Sophie Tarbouriech

Isabelle Queinnec
email: queinnec@laas.fr

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Nonstandard use of anti-windup loop for systems with input backlash

Keywords: Stability analysis, stabilization, backlash, anti-windup loops

Control systems with backlash at the input are considered in this paper. The goal of this work is to characterize the attractor of such nonlinear dynamical systems, and to design anti-windup inspired loops such that the system is globally asymptotically stable with respect to this attractor. The anti-windup loops affect the dynamics of the controllers, and allow to increase the performance of the closed-loop systems. Different performance issues are considered throughout the paper such as robustness with respect to uncertainty in the backlash, and L 2 gain when external disturbances affect the dynamics. Numerically tractable algorithms with feasibility guarantee are provided, as soon as the linear closed-loop system, obtained by neglecting the backlash effect, is asymptotically stable. The results are illustrated on an academic example and an open-loop unstable aircraft system.

Introduction

Backlash operators are nonlinearities present in many mechanical systems. They are involved in mechanical slacks, static friction, elastic displacements and ferromagnetism. Figure 1 shows some examples and [START_REF] Macki | Mathematical models for hysteresis[END_REF] is a good introduction of this kind of nonlinearity. Since many smart actuators and materials contain a backlash, and since they are often used for precise control systems, such as position regulation, it is crucial to take the backlash operator into account in the design or the analysis of the closed loop. Neglecting them can reduce the performance or alleviate the stability objective.

There are many different models for backlash operators, see [START_REF] Macki | Mathematical models for hysteresis[END_REF] for a survey of possible models. Here we will focus on a component-wise model considered in [START_REF] Macki | Mathematical models for hysteresis[END_REF][START_REF] Paré | A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities[END_REF][START_REF] Taware | Control of sandwich nonlinear systems[END_REF][START_REF] Brokate | Hysteresis and phase transitions[END_REF][START_REF] Ouyang | Absolute stability analysis of linear systems with duhem hysteresis operator[END_REF]. The undesirable effect of such a nonlinearity map has been already analyzed using circle criterion in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], assuming that the control system is open-loop stable, an assumption that is not required in this paper. The present work is based on a Lyapunov method for control systems with backlash operators in the loop. The control objective is to reduce the undesirable displacement slack Figure 1: Left: mechanical slack with a longitudinal backlash. Right: rotational backlash. effect of the nonlinearity by adding an extra dynamics. Such a strategy inspired by antiwindup techniques has been already employed for control systems with saturated inputs, see e.g. [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF][START_REF] Zaccarian | Modern anti-windup synthesis[END_REF], where it is shown how to design so called anti-windup gains to improve the performance of saturating closed-loop systems.

To be more specific, our contribution is as follows. We succeed to compute numerically tractable conditions for the design of nonstandard anti-windup (NSAW) gains that modify the controller dynamics and its output for different control objectives. In presence of backlash operators, the most natural control objective is probably the reduction of the attractor that is the reduction of the ultimate bounded set where all solutions converge. In an ideal case where the backslash is not present, the attractor is usually reduced to one point, thus our NSAW technique can be seen as a method to approach as much as possible the ideal case.

Another natural control objective is the robustness issue, that is to guarantee some stability properties in presence of external perturbations or disturbances in the loop and in the backlash operator. The Lyapunov method applies for such cases, and we show how to design NSAW gains guaranteeing the best performance in terms of robustness, with respect to uncertainty in the backlash and with respect to external disturbances. Again our design conditions are numerically tractable and apply as soon as the closed-loop system, without any backlash, is asymptotically stable. On the other hand, no restrictive hypothesis is necessary regarding the open-loop stability of the system. More precisely, our conditions are based on the solution to a convex problem written in terms of Linear Matrix Inequalities that are shown to have a solution allowing to explicitly compute NSAW gains.

The paper is organized as follows. The stability problem and the NSAW gain design problem are introduced in Section 2 for backlash control systems. The main results are given in Section 3 yielding a solution to the stability analysis and allowing to compute numerically tractable conditions for the design of NSAW gains. Some extensions are given in Section 4 when considering both external disturbances and backlash uncertainties. Illustrative examples are given in Section 5 and some concluding remarks are collected in Section 6.

Notation. For two vectors x, y of R n , the notation x y means that x (i) -y (i) ≥ 0, ∀i = 1, . . . , n. 1 and 0 denote the identity matrix and the null matrix of appropriate dimensions, respectively. x ∈ R n + means that x 0. The Euclidian norm is denoted • . A and trace(A) denote the transpose and the trace of A, respectively. He{A} = A + A . For two sym-metric matrices, A and B, A > B means that A -B is positive definite. In partitioned symmetric matrices, the symbol stands for symmetric blocks. λ max (A) (respectively, λ min (A)) denotes the maximal (respectively, minimal) eigenvalue of the matrix A.

Problem formulation 2.1 System description

The class of systems under consideration is described by:

ẋp = A p x p + B p u p y p = C p x p (1)
where The connection between the plant and the controller through its output y c ∈ R m is realized as follows:

x p ∈ R np is the state, u p ∈ R m is
u p = Φ[y c ] (2)
where Φ is a component-wise backlash operator (see, for example, [START_REF] Macki | Mathematical models for hysteresis[END_REF][START_REF] Paré | A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities[END_REF][START_REF] Taware | Control of sandwich nonlinear systems[END_REF][START_REF] Brokate | Hysteresis and phase transitions[END_REF]). We denote the set of continuous, piecewise differentiable functions f : [0, +∞) → R m by C 1 pw ([0, +∞); R m ), that is the set of continuous functions f being, for some unbounded sequence (t j ) ∞ j=0 in [0, +∞) with t 0 = 0, continuously differentiable on (t j-1 , t j ) for all j ∈ N. Given the vector ρ in R m + and L = diag( (i) ), with positive values (i) , i = 1, . . . , m, the operator Φ is defined as follows, for all f ∈ C 1 pw ([0, +∞); R m ), for all j ∈ N, for all t ∈ (t j-1 , t j ) and for all i ∈ {1, . . . , m}:

( ˙ Φ[f ](t)) (i) =            (i) ḟ(i) (t) if ḟ(i) (t) ≥ 0 and (Φ[f ](t)) (i) = (i) (f (i) (t) -ρ (i) ) (i) ḟ(i) (t) if ḟ(i) (t) ≤ 0 and (Φ[f ](t)) (i) = (i) (f (i) (t) + ρ (i) ) 0 otherwise (3)
where 0 = t 0 < t 1 < . . . is a partition of [0, +∞) such that f is continuously differentiable on each of the intervals (t j-1 , t j ), j ∈ N. Thus, Φ is a time-invariant nonlinearity with slope restriction, as in [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]. Note however that it is a memory-based operator, since to compute it, we need to have information about the past values of its input (this is not the case in [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]). Since it is possible to stack several backlash operators into one backlash operator (with the dimension equal to the sum of all dimensions), without loss of generality, the standard form defined here remains valid.

The plant is controlled by the following output dynamical controller

ẋc = A c x c + B c y p + θ 1 y c = C c x c + D c y p + θ 2 (4)
where x c ∈ R nc is the state, y p ∈ R np is the output of the plant, y c ∈ R m is the output of the controller. θ 1 ∈ R nc and θ 2 ∈ R m are input signals to be designed to perform a suitable correction inspired by anti-windup action (see, for example, [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], [START_REF] Zaccarian | Modern anti-windup synthesis[END_REF] in the context of saturation nonlinearities) for mitigating the undesired effects of the backlash. The principle of the nonstandard anti-windup (NSAW) loop considered consists in picking the difference between the output of the nonlinear actuator (Φ[y c ]) and the output of the linearized one (Ly c ), which are available signals for building θ 1 and θ 2 :

θ 1 = E c (Φ[y c ] -Ly c ) θ 2 = F c (Φ[y c ] -Ly c ) (5)
with E c and F c matrices of appropriate dimensions.

Remark 2.1 The design of the controller (4) may be performed with classical techniques to stabilize the plant, disregarding the effects of the backlash nonlinearity (with θ 1 = 0 and θ 2 = 0). In other words, it is assumed that the controller (4) stabilizes the plant (1) through the linear interconnection u p = Ly c (which corresponds to take Φ[y c ] = Ly c ) and therefore the matrix:

A 0 = A p + B p LD c C p B p LC c B c C p A c ( 6
)
is Hurwitz.

Problem formulation

The closed-loop system issued from (1), ( 2), ( 3), ( 4) and ( 5) reads as follows

ẋp = A p x p + B p Φ[y c ] ẋc = A c x c + B c y p + E c (Φ[y c ] -Ly c ) y p = C p x p y c = C c x c + D c y p + F c (Φ[y c ] -Ly c ) (7)
Let us note that according to (3), one gets Φ[y c ](t) ∈ I Φ with

I Φ = {Φ[y c ] ∈ R m ; L(y c + ρ) Φ[y c ] L(y c -ρ)} (8)
One can observe that, by definition, from any initial condition in I Φ , the solution Φ[y c ](t) remains confined in I Φ , ∀t ≥ 0. According to [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF], [START_REF] Corradini | Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator[END_REF], that means that the nonlinearity Φ is active.

The presence of the backlash operator Φ may induce the existence of multiple equilibrium points or a limit cycle around the origin. Furthermore, in a neighborhood of the origin, system (1) operates in open loop. Hence, we are concerned with the asymptotic behavior of the state x = x p x c ∈ R n , n = n p + n c but not of the operator Φ. Therefore, we want to study the stability properties of the following attractor:

A = S 0 ⊆ R n (9)
It is important to emphasize that the proposed technique does not require for the open-loop system to be stable, contrary to [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], [START_REF] Tarbouriech | Stability analysis for linear systems with input backlash through sufficient LMI conditions[END_REF].

Then the problem we intend to solve can be summarized as follows.

Problem 2.1 Characterize the region S 0 of the state space, containing the origin, and design the NSAW gains E c and F c such that system ( 7) is globally asymptotically stable with respect to S 0 , when initialized as in [START_REF] Khalil | Nonlinear Systems[END_REF]. In other words, S 0 is a global asymptotic attractor for the closed-loop dynamics [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], for any initial value of Φ in I Φ .

Closed-loop description and well-posedness

Hence, with the augmented state x = x p x c ∈ R n , the closed-loop system reads:

ẋ = A 0 x + (B + RE c + BLF c )Ψ y c = Kx + F c Ψ ( 11
)
with A 0 defined in ( 6) and

K = D c C p C c ; B = B p 0 ; R = 0 1
When F c = 0, system (11) does no contain any algebraic loop. However, it can be interesting to consider F c = 0 in order to have more degrees of freedom to characterize the attractor set S 0 . A particular attention should be paid to the well posedness of the implicit relation defining y c in [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF]. Indeed, the algebraic loop y c = Kx + F c Ψ is said to be well-posed if there exists a unique solution y c of the second line of [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF] for each Kx + F c Ψ. We have the following well-posedness result:

Proposition 2.1 Under the assumption that 1 + F c L is nonsingular, the algebraic loop in [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF] is well defined, and the system (11) is well-posed.

To prove the proposition, note that from the definition of Φ, one can prove that the implicit function g(y c ) = y c -Kx + F c Ψ = 0 has a solution, and therefore, provided that 1 + F c L is nonsingular, Proposition 2.1 holds. See, for example, [START_REF] Zaccarian | Modern anti-windup synthesis[END_REF] (Chapter 2, page 38) for more details.

3 Main Results

Theoretical conditions

The following result provides a solution to Problem 2.1, by adapting Lemma 1 in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] to system [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF].

Theorem 3.1 Suppose there exist a symmetric positive definite matrix W ∈ R n×n , two diagonal positive definite matrices

S 2 ∈ R m×m , T 3 ∈ R m×m , two matrices E c ∈ R nc×m , F c ∈ R m×m
and a positive scalar τ satisfying the following conditions

M 1 < 0 (12) ρ LT 3 Lρ -τ ≤ 0 (13)
with

M 1 =   He{A 0 W } + τ W (B + RE c + BLF c ) -T 3 -LKA 0 W -LK(B + RE c + BLF c ) -He{(1 + LF c )S 2 }   (14)
Then, for any admissible initial conditions (x(0), Ψ(0)), the closed-loop system ( 11) is globally asymptotically stable with respect to the set S 0 defined as follows:

S 0 = {x ∈ R n ; x W -1 x ≤ 1} (15)
In other words, E c , F c and S 0 are solution to Problem 2.1.

Proof. First note that the satisfaction of relation [START_REF] Ouyang | Absolute stability analysis of linear systems with duhem hysteresis operator[END_REF] guarantees that the matrix 1 + LF c is Hurwitz and therefore nonsingular, which implies that Proposition 2.1 holds.

To prove Theorem 3.1, we consider a quadratic Lyapunov function candidate V defined by V (x) = x P x, P = P > 0, for all x in R n . We want to verify that there exists a class K function α such that V (x) ≤ -α(V (x)), for all x such that x P x ≥ 1 (i.e. for any x ∈ R n \S 0 ), and for all nonlinearities Ψ satisfying Lemma 1 in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF]. By using the S-procedure, it is sufficient to check that L < 0, where

L = V (x) -τ (1 -x P x) -Ψ T 3 Ψ + ρ LT 3 Lρ -2( Ψ + L ẏc ) N 1 Ψ -2( Ψ + L ẏc ) N 2 ( Ψ + (1 -N 3 )L ẏc ) (16)
with τ a positive scalar and T 3 a positive diagonal matrix. Choosing N 1 = 01 and N 3 = 1, from the definition of Ψ in [START_REF] Macki | Mathematical models for hysteresis[END_REF], noting that V (x) = x (A 0 P + P A 0 )x + 2x

P (B + RE c + BLF c )Ψ, it follows that L = L 0 + ρ LT 3 Lρ -τ with L 0 =   x Ψ Ψ   M 2   x Ψ Ψ  
with M 2 defined as follows

M 2 =   He{P A 0 } + τ P (B + RE c + BLF c ) P -T 3 -N 2 LKA 0 -N 2 LK(B + RE c + BLF c ) -He{N 2 (1 + LF c )}   (17)
The matrix M 1 of relation ( 12) is directly obtained by pre-and post-multiplying

M 2 by   W 0 0 0 1 0 0 0 S 2   , where W = P -1 and S 2 = N -1 2 .
The satisfaction of relations ( 12) and ( 13) implies both L 0 < 0 and ρ LT 3 Lρ -τ ≤ 0, and then L < 0, for all (x, Ψ, Ψ) = 0, and for any x ∈ R n \S 0 .

Therefore, the satisfaction of relation [START_REF] Ouyang | Absolute stability analysis of linear systems with duhem hysteresis operator[END_REF] ensures that there exists ε > 0, such that L ≤ -ε

x Ψ Ψ 2 ≤ -εx x. Hence, since by definition one gets V (x) ≤ V (x)τ (1 -x P x) ≤ L, one can also verify V (x) ≤ -εx x , ∀x such that x P x ≥ 1 [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] Furthermore, from [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF], there exists a time T ≥ t 0 + (x(t 0 ) P x(t 0 ) -1)λ max (P )/ such that x(t) ∈ S 0 , ∀t ≥ T , and therefore, S 0 is an invariant set for the trajectories of system [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF]. Hence, in accordance with [START_REF] Khalil | Nonlinear Systems[END_REF], it concludes the proof of Theorem 3.1.

Let us comment the feasibility of conditions of Theorem 3.1.

Proposition 3.1 Theorem 3.1 enjoys the following properties:

1. Given E c = 0 and F c = 0, condition ( 12) is feasible if and only if matrix A 0 is Hurwitz.

2. There always exist E c and F c non null such that condition (12) holds.

Proof. Let us first define matrix M 0 corresponding to M 1 in the case E c = 0 and F c = 0:

M 0 =   He{A 0 W } + τ W B -T 3 -LKA 0 W -LKB -2S 2   (19)
Then, in this case, condition ( 12) corresponds to M 0 < 0. Recall that matrix A 0 is Hurwitz by construction (see Remark 2.1). Then one can show that the inequality M 0 < 0 is always feasible by using 1) the fact that the stability property of A 0 implies the existence of a matrix W = W > 0 such that He{A 0 W } + τ W < 0;

2) the Schur complement to show that all the minors of matrix -M 0 are positive for large enough values of T 3 and S 2 .

Let us now consider the case where E c and F c are non null. In this case, condition (12) can be written as

M 1 = M 0 + He{(   R 0 -LKR   E c 0 1 0 } + He{   BL 0 0 0 -LKBL -L   (1 ⊗ F c ) 0 1 0 0 0 S 2 } (20) Hence, if M 0 < 0 is feasible, it
is always possible to find some value for E c , F c such that relation [START_REF] Teel | A smooth Lyapunov function from a class-kl estimate involving two positive semidefinite functions[END_REF] is feasible, or equivalently such that M 1 < 0 is feasible Remark 3.1 Conditions of Theorem 2 in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] can be deduced from that one of Theorem 3.1. Indeed, as pointed out previously, the part due to N 1 appearing in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] is not useful to solve Problem 2.1 and may be removed. Hence, by setting E c = 0 and F c = 0 in the conditions of Theorem 3.1 one retrieves the conditions of Theorem 2 in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] with N 1 = 0 and P = W -1 .

Computational issues

Theorem 3.1 may be used to solve both the NSAW design Problem 2.1 and the analysis problem (E c and F c given). It may then be noted that there are two sources of nonlinearities in the conditions of Theorem 3.1. The first one is issued from the use of the S-procedure, with the product τ W , and is the unique source of nonlinearity in the stability analysis problem and in the NSAW design problem of E c only (F c = 0 or given). This nonlinearity is easily managed as τ is a simple parameter without much influence on the solution. It may be selected arbitrarily by try-and-error until a feasible condition is found, or thanks to a grid search, or encapsulated in an optimization problem with the Matlab function fminsearch. On the other hand, when F c is a decision variable, another nonlinearity occurs due to the product F c S 2 . An iterative procedure may then be used, where S 2 and F c are alternatively a decision variable, the other one being fixed to the solution to the previous step. The initialization of the iterative process may be done with S 2 solution to the analysis problem. Finally optimization problems may be solved to evaluate the smallest set S 0 , typically described by its volume, proportional to det(W ) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], and related to the trace of matrix W .

The following algorithm can then be considered for the analysis problem or to design the gain E c (F c given, typically equal to 0) such that Theorem 3.1 holds.

Algorithm 3.1 min τ min W,S 2 ,T 3 (,Ec) trace(W )
subject to conditions ( 12), (

Similarly, the following algorithm can be considered to design the gain F c (and eventually E c ) such that Theorem 3.1 holds. In both algorithms, the Matlab function fminsearch is used to find the minimal value of τ allowing to obtain a solution to optimization problems [START_REF] Wu | Anti-windup controller design using linear parameter-varying control methods[END_REF] and [START_REF] Zaccarian | Modern anti-windup synthesis[END_REF].

Extensions

Existence conditions

Note that Theorem 3.1 implies that system (7) is asymptotically stable with respect to the set S 0 × I Φ , as soon as Φ is initialized in I Φ . Then, let us state an existence condition regarding the robustness of the solution to Problem 2.1. To do that, denote respectively the (set-valued) right-hand side of the first line of [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF], the closed unit ball and the closed convex hull of a set by F , B and co. Proposition 4.1 Under the hypotheses of Theorem 3.1, there exists a continuous function δ : R n → R ≥0 which is positive outside S 0 such that the differential inclusion

ẋ ∈ coF (x + δ(x)B) + δ(x)B ( 23
)
is asymptotically stable with respect to the set S 0 × I Φ , for any admissible initial condition, i.e. Φ is initialized in I Φ ,. It means that it exists a class KL-function β such that, all solutions to (23), with Φ initialized in (8), satisfy

x(t) S 0 ≤ β( x(0) S 0 , t) , ∀t ≥ 0.
Proof. To prove this result, we first note that we may include all the dynamics in the two first lines of (7) together with (3) in terms of a differential inclusion of the state x:

ẋ ∈ F (x) (24)
where F is a nonempty, compact, convex and locally Lipschitz set-valued function, by considering the Filippov regularization of [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF]. Moreover the attractor S 0 is compact and, with Theorem 3.1, the system (24) is globally asymptotically stable with the attractor S 0 . Therefore, according to [20, Theorem 1], system [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF] is robustly globally asymptotically stable with the attractor S 0 . Moreover the distance of any point x to the set S 0 is x S 0 . This implies, with [20, Definition 8], the existence of a δ function and of a a class KL-function β as considered in the statement of Proposition 4.1.

Remark 4.1 The notion of global ultimate boundedness with respect to a compact set S 0 (see, for example, [START_REF] Khalil | Nonlinear Systems[END_REF]) could be used, but the use of such a notion does not allow to ensure Lyapunov stability of the considered compact set S 0 . That means that it may exist some trajectories starting close to S 0 , which may not converge to it. However, the property of global asymptotic stability of a compact set S 0 , as guaranteed in Theorem 3.1, allows to inherit robustness property with respect to small perturbations, as developed in Proposition 4.1. Similar facts appear in the context of quantized systems as discussed in [START_REF] Ferrante | On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming[END_REF].

Uncertainty on the backlash

In this section, we address the case where the backlash operator is uncertain, in order to study the impact of the uncertain backlash operator on the asymptotic stability of the closed loop as done for example in [START_REF] Bisoffi | 8: F-8 example (DOF) -Time evolution of the angle of attack (x 3 ). NSAW using the nominal dead-zone ρ N (dashed red line) and the real dead-zone ρ N + µ ([END_REF] for the Coulomb friction. To be more specific, the uncertain parameter of backlash operator is ρ, that is ρ = ρ N + ∆ρ, where ρ N is the nominal part and ∆ρ the uncertain part with:

-µ ∆ρ µ (25)

for some constant vector µ in R m with positives entries.

Remark 4.2 Taking into account explicitly the backlash uncertainty makes sense when associated to an NSAW strategy. Actually, in the case without NSAW, the smallest set S 0 in which the trajectories of the closed-loop system are uniformly ultimately bounded is forced by the worst case ρ = ρ N + µ. On the other hand, one cannot expect to access the true value of the backlash in the NSAW scheme, and it has to be built by considering the a priori nominal value of the uncertain backlash dead-zone ρ.

Then, in the control scheme (4), the NSAW signals are modified as follows:

θ 1 = E c (Φ N [y c ] -Ly c ) θ 2 = F c (Φ N [y c ] -Ly c ) ( 26
)
where Φ N (which corresponds to ρ N ) represents the nominal backlash. Then, the closed-loop system is modified as follows:

ẋ = A 0 x + (B + RE c + BLF c )Ψ N + B(Ψ -Ψ N ) y c = Kx + F c Ψ N z = Cx (27)
with Ψ N = Φ N -Ly c (associated to ρ N ), Ψ = Φ -Ly c (associated to ρ). Note also that:

-2Lρ -Lµ ≤ Ψ -Ψ N ≤ 2Lρ + Lµ.
Problem 2.1 is unchanged, excepted that it now concerns system (27) and a solution to this uncertain problem is given by the following theorem.

Theorem 4.1 Suppose there exist a symmetric positive definite matrix W ∈ R n×n , three diagonal positive definite matrices

S 2 ∈ R m×m , T 3 ∈ R m×m , T 4 ∈ R m×m , two matrices E c ∈ R nc×m , F c ∈ R m×m and a positive scalar τ satisfying M 1 B 1 0 0 -T 4 < 0 (28) ρ LT 3 Lρ -τ + (2ρ + µ )LT 4 L(2ρ + µ) ≤ 0 ( 29
)
with M 1 defined in [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]. Then, for any admissible initial conditions (x(0), Ψ(0)), the closedloop system (27) is well posed and globally asymptotically stable with respect to the set S 0 defined in [START_REF] Prieur | Wave equation with conebounded control laws[END_REF] and E c , F c and S 0 are solution to the uncertain problem 2.1.

Algorithms 3.1 and 3.2 are simply modified by updating the conditions with (28) and (29) instead of ( 12) and ( 13).

External stability

Let us consider that plant (1) is affected by an additive disturbance as follows:

ẋp = A p x p + B p u p + B pw w y p = C p x p z = C pz x p (30)
where w ∈ R q is the additive perturbation and z ∈ R nz is the performance output to be attenuated. The exogenous signal w is supposed to be limited in energy:

∞ 0 w(t) w(t) ≤ δ -1 (31)
with a positive finite scalar δ.

In that case, the plant (30) in closed loop with the controller (4) reads:

ẋ = A 0 x + (B + RE c + BLF c )Ψ + B w w y c = Kx + F c Ψ z = Cx (32)
with

B w = B pw 0 ; C = C pz 0
Problem 2.1 is then modified as follows.

Problem 4.1 Characterize the regions S 0 and S ∞ of the state space, containing the origin, and design the NSAW gains E c and F c such that:

1. When w = 0, the system (32) is globally asymptotically stable with respect to S 0 , when initialized as in [START_REF] Khalil | Nonlinear Systems[END_REF]. In other words, S 0 is a global asymptotic attractor for the closed-loop dynamics (32), for any initial value of Φ in I Φ ;

2. When w = 0, the closed-loop trajectories of system (32) remains bounded in the set S ∞ .

3. When w = 0, characterize the L 2 -gain of the map from w to z. Then in this case we are interested to prove that the map from w to z is finite L 2 -gain stable with

T 0 z(t) z(t)dt ≤ γ 2 T 0 w(t) w(t) + g(x(0)), ∀T ≥ 0 ( 33
)
where g(x(0)) is a bias related to the initial condition

x(0) = x p (0) x c (0) ∈ R n .
Theorem 3.1 is modified to deal with Problem 4.1 as follows.

Theorem 4.2 Suppose there exist a symmetric positive definite matrix W ∈ R n×n , two diagonal positive definite matrices S 2 ∈ R m×m , T 3 ∈ R m×m , two matrices E c ∈ R nc×m , F c ∈ R m×m and two positive scalars τ and γ satisfying ( 13) and the following condition

M 1 M 3 -γ1 < 0 (34)
with M 1 defined in Theorem 3.1, and

M 3 = B w 0 -B w K L CW 0 0 (35)
Then,

1. When w = 0, for any admissible initial conditions (x(0), Ψ(0)), the closed-loop system (32) is globally asymptotically stable with respect to the set S 0 , where the set S 0 is defined in [START_REF] Prieur | Wave equation with conebounded control laws[END_REF].

2. When w = 0, (a) The closed-loop trajectories (32) remain bounded in the set S ∞ defined as

S ∞ = {x ∈ R n ; x P x ≤ γδ -1 + x(0) P x(0)} (36)
for any admissible initial conditions (x(0), Ψ(0)) such that x(0) ∈ R n \S 0 (i.e., x(0) P x(0) ≥ 1).

(b) The map from w to z is finite L 2 -gain stable with

T 0 z(t) z(t)dt ≤ γ 2
T 0 w(t) w(t) + γx(0) P x(0), ∀T ≥ 0 (37)

with P = W -1 .
Proof. The proof of item 1 of Theorem 4.2 is directly obtained from the proof of Theorem 3.1. Indeed, in presence of disturbance we want to verify that there exists a class K function α such that V (x) + 1 γ z z -γw w ≤ -α(V (x)), for all x such that x P x ≥ 1, and for all nonlinearities Ψ satisfying Lemma 1 in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF]. Mimicking the previous arguments, by using the S-procedure, it is sufficient to check that L w < 0, where

L w = V (x) -τ (1 -x P x) -Ψ T 3 Ψ + ρ LT 3 Lρ -2( Ψ + L ẏc ) N 2 Ψ + 1 γ z z -γw w (38)
with τ a positive scalar and T 3 a positive diagonal matrix. Then, it follows that L w = L w0 + ρ LT 3 Lρ -τ with

L w0 =     x Ψ Ψ w     M 1 + C C γ B w P 0 -B w K LN 2 -γ1     x Ψ Ψ w    
By pre-and post-multiplying the matrix in the inequality above by

    W 0 0 0 0 1 0 0 0 0 S 2 0 0 0 0 1     , with W = P -1 and S 2 = N -1
2 , and by using the Schur complement, the satisfaction of relations (34) and ( 13) implies both L w0 < 0 and ρ LT 3 Lρ -τ ≤ 0, and then L w < 0, for all (x, Ψ, Ψ, w) = 0. By integrating L w0 < 0 between 0 and T , one gets:

V (x(T )) -V (x(0)) + 1 γ T 0 z(t) z(t)dt -γ T 0 w(t) w(t) < 0 or still V (x(T )) ≤ V (x(T )) + 1 γ T 0 z(t) z(t)dt ≤ γ T 0 w(t) w(t) + V (x(0))
Therefore one obtains relation (37) and the definition of S ∞ given in (36). That concludes the proof of item 2 of Theorem 4.2. Algorithms 3.1 and 3.2 are updated in this case with the new condition (34) in plus of ( 13), but also considering for the optimization criterion min γ or a mixed criterion min γ + trace(W ).

Remark 4.3

The presence of an additive disturbance has no direct impact on the set S 0 in which converges the trajectory once the disturbance vanished. Then, during the design step, a mixed criterion min γ +trace(W ) consisting in minimizing γ without neglecting the long term convergence to S 0 , allows a trade-off between disturbance rejection and size of the attractor. At the inverse, in the analysis step, it is recommended to separate both optimization problems.

Illustrative examples

Algorithms 3.1 and 3.2 (and their extensions to the uncertain and/or perturbed cases) have been implemented in the Matlab environment, using Yalmip [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF] and Mosek [1].

Academic example

Let us consider to illustrate the approach a small-size example described by (A p , B p , C p , D p ) = (0.1, 1, 1, 0), with the stabilizing PI control (A c , B c , C c , D c ) = (0, -0.2, 1, -2) and a backlash element given by (ρ, L) = (1.5, 1). Let us first compare the global asymptotic attractor S 0 obtained without and with NSAW. Algorithm 3.1 is used both for the analysis of the system without NSAW, and to design an anti-windup gain E c , whose the solution is E c = 0.11. The results are illustrated on Figure 2. The ellipsoidal sets S 0 and trajectories initiated from several initial conditions in the case without backlash (black dotted line), with backlash alone (dashed red line) and with backlash and NSAW (solid blue line) illustrate the positive effect of the NSAW. The simulations without NSAW also illustrate that the system may converge to a fixed point or to a limit cycle around the origin.

Four-dimensional unstable F-8 aircraft

Let us now consider an unstable MIMO plant model with four states and two inputs. It corresponds to the longitudinal dynamics of an F-8 aircraft, slightly modified in [START_REF] Wu | Anti-windup controller design using linear parameter-varying control methods[END_REF] to make it unstable. This model has been frequently used in the past to study the influence of saturating inputs. In this paper, we do not consider saturations but concentrate on the presence of backlash phenomenon in the input. Let us then consider the system defined by the following matrices The state variables correspond to the pitch rate, the forward velocity, the angle of attack and the pitch angle. The elevator and flaperon angles are the input variables. The output measurements are set as the pitch angle and the flight path angle (see [START_REF] Wu | Anti-windup controller design using linear parameter-varying control methods[END_REF] for details). External disturbances are considered as acting at the input of the system, with B pw = B pu , mimicking the effect of noise on the actuator signal.

State-feedback control (SF)

This system has already been used in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] to illustrate the influence of backlash defined as follows L = 1 0 0 1 , ρ = 0.5 0.5

interconnecting the system with a state-feedback controller K = 0.2672 0.0059 -0.8323 0.8089 0.5950 -0.1534 2.1168 0.0525 .

In [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF], the control problem was to compute a new state-feedback controller allowing to minimize the global asymptotic attractor S 0 for the closed-loop system. In the current paper, the control problem is to compute a NSAW gain F c allowing to minimize S 0 , but without modifying the controller gain K. Algorithm 3.2 is used, initialized with τ = 0.5.

• First, Step 1 is performed for the nominal case, with conditions ( 12)- [START_REF] Paré | A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities[END_REF]. It gives the nominal value of S 0 , independently of the presence of additive disturbance (See Remark 4.3). The volume of the ellipsoid obtained without NSAW is v(W 0 ) = det(W 0 ) = 19.8668.

• Secondly, a NSAW F c is computed in Step 2 for the perturbed system with δ = 20, considering conditions ( 13) and (34) in the optimization problem, with the cost function γ + trace(W ). One then obtains for the NSAW gain:

F c = -0.9798 -0.2004 0.1946 -0.9794
Solutions with and without NSAW are compared in two directions.

• Ellipsoidal sets are compared in both cases. The analysis of the system with NSAW is performed for the nominal case (B pw = 0). It gives a volume v(W 1 ) = det(W 1 ) = 0.0852 to be compared to v(W 0 ) = det(W 0 ) = 19.8668. Note that it corresponds more or less to the volume obtained in [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] when building a new state-feedback gain.

• L 2 gains may be also compared in both cases, solving the analysis problem with cost function γ. One obtains:

-without NSAW: γ 0 = 40.8 -with NSAW: γ 1 = 27.1

Results are also illustrated in Figures 3 to 6. The time evolution of the angle of attack (x 3 ) and the pitch angle (x 4 ) are plotted in Figures 3 and4, respectively, for the system without backlash (dotted black line), with backlash alone (solid blue line) and with backlash and NSAW (dashed red line), initiated from x 0 = 0 0 1 0 (angle of attack equal to 1). The backlash characteristics Φ[y c ] of the closed-loop systems without (left side) and with NSAW (right side) are plotted in Figures 5 and6 for the elevator and flaperon angles, respectively.

Dynamic output-feedback control (DOF)

Let us now consider the same F-8 aircraft but controlled by a dynamic output feedback, issued from [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]: Consider also that the exact dead-zone ρ of the backlash is unknown, with an uncertainty of ±10% around the nominal ρ N = 0.15 0.15 and that the system is prone to disturbance limited in energy with bound δ = 20. A NSAW gain E c may be computed by combining the conditions (28), ( 29) and (34). One obtains: The time evolution of the angle of attack (x 3 ) for the system initiated with x 3 (0) = 1 is plotted in Figure 7 for the system without backlash (dotted black line), with backlash alone (blue line) and with backlash and NSAW (red line). Simulations with backlash are provided with ρ = ρ N + µ (solid line) and ρ = ρ N -µ (dashed line). Once again, one can check that the NSAW significantly reduces the effect of the backlash on the response of the closed-loop system. The influence of the backlash uncertainty is illustrated in Figure 8, relative to the time evolution of the angle of attack. It is compared in this figure the evolution of the angle of attack, in presence of a backlash with dead-zone ρ = ρ N + µ, with a NSAW using either the nominal dead-zone ρ N (dashed red line) or the real dead-zone ρ (dashdot green line). One can check that even with an approximate knowledge of the backlash dead-zone, one can expect to reduce its effect on the system response thanks to a non-standard anti-windup strategy.

