization of the laser changes with the applied E-field. Optoelectronic devices were used to analyze the modulation of the polarization and insure the optical-to-electrical conversion of the signal to be then visualized by a spectrum analyzer. The probe measures the radial component of the E-field. It was calibrated outside the MRI to determine the proportionality factor between the EO signal and the E-field. The proximal side of the birdcage corresponds to the plane Z=0. On this plane, we have chosen nine measurement points as shown in Fig 2 . For these nine points, we have measured the E-field every 1cm by moving the probe along z-axis. The measurements points have been used to plot 2D maps of the E-field using Matlab language. For these experiments, a FLASH sequence with hermite shape RF pulse was used.

Results

The simulated results show that the E-field is not uniform within the birdcage. Figure 3 exhibits clearly the positions of the E-field concentrations regarding the geometry of the birdcage. The high-concentrations areas (around 1900 V/m) are mainly located near the lumped port and the capacitors. The concentrations-value gradually decreases along the copper rungs to have the lowest value (100 V/m) in the middle of the birdcage rungs/legs. Experimental maps are shown in the Fig 4 . The E-field high-concentrations are located at Z-values equal to 5 cm and 11 cm for all the slices. The E-field was measured with a factor of proportionality equal to 417714.9 m . The maximum value was around 1900 V/m and decreases progressively by approaching to the center.

Discussion

The E-field measurements and simulations were performed in air in order to determine the distribution of the E-field inside a birdcage and specifically near its walls without any limitations due to the dimensions of a phantom . The experimental maps of the E-field are very close to the simulation in terms of distribution and value. The high concentrations of the E-field are in proximity of the birdcage wall and the weakest concentrations are in the center. The simulation shows that the E-field high-concentrations are located near lumped port and capacitors. The distance between the two rings is 56 mm, therefore the high-concentrations in the experimental maps corresponds exactly to the locations of the capacitors.

Conclusion

We proved that the optical-based probe has the ability to measure in air the E-field inside a birdcage within picoseconds. This probe would also be used inside a phantom or a biological tissue to control the SAR-parameter and could definitely replace thermal sensors commonly used as an alternative method for SAR assessment but cause additional measurement errors due to the conviction and the diffusion of the temperature in the medium nearby. Experimental two-dimensional maps of the E-field inside the birdcage using an optical probe. For the four maps, the distance between the high-concentrations along the z-axis is around 60 mm which corresponds exactly to the locations of the capacitors. Referring to the simulation, the high-concentration at the point (X=2.5,Z=5) of XZ slide is due to the coaxial cable.

  Figures

  Fig 1. a) A photograph of the birdcage used. The birdcage was put inside the 4.7-T MRI as shown in the photograph. b) The design of the birdcage modelled by COMSOL Multiphysics. The RF excitation is linear and transmitted along the x-axis. A lumped port was used to simulate the coaxial cable that transmits the excitation pulse.

Fig 2 .

 2 Fig 2. The black dots indicate the locations of the nine measurement points in xy-plane. Each point is referenced by a (X, Y) coordinate-values (in cm). The E-field was measured every 1cm by moving the probe along the z-axis. The points are used to plot four slices: (2.5,0), (0,0) and (-2.5,0) points are used to plot XZ slice, (0,2.5), (0,0) and (0,-2.5) points are used to plot YZ slice, (1.58,1.58), (0,0) and (-1.58,-1.58) points are used to plot diagonal(1) slice and (1.58,-1.58), (0,0) and (-1.58,1.58) points are used to plot diagonal(2) slice. The spatial resolution of the probe is 5mm.

Fig 3 .

 3 Fig 3. Simulated three-dimensional maps of the E-field distribution within the birdcage (E-field mapping on the surface of a 50 mm-diameter cylinder of air inside the birdcage). a) The E-field concentrates principally near the lumped port modelling the coaxial cable. b) The simulation demonstrates that the E-field concentrates as well close to the capacitors put at the proximal and distal sides of the birdcage. The asterisk and the black dots are used to mark the positions of the lumped port and the capacitors, respectively.

Fig 4 .

 4 Fig 4.Experimental two-dimensional maps of the E-field inside the birdcage using an optical probe. For the four maps, the distance between the high-concentrations along the z-axis is around 60 mm which corresponds exactly to the locations of the capacitors. Referring to the simulation, the high-concentration at the point (X=2.5,Z=5) of XZ slide is due to the coaxial cable.

Acknowledgements

This work was supported by the Rhône-Alpes region and performed within the framework of LABEX-PRIMES (ANR-11-LABX-0063) of Lyon University, and the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency.