Frédéric Mazenc

Michael Malisoff

Jerome Weston

New Bounded Backstepping Control Designs for Time-Varying Systems under Converging Input Converging State Conditions

Keywords: Backstepping, delays, stabilization

We provide new globally stabilizing backstepping controls for single input systems in a partially linear form. Instead of measuring the full state, our feedbacks use current and several time lagged values of a function of the state of the nonlinear subsystem, and have no distributed terms. Other possible advantages are that we do not require differentiability of the available nominal controls for the nonlinear subsystems, and that our controls do not contain Lie derivatives. We also allow input delays. This improves on our earlier work on the special case where the linear part has one integrator, since we now allow an arbitrary number of integrators.

I. INTRODUCTION

This work continues our search for novel backstepping techniques that can apply in cases where the entire current state of the system may not be available for measurement; see [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF], [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF], [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF], and [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] for some of our earlier work. Backstepping is ubiquitous in control engineering, and it usually involves recursively transforming stabilizing controllers (and corresponding Lyapunov functions) for subsystems into globally asymptotically stabilizing controllers for the entire systems; see [START_REF] Freeman | Integrator backstepping for bounded controls and control rates[END_REF], [START_REF] Jiang | A recursive technique for tracking control of nonholonomic systems in chained form[END_REF], [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF], and [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. However, standard backstepping generally requires measuring the entire state and leads to unbounded controls, which can be a serious limitation; see, e.g., Section VI-A below, for an example where we compare the results in this work with a standard backstepping design from [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF].

One important setting where backstepping has been useful is for systems in the partial linearized single input form    ẋ(t) = F(t, x(t), z 1 (t)) żi (t) = z i+1 (t), i ∈ {1, 2, . . . , k -1} żk (t) = u(t) + h(t, x(t), z(t)).

(1) See [START_REF] Isidori | Nonlinear Control Systems, Third Edition[END_REF]Section 9.1] for differential geometric conditions that hold for many nonlinear systems and make it possible to change variables to obtain systems of the form [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF]. For cases where the equations żi (t) = z i+1 (t) are not present (which we call the k = 1 case), our work [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] designed globally asymptotically stabilizing input delayed bounded controllers whose main advantages were that they did not involve distributed terms or Lie derivatives and did Mazenc is with EPI DISCO Inria-Saclay, Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS, Centrale-Supélec, Université Paris-Sud, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France. frederic.mazenc@l2s.centralesupelec.fr. Supported by l'Institut pour le Contrôle et la Décision de l'Idex Paris-Saclay (iCODE).

Malisoff and Weston are with the Department of Mathematics, 303 Lockett Hall, Louisiana State University (LSU), Baton Rouge, LA 70803-4918, USA. malisoff@lsu.edu and jwesto3@lsu.edu. Supported by NSF-ECCS Grant 1408295. not require differentiability of the available nominal control for the x-subsystem. Instead, [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] used a converging input converging state condition, and then [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] gave criteria for this condition to hold. However, there is no obvious analog of [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] and [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] for the general form [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] where k ≥ 2 is arbitrary.

This work presents a generalization of the converging input converging state assumption from [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] and [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF], and uses it to design feedback controllers that drive all solutions of (1) to 0 asymptotically for any k ≥ 2. We also present Lyapunov conditions that facilitate checking our generalized condition. We illustrate our work in two examples, including one where a scaling argument leads to a bounded backstepping controller with arbitrarily small sup norm. This contrasts with other backstepping results, such as [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] (which covers cases with only one integrator and requires certain persistency of excitation conditions that we do not use here), [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] (which uses a very different forwarding approach to cover the one integrator case), [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] (which gives unbounded controls), and the works [START_REF] Tsinias | Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation[END_REF], [START_REF] Tsinias | Backstepping design for time-varying systems and application to partial-static feedback and asymptotic tracking[END_REF], and [START_REF] Tsinias | ISS property for time-varying systems and application to partial-static feedback stabilization and asymptotic tracking[END_REF] (which involve Lie derivatives or measurements of the full state).

II. PRELIMINARIES

We provide definitions, notation, and a key lemma. Unless otherwise noted, the dimensions of our vectors are arbitrary. The Euclidean norm of vectors in R a , and the induced norm of matrices, are denoted by | • |. We always assume that the initial times t 0 ≥ 0 for all trajectories are t 0 = 0, but analogous results can be written for all t 0 ≥ 0. Given any constant T > 0, we let C in denote the set of all continuous functions φ : [-T, 0] → R a , which we call the set of all initial functions. We define Ξ t ∈ C in by Ξ t (s) = Ξ(t + s) for all choices of Ξ and t for which the equality is defined. A function G : [0, ∞) × R a → R p is called uniformly bounded with respect to the first variable provided that there is a nondecreasing continuous function α

: [0, ∞) → [0, ∞) such that |G(t, x)| ≤ α(|x|) holds for all t ≥ 0 and x ∈ R a .
For each constant r > 0, we use B p,r to denote the closed radius r ball in R p centered at 0 ∈ R p . We call a function G : [0, ∞) × R a → R p uniformly locally Lipschitz in the second variable provided that for each constant K > 0, there is a constant L K > 0 such that |G(t, x) -G(t, y)| ≤ L K |x -y| holds for all t ≥ 0 and for all x and y in B a,K . Let K ∞ be the set of all continuous functions γ : [0, ∞) → [0, ∞) such that γ(0) = 0, γ is strictly increasing, and lim s→∞ γ(s) = ∞. For any subset S of a Euclidean space such that 0 ∈ S, a function α : S → [0, ∞) is called positive definite provided α(0) = 0 and α(r) > 0 for all r ∈ S \ {0}. A function

V : [0, ∞) × R a → [0, ∞)
is called uniformly proper and positive definite provided that there exist class K ∞ functions α and ᾱ such that α(|x|) ≤ V (t, x) ≤ ᾱ(|x|) hold for all t ≥ 0 and x ∈ R a . We use the sup norm |φ| I = sup t∈I |φ(t)| for all continuous functions φ : [0, ∞) → R a and all subsets I ⊆ [0, ∞). Throughout this work, we assume:

Assumption 1: The functions F and h in (1) are uniformly bounded in t and uniformly locally Lipschitz in (x, z).

We prove the following in the appendix: Lemma 1: Let q > 0 be any constant. Then there are an invertible matrix P ∈ R k×k and a locally Lipschitz function m : R n × R k → R such that the change of coordinates y = P z satisfies y k = z 1 and transforms (1) into

     ẋ(t) = F(t, x(t), y k (t)) ẏi (t) = -qy i (t) + y i-1 (t), i ∈ {2, . . . , k} ẏ1 (t) = -qy 1 (t) + m(x(t), y(t)) + u(t) (2)
where y = (y 1 , . . . ., y k ) . If, in addition, h = 0, then m is independent of x and has the form m(y) = g 1 y 1 +. . .+g k y k for some nonnegative constant g i 's.

The change of input v(t) = m(x(t), y(t))+u(t) now gives

     ẋ(t) = F(t, x(t), y k (t)) ẏi (t) = -qy i (t) + y i-1 (t), i ∈ {2, . . . , k} ẏ1 (t) = -qy 1 (t) + v(t) (3)
Hence, it suffices to build a feedback control v(t) such that all solutions of (3) asymptotically converge to 0.

Our formula for v will be bounded and only depend on k+ 1 time lagged values of x, so we get a bounded backstepping theorem. Each constant g i from (1) will have a factor q, and so can be made as small as desired by reducing q; see the appendix. Hence, in the special key case of undelayed cases where h = 0, our control u is the sum of a bounded control term, plus a linear term with arbitrarily small coefficients.

III. INPUT DELAYED BOUNDED BACKSTEPPING FOR (3)

This section provides a bounded backstepping result for (3) under input delays T ≥ 0 in the control v; see Section V where we use ideas from this section to build bounded backstepping controls for the original system (1). Assume the following (but see below for ways to check our assumptions):

Assumption 2: There are a locally Lipschitz bounded function ω : R n → R and constants q > 0, τ > 0, and T ≥ 0 such that for all continuous functions δ : R → R that exponentially converge to zero, all solutions of

ẋ(t) = F (t, x(t), G(x t ) + δ(t)) (4)
with the choice

G(x t ) = t t-τ m k-1 m k-1 -τ • • • m1 m1-τ q k e q(m 0 -t) ω(x(m0-T )) (1-e -qτ ) k dm 0 . . . dm k-1
asymptotically converge to zero. Also, ω(0) = 0. We can rescale ω to replace the inner most integrand in (4) by ω(x(m 0 )), but we use the integrand in (4), because

t t-τ m k-1 m k-1 -τ . . . m1 m1-τ q k e q(m 0 -t) (1-e -qτ ) k dm 0 . . . dm k-1 = 1 (5)
holds for all t ≥ 0, which will simplify our analysis below. We prove this delayed bounded backstepping result:

Theorem 1: Let Assumptions 1-2 hold. Then all solutions of (3), in closed loop with the bounded control

v(t) = q k (1-e -qτ ) k k j=0 k!(-1) j e -jqτ j!(k-j)! ω x(t -jτ -T ) , (6)
asymptotically converge to 0.

Proof: Fix any solution (x(t), y(t)) of ( 3), and set

ω(x) = q k ω(x) (1-e -qτ ) k and ω (r, x) = e rq ω(x). (7)
Consider the operators defined by β 0 (t) = ω(x(t -T )),

β 1 (t) = t t-τ ω (m 0 -t, x(m 0 -T ))dm 0 , and β j (t) = t t-τ mj-1 mj-1-τ • • • m1 m1-τ ω (m 0 -t, x(m 0 -T ))dm 0 . . .dm j-1
for all j ∈ {2, . . . , k}, and set b = -e -qτ . Notice for later use that (4) reads ẋ(t) = F(t, x(t), β k (t) + δ(t)), and that

βj (t) = -qβ j (t)+β j-1 (t)+bβ j-1 (t -τ ), 1 ≤ j ≤ k (8)
The j = 1 case of ( 8) follows from the Fundamental Theorem of Calculus, while for j ≥ 2, we get

βj (t) = -qβ j (t) + β j-1 (t) -e -qτ × t-τ t-2τ . . . m1 m1-τ ω (m 0 -t+τ, x(m 0 -T ))dm 0 . . .dm j-2
which gives (8) for all j ∈ {1, 2, . . . , k}.

