V Andrieu

C Prieur

S Tarbouriech

L Zaccarian

A hybrid scheme for reducing peaking in high-gain observers for a class of nonlinear systems 1

For a family of nonlinearly interconnected second order plants, we propose a hybrid modification of the well known high-gain observer design, which leads to guaranteed asymptotic stability properties of the error dynamics, as well as exhibiting significantly reduced peaking. The hybrid modification results in suitable jumps of the observer state, which therefore corresponds to a discontinuous function of ordinary time. The jump conditions depend on the plant output and on extra states incorporated in the observer that evolve according to some suitable combination of the output error and of its integral. Our result is illustrated on a well known case study taken from the literature, and also on a planar robot with rigid links.

Introduction

The observer design problem for nonlinear dynamical systems has attracted the attention of several researchers over the last years (see the books [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]; [START_REF] Besançon | Nonlinear observers and applications[END_REF] and references therein). Different approaches are possible such as high-gain observers [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]; [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]; [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]) nonlinear Luenberger observers (see [START_REF] Andrieu | On the existence of Kazantzis-Kravaris / Luenberger Observers[END_REF]), passivity (as in [START_REF] Ailon | An observer-based set-point controller for robot manipulators with flexible joints[END_REF]) or bounds on the slope of the nonlinearity (see e.g. [START_REF] Arcak | Observer-based control of systems with slope-restricted nonlinearities[END_REF]). High-gain observers design dates back to the late 1980's [START_REF] Nicosia | An approximate observer for a class of nonlinear systems[END_REF]; [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]; [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]) and essentially corresponds to the intuitive idea that a very strong Lyapunov decrease arising from high-gain linear output injection is capable of dominating nonlinear terms satisfying a suitable bound. Due to this intrinsic nature of the high-gain approach, these observers suffer from the so-called peaking phenomenon [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF]; [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]; [START_REF] Sepulchre | Constructive nonlinear control[END_REF]; [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]) wherein the observer error can grow very large during the transient response.

One of the interest of asymptotic observers is to use the 1 This work was supported by the ANR project LimICoS contract number 12-BS03-005-01, and by the University of Trento, grant OptHySYS. state estimation in an output feedback loop to achieve asymptotic stabilization. To protect the system from the destabilizing effect of peaking, high-gain observers have to be followed by saturation as explained in [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]. This strategy has been formalized in [START_REF] Teel | Global stabilizability and observability imply semi-global stabilizability by output feedback[END_REF] in combination with an a priori knowledge of a set of initial conditions to achieve semi global stabilization and in [START_REF] Kaliora | Norm estimators and global output feedback stabilization of nonlinear systems WithISS inverse dynamics[END_REF] in combination with a norm observer. [START_REF] Oliveira | Peaking free outputfeedback exact tracking of uncertain nonlinear systems via dwell-time and norm observers[END_REF][START_REF] Oliveira | Dwelltime and disturbance monitoring for peaking avoidance and performance improvement in high-gain observer based sliding mode control[END_REF] successively exploit a dwell-time control activation to avoid the peaking phenomenon.

In this paper, we focus on the state estimation problem and we don't assume that an estimation of the norm of the state is available. We revisit high-gain observer designs within the hybrid dynamical systems framework of [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF] and, for a family of nonlinearly interconnected second order plants, we propose a hybrid high-gain observer having the novel feature of not exhibiting peaking when the (high) gain of the observer is increased. In particular, the observers that we propose comprise a flow dynamics (continuous-time evolution) which essentially coincides with the original continuoustime high-gain observer of [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]; [START_REF] Nicosia | An approximate observer for a class of nonlinear systems[END_REF]; [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], augmented with a suitable resetting rule enforced on the observer state. Such a scheme ensures that trajectories approaching a region of the state space where peaking occurs, are projected in another region where peaking is absent. We show that the Lyapunov function establishing exponential stability of the error system does not increase, thus preserving the asymptotic state estimation property of the observer. Alternative hybrid observation schemes have been presented in, e.g., [START_REF] Raff | An observer that converges in finite time due to measurement-based state updates[END_REF]; [START_REF] Paesa | Reset adaptive observer for a class of nonlinear systems[END_REF] and reference therein, which don't specifically focus on peaking reduction.

Preliminary results in the directions of this paper have been presented in [START_REF] Prieur | Hybrid highgain observers without peaking for planar nonlinear systems[END_REF]. Here, as compared to those preliminary results, we extend our theory from a single second-order plant to a family of nonlinearly interconnected second-order plant, which allows us to address the case of joint speed estimation in ndegrees of freedom Euler-Lagrange systems. Moreover, we propose different resetting laws and jump and flow sets, to allow for a suitable adaptation of the results to the 2n-dimensional case. Another observer approach to avoid overshoot in the case of planar multidimensional systems is the reduced order observer. More precisely, this framework presented in (Besançon, 2000, Lemma 3.1) computes a reduced order observer for any system for which a high-gain observer has been computed and thus leads to a reduction of the overshoot. In the context of planar observer (that is n = 1 in (1) below), the obtained observer is scalar and has a decreasing state estimation error. However the estimate has a feedthrough term from the measurement output and thus is more sensitive to measurement noise. The hybrid observer that is designed in the present paper can be seen as a tradeoff between these two techniques. Recently in [START_REF] Astolfi | Output feedback stabilization for SISO nonlinear systems with an observer in the original coordinates[END_REF] another approach to avoid peaking has been investigated. It employs a projection approach and requires the knowledge of a convex set in which the state trajectory remains. This is not required in our approach.

The paper is organized as follows. Section 2 is dedicated to revise the high-gain observer by introducing suitable resetting laws for the state of the observer in order to reduce the peaking. Section 3 deals with an application of the results to a robotic example.

Notation: Throughout the paper, notation is standard. |•| stands for the Euclidean norm. For any vector x (resp. matrix A), x T (resp. matrix A T ) denotes its transpose.

A hybrid observer with reduced peaking

Before considering a hybrid observer in Subsection 2.2 below, let us first define the class of nonlinear planar systems and the associated high-gain observer. The peaking phenomenon is also briefly discussed and illustrated.

High-gain observer design and Lyapunov properties

Following the standard approach in, e.g., [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF], we consider a set of n second order nonlin-ear systems whose state is denoted by x = (x 1 , . . . , x n ) ∈ R 2n and x i = (p i , v i ) ∈ R2 , where each substate x i obeys the following second order nonlinear dynamics with nonlinear couplings arising from the terms ψ i and φ i :

ṗi = v i + ψ i (y) , vi = φ i (x, u) y i = p i , (1)
for all i = 1, . . . , n, where y = (y 1 , . . . , y n ) ∈ R n is the measurement output, u ∈ R p is a known control input that is piecewise continuous 2 and ψ i and φ i , i = 1, . . . , n are known nonlinearities establishing a nonlinear coupling among the n planar systems. For example, in the robot example of Section 3, these couplings are caused by strong nonlinear coupling effects between the two joints.

Remark 1 Note that in [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF] an output feedback is designed assuming that the nonlinearities φ i are uncertain. In this paper, since we address the asymptotic observer design problem we consider the case in which dynamics are perfectly known (as [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] or [START_REF] Besançon | Nonlinear observers and applications[END_REF] for instance).

High-gain observers are used to provide an estimate x of the state x. If the following Lipschitz-like condition holds for a suitable constant L δ > 0:

|φ i (x, u) -φ i (x, u)| ≤ L δ |x -x|, ∀i = 1, . . . , n, (2)
for all x and x in R 2n , u in R p , then a high-gain observer can be designed to ensure asymptotic stability of the origin of the feedback system from the observed state.

Remark 2 Lipschitz condition (2) is standard in the context of high-gain observer designs. Note however that it has been relaxed in [START_REF] Lei | A global observer for observable autonomous systems with bounded solution trajectories[END_REF] allowing for a strictly increasing high-gain parameter. We do not follow this route since this approach may lead to undesirable behavior in the context of disturbed measurements. Moreover, this Lipschitz condition has been also slightly relaxed in [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF] employing homogeneous tools. Possible extensions of our tools in these directions are object of future work.

For system (1) we can design the following high-gain observer:

ẋi = ṗi vi = vi + ψ i (y) + k 1 e pi φ i (x, u) + 2 k 2 e pi =: f i (x, e pi , u), (3)
i = 1, . . . , n, where e pi = y i -pi is the i-th output error, k 1 and k 2 are any pair of positive scalars, and is the "high" gain of the high-gain observer. The following dynamics can be easily derived from (1), (3) if one defines the scaled error coordinates e i := (e pi , e vi ) :

= (p i - pi , -1 (v i -vi )): ėi = -k 1 1 -k 2 0 e i + 0 φi(x,u)-φi(x,u) =: A e e i + 0
δ i (x, x, u) , i = 1, . . . , n, (4)
which reveals the potential for the high-gain 1 to dominate over each nonlinear term δ i , i = 1, . . . , n. In particular, from (2), if ≥ 1, we obtain

|δ i (x, x, u)| ≤ L δ |x -x| ≤ L δ |e|. (5)
A possible way to characterize the set of sufficiently highgains ensuring exponential stability of the error dynamics (4) is given in the next proposition, which is given here for completeness and will be useful for our results.

Proposition 1 Consider plant (1) satisfying (2) and observer (3), and pick any matrix P = P T > 0 such that A T e P + P A e < 0. Then, for each ϑ > 0, there exists a high-gain parameter ≥ 1 such that

(A T e P + P A e ) + nϑL 2 δ I 2 P [ 0 1 ] (P [ 0 1 ]) T -ϑ < 0. ( 6
)
Moreover, with such a selection of the high-gain , the origin is globally exponentially stable for the error dynamics (4), as established by the Lyapunov function

e → V (e) := n i=1 e T i P e i = e T (P ⊗ I n )e, (7)
where e = (e 1 , . . . , e n ) and I n ∈ R n×n is the identity matrix.

