Tamas Balogh
email: tamas.balogh@mm.bme.hu

Tamas Insperger
email: insperger@mm.bme.hu

Islam Boussaada
email: islam.boussaada@l2s.centralesupelec.fr

Silviu-Iulian Niculescu
email: silviu.niculescu@l2s.centralesupelec.fr

Towards an MID-based Delayed Design for Arbitrary-order Dynamical Systems with a Mechanical Application

Keywords: dynamical system, delay, spectral properties, inverted pendulum

In this study we consider the stabilization of n th -order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input/output channels. A sufficient condition is given for the dominancy of a real root with multiplicity n + 1 using an integral representation of the corresponding characteristic function. Furthermore, this sufficient condition is analyzed in the case when the characteristic function of the open-loop system is real-rooted, and delay intervals are derived for the set of parameters satisfying stabilizability and dominancy conditions. The efficiency of the proposed controller design is shown in the case of a multi-link inverted pendulum.

INTRODUCTION

Stabilization of unstable equilibria and orbits in the presence of communication delay is an important and challenging task in engineering applications (see, for instance, [START_REF] Michiels | Continuous pole placement for delay equations[END_REF]; [START_REF] Olgac | An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems[END_REF]; Gu et al. (2003); [START_REF] Landry | Dynamics of an inverted pendulum with delayed feedback control[END_REF]). In this context, as pointed out by [START_REF] Ma | Delay margin of loworder systems achievable by PID controllers[END_REF], computing the control parameters allowing stable operation for large (feedback) delays is not a trivial task. Among the existing approaches, the socalled parameter-based approach is largely used in the case of low-order and/or low-complexity controllers (see, e.g., Bhattacharyya et al. (1995); [START_REF] Ackermann | Robust control. The parameter space approach[END_REF]; [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach[END_REF] and the references therein). In particular, the stability diagrams can be used to visualize stability properties of the closed-loop system in the space of control parameters Bellman and Cooke (1963); [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF]; [START_REF] Hassard | Counting roots of the characteristic equation for linear delay-differential systems[END_REF]; [START_REF] Breda | On characteristic roots and stability charts of delay differential equations[END_REF].

It is commonly accepted that the delay in the feedback loop is generally seen as a source of unstable behavior. Furthermore, as the feedback delay gets larger, the stable region in the stability diagrams (charts) gets smaller. In this frame, a challenging task is to find the so-called critical delay or delay margin1 . To the best of the au-thors' knowledge, the idea of exploiting the delay effects in controllers' design was first introduced in [START_REF] Suh | Proportional minus delay controller[END_REF] where it is shown that the conventional proportional controller equipped with an appropriate time-delay performs an averaged derivative action and thus it can replace the proportional-derivative controller. Furthermore, it was pointed out by [START_REF] Niculescu | Delay Effects on Output Feedback Control of Dynamical Systems[END_REF] that, under appropriate assumptions, the presence of some delay in the control law may induce stability in closed-loop. Finally, in the context of mechanical engineering problems, the effect of time-delay was emphasized in [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF] where some concrete applications are studied, such as the machine tool vibrations and robotic systems.

It is worth noting that, in the case of stable linear timedelay system, the rightmost root (or spectral abscissa) of the characteristic function is actually the exponential decay rate of its time-domain solution (see, for instance, [START_REF] Mori | On an estimate of the decay rate for stable linear delay systems[END_REF] for an estimate of the decay rate). Furthermore, to the best of the authors' knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in [START_REF] Hayes | Roots of the transcendental equation associated with a certain difference-differential equation[END_REF]. This Multiplicity-Induced-Dominancy (MID) property is further explored and analytically derived for scalar delay equations in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF]. Next, for second-order systems controlled by a delayed proportional is proposed in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of timedelay systems. application to the control of a mechanical system[END_REF][START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control[END_REF] where its applicability in damping active vibrations for a piezo-actuated beam is proved. Finally, an extension to the delayed proportional-derivative controller case is studied in [START_REF] Boussaada | Toward a decay rate assignment based design for timedelay systems with multiple spectral values[END_REF]; Boussaada et al. (2018a) where the dominancy property is parametrically characterized and proven by using the argument principle.

In this paper, we analyze a general n th -order linear timeinvariant dynamical system with a single delay and show that, under some appropriate conditions, the MID property can be used to assess the delay margin (see, e.g. Boussaada et al. (2015); [START_REF] Boussaada | Toward a decay rate assignment based design for timedelay systems with multiple spectral values[END_REF]; [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of timedelay systems. application to the control of a mechanical system[END_REF]). The novelty of the paper lies on the way we are exploiting the root location of the open-loop characteristic polynomial in order to have the multiplicity-induced-dominancy of the overall system, as done in [START_REF] Boussaada | Multiplicity-induceddominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF] in the particular case of second-order systems.

The remaining paper is organized as follows. The problem statement is presented in Section 2. Section 3 contains some motivating example. Section 4 provides the main ingredients of the dominancy proof, which consists in writing the characteristic function with multiple real roots as an integral operator. The main results are presented in Section 5. An illustrative example is presented in Section 6, where the 5-link inverted pendulum is studied.

PROBLEM STATEMENT

Consider the quasipolynomial D(s) = P (s) + e -sτ Q(s) ,

(1) where the polynomials P (s) and Q(s) are real and have degrees n and n -1, respectively:

P (s) = a n s n + a n-1 s n-1 + ... + a 1 s + a 0 , Q(s) = b n-1 s n-1 + b n-2 s n-2 + ... + b 1 s + b 0 . (2)
Assume that the coefficients a i of P are fixed and known, and that a n > 0.

The problem we are addressing can be resumed as follows: computing the values of the delay parameter τ such that the system (1) is stabilizable 2 if the control parameters b i are tuned such that the characteristic function D(s) has a real root s 0 with multiplicity n + 1?

MOTIVATING EXAMPLE: THE INVERTED PENDULUM

Balancing an inverted pendulum in the presence of feedback delay is a frequently cited example in dynamics and control theory (Atay, 1999;[START_REF] Sieber | Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity[END_REF][START_REF] Li | A frequency-sweeping framework for stability analysis of time-delay systems[END_REF]. Different control methods are often implemented in simple inverted pendulum systems [START_REF] Habib | Nonlinear model-based parameter estimation and stability analysis of an aero-pendulum subject to digital delayed control[END_REF][START_REF] Xu | Balancing a wheeled inverted pendulum with a single accelerometer in the presence of time delay[END_REF][START_REF] Qin | Control experiments on time-delayed dynamical systems[END_REF][START_REF] Cieżkowski | Method for determination of interaction between a two-wheeled self-balancing vehicle and its rider[END_REF]. The inverted pendulum is also a basic concept in human balancing models [START_REF] Maurer | A new interpretation of spontaneous sway measures based on a simple model of human postural control[END_REF][START_REF] Milton | Control at stability's edge minimizes energetic costs: expert stick balancing[END_REF][START_REF] Milton | Acting together, destabilizing influences can stabilize human balance[END_REF][START_REF] Morasso | Quiet standing: The single inverted pendulum model is not so bad after all[END_REF]. The equation of motion of an inverted pendulum controlled by a proportional-derivative (PD) controller reads as:

φ(t) + a 0 ϕ(t) = -b 0 ϕ(t -τ ) -b 1 φ(t -τ ) , (3)
2 i.e. there exist some real parameters b i (i = 0, 1, . . . , n -1) depending explicitly on (a 0 , a 1 , . . . , an, τ ) such that the characteristic roots of (1) have negative real part.

with a feedback delay τ > 0 and a system parameter a 0 < 0. The characteristic function corresponding to (3) is

D(s) = s 2 + a 0 + e -sτ (b 0 + b 1 s) . (4
) The open-loop characteristic function P (s) = s 2 + a 0 has real roots ± √ -a 0 since a 0 < 0. This property proves to be useful in Section 5.

The critical delay (delay margin) of the system (3) is wellknown from the literature [START_REF] Schurer | Zur Theorie des Balancierens[END_REF][START_REF] Stépán | Delay effects in the human sensory system during balancing[END_REF]:

τ crit = - 2 a 0 , (5)
that is, the trivial solution of system (3) can be asymptotically stable if and only if τ < τ crit . In the sequel, we will show that the critical delay ( 5) can be obtained by studying the multiple roots of the characteristic function.

Assume that D(s) has a real root s 0 with algebraic multiplicity at least degP (s) + 1 = 3. Then D(s 0 ) = 0, D (s 0 ) = 0 and D (s 0 ) = 0:

s 2 0 + a 0 + e -s0τ (b 0 + b 1 s 0 ) = 0 , 2s 0 + e -s0τ (-τ (b 0 + b 1 s 0 ) + b 1 ) = 0 , 2 + e -s0τ (τ 2 (b 0 + b 1 s 0 ) -2τ b 1 ) = 0 .      (6)
From ( 6), one gets:

b 0 = e s0τ τ s 3 0 + s 2 0 + a 0 τ s 0 -a 0 , b 1 = -e s0τ τ s 2 0 + 2s 0 + a 0 τ , s 0 = -2 ± √ 2 -a 0 τ 2 τ =: s 0,± .                  (7)
It can be shown that the triple root s 0,+ is negative and dominant for every 0 < τ < τ crit , and therefore the system (3) is asymptotically stable. In particular, at the upper bound τ = τ crit the triple root is s 0,+ = 0 and it is the dominant (rightmost) root of (4) with the control parameters b 0 = -a 0 and b 1 = -a 0 τ crit . The dominancy of s 0,+ may be shown by using the argument principle (see, for instance, [START_REF] Boussaada | Toward a decay rate assignment based design for timedelay systems with multiple spectral values[END_REF]Boussaada et al., 2018a). In the next section, to show the dominancy, we use a method first introduced in Boussaada et al. ( 2016) and based on an integral representation of the characteristic function (see also [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of timedelay systems. application to the control of a mechanical system[END_REF]).

INTEGRAL REPRESENTATION AND A SUFFICIENT CONDITION FOR DOMINANCY

We have the following result: Proposition 1. If the quasipolynomial (1) has a real root s 0 with multiplicity at least n, then it can be written as

D(s) = (s -s 0 ) n a n + 1 0 e -(s-s0)τ t τ R n-1 (s 0 ; τ t) (n -1)! dt , (8)
where the family of polynomials R k (s; τ ) is defined as

R k (s; τ ) = k i=0 k i P (i) (s)τ k-i , k ∈ Z + 0 . (9)
Proof. The quasipolynomial D(s) has a root s 0 with algebraic multiplicity at least n if and only if D (k) (s 0 ) = 0, k = 0, 1, ..., n -1:

P (s 0 ) + e -s0τ Q(s 0 ) = 0 , P (s 0 ) + e -s0τ (-τ )Q(s 0 ) + Q (s 0 ) = 0 , . . . P (k) (s 0 ) + e -s0τ k i=0 k i Q (i) (s 0 )(-τ ) k-i = 0 ,
. . . 10) gives a linear system of equations for the control parameters. Solving (10) for b i enables the integral representation of the form (8).

P (n-1) (s 0 ) + e -s0τ n-1 i=0 n -1 i Q (i) (s 0 )(-τ ) n-1-i = 0 .                                            (10) Equation (
If D(s) has a real root s 0 with multiplicity at least n + 1 then (8) holds and, in addition, D (n) (s 0 ) = 0:

D (n) (s 0 ) = n! a n + 1 0 τ R n-1 (s 0 ; τ t) (n -1)! dt = R n (s 0 ; τ ) = 0 . ( 11
)
Proposition 2. Let s 0 be a real root of R n (s 0 ; τ ) = 0. Assume that the parameters b i satisfy the conditions (10).

If R n-1 (s 0 ; τ t) ≤ 0, ∀t, 0 < t ≤ 1 then the root s 0 with multiplicity n+1 is the dominant root of the characteristic function (8).

Proof. To prove that there exists no root s 1 = γ 1 + iω 1 of (8) such that γ 1 > s 0 , substitute s 1 into (8). Since a n > 0 one can obtain that

a n = 1 0 e -(s1-s0)τ t τ (n -1)! R n-1 (s 0 ; τ t)dt ≤ 1 0 e -(γ1-s0)τ t τ (n -1)! R n-1 (s 0 ; τ t) dt . (12)
Using the condition R n-1 (s 0 ; τ t) ≤ 0, ∀t, 0 < t ≤ 1 (12) can be written as

a n ≤ - 1 0 e -(γ1-s0)τ t τ (n -1)! R n-1 (s 0 ; τ t)dt =: f (γ 1 ) .
(13) For γ 1 = s 0 the function f takes the value

f (s 0 ) = - 1 0 τ (n -1)! R n-1 (s 0 ; τ t)dt = - 1 0 1 n! dR n (s 0 ; τ t) dt dt = - 1 n!     R n (s 0 ; τ ) =0 -R n (s 0 ; 0) =ann!     = a n . (14)
For γ 1 > s 0 the value of the integral in ( 13) is f (γ 1 ) < a n since 0 < e -(γ1-s0)τ t < 1 for γ 1 > s 0 , τ > 0, 0 < t ≤ 1. Therefore, from (13), we obtain that a n < a n which proves the inconsistency of the hypothesis that the characteristic function ( 8) has a root s 1 = γ 1 + iω 1 with γ 1 > s 0 .

OPEN-LOOP SYSTEMS WITH ONLY REAL ROOTS

In this section, we assume that the polynomial P (s) corresponding to the open-loop system has only real roots.

In this case, P (s) has the form P (s) = a n n i=1 (s -s i ), s i ∈ R, s n ≤ s n-1 ≤ ... ≤ s 1 . To apply the sufficient condition in Proposition 2, first, we need to characterize the algebraic properties of the polynomials R k (s; τ ). These properties are outlined and discussed in the sequel.

Interlacing property of polynomials R k (s; τ )

The two-variable polynomials R k (s; τ ), k ∈ Z + have the following properties:

R k (s; τ ) = τ R k-1 (s; τ ) + ∂R k-1 (s; τ ) ∂s , (15)
∂R k (s; τ ) ∂τ = kR k-1 (s; τ ) . ( 16
)
The property (15) allows saying that, for a fixed τ , the polynomials R k (s; τ ) and R k-1 (s; τ ) interlace and R k (s; τ ) has only real roots for s since R 0 (s; τ ) = P (s) has only real roots [START_REF] Fadeev | Problems in Higher Algebra[END_REF]. Polynomials R n (s; τ ) and R n-1 (s; τ ) have n distinct real roots for s if τ = 0. Let s 0,k , k = 1, 2, ..., n denote the roots of R n (s; τ ), τ = 0 with s 0,n < s 0,n-1 < ... < s 0,1 .

It can also be shown that, for a fixed s, R n (s; τ ) has only real roots for τ [START_REF] Pólya | Problems and Theorems in Analysis II: Theory of Functions[END_REF]. Moreover, R k (s; τ ), k = 1, 2, ..., n -1 has only real roots for τ , and R k (s; τ ) and R k-1 (s; τ ) interlace which are direct consequences of the property ( 16) above and Rolle's theorem.

It should be mentioned that, in general, the solutions of ( 15) and ( 16) can be investigated by using computer algebra techniques (see, e.g., [START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF]).

Monotonicity

In the (τ, s)-plane, the algebraic curve R n (s; τ ) = 0 has distinct branches, and every branch is strictly increasing since the derivative of the implicit function R n (s; τ ) = 0 in a point (s, τ ) writes as (using ( 15) and ( 16)):

ds dτ = - ∂Rn(s;τ ) ∂τ ∂Rn(s;τ ) ∂s = - nR n-1 (s; τ ) R n+1 (s; τ ) > 0 , ( 17
)
where the fraction Rn-1(s;τ ) Rn+1(s;τ ) is negative since for a fixed τ = 0 at a root s of the polynomial R n (s; τ ) the function values R n-1 (s; τ ) and R n+1 (s; τ ) are nonzero and have different signs because of the interlacing property. A similar analysis can be done for the algebraic curve R n-1 (s; τ ) = 0.

Asymptotic properties

If τ → ∞ (or τ → -∞) then the roots of R k (s; τ ) for s approach the roots of P (s) (i.e. s n ≤ s n-1 ≤ ... ≤ s 1 ). Similarly, if s → ∞ or s → -∞ then the roots of R k (s; τ ) for τ approach the roots of τ k = 0 (i.e. 0 with multiplicity k). 5.4 Roots of R k (s; 0)

If τ = 0 then R k (s; τ ) = R k (s; 0) = P (k) (s). Therefore, if k = n, then R n (s; 0) = n!a n has no roots for s. If k = n-1, then R n-1 (s; 0) = n! 1! a n s + (n-1)! 0! a n-1
has one root for s:

s a = - 1 n a n-1 a n = 1 n n i=1 s i , (18)
which is the average of the roots of P (s).

Sufficient conditions for dominancy and stabilizability if P (s) is real-rooted

Fig. 1 shows the branches of the algebraic curves R n (s; τ ) = 0 and R n-1 (s; τ ) = 0 corresponding to the interlacing and asymptotic properties on the (τ, s) plane (top), as well as the right half-plane in the neighborhood of the origin. The horizontal asymptotes corresponding to the roots of P (s) are indicated with dashed lines.

Let τ 0 denote the smallest positive root of R n (0; τ ) = 0 for τ . For τ > 0 the first branch of the algebraic curve R n (s; τ ) = 0 corresponds to the greatest s values, and it takes values in the interval ] -∞, s 1 [. Therefore, if s 1 > 0, then τ 0 corresponds to the first branch of R n (s; τ ) = 0. If s 1 ≤ 0 ,then R n (0; τ ) = 0 has no positive roots. In this case, set τ 0 = ∞.

Moreover, let τ a denote the smallest positive root of R n (s a ; τ ) = 0 for τ . If P (s) = a n (s -s 1 ) n then τ a corresponds to the first branch of R n (s; τ ) = 0 since s n < s a < s 1 . If P (s) = a n (s -s 1 ) n , then s a = s 1 and R n (s a ; τ ) = 0 has no roots. In this case, set τ a = ∞.

The curve R n (s; τ ) = 0 gives a connection between the delay τ and the possible values of the real root s 0 with multiplicity n+1, while the curve R n-1 (s; τ ) = 0 is needed to analyze the sufficient condition given in Proposition 2. It is clear that the condition R n-1 (s 0,k ; τ t) ≤ 0, ∀t, 0 < t ≤ 1 can be satisfied if and only if k = 1 and 0 < τ ≤ τ a (i.e. for the greatest s 0 and in a certain delay interval).

These observations are summarized as follows: Proposition 3. Consider the case s a ≥ 0. Then 1. τ 0 ≤ τ a 2. s 0,1 is the dominant root of system (8) if 0 < τ ≤ τ a , and system (8) is stabilizable if 0 < τ < τ 0 .

Proof. If s a > 0 then there is at least one positive root s 1 of P (s), therefore there is a finite τ 0 corresponding to the first branch of R n (s; τ ) = 0. Then the inequality τ 0 ≤ τ a follows from the monotonicity of the curve R n (s; τ ) = 0. Furthermore, if s a = 0 then τ 0 = τ a . Next, the dominancy and stabilizability properties follow from the sufficient condition in Proposition 2 and the fact that s 0,1 ≥ 0 if τ 0 ≤ τ ≤ τ a . Proposition 4. Consider the case s a < 0. Then 1. τ 0 ≥ τ a 2. s 0,1 is the dominant root of system (8) and system (8) is stabilizable if 0 < τ ≤ τ a .

Proof. The proof follows the same lines as the proof of Proposition 3.

MULTI-DEGREE-OF-FREEDOM MECHANICAL EXAMPLE: 5-LINK INVERTED PENDULUM

Consider a 5-link inverted pendulum with rods of equal mass m and length l. The control torque is applied at the first (lowest) rod:

M = - 5 i=1 p i ϕ i (t -τ ) - 5 i=1 d i ϕ i (t -τ ) . (19)

Derivation of the equation of motion

The equation of motion can be derived by using the Euler-Lagrange equations. The generalized coordinates are chosen to be the pendulum angles ϕ i (i.e. angular displacement of the rods from the vertically upward position). The equation of motion linearized around the unstable equilibrium has the form:

Mq + Kq = Q , ( 20
) where the mass matrix M and the stiffness matrix K can be written as

M = 1 6 l 2 m       
26 21 15 9 3 21 20 15 9 3 15 15 14 9 3 9 9 9 8 3 3 3 3 3 2

       , K = - 1 2 glm      
9 0 0 0 0 0 7 0 0 0 0 0 5 0 0 0 0 0 3 0 0 0 0 0 1

      . The generalized force Q reads Q = -      
p 1 p 2 p 3 p 4 p 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

      q(t-τ )-       d 1 d 2 d 3 d 4 d 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       q(t-τ ).
Therefore, the characteristic function has the form ( 1) where the open-loop characteristic function P (s) reads

P (s) = det s 2 M + K . (21)

Stabilizable delay interval

Since the mass matrix M is positive-definite and the stiffness matrix K is negative-definite, then P (s) has only real roots. The average of the roots is s a = 0 since the roots occur in real pairs ±s i . Therefore, we can apply the results of Proposition 3. More precisely, the system (20) is stabilizable if 0 < τ < τ 0 , where τ 0 can be calculated if the system parameters are known. For example, if the parameter a 0 := 3g l = 1 then τ 0 = 0.3816. Some numerical simulations are shown in Fig. 2 for τ = 0.22. Bhattacharyya, S., Chapellat, H., and Keel, L. (1995).

Robust control: the parametric approach. Prentice-Hall PTR, Englewood Cliffs, NJ, USA. Boussaada, I., Morȃrescu, I.C., and Niculescu, S.I. (2015).

Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains. Systems & Control Letters, 82, 1-9. [START_REF] Boussaada | Toward a decay rate assignment based design for timedelay systems with multiple spectral values[END_REF]. On the dominancy of multiple spectral values for time-delay systems with applications. IFAC-PapersOnLine, 51(14), 55-60.

Fig. 1 .

 1 Fig. 1. Roots location of the polynomials R n (s; τ ) and R n-1 (s; τ ) on the (τ, s) plane for the open-loop characteristic function P (s) = (s -2)(s -1)(s + 2). Top: Entire (τ, s) plane. Bottom: Right half-plane near the origin.

Fig. 2 .

 2 Fig. 2. Time-domain simulation for system (20) with 3g/l = 1 and τ = 0.22 < τ 0 .

i.e. the maximum delay, for which the system can still be stabilized by some control law, but for larger delay, the corresponding closedloop system is unstable.

ACKNOWLEDGEMENTS

The research reported in this paper was supported by the Hungarian-French Bilateral Scientific and Technological Cooperation Fund under Grant no. 2017-2.2.5-TT-FR-2017-00007, by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial Intelligence research area of Budapest University of Technology and Economics (BME FIKP-MI), by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). Finally, IB & SIN are partially financially supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissement d'Avenir" program, through the "iCODE Institute project" funded by the IDEX Paris-Saclay.


Keywords: Time-delay systems, frequency-domain approaches, controller design, argument principle

This paper investigates the stability and stabilization of some generic linear secondorder time-invariant retarded system with single delay. It provides an appropriate stability criterion based on the manifold defined by the coexistence of the maximal number of negative spectral values. Next, such ideas are exploited in the context of delayed output feedback by an appropriate "partial" pole placement guaranteeing simultaneously the stability in closed-loop and an appropriate exponential decay rate. To perform such an analysis, the argument principle is explicitly used.

INTRODUCTION

Time-delay is encountered in many fields and applications, such as in Physics, Biology and Economy. A strong and ever-growing interest on time-delay systems has been reported over the years, see, for instance, [START_REF] Niculescu | Delay Effects on Stability, A Robust Control Approach[END_REF], [START_REF] Bellman | Differential-difference equations[END_REF], [START_REF] Hale | Introduction to functional differential equations[END_REF], [START_REF] Stépán | On the stability of linear differential equations with delay[END_REF] and references therein.

In the stability analysis of time-delay systems, many approaches have been developed in both time-and frequencydomain. In particular, a wide range of frequency -omain techniques have been derived, adressing the distribution of the roots of the corresponding characteristic equation and related properties, see for instance [START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF], [START_REF] Michiels | Stability and stabilization of time-delay systems[END_REF], [START_REF] Walton | Direct method for tds stability analysis[END_REF], [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF], [START_REF] Boussaada | Further remarks on spectral analysis of delay differential-algebraic systems[END_REF], [START_REF] Cooke | On zeroes of some transcendental equations[END_REF].

This work is motivated by recent studies Boussaada and Niculescu [2016b], Boussaada et al. [2018], Boussaada and Niculescu [2016b,a] where a property called Multiplicity-Induced-Dominancy (MID) is emphasized, which consists in characterizing the exponential decay rate of the trivial solution. Recently, in Boussaada and Niculescu [2016b], a result by Polya and Szegô is revisited and exploited allowing to explicitly derive a bound for the number of real roots of the quasipolynomial function. Such a bound is nothing else that degree of the quasipolynomial Boussaada et al. [2018], that is the sum of the involved polynomials plus the number of the involved delays. In Boussaada and Niculescu [2018] an analytical proof for the dominancy of the spectral value with maximal multiplicity for second-order systems is explicitly provided. Quite recently, [START_REF] Amarne | On qualitative properties of low-degree quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment[END_REF] showed that the multiplicity of a real root itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control synthesis. As a matter of fact, it is proven that, under appropriate conditions, the coexistence of the maximal number of negative distinct roots guarantees their dominancy. In this case, an adequate factorization is derived in the scalar and second-order delay differential equations with single delay allowing to write the quasipolynomial in an some appropriate integral operator form. Furthermore, if these (real) roots are negative, this guarantees the asymptotic stability of the trivial solution.

In some cases, such a factorization is hard to be established, especially for the quasipolynomials with higher degrees. The stability criteria proposed and developed in [START_REF] Stépán | On the stability of linear differential equations with delay[END_REF] and generalized later in [START_REF] Hassard | Counting roots of the characteristic equation for linear delay-differential systems[END_REF] allow to compute analytically the number of characteristic roots in the right-half complex plane (i.e., unstable roots) for time-delay system of retarded type avoiding the direct calculating the corresponding improper integral. Our study is mainly inspired by [START_REF] Boussaada | Multiplicity-Induced-Dominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF], where the property of MID is investigated for the generic secondorder retarded differential equation that we address in this paper. In other words, the present paper focuses on the application of the Stépàn-Hassard approach to show the dominancy of spectral values which are not necessarily multiple.

The remaining of the paper is organized as follows. In Section 2, we present the problem formulation and we recall some known results on the spectrum distribution of retarded delay systems useful in the forthcoming sections. Next, conditions on the system parameters guaranteeing the co-existence of the maximal number of real roots is carried out in Section 3. In the particular case of equidistributed real roots the argument principle is applied to prove the dominancy of such real spectral values. Some concluding remarks end the paper.

PRELIMINARIES

Problem Statement

Consider the following generic LTI system including one delay:

ẋ(t) = Ax(t) + Bx(t -τ ),
(1) where x(t) ∈ R n and A and B are appropriate n × n real matrices. The corresponding spectral values are characterized by evaluating for s the following characteristic equation:

∆(s) = det(sI -A -Be -τ s ) = 0.
(2) The asymptotic stability of the trivial solution is guaranteed if all roots of (2) lie in the left-half plane. Notice also that real spectral values correspond to non oscillatory solutions of (1).

It is also well known that second-order linear systems capture the dynamic behavior of many natural phenomena and have found numerous applications in a variety of fields, such as vibration, structural analysis and human balancing. In this paper we focus on planar systems (1) (i.e. n = 2) and restrict our analysis to systems where the corresponding quasipolynomials writes under the form: ∆(s, τ ) = P (s) + Q(s)e -sτ with deg(P ) = 2 and deg(Q) = 1. As mentioned in the Introduction, we will explicitly investigate the effect of the coexistence of real roots on the stability of the trivial solution. In other words, we wish to give an answer to the following question: Is the coexistence of sufficiently many negative roots guarantees the location of the remaining roots in the left-half plane?

Prerequisites

In complex analysis, it is well-known that the argument principle is a consequence of Cauchy theorem, and it connects the winding number of a closed rectifiable curve (see, e.g., [START_REF] Conway | Functions of one complex variable[END_REF] for a deeper discussion on the topics) with the number of zeros and poles inside the curve. This result gives insights on the location of zeros and poles of a given meromorphic function. Theorem 1. (Argument principle): let V ∈ C be a bounded domain with smooth boundary Γ positively oriented (counter-clockwise) and let f be a meromorphic function inside the contour Γ, then

1 2iπ Γ f (z) f (z) dz = Z -P, (3)
where P and Z denote the number of poles and zeros of f in V , counted with their multiplicities.

In particular, if f (z) is analytic inside Γ, then the left-hand side of (3) gives the number of zeros of f (z) inside Γ.

We need also to recall the statement of Hassard's Theorem: Theorem 2. (Hassard's Theorem). Let A 1 , ..., A m be real n by n matrices, and let

τ 1 , • • • , τ m be nonnegative reals. Let ∆(s, τ ) = det(sI - m j=1 e -sτj A j ). (4) Let ρ 1 , • • • , ρ J be the positive zeros of R(y) = (i -n ∆(iy)),
counted by multiplicity and ordered so that

ρ 1 ≥ • • • ≥ ρ j > 0. For each j = 1, • • • , J such that ∆(iρ j ) = 0,
assume that the multiplicity of iρ j a zero of ∆(λ) is the same as the multiplicity of ρ j as a zero of R(y). Then, the number of roots of the characteristic equation∆(λ) = 0 which lie in (s) > 0, counted by multiplicity, is given by the formula

n -K 2 + 1 2 (-1) J sgnS (κ) (0) + J j=1 (-1) j-1 sgnS(ρ j ) (5)
where K is the number of zeros of ∆(s, τ ) on (λ) = 0, counted by multiplicity, k is the multiplicity of s = 0 as a root of ∆(s, τ ) = 0, and S(y) = Im(i -n ∆(iy)). Furthermore, the count ( 5) is odd if ∆ (k) (0) < 0 and is even if ∆ (k) (0) > 0. If R(y) has no positive zeros, set r = 0 and omit the summation term in (5). If λ = 0 is not a root of the characteristic equations, set κ = 0 and interpret S(0) (0) as S(0) and ∆(0) (0) as ∆(0).

MAIN RESULTS

Consider now, the second-order system

ẍ(t) + a 1 ẋ(t) + a 0 x(t) = u(t) ( 6
)
where u is the unknown control. Assume that the system (6) is unstable in the uncontrolled case (u(t) = 0). Our aim is to construct an appropriate delayed-state-feedback controller of the form:

u(t) = -α 0 x(t -τ ) -α 1 ẋ(t -τ ), (7)
allowing to guarantee the stability of the system (6) in closed-loop. The characteristic quasi-polynomial function corresponding to the closed-loop system is described as follows: ∆(s, τ ) = P (s) + Q(s)e -sτ = 0, (8) where P (s) and Q(s) are polynomials in s with degree of Q(s) is less then the degree of P (s). In our case, we will have explicitly

P (s) = s 2 + a 1 s + a 0 s and Q(s) = α 1 s + α 0 . ( 9
)
3.1 On qualitative properties of s 1 as a root of (8)

The following proposition gives conditions on the parameter coefficients that allows assigning a maximum number of spectral values of the second-order system (6)-( 7). Proposition 3. The following assertions hold:

• The quasipolynomial (8) admits four distinct real spectral values s 1 , s 2 , s 3 and s 4 with s 4 < s 3 < s 2 < s 1 if and only if the parameters a 1 , a 2 , α 1 and α 0 satisfy

                                                       a 1 (τ ) = 1 Q(τ ) (10)
where

Q(τ ) = 1 s 1 s 1 e -s1τ e -s1τ
1 s 2 s 2 e -s2τ e -s2τ 1 s 3 s 3 e -s3τ e -s3τ 1 s 4 s 4 e -s4τ e -s4τ

.

(11)

• The spectral value s 1 is negative if and only if there exists τ 0 > 0 such that

a 1 (τ 0 ) + s 2 = 0. ( 12
)
Sketch of the Proof:

• According to the G. Pólya and G. Szegö Theorem [START_REF] Pólya | Problems and Theorems in Analysis: Series, Integral Calculus, Theory of Functions; Translation[END_REF], the number of real roots of ( 8) is four, hence we investigate the existence of four distinct real spectral values s 1 > s 2 > s 3 > s 4 . The coefficients of the quasi-polynomial function (8) described by ( 10) and ( 11) are obtained by solving the following system s 2 i + a 1 s i + a 0 + (α 1 s i + α 0 ) exp (-s i τ ) = 0, i = 1 . . . 4, (13) that we can represents under the more suitable form:

    1 s 1 s 1 e -s1τ e -s1τ 1 s 2 s 2 e -s2τ e -s2τ 1 s 3 s 3 e -s3τ e -s3τ 1 s 4 s 4 e -s4τ e -s4τ        a 0 a 1 α 1 α 0    =     -s 2 1 -s 2 2 -s 2 3 -s 2 4     (14)
System ( 13) admits a unique solution because the Vandermonde-type matrix in Q(τ ) is invertible, for every τ > 0. Indeed, its determinant is positive since

τ → Q(τ ) is increasing from 0 to ∞, with Q(0) = 0.
To get the formulas (10), then just apply the Cramer's rule. • The negativity of the root s 1 is shown through the variation of the function τ → a 1 (τ ) + s 2 .

(15) So, for all values of τ ∈ R + * (τ > 0), this later function is continuous and increasing from -∞ to -s 1 , from which we deduce that τ → a 1 (τ ) + s 2 takes a positive values if and only if s 1 < 0. See [START_REF] Amarne | On qualitative properties of low-degree quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment[END_REF].

Due to the computation complexity of these time-delay dependent coefficients, we consider only the case of equidis-tance roots. This allows also to decrease the number of parameters to handle. So, we suppose that the roots of ( 8) are such that:

d = |s 1 -s 2 | = |s 2 -s 3 | = |s 3 -s 4 |.
(16) Substituting the formula ( 16) in ( 10) and ( 11), the above parameter coefficients can be rewritten as follows:

                                       a 1 (τ ) = 1 e dτ -1 2s 1 -5d + de dτ -2s 1 e dτ a 0 (τ ) = 1 (e dτ -1) 2 6d 2 -ds 1 e 2dτ + 6ds 1 e dτ -5ds 1 + s 2 1 e 2dτ -2s 2 1 e dτ + s 2 1 α 1 (τ ) = 2de -2dτ e τ s1 e -dτ -1 α 0 (τ ) = -2de -2dτ e τ s1
(e -dτ -1)

2 3d -s 1 + s 1 e -dτ . (17)
The following theorem gives conditions on the negativeness of s 1 . Theorem 4. The spectral values s 1 is negative if and only if one of the following equivalent conditions is satisfied:

(1) The spectral value s 1 is explicitly given by:

s 1 = -4 d e dτ -1
.

(2) The delay τ takes the value

τ * = 1 d ln 1 s 1 (-4d + s 1 ) .
(3) The distance d between the real roots satisfies

d = 1 4 s 1 - 1 τ LambertW l, τ s 1 4 e τ s 1 4 | l ∈ Z \ 2iπl τ | l ∈ Z ∩ ]0, +∞[ .

Sketch of the Proof:

We use the characterization of the negativeness of s 1 given by equation ( 12). Observe that, as τ → ∞

a 1 (τ ) = 1 e dτ -1 2s 1 -5d + de dτ -2s 1 e dτ → d -2s 1
with the property that τ → a 1 (τ ) is bounded by -s 1 . So if there exists τ * such that a 1 (τ * ) + s 1 -d = 0, we obtain that s 1 < 0, and vis versa. Now, to get the value of s 1 , τ * and d respectively, we just have to solve the equation 4d -s 1 + s 1 e dτ = 0. (18) Easy computation gives the value of s 1 and τ . For the value of d, it can be calculated using the Lambert W function since the unknown (d) appears both outside and inside the exponential function. The Lambert W function is defined as the multivalued function that satisfies x = W (x)e W (x) , for any complex number x. Equivalently, it may be defined as the inverse of the complex function f (x) = xe x . In order to solve equation ( 18) for d we rewrite it as follows:

d - s 1 4 e -τ d = - s 1 4 , this implies that -τ d - s 1 4 e -τ d-s 1 4 = s 1 τ 4 e s 1 τ 4 .
From the later equation, we have LambertW τ s 1 4 e τ s 1 4

= -dτ + τ s 1 4 . ( 19
)
Hence

d = 1 4 s 1 - 1 τ LambertW l, τ s 1 4 e τ s 1 4 | l ∈ Z \ 2iπl τ | l ∈ Z ∩ ]0, +∞[ .
(t) + m(t) = kϑ(t -τ 0 ), ϑ (t) + 2ζωϑ (t) + ω 2 ϑ(t) = ω 2 u(t), (20)
in which m, ϑ, and u represent perturbations of the Mach number of the flow, the guide vane angle, and the input of the guide vane actuator, respectively, with respect to steady-state values. The parameters κ and k depend on the steady-state operating point and are assumed to be constant as long as m, ϑ, and u remain small, and satisfy κ > 0 and k < 0. The parameters ζ ∈ (0, 1) and ω > 0 come from the design of the guide vane angle actuator and are thus independent of the operating point. The time delay τ 0 is assumed to depend only on the temperature of the flow. Notice that, in the absence of control (u(t) = 0), the open-loop system ( 20) is exponentially stable.

