Juan G Rueda-Escobedo

Rosane Ushirobira

Denis Efimov

Jaime A Moreno

Gramian-based uniform convergent observer for stable LTV systems with delayed measurements

Keywords: Observer, Delayed output, Linear Time-varying Systems

In this work, an observer for a linear time-varying system with delayed measurements is developed. The delay is assumed to be unknown, bounded, and it can be time-varying with no restriction on its rate of change. The observer uses auxiliary signals related to the constructibility Gramian of the system and it contains nonlinearities that provide a uniform fixed-time convergence to a bounded region in the estimation error coordinates. This means that the convergence time can be bounded by a positive constant which is independent from the initial conditions and the initial time. This property is new for the addressed class of systems. The ultimate bound of the estimation error depends on the maximum difference between the nominal output and the delayed one, and not directly on the delay size or its time derivative. These properties are illustrated in a numerical simulation.

Introduction

The internal state estimation of a given system is one of the basic tasks in automatic control theory (Astolfi & Marconi, 2008;[START_REF] Besançon | Nonlinear observers and applications[END_REF][START_REF] Crassidis | Optimal estimation of dynamic systems[END_REF]Meurer, Graichen, & Gilles, 2005). To develop such a task, an input-output information of the system is needed. If the information is carried through a network, or transmitted over long distances, it will be available but with a delay. The design of observers to perform the estimation using delayed data has been a topic of recent interest [START_REF] Assche | High gain observer design for nonlinear systems with time varying delayed measurements[END_REF][START_REF] Cacace | Joint state estimation and delay identification for nonlinear systems with delayed measurements[END_REF][START_REF] Cacace | An observer for a class of nonlinear systems with time varying observation delay[END_REF][START_REF] Khosravian | State estimation for invariant systems on lie groups with delayed output measurements[END_REF][START_REF] Vafaei | A chain observer for nonlinear long constant delay systems: A matrix inequality approach[END_REF]. In the case of linear time invariant system (LTI), the approaches based on delayed output error injection have been developed. This strategy can be used, among others, in the case of a known constant delay [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF], an unknown but constant delay [START_REF] Cacace | Joint state estimation and delay identification for nonlinear systems with delayed measurements[END_REF], or a time-varying known delay [START_REF] Cacace | An observer for a class of nonlinear systems with time varying observation delay[END_REF], (Fridman, 2014a)[Sec. 5.2], [START_REF] Kruszewski | A switched system approach to exponential stabilization through communication network[END_REF]. These re-CONTACT: Juan G. Rueda-Escobedo. Email: JRuedaE@iingen.unam.mx sults can also be applied for a certain class of nonlinear systems, as it is shown in the cited works, but there are also specialized works that study this type of systems. Examples of these works are [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF] where conditions for the identifiability of constant delays in nonlinear systems are given, also [START_REF] Ibrir | Adaptive observers for time-delay nonlinear systems in triangular form[END_REF] where an observer is proposed for nonlinear systems in triangular form and the observer gain is adapted by solving an algebraic Riccati equation depending on a dynamic parameter, or [START_REF] Ghanes | Observer design for nonlinear systems under unknown time-varying delays[END_REF] where the authors present an observer that provides a bounded estimation error in the presence of a time-varying, unknown, and bounded delay. The results of this last reference are the closest in essence to the objectives of this work.

Most of these works require to check a linear matrix inequality (LMI) to establish the convergence of the observer. Application of LMIs is common in the study of timedelay systems [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF][START_REF] Sun | A survey on lyapunov-based methods for stability of linear time-delay systems[END_REF]). However, an analogue result for linear time-varying (LTV) systems seems to be not available. This can be related to the scarce results about stability of time-delay LTV systems and the difficulties that arise in their study. Among such works, one can find [START_REF] Alaviani | Delay dependent stabilization of linear time-varying system with time delay[END_REF] where conditions are provided in terms of LMIs involving time-varying matrices, the stability conditions for a class of positive system are given in [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear lyapunov functionals and a positive systems approach[END_REF], or in [START_REF] Zhou | Razumikhin and Krasovskii stability theorems for timevarying time-delay systems[END_REF] where the stability conditions are stated in terms of a Lyapunov function for the nominal case, i.e. without delay.

In this note, a LTV plant with delayed measurements is considered. The delay is assumed to be time-varying and bounded, and no restriction over its speed of variation will be imposed. The upper and lower bounds for the delay are also assumed to be unknown, and they are not needed for the design. Also, it is required that the system with the undelayed output be uniformly completely observable. Under these assumptions, an observer which provides fixed-time convergence of the estimation error to a ball is proposed. An important difference between the approaches mentioned previously and what we are proposing is that the effect of the delay is not introduced in the error dynamics. This allows to study the observer convergence by means of standard techniques used to analyze LTV systems. The observer also includes a nonlinearity, based on a Gramian-like constructions, which is responsible for the accelerated rate of convergence. Additionally, the ball to which the estimation error converges, depends on the difference between the delayed and the nominal output, and not directly on the size of the delay or its time derivative, making the approach suitable for delays that are large in the time scale of the system. Despite the fact that the design of finite and fixed-time convergent observers has been on focus recently [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF][START_REF] Cruz-Zavala | Lyapunov functions for continuous and discontinuous differentiators[END_REF][START_REF] Cruz-Zavala | Asymptotic stabilization in fixed time via sliding mode control[END_REF], 2011;[START_REF] Lopez-Ramirez | Fixed-time output stabilization of a chain of integrators[END_REF][START_REF] Polyakov | Fixed-time stabilization of linear systems via sliding mode control[END_REF][START_REF] Ríos | A hybrid observer for uniform finite-time state estimation of linear systems[END_REF], their advantages in the case of time-delay systems are not fully investigated yet. A preliminary version of this work can be found in (Rueda-Escobedo, Ushirobira, [START_REF] Rueda-Escobedo | A gramianbased observer with uniform convergence rate for delayed measurements[END_REF]. The main difference with the previous version is the extension to LTV systems and a refinement on the ultimate bound for the estimation error.

The paper outline is as follows. In Section 2, the class of system under study and the problem statement are given. Some preliminary results regarding Riccati differential equations are discussed in Section 3. The observer structure and its properties are given in Section 4. The analysis of the estimation error and the proof of the results are developed in Section 5. The properties of the observer are illustrated in a numerical example in Section 6. Some auxiliary lemmas are established in the Appendix.

Notation: Let R > 0 and R ≥ 0 be the sets of positive and non-negative real numbers, respectively; R n denotes the real Euclidean space of dimension n; R n×m is the space of real matrices of n rows and m columns and I n denotes the identity matrix of R n×n . For x ∈ R n and p ≥ 1, x p is the p-norm, defined as ( n i=1 |x i | p ) 1/p . For A ∈ R n×n , A p means the induced matrix norm. Whenever the subscript p is omitted, • refers to the Euclidean norm, i.e. p = 2. For two symmetric matrices

Q 1 , Q 2 ∈ R n×n , Q 1 > Q 2 Q 1 ≥ Q 2 means that Q 1 -Q 2 is positive (semi-) definite. Given x ∈ R and p ∈ R ≥0 , x p denotes |x| p sign(x), if the exponent is omitted, it correspond to p = 1. For x ∈ R n , x p is understood element-wise.

Problem statement and motivation

In this note the state estimation of a linear time-varying system with delayed output will be investigated. To begin with, let us consider the following nominal system:

ẋ(t) = A(t)x(t) + B(t)u(t), ȳ(t) = C(t)x(t), (1)
where x ∈ R n , ȳ ∈ R m , and u ∈ R r are the state, the output, and the input vectors, respectively. The matrices A(t), B(t), and C(t) are assumed to be known, piecewise continuous in t, and uniformly bounded in their norm. The state transition matrix associated to A(t), which maps x(t 1 ) → x(t 2 ) in the absence of inputs, will be denoted by Φ(t 2 , t 1 ). In the following τ (t) : R → [0, τ ] will denote the delay and τ its upper bound. The state, input, and system matrices are assumed to be defined over the interval [t 0 -τ , ∞), where t 0 ≥ 0 represent the process start time. Two kinds of delayed output will be recognized: when the delay affect both, the output matrix C(t) and the state, and when the delay only appears in the state. If only the output matrix is delayed, one can rename it as C(t) = C(t -τ (t)), and address the problem as if there were no delay. We will focus our attention in the first case, when the output is completely delayed, since the second one can be treated by just slightly changing the notation. Then, the system of interest is as follows:

ẋ(t) = A(t) x(t) + B(t)u(t), y(t) = C t -τ (t) x t -τ (t) . (2)
In order to estimate the state of (2), we would ask the following:

Assumption 2.1. There exist positive constants T > 0, α 1 ≥ α 2 > 0 such that

α 1 I n ≥ W(t, t -T ) := t t-T Φ (s, t)C (s)C(s)Φ(s, t)ds ≥ α 2 I n for all t ∈ [t 0 + T, ∞), that is, the pair (A(t), C(t)
) is uniformly completely constructible.

Remark 1. Constructibility is related with the ability to reconstruct the current state from past measurements, whereas observability correspond to the reconstruction of the initial conditions from future data ("Chapter 4 Observability/Constructibility", 1977). In linear continuous-time systems, both properties are equivalent since Φ (t + T, t)W(t + T, t)Φ(t + T, t) is the observability Gramian.