Remark 3 Before proving Proposition 1, let us note that in its context, and recalling the definition of the scaled error, Lemma 3.1 of [START_REF] Besançon | Remarks on nonlinear adaptive observer design[END_REF] applies and leads to the following reduced order observer xi = yi vi for i = 1, . . . , n where vi = w i + p -1 11 p 12 p i and w i has the following dynamics ẇi = φ i (v, u) + p12 p11 vi where P = [ p11 p12 p22 ]. With this observer, the Lyapunov function of the error system is i (v i -vi ) 2 , which has balanced coordinates. This is not the case for Lyapunov function (7) in the original error coordinates. Therefore the reduced order observer prevents in general peaking, but its estimate vi depends directly on p i and is therefore expected to be more sensitive to measurement noise.

Proof of Proposition 1. Regarding exponential stability of (4), we show next that V is a Lyapunov function for dynamics (4). To this aim, from (5) we get

|δ(x, x, u)| := δ1(x,x,u) . . . δn(x,x,u) ≤ √ nL δ |e|. (8)
Then, from (8) and for any ϑ ≥ 0, we can write for all x and x in R 2n ,

e δ(x, x, u) T -ϑnL 2 δ I 2n 0 0 ϑI n e δ(x, x, u ) = ϑ(δ(x, x, u) T δ(x, x, u) -nL 2 δ e T e) ≤ 0. ( 9
)
Now let us rewrite dynamics (4) as follows:

ė = (A e ⊗ I n )e + ([ 0 1 ] ⊗ I n ) δ(x, x, u), (10)
and note that using the well known relation (A⊗B)(C ⊗ D) = AC ⊗ BD, the derivative of the function V in (7) along dynamics (10) can be bounded as follows, where we also use ( 9):

V = 2e T (P ⊗ I n ) (A e ⊗ I n )e + ([ 0 1 ] ⊗ I n ) δ(x, x, u) = e T 2 (P A e ⊗ I n )e + e T 2(P [ 0 1 ] ⊗ I n )δ(x, x, u) ≤ [ e δ ] T ( (P A e + A T e P ) + ϑnL 2 δ I 2 ) ⊗ I n (P [ 0 1 ] ⊗ I n ) T -ϑ ⊗ I n [ e δ ] , < -V (|e| 2 + |δ(x, x, u)| 2 ), (11)
where denotes symmetric entries, δ is a shortcut notation for δ(x, x, u) and where the last inequality holds for some small enough V > 0 from combining the strict inequality in (6) with standard properties of the Kronecker product. Since V is quadratic in e, with (Khalil, 2002, Thm 4.10), this implies exponential stability of the origin for the error dynamics (4).

Remark 4 Since A T e P + P A e < 0, there exist ϑ > 0 and ≥ 1 such that (6) holds. It should be noted that ( 6) is more conservative than the worst case necessary condition for obtaining V < 0 in (11). This, in light of bound ( 5) can be written as

n i=1 e T i (P A e + A T e P )e i + 2 √ nL δ |e T i P [ 0 1 ] ||e| < 0 for all e = 0.
The important advantage of (6) with respect to the above necessary condition is that it is quasi-convex with respect to the unknown variables P > 0, , ϑ and thus defines a quasiconvex domain. It is thus natural to compute the optimal solution to min P >0, ,ϑ , subject to (6), ( 12) so that one then implements the smallest possible gain guaranteeing global exponential stability (GES) of the origin for the error dynamics. The degree of conservativeness of ( 6) with respect to the above mentioned necessary condition is easily understood from the proof of Proposition 1, which is based on an application of the S-procedure (a necessary and sufficient condition) applied to bound (8). Therefore ( 6) is necessary under the worst case condition (8), which is in general more conservative than ( 5), even though it is equivalent to (5) in the case n = 1. For n > 1, the increased conservativeness allows to formulate the quasi-convex optimization problem ( 12).

Example 1 To motivate our work, we consider the following example, inspired by [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF], Example 3) (function φ(•, •) is perfectly known here while it is uncertain in [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]) and corresponding to the case n = 1 so that the state

x = x 1 = (p 1 , v 1 ) = (p, v) is two-dimensional: ṗ = v , v = a 1 sin(p) + b 1 u, (13)
where a 1 = 1.4 and b 1 = 0.8 are two constant values.

As explained in [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], this example illustrates the peaking phenomenon, and shows how high gain can destabilize the closed-loop system (see also (Sepulchre et al., 1997, Sections 1.3.2 and 4.5) where the consequences of peaking are studied). The high-gain observer (3) is designed with the parameter satisfying (6) based on the following condition, which is easily proven to hold with

L δ = 3 2 : |δ(x, x, u)| = |φ(x, u) -φ(x, u)| = |a 1 (sin(p) -sin(p)|, ≤ a 1 |p -p| ≤ 3 2 |p -p| .
Using k 1 = k 2 = 1, we may check that the assumption of Proposition 1 is satisfied with P = 1 -0.5 -0.5 1 and that (6) holds with ϑ = 2 and = 20 (the high gain = 20 is also used 3 in [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]). Therefore the conclusion of Proposition 1 holds and the scaled error system (4) is exponentially stable. Note that exponential stability of ( 4) is guaranteed in this example by using the quadratic Lyapunov function V (e) = e T P e.

To illustrate the peaking phenomenon on simulations, let us consider (as in [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]) the initial condition (p(0), v(0), p(0), v(0)) = (1, 0, 0, 0). The phase portraits of (e p , e v ) and (p -p, v -v) = (e p , e v ) are given in Figure 1. We may check on this simulation that the choice of the high-gain = 20 induces a peaking phenomenon for the error coordinates x -x (right plot) but not for the scaled error coordinates e (left plot).

3 It corresponds to the value ε = 0.05 using the notation of [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]. It is also interesting to observe that the quasi-convex GEVP in (12) leads to the less conservative estimate * = 3.94. The undesirable peaking phenomenon illustrated by Example 1 (see Figure 1) can be understood well in terms of the trend of the norm of the estimation error |x -x| = epi evi . A coarse understanding of the trend of this error can be gained by neglecting the nonlinear terms in the error dynamics (4), because these are dominated by the high-gain linear dynamics (see Proposition 1). Then, neglecting those nonlinear terms, one obtains:

d dt |x -x| 2 = -2 (k 1 e 2 pi + ( 2 k 2 -1)e pi e vi ),
which indicates that the flow dynamics (4) naturally enforces non increase of the norm of the estimation error in the quadrants e pi e vi ≥ 0, whereas the peaking effect (increase of |x-x|) arises in the second and fourth quadrants where e pi e vi ≤ 0. Thus one would expect that a reasonable thing to do when e belongs to the second (or fourth) quadrant is to change the sign of e pi , thus jumping on the opposite side of the vertical axis so that the size of e vi will stop increasing (see also Figure 2). The following result substantiates this intuition.

Lemma 1 Consider the matrices in (4) and assume that (6) holds for some P, ϑ. Given any e satisfying e pk e vk ≤ 0 for some k ∈ {1, . . . , n}, define e + as the vector with the following components:

e + i = e i , if i ∈ {1, . . . , n} \ {k} (-αe pi , e vi ), if i = k, (14)
where α ∈ [0, 1] is a real number. Then the function V defined in (7) satisfies V (e + ) ≤ V (e).

Proof. Since inequality (6) implies A T e P +P A e < 0, then, from the structure of P and A e we have that the elements of P := [ p1 p12

p12 p2 ] satisfy:

A T e P + P A e = # # # 2p 12 < 0,
where # denotes "don't care" entries. Since the diagonal terms must be negative, we have p 12 < 0. Consider now the equation

(e + k ) T P e + k -e T k P e k = e T k (α 2 -1)p 1 -(1 + α)p 12 0 e k , (15)
where denotes symmetric terms. The term (1, 1) in ( 15) is negative and the term (1, 2) is positive. Therefore as long as e pk e vk ≤ 0, it holds (e + k ) T P e + k -e T k P e k ≤ 0 which concludes the proof of Lemma 1.

Hybrid observer equations and properties of solutions

In this section we inherit the continuous-time dynamics (3) of the high-gain observer and introduce suitable resetting laws for the estimate x in such a way to reduce the peaking along the intuitions provided by Lemma 1. This is formally done by considering the following hybrid system and by denoting the overall state

ξ = (x, x, ζ, η) ∈ R 2n × R 2n × R n × R n : ξ ∈ C,        ẋi = f i (x, e pi , u) ζi = 0 ηi = (y i -ŷi ) 2 , (16a)
ξ ∈ D p i ,        x+ i = xi ζ + i = (y i -ŷi ) 2 η + i = 0, (16b) ξ ∈ D m i ,              x+ i = y i -α(p i -y i ) vi ζ + i = α 2 (y i -ŷi ) 2 η + i = 0, (16c)
where i = 1, . . . , n and in the jump rules the unspecified states remain constant across jumps and α ∈ [0, 1]. The flow and jump sets C and D are defined as:

C :={ξ ∈ R 6n : -∆ m ≤ e 2 pi -ζ i + 2 k 1 η i ≤ ∆ p , ∀i}, D := n i=1 D p i ∪ D m i , D p i :={ξ ∈ R 6n : e 2 pi -ζ i + 2 k 1 η i ≥ ∆ p }, D m i :={ξ ∈ R 6n : e 2 pi -ζ i + 2 k 1 η i ≤ -∆ m }, (16d) where ζ = (ζ 1 , . . . , ζ n ), η = (η 1 , . . . , η n ) and ∆ p > 0, ∆ m > 0.
Using the notation in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF], the hybrid dynamics should be interpreted taking the jump set as the union of all the jump sets D m i , D p i (as specified in (16d) and a set-valued jump map corresponding to the outer semicontinuous hull (see (Rockafellar and Wets, 1998, pp. 154-155)) of the 2n jump maps in (16b)-(16c), which may be written as follows:

G(ξ) := i: ξ∈D p i g p i (ξ) ∪ i: ξ∈D m i g m i (ξ) ,
where g p i and g m i , i = 1, . . . , n, denote the functions specified in (16b) and (16c), respectively, augmented with trivial identity functions for the remaining components of the states (see also [START_REF] Phillips | Robust asymptotic stability of desynchronization in impulse-coupled oscillators[END_REF] for this type of jump map constructions with outer semicontinuous hulls). Then when any of the i-th jumps in (16b), (16c) is enabled, the solution jumps following that specific jump rule for component i while the other components remain unchanged across the jump. Then, the overall hybrid closed loop between plant (1) and observer ( 16) is understood as the combination of the flow and jump dynamics in ( 16) with the flow dynamics in (1) and the trivial (omitted) jump dynamics associated to the states that remain constant among jumps. For example, one always has x + = x, resembling the fact that the (continuous-time) plant state x never jumps. Note also that the proposed jump rules all imply ζ + i = (e + pi ) 2 and η + i = 0, so that (e

+ pi ) 2 -ζ + i + k 1 η + i = 0 so that ξ + / ∈ D p i ∪ D m i .
The following proposition establishes well behavedness of the proposed hybrid observer in terms of number of jumps. In particular, since we are interested in continuous-time observation laws, the proposition ensures that all solutions satisfy an average dwell-time condition so that no Zeno solutions may occur and all maximal solutions are complete and have unbounded time domain in the ordinary time direction. Its proof is an extension of the proof of our main result and is therefore given at the end of the next section.