The design of the stabilizing feedback for (20) improving its stability properties has been considered in [START_REF] Manitius | Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation[END_REF], Boussaada et al. [2018]. The design we propose here is a delayed PD controller which can be written u(t) = β 0 x(t -τ 1 ) + β 1 ẋ(t -τ 1 ). In closed-loop system, the corresponding characteristic equation reads: ∆(s, τ 1 ) = (s -a) (sβ 1 + β 0 t)e -sτ1 + 2 ω sζ + ω 2 + s 2 Since the parameter a is fixed then one focus on the second factor only, ∆ 4 (s, τ 1 ) = s 2 + 2 ω sζ + ω 2 + (sβ 1 + β 0 )e -sτ1 .

  of the presented result, we consider the problem of stabilizing the Mach number of a transonic flow in a wind tunnel. The analysis of transonic flows is a challenging problem in compressible fluid dynamics, since a full model of the flow would involve considering the Navier-Stokes equations in a three-dimensional domain and boundary controls for temperature and pressure regulation. A simplified model was considered in[START_REF] Armstrong | An application of multivariable design techniques to the control of the National Transonic Facility[END_REF] in order to analyze the response of the Mach number of the flow to changes in the guide vane angle. Instead of using a PDE model, propagation phenomena are modeled through a time delay, leading to the time-delay system κm

  Figure 1 exhibits the spectrum distribution corresponding to (21).

Fig. 1 .

 1 Fig. 1. Zeros distribution of quasipolynomial (21) with ω = 1.2, ζ = 0.472 and the controller gains given by .


Jihane Rouissi

Robin Arvieu

Arnaud Dubory

Claudio Vergari

Manon Bachy

Raphaël Vialle
email: raphael.vialle@aphp.fr

Intra and inter-observer reliability of determining degree of pelvic obliquity in neuromuscular scoliosis using the EOS-CHAIR® protocol

Keywords: Neuromuscular scoliosis, Eos-chair, Pelvic obliquity, Trunk imbalance, Eos system, CD reformatting

Purpose Scoliosis with pelvic obliquity (PO) could be investigated with the EOS-CHAIR protocol as the most common deformity especially in patients with trunk hypotonia and quadriplegia. However, the intra-observer and inter-observer reliability of various angles assessing PO was not investigated with this new imaging protocol. Methods A retrospective cohort of 36 EOS frontal fullspine acquisitions made in sitting position was used. The sacroiliac pelvic obliquity angle, iliac crest pelvic obliquity angle, and ischiatic pelvic obliquity angle were assessed in an intra-observer and inter-observer study. Results The use of the EOS-CHAIR protocol was implemented satisfactory with a high acceptance rate by all caregivers and patients and their families. Intra-observer and inter-observer reliability was excellent for the three tested angular measurements. Discussion As for idiopathic scoliosis, we postulate the EOS system as being superior to standard radiographs to assess 3D spinal deformities in neuromuscular conditions. The EOS-CHAIR protocol improves preoperative comprehension of the lumbosacral junction anatomy in patients with poor standing or sitting postures. Our results show a very high reliability of three different angular measurements of the frontal pelvic obliquity in sitting position. Then it is possible to use one of these three angles as well as the others to assess frontal pelvic obliquity in neuromuscular patients. This frontal pelvic obliquity protocol in sitting position with the EOS-CHAIR is a validated measurement technique that needs to be used now to measure PO as a critical parameter of the global trunk balance in neuromuscular patients.

Introduction

Spinal deformity is common in children and adolescents with neuromuscular disorders. Incidence is higher than 20 to 25% in non-ambulatory patients and total-body cerebral palsy (CP) patients [1]. Scoliosis with pelvic obliquity (PO) is the most common deformity especially in patients with trunk hypotonia and quadriplegia. The development of a severe trunk deformity associated with the inherent neurological disorder can significantly restrict the patients' functional capacities and maximize the need for nursing care [1][2][START_REF] Vialle | Neuromuscular scoliosis[END_REF]. Sitting tolerance may be increasingly diminished and trunk decompensation can result in back or rib impingement pain, as well as pulmonary complications [START_REF] Brooks | What's new in the management of neuromuscular scoliosis[END_REF]. Such patients often require a long fusion from the upper thoracic spine to the sacrum or pelvis to achieve and maintain correction of the curve and the PO. One of the technical challenges of such surgical strategy is the objective evaluation and measurement of pelvic obliquity in sitting position. Previous published data established that the EOS system 2D/3D images made with the EOS-CHAIR protocol improve preoperative comprehension of the lumbosacral junction anatomy and assessment of the global frontal and sagittal sitting balance [START_REF] Bouloussa | A radiolucent chair for sitting-posture radiographs in non-ambulatory children: use in biplanar digital slot-scanning[END_REF]. However, the intra-observer and inter-observer reliability of various angles assessing PO was not investigated with this new imaging protocol. The aim of this study was to assess the reliability of various PO angular measurements and to propose the best technique to validate the best way to evaluate this very critical radiological data with the EOS-CHAIR solution.

Material and methods

This study was retrospective, using a cohort of 36 EOS frontal full-spine acquisitions made in neuromuscular patients with pelvic obliquity. The EOS-CHAIR protocol [START_REF] Bouloussa | A radiolucent chair for sitting-posture radiographs in non-ambulatory children: use in biplanar digital slot-scanning[END_REF] was used as follows: The patient was placed in the chair by two persons (the parents with the help of radiology department caregivers). Then, the three-point seatbelts were adjusted and closed to prevent accidental fall from the chair. The headrest was adjusted to the appropriate height. The headbands were used for children with poor cervical tonus. Once patients were properly seated, the chair was introduced into the EOS system by its 46-cm-wide access and placed for image acquisition. During image acquisition (approximately 20 s), patients were required to have their arms in upper position. At the end of the procedure, the chair is pulled backward from the EOS system using its handling straps. With this protocol, it was possible to visualize the whole axial skeleton between the head and the pelvis. PO was assessed using three different angular measurements (Fig. 1): & The sacroiliac pelvic obliquity angle [START_REF] Bouyer | Correction of pelvic obliquity in neuromuscular spinal deformities using the BT construct^: results and complications in a prospective series of 60 patients[END_REF][START_REF] Vialle | Surgical treatment of scoliosis with pelvic obliquity in cerebral palsy: the influence of intraoperative traction[END_REF] (SIPOA) was measured by defining a sacroiliac line between the inferior edges of both sacroiliac joints on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the sacroiliac line (Fig. 1a). & The iliac crest pelvic obliquity angle (ICPOA) was measured by defining an iliac crest line between the upper part of both iliac crests on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the iliac crest line (Fig. 1b). & The ischiatic pelvic obliquity angle (IPOA) was measured by defining an ischiatic line between the lower part of both ischiatic bones on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the ischiatic line (Fig. 1c).

First, the intra-and inter-observer reliability of the threeangle measurements were assessed. Computer-assisted measurements of the three different angles were made in the 36 cases by two different observers (JR and RA), on two separate occasions, in random order using the same software (Carestream PACS Carestream Health, Inc. 150 Verona Street, Rochester, NY, 14608, USA). For all comparisons, the reliability between the two series of measures was assessed using Spearman's rank correlation test, a t test and intraclass Fig. 1 The three different angles used to assess the frontal pelvic obliquity. a The sacroiliac pelvic obliquity angle (SIPOA) was measured by defining a sacroiliac line between the inferior edges of both sacroiliac joints on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the sacroiliac line. b The iliac crest pelvic obliquity angle (ICPOA) was measured by defining an iliac crest line between the upper part of both iliac crests on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the iliac crest line. c The ischiatic pelvic obliquity angle (IPOA) was measured by defining an ischiatic line between the lower part of both ischiatic bones on the frontal radiograph, the perpendicular to this line, and the spinous process of T1 to the middle of the ischiatic line correlation coefficient (ICC) [8], and the limits of agreement by Bland and Altman [START_REF] Bland | A note on the use of the intraclass correlation coefficient in the evaluation of agreement between two methods of measurement[END_REF]. The data were analyzed with SPSS V22.0 software (SPSS Inc. Chicago, IL, USA). A P value for Spearman's rank correlation and all t tests were considered significant if less than 0.05. Intraclass correlation coefficients (ICC) [8] of one imply perfect agreement, and values less than one imply less than perfect agreement [8].

