Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

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

Sampled-Data Estimator for Nonlinear Systems with Uncertainties and Arbitrarily Fast Rate of Convergence

Keywords: Estimation, nonlinear systems, delay, stability

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.

Introduction

Estimator design plays a central role in current control systems research; see for instance [START_REF] Borri | Luenberger-like observers for nonlinear time-delay systems with application to the artificial pancreas: the attainment of good performance[END_REF], [START_REF] Buccella | Observer-based control of LCDC/DC resonant converter using extended describing functions[END_REF], Cacace et al. (2014a), Cacace et al. (2014b), and [START_REF] Parikh | Framework for guaranteed convergence of image-based observers with intermittent measurements[END_REF]. One significant criterion for quantifying the performance of an estimator is the sensitivity of the estimation error to measurement or model uncertainty. A standard approach to quantifying sensitivity to uncertainty is input-to-state stability (or ISS), which bounds the norm of the input-to-state stable state by the sum of two terms, namely, a decaying term plus an overshoot term that usually depends on the supremum of the uncertainty from the initial time to the current time. One useful estimator result was Mazenc et al. (2015a), which provides estimators for continuous-time systems of the form ẋ(t) = Ax(t) + µ(y(t), u(t))

y(t) = Cx(t) (1)
where x is the state, A and C are constant matrices of appropriate dimensions, y is the output, µ is a nonlinear function, and the input u can be a control. The estimators in Mazenc et al. (2015a) and [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF] are finite time observers incorporating delays, and their instant Special cases of this work have been published in the conference paper Mazenc et al. (2020a); see Section 1 for a comparison between this work and the conference version.

of convergence can be chosen by the user as an arbitrarily small positive value. In particular, [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF] requires delayed measurements of the output. Being fixed time observers, they possess a potential advantage over traditional observers that are only asymptotically converging. However, Mazenc et al. (2015a) and [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF] only apply when continuous output measurements y(t) are available. This can pose challenges in settings where only discrete noisy output measurements of the form y(t) = Cx(t i ) + δ o (t i ) for all t ∈ [t i , t i+1 ) and i ≥ 0 (2) are available, where the bounded piecewise continuous function δ o represents measurement noise and t i is an increasing sequence of nonnegative values. In such a case, Mazenc et al. (2015a) does not apply. This motivates this work, which helps overcome the preceding challenge by adapting a technique from Mazenc et al. (2015a) to cases where the lengths t i+1 -t i of the sampling intervals are not required to be constant and where the output is discrete. Our method borrows key ideas from the work [START_REF] Karafyllis | From continuoustime design to sampled-data design of observers[END_REF], which redesigns asymptotic observers to accommodate systems with discrete measurements. A key method in [START_REF] Karafyllis | From continuoustime design to sampled-data design of observers[END_REF] is a dynamic extension that predicts the measured variables between consecutive measurements. Here we show that ideas from [START_REF] Karafyllis | From continuoustime design to sampled-data design of observers[END_REF] combined with the method from Mazenc et al. (2015a) yield a state estimate whose most significant feature is that it leads to a useful variant of ISS where the overshoot term has a fading effect, meaning, it only depends on a recent history of the uncertainty instead of the supremum of the uncertainty from the initial time to the current time, and where the decaying term also converges with a rate of convergence that can be made arbitrarily large by reducing the supremum of the sampling intervals.

In the sequel, we study systems of the form 3) are replaced by nonlinearities of the form ϕ(t, x(t), δ s (t)) under additional conditions involving the sample rates). We design continuousdiscrete observers which differ from the ones of Mazenc et al. (2015a) because ours will not converge in finite time, but our observers converge with a rate of convergence of the form c 1 ln(c 2 sup i {t i+1 -t i }) for certain constants c 1 < 0 and c 2 > 0. Hence, the convergence rate can be made arbitrarily large by sampling frequently enough.

ẋ(t) = A(t)x(t) + ϕ(Cx(t), t) + δ s (t) y(t) = Cx(t i ) + δ o (t) for all t ∈ [t i , t i+1 ) and i ≥ 0 (3) where x is valued in R n with n ≥ 2, the output y is valued in R, C ∈ R 1×n
Our study of connections between sample rates in the output and convergence rates of the observer is strongly motivated by electrical engineering and other applications. To obtain the sampled measurements y(t) in (3) in applications, one needs sensors, which are sometimes limited by their sample rate. Therefore, to obtain higher sampling rates, it may be necessary to remove old sensors and replace them by new sensors that may offer higher sampling rates. When observers are used to estimate states for use in controls, increasing the sample rates can improve the performance of controls, which may be implemented on a digital signal processor (or DSP). However, a given DSP may have limited computational capabilities and therefore may only be able to process data at a given rate. In addition to the computational capacity of the DSP, the computational capabilities for control implementations can depend on the complexity of the control. Therefore, by quantifying the rate of convergence of our observer in terms of the sample rate, our work provides guidance on when to replace sensors or DSPs by new sensors or new DSPs that allow the faster sample rates that may be called for by our theory to achieve desired convergence rates.

Our results use a trajectory based approach from [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF], Mazenc and Malisoff (2015), and [START_REF] Mazenc | Stability and control design for time-varying systems with timevarying delays using a trajectory based approach[END_REF]. Since the observer formula of Mazenc et al. (2015a) uses delays, one cannot directly apply [START_REF] Karafyllis | From continuoustime design to sampled-data design of observers[END_REF] to solve this problem.

φ : [-T, 0] → R a . We define Ξ t ∈ C in by Ξ t (m) = Ξ(t + m) (4)
for all choices of Ξ, m ≤ 0, and t ≥ 0 such that t + m is in the domain of Ξ. We use | • | S to denote the essential supremum over a set S, |M| ∞ is the essential supremum of any function M over its domain, and Φ A is the fundamental solution associated with the matrix A in (3) (as defined in (Sontag, 1998, Appendix C.4)).

2 Main result

Assumptions

Our main assumptions on (3) are as follows:

Assumption 1 There are constants T > 0 and T > 0 such that for all integers i ≥ 0, the inequalities

T ≤ t i+1 -t i ≤ T (5)
are satisfied and t 0 = 0.

Assumption 2 The function ϕ is continuous. Also, there is a constant ϕ ≥ 0 such that |ϕ(y a , t)-ϕ(y b , t)| ≤ ϕ|y a -y b | holds for all real values y a and y b and all t ≥ 0.

Assumption 3 There is a constant τ > 0 such that for each t ∈ R, the matrix

Ω(t) =     C CΦ A (t -τ, t) . . . CΦ A (t -(n -1)τ, t)     (6)
has an inverse Ω -1 (t) that satisfies sup t≥0 |Ω -1 (t)| < ∞.

Assumption 3 generalizes the observability condition on (A, C) from the special case of Mazenc et al. (2020a) where A is constant, and can be checked by computing the determinant of Ω(t) for t ∈ [0, τ ] when A has period τ ; see Section 2.3 for more on the connection between Assumption 3 and observability, and for sufficient conditions for our assumptions to be satisfied including cases where A is not periodic, and for ways to compute Φ A to facilitate checking our assumptions and implementing our observer.

We set Ψ(t) = Ω -1 (t) (7) for all t ∈ R, and

λ(τ ) = |CAΨ| ∞ √ n + G(τ ) √ τ + |C|ϕ, (8)
where

G(τ ) = n-1 j=1
jσ 2 j , and

σ j = sup{|CΦ A (r, s)| : 0 ≤ s -r ≤ jτ, r ≥ 0} (9)
for j = 1, . . . , n -1. Our last assumption is as follows, which implies that λ(τ ) must be positive:

Assumption 4 The inequality

T < 1 λ(τ ) (10)
is satisfied.

Since the right side of ( 10) is independent of the constants T and T that we used to bound the intersample times t i+1 -t i in (5), and since we can choose

T = sup i≥0 (t i+1 -t i ) and T = inf i≥0 (t i+1 -t i ),
it follows that we can always satisfy ( 5) and ( 10) by sampling the output faster (and so reducing T ) while maintaining a positive lower bound T on the intersample times t i+1 -t i . In the next section, our ability to choose T as small as we want will play a key role in showing how we can ensure arbitrarily fast convergence of our observer, while ensuring the fading effect in the ISS overshoot that we explained in Section 1.

Estimator Design

Before providing the formulas for our estimator, we provide an informal summary of its structure and the properties of our estimation algorithm. Our estimator design is composed of an interconnection of (a) a continuous-discrete system whose state is an estimator of the unperturbed output and which is reinitialized at the sample times t i using measurements y(t i ) of the perturbed output and (b) an estimator variable that is computed from continuous measurements from the interconnected continuous-discrete system. Using this estimator variable, the estimation error has the required features of ensuring arbitrarily fast convergence (by sampling fast enough) and our variant of ISS whose overshoot only depends on a recent history of the uncertainties at each time.

We use the continuous-discrete system:

         ω(t) = CA(t)x(t) + Cϕ(ω(t), t) for all t ∈ [t i , t i+1 ) and i ≥ 0 ω(t i ) = y(t i ) for all i ≥ 0 x(t) = Ψ(t)U 1 (ω t ) + Ψ(t)U 2 (ω t , t) (11)
with x valued in R n , the perturbed measurements y(t i ) from ( 3), ω valued in R, τ > 0 satisfying Assumption 3, and

U 1 (ω t ) =        ω(t) ω(t -τ ) . . . ω(t -(n -1)τ )        and (12) U 2 (ω t , t) =        0 CΦ A (t -τ, t)∆H 1 (t) . . . CΦ A (t -(n -1)τ, t)∆H n-1 (t)        (13)
and where

∆H j (t) = H j (t)-Φ A (t, t -jτ )H j (t -jτ ) and the R n -valued functions H j solve Ḣj (t)=A(t)H j (t) + ϕ(ω(t), t) (14)
for j = 1, . . . , n -1 and any constant initial functions that are defined over [-(n -1)τ, 0]. Our x formula is reminiscent of the observer of Mazenc et al. (2015a), and the ω-subsystem of our observer is inspired by the one from [START_REF] Karafyllis | From continuoustime design to sampled-data design of observers[END_REF]. Notice that the predicted output ω only requires the perturbed sampled output measurements y(t i ), and in particular, it does not require derivatives of the predicted output. Also, while the formula for U 1 implies that past values of the predicted output are used, these are readily available for implementations because the dynamics for ω only contain known measured quantities.

While we can apply variation of parameters on the intervals [t -jτ, t] to check that

Φ A (t -jτ, t) (H j (t) -Φ A (t, t -jτ )H j (t -jτ )) = t t-jτ Φ A (t -jτ, m)ϕ(ω(m), m)dm (15)
holds for all t ≥ (n -1)τ and j = 1, 2, . . . , n -1 and t ∈ R, we use our expressions ( 13) and ( 14) as a way to get formulas for estimators that do not contain any integrations.