Proposition 2 Consider the hybrid closed loop between plant (1) and observer ( 16). If (6) holds for some P and ϑ, then all maximal solutions are complete and each solution has a hybrid domain satisfying an average dwell-time condition.

Remark 5 Hybrid observer ( 16) produces a hybrid estimate x of the continuous plant state x. By Proposition 2, this estimate x, whose domain dom x is a hybrid time domain, can be projected on the ordinary time axis t, thereby obtaining a discontinuous function of time whose value at the jump instants may be selected in multiple ways depending on the type of projection method (for example imposing that the function be left-or right-continuous). Note that the speed estimates vi , i = 1, . . . , n are continuous functions of the ordinary time, indeed from (16b) and (16c) it is clear that vi does not change across jumps. However, the position estimates pi do experience discontinuities across jumps occurring from D m i (see (16c)), therefore these estimates are discontinuous functions of the ordinary time t. In our preliminary work [START_REF] Prieur | Hybrid highgain observers without peaking for planar nonlinear systems[END_REF], §IV), we proposed an extension of observer ( 16) that makes suitable use of a hybrid logic state q toggling between -1 and 1 to obtain a continuous estimate of the position. We avoid repeating here the same type of generalization that can be obtained by following the steps in (Prieur et al., 2012, §IV). We emphasize, in addition, that the toggling therein suggested is such that when q = -1 one obtains a direct feedthrough term from the output y to the state estimate x (see (Prieur et al., 2012, eqn (15))). Due to this reason, and since we insist here on the advantages arising from the use of a strictly proper observer as compared to an observer with feedthrough terms (see the discussion at the end of Remark 3), we promote the observer dynamics ( 16), which is an enhanced generalization of the preliminary observation law proposed in (Prieur et al., 2012, §III).

Proof of convergence of the error dynamics

In hybrid observer ( 16), for each i ∈ {1, . . . , n}, as long as ξ ∈ D p i , we implement the resetting law suggested by Lemma 1 and corresponding to (14). Indeed, note that the first equation in ( 16c) implies e + pi = y + i -p + i = -αe pi .

In addition, to suitably define the jump and flow sets, hybrid observer ( 16) implements two extra states ζ i and η i . The state ζ i remains constant during flows and, at each jump time, jumps to the value of the i-th squared output error immediately after the jump, namely (e + pi ) 2 (this fact can be easily seen by inspecting (16b) and (16c)). This means that ζ i acts like a sample and hold of the value of e 2 pi immediately after the latest jump. The state η i , instead, is reset at zero at each jump time and then integrates the i-th squared output error e 2 pi during flows. Finally, ∆ p > 0 and ∆ m > 0 are (constant) design parameters, which can be tuned by bearing in mind that smaller values of ∆ m lead to increased observer jumps (and reduced peaking) of the hybrid observer dynamics, while large values of ∆ m induce less resets and possibly no resets at all for very large values.

Based on the above observation, given any hybrid solution ξ to the hybrid closed-loop (1), ( 16), denoting (t j ) the sequence in R such that (t j , j) and (t j , j+1) ∈ dom ξ, and any (t, j) ∈ dom ξ such that, for some i ∈ {1, . . . , n}, η i (t j , j) = 0 and ζ i (t j , j) = e pi (t j , j), we have the following property:

ζ i (t, j) = e 2 pi (t j , j) , η i (t, j) = t tj e 2 pi (τ, j)dτ, (17)
for all i = 1, . . . , n. The next lemma is the main motivation for the introduction of the extra states.

Lemma 2 Consider any solution ξ to the hybrid closed loop between plant (1) and observer ( 16) starting from initial conditions satisfying η(0, 0) = 0 and ζ(0, 0) = e p (0, 0). For each (t, j) ∈ dom ξ, if ξ(t, j) ∈ D m i for some i ∈ {1, . . . , n}, then e pi (t, j)e vi (t, j) ≤ 0.

The following simple claim will be useful for the proof of Lemma 2.

Claim 1 Given any two non-negative reals satisfying 0 ≤ t j < t, a scalar δ ∈ R and a continuous function ϕ(s), s ∈ [t j , t], assume that:

t tj ϕ(τ )dτ ≤ δ, and δ ≤ s tj ϕ(τ )dτ, ∀s ∈ [t j , t), (18) then ϕ(t) ≤ 0.
Proof. First note that t tj ϕ(τ )dτ = lim s→t s tj ϕ(τ )dτ and thus it follows from (18) that t tj ϕ(τ )dτ = δ. Moreover defining the continuously differentiable function Ψ by Ψ : s ∈ [t j , t) → s tj ϕ(τ )dτ , it follows from the second inequality in (18) that Ψ(t) ≤ Ψ(s), for all s in [t j , t). Considering the derivative of Ψ at t, this inequality implies ϕ(t) ≤ 0.

Proof of Lemma 2. For all (t, j) ≥ (t j , j) ∈ dom ξ, s → ξ(s, j) is absolutely continuous and in particular (4) makes sense for all almost all s ∈ [t j , t). By multiplying the first equation in (4) (with i = k) by 2e pi , we get 2e pi ėpi = 2 e pi e vi -2 k 1 e 2 pi , which can be integrated between t j and s ∈ [t j , t] to get e 2 pi (s, j) -e 2 pi (t j , j) = where ϕ(τ ) = 2 e pi (τ, j)e vi (τ, j). Using the identities in (17) (which hold for j ≥ 1), this implies:

s tj ϕ(τ )dτ = e 2 pi (s, j) -ζ i (s, j) + 2 k 1 η i (s, j). ( 19
)
Since we also have ξ(s, j) ∈ C for all s ∈ [t j , t) (otherwise the solution could not flow in this interval), then using the definition of flow set in (16d) and the assumption of the lemma, we have that which, combined with (19), implies equation ( 18) with δ = -∆ m and then by Claim 1 we have ϕ(t) = 2 e pi (t, j)e vi (t, j) ≤ 0, which completes the proof.

e 2 pi (s, j) -ζ i (s, j) + 2 k 1 η i (s, j) ≥ -∆ m , ∀s ∈ [t j , t) e 2 pi (t, j) -ζ i (t, j) + 2 k 1 η i (t, j) ≤ -∆ m ,
In light of Lemma 2, the rationale behind the hybrid observer dynamics ( 16) can now be explained. As long as the solution ξ belongs to the flow set, the observer dynamics coincides with that of the continuous-time highgain observer. However, the extra variables ζ and η keep track of the quantities in ( 17) for each i ∈ {1, . . . , n}. These quantities are used in the jump/flow rules of (16d) to detect (based on Lemma 2) whether a substate e i belongs to the second/fourth quadrant. In particular, once we detect ξ ∈ D m i for some i, then from the last equation in (16d) we have e 2 pi -ζ i + 2 k 1 ξ i ≤ -∆ m and Lemma 2 applies. Such an event triggers the jump rule in (16c) which implies that e + i = (-αe pi , e vi ) and then non-increase of the Lyapunov function follows from Lemma 1. This result is formalized in the next statement which is the main result of this paper.

Theorem 1 Consider plant (1) satisfying ( 2), and observer ( 16). Assume that (6) holds for some P and ϑ. Then there exist positive scalars M, λ such that the estimation error e = x -x satisfies

|e(t, j)| ≤ M exp(-λt)|e(0, 0)|, ∀(t, j) ∈ dom e, ( 20
)
for any solution ξ to the closed loop between plant (1) and observer ( 16) starting from initial conditions satisfying η(0, 0) = 0 and ζ(0, 0) = e p (0, 0).

Remark 6 Due to the expressions of the flow set, it yields that (

x, x, ζ, η) is in C if and only if -∆ m -e 2 pi + ζ i 2 k i ≤ η i ≤ ∆ p -e 2 pi + ζ i 2 k i , ∀i .
In light of Proposition 2 (ensuring that all maximal solutions have unbounded domain in the ordinary time direction t), Theorem 1 states the exponential convergence to zero of e pi . Since along flow ζ i are constant, this implies that η i are also bounded along flow. With the expression of the jump map in (16a)-(16c), it yields boundedness of the variables ζ and η along the solutions to the hybrid system.