We next show that the operators

γ k-s (t) = s j=0 s!b j j!(s-j)! β k-s (t -jτ ) (9)
defined for all s ∈ {0, . . . , k} solve the following system:

γi (t) = -qγ i (t) + γ i-1 (t), i = 1, 2, . . . , k (10)
To this end, notice that for all s ∈ {0, 1, . . . , k -1}, we can use [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] to get

γk-s (t) = s j=0 s!b j j!(s -j)! -qβ k-s (t -jτ ) + β k-s-1 (t -jτ ) + bβ k-s-1 (t -(j + 1)τ ) = -qγ k-s (t) + s j=0 s!b j β k-s-1 (t -jτ ) j!(s -j)! + s+1 j=1 s!b j β k-s-1 (t -jτ ) (j -1)!(s -j + 1)! . ( 11
)
For each j ∈ {1, . . . , s}, the coefficient of b j β k-s-1 (t -jτ ) in the summation terms in (11) add to

s! j!(s-j)! + s! (j-1)!(s-j+1)! = s! s-j+1+j j!(s-j+1)! = (s+1)! j!(s+1-j)!
, and the coefficients of β k-s-1 (t) and b s+1 β k-s-1 (t -(s + 1)τ ) in [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] are both 1, which gives [START_REF]The world's definitive system for modern technical computing[END_REF].

Next, let ξ i (t) = y i (t) -γ i (t) for all i ∈ {1, 2, . . . , k}. Then we can combine (3) with [START_REF]The world's definitive system for modern technical computing[END_REF] and then use the fact that our controller v(t) in ( 6) is γ 0 (t) to obtain this system:

   ẋ(t) = F(t, x(t), ξ k (t) + γ k (t)) ξi (t) = -qξ i (t) + ξ i-1 (t), i ∈ {2, . . . , k} ξ1 (t) = -qξ 1 (t) (12)
The ξ-subsystem of ( 12) is globally exponentially stable to zero, and we can use a corresponding quadratic Lyapunov function for this system (e.g.,

V (ξ) = c 0 ξ 2 k + . . . + c k-1 ξ 2 1
, where c 0 = 1 and c i = (c 2 i-1 /(2q 2 ))+1 for i = 1, 2, . . . , k-1) to prove input-to-state stability (or ISS) for [START_REF]The world's definitive system for modern technical computing[END_REF] with respect to γ 0 (t); see [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] for background on ISS. This leads us to consider ẋ(t) = F(t, x(t), γ k (t)+δ(t)), where δ = ξ k exponentially converges to zero. Since γ k = β k , Assumption 2 now gives lim t→∞ x(t) = 0, so lim t→∞ γ 0 (t) = 0, because ω(0) = 0. Since [START_REF]The world's definitive system for modern technical computing[END_REF] satisfies ISS with respect to γ 0 (t), it follows that lim t→∞ γ i (t) = 0 for all i ∈ {1, 2, . . . , k}, so lim t→∞ y i (t) = lim t→∞ ξ i (t) + lim t→∞ γ i (t) = 0 for all i ∈ {1, 2, . . . , k}, which proves Theorem 1.

IV. WAYS TO CHECK ASSUMPTION 2

As in the special case where k = 1 that we studied in [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF], a key feature of the preceding result for general k > 1 is that we do not require the nominal control ω to render

ẋ(t) = F(t, x(t), ω(x(t))) (13)
uniformly globally asymptotically stable to 0, nor do we require differentiability of ω or Lyapunov functions for the x dynamics. However, we can use [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] to give sufficient conditions for Assumption 2 to hold. One way to do so is as follows, where V t and V x denote the partial derivative with respect to t and the gradient with respect to x, respectively: Assumption 3: The function F admits the control affine form F(t, x, p) = f (t, x) + g(t, x)p where f : [0, ∞) × R n → R n and g : [0, ∞) × R n → R n are continuous and uniformly locally Lipschitz in the second variable. Also, there are a C 1 uniformly proper and positive definite function V : [0, ∞) × R n → [0, ∞); a uniformly continuous positive definite function W : R n → [0, ∞); positive constants r 0 , r 1 , and r 3 ; and a constant r 2 ≥ 0 such that

V t (t, x) + V x (t, x)(f (t, x) + g(t, x)ω(x)) ≤ -W (x), |V x (t, x)g(t, x)| ≤ r 0 W (x), |ω(x)| ≤ r 1 W (x), |f (t, x)| ≤ r 2 W (x)
, and |g(t, x)| ≤ r 3 [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] hold for all (t, x) ∈ [0, ∞) × R n , where ω : R n → R is bounded, satisfies ω(0) = 0, and has a global Lipschitz constant C > 0.

See [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] for general conditions under which Assumption 3 holds. Fix any constants τ > 0 and T ≥ 0. Set k = kτ + T and

M * = 4(Cr 0 k ) 2 2r 2 2 + 5 2 (k r 1 r 3 ) 2 τ 2(k-1) q 1-e -qτ 2k . ( 15
)
When T = 0, we get M * → 0 as τ → 0 + , in which case we can satisfy the condition M * < 1 from the following proposition for any values of the other constants, by choosing τ > 0 small enough:

Proposition 1: Let Assumption 3 hold and M * < 1. Then for all continuous functions δ : [0, ∞) → R that exponentially converge to 0, all solutions of (4) asymptotically converge to zero, so Assumption 2 is satisfied.

Proof: Along all trajectories of (4) for large enough t, the functions V and W from Assumption 3 satisfy

V ≤ -W (x(t)) + V x (t, x(t))g(t, x(t)) δ(t)+ t t-τ m k-1 m k-1 -τ . . . m1 m1-τ δ ω(m 0 , t)dm 0 . . . dm k-1 ≤ -W (x(t)) + r 0 W (x(t)) C t t-k | ẋ( )|d + |δ(t)| ,
where δ ω(m 0 , t) = e q(m0-t) (ω(x(m 0 -T )) -ω(x(t))) and ω is from [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF], and where we used the relations [START_REF] Isidori | Nonlinear Control Systems, Third Edition[END_REF] and

sup m0∈[t-k ,t] |ω(x(m 0 )) -ω(x(t))| ≤ C sup m0∈[t-k ,t] |x(m 0 ) -x(t)| ≤ C t t-k | ẋ( )|d . (16)
Hence, Jensen's inequality gives

V ≤ -1 2 W (x(t)) + r 2 0 |δ(t)| 2 +C 2 r 2 0 k t t-k | ẋ( )| 2 d , (17)
where we also used Young's Inequality to get b W (x(t)) ≤ 0.25W (x(t)) + b 2 twice for suitable choices of b ≥ 0.

We can also use the Young and Jensen inequalities, and the fact that (a + b) 2 ≤ (5/4)a 2 + 5b 2 holds for all a ≥ 0 and b ≥ 0, to upper bound the integrand in [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] by

| ẋ( )| 2 ≤ 2|f ( , x( ))| 2 + 2|g( , x( ))| 2 |δ( )|+ τ k-1 q 1-e -qτ k -k |ω(x(m 0 ))|dm 0 2 ≤ 2r 2 2 W (x( )) + 2r 2 3 5|δ( )| 2 + 5k 4τ τ q 1-e -qτ 2k -k ω 2 (x(m 0 ))dm 0 ≤ 2r 2 2 W (x( )) + 10r 2 3 |δ( )| 2 + 5k r 2 1 r 2 3 2τ τ q 1-e -qτ 2k -k W (x(m 0 ))dm 0 .
Combining the preceding bound with [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] gives the estimate

V ≤ -1 2 W (x(t)) + M 1 t t-2kτ W (x( ))d +M 2 |δ| 2 [t-k ,t] , where (18)
M 1 = (r 0 C) 2 k 2r 2 2 + 5 2 (k r 1 r 3 ) 2 τ 2(k-1) q 1-e -qτ 2k and M 2 = r 2 0 + 10(Cr 0 r 3 k ) 2 .
Also, our bound M * < 1 can be written as 4k M 1 < 1. Hence, we can find a constant λ > 1 such that 2λk M 1 < 1 2 . Then along all trajectories of (4), the function

V 1 (t, x t ) = V (t, x(t)) + λM 1 t t-2k t s W (x( ))d ds satisfies V1 ≤ -1 2 -2λk M 1 W (x(t)) + M 2 |δ| 2 [t-k ,t] +M 1 (1 -λ) t t-2k W (x( ))d ≤ -1 2 -2λk M 1 W (x(t)) + M 2 |δ| 2 [t-k ,t] . (19)
Since the quantity in curly braces in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] is positive and δ exponentially converges to 0, we can integrate [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] to conclude that V 1 (t, x t ) is bounded. Since V is uniformly proper and positive definite, it follows that |x(t)| is bounded. It follows from the structure of the x(t) dynamics that x(t) is uniformly continuous. Recalling that W is uniformly continuous, we conclude that W (x(t)) is a uniformly continuous function of t. Also, since δ ∈ L 2 ([0, ∞), R), the inequality [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] gives

∞ 0 W (x( ))d < ∞.
It follows from Barbalat's Lemma that lim t→∞ W (x(t)) = 0, so since W is positive definite, we get lim t→∞ x(t) = 0, as needed.