One way to approach the posed problem is to propose a delayed observer of the form

ẋ(t) = A(t)x(t) + B(t)u(t) -L(t) C t -τ (t) x t -τ (t) -y(t) , (3)
where L(t) is a piecewise continuous function denoting the observer gain. Defining the estimation error as e(t) = x(t) -x(t), its dynamics results in

ė(t) = A(t)e(t) -L(t)C t -τ (t) e t -τ (t) . ( 4
)
If one is able to design L(t) in the nominal case, it can be expected that this approach work for sufficient small delay. Notice that (3) requires the knowledge of the delay in order to be implemented. The described method has been successfully applied for LTI system with constant delay [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF], and for time-varying one [START_REF] Léchappé | Dynamic observation-prediction for LTI systems with a time-varying delay in the input[END_REF]. In both cases it is required that τ satisfies some size restriction. If the delay is larger, the authors of [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF] has shown that the estimation can be handled by a chain of n observers. Each of the observers is oriented on treatment of an equivalent delay of τ /n, then if n is big enough, the scheme will provide an accurate estimate.

Another manner to approach the problem is to apply an undelayed output error injection, resulting in:

ẋ(t) = A(t)x(t) + B(t)u(t) -L(t) C(t)x(t) -y(t) , (5)
which does no require the value of τ (t), but imposes another source of inaccuracy. The error dynamics produced by this approach revels that it will depend on the difference between y(t) and ȳ(t):

ė(t) = A(t) -L(t)C(t) e(t) + L(t) y(t) -C(t)x(t) = A(t) -L(t)C(t) e(t) + L(t) y(t) -ȳ(t) (6)
Then, the boundedness of y(t)-ȳ(t) implies the boundedness of e(t) . This happens if, for example, the matrix A(t) defines a uniformly asymptotically stable motion. Since the delay is not needed in this case, it can be assumed uncertain, time-varying and bounded. The ultimate bound of the error is given in the following lemma:

Lemma 2.2. Let P (t) : R ≥0 → R n×n be a continuously differentiable matrix function,

Q(t) : R ≥0 → R n×n be a piecewise continuous matrix function, p L,1 I n ≥ P L (t) ≥ p L,2 I n and q L,1 I n ≥ Q L (t) ≥ q L,2 I n with positive constants p L,1 ≥ p L,2
> 0 and q L,1 ≥ q L,2 > 0, and they satisfy the differential Lyapunov inequality for any t 0 ∈ R ≥0 and all t ≥ t 0

ṖL (t) + P L (t) A(t) -L(t)C(t) + A(t) -L(t)C(t) P L (t) ≤ -Q L (t)
for a given piecewise continuous and bounded matrix function L(t) : R ≥0 → R n×r . Then in (6) the error e(t) stays bounded for all t ≥ t 0 and

lim t→∞ e(t) ≤ 2 p L,1 p L,2 • p L,1 q L,2 sup t≥t0 L(t) y(t) -ȳ(t) . (7)
Proof. Consider the Lyapunov function candidate V (t, e) = e P (t)e. Its derivative along (6) can be bounded as follows:

V (t) ≤ -e (t)Q(t)e(t) + 2 e (t)P (t)L(t) y(t) -ȳ(t) ≤ - 1 2 q L,2 e(t) 2 + 2 p 2 L,1 q L,2 L(t) 2 y(t) -ȳ(t) 2 ≤ - q L,2 2 p L,1 V (t) + 2 p 2 L,1 q L,2 L(t) 2 y(t) -ȳ(t) 2 , then V (t) ≤ V (t 0 )exp - q L,2 2 p L,1 (t -t 0 ) + 4 p 3 L,1 q 2 L,2 sup t≥t0 L(t) 2 y(t) -ȳ(t) 2 e(t) ≤ p L,1 p L,2 e(t 0 ) exp - q L,2 4 p L,1 (t -t 0 ) + 2 p L,1 p L,2 • p L,1 q L,2 sup t≥t0 L(t) y(t) -ȳ(t) .
From the last expression, the bound follows.

Both of the described methods has some disadvantages. In the case (3), not only τ (t) has to be known, but the design of L(t) can be really challenging, being this particularly true for LTV systems. In the second case (5), the delay can be unknown at the price of having a bounded error, and if one require to approach this bound faster, such bound will increase because this can only be achieved by increasing L(t).

Based on these observations the second approach seems to be more convenient, first, because there is no an extension of (3) to the LTV case, and second, because it does not require precise knowledge of the delay. To alleviate the problem w.r.t. the convergence rate, in this note, we will provide a methodology to modify (5) by introducing some nonlinearities in order to obtain a uniform rate of convergence, that is, the capability of reaching a bounded region of e(t) = 0 uniformly in t 0 and in the initial error. This will be done under the following hypothesis:

Assumption 2.3. The input is known and uniformly bounded, i.e., u(t) ≤ u M < ∞ for all t ≥ t 0 .

Assumption 2.4. The output of the system is a Lipschitz function of time, that is, there exist a constant γ > 0 such that

y(t 1 ) -y(t 2 ) ≤ γ t 2 -t 1 ∀ t 1 , t 2 ≥ t 0 -τ . Remark 2.
For example, the property in Assumption 2.4 is obtained if the matrix A(t) describes an uniformly asymptotically stable motion. In such case, and because the state remains bounded, the difference y(t) -ȳ(t) can be bounded independently of τ . Also, in the case of a LTI system with one pole at zero and the rest in the open left-half complex plane, where the system is marginally stable, Assumption 2.4 is satisfied.

Remark 3. If a system satisfy the Assumption 2.4, then y(t) -ȳ(t) ≤ γ τ (t) ≤ γ τ . Assumption 2.3 and 2.4 are required in order to keep the error y(t)-ȳ(t) bounded. On the other hand, if A(t) describes an uniformly asymptotically stable motion, one can use a copy of the plant as an observer, without using any kind of correction term. In such approach, there is no control over the convergence velocity, and one has to relay on the intrinsic properties inscribed in A(t), whereas an objective of this work is to increase the rate of convergence. Nevertheless, the price to pay is a bounded error. Finally, we want to remark that, in the case of delay-independent stability, uniform asymptotic stability is necessary for both, LTI [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF] and LTV [START_REF] Zhou | Razumikhin and Krasovskii stability theorems for timevarying time-delay systems[END_REF] systems. In the case of delay-dependent stability, and for LTI systems, asymptotic stability might reduce the difficulty in finding L [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF].

Preliminaries: Riccati differential equations

In this section some properties about a Riccati differential equation (RDE), related to observation, will be discussed together with some properties related to the uniform asymptotic stability of LTV systems. The solution of the aforementioned RDE, which can be computed on-line, will be used to propose a correction term for the observer.

Consider the following RDE:

Ṅ (t) = -A (t)N (t) -N (t)A(t) -N (t)Θ(t)N (t) + C (t)C(t), N (t 0 ) = N 0 = N 0 ≥ 0, (8)
with a piecewise continuous matrix function Θ(t) :

R ≥0 → R n×n satisfying θ 1 I n ≥ Θ(t) ≥ θ 2 I n for some positive constants θ 1 ≥ θ 2 > 0.
This RDE is commonly associated to the Kalman-Bucy filter. It has been proved that under Assumption 2.1 and the bounds imposed to Θ(t), N (t) is uniformly bounded and invertible [START_REF] Anderson | Stability properties of Kalman-Bucy filters[END_REF][START_REF] Bucy | The Riccati equation and its bounds[END_REF]. In particular, it is shown in [START_REF] Anderson | Stability properties of Kalman-Bucy filters[END_REF]) that N (t) and Φ (t 0 , t)N 0 Φ(t 0 , t) + W(t, t 0 ) share the same null space. Furthermore, in [START_REF] Bucy | The Riccati equation and its bounds[END_REF] the following bounds are provided:

N (t) ≤ C -1 (t, t -T ) + λ W(t, t -T ), N (t) ≥ W -1 (t, t -T ) + λ C(t, t -T ) -1 , (9)
for all t ≥ t 0 + T , where C is defined as

C(t 2 , t 1 ) := t2 t1 Φ(t 2 , s)Θ(s)Φ (t 2 , s)ds.
Given θ 1 and θ 2 , there always exist

β 1 ≥ β 2 > 0 such that β 1 I n ≥ C(t, t -T ) ≥ β 2 I n . Taking α 1 ≥ α 2 > 0 as in Assumption 2.1, λ can be chosen as λ = n 2 (α 1 β 1 )/(α 2 β 2 ).
In particular, if N 0 is taken definite positive, H(t) := N -1 (t) exists for all t ≥ t 0 and satisfies:

Ḣ(t) = H(t)A (t) + A(t)H(t) -H(t)C (t)C(t)H(t) + Θ(t), H(t 0 ) = N -1 0 .
This dynamics follows from deriving the relation H(t)N (t) = I n , which results in Ḣ(t) = -H(t) Ṅ (t)H(t). These two matrices, N (t) and H(t), will be of interest along the note.