A c =     -4.
E c =     - 18

Conclusion

Nonlinear control systems, such as those modeled with a backlash operator in the input, have been studied in this paper. Under the assumption that the linear closed-loop system is asymptotically stable, it was shown that some numerically tractable conditions hold so that it is possible to estimate the attractor of the backlash control system. Moreover conditions allow for the design of anti-windup loops to improve some performance, such as sensitivity with respect to backlash uncertainty, or L 2 -gain with respect to external disturbances. Both academic and aeronautical examples illustrate the obtained results. The use of a non-standard anti-windup loop showed its benefit in a similar way as in the presence of saturation.

This work lets some research lines open. In particular, it would be relevant to study the effect of the backlash when this nonlinearity affects the output. For such control problems, it may be necessary to exploit an observer of the nonlinearity map in order to be able to build the NSAW loop. Moreover it could be interesting to see the effect of the backlash on infinite dimensional systems, maybe in a similar way as in [START_REF] Prieur | Wave equation with conebounded control laws[END_REF] for the wave equation with input saturation. Also some works are under progress on a complementary subject, namely when both saturation and backlash nonlinearities are active in the closed loop.

For

  conciseness, throughout the paper, we denote Φ instead of ˙ Φ[y c ], and Φ instead of Φ[y c ]. Let us define the nonlinearity Ψ: Ψ = Φ[y c ] -Ly c

Algorithm 3. 2 • 22 )•

 222 Step 1. Initialization. Given E c = 0 and F c = 0, solve the optimization problem (21). • Step 2. Design. Keep the value of S 2 and solve the following min τ min W,T 3 ,Ec,Fc trace(W ) subject to conditions (12), (13) (Step 3. Analysis. Keep the values of E c and F c and solve (21). • Step 4. Iterate between Step 2 and Step 3 until trace(W ) does not decrease more than a given tolerance > 0.

Figure 2 :

 2 Figure 2: Academic example -Left: Plot of the ellipsoidal set S 0 with (in magenta) and without (in cyan) NSAW. State-space evolution of the system initiated in [1 0] , [-1 0] , [0 1] and [0 -1] without backlash (dotted black line), with backlash alone (solid blue line) and with backlash and NSAW (dashed red line). Right: zoom of the left figure.

Figure 3 :

 3 Figure 3: F-8 example (SF) -Time evolution of the angle of attack (x 3 ). System without backlash (dotted black line), with backlash alone (solid blue line) and with backlash and NSAW (dashed red line).

Figure 4 :

 4 Figure 4: F-8 example (SF) -Time evolution of the pitch angle (x 4 ). System without backlash (dotted black line), with backlash alone (solid blue line) and with backlash and NSAW (dashed red line).

Figure 5 :

 5 Figure 5: F-8 example (SF) -Backlash characteristics Φ[y c ] for the elevator angle (u 1 ). Left: system without NSAW. Right: system with NSAW.

Figure 6 :

 6 Figure 6: F-8 example (SF) -Backlash characteristics Φ[y c ] for the flaperon angle (u 2 ). Left: system without NSAW. Right: system with NSAW.

Figure 7 :

 7 Figure 7: F-8 example (DOF) -Time evolution of the angle of attack (x 3 ). System without backlash (dotted black line), with backlash alone (solid blue line) and with backlash and NSAW (dashed red line).

  the input of the plant, y p ∈ R p is the measured output. A p , B p , C p are matrices of appropriate dimensions. The pairs (A p , B p ) and (C p , A

p ) are supposed to be controllable and observable, respectively. The function u p is a backlashlike nonlinearity.

The positive definiteness constraint of N 1 of Lemma 1 in[START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF] can be relaxed in a positive semi-definiteness constraint without loss of generality. It is the reason why in the current paper we fix N 1 = 0.


Liguo Zhang
email: zhangliguo@bjut.edu.cn

Christophe Prieur

Junfei Qiao

PI Boundary Control of Linear Hyperbolic Balance Laws with Stabilization of ARZ Traffic Flow Models

Keywords: Hyperbolic systems of balance laws, Boundary control, Lyapunov function, Exponential stability, ARZ traffic flow model

This paper investigates the proportional-integral (PI) boundary feedback control for the linear hyperbolic systems of balance laws which control and output measures are located at the boundaries. We address the issue of feedback stabilization by means of PI boundary controllers. By constructing a new weighted Lyapunov function, the sufficient conditions in terms of matrix inequalities are developed for the exponential stability of closed-loop systems. These results are illustrated by the linearized Aw-Rascle-Zhang (ARZ) traffic flow model. We design a PI boundary controller to stabilize the oscillations of the traffic parameters of a freeway segment and evaluate the performance with numerical simulations.

Introduction

In this paper, we are concerned with balance laws governed by linear hyperbolic partial differential equations (PDEs). Such systems have been extensively used to model physical processes and engineering problems. The typical examples include Saint-Venant equation for open channels in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF], Euler equation for gas pipes in [START_REF] Gugat | Gas flow in fan-shaped networks: classical solutions and feedback stabilization[END_REF], and Aw-Rascle-Zhang equation for long roads in [START_REF] Aw | Resurrection of 'second order' models of traffic flow[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF].

Boundary feedback control has been central for stabilization of the hyperbolic balance laws. Many results have established appropriate boundary conditions for linear or quasi-linear systems in L 2 or H 2 topology spaces as in [START_REF] Herty | Feedback boundary control of linear hyperbolic systems with relaxation[END_REF][START_REF] Zhang | Local exponential stabilization of semi-linear hyperbolic systems by means of a boundary feedback control[END_REF][START_REF] Bastin | Exponential stability of semi-linear onedimensional balance laws[END_REF][START_REF] Hasan | Boundary observer design for hyperbolic pde-ode cascade systems[END_REF][START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]. Interested readers are suggested to review literature with [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF].

Proportional-integral (PI) control is one of the mostly used method in engineering since its advantage is to cancel forced oscillations and attenuate load disturbances. Frequency domain is widely used to prove boundary stability of the closed-loop systems. An explicit necessary and sufficient condition is proposed in [START_REF] Bastin | Stability of linear density-flow hyperbolic systems under PI boundary control[END_REF] for linear density-flow systems by using Laplace transformation. Similar results are developed with the operator and semi-group approach in [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF]. Backstepping method is exploited in [START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking[END_REF] to elaborate the PI-based trajectory tracking. Moreover, Lyapunov approach is considered in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF][START_REF] Trinh | Stability and output regulation for a cascaded network of 2×2 hyperbolic systems with PI control[END_REF] for linear, nonlinear and network hyperbolic systems of conservation laws following the idea introduced in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. However, only few results appear in this direction due to the difficulties of constructing an appropriate Lyapunov candidate with PI boundary control.

Aw-Rascle-Zhang (ARZ) model, compared to scalar LWR model [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF], [START_REF] Richards | Shock waves on the highway[END_REF], developed a velocity equation to reveal the anisotropic property of traffic dynamics. In [START_REF] Belletti | Prediction of traffic convective instability with spectral analysis of the Aw-Rascle-Zhang model[END_REF], spectral analysis is applied to design boundary control for the linearized ARZ model. An integrated on-ramping metering and variable speed limit control is proposed in [START_REF] Zhang | Necessary and sufficient conditions on the exponential stability of positive hyperbolic systems[END_REF] to ARZ model by investing the boundary stabilization of positive hyperbolic systems. To deal with the model uncertainties, [START_REF] Zhang | Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control[END_REF] develops the stochastic stability for Markov jumped hyperbolic systems with application to ARZ model. Recently, boundary control are developed in [START_REF] Yu | Traffic congestion control on Aw-Rascle-Zhang model: Full-state feedback[END_REF] by using the backstepping method to reduce stop-and-go oscillations for the congested traffic.

This paper constructs a new weighted Lyapunov function to prove the exponential stability of n-order linear hyperbolic balance laws with PI boundary control. For the best of our knowledge, existing results are focused on 2 × 2 systems. In [10, Section 5.4], PI boundary stabilization of linearized density-flow systems described by balance laws is provided with application to control in navigable rivers. Moreover in this later work, the source term is assumed to be stable. In our paper, we do not need this assumption on the source term. Furthermore we consider a more general linear hyperbolic system of balance laws and a practical application to freeway traffic by the linearized ARZ equation. PI boundary feedback controller is designed to stabilize the system by using integrated on-ramp metering and variable speed control. With respect to [START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF], we do not consider only a scalar conservation law, but rather a hyperbolic system of balance laws.

The paper is organized as follows. The statement of the PI boundary control for linear balance laws and the main result are given in Section 2. In Section 3 we present the proof of the main result by using the Lyapunnov techniques. Section 4 is devoted to the PI boundary stabilization of the linearized ARZ model. Some concluding remarks and possible further research lines are presented in Section 5.

Notation. R n and R n×n denote the set of n-order vectors and n-order matrices. For a matrix A, A denotes the transpose, λ max (A) denotes the largest real part of eigenvalues of A, and for a partitioned symmetric matrix, the symbol stands for the symmetric blocks. Given an integer n, I n is the identity matrix of order n. Given a function g :

[0, L] → R n , we define its L 2 - norm as g L 2 ((0,L);R n ) = L 0 |g(x)| 2 dx where |•| is the Euclidean norm in R n .
We call L 2 ((0, L); R n ) the space of all measurable functions g for which g L 2 ((0,L);R n ) < ∞.