V. BOUNDED BACKSTEPPING CONTROLLERS FOR (1)

A. Undelayed Case A potential advantage of the control v(t) from Theorem 1 is its simple form, which leads to the control u(t) = v(t) -gP z(t) for the original system (1) when h = 0 case, where P and g = (g 1 , . . . , g k ) are from Lemma 1. When h is an arbitrary bounded locally Lipschitz function, we can use an analog of Theorem 1 to find globally bounded (but more complicated) controllers u such that all solutions of (1) asymptotically converge to 0, as follows. We assume that we know a constant τ > 0 and a bounded globally Lipschitz function ω : R n → R such that ω(0) = 0 and such that for each continuous function δ : [0, ∞) → R that exponentially converges to 0, all solutions of ẋ(t) = F(t, x(t), α k (x t )+δ(t)) asymptotically converge to 0, where

α k (x t ) = 1 τ k t t-τ m k-1 m k-1 -τ • • • m1 m1-τ ω(x(m 0 ))dm 0 . . .dm k-1 . ( 20
)
We can provide sufficient conditions for the preceding assumption to hold, using a variant of Proposition 1.

Then we set zi = z i -β i for all i ∈ {1, 2, . . . , k}, where

β i (t) = D (i-1) t α k (x t ) is the (i -1)-st derivative of α k (x t ) in t, so β 1 (t) = α k (x t ). This produces the system    ẋ(t) = F(t, x(t), β 1 (t) + z1 (t)) żi (t) = zi+1 (t), i ∈ {1, 2, . . . , k -1} żk (t) = u(t) + h(t, x(t), z(t)) -βk (t). (21)
Also, results from [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF] provide a nested saturation bounded feedback µ such that the k-dimensional system

ȧi = a i+1 for 1 ≤ i ≤ k -1, ȧk = µ(a) (22)
is globally asymptotically and locally exponentially stable to 0. Hence, using the bounded control u(t) = µ(z(t)) -h(t, x(t), z(t)) + βk (t) in ( 21) produces an exponentially stable z subsystem, so our assumption ensures that lim t→∞ x(t) = 0, so lim t→∞ β i (t) = 0 holds for all i (e.g., by the Dominated Convergence Theorem), which gives lim t→∞ z i (t) = lim t→∞ zi (t) + lim t→∞ β i (t) = 0 for all i.

B. Input Delayed Cases

By replacing the integrand ω(x(m 0 )) in the definition (20) of α k by ω(x(m 0 -T )), we can generalize the preceding bounded backstepping result to cover the systems (1) with h = 0 and any input delay T ≥ 0. This extends the delayed bounded backstepping results from [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] (which were confined to the k = 1 case) to cover any choice k ≥ 2 of the dimension of z. To see how this extension can be done, first note that for any constant delay T ≥ 0, we can find a bounded nested saturation controller µ such that

ȧi (t) = a i+1 (t) for 1 ≤ i ≤ k -1, ȧk (t) = µ(t -T ) (23)
is globally asymptotically and locally exponentially stable to 0. This is done by using a time rescaling t = vs for a suitable constant v > 0, the changes of variables ãi (s) = v i-1 a i (vs), and the new control μ(s

-T /v) = v k µ(vs -T ) to get ȧi (s) = ãi+1 (s), 1 ≤ i ≤ k -1 ȧk (s) = μ(s -T /v), (24)
and then using the nested saturation construction from [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF]Theorem 8] to get a globally asymptotically and locally exponentially stabilizing controller μ for (24) when v is large enough. This implies the desired asymptotic properties of ( 23), and then we choose u(t-T ) = µ(t-T )+ βk (t) to show the stability properties as before, where µ(t) = μ(t/v)/v k . This is a valid choice of the control, because our replacement in the integrand for α k implies that βk (t) only depends on x values on (-∞, t -T ].

VI. EXAMPLES A. Comparison with Standard Backstepping

We can sometimes apply Theorem 1 by checking Assumption 2 through a mixture of Lyapunov and direct trajectory analyses. For instance, consider the three dimensional system

ẋ = x 2 -x 3 + z 1 ż1 = z 2 , ż2 = u. (25)
As noted in [7, pages 593-594], the system (25) is globally asymptotically stabilized by the unbounded control

u(x, z) = -∂V0 ∂z1 (x, z 1 ) + ∂φ ∂z1 (x, z 1 )z 2 -z 2 + ∂φ ∂x (x, z 1 )(x 2 -x 3 + z 1 ) + φ(x, z 1 ), where V 0 (x, z 1 ) = 1 2 x 2 + 1 2 (z 1 + x + x 2 ) 2 and φ(x, z 1 ) = -x -(1 + 2x)(x 2 -x 3 + z 1 ) -(z 1 + x + x 2 ).
By contrast, we can build our control from Theorem 1 for (25), by checking that Assumption 2 holds with ω

(x) = -sin(πx/2)1 [-2,2] (x), where the indicator function 1 [-2,2] is defined to be 1 on [-2, 2] and 0 on R \ [-2, 2].
To verify Assumption 2 for this choice of ω, first note that for each continuous function δ : R → R that exponentially converges to 0 and each initial state x 0 ∈ R, we can use [START_REF] Isidori | Nonlinear Control Systems, Third Edition[END_REF] with k = 2 to find a value T (x 0 , δ) ∈ [0, ∞) such that for all constants q > 0 and τ > 0, the corresponding solution of

ẋ(t) = x 2 (t) -x 3 (t)+ t t-τ m1 m1-τ
q 2 e q(m 0 -t)

(1-e -qτ ) 2 ω(x(m 0 ))dm 0 dm 1 + δ(t) (26) satisfies x(t) ∈ [-0.8, 3/2] for all t ≥ T (x 0 , δ). This can be done by noting that the integral in (26) is bounded by 1, that x 2 -x 3 ≤ -1.125 for all x ≥ 3/2, and that x 2 -x 3 ≥ 1.152 for all x ≤ -0.8, so the right side terms x 2 (t) -x 3 (t) in (26) dominate the other right side terms. Hence, it suffices to check the inequalities ( 14) from Assumption 3 for all x ∈ [-0.8, 3/2], by only considering values of t ≥ T (x 0 , δ).

We now check the estimates from ( 14) for all x ∈ [-0.8, 3/2] using V (x) = 1 2 x 2 , W (x) = x 2 , f (x) = x 2 -x 3 , and g(x) = 1. First note that simple calculations (e.g., using Mathematica [START_REF]The world's definitive system for modern technical computing[END_REF]) give x 2 -x 3 -sin(πx/2) ≤ -x (resp., ≥ -x) for all x ∈ [0, 3/2] (resp., x ∈ [-0.8, 0]) which gives Using the change of feedback that transformed (2) into (3) and the formula for the function m in (3) in the appendix, we obtain the stabilizing controller

∇V (x)(f (x) + ω(x)) ≤ -W (x), |x 2 -x 3 | ≤ 1.
u(t) = q 2 (1-e -qτ ) 2 ω(x(t)) -2e -qτ ω(x(t -τ ))+ e -2qτ ω(x(t -2τ )) -2qz 2 (t) -q 2 z 1 (t), where ω(x) = -sin πx 2 1 [-2,2] (x). (27)
To illustrate our results, we used Mathematica to simulate (25) in closed loop with (27) and the preceding parameter values, with the initial states x = z 1 = z 2 = 1, assuming that x takes the constant value 1 at negative times in order to initialize the controller, and we plotted our results in Figure 1. They illustrate the rapid convergence of the states to 0.

B. Controls of Arbitrarily Small Amplitude

Consider the special case

   ẋ(t) = y 2 (t) ẏ2 (t) = -qy 2 + y 1 (t) ẏ1 (t) = -qy 1 + v(t)
k = 2, F(t, x, p) = p, V (x) = x 0 sat(s)ds, W (x) = εsat 2 (x), C = ε, r 0 = 1/ √ ε, r 1 = √ ε, r 2
= 0, and r 3 = 1 in Assumption 2, and any choices of the constants ε > 0, τ > 0, and q > 0 and the delay T ≥ 0 such that

√ 10τ ε(2τ + T ) 2 q 1-e -qτ 2 < 1. (29)
Then our theorem ensures that all solutions of (28) in closed loop with the control

v(t) = -q 2 ε (1-e -qτ ) 2 sat(x(t -T ))- 2e -qτ sat(x(t -τ -T )) + e -2qτ sat(x(t -2τ -T )) (30)
from ( 6) asymptotically converge to 0. We can allow controls of arbitrarily small sup norm, and any delay T ≥ 0, by choosing ε > 0 small enough. Also, only the value of sat(x) is needed. To illustrate our control (30), we used Mathematica to simulate (28) in closed loop with (30) with the initial state vector (-1, -1, 1) the parameter choices τ = 0.35, q = 0.345, ε = 0.05, and T = 0.7, which satisfy (29), and we report our results in Figure 2. They show convergence of the states and control values to 0.

C. Normal Forms for Underactuated Systems

Normal form systems (corresponding to (1) with h = 0) commonly arise when modeling two degree-of-freedom (or DOF) underactuated systems. For instance, [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF]Chapter 3] shows that the acrobot, inertia-wheeled pendulum, and translational oscillation with an eccentric rotational actuator (or TORA) models admit changes of coordinates that produce ẋ1 = m -1 11 (z 1 )x 2 , ẋ2 = g 1 (x 1 -γ(z 1 ), z 1 ), ż1 = z 2 , ż2 = u

Our dynamics (1) also cover the planar vertical takeoff and landing (or PVTOL) model [START_REF] Gruszka | Tracking control and robustness analysis for PVTOL aircraft under bounded feedbacks[END_REF], [START_REF] Olfati-Saber | Global configuration stabilization for the VTOL aircraft with strong input coupling[END_REF] after suitable changes of coordinates. Due to space constraints, we will not discuss these examples here, but it is worth studying how our novel control designs could also benefit the study of these systems.

