Artem N Nekhoroshikh
email: artem.nekhoroshikh@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Emilia Fridman
email: emilia@tauex.tau.ac.il

Wilfrid Perruquetti
email: wilfrid.perruquetti@centralelille.fr

Igor B Furtat

Andrey Polyakov
email: andrey.polyakov@inria.fr

Practical fixed-time ISS of neutral time-delay systems with application to stabilization by using delays

Keywords: neutral time-delay systems, implicit Lyapunov-Krasovskii functional, nonlinear control, delay-induced output feedback, practical input-to-state stability, linear matrix inequalities

The concept of practical fixed-time input-to-state stability for neutral time-delay systems with exogenous perturbations is introduced. Lyapunov-Krasovskii theorems are formulated in explicit and implicit ways. Further, the problem of static nonlinear output-feedback stabilization of a linear system with parametric uncertainties, external bounded state and output disturbances by using artificial delays is considered. The constructive control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It is shown both, theoretically and numerically, that the system governed by the proposed controller converges faster to the given invariant set than in the case of using its linear counterpart.

Introduction

Stabilization of dynamical systems with a faster than exponential rate of convergence has become one of the main trends in modern control theory [START_REF] Lopez-Ramirez | Fixed-Time Output Stabilization and Fixed-Time Estimation of a Chain of Integrators[END_REF][START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]. Frequently, such an approach allows systems to be stabilized at the origin in a finite time. For example, for homogeneous autonomous systems, a special class of nonlinear ones, the type of convergence is defined by their degree of homogeneity [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]. For perturbed systems this concept can be extended to non-asymptotic input-to-state stability (ISS) [START_REF] Hong | Finite-Time Inputto-State Stability and Applications to Finite-Time Control Design[END_REF] when the steady-state error is upper bounded by the norm of external disturbance. In [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] robustness of homogeneous systems with respect to bounded exogenous perturbations was studied.

However, finite-time stabilization is hard to obtain for time-delay systems [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF][START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF]. For instance, to ensure such a property the delays have to diminish proportionally to the norm of the state vector and vanish at the origin, or time-delay terms have to be multiplied by the instantaneous state vector. But in many applications it is sufficient to stabilize a system in finite time only in the vicinity of the origin, the radius of which depends on the time delay and external perturbations, and following [START_REF] Efimov | Homogeneity of neutral systems and accelerated stabilization of a double integrator by measurement of its position[END_REF] such a problem is investigated in this work. In [START_REF] Efimov | Homogeneity of neutral systems and accelerated stabilization of a double integrator by measurement of its position[END_REF] the homogeneity theory was extended to neutral type systems and it was shown how the convergence can be accelerated by selecting a non-zero degree of homogeneity. Nevertheless, it is worth mentioning that for linear systems any stable set is reachable in a finite time also and the settling time can be reduced by feedback gains increasing. But differently from the delay-free case, this approach has limited use for time-delay systems: for any given delay h sufficiently large gains make the closedloop system unstable, which motivated [START_REF] Efimov | Homogeneity of neutral systems and accelerated stabilization of a double integrator by measurement of its position[END_REF].

Stability analysis of time-delay systems could be done by using different methods [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Hale | Theory of functional differential equations[END_REF][START_REF] Kolmanovskii | Applied Theory of Functional Differential Equations[END_REF]. For example, in [START_REF] Kharitonov | Static output feedback stabilization. Necessary conditions for multiple delay controllers[END_REF] Hurwitz stability of transcendental polynomials has been studied. However, such an approach is difficult to use for the synthesis of control systems with delays due to its complexity. Another conventional tools are Lyapunov-Krasovskii [START_REF] Krasovskii | Stability of Motion[END_REF] or Lyapunov-Razumikhin [START_REF] Razumikhin | On the stability of systems with a delay[END_REF] methods. They impose restrictions on the time derivative of an auxiliary functional or function, respectively, with respect to the differential equation of the system. Being well-developed for analysis, both of them do not provide a constructive way for control design in the nonlinear case. On the contrary, their implicit extensions are free of such a drawback: all stability conditions can be checked directly by analyzing some algebraic equations, which implicitly define Lyapunov functionals (functions) [START_REF] Polyakov | Implicit Lyapunov-Krasovski Functionals For Stability Analysis and Control Design of Time-Delay Systems[END_REF]. Moreover, control parameters can be obtained by solving a system of linear matrix inequalities (LMIs).