Proof of Theorem 1. Consider the Lyapunov function V (e) = e T (P ⊗ I n )e as in ( 7), which satisfies:

a 1 |e| 2 ≤ V (e) ≤ a 2 |e| 2 , ∀e ∈ R 2n , (21)
for some positive scalars a 1 , a 2 . Moreover, since the flow dynamics in (16a) coincides with the continuous-time high-gain observer dynamics, one has from Proposition 6 (see in particular (11) and the scalar V therein introduced):

V (ξ) ≤ -V |e| 2 ≤ - V a 2 V (e), ∀ξ ∈ C. (22)
Regarding the change of V across jumps, notice that from (16b) ξ ∈ D p i leads to e + = x + -x+ = x -x = e, which means V (e + ) -V (e) = 0, ∀ξ ∈ D p i . As for D m i we need to specifically look into solutions and apply Lemmas 1 and 2. In particular, for any solution ξ(t, j), (t, j) ∈ dom ξ, if ξ(t j , j -1) ∈ D m i for some jump time t j , then we have from Lemma 2 that e pi (t j , j -1)e vi (t j , j -1) ≤ 0, which, from Lemma 1 implies V (e + (t j , j -1)) = V (e(t j , j)) ≤ V (e + (t j , j -1)). More specifically, we have for all (t h , h) ∈ dom ξ,

V (e(t h , h)) -V (e(t h , h -1)) ≤ 0. ( 23
)
Consider now a hybrid time (t, j) ∈ dom ξ. and, for simplicity, let us denote t j+1 = t (even though t j+1 is not necessarily a jump time). Along any flowing interval [t h , t h+1 ], h ∈ {0, . . . , j}, we know that the solution belongs to C (because C is closed) and then we may use ( 22) to characterize the continuous variation of V as follows:

d dτ V (e(τ, h)) ≤ - V a 2 V (e(τ, h)), ∀τ ∈ (t h , t h+1 ). ( 24
)
Using standard comparison theory and the fact that V is continuous along flow, we may transform (24) into an exponential bound. Then, concatenating all the bounds from h = j to h = 0 and exploiting recursively (23), we get:

V (e(t, j)) ≤ exp - V a 2 (t -t j ) V (e(t j , j)) ≤ exp - V a 2 (t -t j ) V (e(t j , j -1)) ≤ ... ≤ ≤ exp - V a 2 (t -t 0 ) V (e(t 0 , 0)),
where t 0 = 0 by definition. Finally, using (21) we get

|e(t, j)| 2 ≤ 1 a 1 V (e(t, j)) ≤ 1 a 1 exp - V a 2 t V (e(0, 0)) ≤ a 2 a 1 exp - V a 2 t |e(0, 0)| 2 , (25)
completing the proof with λ = V 2a2 and M = a2 a1 .

Proof of Proposition 2. To show that all (maximal) solutions are complete, let us first note that the Viability Condition (VC) of (Goebel et al., 2012, Prop 6.10) holds and that C ∪ D = R 6n . Moreover, using ( 22) and ( 23), we obtain that function V is nonincreasing along solutions, therefore solutions cannot escape to infinity in finite time. Therefore, with (Goebel et al., 2012, Prop 6.10), all maximal solutions are complete. Moreover it follows from (25) that

|e(t, j)| ≤ a 2 a 1 |e(0, 0)|, ∀(t, j) ∈ dom e. (26)
Then, using (4) and ( 5), we get for almost every t and for all j such that (t, j) ∈ dom e, | ė(t, j)| ≤ |A e ⊗ I n | a2 a1 + nL δ |e(0, 0)| =: M e |e(0, 0)|. Based on the definition of jump set in (16d), we know that the solution does not jump as long as

θ i = |e 2 pi -ζ i +2 k 1 η i | < min{∆ p , ∆ m } =:
δ, for all i = 1, . . . , n. This boundedness property, together with the fact that C ∪ D spans the whole space, implies that solutions can always be continued and then maximal solutions are complete.

Consider now identities (17) and conclude that immediately after each jump time t j with the jump triggered by D p i ∪ D m i , we have θ i (t j , j) = 0. Moreover, using ( 17) and ( 26) we have

| θ| ≤ 4| ė||e|+2 k 1 |e| 2 ≤ 4M e a2 a1 + 2 k 1 a2 a1 |e(0, 0)| 2 =: M θ .
Therefore, for each t ∈ [t j , t j + δ/M θ ), such that (t, h) ∈ dom e for some h ≥ j, we have θ i (t, h) ≤ δ, which implies by definition of δ that ξ(t, h) / ∈ D p i ∪ D m i . Therefore any pair of hybrid times such that the state is in the same jump subset D p i ∪ D m i should be separated by at least δ/M θ flowing time, for each i ∈ {1, . . . , n}. Since there are n instances of subsets D p i ∪ D m i , we conclude that each solution jumps at most n times in any ordinary time interval of length δ/M θ . In other words, following the characterization of (Cai et al., 2008, Prop. 1.1) all solutions to the hybrid observer closed loop satisfy an average dwell-time condition with parameters (M θ /δ, n).

Example 2 Let us reconsider Example 1 and compare the classical high-gain solution with the hybrid solution having the same parameters and with the selection ∆ p = ∆ m = 0.01 and α = 0.5. Starting from the same initial condition as in Example 1, we obtain the phase portraits of (e 1 , e 2 ) and (x 1 -x1 , x 2 -x2 ) given in Figure 2.

By comparing Figures 1 and2, we note that the highgain reduces dramatically the peaking experienced in the linear case. The Lyapunov function V (e) = e T P e decreases exponentially fast to zero when flowing, while it does not increase across jumps, as established, respectively, in ( 22) and ( 23) in the proof of Theorem 1.

In particular, Figure 3 shows the ordinary time evolution of this Lyapunov function for various selections of α within the allowable range. From the top plot it is apparent that smaller values of α lead to a more substantial decrease of the Lyapunov function across jumps (despite the fact that only non-increase is guaranteed by our results). On the other hand, comparable transients are experienced on the norm squared of the unscaled error (x 1 -x1 , x 2 -x2 ) (bottom plot), thus showing that, regardless of the selection of α, the hybrid high-gain ob- server shows a tremendous improvement on the peaking of the standard high-gain solution (the dashed curve in the lower plot reaches a peak value of 11.9). For this specific simulation, the selection α = 0.25 appears to be the most desirable one, and the selection α = 0 surprisingly leads to just a few resets.

Remark 7 As illustrated in Example 2 below, the hybrid high-gain observer proposed here is capable of suppressing the peaking behavior, as illustrated in Figure 3, where the estimation error |x -x| is represented. When ∆ p and ∆ m are selected sufficiently small as compared to the period of the revolving trajectories, once the error enters the "peaking-prone" region e pi e vi ≤ 0, the resetting rule in (16c) changes the sign of e pi , thereby transferring the trajectory to the good region e pi e vi ≥ 0 where the error is decreasing again. Since the sign of e pi e vi is detected by our algorithm based on an integral inequality, then some increase of the norm of the error should be expected (see e.g., the black curves in the bottom plot of Figure 3) but this increase is expected to be significantly smaller than the one experienced from peaking (the dashed curve in the bottom plot of Figure 3 reaches a peak of 11.9).

3 Application to a planar robot example

A relevant class of systems that falls into the setting of equation ( 1) is that of fully actuated Euler-Lagrange systems (see, e.g., [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF]; [START_REF] Sciavicco | Modelling and control of robot manipulators[END_REF]). Denoting by q the generalized positions and by q ∈ R 2 the generalized velocities, the equations of motion of these systems can be compactly represented as (see, e.g., [START_REF] Sciavicco | Modelling and control of robot manipulators[END_REF], §7.1))

D(q)q + C(q, q) q + h(q) = u, (27)
where D(q) is the inertia matrix, which satisfies d min I ≤ D(q) ≤ d max I for all q and for a suitable selection of positive values d min , d max , C(q, q) q comprises centrifugal and Coriolis terms (namely quadratic terms in q), h(q) is the gravitational vector, and u represents the external forces/torques applied on each degree of freedom of the system.

Selecting y = p = q and v = q, model ( 27) can be represented as in (1) with selection ψ(y) = 0 and

φ((p, v), u) = D -1 (q) (u -C(p, v)v -h(p)) . (28)
While we may apply our technique to any such Euler-Lagrange system, we concentrate on a planar two-link rigid robot manipulator with rotational joints, used in many existing works (see, e.g., (Sciavicco and Siciliano, 2000, §7.3.2)). We use the parameters used in [START_REF] Morabito | Measuring and improving performance in anti-windup laws for robot manipulators[END_REF].

A typical control selection for robotic systems is the feedback linearizing controller, which can be implemented as follows in feedback from the measured position and the observed velocity:

u = C(p, v)v + h(p) -D(p)(K p p + K d v), (29)
where we select K p = [ 20 0 0 20 ] and K d = [ 20 0 0 10 ] to induce a desirable and graceful response of the closed-loop system via state feedback (see the top curves in Figure 4). To describe a practical scenario, we carry out simulations by taking into account a disturbance acting on the position measurement p, corresponding to the presence of a quantizer, possibly arising from the use of incremental encoders that measure the joint positions. The quantizer level is chosen as 2π/3600, namely we assume that the encoder has 10 steps per degree. Since it is known that high-gain observers are sensitive to measurement noise, in addition to the quantization level, we add a band limited white noise to the position measurements (after the quantization) having sample time 0.01 and noise power 10 -7 , so that the sensitivity to noise is comparatively assessed. 4 As customary in robotic applications, rather than relying on the conservative bound on arising from Proposition 1, we tune parameter with the goal of obtaining a suitable filtering action for the quantization noise, thereby obtaining the value = 5. We still implement the reduced order observer described in Remark 3 by computing a matrix P e from (6) by maximizing the allowable L δ for this specific value of (similar to Remark 4, this is a generalized eigenvalue problem). We also select ∆ p = ∆ m = 1e -3 so that resets are suitably induced during the transient response of the hybrid observer.

Figure 4 shows the simulation results starting from the initial condition p(0) = (50, 0) degrees and v(0) = (50, 50) degrees/sec. The peaking occurring with the bare high-gain solution is quite evident from the left graph on the third row, while this peaking is removed by both the hybrid high-gain solution and the reduced order solution of the second and fourth rows. Interestingly, most resets of the hybrid observer occur during the initial transient (when peaking must be removed) and only a few resets occur in the remaining part of the response (see the last row of the figure). This reveals that after the initial transient, the hybrid solution essentially produces responses coinciding with those of the high-gain solution (essentially without resets, except for seldom ones caused by noise). The advantage of the hybrid solution versus the reduced order one is evident at the steady state (right graphs in Figure 4), where unpleasant bursts can be inspected in the curves of the second row, corresponding to the reduced order solution. These bursts arise from the fact that the high-gain and hybrid high-gain solutions are both strictly proper observers, thereby providing a filtering action that is impossible to obtain from the reduced order approach, where the position measurement is directly fed to the input of the controller. 5 No disturbance is used in the state feedback simulation (upper curves) to better illustrate the desirable features of the (ideal) undisturbed state feedback response. Note also that in the presence of noise having larger amplitude, one expects a graceful performance degradation of the proposed scheme due to the semiglobal practical stability results proven in (Goebel et al., 2012, Thms 7.20 & 7.21), which hold for our solution because it satisfies the hybrid basic conditions by virtue of the regularity of the data in our solution.