Remark 4. Although bounds (9) are of particular importance to establish stability results, the evaluations that they provide are, in general, very conservative. Now, consider the auxiliary function ψ(t) = N (t)x(t), which being calculated due to the invertibility of N (t) provides an estimate for x(t) immediately. To compute ψ(t), let us write its time derivative. When there is no delay, this results in

ψ(t) = Ṅ (t)x(t) + N (t) ẋ(t) = -A (t) + N (t)Θ(t) ψ(t) + N (t)B(t)u(t) + C (t)ȳ(t). ( 10
)
To preserve the proposed relationship, ψ(t 0 ) should be taken as N 0 x(t 0 ), which require the initial condition of the system. Since ȳ(t) and x(t 0 ) are not available, we propose to compute an estimate ψ(t) of the auxiliary function ψ(t) as

ψ(t) = -A (t) + N (t)Θ(t) ψ(t) + N (t)B(t)u(t) + C (t)y(t), ψ(t 0 ) = ψ 0 , ( 11
)
where the available output is used, and the initial condition is left free. Now, consider the error ∆ ψ (t) = ψ(t) -ψ(t) whose dynamics yields

∆ψ (t) = -A (t) + N (t)Θ(t) ∆ ψ (t) + C (t) ȳ(t) -y(t) . (12)
Then, if the matrix -A (t) + N (t)Θ(t) is uniformly asymptotically stable, ∆ ψ (t)

will remain bounded, meaning that ψ(t) can be used instead of ψ(t) despite the lack of correct information.

Lemma 3.1. Let η 1 I n ≥ N (t) ≥ η 2 I n . Then lim t→∞ ∆ ψ (t) ≤ η 1 η 2 θ 2 sup t≥t0 ȳ(t) -y(t) . (13)
Proof. Consider as a Lyapunov function candidate

V (∆ ψ , t) = ∆ ψ H(t)∆ ψ , which satisfies 1 η2 ∆ ψ 2 ≥ V (∆ ψ , t) ≥ 1 η1 ∆ ψ 2 . Its derivative along (12) results in V (t) = -∆ ψ (t) H(t)C (t)C(t)H(t) + Θ(t) ∆ ψ (t) + 2 ∆ ψ (t)H(t)C (t) ȳ(t) -y(t) ≤ -θ 2 ∆ ψ (t) 2 -C(t)H(t)∆ ψ (t) 2 + 2 C(t)H(t)∆ ψ(t) ȳ(t) -y(t) ≤ -θ 2 ∆ ψ (t) 2 + ȳ(t) -y(t) 2 ≤ -η 2 θ 2 V (t) + ȳ(t) -y(t) 2 .
The last inequality implies that

V (t) ≤ V (t 0 ) exp -η 2 θ 2 (t -t 0 ) + 1 η 2 θ 2 sup t≥t0 ȳ(t) -y(t) 2 ∆ ψ (t) ≤ η 1 η 2 ∆ ψ (t 0 ) exp - 1 2 η 2 θ 2 (t -t 0 ) + η 1 η 2 θ 2 sup t≥t0 ȳ(t) -y(t) .
Taking the limit when t → ∞, the bound of the lemma follows.

In the next section, the term N (t)x(t)-ψ(t) will be used to enhance the convergence rate.

Main Result

The undelayed observer (5) will be taken as a starting point for the nonlinear observer proposed in this section. The results developed in Section 3 will be used to strengthen its convergence rate. Denote by x(t) the estimate of x(t), and define the estimation error as e(t) = x(t) -x(t). Recalling that ψ(t) = N (t)x(t) -∆ ψ (t), we remark that N (t)x(t) -ψ(t) = N (t)e(t) + ∆ ψ (t). Then, the term N (t)x(t) -ψ(t) carries an information about the estimation error and the matrix N (t) is positive definite. Following this observation, we propose the following system as an observer for (2):

ẋ(t) = A(t)x(t) + B(t)u(t) -H(t)C (t) C(t)x(t) -y(t) -Λ N (t)x(t) -ψ(t) p , (14)
where the auxiliary signals N (t), ψ(t), and H(t) are computed following

Ṅ (t) = -A (t)N (t) -N (t)A(t) -N (t)Θ(t)N (t) + C (t)C(t), (15)
ψ(t) = -A (t) + N (t)Θ(t) ψ(t) + N (t)B(t)u(t) + C (t)y(t), (16)
Ḣ(t) = H(t)A (t) + A(t)H(t) -H(t)C (t)C(t)H(t) + Θ(t), (17)
and p > 1 is an exponent to be chosen, whereas the matrix Λ = diag{λ 1 , λ 2 , • • • , λ n } with λ i > 0 contains tuning parameters. In ( 15)-( 17), Θ(t) has to satisfy the bounds θ 1 I n ≥ Θ(t) ≥ θ 2 I n for some positive θ 1 and θ 2 . Finally, the initial condition of N , N 0 , has to be selected positive definite, and H(t 0 ) = N -1 0 . The initial conditions for x and ψ are free.

Denote by ỹ(t) = ȳ(t) -y(t). The error dynamics produced by implementing ( 14) results in

ė(t) = A(t) -H(t)C (t)C(t) e(t) + H(t)C (t)ỹ(t) -Λ N (t)e(t) + ∆ ψ (t) p . ( 18
)
Theorem 4.1. Let Assumptions 2.1, 2.3, and 2.4 be satisfied, p > 1, θ 1 I n ≥ Θ(t) ≥ θ 2 I n , Λ > 0, and N 0 > 0. Then there exist η 1 and η 2 satisfying

η 1 I n ≥ N (t) ≥ η 2 I n .
Furthermore, the estimation error converge uniformly in fixed-time to the region given by

e(t) ≤ ρ η 2 ∀ t ≥ t 0 + T , ( 19
)
where ρ is the unique positive root of

P(v) = θ 2 η 2 2 η 1 v + 2 λ m n 1-p 2 κ 1 η p+1 2 η p+1 2 1 v p+1 2 - γ 2 τ 2 q - 2 λ m κ 2 q sup t≥t0 ∆ ψ (t) p+1 , (20)
with q ∈ (0, 1), λ m = min 1≤i≤n λ i , κ 1 and κ 2 as in Corollary A.2, and where the convergence time is at most T and is given by:

T = 2 η1 η2 θ 2 η 2 (p -1)(1 -q) ln   1 + θ 2 2 λ m n 1-p 2 κ 1 η p-1 2 2 η 1 η 2 p-1 2   .
The uniform fixed-time convergence follows from the fact that T bounds the convergence time for any initial error and initial time and T does not depend on them. The structure of T reveals that the convergence time can be reduced by increasing p, making the condition number of N (t), (η 1 /η 2 ), smaller and by increasing λ m . Although θ 2 seems to be a free parameters, changing it affects both, η 1 and η 2 . The proof of Theorem 4.1 is given in Section 5.

Remark 5. If N 0 ≥ 0 or H(t 0 ) = N -1 0 for some reason, there appears an error that vanishes exponentially fast [START_REF] Kalman | Contributions to the theory of optimal control[END_REF][Sec. 7], leaving the qualitative result of Theorem 4.1 unaltered.

Corollary 4.2. Make the same assumptions as in Theorem 4.1, but with p = 3. Then

e(t) ≤ 1 η 2 n q 1/4 η 1 η 2 1.786 √ γ τ λ 1/4 m + 1.211 sup t≥t0 ∆ ψ (t) ∀ t ≥ t 0 + T , ( 21
)
where q ∈ (0, 1), λ m = min 1≤i≤n λ i , and

T = 1 θ 2 η 2 (1 -q) • η 1 η 2 ln 1 + 10.2 n θ 2 λ m η 2 • η 1 η 2 .
Proof. With p = 3 and using the values of κ 1 and κ 2 in Table A2, the polynomial (20) becomes

P(v) = θ 2 η 2 2 η 1 v + 0.0982 λ m η 4 2 n η 2 1 v 2 - γ 2 τ 2 q -0.2114 λ m q sup t≥t0 ∆ ψ (t) 4 .
A root of this polynomial is

ρ = 5.092 λ m η 2 η 1 η 2 -n θ 2 + n q n q θ 2 2 + 0.3928 λ m γ 2 τ 2 + 0.08304 λ 2 m sup t≥t0 ∆ ψ (t) 4 .
Since the 1-norm is greater that the Euclidean norm (2-norm), we have

ρ ≤ n q • η 1 η 2 2 3.19 γ τ λ 1/2 m + 1.467 sup t≥t0 ∆ ψ (t) 2 .
Then, the norm of the error satisfies:

e(t) ≤ 1 η2 n q 1/4 η1 η2 3.19 γ τ λ 1/2 m + 1.467 sup t≥t0 ∆ ψ (t) 2 ≤ 1 η2 n q 1/4 η1 η2 1.786 √ γ τ λ 1/4 m + 1.211 sup t≥t0 ∆ ψ (t) ∀ t ≥ t 0 + T .
Replacing the values of p and κ 1 in the definition of T we obtain the given estimate.

In Corollary 4.2 we find the root of (20) for a particular choice of p. This allows us to see how the size of the attraction region behaves in this case, and it can give us an intuition on how it probably behaves for other values of p. As can be seen in ( 21), the root can be decreased if η 2 increases, or if the condition number of N (t) decreases. However, these terms cannot be directly adjusted. On the other hand, there are two terms in (20), one depending on τ and other depending on ∆ ψ (t), that also affects the size of the attraction region. The term depending on τ can be decreased by increasing λ m . The other one can be decreased by increasing θ 2 as shown in ( 13). Then, the size of the final error can be modified by varying these two parameters, λ m and θ 2 .