VII. CONCLUSIONS

We introduced a key extension of our converging input converging state assumption from [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF], which allowed us to extend the bounded backstepping results from [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] for systems with one integrator to higher dimensional partially linear systems with an arbitrary number of integrators. We gave Lyapunov-like sufficient conditions for our new assumption to hold. Our control formulas do not require any Lie derivatives, nor do they require any differentiability of the nominal control for the nonlinear subsystems. Our main controls ensure globally asymptotic stability to the equilibrium, and are a sum of (i) a bounded output term involving the state of the nonlinear subsystem and (ii) a linear combination of the states of the linearized part that can be scaled by an arbitrarily small constant. We also covered input delays, using a nested saturation approach that produced globally asymptotically stabilizing bounded controls. We plan to apply our work to the PVTOL and other engineering systems that can be put in triangular form, e.g., by combining our methods with ideas from [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF], [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF].

Fig. 1 .

 1 Fig. 1. Simulating System (25) with Control (27), q = 0.345, and τ = 0.029. All Initial States Set to 1. Upper Left: x(t). Upper Right: z 1 (t). Lower Left: z 2 (t). Lower Right: u(t).

  44|x|, and | sin(πx/2)| ≤ (π/2)|x| when x ∈ [-0.8, 3/2], so we can choose r 0 = 1, r 1 = π/2, r 2 = 1.44, r 3 = 1, and C = π/2. Hence, the sufficient condition M * < 1 from Proposition 1 (for lim t→∞ x(t) = 0 to hold) is satisfied by the choices q = 0.345 and τ = 0.029, since these choices give

Fig. 2 .

 2 Fig.2. Simulating (28) with Control (30), τ = 0.35, q = 0.345, ε = 0.05, and T = 0.7. Initial State Vector: (-1, -1, 1). Upper Left: x(t). Upper Right: y 1 (t). Lower Left: y 2 (t). Lower Right: v(t).

  M = [m ij ] ∈ R 2×2 is the inertia matrix, g 1 (x, z) = -∂V /∂x, Vis the potential energy, and γ(z 1 ) = z1 0 m -1 11 (θ)m 12 (θ)dθ.

  ). Upper Left: x(t). Upper Right: y 1 (t). Lower Left: y 2 (t). Lower Right: v(t).

	a constant. The assumptions of Proposition 1 are satisfied,
	by choosing

We prove Lemma 1. Set y k = z 1 . For all i ∈ {1, 2, . . . , k-1}, we recursively construct linear functions of the form

for some constants r ij > 0, and functions y k-i of z, such that these two equalities hold:

where

), so we can take y k-1 = z 2 + qz 1 and L 1 (z 1 ) = z 1 . Assume that we found L i and y k-i satisfying (A.2) for all i ∈ {1, . . . , } for some < k -1. We show how to satisfy (A.2) for i = + 1. Our requirement (A.2) for i = + 1 is

which allows us to solve for L +1 and y k--1 . Since z 1 = y k , we can also use (A.2) to solve for z in terms of y, so this provides P . Also, by differentiating both sides of (A.2) with the choice i = k -1, we obtain +q(z k + qL k-1 (Z k-1 )).

(A.5)

This proves the lemma.


Shenle Pan
email: shenle.pan@mines-paristech.fr

Damien Trentesaux
email: damien.trentesaux@univ-valenciennes.fr

Yves Sallez
email: yves.sallez@univ-valenciennes.fr

Specifying Self-organising Logistics System: openness, intelligence, and decentralised control

Keywords: Logistics, Self-organisation, Openness, Intelligence, Decentralised control, Holonic and Multi-Agents System

This paper contributes to the development of a novel paradigm in logistics, i.e. self-organising logistics system (SoLS). A SoLS can be considered as a logistics system that can function without significant intervention by human, and without central control by software. It is functioning based on contextual local interactions. By such definition, SoLS could be a powerful solution to manage nowadays logistics that is much larger and complex than ever before. The goal of this paper is to provide a comprehensive and in-depth discussion to specify expected advantages and functionalities of SoLS, supported by some recent emerging concepts and technologies. The paper also attempts to provide a theoretical framework for the future work.

Introduction

Nowadays logistics system is much larger and complex than ever before, and, therefore, difficult to manage as a whole. One may have heard about many emerging technologies, concepts and paradigms that are proposed as solutions to cope with this difficulty, e.g. intelligent product or intelligent logistics [START_REF] Mcfarlane | Intelligent logistics: Involving the customer[END_REF], Holonic and Multi-Agents System [START_REF] Barbosa | Dynamic self-organization in holonic multi-agent manufacturing systems: The ADACOR evolution[END_REF], Cyber-physical System [START_REF] Lee | Cyber physical systems: Design challenges[END_REF], or Physical Internet [START_REF] Ballot | The Physical Internet: The Network of Logistics Networks[END_REF]. One may be interested in the following question: what could be the next era of logistics if all these solutions have a joint effort to develop the logistics? This paper discusses one of the possible eras, namely self-organising logistics system (SoLS hereinafter).

Self-organisation is a popular term in many research fields, for example in computer science, human society and behaviour, and biology [START_REF] Serugendo | Self-organisation: Paradigms and applications[END_REF][START_REF] Aliu | A survey of self organisation in future cellular networks[END_REF][START_REF] Serugendo | Self-organisation and emergence in MAS: An overview[END_REF]. However, in the field of logistics, it is still a developing paradigm. As suggested in [START_REF] Bartholdi Iii | Self-organizing logistics systems[END_REF], a SoLS is a logistics system that "can function without significant intervention by managers, engineers, or software control". The suggested definition implies some potential advantages of such systems to today's large-scale and complex logistics. Based on these works, we attempt to provide a wider and deeper vision to specify SoLS, by defining its main desired functionalities and analysing its advantages in the context of logistics organisation. The advantages are discussed qualitatively from perspectives on logistics performance and on sustainability, to illustrate that SoLS could make logistics more autonomous, efficient and effective, i.e. a more sustainable system.

For the sake of providing an overview and insightful discussion of SoLS, this paper proposes to discus several main functionalities of future SoLS: openness, intelligence, and decentralised control. Openness means that a SoLS should allow actors (e.g. shippers, service providers, or customers) or assets (e.g. trucks, containers, or warehouses) to easily join and leave the system, for the sake of fast reconfiguration. Intelligence emphasises the object-based capability of local real-time communication and activeness [START_REF] Sallez | The Augmentation Concept: How to Make a Product "Active" during Its Life Cycle[END_REF]. Decentralised control focuses on collaborative rules and communication protocols designing, i.e. environment, for SoLS. It should be noticed that decentralised control for SoLS aims to avoid unexpected or disastrous outcomes of the system, but not to optimally plan activities or control objects. Further, we will also discus expected advantages and performance of SoLS with regard to current challenges in logistics.

Relevant Literature

Self-organisation in Logistics

There is limited literature investigating self-organisation in the field of logistics. References [START_REF] Wycisk | Smart parts" supply networks as complex adaptive systems: analysis and implications[END_REF][START_REF] Choi | Supply networks and complex adaptive systems: control versus emergence[END_REF][START_REF] Hülsmann | Autonomous Cooperation-A Way to Cope with Critical Incidents in International Supply Networks (ISN)? An Analysis of Complex Adaptive Logistic Systems (CALS) and their Robustness[END_REF] propose to consider supply network as complex adaptive system where self-organisation is considered as an internal mechanism between agents. Differently, reference [START_REF] Bartholdi Iii | Self-organizing logistics systems[END_REF] considers self-organisation as a organisational paradigm to cope with complex assembly lines, by using "bucket brigade'' assembly lines as illustration. Other studies from the same standpoint can be found in the field of manufacturing, e.g. [START_REF] Barbosa | Dynamic self-organization in holonic multi-agent manufacturing systems: The ADACOR evolution[END_REF][START_REF] Bousbia | Self-organization in distributed manufacturing control: state-of-the-art and future trends[END_REF].

This paper is in line with [START_REF] Bartholdi Iii | Self-organizing logistics systems[END_REF] and extended to investigate SoLS functionalities for effectiveness and efficiency, which is particular important to cope with current and future challenges in logistics, see Table 1. From our point of view, a SoLS is an open, intelligent and holonic logistics system that aims to harmonise and lead individuals within the system towards a system-wide common goal, without significant human intervention from outside. An individual within SoLS can be an object (e.g. a truck), a manufacturer, a service provider (e.g. 3PL), a receiver (e.g. a customer), even a supply chain or a supply network as a whole. Assuming that individuals may have different constraints and objectives, e.g. service rate improvement or cost reduction, we argue that SoLS should have the ability to respect individual's constraints and objectives, meanwhile leading them towards a common goal -the sustainability for exam-ple. To this end, individuals should be coordinated by system-wide well-designed rules, i.e. the system environment [START_REF] Narahari | Game Theoretic Problems in Network Economics and Mechanism Design Solutions[END_REF].

Expected Advantages and Performance

Based on the suggested definition, some advantages of SoLS can be expected. We here discuss qualitatively expected advantages according to six main indicators of performance measurement, as described in Table 1:

Table 1.
Advantages and expected performance of SoLS functions

Description

SoLS expected advantages Effectiveness

How well a goal is adequately met? [START_REF] Mentzer | An efficiency/effectiveness approach to logistics performance analysis[END_REF] Individuals should be able to make autonomous decision to meet a given goal

Efficiency

How well the resources expended are utilised? [START_REF] Mentzer | An efficiency/effectiveness approach to logistics performance analysis[END_REF] Individuals should intelligently use the on-hand resources Agility Ability of a system to rapidly reconfigure [START_REF] Bernardes | A theoretical review of flexibility, agility and responsiveness in the operations management literature: Toward a conceptual definition of customer responsiveness[END_REF] Individuals should have good connectivity to each other and to environment for rapid reconfiguration Flexibility Ability of a system to change status within an existing configuration [START_REF] Bernardes | A theoretical review of flexibility, agility and responsiveness in the operations management literature: Toward a conceptual definition of customer responsiveness[END_REF] Individuals should be able to connect and adapt to each other to provide flexible solution

Resilience

Ability of a system to return to its original state or move to a new, more desirable state after being disturbed [START_REF] Bernardes | A theoretical review of flexibility, agility and responsiveness in the operations management literature: Toward a conceptual definition of customer responsiveness[END_REF] SoLS should be able to dynamically selfreconfigure to adapt to unpredictable and disruptions

Sustainability

Ability of a system to protect, sustain and enhance the human and natural resources that will be needed in the future while meeting a desired goal [START_REF] Labuschagne | Assessing the sustainability performances of industries[END_REF] SoLS should function on the basis of common goal of sustainability, by protecting human and natural resources

Related Concepts and Paradigms

Before illustrating the features and functions of SoLS, it is necessary to provide consensual definition to the emerging concepts and paradigms used in this paper. The related concepts and paradigms are classified here according to their objective: conceptual criteria, technical paradigm, technology and application. Table 2 gives a short description from the literature adapted for the purpose of this paper.