Conclusion

Considering a family of nonlinearly interconnected second order plants, the high-gain observer design has been revised within a hybrid dynamical systems framework, with the objective to avoid the peaking phenomenon.

A hybrid high-gain observer has been proposed by augmenting the original continuous-time high-gain observer dynamics with a suitable resetting rule relying on the observer state. This approach has been illustrated on an example borrowed from the literature, and also by means of simulations on a planar robot. The hybrid high-gain observer is also interesting when compared to a reduced order observer, since both observers reduce peaking, but the hybrid high-gain observer exhibits a few number of jumps and avoids bursts of the input, that are present when closing the loop with a reduced order observer.

Figure 1 .

 1 Figure 1. Phase portrait of the coordinates (ep, ev) (left) and of (p -p, v -v) (right).

Figure 2 .Figure 3 .

 23 Figure 2. Phase portrait of the coordinates (e1, e2) (left) and of (x1 -x1, x2 -x2) (right), with α = 0.5.

Figure 4 .

 4 Figure 4. Output responses for the two link robot arm: state feedback (top), classical high-gain (middle-top), hybrid high--gain (middle-bottom), Remark 3 (bottom).

The assumption on piecewise continuity of input u could be relaxed but special care has to be paid to the hybridization of the corresponding function of time (see, e.g.,[START_REF] Sanfelice | Input-output-to-state stability tools for hybrid systems and their interconnections[END_REF] for details).

We realize that the presence of noise after a quantizer is not physically meaningful but we expressively carry out this test to verify the observers responses in a worst case scenario.

Maybe the bursts, obtained with the reduced order observer, may be avoided by adding an appropriate one-order filter to recover the same dimension of the dynamics as the the hybrid observer. However the tuning of this filter may be problematic.


Provensal

Caroline Zanetti
email: c.zanetti@arbeausolutions.fr

Michel Vennetier

Patrice Mériaux

Mireille Provansal

Plasticity of tree root system structure in contrasting soil materials and environmental conditions

Keywords: tree root system structure, root number, root size, root distribution, trees on dikes

-Introduction

Assessing the structure of tree root systems (root number, size and distribution, root system span, depth and volume) is a difficult task, and their relationship with environmental conditions has been poorly investigated for large adult trees. For many species and environments, research was limited to root biomass distribution according to the distance from the stump in depth or laterally [START_REF] Plourde | Spatial distribution, architecture, and development of the root system of Pinus banksiana Lamb. in natural and planted stands[END_REF]. Indeed, modelling root systems is not easy, because of their complexity and of the multiple parameters influencing their development and structure. However, some models have been developed and implemented for cultivated plants and fruit trees [START_REF] Jourdan | Architecture and development of the oil-palm (Elaeis guineensis Jacq) root system[END_REF][START_REF] Vercambre | Architectural analysis and synthesis of the plum tree root system in an orchard using a quantitative modelling approach[END_REF] because roots can be fully extracted and measured in agricultural soils. Such an extraction is far more difficult for large trees because of the span and depth of their root systems [START_REF] Canadell | Maximum rooting depth of vegetation types at the global scale[END_REF][START_REF] Stone | On the maximum extent of tree roots[END_REF], except in very favourable conditions, such as deep, homogeneous soils with a fine texture (Danjon et al., 2005;[START_REF] Vyskot | Tree Story Biomass in Lowland Forests in South Moravia[END_REF]. This is why in most cases, studies on tree root system architecture deal with small and young trees. One of the more thorough investigations of root system development for many forest tree species in various environments was performed by [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF] but data on root system global structure according to soil material are scarce. We particularly lack descriptions of root systems on coarse materials [START_REF] Devine | Root system morphology of Oregon white oak on a glacial outwash soil[END_REF]. Tree root system and its development processes are partly known by studies carried out in specific contexts, for example in planted forests [START_REF] Plourde | Spatial distribution, architecture, and development of the root system of Pinus banksiana Lamb. in natural and planted stands[END_REF], in agroforestry systems [START_REF] Akinnifesi | Below-ground dynamics in agroforestry systems[END_REF], after storms or avalanches on uprooted systems (Danjon et al., 2005;Nicoll et al., 2006b), in sand dunes [START_REF] Wagner | Cross-sectional interpolation of annual rings within a 3D root model[END_REF] or alluvial environment [START_REF] Rood | Root architecture of riparian trees: river cut-banks provide natural hydraulic excavation, revealing that cottonwoods are facultative phreatophytes[END_REF], and on unstable or steep slopes [START_REF] Reubens | The role of fine and coarse roots in shallow slope stability and soil erosion control with a focus on root system architecture: a review[END_REF]. Some specific methods and tools were progressively developed to accurately describe and model roots and root systems [START_REF] Armengaud | EZ-Rhizo: integrated software for the fast and accurate measurement of root system architecture[END_REF][START_REF] Danjon | Assessing and analyzing 3D architecture of woody root systems, a review of methods and applications in tree and soil stability, resource acquisition and allocation[END_REF][START_REF] Tobin | Towards developmental modelling of tree root systems[END_REF].

Tree root system classification is generally defined by its global structure [START_REF] Atger | Premières données sur l'architecture comparée des systèmes racinaires et caulinaires[END_REF] and its adaptation to the environment [START_REF] Fitter | Architectural analysis of plant root systems 2. Influence of nutrient supply on architecture in contrasting plant species[END_REF][START_REF] Fitter | Architecture and biomass allocation as components of the plastic response of root system to soil heterogeneity[END_REF][START_REF] Foussadier | Les systèmes racinaires des arbres de la ripisylve : effets des contraintes physiquues et exemples[END_REF]. Root systems are differentiated according to their morphology and main types of roots [START_REF] Sutton | Root and root system terminology[END_REF]. The three most used groups [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF] are: (i) the heart root system, which roots occupy the whole space around the stump, with no preferential root angle, (ii) the plate-shape root system, also called the shallow root system, in which all roots remain superficial, and (iii) the taproot root system in which one single or a few large vertical roots dominate. Mixed root systems combining plate-shape and taproot systems are frequent, with two dominant root angles (vertical and horizontal) and no or few oblique roots.

Root system structure is generally considered as a trade-off between the genetic potential of a species and its plasticity in allowing adaptation to local environment [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF]. Variations in soil conditions are known to drive root growth at local scale [START_REF] Desnos | Root branching responses to phosphate and nitrate[END_REF][START_REF] George | Responses of Picea, Pinus and Pseudotsuga roots to heterogeneous nutrient distribution in soil[END_REF][START_REF] Soar | The effect of changing patterns in soil-moisture availability on grapevine root distribution, and viticultural implications for converting full-cover irrigation into a point-source irrigation system[END_REF], as well as at root system scale [START_REF] Schenk | Mapping the global distribution of deep roots in relation to climate and soil characteristics[END_REF]. Therefore, strong soil constraints should result in at least partly similar root systems structure for various tree species.

Hundreds of thousands of km of dikes are found along rivers and canals throughout the world, in addition to countless earth dams built for irrigation, flood control, fishing, tourism, landscape or electricity. The presence of mature trees and even forest stands on such fills is common. Very few scientific articles have been published on tree root development in earth dikes or dams [START_REF] Corcoran | The Effects of Woody Vegetation on Levee Performance[END_REF][START_REF] Haselsteiner | Woody Vegetation on Small Embankment Dams[END_REF][START_REF] Shields | Effects of woody vegetation on the structural integrity of sandy levees[END_REF]. Recently, dike managers in France and other countries started removing these trees, giving us the opportunity to study a large number of tree root systems in contrasting environments.

This study aimed at comparing the structure and development of tree root system according to two main soil materials (fine and coarse), taking advantage of a study of root system structure and growth in earth dikes and dams [START_REF] Zanetti | Characterization of tree root systems in earth dikes[END_REF]. The goal was to relate root system structure to biological and physical parameters: species, tree age and size, water availability, soil material, slope angle and climate. We hypothesized that due to the contrasting physical constraints between the two types of soil, particularly for water availability and nutrients, tree root systems would adapt their morphology to environmental constraints independently from tree species and that resulting root systems would be very different in terms of root size, number and distribution for a given species.

-Material and methods

-Study Sites

Study sites (Fig. 1) were distributed along a wide range of climates from dry and hot Mediterranean plains to humid and cold mountains (Table 1). Access to water resources for trees growing on the dikes varied with fill size and with the distance between the stump and the canal, river or dike toe. In this study, canal dikes and dams with a permanent water head were twice to three times larger than flood protection dikes.

Figure 1: Study sites (rivers) and number of trees per site Table 1 : Characteristics or the 8 study sites

Dikes or their sub-sections were split into two groups according to fill material, coarse and fine. Coarse materials (dominant particle size between 63 mm to 0.63 mm) correspond to the stony to gravely-sandy ones according to GTR NFP 11-300 norm, and fine materials (dominant particle size between 0.63 mm and 0.002 mm) include medium and fine sand, silt, and clay [START_REF] Zanetti | Characterization of tree root systems in earth dikes[END_REF].

Tree sampling design included five criteria: tree species, size, age, position on the dike or dam (crown, toe, lower, middle or upper slope), and soil material (coarse or fine). Twelve species frequently growing on European dikes were studied (table 1): oaks (Quercus ilex L. and Q. pubescens Willd.), maples (Acer campestris L. and A. negundo L.), ash (Fraxinus excelsior L.), poplars (Populus nigra L.and P. alba, L.), locust (Robinia pseudo-acacia L.), willow (Salix alba L.) and alder (Alnus glutinosa L.) in low elevation sites, larch (Larix decidua Mill.) and Scots pine (Pinus sylvestris L.) in mountains.

A total of 243 root systems of mature trees were extracted. Mature means that most of these trees were not far from the maximum height they can reach on studied sites. The study was divided into two phases. In the first, exploratory phase, 87 tree root systems were described in broad terms (global structure, span, size, direction and depth of large roots [proximal diameter > 5 cm], root number, distinctive features). In the main study phase, the remaining systems (156), were described in the same way together with accurate measures of individual roots.