Convergence analysis and proof of claims

Consider as a Lyapunov function candidate V (e, t) = e N (t)e, which is a valid candidate given the existence of η 1 and η 2 . Its derivative along (18) yields Using the bounds for N (t), Θ(t) and the result of Corollary A.2, the time derivative of V can be estimated as

V (t) = e (t) -N (t)Θ(t)N (t) -C (t)C(t) e(t) + 2 e (t)C (t)ỹ(t) -2 e (t
V (t) ≤ -θ 2 η 2 2 e(t) 2 -2 λ m n 1-p 2 κ 1 N (t)e(t) p+1 + 2 λ m κ 2 ∆ ψ (t) p+1 + γ 2 τ 2 ≤ -θ 2 η 2 2 e(t) 2 -2 λ m n 1-p 2 κ 1 η p+1 2 e(t) p+1 + 2 λ m κ 2 ∆ ψ (t) p+1 + γ 2 τ 2 .
κ 1 and κ 2 can be chosen following tables A1 and A2. Since V (t) ≤ η 1 e(t) 2 , the previous inequality can be transformed into a differential inequality in terms of V (t):

V (t) ≤ -θ 2 η 2 η 2 η 1 V (t) -2 λ m n 1-p 2 κ 1 η p+1 2 2 η 2 η 1 p+1 2 V p+1 2 (t) + 2 λ m κ 2 ∆ ψ (t) p+1 + γ 2 τ 2 ≤ -k 1 V (t) -k 2 V p+1 2 (t) + ∆, with ∆ := γ 2 τ 2 + 2 λ m κ 2 sup t≥t0 ∆ ψ (t) p+1 , k 1 := θ 2 η 2 η 2 η 1 , k 2 := 2 λ m n 1-p 2 κ 1 η p+1 2 2 η 2 η 1 p+1 2
.

The constant ∆ is finite due to Lemma 3.1 and assumptions 2.3 and 2.4. Let q ∈ (0, 1) and denote by ρ the unique positive root

1 of k 1 v + k 2 v p+1 2
-1 q ∆. Then, we have that (Moreno, 2012, pp. 134), the change of variable w(t) = exp (1 -q)k 1 (t -t 0 ) z(t) transform the equation into:

V (t) ≤ -(1 -q) k 1 V (t) + k 2 V p+1 2 (t) < 0, V (t) ≥ ρ. Consider the differential equation ż(t) = -(1 -q) k 1 z(t) + k 2 z p+1 2 (t) , z(t 0 ) ≥ 0. As in
ẇ(t) = -(1 -q)k 2 exp - 1 2 (p -1)(1 -q)k 1 (t -t 0 ) w p+1 2 (t).
By separation of variable, the solution for w(t) and z(t) results in:

w(t) = 1 w p-1 2 (t 0 ) + k 2 k 1 1 -exp - 1 2 (p -1)(1 -q)k 1 (t -t 0 ) -2 p-1 , z(t) = 1 z p-1 2 (t 0 ) + k 2 k 1 exp 1 2 (p -1)(1 -q)k 1 (t -t 0 ) - k 2 k 1 -2 p-1
.

Then, by the Comparison Lemma (Khalil, 2002, Lem. 3.4) we have

V (t) ≤ 1 V p-1 2 (t 0 ) + k 2 k 1 e 1 2 (p-1)(1-q)k1(t-t0) - k 2 k 1 -2 p-1
.

Without loss of generality, assume that

V (t 0 ) > ρ. Then, V (t) ≤ ρ for t -t 0 ≥ 2 k 1 (p -1)(1 -q) ln V (t 0 ) ρ p-1 2 k 1 + k 2 ρ p-1 2 k 1 + k 2 V p-1 2 (t 0 ) ≥ 2 k 1 (p -1)(1 -q) ln    k 1 + k 2 ρ p-1 2 k 2 ρ p-1 2 + k 1 ρ V (t0) p-1 2    .
Taking the limit when

V (t 0 ) → ∞ it yields t -t 0 ≥ 2 k 1 (p -1)(1 -q) ln 1 + k 1 k 2 ρ p-1 2 = 2 η1 η2 θ 2 η 2 (p -1)(1 -q) Ln   1 + θ 2 2 λ m n 1-p 2 κ 1 η p-1 2 2 η 1 η 2 p-1 2   =: T .
This bound represents the amount of time which guarantees that the level set V (t) ≤ ρ is reached. This bound is finite and does not depend on the initial time nor on the initial value of V , then the level set is reached in finite time, uniformly in t 0 and in the initial condition. Finally, since V (t) ≥ η 2 e(t) 2 , e(t) ≤ ρ/η 2 for all t ≥ t 0 + T .

Numerical example

To exemplify the proposed observer, the following system is considered:

ẋ(t) =   0 1 0 0 A 22 (t) 0   x(t) +  0 1 1   u(t) = A(t)x(t) + b u(t), y(t) = 1 0 0 x(t -τ (t)) = C x(t -τ (t)) = x 1 (t -τ (t)), with A 22 (t) = -ω(t)ω (t), ω (t) = [cos(3 t), 1], u(t) = cos(5 t) + 1.
It is well known that if ω(t) is of persistent excitation, -ω(t)ω (t) describes a uniformly asymptotically stable motion [START_REF] Anderson | Exponential stability of linear equations arising in adaptive identification[END_REF]. Then, the dynamics of x 2 (t) and x 3 (t) is uniformly asymptotically stable. However, x 1 (t) integrates x 2 (t), resulting unbounded, but a Lipschitz time function, fulfilling Assumption 2.4. For the simulation, the initial conditions of the system are set in x 1 (t 0 ) = 0, x 2 (t 0 ) = 10, and x 3 (t 0 ) = 5. For this example, the delay is selected as τ (t) = α + 2 α cos(6 t)/3 with alpha taking the values {0.1, 0.3, 0.6, 1.2}. In the design and implementation of the observer, the value of τ (t) is not needed. The observer parameters were chosen as follows:

Θ(t) = diag 10, 6, 6 , Λ = 30 I 3 , p = 2, N 0 = I 3 .
The initial conditions for the observer were set to zero. For these conditions, the norm of the estimation error for the different values of α is shown in Figure 1. It can be seen that the error bound grows with the size of α. This can be attributed to the fact that the difference |y(t) -ȳ(t)| also grows with α, as is shown in Figure 2. Now, to show the fixed-time convergence, the initial condition of the observer was increased to induce initial errors of 10 3 , 10 5 and 10 7 , while keeping α = 1.2. The evolution of the estimation norm, under these circumstances, is shown in Figures 4 and5 in a logarithmic scale. In Figure 4 the fast attraction it is shown. In this figure, a logarithmic scale is also used for the time axis since the convergence was really fast. In Figure 5 it is shown that the three trajectories reach the same region in almost the same time despite the difference between the orders of magnitude in the initial condition. In the previous scenario, the size of |y(t) -ȳ(t)| increases with α given the marginal stability of the system. To shown how the observer behaves when the systems is uniformly asymptotically stable, we will consider only the dynamic of x 2 (t) and x 3 (t). The output will be y(t) = x 2 (t -τ (t)), leaving τ (t) as before. For this setting, the observer was configured as follows:

N 0 = I 2 , Θ(t) = 10 I 2 , Λ = 30 I 2 , p = 2.
The initial conditions for the observer state and ψ(t) were set in zero. The difference between y(t) and ȳ(t) in this case is shown in Figure 7. Since the nominal output is bounded, the difference between it and the delayed one does not grow with α. This is reflected in the fact that the size of the convergence region is maintained despite the increase in α, as it is demonstrated in Figure 6. This contrasts with the previous situation where the error grew with the delay.

As before, to show the fixed-time convergence, the initial error was increased in orders of magnitude of 10 3 , 10 5 , and 10 7 while keeping α = 1.2. In Figure 8 it is shown how the three trajectories converge to the same. Logarithmic scale were used in both axes to show the transient phase. In Figure 9 it can be observed how in the three cases, the same error is achieved.

With these two examples, we have illustrated the two main properties of ( 14). First, the delay size and its rate of variation does not affect directly the ultimate bound for the estimation error. For a system where the output asymptotically remains in a compact region, the error bounds will depend on the size of that region. Second, when the estimation error is large, its rate of convergence is accelerated due to the nonlinearity. These properties make the algorithm suitable when the delay and the initial error are uncertain and large.

In the case of a linear time-invariant system, a comparaison of the proposed observer with a delayed Luenberger observer (3) and the undelayed version (5) can be found in [START_REF] Rueda-Escobedo | A gramianbased observer with uniform convergence rate for delayed measurements[END_REF].