The goal of this work is to extend the exponential ISS concept for neutral time-delay systems to its fixed-time analog. Both, Lyapunov-Krasovskii theorem and its implicit counterpart, are introduced. Then the proposed approach is applied to static nonlinear output-feedback stabilization of a non-delayed linear system in the controllable canonical form with parametric uncertainties, external bounded state and output disturbances. To this end, the unmeasured states are approximated by finite differences [3,9,10], i.e., an artificial delay is induced. In [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF] it was shown that in this case closed-loop system has a neutral time-delay representation. Moreover, since no observers/predictors are introduced, the control law is static, which essentially simplifies its practical implementation. Differently from [START_REF] Efimov | Homogeneity of neutral systems and accelerated stabilization of a double integrator by measurement of its position[END_REF], in this paper 1) the homogeneity is not used to prove non-asymptotic rate of convergence, 2) the designed control system is practically fixed-time stable and 3) feedback gains are explicitly calculated. In a conference version of this paper [START_REF] Nekhoroshikh | On output-based accelerated stabilization of a chain of integrators: Implicit Lyapunov-Krasovskii functional approach[END_REF] fixed-time stability has not been considered and the influence of parametric uncertainties, external state and output disturbances has not been studied.

The outline of this work is as follows. Notation and auxiliary lemmas are given in Section 2. Practical fixed-time ISS concept of neutral time-delay systems and Lyapunov-Krasovskii theorems are introduced in Section 3. Output stabilization of a linear perturbed system is considered in Section 4. Results of numerical simulations and comparison with a linear analog of the proposed controller are discussed in Section 5. Finally, all the proofs can be found in the Appendices.

+ = {x : x > 0}, R + = R ⋆ + ∪ 0. A series of natural numbers up to n is defined as 1, n. 2) Spaces: L m
∞ is the space of Lebesgue measurable essentially bounded functions d : [0, +∞) → R m with the norm ∥d∥ ∞ := ess sup t∈[0,+∞) ∥d(t)∥ < +∞, where ∥ • ∥ is the Euclidean norm in R n . For ℏ > 0 denote the space of Lebesgue square integrable functions χ :

[-ℏ, 0] → R n with the norm ∥χ∥ 2 := 0 -ℏ ∥χ(τ )∥ 2 dτ < +∞ by L 2 ℏ . The Banach space W 1 ℏ of absolutely continu- ous functions χ : [-ℏ, 0] → R n has the norm ∥χ∥ W := max τ ∈[-ℏ,0] ∥χ(τ )∥+∥ χ∥ 2 . W 1,0 ℏ = {χ ∈ W 1 ℏ : χ(0) = 0} is a subspace of W 1 ℏ .
3) Matrices: For symmetric matrices P ∈ R n×n notations P ≻ 0 (P ≺ 0) and P ≽ 0 (P ≼ 0) mean that P is positive (negative) definite and semidefinite, respectively. The minimal and maximal eigenvalues of a symmetric matrix are symbolized by λ min (P ) and λ max (P ). Block diagonal matrices are indicated as diag{λ i } n j=1 or diag{λ 1 , . . . , λ n } with λ i ∈ R ni×ni . Identity and zero n × n matrices are marked as I n and O n , respectively. A zero column is denoted by o n ∈ R n×1 .

4) Functions:

Denote by C i a class of i times continuously differentiable functions R ⋆ + → R.

Comparison functions

A continuous function w : R + → R + belongs to the class K if it is strictly increasing on R ⋆ + and w(0) = 0; if additionally it is unbounded then w belongs to K ∞ . A continuous function w : R + → R + is said to be a generalized class-K function (GK function) if it is strictly increasing on (s 0 , +∞) and w(s) = 0 for all s ∈ [0, s 0 ] for some

s 0 ∈ R + . A function ν : R + × R + → R + is a generalized class-KL function (GKL function) if for each fixed t ≥ 0 the function ν(•, t
) is a class-GK function, and for each fixed p ≥ 0 the function ν(p, •) is continuous, strictly decreasing and there exists some T (p) ∈ R + such that ν(p, t) → 0 as t → T .

Definition 1 ([25]

) A function q : R ⋆2 + → R, (ρ, s) → q(ρ, s) is said to be of the class IK ∞ if and only if: 1) q is continuous on R ⋆2 + ; 2) for any s ∈ R ⋆ + there exists ρ ∈ R ⋆ + such that q(ρ, s) = 0; 3) for any fixed s ∈ R ⋆ + the function q(•, s) is strictly decreasing on R ⋆ + ; 4) for any fixed ρ ∈ R ⋆ + the function q(ρ, •) is strictly increasing on R ⋆ + ; 5) lim s→0 + ρ = 0, lim ρ→0 + s = 0 and lim

s→∞ ρ = ∞ for all (ρ, s) ∈ Γ = {(ρ, s) ∈ R ⋆2 + : q(ρ, s) = 0}.
In other words, Definition 1 states that there exists a unique function ρ ∈ K ∞ such that q(ρ(s), s) = 0 for all s ∈ R ⋆ + .