In terms of the constant G(τ ) from ( 9) and the function

M(t)=|δ o | [t-T ,t] + T |C|+ + G(τ ) √ τ |CAΦ| ∞ |δ s | [t-T -(n-1)τ,t] (16)
and the constant

σ * = sup{|Φ A (r, s)| : r ≥ 0, s ∈ [r, r + (n -1)τ ]}
and τ from Assumption 3, our main result is:

Theorem 1 Let (3) satisfy Assumptions 1-4. Then for all constant initial functions for (3) and ( 11) and all uncertainty functions (δ s , δ o ) in (3), and with the choice

c * = √ n|Ψ| ∞ 1 + |C|(n -1)ϕτ σ * , (17)
we have that

|ω(t) -Cx(t)| ≤ M(t) 1-T λ(τ ) + sup ∈[0,T +(n-1)τ ] |ω( ) -Cx( )|e ln(T λ(τ )) T +(n-1)τ (t-T -(n-1)τ ) (18)
for all t ≥ T + (n -1)τ and

|x(t) -x(t)| ≤ √ n|Ψ|∞|M| [t-(n-1)τ,t] 1-T λ(τ ) + c * sup ∈[0,T +(n-1)τ ] |ω( )-Cx( )|e ln(T λ(τ )) T +(n-1)τ (t-T -2(n-1)τ ) + √ n|Ψ| ∞ |C|(n-1)τ σ * |δ s | [t-(n-1)τ,t] (19)
for all t ≥ T + 2(n -1)τ .

Remarks on Theorem 1

Before proving Theorem 1, we provide several remarks concerning the feasibility of checking its assumptions, the implementability of the observer, and the rate of convergence guarantees that are provided by the theorem.

Structure of Observer

Our observer ( 11)-( 14) calls for resetting ω at each time t i when a new output value becomes available. This implies that at each time t, the future sampling times can be uncertain. Theorem 1 gives a convergence rate to zero of

r = c 1 ln(c 2 T ) (20)
of the second right side term in (19), where

c 1 = - 1 T +(n-1)τ < 0 and c 2 = λ(τ ) > 0. ( 21
)
Since r diverges to ∞ as T → 0, the continuous time case (where the state is computed exactly in finite time) can be interpreted as a limiting case as T → 0. This contrasts with works such as Mazenc et al. (2015b) on continuous-discrete observer designs that do not provide methods to adjust the sample times to achieve arbitrarily fast convergence.

Our observer formulas also call for computing the Φ A values in the formulas for Ψ and U 2 in ( 11) and ( 13). This can be done using the fact (e.g., (Malisoff, 2020, Section 4

.3)) that Φ A (t, s) = α A (t)β A (s) (22
) for all real t and s, where α A and β A are the unique solutions of the matrix differential equations 18)-( 19) converge to 0 as t → ∞ if the uncertainties decay to 0. As noted above, the fading effect (where only a recent history of the uncertainties is used to define the overshoot terms on the right sides) contrasts with standard ISS results where the suprema would be over [0, t]. Hence, we believe that our novel observer design adds significant value relative to the literature. where the uncertainties are 0, here we obtain better results in terms of performance in such cases. Theorem 1 can also provide a tighter estimate than traditional ISS estimates, because instead of suprema of (δ s , δ o ) over an interval from the initial time T + (n -1)τ or T + 2(n -1)τ to the current time t, the suprema are only over a recent history of (δ s , δ o ). Hence, the conclusions of Theorem 1 are less restrictive than ISS. Moreover, as the measurement step converges to zero, our theorem gives the limiting version

αA (t) = A(t)α A (t) and βA (t) = -β A (t)A(t) (23) that satisfy α A (0) = β A (0) = I.

Comparison with

|x(t) -x(t)| ≤ √ n|Ψ|∞|M| [t-(n-1)τ,t] 1-T λ(τ ) + + √ n|Ψ| ∞ |C|(n-1)τ σ * |δ s | [t-(n-1)τ,t] (25)
of t ( 19) for all t ≥ T + 2(n -1)τ . An analogous limiting estimate holds for the output estimation error (18).

Checking Assumption 3

When A is constant, we have Φ A (t, s) = e A(t-s) for all real t and s. Therefore, when A is constant, our function Ω from ( 6) is the constant matrix

Ω =        C Ce -Aτ . . . Ce -(n-1)Aτ        (26)
which allows us to check Assumption 3 by computing eigenvalues. Also, (6) has period τ when A has period τ . Therefore, when A has period τ , we can check Assumption 3 by checking for the invertibility of Ω on [0, τ ] (since the boundedness from Assumption 3 will then follow because of the continuity of Ω -1 ). Also, by Mazenc et al. (2020a), when A is constant and (A, C) is observable, we can satisfy Assumption 3 by choosing τ > 0 such that (-A , C ) is τsampled controllable, using [START_REF] Sontag | Mathematical Control Theory[END_REF], Theorem 4 and Lemma 3.4.1) and the fact that the observability of (A, C) implies that (-A , C ) is controllable.

We can also satisfy Assumption 3 when A is not periodic but has the form A(t) = A 0 + ∆ A (t) where (A 0 , C) is observable, when the sup norm of the continuous time varying part ∆ A is small enough. To find a bound on the allowable sup norms of ∆ A , we can use the following three step procedure. First, we can choose τ such that the constant matrix Ω from ( 26) is invertible when A is replaced by A 0 , using the reasoning from the preceding paragraph. Second, for the preceding choice of τ and with the choice Ψ = Ω-1 , we can choose a constant ¯ > 0 such that |M -1 | ≤ 2| Ψ| for all matrices M that satisfy |M -Ω| < ¯ . Finally, for the preceding choices of τ and ¯ > 0, we can use the estimates

Φ A (t -jτ, t) -e -jτ A0 ≤ jτ |∆ A | ∞ e (|A0|+|∆ A |∞)jτ (27)
for j = 1, . . . , n-1 (e.g., from [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF], Lemma 1)) to compute a constant δ > 0 such that |Ω(t) -Ω| ∞ < ¯ for all ∆ A 's that satisfy |∆ A | ∞ < δ. Then Assumption 3 will hold if |∆ A | ∞ < δ. Moreover, we can use the estimate

|Φ A (t, s)| ≤ e |t-s|(|A 0 |+|∆ A |∞) (28)
from [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF], Lemma 1) for all real t and s to compute upper bounds on the functions σ j from (9) to check Assumption 4 when A is nonconstant.

Proof of Theorem 1

First note that Assumptions 1-2 ensure that the system consisting of (3) and ( 11) is forward complete. By applying variation of parameters to (3) over any interval [s, t] with 0 ≤ s ≤ t, we have

x(t) = Φ A (t, s)x(s) + t s Φ A (t, m)[ϕ(Cx(m), m) + δ s (m)]dm. (29)
Thus, we can use the semigroup property of Φ A to check that for all p ∈ {0, ..., n -1}, we have

CΦ A (t -pτ, t)x(t) = Cx(t -pτ ) + C t t-pτ Φ A (t -pτ, m)ϕ (m)dm (30)
for all t ≥ pτ , where

ϕ (m) = ϕ(Cx(m), m) + δ s (m). ( 31
) Setting ∆ 1 (x t ) = Ψ(t)        Cx(t) Cx(t -τ ) . . . Cx(t -(n -1)τ )        and (32) ∆ 2 (x t ) = Ψ(t)        0 C t t-τ Φ A (t -τ, m)ϕ (m)dm . . . C t t-(n-1)τ Φ A (t -(n -1)τ, m)ϕ (m)dm        , (33)
we deduce from (30) that

x(t) = ∆ 1 (x t ) + ∆ 2 (x t ) (34)
for all t ≥ (n -1)τ .

We next use the variables e ω (t) = ω(t) -Cx(t) and e x (t) = x(t) -x(t)

and we choose i to be the integer such that

t i-1 < (n -1)τ and t i ≥ (n -1)τ. ( 36
)
Using the functions

D 3,i (x t , ω t ) = ω(t -iτ ) -Cx(t -iτ ) (37)
for i = 0, . . . , n -1 and

D 4,i (x t , ω t ) = C t t-iτ Φ A (t -iτ, m)[ϕ(ω(m), m) -ϕ (m)]dm, (38)
one can use ( 15) and ( 11)-( 13) to check that

                                   ėω (t) = CA(t)e x (t) + C[ϕ(ω(t), t) -ϕ(Cx(t), t)] -Cδ s (t) for all t ∈ [t i , t i+1 ) e ω (t i ) = δ 0 (t i ) e x (t) = Ψ(t)U 1 (e ω,t ) + Ψ(t)        0 D 4,1 (x t , ω t ) . . . D 4,n-1 (x t , ω t )        (39)
for all i ∈ N with i ≥ i, where we use the definition of U 1 from (12) (with ω t replaced by e ω,t in the U 1 formula in (12)). For all i ∈ N with i ≥ i, this gives

                                   ėω (t) = CA(t)Ψ(t)U 1 (e ω,t ) -Cδ s (t) +CA(t)Ψ(t)        0 D 4,1 (x t , ω t ) . . . D 4,n-1 (x t , ω t )        +C[ϕ(ω(t), t) -ϕ(Cx(t), t)] if t ∈ [t i , t i+1 ) e ω (t i ) = δ o (t i ) (40)
By integrating (40) over the interval [t i , t), we obtain

e ω (t) = C t ti ϕ(ω( ), ) -ϕ(Cx( ), ) -δ s ( ) d + δ o (t i ) + U 3 (t, x t , ω t ) + U 4 (t, x t , ω t ) (41)
for all t ≥ 0 and i ∈ N such that t ∈ [t i , t i+1 ), where

U 3 (t, x t , ω t ) = n j=1 t ti A j (m)e ω (m -(j -1)τ )dm and U 4 (t, x t , ω t ) = n j=2 t ti A j ( ) -(j-1)τ CΦ A ( -(j -1)τ, m)∆ϕ(m)dmd (42)
where ∆ϕ(m) = ϕ(ω(m), m)-ϕ (m) (43) and A j is the jth component of the vector valued function A = CAΨ for 1 ≤ j ≤ n when t ≥ T + (n -1)τ . To use our trajectory based method, we find a suitable upper bound for the right side of (41).

To this end, note that the Cauchy-Schwarz inequality followed by Hölder's inequality give

|U 3 (t, x t , ω t )| ≤ t ti |A (m)||U 1 (e ω,m )|dm ≤ t ti n j=1 A j (m) 2 dm × t ti n j=1 e 2 ω (m -(j -1)τ )dm ≤ √ T |CAΨ| ∞ n-1 j=0 t-jτ ti-jτ e 2 ω (m)dm ≤ T |CAΨ| ∞ √ n|e ω | [t-T -(n-1)τ,t] (44)
and similarly,

|U 4 (t, x t , ω t )| ≤ t ti n i=2 A j (m) 2 dm × t ti n-1 j=1 -jτ |CΦ A ( -jτ, m)||e ω (m)|dm 2 d ( 45
)
and so also

|U 4 (t, x t , ω t )| ≤ √ T |CAΨ| ∞ n-1 j=1 t ti jτ d σ 2 j |e ω | 2 [t-T -(n-1)τ,t] ≤ T |CAΨ| ∞ n-1 j=1 jτ σ 2 j |e ω | [t-T -(n-1)τ,t] ≤ T √ τ |CAΨ| ∞ G(τ )|e ω | [t-T -(n-1)τ,t] (46)
if t ≥ T + (n -1)τ , where we used the notation ( 9) and

e ω (m) = ϕ|e ω (m)| + |δ s (m)|, ( 47
)
and where the first inequality in ( 46) applied Jensen's inequality to the inside integral in the double integral. By using our bounds ( 44)-( 46) to bound the last two right side terms in ( 41) and recalling our definitions of ϕ and T , and by defining the intervals

I(t) = [t -T -(n -1)τ, t], we get |e ω (t)| ≤ |C|T ϕ|e ω | [t-T ,t] + |δ o | [t-T ,t] +|C|T |δ s | [t-T ,t] +T |CAΨ| ∞ √ n|e ω | I(t) +T √ τ |CAΨ| ∞ G(τ )|e ω | I(t) ≤ T λ(τ )|e ω | I(t) + M(t) (48)
for all t ≥ T +(n-1)τ , where λ is as defined in ( 8) and M is defined in ( 16). Then ( 10 T +(n-1)τ

(t-T -(n-1)τ ) + M(t) 1-T λ(τ ) (49)
for all t ≥ T + (n -1)τ . This concludes the proof of the first conclusion of the theorem. The second conclusion now follows from the formula for e x in (39).