Conclusion

In the note, an observer for marginally stable time-varying systems with delayed measurements is presented. The observer provides an estimate that converges to the internal state of the system up to a bounded error. The convergence time can be bounded by a constant independent from the initial error and the initial time, meaning that the ultimate bound of the error is reached in uniform fixed-time. The proposed observer proves to be useful when the knowledge about the delay is scarce, or when it is large with respect to the time-scale of the system. Also, if the convergence time of the observer is crucial for the application, the proposed approach is helpful due to the fixed-time convergence since it guarantees the time needed to trust the estimate. This time can be adjusted by means of the observer parameters. consider the case with x = 0 and δ = 0. Now, by homogeneity we have:

x x + δ p = 1 p+1 ( x) ( x) + ( δ) p κ 1 |x| p+1 -κ 2 |δ| p+1 = 1 p+1 κ 1 | x| p+1 -κ 2 | δ| p+1 ,
for any > 0. Set = 1/|δ| and define z = x/|δ|. The inequality then is equivalent to z z + δ 0 p ≥ κ 1 |z| p+1 -κ 2 . We will only consider the case δ 0 = 1 since the other one is analogous. For δ 0 = 1 we have z z

+ 1 p ≥ κ 1 |z| p+1 -κ 2 , or z z + 1 p -κ 1 |z| p+1 ≥ -κ 2 . (A1)
This reduces the problem to prove that the left hand side of the inequality has a lower bound. For z > 0, we have that z + 1 0 = 1 and |z + 1| > |z|, then z z + 1 p > |z| p+1 . Since κ 1 < 1, z z + 1 p -κ 1 |z| p+1 > 0 and (A1) holds for any κ 2 ≥ 0 on this interval. Now, for z ∈ (-1, 0), we must show -|z| |z + 1| p -κ 1 |z| p+1 ≥ -κ 2 . In this interval, we have that |z| < 1 and |z + 1| < 1, then -|z| |z + 1| p -κ 1 |z| p+1 ≥ -1 -κ 1 , which implies that (A1) holds with κ 2 ≥ 1 + κ 1 . Last, we consider the interval z ∈ (-∞, -1], where now we must check |z| |z + 1| p -κ 1 |z| p+1 ≥ -κ 2 . Notice that |z| > |z + 1| in this case. To find a lower bound, consider the following auxiliary function:

|z| |z + 1| p -κ 1 |z| p+1 ≥ |z + 1| p+1 -κ 1 |z| p+1 := g(z)
.

Now, we proceed to look for the minimum of g(z). Taking its derivative, this results in g (z) = -(p + 1) κ 1 |z| p -|z + 1| p , which has a unique zero at z 0 = -1/(1 -κ 1/p 1 ). The second derivative of g(z) evaluated at z 0 is positive, reveling that g(z) has a minimum at this point. Then g(z 0 ) can be taken as κ 2 . This gives us κ 2 ≥ κ 1 /(1-κ 1/p 1 ) p . Finally, looking at the three conditions we get that

κ 2 ≥ max κ 1 + 1, κ 1 1 -κ 1/p 1 p .
Remark 6. In the proof of Lemma A.1, a value for κ 2 is given analytically as a function of κ 1 and p. However, the ratio κ 2 /κ 1 can be really large for κ 1 close to one and p >> 1. A sharp value for κ 2 can be found numerically by looking at the minimum of z z + 1 p -κ 1 |z| p+1 on z ∈ (-∞, 0). This is discussed latter in this appendix.

Corollary A.2. Let x, δ ∈ R n and p > 0. Then, for any κ 1 ∈ (0, 1) there is κ 2 > 0 such that

x x + δ p ≥ κ 1 x p+1 p+1 -κ 2 δ p+1 p+1 .
In particular, for p > 1 we have

x x + δ p ≥ κ 1 n 1-p 2
x p+1 -κ 2 δ p+1 .

In both cases, κ 2 can be taken as in Lemma A.1.

Proof. First, the product x x + δ p is developed and bounded using Lemma A1: Each of the sum represent the (p + 1)-norm raised to p + 1. From here, the first statement of Corollary A.2 follows. Now, p > 1 =⇒ p + 1 > 2. Then, using the equivalence between norms in R n , we have

x x + δ p = n i=1 x i x i + δ i p ≥ κ 1 n i=1 |x i | p+1 -κ 2 n i=1 |δ i | p+1 .
κ 1 x p+1 p+1 -κ 2 δ p+1 p+1 ≥ κ 1 n 1-p 2
x p+1 -κ 2 δ p+1 .

This concludes the proof.

A.1. Discussion

To obtain a sharp value for κ 2 , the minimum of f (z) := z z + 1 p -κ 1 |z| p+1 on the interval (-∞, 0) is required. This can be done by looking at its derivative: Since |z + 1| p , |z| p , and |z| |z + 1| p-1 are monotonically increasing functions of z, there are two critical points, one in each sub-interval. The zeros of f (z) can be found numerically and then used to find the minimum for f (z). In Figure A1, we present the values of κ 2 found using this approach for some values of p.

f (z) = z + 1 p + p z z + 1 p-1 -(p + 1)κ 1 z p , f (
Beside the value of κ 1 and κ 2 , one may be interested in the best selection of these parameters for a given p. To chose some κ 1 and its associated κ 2 , we propose to take the ones that minimize the ratio κ 2 /κ 1 1 p+1 . From the data generated for Figure A1 we got the values presented in Table A1 andA2.

Figure 1 .Figure 2 .

 12 Figure 1. Evolution of the estimation error norm for the different values of α.

Figure 3 .

 3 Figure 3. Tracking of the nominal output ȳ(t) for the different values of α.

Figure 4 .

 4 Figure 4. Time evolution of the error norm for different initial errors. In the plot, the fast attraction to the ultimate bound is shown in a logarithmic scale in both axes.

Figure 5 .Figure 6 .

 56 Figure5. Three trajectories reach the same bounded region despite the differences in the initial condition.

Figure 7 .Figure 8 .

 78 Figure7. Plot of the difference |y(t) -ȳ(t)| when the system is UAS. Increasing the value of α does not affect significantly this difference.

Figure 9 .

 9 Figure9. Three trajectories reach the same bounded region despite the differences in the initial condition when the system is UAS.

Figure A1 .

 A1 Figure A1. Values of κ 2 for different values of κ 1 and p.

  z) = |z + 1| p + (p + 1)κ 1 |z| p -p |z| |z + 1| p-1 , z ∈ (-1, 0), f (z) = (p + 1)κ 1 |z| p -|z + 1| p -p |z| |z + 1| p-1 , z ∈ (-∞, -1].

Table A1 .

 A1 Suggested values of κ 1 and κ 2 for some p.

	p		1.2	1.4	1.6	1.8	2	2.2	2.4	2.6	2.8
	κ 1		0.297 0.219	0.173	0.1407 0.1162	0.097	0.08134 0.0685 0.0579
	κ 2		0.300 0.232 0.1946 0.1703 0.1529 0.1396	0.1288	0.1199 0.1123
		1
	(κ 2 /κ 1 )	p+1	1.005 1.024	1.046	1.071	1.096	1.121	1.145	1.168	1.190

Table A2 .

 A2 Suggested values of κ 1 and κ 2 for some p, continuation.

	p		3	3.4	3.8	4	4.4	4.8	5
	κ 1		0.049	0.0353 0.0256 0.0218 0.0159 0.0117 0.00997
	κ 2		0.1057 0.0946 0.0858 0.0819 0.0757 0.0697	0.0671
		1
	(κ 2 /κ 1 )	p+1	1.212	1.251	1.286	1.303	1.333	1.361	1.374

For v ≥ 0, k 1 v + k

v p+1 2is a strict positive monotonic function of v, then its image is R ≥0 .

Acknowledgment

This work is supported in part by HoTSMoCE Inria associate team program; by the Government of Russian Federation (Grant 074-U01); by the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031); by the financial support from PAPIIT-UNAM, project IN113614; Fondo de colaboración II-FI UNAM, Project IISGBAS-100-2015; CONACyT, project 241171; and CONACyT CVU 491701.

Appendix A. Some inequalities Lemma A.1. Let x, δ ∈ R and p > 0. Then, for any κ 1 ∈ (0, 1) there exists κ 2 > 0 such that

In particular, one can select

Proof. For x = 0 or δ = 0, the inequality is satisfied trivially with any κ 2 ≥ 0, then


Steven M Drucker

Samuel Huron

Robert Kosara

Jonathan Schwabish

Nicholas Diakopoulos

Communicating Data to an Audience

come

Introduction

Communicating data in an effective and efficient story requires the content author to recognize the needs, goals, and knowledge of the intended audience. Do we, the authors, need to explain how a particular chart works? It depends on the audience. Does the data need to be traced back to its source? Depends on the audience. Can we skip obvious patterns and correlations and dive right into the deeper points? Depends on the audience. Do we need to explain what the findings in the data mean in terms of what the data represents? Depends on the audience. There are many more questions for which this is true.

It appears reasonable, then, to know who that audience is and what they know. In addition, we might also want to know what their expectations and needs are: What does the audience want to get out of the story? A single story cannot possibly address all possible different audiences and their needs. And any well-designed story will be tailored not just to its data and the intended message, but to its audience.

That is the theory, at least. In practice, this is quite difficult to achieve. A large audience will consist of people with different backgrounds and knowledge levels that are impossible to target at the same time; in breaking news and media production, short deadlines often make it impractical to create multiple versions of a news graphic that will work on different devices; in other fields, personnel or financial considerations may constrain the ability to create a product that targets the correct audience. Knowledge about the audience is also often quite limited. It tends to be general and broad, and usually not specific enough to target visualizations to individuals.

Despite these limitations, considering the audience, even broadly, can help guide the design of visual stories. The results are more appealing, effective, and meaningful to the intended recipients.

In some cases, content authors know precisely who a presentation or report will be given to: what they know about the data, the background, etc. In other cases, the story author may at least have an expectation or an imagination of who they think the audience may be [START_REF] Litt | Knock, Knock. Who's There? The Imagined Audience[END_REF], such as the general level of data and visualization literacy of a particular publication's target audience. Such audience expectations inform the editorial decisions that need to be made about visualizations and news graphics. For instance, if the intended audience for a story is teenagers with limited statistical or visualization literacy, then additional explanation or context might be needed for less common chart types so that they can be accurately interpreted.

Different audiences not only have different levels of familiarity and literacy with visualization, they also have different goals, expectations, attention spans, and cognitive abilities. A general reader of a newspaper has very different needs and goals than an academic researcher reading a scholarly journal or a policymaker reading a briefing memo. A television audience will expect a different kind of presentation than a newsprint reader or a news website user, who in turn will have different expectations than those in a meeting with colleagues.