Auxiliary lemmas

Lemma 1 (Jensen's inequality [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF])

Let ϕ : [a, b] → R and ϖ, ϑ : [a, b] → [0, ∞) be
g β (s) ≤ ḡ(s, β) := max{g β (β 1/(1-β) ), g β (s)}.
ẋ(t) = f (x t , ẋt , d(t)), t > 0, x(τ ) = Φ(τ ), τ ∈ [-ℏ, 0], (1)
where x(t) ∈ R n is the instantaneous state;

x t ∈ W 1 ℏ is the state function defined by x t (τ ) := x(t + τ ), τ ∈ [-ℏ, 0] with ẋt ∈ L 2 ℏ ; d(t) ∈ R m is the external distur- bance, d ∈ L m ∞ . The continuous operator f : W 1 ℏ × L 2 ℏ × R m → R n is
Lipschitz in the second variable with a constant smaller than one, ensuring forward uniqueness and existence of the system solutions at least locally in time [START_REF] Kolmanovskii | Stability of Functional Differential Equations[END_REF]. Assume that the origin is an equilibrium point of the system (1), i.e., f (0, 0, 0) = 0. A solution of the system (1) with the initial function Φ ∈ W 1 ℏ is denoted by

x(t, Φ, d) ∈ R n or x t (Φ, d) ∈ W 1 ℏ .
Following [START_REF] Hong | Finite-Time Inputto-State Stability and Applications to Finite-Time Control Design[END_REF], we present the concept of practical fixedtime ISS stability of neutral time-delay systems with external inputs.

Definition 2

The system (1) is called (γ, κ)-practically locally fixed-time ISS 1 , if there exist a constant v ≥ 0 and functions w ∈ K, ν ∈ GKL with the settling time estimate T := sup p<γ T (p) < +∞ such that:

∥x(t, Φ, d)∥ ≤ ν(∥Φ∥ W , t) + v + w(∥d∥ ∞ ), ∀t ≥ 0, (2)
for all Φ ∈ X := {Φ ∈ W 1 ℏ : ∥Φ∥ W < γ} and d ∈ D := {d ∈ L m ∞ : ∥d∥ ∞ < κ}.
If v = 0, then system (1) is called (γ, κ)-locally fixedtime ISS. If additionally γ = κ = +∞, then system (1) is called fixed-time ISS.

The following theorem (see the proof in Appendix A) provides sufficient conditions to check (γ, κ)-practical local fixed-time ISS property (2) by using Lyapunov-Krasovskii functionals. For any χ ∈ W 1 ℏ and d ∈ R m we define the upper-right Dini derivative of functional

V k : W 1 ℏ → R + , k = 1, 2
, with respect to equation (1) as follows:

D + V k (χ, d) := lim sup ∆t→0 + V k (x∆t(χ, d))-V k (χ) ∆t
, where x ∆t (χ, d) is the solution of (1) with initial conditions χ ∈ W 1 ℏ and the input d = d for all t ∈ [0, ∆t).

Theorem 1 Let there exist constants v ∈ [0, 1), γ > 1, µ 1 ∈ (-1, 0), µ 2 > 0, θ k > 0, functions ρ 1,k , ρ 2,k ∈ K ∞ , w ∈ K and continuous functionals V k : W 1 ℏ → R + , k = 1, 2, such that for all χ ∈ W 1 ℏ and d ∈ D: ρ 1,k (∥χ(0)∥) ≤ V k (χ) ≤ ρ 2,k (∥χ∥ W ); (3a) V 1 (χ) ≤ 1 ⇔ V 2 (χ) ≤ 1; (3b)
1 Hereinafter, ISS also stands for "input-to-state stable".

max{v, w(∥d∥

∞ )} < V 1 ≤ 1 ⇒ D + V 1 (χ, d) ≤ -θ 1 V 1+µ1 1 (χ); (3c) max{1, w(∥d∥ ∞ )} < V 2 < γ ⇒ D + V 2 (χ, d) ≤ -θ 2 V 1+µ2 2 (χ). ( 3d
)
Then the system (1) is (γ, κ)-practically locally fixed-time ISS (2) with γ, κ, T , v, w(s) and ν(p, t) given by

γ = ρ2,2 (γ), κ = w(γ), T = 1 µ2θ2 + 1 |µ1|θ1 , v = ρ1,1 (v), w(s) = ρ1,1 ( w(s)), if w(s) < 1, ρ1,2 ( w(s)), if w(s) ≥ 1, ν(p, t) =    ν 2 (p, t), t ∈ [0, T 2 (p)), ν 1 (p, t), t ∈ [T 2 (p), T 2 (p) + T 1 (p)), 0, t ≥ T 2 (p) + T 1 (p), (4)
where functions w, ρ1,1 , ρ1,2 and ρ2,2 are inverse of w, ρ 1,1 , ρ 1,2 and ρ 2,2 , respectively, and

ν 1 (p, t) = ρ1,1 ((µ 1 θ 1 (t -T 2 (p) -T 1 (p))) -1/µ1 ), ν 2 (p, t) = ρ1,2 ((µ 2 θ 2 (t -T 2 (p)) + 1) -1/µ2 ), T 1 (p) = max{0, (min{1, ρ -µ1 2,1 (p)})/(-µ 1 θ 1 )}, T 2 (p) = max{0, (1 -ρ -µ2
2,2 (p))/(µ 2 θ 2 )}. One can see that conditions (3c) and (3d) in general are hard to check, especially in a control design scenario. As it has been shown in [START_REF] Polyakov | Implicit Lyapunov-Krasovski Functionals For Stability Analysis and Control Design of Time-Delay Systems[END_REF], this problem can be overcome by defining functionals V k in Theorem 1 implicitly. To this end, we first need to introduce Fréchet derivatives.