Table 2. Concepts and paradigms related to SoLS

Classes Short description

Conceptual Criteria

Openness: system boundary is unset but variable and flexible [START_REF] Ballot | The Physical Internet: The Network of Logistics Networks[END_REF]. Adaptation: objects adapt to each other for coordination of activities and to leads to more efficient processes [START_REF] Jahre | Adaptation and adaptability in logistics networks[END_REF]. Reconfiguration: add, remove and modify logistics activities and functions [START_REF] Koren | Reconfigurable manufacturing systems[END_REF].

Self-organisation: a system can function without significant intervention by managers, engineers, or software control [START_REF] Bartholdi Iii | Self-organizing logistics systems[END_REF].

Technical Paradigm

Cyber-Physical System: integrations of computation with physical processes [START_REF] Lee | Cyber physical systems: Design challenges[END_REF].

Intelligent and active products: a product able to self-identify its state, and able to send information once certain pre-conditions are met [START_REF] Sallez | On the activeness of intelligent Physical Internet containers[END_REF].

Multi-agents System: system to coordinate intelligent behaviour among a group of agents which are autonomous and flexible computational systems [START_REF] Botti | ANEMONA: A multi-agent methodology for Holonic Manufacturing Systems[END_REF].

Holonic System: a system composed of holons. Holons are autonomous, cooperating and potentially recursive decisional entities, which can simultaneously be part or sub-whole of the system. A holon is most of the time composed of a physical part along with a digital one [START_REF] Giret | Holons and agents[END_REF].

Complex Adaptive System: a system that emerges over time into a coherent form, and adapts and organises itself without any singular entity deliberately managing or controlling it [START_REF] Choi | Supply networks and complex adaptive systems: control versus emergence[END_REF].

Technology

IoT: Internet of Things is the network of physical objects that are connected via Internet and technologies providing unique addressing schemes, such as RFID tags, sensors etc. [START_REF] Atzori | The Internet of Things: A survey[END_REF] ICT: Information and communications technology for the communication between objects or systems [START_REF] Shen | Applications of agent-based systems in intelligent manufacturing: An updated review[END_REF] Embedded systems: combination of computer hardware and software, and perhaps additional mechanical or other parts, designed to perform a dedicated function inside a single identified system [START_REF] Barr | Embedded Systems Glossary[END_REF].

Application

Intelligent logistics: a will to plan, manage or control logistics activities in a more intelligent way [START_REF] Mcfarlane | Intelligent logistics: Involving the customer[END_REF]. It may rely on intelligent products, data techniques etc. Physical Internet: the network of logistics networks using the internet analogy [START_REF] Ballot | The Physical Internet: The Network of Logistics Networks[END_REF].

After this short literature review, the next section introduces specifications of the SoLS.

Specifying Self-organising Logistics System

To construct an effective and efficient SoLS, and to avoid some unexpected or undesirable outcomes, we propose several functionalities for the system. This part discusses three that are identified as crucial factors: openness, intelligence and decentralised control.

Openness

Openness in SoLS means the boundary of the system is unset and open so that individual (actors, assets, supply chain, etc.) can easily join and leave the system. An illustrative example can refer to connecting a computer to the Internet from all over the world for different purpose, like searching information or communicating with others. Individuals join the SoLS to provide, procure or share logistics assets (truck, facilities etc.) and logistics service (delivering, planning etc.). By that, SoLS becomes fast extendable and reducible to answer highly flexible logistics requirements, as shown Fig. 1. Here we discuss several functions that are essential to this end.

Function 1.1 Connectivity: this function means that individual can easily connect

with others or and the environment. For example in Fig. 1, individuals outside can easily connect to the system or other individuals within, if rules and standards are respected. System-wide modularisation and standardisation of physical assets (e.g. Physical Internet Container [START_REF] Sallez | On the activeness of intelligent Physical Internet containers[END_REF]), information systems (e.g. ICT) and organisation models is crucial to enable good connectivity. Examples can refer to Physical Internet [START_REF] Ballot | The Physical Internet: The Network of Logistics Networks[END_REF][START_REF] Montreuil | Toward a Physical Internet: meeting the global logistics sustainability grand challenge[END_REF]. Notions of interoperability must equally be considered to allow an adequate communication between the different actors.

Function 1.2 Reconfiguration:

Once the system receives new logistics requests (transport, stocking etc.), it should be able to add or change some current functions inside to fulfil the requests. The reconfiguration function is particularly important to cope with disruptions, i.e. the system can rapidly self-reconfigure [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF].

Function 1.3 Adaptation:

individuals should adapt to each other and to environment, in order to rapidly build up effective and efficient coordination between them [START_REF] Choi | Supply networks and complex adaptive systems: control versus emergence[END_REF].

Intelligence

Intelligence in SoLS means that every individual within the system is able to make and execute autonomous decision, and to interact with other individuals and with the environment. For that, they should have the ability to collect, store and process information from other individuals (their state and decision) and from environment (rule modification). We propose that IoT technology is a fundamental support to such requirements. Further, two important functions are identified for such a functionality.

Function 2.1 Activeness:

individuals should have to ability to collect information from other individuals (their state and decision) and from environment (rules and any modifications), then to store and process the information for autonomous decisionmaking. Further, they should also be able to send information to others, to advertise the changes of their state and to take decision. Each event can be seen as a change to the system, e.g. a warehouse is disrupted, a truck is full, or an order is cancelled. Activeness is therefore an important functionality for SoLS.

Adaptation

Function 2.2 Autonomous acting: once a decision has been made, systems should also be able to apply and execute the decision. This function could be relying on embedded systems of autonomous operating assets, autonomous supply chain or autonomous network, including sensors and actuators inside decisional loops. Fig. 2 illustrates some existing and emerging examples of intelligence applied in logistics. It is obvious that some of the three functions are questionable under current organisation and technologies in logistics. However, emerging innovative technologies may help us to project the future of intelligence in SoLS, e.g. drone, autonomous operating truck, or Kiva.

Decentralised Control

As stated in [START_REF] Choi | Supply networks and complex adaptive systems: control versus emergence[END_REF], "imposing too much control in a complex system detracts from innovation and flexibility; conversely, allowing too much emergence can undermine managerial predictability and work routines". In this paper we propose to specify SoLS as rule-based decentralised control system, for the sake of effectiveness and efficiency [START_REF] Bakule | Decentralized control: An overview[END_REF]. More precisely, rule-based decentralised control in SoLS aims to help the autonomous decision-making of individuals within the system. It has a twofold objective: to avoid unpredictable and undesirable outcomes, and to lead individuals towards a system-wide common goal. We propose that decentralised control should rely on system-wide well-designed rules and individual-wide protocols, which should be respected when local autonomous decisions occur [START_REF] Bakule | Decentralized control: An overview[END_REF]. In such a context, we propose that the decentralised control in SoLS should be modelled as a holonic and multi-agent system [START_REF] Serugendo | Self-organisation and emergence in MAS: An overview[END_REF][START_REF] Botti | ANEMONA: A multi-agent methodology for Holonic Manufacturing Systems[END_REF], with the following functions: Function 3.1 Holonic multi-agent system: we propose to model SoLS as holonic multi-agents system (HMAS), since controlling in such a system should be at individual level, i.e. at holon level (whole, part, or sub-whole). Each holon is composed of a physical part mirrored by an agent in the model. Each agent is self-controlling so that no controlling agent is necessary in the HMAS. Function 3.2 Rule-and protocol-based: rules here represent the common regulations of the system that every individual should respect; and protocols stand for communication protocols that allow individual communicate to each other, i.e. interoperability. Take digital internet as an example, in which rules are law and protocol is internet protocol like TCP/IP. In other words, rules are designed for meeting common goal; and protocols for effective and efficient communication.

Intelligence

Fig. 3 presents system architecture of rule-and protocol-based HMAS. We consider here a transport request as an example of logistics activities. As per current organisation, driver (P1) and transportation means (P2) are essential for it. In SoLS scenario, a HMAS model is employed to help P1 and P2 to make decisions. More precisely, each of P1 and P2 is mirrored by an agent in the MAS model. Rules of logistics organisation (e.g. speed limit, preconizing multi-modal transport, etc.) and protocols between objects (e.g. driving instruction, autonomous driving control etc.) are translated as the environment of the MAS model. By that, P1 (or V1) and P2 (or V2) should respect the environment when they make autonomous decisions. The MAS model here thus aims to coordinate and cooperate the agents -physical objects, but not to control.