A television audience's attention span might be on the order of a few seconds. For web audiences, some data show that 55 percent of users spend less than 15 seconds on an item of news content [START_REF] Haile | Visualization Rhetoric: Framing Effects in Narrative Visualization[END_REF]. By contrast, communicating data and stories to people who need the information to make decisions or change policies -for example, executive managers or policymakers -attention may be substantially longer because the information being communicated is essential to their performance. Other audiences may want to accomplish different things by reading a data story, such as being entertained, educated, or satisfying their general curiosity.

To effectively communicate ideas and concepts, content authors need to think carefully about how their work best fits the needs of the audience. In this chapter, we explore design considerations relating to audience knowledge and goal contexts, and consider the difference between the theory of what we might know and the reality of what we can know. We discuss some approaches that allow us to tailor a piece to the audience with little or no knowledge about them. Finally, we describe a few specific design contexts and their particular requirements, such as news graphics and broadcast television.

What Does the Audience Know?

Identifying the background knowledge of an audience for any given project is a challenging endeavor. In this section we identify some of the audience knowledge characteristics that are important when communicating information driven by data and told through text, images, and data visualizations.

Through discussion of our experiences coming from different backgrounds and past work experiences, we have identified three general areas where content producers could know more about their audience. Such knowledge would enable more effective communication of data-driven content.

1. How literate is the audience in terms of data and visualization? And what can be done to increase it? The answer in news graphics tends to be an annotation layer . 2. How knowledgeable is the audience in terms of jargon, domain expertise, and other background knowledge? 3. What are audience expectations about the design of visualization, such as style, tone, or use of iconography?

Data and Visualization Literacy: The Annotation Layer Definitions of visualization literacy describe the ability to interpret visual patterns of the underlying data, and to confidently use a visualization to answer questions about the data [START_REF] Boy | A Principled Way of Assessing Visualization Literacy[END_REF]. As the value and availability of data continues to grow, so does the demand for understanding data and graphs --of increasing visualization literacy. Data science and data visualization courses and training in the private sector, in post-secondary schools, and in online training programs have increased substantially over the past few years [START_REF] Womack | Data Visualization and Information Literacy[END_REF]. The onus isn't just on the audience though, data storytellers have an ethical obligation to ensure that their visualizations do not skew or mislead interpretations unnecessarily (see Chapter 10: "Ethics in Data-Driven Visual Storytelling"), and this intersects with the literacy level of the expected audience. Increasing an audience's understanding of different data and graph types may come from a variety of sources, some within the content-producers' control, others not.

In news graphics, the solution is what is usually called the annotation layer It adds explanations and descriptions to introduce the graph's context, which is important for almost any audience. It also explains how to read the graph, which helps readers unfamiliar with the graph -whether a simple line chart or an advanced technique like a treemap or scatterplot. When done right, it will not get in the way for experienced ones.

Annotations and explanations have a long history. Figure 9.1 shows a line chart in The New York Daily Tribune in 1849 in which the addition of an explanatory caption and annotations help guide the interpretation of how to read the then still unfamiliar line chart. Amanda Cox, the editor of the Upshot at the New York Times, is famously quoted as saying, "The annotation layer is the most important thing we do...otherwise, it's a case of here it is, you go figure it out" [START_REF] Cox | Shaping Data for News[END_REF].

Annotation goes beyond just labeling points or lines on a chart: the bubble plot shown in Figure 9.2 from the Los Angeles Times, for example, expertly combines annotation that explains how to read the chart with what the content means. The chart shows the relationship between the change in violent crime rate (horizontal axis) and the property crime rate (vertical axis) in about 30 cities in California. For readers familiar with this chart type, it is immediately clear how to glean conclusions from the data. The average LA Times reader may not be familiar with this chart type, so upon first viewing the graphic, there is a big, red box in the top-right with big, red text that says "Worse" and a big, blue box in the bottom-left with big, blue text that says "Better". Then, each quadrant has a small headline in boldface text with a sentence below to deliver the content. Thus, even for the reader who has never seen a bubble plot before, the annotation instructs them on how to read the chart, and then delivers the content through the additional explanatory text. The audience for a small local publication is likely to have knowledge of the local area, whereas the same couldn't be assumed for the audience of a large national or international publication. Data storytelling needs to take into account the baseline of information and knowledge that the intended audience is expected or assumed to have.

Different levels of expertise and knowledge should encourage content-producers to think carefully about how their products will best meet the needs of a diverse audience. Niche audiences may be more comfortable with the jargon of their group, but designers may consider whether removing jargon would serve a larger audience just as easily while making the content more broadly accessible. Universal design is an aspiration that can allow data stories to appeal to and be used by a broad array of people [START_REF] Shneiderman | Designing the User Interface: Strategies for Effective Human-Computer Interaction[END_REF], but design is also about tradeoffs that may make a data story less appealing to some, while simultaneously much more appealing and useful to others. In such cases different versions of the story might be produced for different audiences, each targeting a unique range of knowledge, expertise, and other factors discussed in this chapter.

An individual in the audience brings their own viewpoints, backgrounds, and experience to each and every data-driven story they consume. Whether driven by their cultural background, social position, education, or other demographic characteristics, readers carry with them their own unique set of knowledge and biases. While it is impossible for content-producers to be aware of each individual's particular experience--to know in advance what they already know--it may be possible to group individuals (i.e. to segment the audience) in ways that are useful to guiding design.

Some examples may be illustrative here. The first comes from the Congressional Budget Office (CBO), which is the budget arm of the U.S. Congress. CBO reports and numbers are regularly used and referenced by Congress, as is often dictated by law. Their audience of Members of Congress and their staffs is well-defined and well-known. That audience wants (and needs) headline numbers, facts, and statistics that they can use to communicate with their colleagues and their constituents. In June 2012, CBO published its Long-Term Budget Outlook , a 109-page report about the budget outlook for the federal government (CBO 2012). Paired with that report was a one-page infographic that highlighted the top-level items, facts, and patterns (see Figure 9.3). In a congressional hearing about the long-term budget outlook, Ranking Member Congressman Chris van Hollen (D-MD) held up that infographic as an exhibit to support his point. From the perspective of CBO, this is arguably recognizing the needs of their audience and how a member of Congress could effectively use their analysis. Another example comes from the Urban Institute, a nonprofit research organization in Washington, DC where one of the authors currently works. The Urban Institute is interested in communicating with policymakers, but is also interested in communicating their work to a broader audience of local and state policymakers, other researchers, practitioners, the press, and the public more broadly (see, also, Chapter 8: "Organizing the Work of Data-Driven Visual Storytelling"). Thus, in some ways, Urban is similar to the news organizations mentioned above, but communicating to an audience that is particularly interested in the kind of in-depth social and public policy research they conduct.

In August 2013, with the release of a report on the potential policy responses to long-term unemployment in the U.S., Urban published a "long form narrative" story on its website that combined the highlights from the report with interactive data visualizations, photographs, and sound clips from interviews with people who were unemployed [START_REF] Acs | Responding to Long-Term Unemployment[END_REF]. Recognizing that the traditional reports may not always have impact or reach a broad(er) audience, this type of data-driven story--which is now common for news organizations--is (still) fairly uncommon in the social science and non-profit research space.

Design Expectations

Basic data visualization best practices and strategies are needed across multiple domains to help content-creators find better ways to communicate their work to their audiences. Many people who have experience working with data and conducting in-depth analysis may not have the knowledge or experience in how to communicate their work. The field of data visualization--a broad field that can be considered to encompass data journalism, data science, data visualization scientific research, as well as general practitioners--continues to evolve in the face of new methods, tools, and data. A growing number of books, blogs, and articles have been published just over the past few years dedicated to helping people improve their visualizations.

A consistent design approach (one might call it "visual branding") can help an audience relate to a producer's work and entice them to return. Consistency across color, font, layout, and type of interactivity, for example, can help the audience better identify what the producer is going to deliver, when, and for what level of expertise. Conventions used in presentation can also reinforce good data hygiene and inform the audience as to the quality of a visualization such as when source and designer credits or methodologies are consistently incorporated into the story delivery (Hullman and Diakopoulos, 2011; Schwabish, 2016a).

Graphics from outlets such FiveThirtyEight , Vox , New York Times , The Economist , and others each have a specific design aesthetic. The inclusion of their brands' colors and fonts, a specific text or annotation style, and method of publication (e.g., daily, blogs, etc.) all help audiences recognize and relate to the producer's content. Practically, a design aesthetic is enculturated by having style guides that designers working for a given publication use to help choose colors, fonts, and layout. Non-profit organizations such as the Urban Institute (See Figure 9.4) as well 1 as news outlets such as the Dallas Morning News have even published their style guides 2 publicly. These guides help enforce a consistency of visual output (see Schwabish 2016b for a collection of other style guides). Another dimension that informs design expectations are cultural differences and viewing codes.

For instance the layout of design elements such as the title and credits in Western cultures may not meet expectations in cultures that read from right to left. Such differences may be particularly relevant for complex designs that can evoke a layer of meaning via connotation. For instance, colors can suggest very different meanings in the east versus west: in Asia white is often the color for mourning, whereas it is black in most of the west [START_REF] Ware | Visual Thinking for Design[END_REF]. This complicates the design process for data stories that are oriented towards a global audience such as may be the case for international publications like The Economist .