Definition 3 An operator F : U → V is called Fréchet differentiable at χ ∈ U if there exists a bounded linear operator DF χ : U → V such that: lim ∆χ→0 ∥F (χ+∆χ)-F (χ)-DFχ(∆χ)∥ V ∥∆χ∥ U = 0,
where ∥ • ∥ U and ∥ • ∥ V are norms in the Banach spaces U and V, respectively.

Denote by

Q ′ V,k (V k , χ) and Q ′ t,k (V k , χ, d) derivatives of functions V k → Q k (V k , χ) and t → Q k (V k , x t (χ, d))
, where x(t) satisfies (1) with initial conditions Φ = χ, respectively.

Theorem 2 Let there exist constants v ∈ [0, 1), γ > 1, µ 1 ∈ (-1, 0), µ 2 > 0, θ k > 0, functions q 1,k , q 2,k ∈ IK ∞ , w ∈ K and continuous functionals Q k : R ⋆ + × W 1 ℏ → R, k = 1, 2 such that: C1) Q k (V k , χ) are continuously Fréchet differentiable for all V k ∈ R ⋆ + and χ ∈ W 1 ℏ ; C2) for any χ ∈ W 1 ℏ there exist V k ∈ R ⋆ + such that Q k (V k , χ) = 0; C3) Q ′ V,k (V k , χ) < 0 for all (V k , χ) ∈ Ω k = {(V k , χ) ∈ R ⋆ + × W 1 ℏ : Q k (V k , χ) = 0}; C4) for all (V k , χ) ∈ Ω k and d ∈ D: q 1,k (V k , ∥χ(0)∥) ≤ Q k (V k , χ), ∀χ ∈ W 1 ℏ \ W 1,0 ℏ , Q k (V k , χ) ≤ q 2,k (V k , ∥χ∥ W ), ∀χ ∈ W 1 ℏ \ {0}; (5a) Q 1 (1, χ) = Q 2 (1, χ); (5b) max{v, w(∥d∥ ∞ )} < V 1 ≤ 1 ⇒ Q ′ t,1 (V 1 , χ, d) ≤ θ 1 V 1+µ1 1 Q ′ V,1 (V 1 , χ); (5c) max{1, w(∥d∥ ∞ )} < V 2 < γ ⇒ Q ′ t,2 (V 2 , χ, d) ≤ θ 2 V 1+µ2 2 Q ′ V,2 (V 2 , χ).
(5d)

Then the system (1) is (γ, κ)-practically locally fixedtime ISS (2) with γ, κ, T , v, w(s) and ν(p, t) given by (4), where functions ρ i,k (s) implicitly defined by

q i,k (ρ i,k (s), s) = 0, respectively, i, k = 1, 2.
The proof of Theorem 2 can be found in Appendix B.

Despite the seeming complexity of conditions (5c) and (5d), in the next section we will show how Theorem 2 can be successfully applied to design a control law.

4 Nonlinear delay-induced control

Problem statement

Consider a system in the controllable canonical form with a relative degree n ≥ 2, matched parametric uncertainties, state disturbances and output perturbations:

ẋ(t) = Ax(t) + B u(t) + d 1 (t) + ax(t) , y(t) = Cx(t) + d 2 (t), (6)
where

x(t) ∈ R n is the state vector; u(t) ∈ R is the con- trol input; y(t) ∈ R is the output available for measure- ments; d 1 (t) ∈ R and d 2 (t) ∈ R are the external state and output disturbances, respectively, d = [d 1 , d 2 ] ⊤ ∈ D := {d ∈ L 2 ∞ : ∥d∥ ∞ < κ}; a ∈ R 1×n is the vector of unknown coefficients such that aa ⊤ ≤ ϵ; A = o n-1 I n-1 0 o ⊤ n-1 , B = o n-1 1 , C = 1 o ⊤ n-1 .
Note that all linear single-input single-output controllable systems with a relative degree n can be rewritten in the canonical form ( 6) by applying a linear coordinate transformation. Moreover, for many nonlinear systems, such as a pendulum (n = 2), a magnetic suspension system (n = 3) or a single link manipulator with flexible joints and negligible damping (n = 4), there is a change of variables that transforms them into the form (6) [START_REF] Khalil | Nonlinear systems; 3rd ed[END_REF].

The goal is to design a static output-feedback control practically stabilizing the system (6) with the rate of convergence faster than exponential.