4 More General Systems

Statement of Result

Under alternative conditions involving T , we can generalize the preceding work to cover systems of the form Fixing a constant ϕ that satisfies Assumption 5, and also using the constants

ẋ(t) = A(t)x(t) + ϕ(t, x(t), δ s (t)) y(t) = Cx(t i )+δ o (t) for all t ∈ [t i , t i+1
λ a = |CAΨ| ∞ √ n, λ b = √ τ |CA|Ψ| ∞ G(τ ) + |C|ϕ, λ c = √ n|Ψ| ∞ |C|τ (n -1)|Φ A | ∞ ϕ, and λ d = |Ψ| ∞ √ n, (51)
we then replace Assumption 4 by the following:

Assumption 6 The inequalities

0 < T λ a < 1, 0 < λ c < 1, and 0 < T λ b λ d (1-T λa)(1-λc) < 1 (52) are satisfied.
We also replace the continuous-discrete system (11) by

       ω(t) = CA(t)x(t) + Cϕ(t, x(t), 0) for all t ∈ [t i , t i+1 ) and i ≥ 0 ω(t i ) = y(t i ) for all i ≥ 0 x(t) = Ψ(t)U 1 (ω t ) + Ψ(t)U 2 (ω t , t) (53)
and we replace ϕ(ω(t), t) by ϕ(t, x(t), 0) in ( 14). Then we have this analog of Theorem 1 for ( 50) and ( 53):

Theorem 2 Let (50) satisfy Assumptions 1, 3, 5, and 6.

Then we can construct positive constants ci for i = 1, 2, 3 such that with the choices

D(t) = max{|ω(t) -Cx(t)|, |x(t) -x(t)|} (54)
and T * = T +(n-1)τ , and for all constant initial functions for ( 50) and ( 53) and all measurable essentially bounded functions (δ s , δ o ) in ( 50), we have

D(t) ≤ c1 e -c2t |D| [0,T * ] + c3 (|δ s | [0,t] + |δ o | [0,t] ) (55)
for all t ≥ 0.

Remark 1 It is tempting to surmise that Theorem 2 is less restrictive than Theorem 1, because the system (50) includes (3) as a special case. However, that is not the case, because as we illustrate in Section 5.2, Assumption 6 can lead to a smaller (and therefore more conservative) allowable upper bound T on the sample rate as compared with Assumption 4 when the nonlinearity ϕ only depends on the t and the output y and when the uncertainty δ s is additive. Therefore, Assumption 6 is the price to pay to allow the more general nonlinearities and uncertainties in (50). For simplicity, we assume in this section that δ s has the same dimension as the state. However, the results remain true if δ s and the state x have different dimensions. Also, we will see in the proof of Theorem 2 that the constants ci in Theorem 2 can be constructed using small gain arguments, e.g., those from the proof of (Liu et al., 2020, Theorem 2.6).

Proof of Theorem 2

We indicate the changes needed in the proof of Theorem 1 to prove Theorem 2. In equations ( 29) and the formula (30) for ϕ , we replace ϕ(Cx(m), m) + δ s (m) by ϕ(m, x(m), δ s (m)). Then, in the formula (38) for D 4 , we replace ϕ(ω(m), m) by ϕ(m, x(m), 0). Now each D 4,i only depends on x t and xt , so we can write it as D 4,i (x t , xt ). Then we replace the terms C[ϕ(ω(t), t) -ϕ(Cx(t), t)] -Cδ s (t) in ( 39) and ( 40) by

C[ϕ(m, x(m), 0) -ϕ(m, x(m), δ s (m))].
Also, we replace the integrand ϕ(ω( ), ) -ϕ(Cx( ), )δ s ( ) in the formula for e ω (t) in ( 41) by ϕ(m, x(m), 0)ϕ(m, x(m), δ s (m)), and we replace ϕ(ω(m), m) in the formula (43) for ∆ϕ(m) by ϕ(m, x(m), 0). Then U 4 is a function of (t, x t , xt ), so we write it as U 4 (t, x t , xt ). Then, we replace e ω by e x in the upper bounds for |U 4 (t, x t , xt )| in ( 45)-( 46), where e x (m) = ϕ |e x (m)| + |δ s (m)| . We also use the function

M * (t) = |δ o | [0,t] + ϕT |C| + √ τ |CAΨ| ∞ G(τ ) |δ s | [0,t] . (56)
Then, since (52) gives T λ a ∈ (0, 1), we can use the reasoning of the last part of the proof of Theorem 1 to get

|e ω (t)| ≤ |C|T ϕ(|e x | [t-T ,t] + |δ s | [t-T ,t] ) +T |CAΨ| ∞ √ n|e ω | I(t) + |δ o | [t-T ,t] +T √ τ |CAΨ| ∞ G(τ )|e x | [0,t] ≤ T λ a |e ω | I(t) + T λ b |e x | [0,t] + M * (t) (57)
for all t ≥ T * and therefore also

|e ω (t)| ≤ |e ω | [0,T * ] e ln(T λa ) T * (t-T * ) + T λ b |ex| [0,t] 1-T λa + M * (t) 1-T λa (58)
for all t ≥ 0, by applying (Mazenc et al., 2017, Lemma 1), where we also used the fact that the first term on the right side of (58) upper bounds |e ω (t)| for all t ∈ [0, T * ], in order to ensure that (58) holds for all t ≥ 0. Also, with the preceding change in D 4 , we can combine the last equation in (39) with our definitions of λ c and λ d from (51), to obtain

|e x (t)| ≤ λ d |e ω | [0,t] + λ c |e x | I(t) + λ c |δ s | [0,t] (59)
for all t ≥ 0. Recalling from Assumption 6 that λ c ∈ (0, 1), we can now again apply (Mazenc et al., 2017, Lemma 1), this time to (59) (in a similar way to the approach that we used to obtain (58)) to conclude that

|e x (t)| ≤ |e x | [0,T * ] e ln(λc )t T * + λ d |eω| [0,t] 1-λc + λc|δs| [0,t] 1-λc (60)
for all t ≥ 0. By ( 58) and ( 60) and the negativeness of ln(T λ a )/T * and ln(λ c )/T * , the conclusion now follows from the last inequality in Assumption 6 and small gain arguments, e.g., the proof of (Liu et al., 2020, Theorem 2.6).

Illustrations

Pendulum with Horizontal Acceleration

Consider a pendulum of mass M and length L whose suspension point is subjected to an unknown time varying piecewise continuous bounded horizontal acceleration h(t) and having a time varying friction k(t) and a torque input T c . As noted in (Khalil, 2002, p.627), this produces

M L θ + M g sin(θ) + k(t)L θ = T c L + M h(t) cos(t), ( 61
)
where g = 9.81 is the gravitational constant, and where we assume that sampled perturbed measurements y(t i ) = c o θ(t i ) + δ o (t i ) are available at sample times t i that satisfy our Assumption 1 for some constants T > 0 and T > 0, where c o > 0 is a known constant whose effect we will study in what follows. Then (61) and its output measurements can be written in the form (3) with the choices n = 2,

A(t) = 0 1 0 -k(t) M , x = θ θ , C = c o 0 , ϕ( , t) =   0 -g L sin co + 1 M L 2 T c (t)   , and
δ s (t) = 1 1 L h(t) cos(θ(t)) . ( 62
)
We assume that L = 1, and that k(t) = 0.1 + ∆ k (t) where the function ∆ k : [0, ∞) → [0, 0.1] can represent the increase in friction over time when ∆ k is increasing. This differs from the case in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] which did not allow sampled outputs, uncertainties, or time varying friction.

We choose ϕ = g/c 0 . We next compute bounds on the allowable supremum T under which Theorem 1 applies.

To this end, notice that in terms of our notation from Theorem 1, our choices (62) give

Φ A (t, s) =   1 t s e - r s k( )/M d dr 0 e - t s k( )/M d   (63)
so with the choice (6), the matrix Ψ = Ω -1 has the form

Ψ(t) =   t-τ t e - r t k( )/M d dr 0 -1 1   1 co t-τ t e - r t k( )/M d dr ( 64
)
and so is bounded for any choice of τ > 0. For simplicity, we choose τ = 1 in the rest of this subsection. Then

CA(t)Ψ(t) = (-1 1) 1 t-1 t e - r t k( )/M d dr (65)
so our lower bound of 0.1 on k(t) gives

|CAΨ| ∞ ≤ √ 2 10M (e 0.1/M -1) -1 . (66)
Also, since

G(1) = c o sup    1 + r s e - q s k( )/M d dq 2 : s ≥ r ≥ 0, s -r ≤ 1 , (67)
our upper bound of 0.2 on k gives the upper bound

G(1) ≤ c o 1 + e 0.04/M 2 . ( 68
)
Since c o > 0 can be taken to be arbitrarily small, the assumptions of Theorem 1 will then be satisfied if

T satisfies √ 2T |CAΨ| ∞ + T g < 1. which holds if T 1 5M e 0.1/M -1 -1 + 9.81 < 1. ( 69
)
When M = 1, this gives the approximate upper bound of 0.08538768 on the allowable T values, and a limiting upper bound of T < 1/9.81 = 0.101936799 as M → +∞.

Comparison with Mazenc et al. (2020a)

Theorem 1 provides less restrictive conditions on T than Mazenc et al. (2020a), even when A is constant and there are no uncertainties. To illustrate this point, we revisit the example from Mazenc et al. (2020a), which is the particular case where n = 2, C = (1, 0) and

A = 0 1 -1 0
, and ϕ(y, t) = sin(y) + u 2 (t)

-u(t) (70)
for any constant > 0, where u represents a control input. Then (A, C) is observable, so Assumptions 1-3 can be satisfied with ϕ = ; see Section 2.3.4. By noting that

e -Am = cos(m) -sin(m) sin(m) cos(m) (71)
for all m ∈ R and using our notation from Section 2.3, we get

Ω = C Ce -Aτ = 1 0 cos(τ ) -sin(τ ) (72)
and for any τ ∈ 0, π 2 , we have

Ψ = 1 0 cos(τ ) sin(τ ) -1 sin(τ ) . (73)
Using our formula (24) and τ = π/2, this gives

Ψ = 1 0 0 -1 and λ T , π 2 = √ 2 1 + e π 2 T + π 2 + . (74)
Choosing = 1 6 as in Mazenc et al. (2020a), we get

λ T , π 2 = √ 2 1 + e π 2 6 T + π 2 + 1 6 , (75)
and then the requirement T λ(T , π/2) < 1 from Mazenc et al. (2020a) produces the requirement

T ≤ 0.27242. ( 76
)
Also, the supremum of the set of all T 's that satisfy the requirements (52) of Theorem 2 would be 0.28. By contrast, if we use Theorem 1 with the preceding choices of the model parameters, then we get the requirement

T < 1 √ 2 + π/2 + 1/6 = 0.352834 (77)
which is a 29% increase in the upper bound as compared with (76). Hence, in this special case, our result is significantly less conservative than Mazenc et al. (2020a).