Preliminaries and main results

Consider the one-dimensional n×n linear hyperbolic systems of balance laws

ξ t + Λξ x = Mξ, t ∈ [0, ∞), x ∈ (0, L), (1)
where

ξ : (0, L) × [0, ∞) → R n , Λ ∈ R n×n and M ∈ R n×n .
Without loss of generality, we may assume that Λ is a diagonal matrix with non-zero real diagonal entries such that

Λ = diag{Λ + , -Λ -}, in which Λ + = diag{λ 1 , . . . , λ m }, Λ -= diag{λ m+1 , . . . , λ n }, 1 ≤ m ≤ n,
and λ i > 0 for all i ∈ {1, . . . , n}.

For simplicity, we use the notations

ξ in (t) ξ + (0, t) ξ -(L, t) , ξ out (t) ξ + (L, t) ξ -(0, t) ,
to denote the input and the output of the system (1) on the left and the right boundaries, respectively, where ξ + , ξ -are defined as

ξ + = [ξ 1 , . . . , ξ m ] ∈ R m and ξ -= [ξ m+1 , . . . , ξ n ] ∈ R n-m .
In this paper, our concern is to analyze the exponential stability of the system (1) under boundary conditions of the PI feedback types, i.e.,

ξ in (t) = K p ξ out (t) + K I t 0 ξ out (τ)dτ, (2)
where K p ∈ R n×n and K I ∈ R n×n are matrix gains, and an initial condition of the form

ξ(x, 0) = ξ 0 (x), x ∈ (0, L). (3)
The well-posedness and the existence of the classical maximal solutions of C 0 ([0, ∞); L 2 ((0, L); R n )) of the Cauchy problem (1)-(3) are easily adapted from [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF]Appendix Theorem A.4]. The definition of the exponential stability for the system (1) is then given as follows.

Definition 1. The linear hyperbolic system (1)-( 2) is exponentially stable if there exist ν > 0 and C > 0 such that, for every ξ 0 ∈ L 2 ((0, L); R n ), the solution to the Cauchy problem (1)-( 3) satisfies

ξ(•, t) L 2 ((0,L);R n ) ≤ Ce -νt ξ 0 L 2 ((0,L);R n ) , (4)
for all t ∈ [0, ∞).

We then have the following stability theorem.

Theorem 1. The linear hyperbolic system (1)-( 2) is exponentially stable if there exist a diagonal matrix P 1 ∈ R n×n , a symmetric matrix P 2 ∈ R n×n , and a matrix P 3 ∈ R n×n , a real constant µ, such that the following matrix inequalities hold, for all x ∈ [0, L],

(i) P = P 1 P 3 P 2 > 0, (5)
(ii) Ω e (x) =           Ω e 11 (x) Ω e 12 (x) P 3 (x) Ω e 22 Ω e
23

Ω e 33           < 0, (6)
where

Ω e 11 (x) = M P 1 (x) + P 1 (x)M -2µ|Λ|P 1 (x), Ω e 12 (x) = M P 3 (x) -µ|Λ|P 3 (x), Ω e 22 = 1 L (K I |Λ|P 1 E 1 K I + K I |Λ|P 3 E 3 + E 3 P 3 |Λ|K I )
,

Ω e 23 = 1 L (K I |Λ|P 1 E 1 K p -|Λ|P 3 E 4 + E 3 P 3 |Λ|K p ) + P 2 , Ω e 33 = 1 L (K p |Λ|P 1 E 1 K p -|Λ|P 1 E 2 ), with P 1 (x) = P 1 diag e -2µx I m , e 2µx I n-m , P 3 (x) = P 3 diag {e -µx I m , e µx I n-m }, |Λ| = diag{Λ + , Λ -}, and E 1 = diag I m , e 2µL I n-m , E 2 = diag e -2µL I m , I n-m , E 3 = √ E 1 , E 4 = √ E 2 .
Remark 1. It can be noticed that Theorem 1 provides a numerical tractable method to determine the gains K p and K I of the PI controller by solving the set of matrix inequalities ( 5), ( 6). This result is also consistent with the classical dissipative boundary condition for the linear hyperbolic systems of balance laws in [START_REF] Diagne | Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[END_REF], in which only the proportional feedback control is considered, as setting K I = 0. Note that the deadbeat control obtained by letting K P = K I = 0 stabilizes the linear hyperbolic system (1)-( 2) in finite time but it is not robust with respect to small errors, in particular when the matrix M is not stable. To be more specific, any small error, when numerically computing the controller, or when implementing it, destabilizes the system. In contrast, our approach is based on a Lyapunov function which guarantees a robustness and a performance of the closed-loop system.

Exploiting the linearity we can differentiate (1) with respect to x and the boundary condition [START_REF] Gugat | Gas flow in fan-shaped networks: classical solutions and feedback stabilization[END_REF] with respect to t to study the stability of the linear hyperbolic system (1)-( 2) in H 1 norm. By doing so we would get similar conditions for the stability analysis in H 1 Sobolev space.

•

Inspired from [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], we construct a new weighted Lyapunov function to prove the exponential stabilization of the closedloop system (1)-( 2) with the PI controller.

The Lyapunov function candidate

V : L 2 ((0, L); R n ) × R n → R has the following form V(ξ, ζ) = L 0 ξ P 1 (x)ξ + ξ P 3 (x)ζ + ζ P 3 (x)ξ dx +Lζ P 2 ζ = L 0 ξ ζ P 1 (x) P 3 (x) P 2 ξ ζ dx, (7)
where ζ : [0, ∞) → R is the integral of the output of the system (1), i.e.,

ζ(t) = t 0 ξ out (τ)dτ. ( 8
)
Remark 2. The weighted Lyapunov function V(ξ, ζ) is an extension of the one V(ξ) in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], in which only the proportional feedback control is considered. The functions of the added two terms in V(ξ, ζ), which involve the integral variable ζ, are to measure the energy change introduced by the integral controller.

•

The proof of Theorem 1 is provided in the next section.

Proof of the main result

Proof in the case where m = n

To clarify the demonstration, we first prove Theorem 1 in the special case where m = n, which means that the characteristic matrix Λ = Λ + with λ i > 0, for all i = 1, . . . , n. In this case, the PI boundary controller (2) is simply rewritten as

ξ(0, t) = K p ξ(L, t) + K I t 0 ξ(L, τ)dτ. (9)
Moreover, condition (ii) of Theorem 1 is turning into the single matrix inequality as

(ii ) Ω =           Ω 11 Ω 12 P 3 Ω 22 Ω 23 Ω 33           < 0, (10)
where

Ω 11 = M P 1 + P 1 M -2µΛP 1 , Ω 12 = M P 3 -µΛP 3 , Ω 22 = 1 L (K I ΛP 1 K I + K I ΛP 3 + P 3 ΛK I ), Ω 23 = 1 L (K I ΛP 1 K p -e -µL ΛP 3 + P 3 ΛK p ) + P 2 , Ω 33 = 1 L (K p ΛP 1 K p -e -2µL ΛP 1 ).
We assume for the time-being that the solution ξ to (1) with the boundary condition [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] and the initial condition (3) is of class C 1 on [0, L] × [0, ∞). Under this assumption (that will be relaxed later on) the function V defined in ( 7) is of class C 1 , and the first step of the proof is to compute the estimation of the time-derivative V along the solution (ξ, ζ). It yields the following:

V = L 0 2ξ t P 1 ξe -2µx + ξ t P 3 ζe -µx + ζ P 3 ξ t e -µx +ξ P 3 ξ(L, t)e -µx + ξ (L, t)P 3 ξe -µx dx +2Lξ (L, t)P 2 ζ = V1 + V2 + V3 with V1 2 L 0 ξ t P 1 ξe -2µx dx, V2 L 0 ξ t P 3 ζ + ζ P 3 ξ t + ξ P 3 ξ(L, t) +ξ (L, t)P 3 ξ e -µx dx, V3 2Lξ (L, t)P 2 ζ.
Using the PDE (1) and an integration by parts for V1 and V2 , we obtain

V1 = 2 L 0 -Λξ x + Mξ) P 1 ξe -2µx dx = -ξ ΛP 1 ξe -2µx L 0 -2µ L 0 ξ ΛP 1 ξe -2µx dx + L 0 ξ M P 1 + P 1 M ξe -2µx dx (11)
and