Control design

Inspired by [START_REF] Lopez-Ramirez | Fixed-Time Output Stabilization and Fixed-Time Estimation of a Chain of Integrators[END_REF], we will define a nonlinear control law in the following form:

u(ỹ) = n j=1 K j ⌈ỹ j ⌋ αj (∥ỹ∥) , (7a)
α j (∥ỹ∥) =      1 r2,j , if ∥ỹ∥ ≥ b 2 , 1 r1,j , if ∥ỹ∥ ≤ b 1 , r1,j -r2,j r1,j r2,j ∥ỹ∥-b1 b2-b1 + 1 r1,j , otherwise, (7b)
r k,j (µ k ) = 1 -(n + 1 -j)µ k , k = 1, 2, (7c)
where ỹ ∈ R n with ỹ1 (t) := y(t), ỹi+1 (t) is the approximation of the i-th output derivative y

(i) (t), i = 1, n -1, µ 1 = -µ and µ 2 = µ are degrees of nonlinearity with µ ∈ (0, 1/n), b 1 > 1 and b 2 > b 1 are switch thresholds, K j < 0, j = 1, n are feedback gains, K := [K 1 , . . . , K n ], ⌈•⌋ α := sign(•)| • | α is the signed power.
Instead of introducing a state observer, in this paper, we approximate the output derivatives by finite differences ỹi+1 (t) ≈ y (i) (t), i = 1, n -1:

ỹi+1 (t) := ỹi(t)-ỹi(t-h) h = 1 h i i s=0 (-1) s i! s!(i-s)! y(t -sh), (8)
where h > 0 is a time delay. Since the value of y(t -sh) is undefined for t ∈ [0, sh), then we set it equal to y(0).

Selection of approximation ( 8) follows from the wellknown fact: if h → 0 then ỹi+1 (t) → y (i) (t). It is worth noting that the proposed scheme is similar to a highgain observer [START_REF] Khalil | Nonlinear systems; 3rd ed[END_REF], since only for sufficiently small delays h > 0 derivative estimates ỹi+1 (t) can be used in stabilizing feedback [START_REF] Fridman | Delay-induced stability of vector second-order systems via simple Lyapunov functionals[END_REF][START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF]. But differently from the conventional observer-based control, approximation ( 8) is fully static and, therefore, easy to implement. Nevertheless, to apply Theorem 2, first we have to present ỹi+1 (t) in a different form.

Proposition 1 ([27]) If y ∈ C i and y (i) is absolutely continuous, i ∈ N, then ỹi+1 (t) defined in (8) satisfies: ỹi+1 (t) = y (i) (t) - t t-ih φ i t-s h y (i+1) (s)ds, (9)
where φ 1 (ξ) := 1 -ξ and for i ∈ N \ {1}:

φ i (ξ) :=      ξ 0 φ i-1 (λ)dλ + 1 -ξ, ξ ∈ [0, 1], ξ ξ-1 φ i-1 (λ)dλ, ξ ∈ (1, i -1), i-1 ξ-1 φ i-1 (λ)dλ, ξ ∈ [i -1, i]. (10)
Since

x 1 ∈ C n , x (n)
1 is absolutely continuous and approximation ( 8) is linear, it follows from (9) that ỹi+1 (t) = x i+1 (t) + δ i (t) + d2,i+1 (t), i = 1, n -1, where

δ i (t) : = - t t-ih φ i t-s h ẋi+1 (s)ds, (11a) d2,i+1 (t) : = d2,i(t)-d2,i(t-h) h (11b)
with d2,1 (t) := d 2 (t). Therefore, the closed-loop system (6), ( 7) is in the form (1) and Theorem 2 can be applied. To this end, introduce two implicit Lyapunov-Krasovskii functionals (ILKFs)

Q k (V k , χ), k = 1, 2
, by the equality:

Q k (V k , χ) := -1 + χ ⊤ (0)Λ -r k V k P Λ -r k V k χ(0) + n-1 i=1 i 2Si V -2r k,i+2 +µ k k 0 -ih ψ i ( -τ h ) χ2 i+1 (τ )dτ , (12)
where For the following Lyapunov-Krasovskii analysis we will utilize some characteristics of the functions φ i (ξ) (see Fig. 1) and their integrals ψ i (ξ) := i ξ φ i (λ)dλ which are summarized below (see the proof in Appendix C).

P = P ⊤ ≻ 0, Λ -r k V k := diag{V -r k,j k } n j=1 , S i > 0, i = 1, n -1. Note that in a linear case (µ k = 0), equation Q k (V k , χ) = 0 defines a Lyapunov-Krasovskii functional V k (χ)= χ ⊤ (0)P χ(0)+ n-1 i=1 i 2Si 0 -ih ψ i ( -τ h ) χ2 i+1 (τ )dτ .