Conclusion

This paper aims to contribute to specify advantages and functionalities of selforganising logistics system (SoLS), which is a novel paradigm in logistics. Openness, intelligence and decentralised control are the three main functionalities proposed and discussed in this paper as vital functionalities of SoLS to cope with the current and future challenges in logistics. Real world examples can refer to the recent DHL research project -Parcelcopter SkyPort 1 . Basically, the idea is to conjointly use autonomous drones (Parcelcopter) and automated parcel stations (Packstations) to cope with demanding logistics challenges such as "same day delivery", especially in rural zones. Drones and Packstations together can be seen as a SoLS. Once a package to be delivered is arrived at Packstation with known constraints (volume, delivery location and time etc.), the system will autonomously plan and optimise the delivery. Only monitoring is necessary from outside. We may assume that, supported by the emerging technologies and techniques, more and more examples of SoLS would be found and realised.

The paper provides also an insightful overview and theoretical framework for the future work. For example, organisational and technical issues related to SoLS, such as protocol design, coordination or security issues, can be investigated through mechanism design theory for example. Quantitative researches through simulation and optimisation approaches are also necessary to investigate the performance and viability of SoLS in real-life cases.

Fig. 1 .

 1 Fig. 1. Open logistics system

Fig. 2 .

 2 Fig. 2. Technologies and Techniques of intelligence applied in logistics

Fig. 3 .

 3 Fig. 3. Modelling SoLS as rule-and protocol-based holonic multi-agent system

  1 http://www.dpdhl.com/en/media_relations/specials/parcelcopter.html

	P: physical objects V: virtual (Intelligent) agents	L: logistics activities
	P1	V1	L1
	P2	V2	L2
	.	.	.
	.	.	.
	.	. Multi	.
	. . . Holon System	Agent . System . . (MAS)	. . . Logistics Organisation
	.	.	.
	.	.	.
	Pn	Vn	Ln
	Protocols	Environment	Rules (and Requests)

Acknowledgement

The authors like to thank the French National Research Agency (ANR) for the support to Project PI-NUTS (ANR-14-CE27-0015).


I Sochet

• S Eveillard

J Y Vinçont

P F Piserchia

• X Rocourt

Influence of the geometry of protective barriers on the propagation of shock waves

Keywords: Protective barrier, Explosion effect, Shock wave, Blast load, Detonation

The protection of industrial facilities, classified 1 as hazardous, against accidental or intentional explosions 2 represents a major challenge for the prevention of personal 3 injury and property damage, which also involves social and 4 economic issues. We consider here the use of physical barri-5 ers against the effects of these explosions, which include the 6 pressure wave, the projection of fragments and the thermal 7 flash. This approach can be recommended for the control 8 of major industrial risks, but no specific instructions are 9 available for its implementation. The influence of a protec-10 tive barrier against a detonation-type explosion is studied 11 in small-scale experiments. The effects of overpressure are 12 examined over the entire path of the shock wave across the 13 barrier and in the downstream zone to be protected. Two 14 series of barrier structures are studied. The first series (A) 15 of experiments investigates two types of barrier geometry 16 with dimensions based on NATO recommendations. These 17 recommendations stipulate that the barrier should be 2 m 18 higher than the charge height, the thickness at the crest 19 should be more than 0.5 m, while its length should be equal 20 to twice the protected structure length and the bank slope 21 should be equivalent to the angle of repose of the soil. The 22 second series (B) of experiments investigates the influence 23 of geometrical parameters of the barrier (thickness at the 24 crest and inclination angles of the front and rear faces) on 25 its protective effects. This project leads to an advance in 26 our understanding of the physical phenomena involved in

Introduction

At industrial sites, whether public or private, one of the major concerns in modern society is the safety of goods and people with respect to the risks associated with explosions of either accidental or malicious origin. The disasters at AZF in 2001 (Toulouse, France), at the Nitrochimie dynamite factory in 2003 (Billy-Berclau, France), at a fireworks storage facility at Kolding in 2004 (Denmark) and at the West Fertilizer Company plant in 2013 (Texas, USA) are examples showing that "zero risk does not exist". To limit the occurrence of new accidents, companies have a panoply of safety measures involving prevention or protection against the risks inherent in any accident.

The detonation of an explosive charge causes mechanical effects, such as overpressure, heating and possible effects related to the projection of fragments. The presence of a protection barrier (walls, fill and slope) ensures the easy protection of installations and people against the heating effects of an explosion and the projection of fragments. However, protection from the effects of overpressure is not guaranteed simply by the presence of a physical protection barrier of unspecified form. Indeed, the interaction of a shock wave R e v i s e d P r o o f I. Sochet et al. with a structure is difficult to predict and depends on many parameters.

To our knowledge, few studies have been carried out on the optimization of protective barriers. With this objective in mind, Zhou and Hao [START_REF] Zhou | Prediction of airblast loads on structures behind a protective barrier[END_REF] used digital simulations to study the effectiveness of a protective wall placed in front of a building. Their study demonstrated that the reduction of blast load does not depend solely on the height of the protective wall, the distance between the centre of the explosion and the barrier, the distance between the barrier and the building or the height of the building. The effect of wall thickness was studied but did not contribute significantly to the blast load behind the wall.

The medium-scale experimental study carried out by Allain [START_REF] Allain | Barricade Influence on Blast Wave Propagation[END_REF] comprised barriers with two inclined slopes of 45 • without a flat crest and using a height of 1.5 m. The tests were conducted using spherical charges of TNT (8 and 37 kg) and composition B (50 kg). The distance between the charge and the obstacle (d) varied from 0.75 to 17 m/kg 1/3 . These medium-scale tests demonstrated that a barrier, according to its geometry and form factors, can lead to various flow modes. The protective barrier considered in this case accentuated the positive overpressure of the shock wave and thus did not show a protective effect. These results have been confirmed by the simulations of Borgers [START_REF] Borgers | Blast walls reviewed[END_REF], who noted that the relaxation on the rear face of a Mach stem results from reflection on the front face or from an incident wave (for a regular reflection) for certain configurations. This can lead to an accentuation of the reflection of the shock wave on the ground downstream of the obstacle (according to the nature of the wave and the angle of inclination of the wall).

Thus, the recommendations of NATO evoked in the "Guide to good practices in pyrotechnics" [START_REF]Guide SFEPA n • 9. Guide des Bonnes Pratiques en Pyrotechnie-Edition Syndicat des Fabricants d'Explosifs[END_REF] estimate a minimal thickness of 0.5 m (e > 0.5 m) and specify that the height of the barrier must be more than 2 m higher than the highest point of the charge.

Therefore, the objective of the present study was to evaluate the protective effects related to the overpressure of barriers according to their form and size. The barriers represent protective obstacles placed in the path of the shock wave that are intended to mitigate its effects (such as overpressure and impulses). The obstacle is assumed to be infinitely rigid, so the reflection is considered as "perfect" over all its surface.

In this study, we consider a generic barrier typology (Fig. 1) with the following preset parameters: mass charge (W), height and width at crest of the obstacle (H, e), slope angles of the barrier faces with respect to the ground (α), distance between charge and barrier (d). In this study, the explosive charge is placed only at ground level.

Small-scale experiments are carried out using three types of barrier model. Small-scale tests have many advantages. Indeed, their cost is low and accurate laboratory methods can be applied. Moreover, the test conditions are well controlled and independent of the weather, the reproducibility of test 111 conditions can be readily ensured, and it is easy to establish 112 parametric studies and vary the geometric dimensions of the 113 studied structures. The experiments are conducted at a small scale on a test 117 bench [START_REF] Éveillard | Towards a fastrunning method for blast-wave mitigation by a prismatic blast wall[END_REF][START_REF] Sochet | Effect of a gas charge explosion at the closed end of a gas storage system[END_REF][START_REF] Trélat | Strong explosion near a parallelepipedic structure[END_REF].

118

The explosive charges are made up of a stoichiometric 119 mixture of propane and oxygen gases. The hemispherical 120 charges used are positioned on the ground, initially confined 121 in a soap bubble. Two charge radii are used: R 1 = 0.06 m 122 and R 2 = 0.03 m. The explosive charge is initiated by an 123 exploding wire [START_REF] Sochet | Effect of a gas charge explosion at the closed end of a gas storage system[END_REF]. In the analysis of results (next section), 124 the charge radius will be noted by R0 with its specified value. 125 The zone of experimentation (Fig. 2) is divided into two 126 sectors relative to the centre of the explosive charge [START_REF] Éveillard | Towards a fastrunning method for blast-wave mitigation by a prismatic blast wall[END_REF]: the 127 free field zone and the zone of investigation. In the free field 128 zone, the incident shock wave resulting from detonation of 129 the explosive charge propagates without interaction with the 130 structure. The pressure sensors placed in this zone, called 131 "reference sensors", are used to check the reproducibility 132 of the detonations. The structure is placed in the zone of 133 investigation along with the explosive charge. Within this 134 zone of investigation, an additional zone can be identified in 135 which the protection barrier prevents arrival of the wave.

136

The dimensions and positioning of the protection barrier 137 depend on the studied configuration (series 1 and 2). Pres-138 sure sensors (piezoelectric, PCB) are placed flush with the 139 structure to detect possible couplings of the various physi-140 cal phenomena (reflection, relaxation and recombination of 141 shock waves) as well as downstream from the barrier to study 142 the protective effect. Each position is identified by a distance 143 in direct line with the charge, which is defined by the dis-144 tance between the position of the sensor and the centre of the 145 explosive charge.

R e v i s e d P r o o f

Influence of the geometry of protective barriers on…

Tested configurations

The geometrical configurations and dimensions of the barriers studied in small-scale experiments are designed to analyse several physical phenomena (reflection, relaxation and recombination of shock waves) as well as the protective effect of barriers according to their geometry. The test campaigns included two series of barriers and a configurationfree field (without obstacles), allowing characterization of the evolution of various mechanical wave parameters for a gas load as a function of the distance travelled by the wave. The first series of barriers (A: 1A, 2A) is designed to study two barrier geometries based on the recommendations of NATO [START_REF]Guide SFEPA n • 9. Guide des Bonnes Pratiques en Pyrotechnie-Edition Syndicat des Fabricants d'Explosifs[END_REF]. The second series (B: 1B, 2B) is designed to study the influence of two geometrical parameters of the barrier (thickness at crest and slope angles of the front and rear faces) on the protective effect of the barrier. In this study, the impact of bypassing waves is not analysed, which means that, for these two experimental series, we assume a protective barrier of infinite length.