The comparisons between the two types of material were computed only for 191 trees growing on homogeneous materials after exclusion of heterogeneous sites and root systems severely constrained by artificial obstacles (masonry, rockfill, lining or wall).

-Root system extraction

At the start of sampling, the tree characteristics were noted: tree base diameter (TBD), position on the fill, and side (river-canal side or landside). Then the selected trees were cut at one metre from the base to allow slinging them up.

Root system excavation was performed in several stages: (1) superficial roots were first free and bared, followed by progressively deeper roots, with a mechanical shovel and by hand, from the stump towards the periphery and when possible up to their tip (limit: 0.2 cm in diameter). Once the stump had been cleared from ground materials as much as possible, it was slowly lifted in a sling to observe remaining deep and vertical roots and finally fully uprooted. If some large roots remained in the soil at that stage, they were extracted immediately. This excavation method was used because of the great number and large size of studied trees. It was impossible in many cases to remove all material around root systems, particularly very coarse material, using water or compressed air. In many cases, we dug throughout the whole fill down to the foundation or to the natural soil, which allowed an accurate measurement of the position and distance from the stump to available resources such as a water table or buried organic layers. Tree age was obtained by counting tree rings on the stump, or on the main roots for coppiced trees.

-Root systems measurements

During the excavation process, the root structure was described and measured (root number, orientation, proximal diameter and length), photographed, and drawn. Proximal diameter is the initial diameter at root connection with the stump or another root. Root orientation was noted according to the slope: height sectors of 45 degrees each, four upwards and four downwards. Vertical angle was noted according to soil surface: parallel, oblique or vertical. When roots had to be cut during extraction, their position on the stump was spotted, their direction and angle were noted and they were kept aside for later measurement.

After the extraction, stumps were stored upside down. Root number was counted separately for the upward and downward side when trees were growing on a slope. We counted only root larger than 2 cm in diameter at a distance of 1 m from the stump periphery. This distance was chosen (1) to eliminate the zone of rapid taper [START_REF] Eis | Root system morphology of Western Hemlock, Western Red Cedar, and Douglas-Fir[END_REF], linked to buttresses and the many anastomosis present close to the stump of large trees, and (2) to take fast changes in root orientation into account, these changes being common in this first meter. Roots with a proximal diameter of more than 5 cm were considered as large roots. All vertical roots were counted and measured separately. Root system volume was computed according to the width and depth of coarse roots.

-Statistical analyses

Root proximal diameter (mean and maximum), root length and root number, were computed for each soil material, root system type, and species. Their correlation with tree base diameter (TBD) was computed. On each material, results were successively computed for all trees and for three classes of TBD, common to both soil materials: small, intermediate and medium size trees corresponding respectively to TBD of 25-35 cm, 35-45 cm and 45-55 cm. Only a few large trees (over 55 cm) were found on coarse material so it was not possible to make comparisons with fine material for this class. Because of low numbers in certain TBD classes for some species, multiple comparisons between species were possible only for small trees on coarse material and medium to large trees on fine material. The correlation between root length and root proximal diameter was computed only for entire roots. Root size and number per material, root system type, root orientation or species were compared by analysis of variance in the case of normal distributions, or a Kruskal-Wallis test followed by a Nemenyi test [START_REF] Nemenyi | Distribution-free multiple comparisons[END_REF] in the case of non-normal distributions. The number of roots per orientation (upwards/downwards) was compared with a pairwise nonparametric signed ranks test. Statistics on root diameter were computed with and without large vertical roots (taproots) to test the possible impact of this particular type of roots.

-Results

-Root system type according to soil material, species and position on the fill

Four root system types were found (Fig. 2): shallow (2a), heart (2b), mixed (2c) and tap root system (2d). While heart root systems were limited to fine material, mixed root systems developed mainly (96%) on coarse material. Shallow root systems were found on both materials. Only four tap root systems were observed, all of them on coarse material (Fig. 3).

The distribution of root system types by soil material for individual species or genera was consistent with the general trend (Table 2). When they were found on both materials, none of the studied species or genus developed a specific root system structure. Except for alder, which was always found very close to canals or rivers, all of them presented at least two out of the four root system types, and many of them (e.g. poplars, locust, oaks, willow and ash) had three to four types.

On large dikes made of coarse material, root systems were mainly of the shallow type, and less than 50 cm in depth, located on the crown and the upper part of the slope (75-95%). Among these shallow root systems, many had a short (<50 cm) and weak vertical root corresponding to a dead or aborted taproot. Root systems were mostly of mixed or taproot types in the middle and the lower part of the slope while all types except the heart root system were found at the dike or dam toe. On large dikes made of fine material, the heart root system type was found at all positions and the shallow root system was found only at the dike toe.

Zanetti C., Vennetier M., Mériaux, Provansal M. (in press). Plasticity of tree root system structure in contrasting soil materials and environmental conditions. Plant and Soil. On line 2014; DOI: 10.1007/s11104-014-2272-9

-5 -

-Number of roots per root system

The number of roots per root system depended on soil material and partly on tree base diameter (TBD) and root system type (Table 3a and Fig. 4). Taking all species and root system types together, roots were three to four times more numerous in fine material than in coarse material. The difference was significant for all TBD classes (P<0.001; table 3.a), and increased with TBD. In fine material, the root number was always over 80 for trees with TBD over 25 cm, and increased with TBD, reaching more than 300 for the largest trees. In coarse material, root number varied from 25 to 60, and no trend was found with TBD over 20 cm even for the largest trees. No differences in root number were found between root system types in fine material. Conversely, root number was higher for mixed and tap root systems than for shallow root systems in coarse material (P<0.01) for small and intermediate TBD classes. Within the same TBD classes, roots of shallow root systems were three times more numerous in fine than in coarse material (P<0.001), matching the general trend.

For a given type of material and for the same class of TBD, no significant differences in root number were found between species (Table 3b), except for locust which had a slightly and significantly higher number of roots than other species but only on coarse material.

-Root size

As a whole (Table 3c), although sampled trees were larger on fine material, the mean root diameter was significantly higher on coarse material (6.3 cm versus 3.7 cm; P<0.001). This difference was independent from TBD, as illustrated by the ratios between mean or largest root diameter and TBD, which were two to three times higher for coarse material. In fine material, mean root diameter did not increase with TBD over 25 cm, never exceeding 5 cm (Fig. 5a, table 3.c). Conversely, it increased significantly following a logarithmic trend in coarse material (r² = 0.53, P<0.001), up to 9 cm for the largest trees. The diameter of the largest root by root system (Fig. 5b) was equal on both materials for small trees. It increased with TBD only for trees over 50 cm on fine material, far slower than on coarse material. Consequently, either for the mean or for the largest root diameter, the difference between coarse and fine material increased regularly with TBD and was significant over 30 and 40 cm respectively. We found no differences in the results with and without vertical roots, the average as well as the maximum root diameter being reduced by less than 5% when excluding vertical roots and in the same proportion for each material and species. Vennetier M., Mériaux, Provansal M. (in press). Plasticity of tree root system structure in contrasting soil materials and environmental conditions. Plant and Soil. On line 2014;DOI: 10.1007DOI: 10. /s11104-014-2272-9 -9 -7 -

The mean root diameter was not significantly different between root system types for a given material and similar TBD (Table 3d). It was significantly smaller for shallow root systems in fine material compared to coarse material, for small and intermediate TBD classes, varying consistently with the general difference between materials. The low numbers of mixed root systems on fine material did not allow a comparison with coarse material.

On coarse material, many poplars and locust, and some willows, developed a few very large roots per stump, sometimes out of proportion compared to the average of the system. The same trend could be found in fine material, but to a lesser extent. All roots over 30 cm in diameter were found on poplars stumps on coarse material. This trend is illustrated by the ratio of maximum/mean root diameter (Table 3b) which was 50% higher for poplars on coarse material and for poplars and locust on fine material. Other species did not present such a wide range of root diameter within a single root system whatever their size. Table 3b presents values from small trees on coarse material, and large trees on fine material, respectively, for which all species could be compared. Results are similar for intermediate size trees, but all species were not found in this size class on both materials (data not shown).

Roots were shorter in fine than in coarse material (average of 127 versus 166 cm, P<0.01). This difference was strongly related (Table 3e) to root diameter decreasing rate (DDR): it was highly significant for roots with DDR < 3.1 cm/m but not over this limit. No significant differences were found in length or decreasing rate between species or root system types, whatever the root type or size class. Taking all roots together, the ratio proximal root diameter/DDR was similar whatever the species (Fig. 6), material and root system type.

Table 3: Number and size of roots and root system volume according to root system type, soil material, species and TBD classes (Large

3.a: Number of roots per root system type and material for all trees and different TBD classes

3.b: Root number, mean and largest root diameter per root system according to species, for small trees on coarse material and for large trees on fine material. Values in bold are significantly different from others for a given material. -8 -

3.a

3.c: Mean root diameter per root system type and material for different TBD classes

3.d: Mean root diameter, root system volume and ratio mean and maximum root diameter / TBD for coarse and fine material

3.e: Mean root length (cm) and decreasing rate (cm/m) according to two types of roots and material. The probability is given for the difference between fine and coarse material. NS = non significant (P>0.05).

Figure 4: Relationship in bold line between root number per root system and tree base diameter according to soil material. Thin upper and lower lines = span of observations. Root number higher in fine material than in coarse material.