On the other hand, Theorem 1 also allows uncertainties, which were not allowed in Mazenc et al. (2020a). We next present some Mathematica simulations, which illustrate the effects of the uncertainties on the performance of the observers from Theorem 1 when applied with the choices (70). We use the nonuniformly spaced sampling instants t i = 0.9 + 0.1 sin(iπ/2) iJT whose sampling intervals alternate between lengths 0.8JT , 0.9JT , and JT (but similar reasoning applies for other nonuniform sampling) with T chosen as the upper bound in (76) and the input u(t) = sin(t) in (70), with the choice J = 1, then with J = 1/2, and finally with J = 0.25. Also, we used the initial conditions x 1 (0) = x 2 (0) = 1 and ω(0) = 0.

We plot the resulting observation error values of x2 (t)x 2 (t) for the three different choices of the sampling rate, in Figures 123456. Since the figures show rapid convergence of the errors to 0, with faster convergence as j decreases, when the uncertainty is 0, and an ISS like property for nonzero values of the uncertainty δ 0 or δ s , they help illustrate Theorem 1 for the special case of (70).

Conclusions

This paper extended the observer design method from Mazenc et al. (2015a) to allow discrete measurements, model uncertainty, time varying coefficients, and sensor noise. We used a novel trajectory based approach that en- sured arbitrarily fast convergence of the observation error to zero, as long as the sampling in the output is frequent enough. Our trajectory based contractivity condition was used in place of standard Lyapunov function methods. We illustrated the value of our method in a pendulum example whose novel features included a time varying horizontal acceleration and a time varying friction. When model and measurement uncertainties are present, we provided a variant of input-to-state stability that quantifies the effects of the uncertainties in an overshoot term in our bound on the estimation error.

Our work was motivated by the increasing speeds of available sensors and DSPs, which call for studies of trade-offs between sampling rates in outputs measurements and convergence rates of observers, such as this work. Another motivation was the usefulness of observers in state feedback design, which would entail replacing state values in the stabilizing feedback controls by the estimated values x(t) from our estimator construction (11)-( 13); see, e.g., [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF], where such a replacement was done using finite time observers that required continuous unperturbed output measurements (instead of only the discrete perturbed output measurements as required here). We hope to develop analogs for systems governed by hyperbolic PDEs or reaction-diffusion equations, e.g., for the dynamics in Selivanov and Fridman (2019); [START_REF] Katz | Boundary delayed observer-controller design for reaction-diffusion systems[END_REF]

  Mazenc et al. (2020a) Even in the special case fromMazenc et al. (2020a) where A is constant, Theorem 1 is less restrictive than the main result fromMazenc et al. (2020a), because Mazenc et al. (2020a) required T λ(T , τ ) < 1 with the choices λ(T , τ ) = F(τ ) √ n 1 + |C|ϕe |A|(n-1)τ (T + (n-1)τ ) +|C|ϕ (24) and F(τ ) = |CAΨ|, where (24) was obtained using the conservative bounds |CA| ≤ |C||A| and |e M | ≤ e |M | for suitable square matrices M ; see Section 5.2 below. Therefore, rather than recovering the same performance as Mazenc et al. (2020a) in the special case where A is constant and

  ) in Assumption 4 and (Mazenc et al., 2017, Lemma 1) (which we also include in the appendix below) applied to the function w(t) = |e ω (t+T +(n-1)τ )| and the constant T * = T + (n -1)τ ensure that |e ω (t)| ≤ sup ∈[0,T +(n-1)τ ] |e ω ( )|e ln(T λ(τ ))

  2 and 4 by the two assumptions that we give next, where we continue the other notation from above. First, we replace our Assumption 2 by: Assumption 5 The function ϕ is continuous. Also, there is a constant ϕ ≥ 0 such that |ϕ(t, x a , d a ) -ϕ(t, x b , d b )| ≤ ϕ(|x a -x b | + |d a -d b |) holds for all x a , x b , d a , and d b in R n and all t ≥ 0.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Tracking Error x2(t) -x2(t) Converging to 0 over Time with J Values 1 (Red), 0.5 (Green), and 0.25 (Blue) with δ0 = 0 and δs = 0 after 4 Second Transient Period

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Tracking Error x2(t) -x2(t) over Time with J Values 1 (Red), 0.5 (Green), and 0.25 (Blue) with δ0(t) = 0.1 sin(t) and δs = 0 Converging to Oscillation after 4 Second Transient Period.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Tracking Error x2(t) -x2(t) over Time with J Values 1 (Red), 0.5 (Green), and 0.25 (Blue) with δ0(t) = 5[1 -0.98(20t/(1 + 20t))] and δs = 0 Zoomed to Transient Period.

  is a nonzero constant matrix, the matrix valued function A : R → R n×n is bounded and continuous, ϕ is a nonlinear function, the sampling times t i form an increasing sequence with t 0 = 0 and lim i→+∞ t i = +∞ and are not required to be evenly spaced, the locally bounded piecewise continuous function δ s represents model uncer-

tainty (and in particular can model uncertainties in the nonlinearity ϕ), and δ o is as above (but see Section 4 where the terms ϕ(Cx(t), t) + δ s (t) in (

  and model and sensor uncertainty, and including the new extension (in Section 4 below) that allows more general nonlinearities, and by providing a new application to a model of a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation and whose friction is timevarying. This contrasts withMazenc et al. (2020a), which did not consider uncertainties and required constant A's.

	On the other hand, even in the special case where A is
	constant and the uncertainties δ s and δ o are the zero func-
	tions, this paper improves the results derived in Mazenc
	et al. (2020a), by providing less restrictive conditions on
	the sample rates; see Section 2.3.3 below. Moreover, al-
	though the formula in our main trajectory based lemma
	shares features with exponential ISS, our trajectory based
	approach makes it possible to prove our novel variant of
	the ISS property as indicated above, where the supremum
	of the uncertainties is only over a recent history of uncer-
	tainty values. This sets our work apart from earlier works
	that did not use the trajectory based approach and so led
	to a classical ISS estimate that lacks this key fading effect
	of only depending on recent uncertainty values.
	We use the following notation. The dimensions of our Eu-
	clidean spaces are arbitrary, unless otherwise noted. The
	standard Euclidean 2-norm and the corresponding matrix
	norm are denoted by | • |, and I is the identity matrix. For
	any constant T > 0, C in is the set of all continuous func-
	tions

This paper improves on our conference paper Mazenc et al. (2020a) by allowing time-varying coefficients A,

  The observer calls for storing values of the product Φ A (t, s) = α A (t)β A (s) (but not the values of the factors α A and β A themselves) on an interval of length (n -1)τ . We found this storage to be feasible in practice, and we encountered no difficulties with divergence of α A or α B . However, in practice one could reinitialize the dynamics (23) periodically at times ki for integers i ≥ 0 and a large enough value k > 0. ] is in a sense arbitrary, since τ and T are chosen by the user. Other observer approaches for (3) (such as those of[START_REF] Andrieu | Observer design for Lipschitz systems with discrete-time measurements[END_REF],[START_REF] Dinh | Continuous-discrete time observer design for Lipschitz systems with sampled measurements[END_REF], andMazenc et al. (2015b)) would either use high gain or do not achieve the arbitrarily fast convergence property from Theorem 1. Also, since the suprema of the uncertainties δ s and δ o are only over recent histories of the state, it follows that all of the right side terms of (

	2.3.2 Discussion of (18)-(19)
	Although we can compute an upper bound for |ω(t)-Cx(t)|
	over the interval [0, T + (n -1)τ ] (which would depend on
	the initial conditions), we omit this upper bound compu-
	tation because we believe that it has no significant interest
	from a practical point of view. This is because [0, T + (n -
	1)τ

Then we can interconnect our observer design (with Φ A (t -jτ, t) in (

13

) and Ω replaced by α A (t -jτ )β A (t) for j = 1, . . . , n -1) with (

23

) to compute the required fundamental solution matri-ces.

Acknowledgements

The work of M. Malisoff was supported by US National Science Foundation Grant 1711299. We thank anonymous reviewers for their constructive comments that helped the authors to improve the paper.

Appendix: Statement of (Mazenc et al., 2017, Lemma 1) We provide a statement of (Mazenc et al., 2017, Lemma 1), which we used at the end of our proof of Theorem 1 above: holds for all t ≥ 0.


Fathia Ben Slama
email: fathia.benslama@altys-tech.net

Kanaan Abdo
email: kanaan.abdo@altys-tech.net

Eliot Vignaud
email: eliot.vignaud@altys-tech.net

Alexandre Simonin
email: alexandre.simonin@altys-tech.net

Elena Simona Lohan

Hadeel S Obaid

Taylan Yesilyurt

P A D Nirmal Jayawardana

Bo Tan
email: bo.tan@tuni.fi

Christophe Morlaas

Alexandre Duchevet
email: alexandre.duchevet@enac.fr

Mohamed Ellejmi
email: mohamed.ellejmi@eurocontrol.int

Use of 5G and mmWave radar for positioning, sensing, and line-of-sight detection in airport areas

Keywords: 5G, Angle of Arrival (AoA), Air Traffic Management (ATM), Artificial Intelligence (AI), Communications, Line of Sight (LOS) detection, Machine Learning (ML), Millimetre Wave, Positioning, Sensing, Surveillance, Time of Arrival (ToA), Vector Antenna (VA), Synthetic-aperture Radar

This paper explores innovative low-cost technologies, widely used outside of Air Traffic Management (ATM), for use in airport surface surveillance. These technologies consist of a 5G-signal-based surveillance solution and a millimeter wave (mmWave) radar augmented with artificial intelligence (AI). The 5G solution is based on the combination of 3D Vector Antenna, innovative signal processing techniques, and hybridization techniques based on time-of-arrival and angle-ofarrival estimates with uplink and downlink 5G signals, as well as Machine Learning (ML)-based Line of Sight (LOS) detection algorithms. The mmWave solution is based on mmWave radar for non-cooperative target's positioning and sensing, combined with deep learning for objects classification. Standalone 5G positioning accuracy reaches m-level accuracy in LOS scenarios and it is better with downlink reference signals than with uplink ones, while it deteriorates quite drastically in NLOS scenarios. LOS detection accuracies above 84% average accuracy can be achieved with ML. The mmWave radar is tested in different scenarios (short, medium and long range) and it provides cost-effective surface surveillance up to few hundred meters (depending on the object radar cross section RCS) with ±60°field of view. The work is being conducted within the H2020 European-funded project NewSense and it delves into the 5G, Vector Antennas, mmWave, and ML/AI capabilities for future ATM solutions.

I. INTRODUCTION AND MOTIVATION

The increasing growth in air traffic is putting pressure on airports for greater capacity and increased safety, which is pushing airports to be equipped with an advanced surface movement guidance and control system (A-SMGCS). While most of the world's largest airports have initiated the implementation of an A-SMGCS, small and medium-sized airports cannot afford such systems due to the high costs of the necessary infrastructure (multilateration deployment (MLAT) and surface movement radar (SMR)). Alternative low-cost solutions for airport surface surveillance such as cameras and LIDAR (Laser Imaging Detection And Ranging) have been assessed in SESAR 2020 Exploratory Research ENVISION project and SESAR 2020 Industrial Research project EARTH (solution PJ02.06) and results demonstrated that the optical sensors have limited performances in low visibility conditions that could be caused by rain, frog, snow, darkness, dazzling sunlight, and other meteorological constraints.