Configurations with barriers-series A

The first series of protective barrier geometries is designed based on the recommendations of NATO for two gas loads (R 1 and R 2 ) and using the Hopkinson scale [START_REF] Kinney | Explosive Shocks in Air, 2nd edn[END_REF] for a scaling factor k (k = 15). The protective barriers, 1A and 2A, are dimensioned according to the recommendations for the two tested gas loads.

The charge radius (R 1 ) is 0.06 m, and its detonation releases an energy of 13.75 × 10 -3 MJ (E k on scale 1/k). Thus, for example, for a dimensional scaling factor k of 15, the released energy on the real scale is 46.41 MJ (E 1 on a scale of 1/1).

The distance between the centre of the charge and the obstacle varies between 0.07 and 0.10 m (or between 1.05 and 1.50 m on the real scale). The dimensions of the first protective barrier (1A) follows the recommendations of NATO for a gas load of radius R 1 . The height of the barrier can be 182 calculated from the following equation ( 1):

183 H 1/k = 2 k + Radius or H 1/k ≈ 0.19 m and H 1/1 = 2.85 m, 184 charge radius (R 1 ). ( 1
)
185

The thickness at the crest of the protection barrier can be 186 estimated from the following equation ( 2):

187 e min,1/k = 0.5 k or e min,1/k ≈ 0.03 m. (2) 188
The minimal width of the protective barrier at the top of 189 the explosive charge can be evaluated from the following 190 equation ( 3):

191 e charge,1/k = 0.9 k or e charge,1/k ≈ 0.06 m. ( 3
) 192
The first protective barrier (1A) is dimensioned based on the 193 recommendations for a gas blast load of radius R 1 (Fig. 3). 194

The length of barrier 1A is fixed at 0.80 m (12 m on real 195 scale).

196

The second protective barrier (2A) is dimensioned based 197 on the height recommended by NATO for a gas load of radius 198 R 2 = 0.03 m. To allow a proper comparison of the two 199 geometries, the wave path length (i.e., distance travelled by 200 the shock wave) over barrier 2B must be almost identical 201 or close to that for barrier 1A (Fig. 4). The height is thus 202 estimated at 0.16 m, and the thickness is fixed at 0.06 m (dis-203 tances travelled over the trapezoidal profile are 0.527 m for 204 barrier 1A and 0.513 m for barrier 2A, with a difference of 205 3 %). This second geometry allows us to analyse the influ-206 ence of barrier height and thickness on the protective effect 207 for two explosive loads (R 1 and R 2 ). Barriers 1A and 2A 208 have the same length: L = 0.80 m.

209

The distances travelled by the wave passing over the top 210 crest of the barriers 1A (Fig. 3) and 2A (Fig. 4) are very close, 211 0.527 and 0.513 m, respectively. Barrier 1B has a front face at 45 • and a rear face at 90 • (Fig. 5). The distance of the path of the shock wave over the barriers 1B and 2B is fixed at 0.80 m. Barrier 2B is built with an inclination angle of 90 • for the 229 front face and 45 • for the rear face (Fig. 6). We first examine the variation of overpressure as a function 235 of the scaled distance of the path of the shock wave over the 236 barrier (Fig. 8). The reduced distance Z is defined by

237 Z = R 3 √ m , Z = [m kg -1/3 ], (
P + P 0 = 808 × 1 + Z 4.5 2 1 + Z 0.048 2 × 1 + Z 0.32 2 × 1 + Z 1.35 2 252 ( 5
)
253

For a given gas load (stoichiometric propane-oxygen com-254 bustion), the evolution of the overpressure of a blast wave 255 ( P + ) can be estimated from equations ( 5) and ( 6) for an (7)

Hence, Fig. 8 shows clearly that all overpressure values on the front face are higher than the overpressures corresponding to the free field. The divergent spherical incident wave is reflected on the front face of the protective barrier. The incident Mach number (M 1 ) of the shock wave can be obtained from the maximum of incident overpressure ( P + ) and the initial pressure (P 0 = 101,325 Pa) and γ = 1.4 by using equation ( 7):

M 1 = 1 2 × γ × (γ + 1) × P 0 + P + P 0 + (γ -1) . ( 8
)
The reflection mode (regular reflection or Mach reflection) can vary according to the position and dimensions of the barrier (d, H, α 1 ). The point of transition between these reflection modes can be determined by the simplified relation (8) due to Kinney [START_REF] Kinney | Explosive Shocks in Air, 2nd edn[END_REF], expressed as a function of the Mach number of the incident wave (M 1 ): This physical phenomenon leads to an attenuation of the maximum overpressure in the field close to the edge between the front face and the top of the protective barrier for Z = 3-3.3 m kg -1/3 for R 2 loads (0.06 m) and Z = 5.9-6.4 m kg -1/3 for R 1 loads (0.03 m) (Fig. 8). The maximum overpressure values are less than the free field values, thus contributing to the appearance of a protective effect downstream from the barrier. During wave propagation on the top part of the barrier, maximum overpressure is attenuated by the distance covered by the wave (network of relaxation waves downstream from the shock front).

β 1lim = 1.75 M 1 -1 + 39. ( 9
)
The variation in maximum overpressure between the three configurations arises from the intensity of the incident wave at the top of the barrier.

In this zone of interest, the difference between the two obstacles (barriers 1A and 2A) corresponds to the thickness of the barrier (greater thickness for barrier 2A, e 2A = e 1A ×2, the attenuation effect per distance covered is thus slightly more marked), as shown in Fig. 8.

Relaxation on the downstream face of barriers 1A and 2A

During the passage of the shock wave between the top and the rear face downstream of the barrier, the wave is subject to a second relaxation.

R e v i s e d P r o o f

Influence of the geometry of protective barriers on… In the case of a regular reflection, a wave is formed on the surface and is propagated in the opposite direction to the incident wave (thus, in the direction of the blast load). This reflected wave results from the reflection of the incident wave on the barrier and interacts with the interface between the air and the detonation products shortly after the end of the detonation. Resulting overpressures are higher in the case of barriers with a vertical face with respect to the explosion than in the case of barriers with an inclined face of 45 • .

Figure 10 shows the evolution of the maximum reflected overpressure for the various configurations of barriers 1B and 2B.

Relaxation on the top of barriers 1B and 2B

The Mach stem resulting from reflection of the shock wave on the front face of the barrier undergoes a relaxation at the top of the wall. The angle of deviation of this first relaxation varies according to the inclination angle of the front face of the barrier: 45 • for barrier 1B and 90 • for barrier 2B.

A P = P + r P + i , ( 12
)
405
where P + i is the maximum incident overpressure in the 406 free field [START_REF] Sochet | Effect of a gas charge explosion at the closed end of a gas storage system[END_REF] and P + r is the maximum overpressure in the 407 presence of the protection barrier. Thus, if A P tends towards 408 zero, then the maximum protective effect is characterized by

409 a new scaled distance R barrier [m MJ -1/3
] defined as follows:

410 The energy released by the propane-oxygen reaction is 421 obtained by multiplying the energy per unit volume E v by the volume V of the spherical charge: E = E v × V . By considering the density ρ of the gas mixture, the relationship between R barrier and Z (4) can be derived as follows:

R barrier = R E 1 -d √ d 2 +S 1/3 , (13) load
R barrier = Z ρ E v 1/3 1 1 -d √ d 2 +S 1/3 (14)
Nevertheless, the normative distance presented here is not appropriate for a wall that is infinitely high and infinitely thin.

Attenuation factor

The wave that passes over the top of the protective barrier is reflected on the ground downstream from the barrier (Fig. 11). This physical phenomenon leads to an increase in the maximum overpressure downstream from the barrier.

Figure 11 shows the evolution of the attenuation factor for the four analysed geometries (barriers 1A, 2A, 1B and 2B) and for the two studied loads (R 1 and R 2 ).

Figure 11 shows the following:

-The expression of R barrier allows us to highlight the effects of the type of barrier and the charge volume. Hence, we obtain four groups of curves, for different values of studied load R0 and distance d: two of the groups correspond to barrier A, while the other two correspond to barrier B.

-In the case of barrier A, the attenuation may become greater than 1 if the distance d tends toward 0.14 m. Consequently, these configurations lead to the opposite effect than that expected. This situation is never present in the case of barrier B.

-The Mach stem resulting from the reflection phenomenon on the front face of the barrier B relaxes at the top of the barrier at an angle of 45 • for barrier 1B and 90 • for barrier 2B. The Mach stem relaxes again on the face downstream of the barrier at two different angles: at 90 • and 45 • for barriers 1B and 2B, respectively. This dimensioning also assigns the angle of incidence to ground level downstream from the structure: barrier 1B, β 1B = 0 • and barrier 2B,

β 2B = 45 • .
-The slope of the walls must be dimensioned according to the size of the protection zone; for example, due to the relaxation phenomenon, α 2 contributes to a slight attenuation, as well as a possible reflected overpressure on the ground (possible formation of a Mach stem, β > 40 • ) and a less marked protective effect over a larger proximal but visible field (and conversely for barrier 1B).

- Thus, the user should optimize the dimensioning of the barrier based on three sets of geometrical parameters: height (H) -thickness (e), inclination angles of the front and rear faces (α 1 and α 2 ), as well as the positioning of the barrier with respect to the load (d).