V2 = L 0 -Λξ x + Mξ P 3 ζe -µx dx + L 0 ζ P 3 -Λξ x + Mξ e -µx dx + L 0 ξ P 3 ξ(L, t) + ξ (L, t)P 3 ξ e -µx dx, = -ξ ΛP 3 ζe -µx -ζ P 3 Λξe -µx L 0 -µ L 0 ξ ΛP 3 ζ + ζ P 3 Λξ e -µx dx + L 0 ξ M P 3 ζ + ζ P 3 Mξ e -µx dx + L 0 ξ P 3 ξ(L, t) + ξ (L, t)P 3 ξ e -µx dx. ( 12
)
Then, substituting the PI boundary condition ( 9) into ( 11) and [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF], we have the time-derivatives

V1 = -ξ (L, t)ΛP 1 ξ(L, t)e -2µL + K p ξ(L, t) + K I ζ ΛP 1 K p ξ(L, t) + K I ζ + L 0 ξ M P 1 + P 1 M -2µΛP 1 ξe -2µx dx = ζ ξ(L, t) K I ΛP 1 K I K I ΛP 1 K p K p ΛP 1 K p -e -2µL ΛP 1 ζ ξ(L, t) + L 0 ξ M P 1 + P 1 M -2µΛP 1 ξe -2µx dx = L 0           ξe -µx ζ ξ(L, t)                     M P 1 + P 1 M -2µΛP 1 0 0 K I ΛP 1 K I L K I ΛP 1 K p L K p ΛP 1 K p L -e -2µL ΛP 1 L                       ξe -µx ζ ξ(L, t)           dx, (13) and V2 = -ξ (L, t)ΛP 3 ζe -µL -ζ P 3 Λξ(L, t)e -µL + K p ξ(L, t) + K I ζ ΛP 3 ζ + ζ P 3 Λ K p ξ(L, t) + K I ζ -µ L 0 ξ ΛP 3 ζ + ζ P 3 Λξ e -µx dx + L 0 ξ M P 3 ζ + ζ P 3 Mξ e -µx dx + L 0 ξ P 3 ξ(L, t) + ξ (L, t)P 3 ξ e -µx dx = ζ ξ(L, t) K I ΛP 3 + P 3 ΛK I P 3 ΛK p -e -µL ΛP 3 0 ζ ξ(L, t) + L 0           ξe -µx ζ ξ(L, t)           ×           0 M P 3 -µΛP 3 P 3 0 0 0                     ξe -µx ζ ξ(L, t)           dx = L 0           ξe -µx ζ ξ(L, t)                      0 M P 3 -µΛP 3 K I ΛP 3 L + P 3 ΛK I L P 3 P 3 ΛK p L -e -µL ΛP 3 L 0                      ξe -µx ζ ξ(L, t)           dx. (14)
Moreover, the time-derivative V3 may be rewritten in the integral form as

V3 = L 0           ξe -µx ζ ξ(L, t)                     0 0 0 0 P 2 0                     ξe -µx ζ ξ(L, t)           dx. (15)
Combining V1 in [START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking[END_REF], V2 in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF], and V3 in [START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF] together, the time-derivative of V(ξ, ζ) satisfies

V = L 0           ξe -µx ζ ξ(L, t)           ×           Ω 11 Ω 12 P 3 Ω 22 Ω 23 Ω 33                     ξe -µx ζ ξ(L, t)           dx. (16)
It follows from the condition (ii ) that Ω < 0, then there exists a constant ν = -λ max (Ω) λ max (P) > 0 such that

V ≤ λ max (Ω) L 0           ξe -µx ζ ξ(L, t)                     ξe -µx ζ ξ(L, t)           dx ≤ -νV(ξ, ζ). ( 17
)
By remarking the definition of V(ξ, ζ), and coming from the condition (i) that P > 0, we have V(ξ, ζ) > 0 for any classical solutions of ( 1)-( 2). On the other hand, there exist β > 0 (depending on P i , i = 1, 2, 3) such that the following inequalities hold

1 β L 0 |ξ| 2 dx + |ζ| 2 ≤ V(ξ, ζ) ≤ β L 0 |ξ| 2 dx + |ζ| 2 . ( 18
)
Estimates ( 17) and ( 18) are obtained under the assumption that ξ is of class C 1 on [0, L] × [0, ∞). But the selection of β and ν does not depend on the C 1 -norm of ξ: they depend only on the C 0 ([0, ∞); L 2 ((0, L); R n ))-norm of ξ. Hence, using a classical density argument (see e.g., [10, Section 2.1.3]), the estimates ( 17) and ( 18) remain valid in the distribution sense if ξ is only of class C 0 .

Let ξ ∈ C 0 ([0, ∞); L 2 ((0, L); R n )) be the maximal solution of the Cauchy problem ( 1)-( 2). Using estimates ( 17) and ( 18) for all t ∈ [0, ∞), we get that

ξ(•, t) L 2 ((0,L);R n ) ≤ L 0 |ξ| 2 dx + |ζ| 2 ≤ βV(ξ, ζ) ≤ β 2 e -νt L 0 |ξ 0 | 2 dx + |ζ(0)| 2 = β 2 e -νt ξ 0 L 2 ((0,L);R n ) , (19)
Consequently, the hyperbolic system (1)-( 2) is exponentially stable in L 2 -norm by choosing C = β 2 . This concludes the proof of Theorem 1 in the case m = n.

Remark 3. Depending on the relaxation matrix M being stable or not, the variable µ included in the matrix Ω might be positive or negative in Theorem 1. In fact, it is helpful to reduce the conservative of the design of the boundary feedback controller by sufficiently exploiting the relaxation structure of source terms as considered by [START_REF] Zhang | Local exponential stabilization of semi-linear hyperbolic systems by means of a boundary feedback control[END_REF]. • 3.2. Proof in the case where 0 < m < n.

In this section, we modify the proof of Theorem 1 in Section 3.1 to deal with the case 0 < m < n. The case m = 0 is equivalent to the case m = n by considering ξ(Lx, t) instead of ξ(x, t).

The major difference lies in the time-derivatives V1 , V2 which are now rewritten, respectively, as follows:

V1 = 1 0           ξ ζ ξ out                     M P 1 (x) + P 1 (x)M -2µ|Λ|P 1 (x) 0 0 K I |Λ|P 1 E 1 K I L K I |Λ|P 1 E 1 K p L K p |Λ|P 1 E 1 K p L -|Λ|P 1 E 2 L                       ξ ζ ξ out           dx, (20) and V2
= 1 0           ξ ζ ξ out                      0 M P 3 (x) -µ|Λ|P 3 (x) K I |Λ|P 3 E 3 L + E 3 P 3 |Λ|K I L P 3 (x)
E 3 P 3 |Λ|K p L -|Λ|P 3 E 4 L 0                      ξ ζ ξ out           dx, (21)
where the matrices E i , i = 1, 2, 3, 4, are defined in the statement of Theorem 1. Since P > 0 and Ω e (x) < 0, it may be checked that Theorem 1 can be established for the case 0 < m < n in a manner completely parallel to the one that we have followed in the case m = n.

PI boundary control for ARZ traffic flow model

ARZ Model and Linearization

The traffic dynamics in a freeway segment are given by the so-called Aw-Rascle-Zhang equations, x ∈ (0, L)

∂ t ρ + ∂ x (vρ) = 0 ∂ t v + (v -ρp (ρ))∂ x v = V(ρ)-v τ ( 22
)
where ρ(x, t) is the vehicle density, v(x, t) is the average speed, τ is the relaxation time related to driving behavior. The variable p(ρ) is the traffic pressure defined as

p(ρ) = v f -V(ρ), (23)
where v f is the free flow speed. The speed-density fundamental diagram V(ρ) is given in the form of Greenshields model in [START_REF] Greenshields | A study of traffic capacity[END_REF]

V(ρ) = v f 1 - ρ ρ m γ , (24)
where ρ m is the maximum density and constant γ > 0.

Let z = v, w = v + v f ρ ρ m γ
, then the nonlinear equation ( 22) may be written in the characteristic Riemann coordinates as

∂ t w + z∂ x w = v f -w τ ∂ t z + (1 + γ)z -γw ∂ x z = v f -w τ ( 25
)
where the sign of the velocity (1 + γ)zγw (positive or negative), indicates the freeway traffic lies in the free-flow or in the congested mode, as in [START_REF] Zhang | Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control[END_REF]. We assume that the system ( 25) is strictly hyperbolic. Denote (w * , z * ) being the steady state of the system (25), the corresponding (ρ * , v * ) of the system [START_REF] Zhang | Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control[END_REF], satisfying w * = v f or function v * = V(ρ * ). The deviations from the nominal states (w * , z * ) are defined as w(x, t) = w(x, t)w * ,

(x, t) = z(x, t) -z * . ( (26) z
) 27
We then obtain the linearized model

∂ t w z + z * 0 0 (1 + γ)z * -γw * ∂ x w z = -1/τ 0 -1/τ 0 w z . (28)

PI boundary feedback control

As the freeway traffic lies in the congested regime, i.e. (1 + γ)z * -γw * < 0, for the boundary feedback control, the inflow speed v(0, t) and outflow density ρ(L, t) are measured, and the inlet on-ramp metering r(t) and the outlet speed limit v(0, t) are control units.