Results

The use of the EOS-CHAIR protocol was implemented satisfactory with a high acceptance rate by all caregivers and patients and their families. We did not experience adverse effects using this technique. The quality of the EOS images allowed assessing preoperative trunk collapse and pelvic obliquity as well as postoperative corrections in all planes (Fig. 2).

Intra-observer reproducibility

Intra-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for ischiatic pelvic obliquity angle (IPOA) (R = 0.913, P < 0.0001) and the paired t test was again significant (t = 0.913, P < 0.0001). The ICC was 0.954 (0.927-0.972), and the graphic study by Altman and Bland showed a statistically significant difference between two measurements out of 72.

Intra-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for sacroiliac pelvic obliquity angle (SIPOA) (R = 0.932, P < 0.0001) and the paired t test was again significant (t = 0.955, P < 0.0001). The ICC was 0.977 (0.964-0.986), and the graphic study by Altman and Bland showed a statistically significant difference between one measurement out of 72.

Intra-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for iliac crest pelvic obliquity angle (ICPOA) (R = 0.931, P < 0.0001) and the paired t test was again significant (t = 0.955, P < 0.0001). The ICC was 0.976 (0.962-0.985), and the graphic study by Altman and Bland showed a statistically significant difference between one measurement out of 72.

Inter-observer reproducibility

Inter-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for ischiatic pelvic obliquity angle (IPOA) (R = 0.771, P < 0.0001) and the paired t test was again significant (t = 0.820, P < 0.0001). The ICC was 0.898 (0.927-0.972) and the graphic study by Altman and Bland showed a statistically significant difference between two measurements out of 72.

Inter-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for sacroiliac pelvic obliquity angle (SIPOA) (R = 0.845, P < 0.0001) and the paired t test was again significant (t = 0.898, P < 0.0001). The ICC was 0.944 (0.910-0.965), and the graphic study by Altman and Bland showed a statistically significant difference between two measurements out of 72.

Inter-observer reproducibility of the manual method Spearman's rank correlation showed significant correlation between data sets for iliac crest pelvic obliquity angle (ICPOA) (R = 0.868, P < 0.0001) and the paired t test was again significant (t = 0.932, P < 0.0001). The ICC was 0.965 (0.944-0.978), and the graphic study by Altman and Bland showed a statistically significant difference between one measurement out of 72.

The summary of statistical analysis is reported in Table 1.

Discussion

Scoliosis with pelvic obliquity (PO) is the most common deformity especially in patients with trunk hypotonia and quadriplegia [START_REF] Vialle | Neuromuscular scoliosis[END_REF]. Progressive spinal deformity causes difficulty with daily care, walking, and sitting and can lead to back and rib pain, cardiac and pulmonary complications, altered seizure thresholds, and skin compromise [START_REF] Roberts | Factors influencing the evaluation and management of neuromuscular scoliosis: a review of the literature[END_REF]. Interest in the potential clinical relevance and impact of 3D of trunk and chest deformities is growing. Formerly, CT scanning was the most frequently used imaging tool for such 3D analysis but at the expense of high radiation exposure, and with the limitation of being performed in the supine position [START_REF] Melhem | EOS(®) biplanar X-ray imaging: concept, developments, benefits, and limitations[END_REF]. These drawbacks can be avoided in ambulatory patients by the use of the EOS system low-dose biplanar radiographs. In standing adolescent idiopathic scoliosis patients, the validity of routine 3D reconstructions has been established at present [START_REF] Courvoisier | The change on vertebral axial rotation after posterior instrumentation of idiopathic scoliosis[END_REF]. As for idiopathic scoliosis, we postulate the EOS system as being superior to standard radiographs to assess 3D spinal deformities in neuromuscular conditions. Thanks to our EOS-CHAIRvalidated protocol [START_REF] Bouloussa | A radiolucent chair for sitting-posture radiographs in non-ambulatory children: use in biplanar digital slot-scanning[END_REF], the EOS system 2D/3D images improve preoperative comprehension of the lumbosacral junction anatomy in patients with poor standing or sitting postures. However, as PO is commonly associated with scoliosis in cerebral palsy making sitting difficult [START_REF] Bouyer | Correction of pelvic obliquity in neuromuscular spinal deformities using the BT construct^: results and complications in a prospective series of 60 patients[END_REF], we consider as crucial to investigate pelvis positioning in sitting position. Our results show a very high reliability of three different angular measurements of the frontal pelvic obliquity in sitting position. Compared to the well-known sacroiliac pelvic obliquity angle [START_REF] Bouyer | Correction of pelvic obliquity in neuromuscular spinal deformities using the BT construct^: results and complications in a prospective series of 60 patients[END_REF][START_REF] Vialle | Surgical treatment of scoliosis with pelvic obliquity in cerebral palsy: the influence of intraoperative traction[END_REF], the angular measurements based on ischiatic and iliac crest landmarks are also reliable and reproducible. Then it is possible to use one of these three angles as well as the others to assess frontal pelvic obliquity in neuromuscular patients. This frontal pelvic obliquity protocol in sitting position with the EOS-CHAIR is a validated measurement technique that needs to be used now to measure PO as a critical parameter of the global trunk balance in neuromuscular patients. In further clinical studies, we stress the importance of such simple and fast technique in saving time and maximizing inter-and intra-observer repeatability of angular measurements. We fully recognize that the time gain was not clearly studied in the current study, but only a few minutes were necessary to obtain the three PO values. Despite ischiatic landmarks may be more difficult to identify in some patients, iliac crest or sacroiliac lines could be used without any risk in term of reliability. Additionally, these three easy ways to evaluate the frontal pelvic obliquity could be useful to design promising 3D pelvis reformatting tools to be associated with 3D spine reformatting models in the future.

Compliance with ethical standards

Fig. 2

 2 Fig. 2 Example of a young patient with a severe trunk deformity due to cerebral palsy. The EOS-CHAIR protocol gives the opportunity to visualize trunk deformity and pelvic obliquity in sitting position


Ariane Chabert
email: ariane.chabert@ensat.fr

Damien Marchand

Jean-Pierre Sarthou

Data from extensive comparative measurements of conventional, conservation and organic agricultures in southwestern France

Keywords: agronomic performance, phytopathology, landscape data

yields, crop health, mycorrhiza, soil parameters and landscape

This article present observational and experimental data describing a range of biotic and abiotic parameters that can be related to ecosystem services under contrasted types of crop management: conventional, conservation and organic agricultures. Ninety fields, either cultivated with winter wheat or fava bean, located in Southwestern France, near Toulouse, were monitored for two growing seasons (2014)(2015)(2016). The dataset encompass data about crop pests (aphids, grain borer, bean beetles, slugs), crop pest natural enemies (hoverflies, parasitoids, predators), soil sensitivity to erosion, crop productivity, pathogenic fungal infection and root colonization by mycorrhiza. This article present detailed protocols applied for each measurement and data collected to describe the context of each field: soil structure, landscape and crop management indicators. The data presented here can be found in Portail Data INRA repository (DOI: 10.15454/KEW1GK) and were exhaustively used and discussed in the research article Conservation agriculture as a promising trade-off between conventional and organic agriculture in bundling ecosystem services [START_REF] Chabert | Conservation agriculture as a promising trade-off between conventional and organic agriculture in bundling ecosystem services[END_REF] .

Value of the data

This dataset provides an unprecedented broad range of simultaneous measurements of biophysical, ecological and agronomic parameters, for the study of agroecological performances of conservation agriculture in France, with comparison with organic and conventional production systems. It comes from a collaborative work with farmers allowing reliable in-field assessment rather than experimental plots. This dataset provides a priceless reference for future researcher in agronomy and ecology seeking reference for agroecological performances of early adopters of Conservation Agriculture in France. It also provides in-fields values that are still lacking to support studies on ecosystem services bundles in agroecosystems. This dataset can be reused for meta-analyses, either by focusing on one specific ecosystem service or agronomic performance or by appreciating bundles of them. Furthermore, methods used were rapid and easy to implement, allowing replication of them in future studies, using this dataset as a reference point or a meaningful complement.

Data

In this data article, we are sharing an extensive recollection of data resulting from two years monitoring of winter wheat, fava bean and mix fava bean-wheat fields distributed among 52 farms.

Data encompass several separate measurements presented in 16 separate excel sheets in the file Full_dataset.xlsx. In the sheet 'Cropping_System', a set of crop management indicators to describe each farm is presented, including tillage, use of chemical inputs, diversity of production and semi-natural habitats. The sheet 'Soil_Parameters' presents a soil analysis (texture, pH, organic matter) for the wheat field of each farm. A coarse description of the surrounding landscape, within 1.5 km around the wheat field of each farm, can be found in the sheet ' Land-scape_1500 m' and the area of semi-natural habitats that can be found on the farm are listed in the sheet ' Semi_Natural_Habitat'.