What Does the Audience Want?

Like any other designed experience, data stories are more impactful and useful when they are crafted with the motivations, goals, and tasks of users in mind --in this case the audience for the story. The following subsections explore several typical wants and needs from media consumption that may impact design, as well how interactivity can be leveraged to increase the adaptability of data stories to different user needs and contexts.

Media Wants and Needs

The uses and gratifications framework describes a set of dimensions that can influence why people consume media [START_REF] Ruggiero | Uses and Grats theory in the 21st century[END_REF]. The idea is that people receive certain gratifications from the media they consume that help satisfy various underlying social or psychological needs.

For instance people may seek out data-driven stories for a variety of reasons such as to inform themselves on an issue of personal (e.g., health) or civic importance (e.g., elections), or to casually lean back and be entertained or pass time in a non goal-oriented fashion. Uses and gratifications theory suggests that there are four primary underlying motivations for media consumption:

(1) staying informed about current information, satisfying curiosity and self-education;

(2) gaining insight into oneself or seeing models for others' behavior;

(3) socially interacting and connecting with others, or as a basis for social conversation, or developing empathy; and (4) as relaxing entertainment or diversion. A data-driven story can be designed differently depending on whether the audience is expected to approach the experience from a more social, entertainment, or strictly informational motivation. An entertaining treatment might take more license with visual novelty and not focus as closely on efficiency in decoding visual information, for example. A social treatment might emphasize user comments and responses to a data story, whereas an information-oriented design may focus on well-researched and validated background context provided in an annotation layer. The goal as a designer is to explore the design space in order to try to match the data story to the expected media need of the audience. An individual's needs from a piece of media like a data story can also vary with the intended interface. For instance, if a story is likely to be consumed on a mobile device, the user could be in a loud or distracting environment that might undermine the use of audio in the narrative (written annotations as backup is a nice design strategy). The specific location of a user on a GPS-enabled device can be interesting context that can be used to inform the way a story is presented [START_REF] Kim | Generating Personalized Spatial Analogies for Distances and Areas[END_REF]. In contexts that are more casual than work-oriented, users may have a higher degree of interruptibility, which could mean that a data story is consumed in short bursts of attention. Presenting information in "chunked" patterns might be more effective than as a single, long-form narrative in such cases. Finally, it's worth considering that boredom may be a driving factor for interaction in contexts where the user is consuming a data story while waiting (e.g., for a bus or train), or splitting attention between devices and screens (e.g., as many people do when watching television).

Tailoring to the Audience Without Knowing It

Even if we don't know anything about the audience, we can make it appear as if we did. The audience knows itself: they can pick their city in a story about home prices, their age and gender in a story on health, and their education and income levels in a story about taxes or inequality. This self-identification in a visualization may be why maps are such a popular tool in data visualization. While maps can often present data inaccurately, or even mislead a reader, our familiarity with maps makes them a convenient and familiar visualization choice [START_REF] Wiseman | When Maps Lie[END_REF]. Alternatives--such as cartograms, which distort geographic areas to correspond to the data values--sometimes have the advantage of portraying the data in more accurate ways, but the distortion makes the visualization less familiar to the reader.

A variety of other projects and research have tested or demonstrated the importance of allowing the audience to "find themselves" or make content more familiar in a visualization. For example, research by [START_REF] Chevalier | Using Concrete Scales: A Practical Framework for Effective Visual Depiction of Complex Measures[END_REF] attempt to build a framework to help people think about the best ways to communicate large numbers. More specifically, [START_REF] Kim | Generating Personalized Spatial Analogies for Distances and Areas[END_REF] construct an online interactive application that enables users to re-express spatial measurements in localized terms--in other words, users are able to build a map that makes a distance relevant to their area or experience.

Even outside of maps, readers locating themselves in the data is likely to make them more engaged. A piece breaking down the unemployment rate by race, gender, age group, and education level [START_REF] Carter | The Jobless Rate for People Like You[END_REF] allows the reader to compare themselves to the average and to other specific groups. While not all possible criteria are covered, this piece gets much closer to an individual than just the unemployment rate that averages across everybody. Similarly, a line chart showing the Case-Shiller Home Price Index for 20 cities lets the reader pick their city and see how housing prices have changed over the years (Carter and Quealey, 2014).

Directly Engaging the Audience

A somewhat similar approach asks the audience questions to involve them in what is shown. This can help engagement by allowing them to place themselves in a context (and at the same time also collect data to be shown to everybody), and it has recently also been found to help people remember what they have seen.

In 2011, The New York Times published The Death of a Terrorist: A turning point? [START_REF] Huang | The Death of a Terrorist: A Turning Point? The New York Times[END_REF] where readers were invited to position their opinions on the death of Osama bin Laden in a two dimensional matrix: whether their response was positive or negative overall, and whether they thought that this event was going to be significant in the war on terror or not. The resulting heatmap of responses showed the landscape of people's responses.

Similar pieces have since appeared in different media, such as the Times' You Draw It: How Family Income Predicts Children's College Chances , which allowed users to draw their idea of the correlation between parents' income and their children's ability to attend college [START_REF] Aisch | You Draw It: How Family Income Predicts Children's College Chances[END_REF]. Similar pieces have appeared to let readers predict price changes for common goods to match an inflation goal [START_REF] Canipe | Beat the Fed[END_REF], draw the border between Western Germany and the German Democratic Republic from memory [START_REF] Keseling | Wissen Sie noch, wo Deutschland geteilt war? (Do you still remember where Germany was separated?) Berliner Morgenpost[END_REF], or guess the rate of teenage pregnancies in the UK (ONS, 2016) (Figure 9.5).

Figure 9.5. In this interactive piece by the UK's Office of National Statistics, readers are first asked to make an estimate using the slider at the bottom, which fills in as many people as they are guessing (girls under 18 becoming pregnant, in this case). When they submit their guess, the difference between it and the real value is illustrated by a small animation and the numbers are shown on the right (ONS, 2016).

Interactive pieces like this are still relatively rare, but they are slowly gaining momentum. A study recently tested the effect of asking readers to make estimates before showing the actual data on memory [START_REF] Kim | Explaining the Gap: Visualizing One's Predictions Improves Recall and Comprehension of Data[END_REF]) (Figure 9.6). It showed that asking participants to reflect on their prior knowledge by guessing numbers or explaining them helped readers to with both comprehension and memory of the data.

Engaging an audience is possible even without much knowledge about its members' backgrounds, level of domain knowledge, etc. Personalization as discussed in the previous section and engagement through questions have both been tried and found to be quite effective.

Specific Design Contexts

In addition to the audience itself, the media used to reach it are an important factor that shape what is possible in practice and what people expect. A detailed news graphic designed for print will not work on television, and a TV piece will look out of place on the web.

The Reality of the Newsroom

Over the last several years news media have created many of the most exciting examples of data-driven storytelling. News media, like other organizations, have struggled with different device types, browser idiosyncrasies, screen sizes, as well as different technologies and programming languages (such as the switch from Flash to HTML 5).

Designing for devices means not only taking their idiosyncrasies into account, but taking advantage of them. Trying to design across many devices typically ends up with a sort of lowest common denominator that works but is not as effective as it could be on any specific device.

See Chapter 6: "Watches to Augmented Reality: Devices and Gadgets for Data-Driven Storytelling" for more details.

Recently, many newsrooms have scaled back those efforts. New York Times Graphics Editor Gregor Aisch described the problem as a cost-benefit tradeoff [START_REF] Aisch | Data visualization and the news[END_REF]. Many interactive news pieces see very little interaction--only 10-15% of visitors interact with them. Even a simple secondary interaction, like left-right swiping to reveal more information when the main flow of the story is swiping down, is rarely used. This is despite efforts like adding prompts.

Designers have devised a number of workarounds to make large graphical pieces work on small screens. For maps, a convenient one is to zoom the map into the location of the user, since that is likely what they are interested in. However, that often means that there is no way to zoom out to see it in its entirety. Aisch therefore proposes a variation of Shneiderman's famous information-seeking mantra: " Overview first, zoom and filter, then details on demand " [START_REF] Shneiderman | The Eyes Have It: A Task by Data Type Taxonomy for Information Visualizations[END_REF] for the reality of news graphics on mobile devices: " Details first, no zoom and filter, overview on desktop only."

Newsrooms' resources are limited, and they often produce news stories on tight deadlines (see also Chapter 8: "Organizing the Work of Data-Driven Visual Storytelling"). In light of this, it becomes necessary to consider not just the potential information value, but also the effort to make something interactive when so few readers ever get a benefit from it. Similar to Aisch, Tse (2016) argues that interactive data stories (at least in the newsroom) may not be the best approach. He makes a three-point argument:

1. If you make the reader click or do anything other than scroll, something spectacular has to happen. 2. If you make a tooltip or rollover, assume no one will ever see it. If content is important for readers to see, don't hide it. 3. When deciding whether to make something interactive, remember that getting it to work on all platforms is expensive. Tse's argument rests on changes in technology, and the changing business structure of the media industry (e.g., advertising efforts, subscription packages, etc.).

It is not clear what the solution is to these issues. One approach is certainly better tools that make it easier to create pieces in the first place and allow the author to target different platforms, either automatically or at least with some support. Another is to show potential interactions with little animations. This has been tried in project such as gapminder, but we are not aware of any studies to show if it such animations would actually increase interaction. Perhaps the most important issue is still training the user: if readers are not used to charts responding and allowing them to interact, they won't try hovering over elements, etc. By creating more interactive pieces, media can train their readers to expect everything to be interactive, and thus increase interaction over time (whether this works or not will have to be studied as well, of course).