Precisely, we introduce the PI boundary feedback law:

r(t) = r * + k r p (ρ(L, t) -ρ * ) +k r i t 0 (ρ(L, τ) -ρ * )dτ, (29) v(L, t) = v * + k v p (v(0, t) -v * ) +k v i t 0 (v(0, τ) -v * )dτ, ( 30
)
where r * is the nominal flux rate that satisfies the equilibrium relationship p in + r * = ρ * v * , and k r p , k r i , k v p and k v i are tuning gains.

Let ρ = ρ-ρ * , ṽ = v-v * , and from the PI controller ( 29)-( 30) directly, we have

r(t) = r * + k r p ρ(L, t) + k r i t 0 ρ(L, τ)dτ, (31)
ṽ(L, t) = k v p ṽ(0, t) + k v i t 0 ṽ(0, τ)dτ. ( 32
)
Using the flow-conserving condition at the inlet of the upstream boundary, i.e., x = 0, we have

p in + r(t) = ρ(0, t)v(0, t). ( 33
)
After the linearization of the flow-conserving condition (33) with the integrating of the feedback control laws (29)-(30), we have the following boundary condition:

k r p ρ(L, t) + k r i t 0 ρ(L, τ)dτ = v ρ(0, t) + ρ ṽ(0, t). (34)
As further assume γ = 1, and α = v f ρ m , and deleting the highorder terms, we could rewrite the condition (34) in the Riemann coordinates as w(0, t) = ṽ(0, t) + α ρ(0, t)

= 1 - αρ v ṽ(0, t) + αk r p v ρ(L, t) + αk r i v t 0 ρ(L, τ)dτ = k r p v w(L, t) + k r i v t 0 w(L, τ)dτ + 1 - αρ v - k r p k v p v ṽ(0, t) - k r p k v i v + k r i k v p v t 0 ṽ(0, τ)dτ. (35)
Combining ( 30) and (35) together for the system (28), we have the following PI boundary feedback control w(0, t) z(L, t) = K p w(L, t) z(0, t)

+ K I t 0 w(L, τ) z(0, τ) dτ, (36)
where

K p = k r p v 1 -αρ v - k r p k v p v 0 k v p , (37)
and

K I = k r i v -k r p k v i -k r i k v p v 0 k v i . (38)

Simulations

The developed PI boundary feedback control laws (29)-(30) for the stabilization of freeway traffic are now tested with the numerical simulation.

To this end, we consider a freeway segment whose road parameters are given, respectively, as ρ m = 200 veh./km, v f = 150 km/hour, a = 0.75, p in = 6600 veh./hour, r * = 1800 veh./hour. The total road length L = 1 km and the relation time τ = 60 second.

Given the steady-state (ρ * , v * ) = (120, 70), which satisfies the ARZ equations [START_REF] Zhang | Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control[END_REF]

To compute the numerical solutions of the system (28), we discretize them using the two-step variant of Lax-Wendroff method in [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF]. The initial deviations from the steady-state (ρ * , v * ) are given as ρ(x, 0) = ρ * + 8 sin(5πx) v(x, 0) = v * + 12 cos(5πx) (44)

Figs. 1 and2 show the time evolutions of the derivative components w, z, of the linearized system (28), respectively. Figs. 3 and4 show the evolution of the vehicle density ρ(x, t) and the average speed v(x, t) of the nonlinear ARZ model [START_REF] Zhang | Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control[END_REF] with PI boundary feedback control (29)-(30). It is observed that both variables converge to their steady-state ρ * = 120veh./km, and v * = 70km/hour, and almost arrive in the finite time about 0.2 hour, as expected from Theorem 1.

Conclusion

We have addressed the issue of the PI boundary control of the linear hyperbolic balance laws. The main contribution is the Lyapunov stability analysis of the proposed closed-loop system. In Theorem 1, the sufficient conditions given as a set of matrix inequalities can be used to choice the proper tuning parameters of the boundary condition. The efficiency of the theory contribution is illustrated with the boundary stabilization of the linearized ARZ model. The future research will focus on the extension of Theorem 1 to the nonuniform case and quasilinear hyperbolic systems in the H 2 space.

Figure 1 :

 1 Figure 1: The evolution of the deviation w of the linearized system (28) with PI boundary control (36).

Figure 2 :

 2 Figure 2: The evolution of the deviation z of the linearized system (28) with PI boundary control (36).

Figure 3 :

 3 Figure 3: The evolution of ρ of the nonlinear ARZ model (22) with PI boundary control (29)-(30) and steady-state (ρ * , v * ) = (120, 70).

Figure 4 :

 4 Figure 4: The evolution of v of the nonlinear ARZ model (22) with PI boundary control (29)-(30) and steady-state (ρ * , v * ) = (120, 70).

  with λ * 1 = -20 > 0. Choose the tuning parameters in (29)-(30) ask v i = -0.2, k v p = 0.1, k r i = -2.5, and k r p = 7. The associated boundary condition matrices K p and K i are given, respectively, as Taking µ = 0.01 and solving the inequality conditions (i)-(ii) of Theorem 1, we obtain the following diagonal matrix P 1 , symmetric matrix P 2 , and matrix P 3

	K p =	0.1000 -0.0814 0 0.1000	,		(39)
	K I =	-0.0357 0.0236 0 -0.2000	.	(40)
	P 1 =	0.0610	0 0.7755	,		(41)
	P 2 =	2.1817 0.6366 1.3823	,		(42)
	P 3 =	0.0375 0.0094 0.1604 0.3192	.

$ This work is supported by the National Natural Science Foundation of China (NSFC, grant No. 61374076, No. 61533002), and the Beijing Natural Science Foundation (grant No. 1182001).


J C Goebel

E Roeder

A L Perrier

D Grenier

P Gillet

A Pinzano

O Beuf

Bio-integration assessment of rat knee cartilage repair using in vivo MRI at 7T

Quality of HR-MR images obtained with the developed array coils allowed the segmentation of cartilage compartments and measurements of cartilage thickness and volumes. HR-MRI allowed the follow up of cartilage to evaluate in vivo therapeutical responses after chondrogenic

biomaterial grafting and cartilage regrowth seems to be helped by the implant but additional information is required to estimate the real biofunctionality of the result.

Objectives:

The aim of this study was to follow using 7T MRI the biointegration of biomaterials colonized by mesenchymal stem cells or not after implantation in a calibrated full thickness focal lesion in the rat knee.

Material and method

Animal Protocol:

Ethics guidelines for experimental investigations in animals were strictly followed after acceptance by the animal experimentation ethics committee from Université de Lyon. The knee joint were exposed by medial parapatellar incision, and the trochlear groove were exposed by lateral dislocation of the patella. A 1.8 mm outer diameter drill was used to create osteochondral defects (1.8 -2.0 mm). 24 adult male lewis rats (7-week-old animals) et 12 adult male nude rats underwent surgically a focal calibrated defect of their right femoral articular cartilage on the medial side according to a validated procedure:

• 12 Lewis rats underwent a parapatellar incision without defect • 12 Lewis rats underwent a full-thickness cartilage defect • 12 nude rats underwent a full-thickness cartilage defect treated by collagen sponges seeded with human mesenchymal stem cells.

MR exploration of rat knee was performed at D7, D14, D28 and D120. After MR exploration, 3 rats were sacrificed at each time course and knees were fixed for further histological confrontation.

MRI acquisition:

A four-channel surface coil designed for the simultaneous acquisition of both knee of a rat on a Bruker 7T Biospec MR system was used (Fig. 1). HR-MR images of the rat knees were obtained using a 3D FLASH sequence with the following parameters:

  Magnetic Resonance systems allowing high-resolution Magnetic Resonance Imaging (HR-MRI) is a powerful research tool to visualize and examine hyaline cartilage of small joints non-invasively. Different studies have shown that qualitative assessment of degenerative joint disease, derived from MR images, was reliable (1). The capability to show pathologic changes throughout the time course of the disease from three-dimensional (3D) datasets has also been demonstrated. However, in vivo quantitative imaging for an accurate determination of cartilage thickness and volume is challenging due to small size of rodent joints and dedicated coils are mandatory (2-3). Surgical articular cartilage repair therapies for cartilage defects such as osteochondral autograft transfer, autologous chondrocyte implantation or matrix associated autologous cell transplantation require noninvasive technique to evaluate longitudinally the biointegration of the graft.

Acknowledgements:

  This work was funded by ANR with grant ref. ANR-09-BLANC-0150-01 and was performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Fig. 1 :Fig. 2 :

 12 Fig. 1: Four-channels array coil designed for simultaneously image both knees*

Fig. 4 :

 4 Fig. 3: Segmented cartilage compartments of both knees in control rat.

*

  More information about the array coil can be find at poster # 1833 entitled : "Design of a four-channel array coil for dual high-resolution rat knee MR Imaging" olivier.beuf@creatis.insa-lyon.fr