The recommendations of NATO [START_REF]Guide SFEPA n • 9. Guide des Bonnes Pratiques en Pyrotechnie-Edition Syndicat des Fabricants d'Explosifs[END_REF] appear robust and useful for promoting "good practices" in the dimensioning of protective barriers. These recommendations allow consideration of a minimal height and thickness to ensure a protective effect downstream from the barrier. Nevertheless, the choice of maximum possible height and thickness according to the available resources can be used to enhance the phenomenon of attenuation by increasing the distance covered by the shock wave over the structure. In addition, the tests conducted by Allain [START_REF] Allain | Barricade Influence on Blast Wave Propagation[END_REF] clearly show the limited effect of this type of barrier geometry, as indicated by the small-scale experiments (barrier A).

The choice of the inclination angles of the front and rear faces also depends on the means available. Indeed, an inclination angle of 90 • should be used to enhance attenuation of the maximum overpressure caused by the presence of strong relaxations on the edges of the barrier. Moreover, the use of a vertical barrier face prevents the rapid formation of a Mach stem upstream (front face) and downstream from the barrier. Formation of a Mach stem leads to a recompression of the shock wave, thus reducing the protective effect of the barrier. The experimental results obtained in this study clearly demonstrate that barrier B with an inclination angle of 90 • is more efficient in terms of overpressure attenuation than barrier A.

However, this type of dimensioning (α 1 = α 2 = 90 • ) also implies major constraints affecting the resistance of the structure, with the risk of projection of new fragments from the barrier (maximum considered overpressure on the front and rear faces of the barrier).

The positioning of the protective barrier relative to the explosive charge depends on the geometry of the selected barrier and the position or size of the downstream zone to be protected. Indeed, according to the slope of the wall, the flow mode can be modified by the formation of a Mach stem upstream and downstream from the protective barrier.

-If the angle of inclination is high (α 1 near to 90 • ), a protective barrier placed in the field close to the blast load offers a strong protective effect downstream from the barrier (important screen effect [START_REF] Borgers | Blast walls reviewed[END_REF]. Nevertheless, this dimensioning also implies high overpressure on the upstream barrier face, in particular by deformation of the reflection due to the presence of an interface between air and detonation products in a field close to the wall and shock wave (for a gas load (stoichiometric propane-oxygen combustion), d < 0.58 m/MJ 1/3 ; for a condensed chemical charge (TNT), d < 0.88 m/kg 1/3 ).

Fig. 1

 1 Fig. 1 Schematic diagram of a protection barrier-W charge mass (kg of TNT), d distance between centre of charge and the front face (m), e thickness at crest of obstacle (m), H height of barrier (m), α 1 angle of inclination of front face ( • ), α 2 angle of inclination of rear face ( • )

Fig. 2

 2 Fig. 2 Schematic diagram of the experimental setup

Fig. 3 Fig. 4 0 Fig. 5

 3405 Fig. 3 Dimensions of barrier 1A with gauge positions-view in the plane z = 0 Fig. 4 Dimensions of barrier 2A with positions of sensors-view in the plane z = 0

230

  Barriers 1B and 2B have identical sizes, with equal dis-231 tances covered by the wave passing across the top face 232 (0.649 m). The distances of sensors are summarized in Fig. 7. 233 3 Analysis of phenomena on barrier A 234

Fig. 6 Fig. 7 242 3 . 1

 6731 Fig. 6 Dimensions of barrier 2B with positions of sensors-view in the plane z = 0

  256 energy-scaled distance to the point of combustion of a spherical gas load (λ in m/MJ 1/3 ) [5]: ln P + P 0 = 0.0895 -1.7633 × ln(λ) + 0.1528 × ln(λ) 2 -0.0066 × ln (λ) 3 -0.0021 × ln (λ) 4 , (6) or according to the distance scaled with respect to the cubic root of the mass of the explosive gas load (Z in m kg -1/3 ) [6]: ln P + P 0 = 1.486 -1.782 × ln (Z ) -0.104 × (ln (Z)) + 0.115 × (ln (Z )) 3 -0.017 × (ln (Z )) 4 .

Fig. 8 Fig. 9 289 3 . 2

 8932 Fig. 8 Overpressure as a function of scaled distance on barrier A

Fig. 10

 10 Fig. 10 Overpressure as a function of scaled distance on barrier B

332 4

 4 Analysis of phenomena on barrier B333We now examine the variation of overpressure as a function 334 of the scaled distance Z defined by the relation (4) on the 335 path of the shock wave over the barrier (Fig.10).

336 4 . 1

 41 Reflection on barrier 2B 337 Barrier 2B has the same downstream face as barrier 1A. The 338 physical phenomena on this barrier surface are thus of com-339 parable nature, i.e., reflection of the incident wave on a plane 340 inclined at 45 • and appearance of a Mach stem. These two 341 geometries are only different at the top of the protective bar-342 rier, with a much greater thickness at the top of barrier 2B 343 (e 2B = 0.19 m e 1A = 0.03 m). 344 The front face of barrier 2B is inclined at 90 • and has a 345 height of 0.19 m. 346 The incident divergent spherical wave resulting from det-347 onation of the gas load is reflected on the front face of barrier 348 2B. The surface is inclined at 90 • , and the angle of incidence 349 between the wave and wall varies from 0 • to 70 • along this 350 surface. This variation of the angle of incidence leads to an 351 evolution of the reflection mode, with a changeover from reg-352 ular reflection towards Mach reflection. All configurations of barrier 2B lead to the formation of a Mach stem on the front face of the structure near the top.

4 . 3

 43 The phenomenology on the top of barrier 1B is identical to that for barriers 1A and 2A. The level of overpressure decreases rapidly at the foot of the wall because of relaxation induced by the change of slope at the top. The geometry of barrier 1B is different from the barriers of series 1 because of the greater thickness at the top (e = H ), thus enhancing the attenuation per distance covered on this surface. The upstream face of barrier 2B has an angle of inclination of 90 • (α 1 = 90 • ). This leads to an increase in the angle R e v i s e d P r o o f I. Sochet et al. of deviation of the first relaxation of the Mach stem on the 383 top of the barrier. The increase in the angle of deviation thus 384 increases the attenuation of the maximum overpressure dur-385 ing relaxation. The maximum overpressure then decreases more rapidly than in the case of barrier 1B, which has a front 387 face inclined at 45 • . 388 Relaxation on the downstream face of barriers 1B 389 and 2B 390 The shock wave propagated over the top of the barrier 391 undergoes a second relaxation during its passage over the 392 downstream face of the barrier: "relaxation in two stages" 393 (non-zero thickness at the top, e = 0 m). 394 For barrier 2B, the rear face is inclined at 45 • (θ = 45 • ); 395 thus, the slope angle at the top of the rear face is less than that 396 for barrier 1B (θ = 90 • ). On the rear face of the barrier, the 397 maximum overpressure undergoes less attenuation compared 398 to barrier 1B (Fig. 10).

399 5 5 . 1

 551 Protective effect of barriers A and B 400 Normative distance 401 The attenuation factor allows us to evaluate the protective 402 effect of the barrier compared to a configuration-free field 403 (without structure), as shown in equation (11):

  404

  [MJ], d the distance between the centre of the charge 414 and the lower point on the front face of the barrier [m] and 415 S the cross section [m 2 ]. This new parameter corresponds 416 to a normative distance which offers the major advantage of 417 considering the form of the barrier rather than the classical 418 parameter n defined by the ratio of the ground distance behind 419 the barrier to the barrier height [4].

  420

  Fig. 11 Evolution of the attenuation factor downstream from barriers A and B for the two studied gas loads and at various distances d between the centre of the explosive charge and the foot of the barrier

482 5 . 3 Comparison at medium scale 483 --[ 2 ]515

 534832 The presence of some thickness at the top of the bar-484 rier (e = 0) allows relaxation "in two stages", with a 485 Mach stem resulting from reflection on the front face. 486 This recommendation of the NATO report represents are expressed versus the normative distance on Fig.12.492The barriers considered by Allain had two inclined slopes 493 of 45 • without any thickness at the top and a height of 494 1.5 m, using two types of TNT charges (8 and 37 kg).495 The experimental results of Allain[START_REF] Allain | Barricade Influence on Blast Wave Propagation[END_REF] reported here 496 highlight the amplifying effect of "overpressure" due to 497 the barrier. This phenomenon can be explained by the 498 hypothesis that maintenance of the Mach number of the 499 shock front (weak attenuation of overpressure) is caused 500 by diffraction and relaxation on the rear face of the Mach 501 stem (resulting from reflection on the front face) or affect-502 ing the incident wave. In turn, this imposes an angle of 503 incidence close to 45 • that can approach the requirements 504 for the formation of a new Mach stem at the end of the 505 rear face of the barrier for certain configurations. This 506 possible relaxation is accompanied by a reflection on the 507 ground, possibly leading to the creation of a stronger 508 Mach stem on overpressure. 509 -Consequently, the geometry of the barrier used by Allain 510 appears to be unsuitable for the protection of peo-511 ple, equipment and structures, thus supporting our results 512 obtained at small scale with barrier A. The study of various protective barrier configurations leads 516 to an analysis of the interaction of shock waves with barriers 517 according to their geometrical parameters and an assessment 518 of their impact on the protective effect. 519 The ideal protective barrier is a parallelepiped with sig-520 nificant height and thickness. Indeed, this geometry allows 521 enhancement of the attenuation of the maximum overpres-522 sure by increasing the distance covered (Taylor waves) and 523 favouring the presence of "strong" relaxations (angles of 524 deviation (θ ) close to 90 • ). Nevertheless, according to the 525 additional constraints of dimensioning (such as limited space 526 and financing), this type of geometry may be difficult to 527 implement and can be "oversized" compared to the needs 528 of the user ( P + downstream 0.020 bar, threshold 529 of the last affected zone (Z 5 )). The optimal dimension-530 ing of a protective barrier thus depends on the available 531 resources and dimensions of the configuration of interest 532 (position of the zone to be protected with respect to the blast 533 load).

  534

Acknowledgments

This study is part of the BARPPRO ANR research 608 programme (dimensioning physical protective barriers against propa-

where R is the distance between the centre of the gas load and