The potential of radio frequency (RF) signals and Single/Multiple Input Multiple Output (SIMO/MIMO) systems in measuring an object's position and velocity seamlessly and independently of visibility conditions have proved their efficiency in automotive and industrial applications [START_REF] Xiangyu | Ramp-cnn: A novel neural network for enhanced automotive radar object recognition[END_REF] [2] [START_REF] Siddharth | Target classification by mmwave fmcw radars using machine learning on range-angle images[END_REF] [4] [START_REF]positioning and sensing for vehicular safety applications in 5g and beyond[END_REF]. In one hand 5G networks, already in commercial phase and expected to be large-scale deployed in the next decadesthus offering the opportunity to use 5G network infrastructure -, in the other hand the increasing use of mmwave radar for automotive and industrial applications, and the recent interest in applying deep learning algorithms to radar data should be considered when looking for alternative affordable surveillance solutions for the future ATM. Thus, the NewSense project proposes innovative surveillance solutions to improve airport surface surveillance and situational awareness in all weather and visibility conditions. These positioning and sensing solutions, completely new for ATM context, are based on timing and angle measurements from 5G RF signals, mmWave radar augmented with artificial intelligence (AI), and 5G RF-based radar as illustrated in Fig. 1. The novel work done in the scope of this research includes:

• The design of a 5G-signal-based surveillance solution for the airport surface including a 3D vector antenna (3D VA), a 5G positioning function to measure cooperative targets position using their transmitted 5G RF signals, and a MLbased LOS detection mechanism • The design of a radar-like system relying on 5G signals to calculate non-cooperative targets position from reflected 5G base station RF signals.

• The evaluation of a low-cost mmWave radar as a noncooperative surveillance solution to position and classify targets using reflected mmWave radar signals and deep learning methods. This paper is elaborated in continuity of the previous work detailed in [START_REF] Bo | Improved sensing and positioning via 5g and mmwave radar for airport surveillance[END_REF], in which we described in the 5G framework the working principle of the 3D-VA, the 5G reference signals, the angle-delay estimation algorithms, and the imaging algorithms. In the mmWave radar framework, we described FMCW mmWave radar working principle and the signal processing chain applied to mmWave signals. We provided also the preliminary results for both frameworks.

In this paper, we will describe the implemented channels models with 5G signal and the obtained results in Section II. The simulations with 3D-VA are provided in Section III. In Section IV, mmWave radar setup, mmWave measurements scenarios, ML algorithms and obtained results are described. In section V LOS detection with 5G signals is explained.The measurement setup using 5G signals and the 3D-VA, and the corresponding results are provided in Section VI At last, the conclusion and future work are given in the Section VII.

II. SIMULATION TOOLBOXES FOR 5G SIGNALS

The main idea of 5G-signal-based positioning algorithms is exploiting the always-on reference signals of 5G NR to estimate the position of user equipment (UE). Our previous work in [START_REF] Bo | Improved sensing and positioning via 5g and mmwave radar for airport surveillance[END_REF] has introduced the 5G reference signals used in positioning and the time-based and angle-based estimators. We have now developed an in-house Matlab-based simulator for uplink and downlink 5G signals that is based on Matlab 5G toolbox and covering three types of positioning-related reference signals, namely Sounding Reference Signals (SRS) for uplink (UL) and Positioning Reference Signals (PRS) and Channel State Information Reference Signals (CSI-RS) for downlink (DL). Our simulator also comprises four types of channel models: the Additive White Gaussian Channel (AWGN) model taken as benchmark, the 3GPP Tapped Delay Line (TDL) and Cluster Delay Line (CDL) models, and the Matlab WINNER II channel model. AoA-based and ToAbased estimators for the three types of reference signals and a selected NLOS channel (CDL-A channel) are shown in Fig. 2. Similar behaviour was observed for LOS channels. Clearly, time-based estimates are much more sensitive to the type of the reference signal than angle-based estimates, with PRS providing the best performance and CSI-RS providing the worst performance. AoA-based estimates are also much more accurate than our ToA-based estimates, no doubt due to the fact that a high-resolution MUSIC algorithm was used for AoA, while a simple correlation-based estimator was used for ToA results. Fig. 3 also shows the positioning error performance under five channel types: AWGN, two LOS channels (TDL and CDL), and two NLOS channels (TDL and CDL). The left-hand plots are based on AoA estimators and the right-hand plots on ToA estimator. Clearly, the performance in AWGN and LOS channels is significantly better than in NLOS channels. The saturation point in AWGN channels and ToA-based estimation is due to the sampling frequency that defines the maximum resolution in time domain; we used a sampling frequency of 15.36 MHz, which limits the time-domain resolution to 19.5 m. Higher time resolutions can be achieved with higher sampling frequency, at the expense of a higher complexity and longer simulation times. WINNER II channels are the most challenging for a ToA-based estimator, followed by TDL channels, while TDL channels are the most challenging for AoA estimators, followed by CDL channels. AoA-based estimators also have better results than ToA-based estimators for all the considered wireless channels.

III. SIMULATIONS WITH 3D VECTOR ANTENNA

A. Vector antenna description

The 3D-VA is constituted from an arrangement of Vivaldi antennas. it is particularly interesting for this application since it exhibits a directional radiation pattern with a stable gain over a wide bandwidth and beamwidth. In addition to that, it presents a good polarization discrimination allowing DoA capability with physical characteristics of low-profile, lightweight, and low-cost.

The proposed 8-port vector antenna is here supposed to retrieve independently the six components (E x , E y , E z , H x , H y , H z ) of an incoming EM wave in the entire 3-D space. The corresponding sets of weighting coefficients and EM components are reported in Table I. The corresponding set of six radiation patterns will be called RPC0. In addition to that, additional set of radiation patterns (RPCs) can be used in order to improve the DoA performances. The additional weighting coefficient used to define these additional RPC's are obtained from a parametric analysis [START_REF] Duplouy | Wideband and reconfigurable vector antenna using radiation pattern diversity for 3-d direction-of-arrival estimation[END_REF]. We will focus particularly in two RPCs called RPC1 and RPC2 that consists of a set of different weighting coefficients that provide same radiation patterns with a rotation of 90 o along the azimuth angle ϕ. The RPC1 is chosen to improve the vertical polarized DoA while the RPC2 to improve the horizontal polarized DOA. Only port 1 to 4 are used for RPC1 while Port 5 to 8 are used for RPC2.

0 0 0 0 1 1 -1 -1 0 0 0 0 1 -1 -1 1 1 1 1 1 0 0 0 0 0 -1 0 1 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 1 -1 1 -1 1 1 1 1 -1 0 0 0 0 1 1 -1 1 0 0 0 0 1 -1 1 1 0 0 0 0 -1 1 1 1 0 0 0 0 2 0 0 0 0 1 1 1 -1 0 0 0 0 1 1 -1 1 0 0 0 0 1 -1 1 1 0 0 0 0 -1 1 1 1

B. Simulation experiments

In this section, the DoA arrival accuracy is assessed at 3.5 GHz according to the RPCs used. The AoA estimation accuracy refers to the angular error between the estimated and expected angles of an incoming EM wave. The AoA accuracy depends on the signal to noise ratio which is defined in the MUSIC algorithm and the propagation effects of the corresponding EM wave. This parameter is usually given by Root-Mean-Square (RMS) values. More specifically, the AoA angular accuracy may be defined with

ϵ RM S (Ω) = 1 L L i=1 |Ω -Ω| 2 (1)
where Ω is the expected angle (along the azimuth ϕ or elevation angle θ), Ω is the estimated angle, and L is the number of trials.

On the other hand, the angular distance ∆ α can be defined as the minimal distance between the expected angle set (θ, ϕ) and estimated direction ( θ, φ) given by

∆ α RM S = 1 L L i=1 |∆ α (θ, ϕ)| 2 (2)
with,

∆ α (θ, ϕ) = cos -1 (cos θ cos θ + sin θ sin θ cos (ϕ -φ)) (3)
Table II shows the conditions applied for the scenario under test in order to compute the angular distance associated with the θ, and ϕ angles of the proposed VA. The figure 4 depicts the simulated angular distance RMS error for θ, and ϕ and the angular distance (∆ α RM S ) using the MUSIC algorithm for an azimuth angular range of 360 • and an elevation angular range of 120 • i.e. -30 • toward the bottom direction. As a recall the 0 • elevation angle is toward the zenith. The six components derived from the proposed VA (RPC0) are considered only. The maximal error is around 1.5 • for elevation angle in a range of 80 • to 1000 • and around 30 • . The ∆ α RM S error is due to a mixed error on the ϕ and θ angle error.

In figure 5 the six components derived from the proposed VA (RPC0) and the additional RPC's 1 and 2 are considered. The error for both angles is reduced. The VA is now capable to accurately estimate sources in the bottom half-space until -30 • . A small error persist around 30 • and 100 • for elevation angle.

IV. MMWAVE RADAR AUGMENTED WITH AI

In addition to the assessment of a 5G-signal based positioning solution, the objective in this study is to localize noncooperative targets on the airport surface and identify their type (mainly aircraft, vehicle and person) using Frequency Modulated Continuous Wave (FMCW) mmWave radar and AI. Understanding the technology concept of this type of radar is essential to properly configure the mmWave sensor and optimize its use. The mmWave technology description, the operational principles of an FMCW mmWave radar and the data processing chain to detect and calculate the target's range, angle and velocity have been detailed in [START_REF] Bo | Improved sensing and positioning via 5g and mmwave radar for airport surveillance[END_REF].

A. mmWave radar configuration

Three scenarios have been used to configure the radar and the parameters have been computed in order to have the best precision depending on the targeted area. On the airport surface, we distinguish three areas that should be covered by the radar: the parking, the taxiway and the runway. The first scenario is the Long-Range Radar (LRR) that will cover a section of the runway. The second scenario is the Medium Range-Radar (MRR) that will cover the taxiway and the parking area. Finally, the Short Range-Radar will cover mainly the parking area. Configuration parameters and corresponding range and velocity values are given in Table III.

B. Testbed description

The radar testbed is composed of the mmWave radar AWR2243BOOST EVM [START_REF]Awr2243 evaluation module (awr2243boost) mmwave sensing solution, user's guide[END_REF] combined with the data capture module DCA1000EVM [START_REF]Dca1000evm data capture card, user's guide[END_REF] mounted on a tripod. A host computer is used to record received data from the DCA1000EVM for post processing purposes. A camera is also used in order to keep the ground truth image corresponding to radar data. Finally, we use a portable battery to power the devices as most of the testing is done outdoors.

C. Positioning results

Several measurements with the radar have been conducted in laboratory, underground and external parkings with persons and vehicles, and in real air-side environment (Muret-Lherm Aerodrome, France). Measurements done in external parking and in airfield are very interesting for the study as they make possible to assess the sensor in an environment very close to the targeted deployment environment. Measurements for SRR, MRR and LRR scenarios have been conducted in an external parking for moving person and vehicle. These scenarios have been also tested at Muret airfield for persons in the parking area and for General Aviation (GA) aircraft in the parking, taxiway, and a segment of the runway.