Several measurements of pests and natural regulations were made: counts of grain aphids on wheat ears, are in the sheet ' Aphids' , and counts of pupae of one of their potential predators, the long hoverfly, are in the sheet ' Hoverflies'; counts of wheat stubbles obstructed by grain borer, are in the sheet ' Grain_borer'; number of fava bean grains parasitized by bean beetles and assessment of bean beetle parasitism by Triaspis thoracicus are presented un the sheet 'Bean beetles'; Number and species of slugs counted in wheat fields at different seasons (spring and fall) are presented in the sheet ' Slugs' and number of ground predators (including spiders, carabids and rove beetles) counted simultaneously to the slug assessment are presented in the sheet ' Ground Predators'. Other agronomic performances are presented in this dataset, such as speed of water infiltration and soil surface aggregate stability as indicators of soil sensitivity to erosion, in the sheet ' Erodibility' , crops yields, respectively in the sheets ' Wheat_Yields' and ' Fava_bean_Yields' , and visual assessment of Leaf diseases, in sheets ' Wheat_Diseases' and ' FavaB_Diseases' . The standardized scale used for leaf diseases assessment can be found as supplementary material of this article. Concentrations of several mycotoxins in wheat grains were analyzed and also presented in the sheet 'Wheat_Diseases' . Finally, descriptors of mycorrhiza colonization of roots are presented in the sheet ' Mycorrhiza'.

The second file, Labels_descr_dataset.xlsx, gives a comprehensive list of sheet and column labels that appear in Full_dataset.xlsx with their description and unit, as well as indication of the fields the data came from (NTW: not treated wheat, OW: organic wheat, W: wheat, F: fava bean, WF: wheat + fava bean mix) and either data referred to the first (2014-2015) or the second (2015-2016) year of experiment (see experimental design below).

On every sheet of the Full_dataset.xlsx , a code to identify the farm ( 'Farm_code' ) is given on the first column along with date of sampling and identification of the field ( 'Field_code' : NTW, OW, W, F or WF) when relevant.

Data were collected around Toulouse, Southwestern France, from September 2014 to July 2016. Farms were distributed along a 200 km line intersecting Toulouse, from 43 °N, 0 °E to 44 °N, 2 °E, crossing three French counties: Gers, Haute Garonne and Tarn. Our agreement with farmers did not allow us to publicly identify their fields with geographic coordinates. However, county of each field can be found in the sheet Landscape_1500 m of the dataset and a map along with more details about location can be found in Chabert and Sarthou [START_REF] Chabert | Practices of conservation agriculture prevail over cropping systems and landscape heterogeneity in understanding the ecosystem service of aphid biocontrol[END_REF] .

Experimental design, materials, and methods

Experimental design

A total of 90 fields were monitored: 52 winter wheat fields (14 were label with organic agriculture certification), 38 of which were paired with a fava bean field. Fava bean and wheat fields were always close-by, with a common border in most cases. Field size was 1 ha for winter wheat and 0.5 ha for fava bean. All fields had the same crop cultivars and seed origin. No insecticides, including seed treatment, were applied during the cropping season, and crops were not irrigated. Half of each winter wheat field did not receive fungicide treatment (to standardize disease measurements). All other cropping practices were left to the farmer to decide and were closely monitored (see crop management monitoring below).

Measuring a large number of parameters in the same cropping season on many farms requires extensive labor and time. Measurements were thus chosen to be simple and quick to implement. Data presented here were issued from an experiment designed in cooperation with farmers, in particular for the choice of a meaningful set of measurements. Thus, although that all the measurements were performed by scientific operators to assure homogeneity, they were chosen so that farmers and advisers could reproduce them in the future, mainly for educational purposes.

Crop management monitoring

Information about the field was recorded for tillage, chemical inputs, intercrop management and plant diversity.

Concerning tillage, all mechanical operation in the field were monitored for 3 years (Aug. 2012 -Aug. 2015) and we considered the flowing three types of tillage management: direct seeding ( 'D' , the only mechanical soil disturbance is the opening and closing of the seed line), reduced tillage ( 'RT' , superficial soil tillage without soil inversion) and plowing ( 'P' , use of plow, i.e. soil inversion). To refine tillage description, we also recorded the number of years since last tillage and last plowing in 2015 ( Last_tillage, Last_plow ), and an indicator of tillage intensity calculated as the sum of the depth of all operations for the cropping of both crops in 2014-2015 ( Depth_cum ). A coefficient of 2 was applied for depth associated to plowing operations and 3 for the use of rotary power harrow (shallow but very disruptive).

Concerning chemical inputs, we recorded the amount of mineral and organic nitrogen ( Nmin, Norg, kg.ha -1 ) applied for the cropping of both crops in 2014-2015 and calculated the frequency treatment index for insecticides, herbicides and fungicides taken together ( TFI ), then for herbicides only ( TFI-Herb ) for winter wheat and fava bean respectively. We also indicated if the field was label with French Organic Certification for 3 years of more in 2015 ( Organic ) and if the farming system was specialized in crop production or if a livestock production of any kind exists on the farm ( Farm_type ).

Concerning intercrop management, we recorded whether the soil was bare, covered with crop residues or covered with cover crop during intercrop period before winter crops ( InterCrop ). We completed with the percentage of agricultural area of the farm where cover crops were used during 2013-2014 season ( CoverCrop_% ).

Concerning plant diversity, a Simpson Index was calculated on families of crop cultivated in 2014 (1/ i (n i /N) 2 ; n i : number of crop species in family i cultivated on the farm; N: total number of crop species cultivated) and the area they covered ( CropDiv ). We also recorded the mean length of crop rotation ( Rotation , mean number of years), the percentage of the agricultural area of the farm covered with legume crops ( Legum_percent ) and a Shannon diversity index on semi-natural habitats (woody, herbaceous, fallow, lithic or water habitats) existing on the farm ( SNHDiv;i (n i /N).log 2 (n i /N); n i : area in square meter of semi-natural habitat type i; N: total area of semi-natural habitat, in square meter). Semi-natural habitats were identified with farmers from photographs in their 2014 European Union Common Agricultural Program forms.

Soil parameters

Organic carbon content ( C_organic and Organic_Matter ), pH (pH _water ) and proportion of each size class (5 fractions without decarbonation: Clay, Silt_fine, Silt_coarse, Sand_fine and Sand_coarse ) were measured from a homogenous mix of five soil cores (30 cm deep) collected from the treated half of each winter wheat field, taking care to avoid the 10m-wide border of the field. Soil analyses were performed by an approved laboratory ( http://www.galys-laboratoire.fr/ ).

Landscape context

The landscape context was determined with a rapid assessment of land use type following Chabert et al. [START_REF] Chabert | Assessing landscape composition using visual assessment: accuracy of rapid description compared to digital mapping[END_REF] from aerial photographs (© IGN -BD Ortho R 2014 edition) and expressed as percentages of the area within a 1.5 km radius from the center of the wheat field. We considered six land use types ( Wood, Cult, Fallow, Human, Hedg, Water ): woodlands, cultivated fields (crops or temporary grasslands, to avoid the difficulty to distinguish them on photographs, see [START_REF] Chabert | Assessing landscape composition using visual assessment: accuracy of rapid description compared to digital mapping[END_REF] ), fallow lands (including natural grassland), human-modified areas (e.g. buildings, roads, quarries), hedges and water surfaces (e.g. lakes, ponds, rivers).

Grain aphids, Sitobion avenae

Abundance of Sitobion avenae in each winter wheat field, was estimated during population growth peak, at winter wheat flowering (June 3-6, 2015). Thirty wheat ears were randomly collected in each field, taking care to avoid the 10 m-wide border of the field, and stored in hermetic containers. Containers were brought back to the lab and stored in a temperate room away from direct light. Within two days, containers were emptied over a counting grid and each wheat ear was carefully brushed to retrieve all aphids. Aphids were counted for the 30 ears together, i.e. one value per field. In some fields parasitism by parasitoids occurred and some aphids were dead and mummified. We counted mummified aphids in a separate count. Ears were also re-counted to ensure homogeneity of the counting.

Long hoverfly pupae, Sphaerophoria scripta

To assess the abundance of the aphidophagous larvae of the long hoverfly Sphaerophoria scripta (i.e. estimate of regulation of Sitobion avenae by Sphaerophoria scripta , see Chabert and Sarthou [START_REF] Chabert | Practices of conservation agriculture prevail over cropping systems and landscape heterogeneity in understanding the ecosystem service of aphid biocontrol[END_REF] ) in each winter wheat field, long hoverfly pupae were counted on wheat ears at wheat maturity. Sampling was done immediately before harvest (June 26 -July 2, 2015), after S. scripta metamorphosed on the beards of wheat ears, leaving the empty shell of its pupal stage. In each field, wheat ears from 20 random plots were sampled using a hoop with a radius of 19.25 cm, taking care to avoid the 10m-wide border of the field. Pupae or pupal shells on the beards were counted by placing ears on a light surface. In some rare cases, another hoverfly pupae ( Episyrphus balteatus ) or coccinellidae larvae were observed. They were counted separately. Wheat ears from 6 to 12 plots (depending on field heterogeneity) were then pooled and stored in paper bags to be use for measurements of yields components.

Longhorn grain borer, Calamobius filum

Abundance of Calamobius filum (Coleoptera, Cerambycidae) was assessed after harvest (between July 3-July 23, 2015) by counting wheat stubbles clogged with shredded fiber by C. filum larva. The assessment was based on that particular behavior of the larva, which bore down the stem shortly before harvest and overwinter in the stubble, after clogging the opening to protect itself from the weather. Fiber clogs made by C. filum are easy to see on freshly cut stubbles and cannot be mistaken for some other natural clogs. Additionally, never more than one larva occupies a single stem, making fiber clogs count a rapid assessment of C. filum abundance in the field. Using the same sampling hoops that for hoverflies and yields, number of stubbles obstructed by fiber and total number of stubbles were counted at 20 random plots in each fields, taking care to avoid the 10m-wide border of the field.