3.c

Mean

Figure 5: Relationship in bold lines between (5a) root system mean root diameter or (5b) diameter of root system largest root and tree base diameter according to soil material. Thin upper and lower lines = span of observations. Mean and Max root diameter higher in coarse material than in fine material. Figure 6: Relationship between root diameter decrease rate (cm/m) and root proximal diameter (cm) by species all roots together and by material (logarithm scale; -1 = 0.1 cm and 1= 10 cm). No significant differences between species nor between material

-Root system volume according to material and root system type

Root system volume varied from 3 to 8 m 3 for small trees and from 6 to 22 m 3 for the largest ones (Table 3c). Root system volume was only weakly correlated with TBD or tree age on the whole sample (r² = 0.3, P<0.01) due to the differences between root system types and material. The correlation with TBD was good for shallow root systems (r 2 = 0.80), and remained significant, although far weaker, for mixed and tap root systems (r² = 0.53) and heart root systems (r² = 0.47). For the same class of TBD, (30-35 cm) shallow root systems averaged about 50% of the volume of mixed root system (5.4 versus 10.5 m 3 , p=0.025); the same difference was observed when combining tap root and mixed root systems. Heart root systems had about the same mean volume as mixed root systems (10 and 10.8 m 3 resp.), although their average TBD was much larger (61 versus 34 cm). Thus heart root systems seemed to have a far smaller spatial development, consistent with their shorter roots. These differences between root system types may be linked to material, because heart root systems were found only on fine materials while mixed and tap root systems developed mainly on coarse material. Root system mean volumes were identical on fine and coarse materials (8.7 m 3 ), whereas the average TBD was by far larger on fine material (58 versus 42 cm).

For trees which had been coppiced once or several times, root system volume depended mainly on stump age and not on TBD for given material and species. However stump age was rarely known for trees coppiced several times and on old dikes, so no precise relation could be established.

Zanetti C., Vennetier M., Mériaux, Provansal M. (in press). Plasticity of tree root system structure in contrasting soil materials and environmental conditions. Plant and Soil. On line 2014;DOI: 10.1007DOI: 10. /s11104-014-2272-9 -9 -10 -

-Influence of slope and water resources

As a whole, root number was higher downslope than upslope (67 versus 53, p = 0.025). However, only 57% of the trees had a higher root number downwards (mean difference in root number = + 67%) while 25% had a higher root number upward with a smaller mean difference (+21%) and 18% the same number (±10%).

Root systems with a privileged access to water were asymmetric. The largest roots systematically developed in the direction of the closest water resource when available, thus downslope to reach a water channel or vertically (taproots) to reach the water table. The mean root diameter was 65% larger downslope or vertically than upslope (P < 0.001). In coarse material, such roots reaching a permanent water resource were larger, from 2-3 times (P<0.001), than the mean diameter of other large roots. Mixed or tap root systems were found in coarse material when water or nutrient resources were less than 3 m in depth. The largest vertical roots (proximal diameter up to 40-60 cm) were found in such conditions on poplar stumps.

Climate influenced these general patterns. On the crown and upper slopes of the Castérino dike, a large dike made of coarse material in a cold and humid mountain climate, all seven trees had one or several taproots and several other roots growing throughout the embankment. The permanent availability of water in the fill attenuated the upward/downward dissymmetry and allowed vertical roots to develop through the coarse material. Conversely on the Donzère and Montélimar dikes in a Mediterranean climate with a long summer drought, 95% of the trees (35/37) growing at the same position on the dike had shallow root system and the dissymmetry was more pronounced.

With only one exception (alder), roots never penetrated more than a few centimetres in permanent water tables. Root penetration was also limited in hydromorphic horizons, all the more since the hydromorphy was severe. Thus, root systems were often shallow close to the toe of canal dikes or on the lower banks of rivers, being limited in depth by a permanent or temporary water table. In fine material (pseudogleys) at dike toes, old poplars developed dozens of small vertical roots (sinkers) 5 to 15 cm in diameter and up to 2 m long, growing out from horizontal roots, densely lining up into soil cracks. These additional roots were found mainly under the stump and in a radius of three metres around it. They contributed to increase considerably the volume of these root systems and made them difficult to extract.

-Discussion

Root system structure results from an interaction between genetic heritage and environmental constraints [START_REF] Atger | L'architecture racinaire est-elle influencée par le milieu? L'arbre[END_REF][START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF]. In this study, however, the type of soil material was one of the strongest environmental factors influencing root system structure. In coarse materials, trees develop only few roots but they are large and rather long, whereas fine materials favour denser systems with smaller diameters (less than 5 cm) and shorter roots (mean length < 2 m). Roots tend to proliferate in nutrient-rich pockets and horizons when the soil is heterogeneous [START_REF] George | Responses of Picea, Pinus and Pseudotsuga roots to heterogeneous nutrient distribution in soil[END_REF][START_REF] Hodge | The plastic plant: root responses to heterogeneous supplies of nutrients[END_REF]. In fine materials, roots multiplied over a TBD of 25 cm while the mean root diameter was stable and the maximum root diameter increased slowly. This finding suggests either that trees produce continuously new roots in the core of the root system, directly on the stump and from older roots, or that pre-existing small roots are slowly growing in length and diameter. These new small roots reaching the minimum size limit (2 cm) can compensate for the growth of the older ones, stabilizing the mean root diameter. Such developmental instability is considered as an adaptive trait [START_REF] Forde | Is it good noise? The role of developmental instability in the shaping of a root system[END_REF], useful for plasticity to be able to respond to potentially changing conditions. The permanent production of roots is possible thanks to the absence of strong constraints from the environment. The presence of abundant resources (nutrients and water) in fine materials allows trees to feed in a small volume around the stump. This may explain the reduced length and depth of roots as observed by [START_REF] Tatarinov | Application of "clump technique" for root system studies of Quercus robur and Fraxinus excelsior[END_REF] on clayey alluvial soils: the average radius of root systems for large oaks and ash trees with diameters up to 65 cm were smaller than seven metres. The sharing between many roots, of soil resources in a small volume around the stump, as well as of energy and substances elaborated in the crown, probably contributes to the limited root diameter in these systems.

In water or nutrient-limited environments, particularly those with little organic matter, in sandy soils or in arid climates, tree root systems tend to be large compared to aerial parts, and they reach greater depths [START_REF] Schenk | The global biogeography of roots[END_REF]. In Mediterranean forests, Quercus ilex L. develops roots up to 33 m-long spanning up to seven times the surface of its canopy [START_REF] Moreno | Fine root distribution in Dehesas of Central-Western Spain[END_REF]. Accordingly in dikes made of coarse material, the root system span was far higher and vertical roots were longer and larger than in fine materials. In coarse materials, resources are limited and sometimes far from the stump, particularly on large and thick fills. Moreover, on very coarse material, root length and diameter growth is physically constrained by pebbles, stones and blocks, obliging the growing roots to skirt around hundreds of obstacles and worm their way into narrow and winding chinks. However, roots are able to take advantage of small pockets of better soil and to adapt at small scale to substrate heterogeneity [START_REF] Desnos | Root branching responses to phosphate and nitrate[END_REF], while sending remote signals to the plant to favour their development [START_REF] Forde | The nutritional control of root development[END_REF]. It could be hypothesized that roots probably explore and exploit most of the rare possible routes close to the stump from the very beginning of tree settlement, and only the few successful ones finding water or nutrient resources survive and develop. As most of the supplying of the tree depends on these few roots, they drain all resources from the soil but also those coming from the crown, and grow rapidly. This may be especially true for vertical roots or long shallow roots which have reached a water resource, and become out of proportion compared to other main roots. In such conditions, it would be difficult for new roots to encounter additional resources or unexplored routes. Thus, in contrast to fine material, the root number remains stable with tree size, while mean and maximum root diameter increase rapidly with TBD.

The position of a tree on the dike and dike size act on two parameters which influence root system structure: slope and access to water and nutrient resources. On slopes, roots are generally more numerous and larger upwards in order to secure tree anchoring [START_REF] Iorio | Root system architecture of Quercus pubescens trees growing on different sloping conditions[END_REF]Nicoll et al., 2006a;[START_REF] Schwarz | Quantifying lateral root reinforcement in steep slopes -from a bundle of roots to tree stands[END_REF]. In this study, this general pattern was rarely observed, due to the fact that resources were unevenly distributed around the stump. The presence of a channel, river or water table, a layer of fine and richer materials under a coarse fill, and natural soil easy to reach at dike toe clearly bias root system development towards the richest and wettest targets downslope. This result is consistent with observations on experimental irrigation or fertilization devices [START_REF] George | Responses of Picea, Pinus and Pseudotsuga roots to heterogeneous nutrient distribution in soil[END_REF][START_REF] Soar | The effect of changing patterns in soil-moisture availability on grapevine root distribution, and viticultural implications for converting full-cover irrigation into a point-source irrigation system[END_REF][START_REF] Sokalska | Spatial root distribution of mature apple trees under drip irrigation system[END_REF] that show a strong asymmetry in root system development in the direction of resources.

Many tree species are able to explore the soil and rock layers deeper than 5 m and for some of them as far as 20 m [START_REF] Canadell | Maximum rooting depth of vegetation types at the global scale[END_REF]. Deep rooting depths are generally found in water-limited environments [START_REF] Schenk | The global biogeography of roots[END_REF], particularly those with little organic matter, in sandy soils or in arid climates. Surprisingly in this study, root systems on coarse materials were of mixed or taproot types only when trees had a vertical access to a water table or to a more humid layer at less than 3 m in depth. However, deep rooting depth was reported in the literature mainly for natural old stands [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF][START_REF] Plourde | Spatial distribution, architecture, and development of the root system of Pinus banksiana Lamb. in natural and planted stands[END_REF]. Indeed, most of the largest and highest dikes of this study, where the 3 m limit appeared, are less than 100-years old and often much younger. Such a short time may have limited the ability for roots to find all possible routes or to take advantage of an especially long humid period to cross the critical 1-3 m depth. In fact, very deep rooting seems possible in materials and soils where roots can find some resources and particularly water all along their way down, for example, in the cracks of a karst [START_REF] Jackson | Ecosystem rooting depth determined with caves and DNA[END_REF].

On the crown and upper slope of large dikes or river banks made of coarse, well-drained materials, there are no vertically accessible nutrient resources and the heart of the dike is extremely dry in summer. Accordingly, [START_REF] Devine | Root system morphology of Oregon white oak on a glacial outwash soil[END_REF] found only shallow root systems for Oregon white oak on coarse and poor glacial outwash soils, one of the rare studies on coarse material similar to that of some dike fills. As shown by the presence of small vertical roots (initial taproot) on the youngest stumps and their remainder in older ones, vertical roots appear at an early stage of root system development, which is common for many species [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF]), but remain weak or abort later as they cannot find good growing conditions or water in depth [START_REF] Hipondoka | Root system of Terminalia sericea shrubs across rainfall gradient in a semiarid environment of Etosha National Park, Namibia[END_REF]. Conversely, shallow roots may take advantage of rain water, of fine material deposits at soil surface or of a superficial humus layer formed by vegetation. Moreover, physical constraints may be partially released close to the surface, where roots can more easily move stones while they grow in diameter.