The radar is able to detect a person to up to 66 meters, a vehicle to up to 113 meters and a GA aircraft to up to 83 meters. A truck has also been detected at a distance of 138 meters. Some measurements results are shown in Fig. 6. Within the same configuration (transmit power, SNR, RX and TX antenna gain) and as commercial aircraft have an higher Radar Cross Section (RCS) than targets used within this study, we could estimate that the radar may detect commercial aircraft up to few hundred meters. The maximum range of the radar could be increased if the radar transmit power increases. As examples, the maximum transmit power of an ATC primary surveillance radar (PSR) is 100 kWatt [START_REF]Primary surveillance radar (psr); harmonised standard for access to radio spectrum; part 2: Air traffic control (atc) psr sensors operating in the frequency band 2 700 mhz to 3 100 mhz (s band)[END_REF] and typical Surface Movement Radar has a transmitting power superior than 180 Watt. In the scope of this study we used an evaluation module for the mmWave radar with a transmit power up to 0.016 Watt.

The range resolution obtained by the mmWave radar makes possible to distinguish two close objects. Fig. 6 shows a moving person detected near the aircraft and the radar was capable to distinguish it from the aircraft at a distance less than 1 meter. 1) Classification using the Micro-Doppler Signature: A Micro-Doppler Signature of a target results from the additional motions of the target (rotation, vibration, etc.), other than main mass translation, which generates a time varying frequency modulation imparted in the echo. This signature contains target's features and therefore can be used for classification. The Micro-Doppler is calculated using the Short-Time Fourier Transform (STFT) of the signal during a short laps of time (in our case, 1 second). Fig. 7 shows examples of STFT heatmaps (in the third column). STFT heatmaps contain information about how the object is moving as a whole, and how individual part within the object are moving. For example, with a propeller plane, the STFT heatmap will display both the Doppler shift of the moving plane, as well as the Micro-Doppler shifts of the propeller. For a moving person, it shows his velocity and also the Micro-Doppler of his arms and legs.

We chose EfficientNetV2 [START_REF] Mingxing | Efficientnetv2: Smaller models and faster training[END_REF] as the model for the classification for 3 main reasons:

• At the time of the model selection, EfficientNetV2 was amongst the best neural network on several image classification benchmarks.

•

As discussed earlier, our dataset will be smaller than images dataset. Moreover, each sample is longer to process, since the dimension of the last channel is greater than the traditional 3 for Red Green Blue images. EfficientNetV2 was designed to have less weights (thus needing less data) and shorter training time than other state-of-the-art models.

• EfficientNetV2 is widely used, which makes it accessible with pre-trained weights in several libraries. This means that accuracy alone cannot be used to evaluate different models, because a model that always answer "aircraft" would have an accuracy of 83% in our case, while giving nothing of value. To ensure that our model does not act that way, we train it with a weighted loss function. This means that misclassifications during training are penalized more the less a class is present in the dataset. We trained several models while varying the degree to which the minority classes were favorized, and even in the most favorable case, planes were detected correctly with an accuracy of 93,6%, vehicle with an accuracy of 61,5% and pedestrian with an accuracy of 34,0%.

We see two solutions as the most promising to improve the model further. The first one simply consists of gathering more data, especially for vehicles and pedestrians. The second is to use data augmentation, to create new examples from our current ones. It could be done at two levels: at the STFT heatmaps level or directly at the raw radar data level. Radar data augmentation is a less studied field than image data augmentation, but new research tries to tackle this issue [START_REF] Gao | Ramp-cnn: A novel neural network for enhanced automotive radar object recognition[END_REF]. In order to ameliorate classification results with Micro-Doppler data, future work will consist of creating new STFT heatmaps from our current radar data by simply shifting the time window used to generate the heatmaps.

2) Classification using the Range-Angle heatmap: The previous method was showing a lot of promise, but the reliance on the Doppler effect made it unusable when a target was not moving at all. For that reason, we tried an alternate method, based on the Range-Angle heatmap. The proposed method would use image detection (i.e., means classification and localization of the object of interest in the image) instead of image classification. The image detection method used is YOLOv4 [START_REF] Bochkovskiy | Yolov4: Optimal speed and accuracy of object detection[END_REF]. Like EfficientNetV2, YOLOv4 was chosen for his performance on image detection benchmark and for his wide use in the industry. At the beginning, we tried to train the model using the Range-Angle heatmap obtained using the 3FFT [START_REF] Bo | Improved sensing and positioning via 5g and mmwave radar for airport surveillance[END_REF]. However, the high-level of noise and the low resolution in these heatmaps make it hard to create the image that YOLO would be used on. We have tried then another direction-of-arrival estimation method: the highresolution MUSIC algorithm. Thus, we have obtained better results for the Range-Angle heatmaps as illustrated in Fig. 8. Future work will be to train the model and evalutate the performance of this model in objects' detection. The accuracy of positioning is highly deteriorating in the presence of NLOS, therefore the classification into LOS and NLOS scenarios is a fundamental block towards a comprehensive 5G-based positioning solution. There are typically two main classes of LOS detection algorithm: one based on signal processing models, where different characteristics of the received signal are extracted from measurement data and compared to a threshold, and another one based on ML, and relying on some training phase, where relevant features are extracted from the available data and then ML algorithms are applied to classify the data based on those features. Examples from the first category are: kurtosis-based detection [START_REF] Zhang | Analysis of kurtosis-based los/nlos identification using indoor mimo channel measurement[END_REF], [START_REF] Xie | Identification of nlos condition in different scenarios based on wireless propagation features[END_REF], cross-polarization discrimination [START_REF] Morita | Identification of line of sight by cross polarization characteristic[END_REF], channel impulse response entropy [START_REF] Alsindi | Entropy-based non-line of sight identification for wireless positioning systems[END_REF], Rician-factor-based detection [START_REF] Xie | Identification of nlos condition in different scenarios based on wireless propagation features[END_REF] or time-space-frequency correlation [START_REF] Li | Channel correlation based identification of los and nlos in 3d massive mimo systems[END_REF]. Examples from the second, ML-based, category for LOS detection can be found in [START_REF] Bregar | Improving indoor localization using convolutional neural networks on computationally restricted devices[END_REF]- [START_REF] Flueratoru | Self-learning detection and mitigation of non-line-of-sight measurements in ultra-wideband localization[END_REF]. ML-based algorithms are typically understood to exhibit better performance than the model-based algorithms and they have been poorly studied so far in the context of 5G signals. Therefore, our focus has been on ML-based LOS-detection, by using our in-house 5G signal simulator (Matlab-based) and two classes of wireless channels: Tapped Delay Lines (TDL) and Clustered Delay Lines (CDL) channel models. Each class had five sub-categories (denoted A to E), with categories A-C modeling NLOS cases and categories D-E modeling LOS cases. More details about TDL and CDL channels can be found, for example, in [START_REF] Barb | On the influence of delay spread in tdl and cdl channel models for downlink 5g mimo systems[END_REF] and they are widely known in the 5G research community. In our analysis, 1000 random samples were generated per channel category (total 5000 samples); 80% of data was used for training and 20% of data for testing. We studied both downlink and uplink configurations, but since the obtained results were similar, for clarity purpose we only show here the downlink results based on PRS reference signals. An example of channel impulse response (CIR) obtained from the correlations with PRS downlink signals under LOS and NLOS scenarios is shown in Fig. 9 for CDL channels. The results in Table IV show that time-domain features are the strongest among the other investigated features and combining several features together cannot surpass the performance with a single feature only when time-domain correlation is used as the single feature. In addition, LOS detection probabilities were slightly better for CDL channels than for TDL channels, no doubt due to the higher richness of timedomain features in a CDL compared to TDL. Different ML classifiers have also been investigated and an example of results is shown in Table V for five selected classifiers and time-domain features. As the selected classifiers are wellknown from the ML literature, their exact description is not added here for lack of space. The SVM classifier gave the best results, followed by XGboost and NN classifiers. Again, the results based on CDL channels were slightly better than with TDL channel. Snapshot examples of AoA estimates under static and moving scenarios are depicted in Fig. 13 and Fig. 14, respectively and show perfect estimates under noiseless scenarios. Part of our AoA measurements with this setup are now available in open source at [START_REF] Obaid | Angle measurements with 3d vector antenna for localization purposes -openaccess datasets[END_REF]. The LOS detection algorithm testing via the in-lab set-up is still on-going; part of the measurements have already been collected and an example of correlation outputs under LOS and NLOS scenarios with SRS 5G signals is given in Fig. 15. Compared to the theoretical LOS/NLOS models of Fig. 9, the measured NLOS channel has a lower amount of multipath; the noise floor is higher in LOS compared with LOS case in both simulated and measured data.

VII. CONCLUSION

In this paper, we demonstrate that positioning and sensing using angle and/or timing information combined with 5G reference signals and mmWave radar are promising complementary tools in future airport surveillance. The performances of 3D-VA have been evaluated in the context of AoA estimation. We have addressed the problem of LOS detection through ML algorithms. The testbed setup with the 3D-VA was described. Current 5G-based results show that a receiver can achieve promising accuracy (order of few meters positioning error and close to zero direction-estimation errors with suitable antennas) when receiver is in LOS connection with at least 3 base stations. We have also shown that NLOS scenarios can be detected wit accuracies as high as 90% with machine learning algorithms if enough training data is available. In order to further improve the positioning and direction-estimation results also in NLOS scenarios, the 5Gbased surveillance solution, encompassing positioning, LOS detection, and radar-like sensing is to be further investigated through hybridization with additional sensors and possible map information. Measurements using the FMCW mmWave radar and obtained results prove the potential of using this type of radar for airport surveillance. We have assessed the mmWave radar performances in terms of range, angle and velocity calculation. We have also applied Machine Learning methods to classify targets and we have obtained an accuracy up to 90%. We encountered some constraints during the measurements such as the test environment not large enough (e.g. in external parking, the maximum depth is 130 meters), no control on the target presence and trajectory (e.g. for aircraft at Muret airfield which were sometimes out of reach of the radar) and limited collected data as outdoor measurements have been done during 3 days (1 day in external parking and 2 days at Muret airfield). This is different from measurement campaigns when the sensor is installed 24/7. Thus, we assume that positioning results could be ameliorated in more favourable testing environment, especially if the radar is adapted and trained for a specific airport. Future work will consist of a) completing the 5G-signal-based solution measurements with the 3D-VA and investigating MUSIC AoA estimates under various transmit powers and noise conditions, b) enhancing the ML applied to mmWave data, c) evaluating the system performance against equivalent A-SMGCS standards. This work will also feed the EUROCONTROL A-SMGCS Task force and the EUROCAE Working Group 41 on integrating new technical solutions into the A-SMGCS surveillance service and its performance framework.

ACKNOWLEDGMENT

This work was supported by SESAR Joint Undertaking (SJU) in project NewSense (grant 893917). The opinions expressed herein reflect the authors' view only. Under no circumstances shall the SJU be responsible for any use that may be made of the information contained herein. This work was also partly supported by the Academy of Finland, under the projects ULTRA (328226) and ACCESS (339519).

••Figure 1 .