Broad bean beetle, Bruchus rufimanus , and their parasitoid, Triaspis thoracicus

Broad bean beetles pressure on fava bean and their parasitism by a specialized parasitoid was estimated through observations on the grains. Immediately before harvest (June 26-July 2, 2015), 20 fava bean plants were randomly selected, taking care to avoid the 10m-wide border of the field. All beans were collected manually and pooled into emergence containers, i.e. 1 L plastic jars, approximately 30 cm-long and pierced with 8 holes (3 cm in diameter) that were closed with 0.5 mm sieve. Emergence containers were stored outside protected from the sun, strong wind and rain. Beans were examined 5 months later to ensure that all beetles and parasitoids had emerged. To avoid misinterpretation due to possible insect evasion at the opening of the container, insects were not kept and estimation were rather made on the number of emergence holes left on beans. Due to the significant size difference of B. rufimanus and its parasitoid T. thoracicus , emergence holes left by either bean beetle (neat round 4-5 mm diameter holes) or parasitoids (1.5-2.0 mm diameter holes) were different enough to enable estimation of bean beetle and parasitoid abundances based on number of small or big holes. Because some grains had more than one hole, number of undamaged grain and total of grains were also recorded to allow calculation of percentage of parasitized grains for instance.

Slugs community

Rather than an exhaustive slug community measurement, we focused on the actively feeding slug community and used a non-capture, non-lethal bait method using wheat bran. Measurements were conducted during wheat emergence (November 19-23, 2014) and at activity resumption after winter (April 14-20, 2015) the first year, and twice during wheat emergence (October 28-November 08, 2015 and November 26-December 05, 2015) the second year due to very dry conditions during the first period. Approximately 30 cl of organic wheat bran was left on the soil and spread over an area of 20 cm in diameter. The bait was left in the open on the soil for 36 h without any trapping system. Such baits were placed at least 15 m apart from each other on three random plots in each winter wheat field, avoiding the 15 m-wide border of the field. Plots were inspected for slugs at night between 10:30pm and 4:00am, which was identified as the period of peak feeding activity of the pest slugs. All slugs within a 50 cm square area around each bran bait were identified and counted in field. Taxonomic group or species that were possible to identify in fields were Deroceras reticulatum, Arion hortensis, Milax gagates, Lehmmania valenciana, Limax maximus, Arion gr. ater, Arion gr. hortensis, Arion gr. fasciatus and Arion gr. subfuscus . No sampling of specimens was performed.

Generalist predators

We estimated the number of opportunistic ground predators using simplified pitfall traps constructed from 14 cm-long cones with a 6 cm circular opening (a hard plastic beach ashtray with the cover removed). Traps were set in three sampling plots in winter wheat fields for 36 h, simultaneously with the slugs' community measurements (two periods: April 14-20, 2015 and October 28-November 08, 2015). Traps were filled with soapy water and salt to prevent escape and preserve trapped insects. They were set 5 m away from wheat bran baits for slugs and 15 m apart from each other and from the border of the field. Traps were collected by night and pooled within a single sample per field, stored in 70 °ethanol. Ground beetles (Coleoptera, Carabidae), spiders (Araneae) and rove beetles (Coleoptera, Staphylinidae) were sorted and counted in the lab. No identification to genus or species were done, but special care was taken to identify and remove from counting potential individuals of genera Zabrus and Amara , most common phytophagous species of ground beetles in fields in this area. None were actually found for this dataset. Other invertebrates that may have been trapped were counted and coarsely identify as a comment in the dataset.

Soil erodibility

Soil erodibility was assessed through structural stability of soil aggregates and water infiltration rate. The later was measured as the time needed for the soil to absorb 125 cl of water poured rapidly into a 14 cm diameter metal cylinder sunk approximately 3 cm into the ground. Measurements were performed in April 2016 at three random plots within the wheat field, taking care to avoid the 10 m-wide border of the field. To ensure similar soil moisture when measuring infiltration, a first dose of 125 cl of water was poured into the cylinder and allowed to infiltrate the soil before the second dose of 125 cl of water was poured and timed. In cases where infiltration was very low, the experiment was stopped after 3 min and height of remaining water was measured to calculate the volume of infiltrated water. In this dataset, raw values of actual volume infiltrated and corresponding time of infiltration are given for the first ( Vol_1, Time_1 ) and second pouring ( Vol_2, Time_2 ). Structural stability of soil aggregates was assessed based on a method for measuring water turbidity using a Secchi disk [4] . A small version (45 mm diameter) of the standard Secchi disk was plunged into the metal cylinder immediately after the second pouring of water and the depth of clear water from the water surface was immediately measured. This method relies on the assumption that measured turbidity is attributable to the quantity of fine particles that are likely to be detached from the soil due to rain and runoff. In cases where water infiltration was too rapid to allow measurement, additional water was poured to completely fill the cylinder. In the dataset, turbidity is given as the raw value of depth of clear water in the cylinder ( Turbidity ).

Crop yields

Winter wheat and fava bean yields were estimated by collecting grains immediately before harvest, simultaneously with Long hoverfly pupae and Broad bean beetle parasitoid sampling in 2015. To allow expression of yields in quintals per hectare for both crops, fava bean plant density was estimated by counting plants in three random plots of 1 m 2 and 20 plants were randomly sampled for weighting grains. For wheat, sampling effort was not homogenous among all fields because of very scarce development of wheat in some fields (6 to 15 for the samplings for hoverflies were kept for yields). Yield values are expressed per hectare, taking into account the number of hoop area sampled, yet to keep track of the sampling effort the number of samplings is recorded in the dataset ( Sampling_nber ). The harvested wheat ears were threshed using a square sieve (10 mm) threshing machine and grains were cleaned using a densimetric column separator (type INRA 240). Fava beans were manually shelled. For both crops, samples were then dried at 80 °C for 48 h and weighed immediately (weight at 0% humidity). For wheat, a subsample was weighed and the number of grains determined with a grain counter to calculate the weight of one thousand kernels. Same protocols were applied in 2016 for both crops.

Fungal diseases

We estimated fungal leaf disease infection on both crop with a standardized six-level scale (See Supplementary Materials) evaluating leaves coverage by symptoms of diseases. After thorough inspection of plants in two diagonal transects, fields were scored from 0 to 5, corresponding to the predominant score of observed plants. One score was given for each field, all leaf diseases combined (mainly Septoriosis, rusts and Helminthosporiosis for wheat; Anthracnose and Botrytis for fava bean). For informational purposes, the proportion of symptoms attributable to each disease was coarsely estimated and noted down. Grain samples from wheat yield measurements were also analyzed at INRA laboratory UR1264 MYCSA with high performance liquid chromatography and mass spectrometry (HPLC-MS) for concentrations (ng. g -1 ) of Zearalenone and 5 type B trichothecene mycotoxins: Deoxynivalenol, 15-Acetyl-deoxynivalenol, 3-Acetyl-deoxynivalenol, Nivalenol and Fusarenone X.

Mycorrhiza

Winter wheat roots colonization by mycorrhiza was assessed during 2015-2016 growth season. Twenty wheat plants were collected in May 2016 at 5 random plots in the field, taking care to avoid the 10m-wide border of the field. Whole plant and its roots were collected. Roots were carefully separated in the lab and immediately frozen (-20 °C) for conservation. Samples were defrosted in water at ambient temperature, twelve roots were randomly chosen and sliced into 1 cm fragments. Root fragments were colored following Vierheilig et al. [START_REF] Vierheilig | Ink and vinegar, a simple staining technique for arbuscular-mycorrhizal fungi[END_REF] method and observed under microscope to determine parameters of mycorrhizal colonization (adapted from Trouvelot et al. [START_REF] Trouvelot | Physiological and genetical aspects of mycorrhizae[END_REF] ): frequency of mycorrhiza ( Mycor_freq ) and arbuscules ( Arb_freq ) in the root system, intensity of the mycorrhizal colonization ( Mycor_inten ) and mean number of arbuscules ( Arb_nb ) in the root system, intensity of the mycorrhizal colonization ( Mycor_inten_frag ) and arbuscular structures ( Arb_inten_frag ) in the root fragments. With following calculation:

Mycor_freq (%) = (number of fragments with mycorrhiza/total number of fragment) * 100 Arb_freq (%) = (number of fragments with arbuscule/total number of fragment) * 100 Mycor_inten (%) = (95 * n5 + 70 * n4 + 30 * n3 + 5 * n2 + n1)/(number total of fragment) where n5 = number of fragments rated 5; n4 = number of fragments 4 etc. (see Trouvelot et al. [START_REF] Trouvelot | Physiological and genetical aspects of mycorrhizae[END_REF]) Arb_nb = total number of arbuscule/total number of fragment Mycor_inten_frag (%) = Mycor_inten * (number total of fragments)/(number with mycorrhiza) Arb_inten_frag = total number of arbuscule/number of fragment with arbuscule

Acknowledgments

This work was supported by ANR TATA-BOX project (ANR-13-AGRO-0 0 06).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.dib.2020.105827 .