Proposition 2

The functions φ i (ξ) defined in [START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF] and their integrals ψ i (ξ) possess the following properties:

P1) φ ′ i (ξ) < 0 on ξ ∈ (0, i); P2) 0 ≤ φ i (ξ) ≤ 1 for all ξ ∈ [0, i]; P3) φ i (ξ) + φ i (i -ξ) = 1 for all ξ ∈ [0, i]; P4) φ ′′ i (ξ) < 0 on ξ ∈ (0, i/2) and φ ′′ i (ξ) > 0 on ξ ∈ (i/2, i) for i ≥ 2; P5) ψ i (0) = i/2 and ψ i (i) = 0; P6) ψ i (ξ) ≤ (i/2)φ i (ξ) for all ξ ∈ [0, i];
P7) for all i ∈ N the following integral is well-defined:

ζ i := i 0 ψ -1 i (ξ)φ 2 i (ξ)dξ. ( 13
)
Remark 1 It is worth mentioning that parameters ζ i are independent of time delay h > 0 and, thus, can be calculated in advance. For example, direct computation of ζ 1 gives a quite simple result: ζ 1 = 2. The other values of ζ i can be found by numerical integration (see Table 1).

Now we are ready to present the restrictions on constructive selection of adjustable parameters µ, h, b 1 and b 2 such that Theorem 2 holds for ILKFs [START_REF] Gu | Stability of Time-Delay Systems[END_REF] with respect to the system (6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] (see the proof in Appendix D).

Theorem 3 Given ϵ > 0, let there exist µ ∈ (0, 1/n), h > 0, b 1 > 1, b 2 > b 1 such that the system of LMIs:

0 ≺ XH r k + H r k X ≼ 2ω k X, (14a) max{∥σ∥, b 0 }I n ≼ X ≼ I n /2, (14b)     Ξ 11 Ξ 12 Y ⊤ * Ξ 22 Ξ ⊤ 12 B * * -4Sn-1 (n-1) 2     ≼0, Z X * M ≽0, N N ϱ * Xϱ ≽0, ( 14c
)
where is feasible for some c l > 0, 0

H r k := diag{r k,j } n j=1 , ω 1 := 1+(n+1)µ, ω 2 := 1, ϱ := h 1-√ µ , b 0 := b 2 1 -1, ∥σ∥ := max k=1,2 ∥σ k ∥, σ 1,j := ḡ(b 1 , 1/r 1,j ), σ 2,j := ḡ(b 1 , 1/r 2,j ) + ḡ(b 1/r2,j 2 , r 2,j /r 1,j ), Ξ 11 := XA ⊤ + Y ⊤ B ⊤ + AX + BY + Z + 2 n-1 X, Ξ 12 := BY √ ϱ, BY ∥σ∥, BY √ b 0 , B, B √ ϵ , Ξ 22 := -1 n-1 diag (1 -ϱ)X, 1 2 X, 1 2 X, c1 4 , 1 2 , M := diag c 2 , c 3 , 4Si i 2 n-2 i=1 , N := 1 n-1 diag c 4 , S i n-1 i=1 ,
< S i ≤ i 4ζiϱ , X = X ⊤ ∈ R n×n , Y ∈ R 1×n , Z = Z ⊤ ∈ R n×n ,
with ζ i and r k,j defined by ( 13) and (7c),

respectively, i = 1, n -1, j = 1, n, k = 1, 2, l = 1, 4.
Then the closed-loop system (6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] with K = Y X -1 is (γ, κ)-practically locally fixed-time ISS (2) with γ, κ, T , v and w(s) given by

γ = h -r 2,1 / √ µ √ max{∥σ∥ -1 , h √ µ S0} , κ = h -r 2,1 / √ µ η , T = 1 4(n-1)µ 1 ω1 + 1 ω2 , v = h r 1,n / √ µ √ 2 , w(s) = 1 √ 2 (ηs) r1,n/r1,1 , if ηs < 1, (ηs) r2,n/r2,1 , if ηs > 1,
where

S 0 := max i=1,n-1 i 2 4Si , η := max{c 1 , (2/h) 2n -1 (2/h) 2 -1 /b 2 0 }.
Let us give some comments on the choice of tuning parameters. Firstly, LMIs ( 14 Secondly, it follows from Theorem 3 that the settling time T is inversely proportional to parameter µ. Thus, the best strategy of parameter tuning consists in maximizing µ, for which LMIs ( 14) are feasible for given ϵ.

On the other hand, note that γ = γ(h) and v = v(h) are the functions of the time delay h for the fixed nonlinear degree µ. Obviously, γ (v) can be enlarged (decreased) by reducing time delay h and in the limit case: γ → +∞ (v → 0 + ) as h → 0 + . However, in practice, time delay h cannot be chosen arbitrarily small due to related implementation problems.