 1 Figure 1. Illustration of the integrated-concept architecture for airport surveillance • The validation through in-lab measurements of some of the proposed solution. Part of our measuremet data has also been made available in open-access on Zenodo repository [6].This paper is elaborated in continuity of the previous work detailed in[START_REF] Bo | Improved sensing and positioning via 5g and mmwave radar for airport surveillance[END_REF], in which we described in the 5G framework the working principle of the 3D-VA, the 5G reference signals, the angle-delay estimation algorithms, and the imaging algorithms. In the mmWave radar framework, we described FMCW mmWave radar working principle and the signal processing chain applied to mmWave signals. We provided also the preliminary results for both frameworks.In this paper, we will describe the implemented channels models with 5G signal and the obtained results in Section II. The simulations with 3D-VA are provided in Section III. In Section IV, mmWave radar setup, mmWave measurements scenarios, ML algorithms and obtained results are described. In section V LOS detection with 5G signals is explained.The measurement setup using 5G signals and the 3D-VA, and the corresponding results are provided in Section VI At last, the conclusion and future work are given in the Section VII.

Figure 2 .Figure 3 .

 23 Figure 2. Root Mean Square positioning Errors (RMSE) for CSI-RS, SRS, and PRS signals in a a CDL NLOS channel. Left: AoA estimator; right: ToA estimator.

Figure 4 .

 4 Figure 4. Angular distance, θ, and ϕ error for SN R = 9 dB and with RPC0 used.

Figure 5 .

 5 Figure 5. Angular distance, θ, and ϕ error for SN R = 9 dB and with RPC0, RPC1, and RPC2 used.

Figure 6 .

 6 Figure 6. mmWave radar positioning results, no tracking filter is applied

Figure 7 .

 7 Figure 7. Camera image, Range-Velocity heatmap and STFT heatmap of person, a car, and GA aircraft

Figure 8 .

 8 Figure 8. Examples of Range-Angle images on the left and zoom on targeted object on the right

Figure 9 .

 9 Figure 9. Examples of correlation outputs under LOS and NLOS scenarios with CDL channels (different curves show different realizations of the channel). SNR= 0 dB; PRS downlink signal. Based on the literature search, several features were investigated for use with ML, namely time-domain correlation, frequency-domain spectrum, and kurtosis of the received signal. Results comparing the LOS detection probability under different features are shown in Table IV with CDL and TDL channels and with Support Vector Machine (SVM) classifier with medium Gaussian kernel.

Figure 10 .

 10 Figure 10. The hardware configuration schematic.

  VI. MEASUREMENTS WITH 3D VECTOR ANTENNA The in-lab validation of the 5G-based algorithms has been done based on the block diagram in Fig. 10. The in-lab set-up comprised one PC (Intel i7-8700, with CPU at 3.20 GHz, 32 GB RAM, and 1TB SSD) to host the LabVIEW to run the AoA and time delay estimation algorithm; one cabled PCI Express Switch Box x4 10 Port ( CPS-8910) to connect the three USRPs to the PC's PCIe adapter; one PCIe adapter used in motherboard-level connections as an expansion card interface; several SMA cables for 10 MHz clock and PPS distribution for synchronization and another group for RF signal connection; a four-Way DC Pass Power Splitter (ZN4PD-272-S+) in the frequency range 500 -2700 MHz at 50Ω to deliver multiple output signals with specific output phase and amplitude characteristics; three RIO USRPs (two NI USRP-2954R for receiving signals and one NI USRP-2953R for transmitting signal; a 3D vector antenna connection with 8 ports, among which 4 electric ports were used connect to the two receiver USRPs; one OctoClock CDA-2990 with 10MHz reference clock and PPS output to build multi-channel systems that are synchronized to a common timing source; and a GPS antenna to drive the external clock of the OctoClock. Mesaurements were performed at 2.1 GHz carrier frequency.Several transmitter antennas have been used (e.g., monopole, Yagi); an example of the in-lab set-up with a Yagi antenna with 5dBi antenna gain and a frequency range between 700MHz and 6GHz is shown in Fig.11. Data was collected under both LOS and NLOS scenarios. The obstacle used for NLOS measurements is also illustrated in the lower plot of Fig.11.

Figure 11 .

 11 Figure 11. lEFT plot: Lab picture with the Yagi antenna at the TX and the 3D VA at the RX. Left: LOS scenario; right plot: Obstacle used for NLOS scenario in the wave path.The AoA algorithm has been implemented in LabVIEW 2021, as illustrated in Fig.12.

Figure 12 .

 12 Figure 12. 3D-VA MUSIC Algorithm steps in LabVIEW.

Figure 13 .Figure 14 .

 1314 Figure 13. AoA estimation with 3D-VA at 100°.

Figure 15 .

 15 Figure 15. Examples of LOS and NLOS correlation outputs based on measured data with 5G SRS signals and 3D VA.

TABLE I .

 I SET OF WEIGHTING COEFFICIENTS ASSIGNED TO THE SIGNALS RECEIVED AT THE EIGHT ANTENNA PORTS FOR SYNTHESIZE THE SIX MAIN RADIATION PATTERNS AND ADDITIONAL RPCS RPC Port 1 Port 2 Port 3 Port 4 Port 5 Port 6 Port 7 Port 8 0

TABLE II .

 II PARAMETERS USED FOR ANGULAR DISTANCE ESTIMATION

	Number of incoming EM fields	1
	Polarization of the incoming EM field	Vertical
	Angular coverage ϕ ∈ [0 Angular resolution ∆ϕ = 5 • ∆θ = 5 •
	Incoming EM field power density	-91 dBW.m -2
	Noise power level	-111 dBm
	Mean SNR	9 dB
	Number of DoA estimation	20
	Snapshots per DoA estimation	50
	Frequencies of interest	3.5 GHz

• ; 360 • ] θ ∈ [0 • ; 120 • ]

TABLE III .

 III AWR2243[START_REF]Awr2243 evaluation module (awr2243boost) mmwave sensing solution, user's guide[END_REF] CONFIGURATION FOR LRR, MRR AND SRR

	SCENARIOS
	Parameter	LRR	MRR	SRR
	Idle time/Inter-chirp duration (µs)	2.5	3	3
	Number of ADC samples	128	256	256
	ADC sampling frequency (Msps)	22.50	14.40	7.00
	Chirp duration active/Ramp end time (µs) 9	21	40
	Number of chirps per profile	255	128	128
	Frequency slope (MHz/µs)	10.042 14.001 15.015
	Number of RX channels	4	4	4
	Number of TX channels	2	2	2
	Frequency Limit Low (GHz)	79.50	79.50	79.50
	Maximum theorical range (m)	302.27 138.75 62.89
	Range resolution (m)	2.624	0.602	0.273
	Maximum velocity (Km/h)	147.51 70.60	39.33
	Velocity resolution (m/s)	0.321	0.306	0.171

TABLE IV .

 IV LOS DETECTION PROBABILITIES WITH SVM CLASSIFIER AND

	DIFFERENT NUMBER OF FEATURES
	Features	TDL CDL
	1 feature (time)	84%	90%
	1 feature (frequency)	77%	85%
	1 feature (kurtosis)	69%	71%
	2 features (time + frequency)	83%	88%
	2 features (time + kurtosis)	78%	86%
	2 features (frequency+kurtosis)	83%	86%
	3 features (time + frequency + kurtosis) 84%	88%

TABLE V .

 V LOS DETECTION PROBABILITIES WITH SVM CLASSIFIER AND DIFFERENT NUMBER OF FEATURES

	Classifiers	TDL CDL
	SVM, medium Gaussian	84%	90%
	XGboost	80%	89%
	Random Forest	79%	87%
	Neural Networks (NN)	81%	85%
	Cosine Nearest Neighbours 79%	85%


Nathalie Belhoste

Rachel Bocquet

Véronique Favre-Bonté

Export support services and SME internationalization process

INTRODUCTION

Small and medium-size enterprises (SMEs) are key to the competiveness of advanced industrialized countries. Their growth is driven by many factors among which internationalization plays a significant role [START_REF] Wright | Joining the dots: Building the evidence base for SMEs growth policy[END_REF][START_REF] Love | SME innovation, exporting and growth: A review of existing evidence[END_REF]. International business and management studies frequently investigate SMEs' internationalization [START_REF] Dominguez | Internationalization stages of traditional SMEs: increasing, decreasing and re-increasing commitment to foreign markets[END_REF][START_REF] Johanson | The Uppsala internationalization process model revisited: from liability of foreignness to liability of outsidership[END_REF][START_REF] Johanson | The internationalization of the firmfour Swedish cases[END_REF][START_REF] Vahlne | The Uppsala model on evolution of the multinational business enterprise-from internalization to coordination of networks[END_REF], a complicated process that demands significant resources, especially if the goal is to reach distant markets [START_REF] Ojala | Internationalization of knowledge-intensive SMEs: The role of network relationships in the entry to a psychically distant market[END_REF][START_REF] Zhang | What drives the internationalisation of Chinese SMEs? The joint effects of international entrepreneurship characteristics, network ties, and firm ownership[END_REF]. Among these resources, those from public or private support services (i.e. specialized organizations or actors that help companies expand internationally by providing advice, contacts, training or support) have received increasing attention in the literature. However, results are still contradictory or contrasted, some consider that their efficiency on the internationalization performance is real [START_REF] Leonidou | National export promotion programs as drivers of organizational resources and capabilities: effect on strategy, competitive advantage and performance[END_REF][START_REF] Gençtürk | The effect of export assistance program usage on export performance: A contingency explanation[END_REF]Sousa and Brad-ley, 2009;[START_REF] Brouthers | Trade promotion and SME export performance[END_REF] while others show that their effect remains limited (Ramsden and Bennet, 2005;[START_REF] Francis | Impact of export promotion programs on firm competencies, strategies and performance: The case of Canadian high-tech SMEs[END_REF].

To clarify how public and private support services might encourage internationalization efforts by SMEs, we propose that specifying the type and stages of SMEs' internationalization processes might be insightful. We thus consider that the way SMEs use support services regarding their specific processes (determined by their type and stage) is of utmost importance because it adds a complementary perspective to the debate regarding the efficiency of these services: this efficiency is not only intrinsic to the support service itself but might be enhanced if it is developed or used in conjunction with the process undertaken by the SME.

SMES' INTERNATIONALIZATION SUPPORT SERVICES: WHAT WE KNOW SO FAR

We define internationalization support services as specialized organizations or actors that help companies expand internationally by providing advice, contacts, training or support [START_REF] Freixanet | Export promotion programs: Their impact on companies' internationalization performance and competitiveness[END_REF][START_REF] Holanda Schmidt | Export assistance: A literature review and challenges for future research[END_REF]. They may be organized by the State (such as national export promotion programs or assistance, Chambers of commerce services…) or by private actors, and their focus moves beyond solely export services to include overall internationalization processes. Many SMEs rely on these actors to facilitate or perform their sales, logistics, marketing, and service activities [START_REF] Balabanis | Factors affecting export intermediaries' service offerings: The British example[END_REF][START_REF] Peng | Export intermediary firms: A note on export development research[END_REF], because the support service providers have superior local market knowledge or connections with relevant intermediaries. While the usefulness of export promotion programs or export performance assistance has been widely debated (for an updated review, see [START_REF] Coudounaris | Export promotion programmes for assisting SMEs[END_REF], the results remain contradictory or contrasted. Moreover, the question of the use of these services during the internationalization process is still an issue.