On fine materials the position of trees on the dike seems much less important for root system structure. Resources are more regularly distributed and root development has no severe constraints in any direction. However, permanent water resources also lead to asymmetric development of root systems.

The tap root systems were scarce in this study as well as in the literature [START_REF] Köstler | Die Wurzeln der Waldbäume. Untersuchung zur Morphologie der Waldbäume in Mitteleuropa[END_REF]. Indeed, even when they develop one or several large vertical roots, trees generally produce a set of significant shallow roots to secure their anchoring, especially for support against wind (Danjon et al., 2005;[START_REF] Kamimura | Root anchorage of hinoki (Chamaecyparis obtuse (Sieb. Et Zucc.) Endl.) under the combined loading of wind and rapidly supplied water on soil: analyses based on tree-pulling experiments[END_REF]Nicoll et al., 2006a;[START_REF] Tamasi | Influence of wind loading on root system development and architecture in oak (Quercus robur L.) seedlings[END_REF], and to extract nutrients. Sheer tap root systems tend to occur only when a lack of shallow resources is combined with a permanent deep water table.

Roots of most tree species cannot develop directly in a permanent water table, as abundantly documented in the scientific literature [START_REF] Imada | Water table depth affects Populus alba fine root growth and whole plant biomass[END_REF][START_REF] Nieuwenhuis | The effect of soil cultivation methods on rooting depth of young Sitka spruce (Picea sitchensis (Bong.) Carr.) trees on wet mineral soils in Ireland[END_REF]. The higher proportion of shallow systems on fine material for alder and willow compared to other species in this study is explained by their natural trend to grow very close to the river or canal, and therefore to the water table. Root development is also strongly constrained by soil compaction as well as hard pan layers, presenting unfavourable mechanical proprieties [START_REF] Danjon | L'arbre et son enracinement[END_REF][START_REF] Unger | Soil Compaction and Root Growth: A Review[END_REF]. Accordingly, shallow root systems were found irrespective of the material when a permanent or semipermanent water table existed in the first 50 cm or when dike structure included a compacted silty or clayey core. Each species has a specific tolerance to hydromorphy. Poplars proved to be able to colonize part of a pseudogley up to 2 m in depth, with dozens of small or medium-size vertical roots filling and following cracks. Such root systems should be considered as a specific type, very different from the typical mixed root systems with only one or a few large taproots. Although the soil initially presents less anoxic conditions along these cracks thanks to evaporation, water uptake by the roots entering these cracks may rapidly contribute to expand the width of the desiccation zone around the cracks and, as a feedback, accelerate their colonisation by roots. Unfortunately, the studied sites offered no opportunities to study root system development of other species in the same conditions.

Root system volume is partly linked to the development of its main roots (Table 3c). The good correlation between volume and TBD for shallow root systems can be explained by the direct relation between the area colonized by roots at low depth and root system volume. Shallow root systems logically have a smaller volume than mixed ones because the latter have similar long shallow roots but they have large and long vertical roots also. Heart root systems and root systems on fine material have a smaller volume than mixed root systems or those on coarse material, because of the reduced root length. Similar differences in root plate volume and span according to root system type were found by [START_REF] Ray | The effect of soil water-table depth on root-plate development and stability of Sitka spruce[END_REF] in Sitka spruce uprooting experiments. Root system volume also depends on the distribution of small and fine roots, which can be highly irregular, leaving large gaps between main roots [START_REF] Tatarinov | Application of "clump technique" for root system studies of Quercus robur and Fraxinus excelsior[END_REF]. As tree anchorage is strongly linked to these characteristics, measuring them more precisely may allow assessing the volume of the root plate which could be lifted when the tree is uprooted, this volume varying with material types and soil humidity [START_REF] Kamimura | Root anchorage of hinoki (Chamaecyparis obtuse (Sieb. Et Zucc.) Endl.) under the combined loading of wind and rapidly supplied water on soil: analyses based on tree-pulling experiments[END_REF].

Root length is significantly correlated to root proximal diameter (Table 3e), as already found in other contexts (Van Noordwijk et al., 1994). Thus, for a given environment and soil material, a validated and calibrated predictive relationship could be used to infer the potential root length from a simple proximal diameter measure close to the tree without fully extracting the stump. As diameter decreasing rates are significantly different between fine and coarse materials, variations in soil composition must be taken into account for a practical use of this relation in field expertise. A detailed analysis of individual root architecture recently confirmed the influence of material on root length and decreasing rate (Vennetier et al., in press) as well as on other architectural characteristics as branching patterns. Long roots with low decreasing and branching rates and short roots with higher values for these two parameters were identified. This may help for recognizing contrasting root types close to the stump. However, the proportion of such root types is unknown for a given root system, and it is difficult to determine except close to the soil surface. Thus assessing root potential length from proximal diameter may only be possible for the root system as a whole, not for all individual roots.

-Conclusion and prospects

The global structure of tree root systems seems opportunistic, mainly influenced by environmental conditions, and particularly by soil material, and to a lesser extent by the tree species. This architectural plasticity offers trees a high capacity of adaptation to various conditions. This study gives a new insight in root development: it will help predict tree root growth in various environments and more specifically on dikes. Material type and water availability allow to presage root system structure.

Although these results are reliable, based on one of the largest set of trees and species ever extracted to study root systems, they focus on locally homogeneous and quite deep soils. The response of tree species to more heterogeneous or extreme conditions or with severe physical or chemical constraints may have been more contrasting due to the specific tolerance of some species to these constraints. Additional research is required to confirm them in other environments, on other soil materials, and for other species.

Figure 3 :

 3 Figure 2: The four types of root systems (a) Shallow root system Maple at dike toe Loire River Fine material

  = TBD >55 cm, Medium = 45-55 cm, Int = 35-45 cm, Small = 25-35 cm. Standard deviation is given between brackets.

  Soil samples were collected in the excavation hole at the level of each tree and for each different layer. After separation and volume measurement of the largest blocks and stones, grain size distribution was determined by wet and dry screening (screens 20 mm to 80 µm -French norm 94-056 / International norm ISO 14688) and sedimentation analysis (French norm 94-057 / International Norm ISO 14688). These materials could be clearly split into coarse and fine types[START_REF] Zanetti | Characterization of tree root systems in earth dikes[END_REF]; no site with intermediate material was considered in this study.

	Study sites characteristics	Montélimar, Donzère	Lyon, Roussillon, St Vallier	Grenoble	Cosne-sur-Loire	Pertuis	Castérino	Chambéry	Montjoux
	River	Rhône	Rhône	Isère, Drac Romanche	Loire	Durance	Bieugne	Leysse	Gervonde
	Altitude (m)	85	200	250	150	200	1 500	270	254
	Rainfall (mm/yr)	700	800	934	690	640	1 200	1 136	946
	Temperature (°C) Min/Mean/Max	9.2/13.8/18.5 8.1/12.8/17 6.3/12/16.2	6.3/10.1/15. 4	7.2/13/18.8 0/6.1/12.6	6.7/12/18	6.6/10.6/17
	Type of dike	Canal	Canal	Flood protection	Flood protection	Flood protection	Flood protection	Flood protection	Dam
	Water head	Permanent	Permanent	Temporary	Temporary	Temporary	Temporary	Temporary	Permanent
	Dike dimension (m) (Height / Width)	10 / 100	8-12 / 80 -100	5 / 20	7 / 30	4 / 15	4 /10	3 / 15	10/50
	Fill slope	20%	20%	40-60%	40-60%	40-60%	40-60%	40-60%	20%
	Studied species	Oak, Locust, Poplar	Oak, Locust, Poplar, Ash, Maple, Willow, Adler	Oak, Locust, Poplar, Ash, Willow	Oak, Locust, Poplar, Ash	Poplar, Oak	Larch, Scots pine	Poplar, Ash, Adler Willow,	Poplar, Ash

Table 2 : Number of sampled trees per root system type according to soil material for each species or genus.

 2

										Total root
		Shallow	Mixed		Taproot	Heart		system number
		Coarse	Fine	Coarse	Fine	Coarse	Fine	Coarse	Fine
	Locust	18	0	18	0	0	0	0	8	44
	Poplar	14	3	29	0	3	0	0	10	59
	Oak	2	0	11	0	1	0	0	8	22
	Ash	1	7	6	0	0	0	0	6	20
	Maple	2	2	0	0	0	0	0	4	8
	Willow	1	5	5	3	0	0	0	2	16
	Alder	0	5	4	0	0	0	0	0	9
	Scots pine	2	0	4	0	0	0	0	0	6
	Larch	0	0	7	0	0	0	0	0	7
	Total root system
	number	40	22	84	3	4	0	0	38	191

C = Coarse material, F = Fine material

Zanetti C.,

Vennetier M., Mériaux, Provansal M. (in press)

. Plasticity of tree root system structure in contrasting soil materials and environmental conditions.

Plant and Soil. On line 2014; DOI: 10.1007 DOI: 10. /s11104-014-2272-9 -9

-6 -

ACKNOWLEDGEMENTS

This study was funded by IRSTEA, Provence Alpes Côte d'Azur region, French National Research Agency (ANR-ERINOH project), by European Union (Interreg IV A France-Suisse 2007-2013) and by dike managers (EDF, CNR, AD Isère-Drac-Romanche, DDT Nièvre, SMAVD, Chambery Métropole, Conseil Général de l'Isère) who hired the technical staff and material to log trees, extract root systems and remediate damages to studied sites. The authors are indebted to many colleagues, technicians and students who contributed in data collection during sometimes difficult field campaigns, and particularly to Willy Martin, Roland Estève, Olivier Chandioux, Gaylord Doirat, Pierre Jean Moundy and David Fiorese. They thank several anonymous reviewers for their useful suggestions, as well as Maureen K. Corcoran and Pete Fulé for their comments, corrections and their help in language editing.