Remark 2 Note that, similar to high-gain observers, approximation (8) is sensitive to high-frequency output perturbations [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]. In order to show this, let us assume that d 2 (t) is a Lipschitz continuous function of time, i.e., there exists a positive constant L such that |d 2 (t 1 ) -

d 2 (t 2 )| ≤ L∥d 2 ∥ ∞ |t 1 -t 2 | for all t 1 , t 2 ∈ R.
Taking into account (11b), it can be shown that in this case

η = max{c 1 , 1 + L 2 (2/h) 2(n-1) -1 (2/h) 2 -1
/b 2 0 }, which coincides with the one given in Theorem 3 if hL = 2. Thus, the slower the output disturbance d 2 changes (the smaller L), the smaller the steady-state error. Nevertheless, the problem of making approximation (8) more robust to high-frequency output perturbations (e.g., by introducing low-pass filters [START_REF] Furtat | Robust stabilization of linear plants under uncertainties and high-frequency measurement noises[END_REF][START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]) is out of the scope of this work.

Let us show what is the main advantage of the proposed control law (7) compared to its linear analog (µ = 0) with the same gains K.

Proposition 3 Let the conditions of Theorem 3 be fulfilled. Then there are h 0 ∈ (0, h] and γ 0 ∈ (0, γ(h 0 )] such that for all Φ ∈ X 0 := {X : ∥Φ∥ W ≥ γ 0 } and d ∈ D the system (6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] with time delay h 0 converges faster to the set A := {x ∈ R n : ∥x(t, Φ, d)∥ ≤ v + w(∥d∥ ∞ )} than its linear counterpart (µ = 0). The proof of Proposition 3 is given in Appendix E.

In other words, for sufficiently large initial conditions or sufficiently small perturbations the proposed control system always converges faster to the vicinity of the origin than its linear analog. 1), ( 7) has been done in MATLAB Simulink by using the explicit Euler method with a state-dependent step [START_REF] Efimov | Discretization of homogeneous systems using Euler method with a statedependent step[END_REF]. The basic and minimum discretization steps, the maximum number of iterations and the homogeneous norm have been defined as ∆t 0 = 10 -2 , ∆t min = 10 -4 , N max = 2 • 10 4 and ∥x∥ hom := ( n j=1 |x j | αj /α1 ) α1 , respectively. First, we will show that the proposed control scheme ( 6), ( 7) is indeed (γ, κ)-practically fixed-time stable. To this end, we will compare it with its linear analog (µ = 0) when ∥d∥ ∞ = 0. Choosing initial conditions as Φ(τ ) = [0, 1, -0.5] • 10 8-2i , i = 0, 3 for all τ ∈ [-0.04, 0], we guarantee that ∥Φ∥ W < γ. The norm of the solutions x(t, Φ, 0) is depicted in Figure 2a in the logarithmic scale, where solid lines corresponds to the proposed control law [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] and dashed ones represent its linear counterpart (µ = 0). The dotted magenta line defines the radius of the set A. The results illustrate Proposition 3: the solutions of the nonlinear system (6), ( 7) converge faster to the set A than its linear analog. However, the superiority of the proposed control over its linear counterpart is not so evident due to the smallness of µ. Recall that this parameter should be chosen as large as possible to ensure the feasibility of LMIs [START_REF] Hong | Finite-Time Inputto-State Stability and Applications to Finite-Time Control Design[END_REF]. Since our Lyapunov analysis is rather conservative, one might expect that the closedloop system ( 6), ( 7) remain fixed-time stable even for larger µ. To demonstrate this, we chose µ = 0.1 and kept other control parameters the same. The results of this numerical comparison are depicted in Figure 2b. Clearly, the proposed control significantly does outperform the linear one. Now we compare performance of the proposed control system ( 6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] with its linear counterpart in the presence of the state disturbance d 1 (t) = cos(t) and the output perturbation d 2 (t) = 0.1 sin(10t). As a result, w(∥d∥ ∞ ) = (η √ 1.01) r2,n/r2,1 = 7 • 10 6 . The norm of the solutions x(t, Φ, d) is depicted in Figure 3a in the logarithmic scale, where the initial conditions Φ are chosen the same as for the disturbance-free case. Again the obtained results go with Proposition 3. As well as in the disturbance-free case, for larger values of µ the difference between nonlinear and linear approaches becomes clearer (see Fig. 3b).

Conclusion

The paper introduces the concept of practical fixedtime input-to-state stability for neutral time-delay systems with exogenous perturbations. Related Lyapunov-Krasovskii theorems have been formulated explicitly and implicitly. The latter has been applied to solve the problem of static output-feedback delay-induced stabilization of a linear system in the controllable canonical form with parametric uncertainties, bounded state and output disturbances. The control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It has been shown that for sufficiently large initial conditions or sufficiently small perturbations the proposed control scheme converges to the stable set faster than its linear counterpart. The numerical simulation has verified the theoretical results. One of the directions for future research may be the search for new, less conservative LMI constraints on nonlinear degree µ and time delay h.