The usefulness and the use of internationalization support services

On a general basis, studies focus more frequently on the frequency of use than on the use itself [START_REF] Ayob | Insight into public export promotion programs in an emerging economy: the case of Malaysian SMEs[END_REF]. To describe how firms use these programs, most studies address the degree of adaptation to the firm needs [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF][START_REF] Naidu | Public sector promotion of exports: A need-based approach[END_REF] or specific firm characteristics, such as its international experience (from non-exporter to advanced exporter) or size [START_REF] Buckley | Government-industry relations in exporting: Lessons from the United Kingdom[END_REF][START_REF] Freixanet | Export promotion programs: Their impact on companies' internationalization performance and competitiveness[END_REF]. For example, [START_REF] Buckley | Government-industry relations in exporting: Lessons from the United Kingdom[END_REF] and [START_REF] Sbrana | Italian exporting SMFs and their use of support services[END_REF] show that SMEs use support services less than larger firms. However, given their budget constraints, they are more dependent than large firms on the information generated by business associations [START_REF] Gashi | Export behavior of SMEs in transition countries[END_REF]. [START_REF] Kedia | Factors inhibiting export performance of firms: An empirical investigation[END_REF] and [START_REF] Ahmed | Export promotion programs of Malaysian firms: International marketing perspective[END_REF] assert that more experienced companies are more likely to be clients of international support service providers. [START_REF] Koksal | Organizational and exporting determinants affecting export promotion program awareness, utilization, and usefulness level[END_REF] combines these aspects to argue that larger, more experienced companies are more familiar with the various support services available from public authorities, so they can better understand and rely on the assistance options that suit their needs.

Experience abroad and the stage of export development also might determine the link between support service use and export performance, though prior research offers some conflicting results. [START_REF] Denis | Market knowledge, diversification, and export expansion[END_REF] and [START_REF] Seringhaus | The use of trade missions in foreign market entry[END_REF] indicate that for the most internationally experienced companies, export services mostly support their pre-export activities. [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF] instead finds that the most experienced firms are more aware of these programs, so they use them more. In general, firms with different degrees of internationalization expertise likely have different needs [START_REF] Diamantopoulos | Understanding the role of export marketing assistance: empirical evidence and research needs[END_REF][START_REF] Spence | Evaluating export promotion programmes: U.K overseas trade missions and export performance[END_REF]. [START_REF] Singer | Factors associated with effective use of export assistance[END_REF] use export stage as an antecedent to understand the relation between service use and export outcomes. According to their findings, the type of service/product sold and the company's experience abroad do not strongly influence its use of support services, whereas management commitment does. [START_REF] Czinkota | Why national export promotions? International trade forum[END_REF] also shows that firms have a greater need of export training services in their early export phase while they need more support for sales logistics when they gain experience and reach a most advance export level. [START_REF] Costa | Institutional networks for supporting the internationalisation of SMEs: the case of industrial business associations[END_REF] explain that most industrial business associations (such as Chambers of commerce) seem to provide a continuous support for the international operations of SMEs and not only for the initial stages. [START_REF] Freixanet | Export promotion programs: Their impact on companies' internationalization performance and competitiveness[END_REF] instead shows that the level of knowledge and uses of various programs vary with the type of assistance offered and the firm's international experience; a proposed segmentation defines the firm's level of internationalization involvement and skills using criteria such as export volume, size of the export department, creation of permanent establishments abroad, and creation of production subsidiaries.

Type and variety of internationalization support services.

Few of these studies take an in-depth look at the nature of the assistance though, or else they focus on only one type. For example, [START_REF] Seringhaus | The use of trade missions in foreign market entry[END_REF] describes trade missions that support foreign market entry. Observing the specific role of chambers of commerce, Leonidou and Theodosiou (2004) show that they provide first-hand information but that this information is not regarded as useful for firms. When gathering information for internationalization, Leonidou and Katsikeas (1997) observed that companies relied more on their personal network than on this institutional network. [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF] details the kind of assistance requested, such as the need for information or financial support, and Freixanet (2012) uses precise criteria to identify the nature of assistance programs and proposes measuring performance according to firms' international experience. Furthermore, we note that most internationalization support services and programs are developed by state agencies, but not exclusively. Studying the direct and indirect effects (via geographic scope) of public support on the export intensity of European SMEs, [START_REF] Ciszewska-Mlinaric | Export intensity, geographic diversification and the role of public support: The evidence from old and new Europe SMEs[END_REF] shows that only financial support (vs. non-financial one) is positively, directly and indirectly associated with their export intensity. Beyond public actors, private companies have gradually emerged as important actors, though extant research mostly ignores their role. Among the few studies that analyze the impact of private-sector involvement in international trade policy [START_REF] Singh | Communication and competence in private sector involvement in international trade policy[END_REF], [START_REF] Bartoli | Bank support and export: evidence from small Italian firms[END_REF] identify an important influence of banks for supporting export efforts by SMEs. Finally, most studies involve traditional exporters that take a gradual internationalization approach. [START_REF] Catanzaro | Impacts of export support: a conceptual model for export start-ups[END_REF] offer some conceptual insights, but they focus on the effectiveness of international assistance programs for specific stages of traditional internationalization processes. This literature review thus reveals two key gaps. First, most studies center specifically on the decision to start exporting. When studies expand to include support services applied to different stages, the results are still contradictory. Besides, with the notable exception of [START_REF] Freixanet | Export promotion programs: Their impact on companies' internationalization performance and competitiveness[END_REF], descriptions of internationalization stages tend to be limited, such as describing pre-export and export stages. Even Freixanet's (2012) classification does not address how each stage of the process aligns with support service uses, despite evidence that SMEs' needs shift with their growth and progress (skills and resources). Second, research predominantly focuses on the role of national export programs. We lack clear distinctions among the types of support provided and about the role of private support services. 2012) called for longitudinal analyses that could confirm the causal effects of export programs. In line with this suggestion, [START_REF] Belhoste | How do SMEs use support services during their internationalisation process: A comparative study of French traditional SMEs and INVs in Asia[END_REF] analyze SMEs' choice to use (or not) internationalization assistance supports during the entire internationalization process. In line with [START_REF] Hilmersson | Time, temporality, and internationalisation: The relationship among point in time of, time to, and speed of international expansion[END_REF] and Efrat and Shoham (2012), they define SME internationalization process according to two main dimensions: 1) time to internationalization (i.e. the time that has elapsed from a firm's inception to its first sales abroad) and 2) the stage of their internationalization process (i.e. early stage and later stage of international operations). Such temporal aspects of internationalization are used to discriminate between the type of SME process of internationalization (gradual or rapid) and the stage (entry or intensification). Based on a qualitative study of 32 French SMEs that have all reached Asian markets, either gradually (19 referred as traditional SMEs) or rapidly (13 referred as International New Ventures), they identify different configurations of the use of internationalization support services by SMEs, challenging some previous assumptions and the conventional ways of studying export services designed to support SMEs.

Type of support services and time to internationalization.

The results from this study show that support services help traditional SMEs to build knowledge, extend networks, reduce psychic distance [START_REF] Johanson | The mechanism of internationalization[END_REF][START_REF] Ojala | Internationalization of knowledge-intensive SMEs: The role of network relationships in the entry to a psychically distant market[END_REF] with Asia and therefore, support gradual international involvement [START_REF] Johanson | The mechanism of internationalization[END_REF]. This contrasts with INVs that use fewer support services given their lack of resources. Traditional SMEs also look for specific private support services and are ready to pay more than INVs and rely less on fundinga specificity of public support services. We also observe high levels of disappointment from traditional SMEs regarding public support services while INVs found some value in those services. This study also constitutes an attempt to resolve some conflicting results in the literature. [START_REF] Ahmed | Export promotion programs of Malaysian firms: International marketing perspective[END_REF], [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF] and [START_REF] Kedia | Factors inhibiting export performance of firms: An empirical investigation[END_REF] argue that most experienced companies are more likely to use international support structures. The findings indicate, contrary to [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF], that the use of internationalization assistance services declines as the firm gains experience abroad if it is a traditional SME, but the same finding does not hold for INVs.

Type of support services and stage of internationalization.

Previous literature had partially identified that the export phase (early export phase to more advanced) could impact the use of support services. [START_REF] Czinkota | Why national export promotions? International trade forum[END_REF] and [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF] showed that firms have a greater need of export training services in their early export phase and that the type of support may vary. [START_REF] Belhoste | How do SMEs use support services during their internationalisation process: A comparative study of French traditional SMEs and INVs in Asia[END_REF] go one step further and complement [START_REF] Czinkota | Why national export promotions? International trade forum[END_REF] and [START_REF] Crick | UK SMEs' awareness, use, and perceptions of selected government export assistance: An investigation into the effect of the internationalization process[END_REF]'s results showing that SMEs have a greater need for export training services in their early export phase but only if they are in a gradual process of internationalization. In contrast, most INVs realise they need help after their first steps locally. Furthermore, as the use of internationalization assistance services varies with experience, the content of this support also differs. [START_REF] Singer | Factors associated with effective use of export assistance[END_REF] hypothesize that services providing objective knowledge (e.g. workshops, advice) are more useful during pre-export activities, whereas experiential knowledge services (e.g. trade shows abroad) are more useful at advanced stages. [START_REF] Belhoste | How do SMEs use support services during their internationalisation process: A comparative study of French traditional SMEs and INVs in Asia[END_REF] does not find such a clear distinction; rather, the use of these services varies with the time to internationalize and stage of internationalization. Their findings also reveal distinct, but complementary, uses of private and public support services. Consistent with [START_REF] Zahra | International entrepreneurship: The current status of the field and future research agenda[END_REF], they show that INVs seek tips and free services, given the scarcity of financial and human resources, but only during the intensification stage. The use of private support actors is less important during this stage, unless traditional SMEs seek them after being disappointed by public services. Because traditional SMEs know what services are available, they pursue an optimal mix of private and public services, next to their business networks.

CONCLUSION

We think that this chapter could help precise and observe the efficiency of support services differently in two different ways. First, in better distinguishing the type of help and support services actors used. Second, in measuring the efficiency regarding the internationalization process and stage rather than solely considering the export intensity as a dependent variable. Efficiency, in order to be evaluated, needs to consider not only the service by itself but also the type of firm that use it according to its internationalization stage and process. We are convinced that this combination could help deepen the debate on efficiency of export support services whether they are public of private.

These findings have several important managerial implications. In particular, support services (public or private) can use them to target their offer more precisely, according to the different types of SMEs they encounter and their internationalization process stages. Furthermore, SMEs can clarify their needs, according to their current stage in their internationalization process. In turn, they can determine which tailor-made proposals offered by private and public actors are most applicable to their internationalization efforts. It also paves the way for several research perspectives. Extensions to distinct geographic contexts might reveal whether the SME's home country affects its use of support services; we anticipate that the economic, institutional, and cultural characteristics of the home country likely influence how public and private support services develop their offers. Other factors of heterogeneity among the SMEs (Ingley et al., 2016) could be observed in order to study the use of support services by SMEs for their internationalization. In particular, since networks are particularly important in the decision to enter a new market [START_REF] Castellaci | Service firms heterogeneity, international collaborations and export participation[END_REF], future research could investigate their (differentiated) role in the internationalization processes of SMEs in relation to support services. Finally, in line with [START_REF] Catanzaro | Impacts of export support: a conceptual model for export start-ups[END_REF], we call for more studies that investigate the impact of support services on SMEs' performance, according to whether their internationalization process is gradual or rapid.

  WHAT WE NEED TO KNOWIn his seminal work,Freixanet (