While news media tend to have a general idea of their audience, it tends to be vague and hard to use for practical decision-making. There are still considerable hurdles to using what is known to target the audience more specifically and precisely. There are many open questions here and ample need (and opportunity) for more research that could help improve the way news pieces are built and how they work.

Visualization for Television

Television is an unusual use case for visualization, even though its visual nature seems to lend itself well to it. This case study is based on the program Le Grand Webzé , which aired in February 24th, 2012 on the France 5 channel [START_REF] Huron | Bubble-TV: Live Visual Feedback for Social TV Broadcast[END_REF].

A TV audience is highly diverse: almost any age, any level of education, any social position, economic role, and cultural interest is represented. Ratings agencies have been tracking audiences for decades, with age groups starting as young as 4 years old, with segments by age, gender, economic status, and if they are living in urban or rural areas. Television audiences are also fickle, often switching programs and channels within less than a second.

The research and development services of the French National Television Network contacted Huron and his colleagues to create a system that would engage a TV audience with online activity during a TV show. The program is focused on web culture and presents a review of the latest web buzz as well as interviews of famous bloggers or web artists. The goal was to allow viewers to interact with each other and to see a visual representation of that interaction on the program.

The design was made for the medium and the way it is consumed. To show vote tallies, bubbles would drop and form into bar charts ( See Fig 9.7.) . The bubbles included small images of the people who had voted, making for a selfie moment and improving the visual interest of the graphic. Animation was used throughout to blend in with the always-moving nature of television.

Television presents unique design challenges for visualization: the amount of detail is very limited, there is no interaction, motion and animation are crucial, TV graphics are often three-dimensional, etc. Despite its prevalence in popular culture, television as a medium is still almost completely overlooked in visualization research.

Conclusions

When presenting data and telling stories, it seems obvious to tailor the presentation to the audience. In practice, unfortunately, there are many reasons why that is not easy or even possible. It could be due to financial or personnel constraints, deadlines, uncertainty about the targeted audience, or even ignorance about audiences capacities. Despite such uncertainty, one way to think about the audience could be to consider their data and visual literacy level, their attention span, background knowledge, and domain expertise.

The inability to know enough of the audience in many cases has led to some interesting approaches that have proven to be helpful beyond just tailoring a visualization to the reader or viewer. Personalization is an effective means not only to tailor a news piece to a particular reader in more detail than could be done otherwise, but also a way to help engage that reader. Asking for the person's background knowledge by having them estimate a value first is necessary when the visualization designer doesn't know the reader, but this has proven helpful with memory and comprehension of the data being shown as well.

In an ideal world, the designer of a data story would know all the relevant information about her audience and be able to put that into practice. Some things are easier to measure or assess, like a person's visualization literacy or familiarity with a particular chart type. Others are harder or close to impossible, like a person's background knowledge about a topic or question they might have. Even without being able to answer these questions, a well-designed story brings in many different viewers by providing pieces of context that are likely relevant to most users without cluttering up the display. Going further, it is conceivable that a story would measure a viewer's visualization literacy in an unobtrusive (and clandestine) way and adapt the amount of information being shown, as well as the types of visualization, etc.

Smart design takes available knowledge about the audience into account, but does not depend on it being available. Many design decisions are also driven more by general ideas about effective presentation than specifics about a particular audience. A better understanding of the audience will likely lead to better data stories, but the current state of the art in new graphics, television, and other areas shows that good designers are able to create compelling stories even with barely any knowledge of the audience at all.

Figure 9 . 1 .

 91 Figure 9.1. A line chart printed in The New York Daily Tribune, September 29, 1849 would not have been familiar to many of the newspaper's readers. As a result, extensive instructions are given for how to decode and understand the chart.

Figure 9 . 2 .

 92 Figure 9.2. A bubble plot from the Los Angeles Times from 2013 expertly annotates both how to read the chart and the story the authors deliver. Source: http://graphics.latimes.com/california-crime-2013/

Figure 9 . 3 .

 93 Figure 9.3. The 2012 Long-Term Budget Outlook infographic from the Congressional Budget Office. Source: https://www.cbo.gov/publication/43289

Figure 9 . 4 .

 94 Figure 9.4. From the "Chart Parts" section of the Urban Institute Data Visualization Style Guide. Source: http://urbaninstitute.github.io/graphics-styleguide/

Figure 9 . 6 .

 96 Figure 9.6. User interface used in a study to test the effectiveness of engaging readers by asking them to make guesses about the data first and then showing them how close (or far off) they were (Kim et al, 2017).

Figure 9 . 7 .

 97 Figure 9.7. Photograph taken of the studio monitoring screen during the Le Grand Webzé TV program. The visualization shows the votes of the show audience about the host's popularity. Each circle represents a tweet sent during the last few seconds, the bar chart represents the total number of tweets for each hashtag. The visualization is updated continuously.

http://urbaninstitute.github.io/graphics-styleguide/

https://knightcenter.utexas.edu/mooc/file/tdmn_graphics.pdf


Joel Gilbert

Sylvain Maugeais

Christophe Vergez

From the

From the bifurcation diagrams to the ease of playing of reed musical instruments. Application to a reed-like instrument having two quasi-harmonic resonances Abstract A reed-like instrument having two quasi-harmonic resonances, represented by a 4-dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches.

Wind musical instruments are nonlinear dynamical systems with a large diversity of oscillating behaviours. Among these, musicians know how to control their instrument in order to select periodic oscillating regimes, which correspond to notes when their frequency is properly adjusted. Understanding the conditions required to obtain the desired oscillating regime, in terms both of control by the musician and of the acoustic properties of the instrument is an interesting and intricate problem of nonlinear dynamics. In this study we focus on reed instruments. The use of the AUTO continuation package allows to revisit and extend the analytical approach proposed in [START_REF] Dalmont | Reed Instruments, from Small to Large Amplitude Periodic Oscillations and the Helmholtz Motion Analogy[END_REF].

Reed musical instruments can be described in terms of conceptually separate linear and nonlinear mechanisms: a localized nonlinear element (the valve effect due to the reed) excites a linear, passive acoustical multimode element (the musical instrument usually represented in the frequency domain by its input impedance). The linear element in turn influences the operation of the nonlinear element. The reed musical instruments are self-sustained oscillators. They generate an oscillating acoustical pressure (the note played) from a static overpressure in the player's mouth (the blowing pressure).

A reed instrument having N acoustical modes can be described as a 2N -dimensional autonomous nonlinear dynamical system [START_REF] Chaigne | Acoustics of Musical Instruments[END_REF]. For instance a reed-like instrument having two quasi-harmonic resonances, represented by a 4-dimensional dynamical system, is considered in this study, in order to be able to use the AUTO continuation method. The modulus of the corresponding acoustic input impedance of the resonator is shown in figure 1. The acoustic input impedance is defined in the frequency domain by the ratio between the pressure and the volume flow at the input of the instrument, so that :

P (ω) = Z(ω)U (ω),
where ω is the angular frequency.

On the other hand, the reed-valve nonlinear behaviour can be modelled by the following polynomial nonlinearity in the time domain, where the volume flow u(t) is defined as a function of the acoustic pressure p(t) [START_REF] Chaigne | Acoustics of Musical Instruments[END_REF]:

u = u 0 + Ap + Bp 2 + Cp 3 , ( 2
)
where u 0 is the mean volume flow and A, B and C are real numbers that depend on the control of the musician. Bifurcation diagrams are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches (see examples on figure 2). The ratio between the two acoustic resonance frequencies of the instrument (also called inharmonicity) appears to be of crucial importance.

Figure 2: Bifurcation diagrams corresponding to the case of an input instrument with a 1 st peak 50% larger than the 2 nd one, like in figure 1. The ratio between the two resonance frequencies is 2.024, inharmonicity 0.012 (left) or 2.072, inharmonicity 0.036 (right). Top plots represent the amplitude of the periodic oscillation branches with respect to the blowing pressure. Bottom plots represent the frequency of the corresponding periodic solutions with respect to the blowing pressure. Green (red) lines correspond to periodic oscillations resulting from the instability of the first (second) acoustic resonance and are called 1 st and 2 nd registers respectively. Blue lines correspond to periodic oscillations of the 1 st register resulting from the period doubling of the 2 nd register.

The oral presentation will show how some of these results can be interpreted in terms of the ease of playing of the reed instrument.

Figure 1 :

 1 Figure 1: Modulus of the dimensionless input impedance of the resonator with respect to frequency. The two resonances are quasi-harmonic: first resonance frequency 229Hz (left red vertical line), second resonance frequency 463.5Hz slightly higher than the octave 458Hz (right red vertical line), corresponding to an inharmonicity of 0.012.

th International Conference on Nonlinear Vibrations, Localization and Energy Transfer July 1-4, 2019 -Marseille, France

Laboratoire d'Acoustique de l'Université du Mans -UMR 6613 CNRS Le Mans, France joel.gilbert@univ-lemans.fr

Laboratoire Manceau de Mathématiques -Le Mans Université Le Mans, France sylvain.maugeais@univ-lemans.fr

Aix-Marseille Univ, CNRS, Centrale Marseille, LMA, UMR 7031 Marseille, France vergez@lma.cnrs-mrs.fr