A Proof of Theorem 1

Let x t = χ, satisfying (1). If w(∥d∥ ∞ ) < 1, then applying the Comparison Lemma (Lemma 3 in [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF]) to the function V2 (t) := V 2 (x t ) from (3d) on interval t ∈ [0, T 2 ), where

T 2 = inf{t ≥ 0 : V2 (t) ≤ 1}, we get V2 (t) ≤ (µ 2 θ 2 t + V -µ2 2 (0)) -1/µ2 . Obviously, T 2 ≤ (1 -V -µ2 2 (0))/(µ 2 θ 2 ). Hence, if ρ 2,2 (∥Φ∥ W ) ≤ 1, then (3a) implies V2 (t) ≤ 1 and T 2 = 0. Oth- erwise, ∥x(t)∥ ≤ ρ1,2 ( V2 (t)) for t ∈ [0, T 2 ) and V2 (0) ≤ ρ 2,2 (∥Φ∥ W ) due to (3a). On the other hand, if w(∥d∥ ∞ ) ≥ 1, then there exists a moment of time T ′ 2 ∈ [0, T 2 ) such that V2 (t) ≤ w(∥d∥ ∞ ) for t ≥ T ′ 2 . Thus, one can conclude that ∥x(t)∥ ≤ ν 2 (∥Φ∥ W , t) + w(∥d∥ ∞ ) for all t ∈ [0, T 2 ). Moreover, V2 (0) < γ if ∥Φ∥ W < ρ2,2 (γ). If w(∥d∥ ∞ ) < 1, then (3b) implies V1 (t) := V 1 (x t ) ≤ 1 for t ≥ T 2 . Assume first that max{v, w(∥d∥ ∞ )} = 0. Applying the Comparison Lemma to the func- tion V1 (t) from (3c) on interval t ∈ [T 2 , T 2 + T 1 ), where T 2 + T 1 = inf{t ≥ 0 : V1 (t) = 0}, we get V1 (t) ≤ (µ 1 θ 1 (t -T 2 ) + V -µ1 1 (T 2 )) -1/µ1 . It is clear that T 1 ≤ V -µ1 1 (T 2 )/(-µ 1 θ 1 ), where V1 (T 2 ) ≤ 1 if T 2 > 0 or V1 (T 2 ) ≤ ρ 2,1 (∥Φ∥ W ) if T 2 = 0. Hence, ∥x(t)∥ ≤ ρ1,1 ( V1 (t)) ≤ ν 1 (∥Φ∥ W , t) for t ∈ [T 2 , T 2 + T 1 )
and ∥x(t)∥ = 0 for t ≥ T 2 + T 1 due to (3a). Now assume that 0 < max{v, w(∥d∥ ∞ )} < 1. Then there exists a moment of time

T ′ 1 ∈ [T 2 , T 2 + T 1 ) such that V1 (t) ≤ max{v, w(∥d∥ ∞ )} for t ≥ T ′ 1 . Thus, ∥x(t)∥ ≤ ν 1 (∥Φ∥ W , t) + v + w(∥d∥ ∞ ) for all t ≥ T 2 . 2 B Proof of Theorem 2
In order to prove the theorem it is sufficient to show that there exist functionals V k : W 1 ℏ → R + , satisfying conditions of Theorem 1. Indeed, C2) and C3) guarantee existence of unique functionals [START_REF] Polyakov | Implicit Lyapunov-Krasovski Functionals For Stability Analysis and Control Design of Time-Delay Systems[END_REF] and C1) guarantee that functionals V k are continuously Fréchet differentiable on W 1 ℏ \ {0}.

V k : W 1 ℏ \{0} → R ⋆ + such that Q k (V k (χ), χ) = 0 for any χ ∈ W 1 ℏ \ {0}. Moreover, Theorem 1 from
From (5a) it follows that q 1,k (V k (χ), ∥χ(0)∥) ≤ Q k (V k (χ), χ) = 0 = q 1,k (ρ 1,k (∥χ(0)∥), ∥χ(0)∥) for all χ ∈ W 1 ℏ \ W 1,0 ℏ and q 2,k (ρ 2,k (∥χ∥ W ), ∥χ∥ W ) = 0 = Q k (V k (χ), χ) ≤ q 2,k (V k (χ), ∥χ∥ W ) for all χ ∈ W 1 ℏ \ {0}. Due to properties of IK ∞ functions, the obtained inequalities imply ρ 1,k (∥χ(0)∥) ≤ V k (χ) for all χ ∈ W 1 ℏ \ W 1,0 ℏ and V k (χ) ≤ ρ 2,k (∥χ∥ W ) for all χ ∈ W 1 ℏ \ {0}
. Thus, the functional V k (χ) can be extended by continuity to W 1 ℏ as follows V (0) = 0. Taking into account that 0 = ρ 1,k (∥χ(0)∥) < V k (χ) for all χ ∈ W 1,0 ℏ \ {0}, we finally derive condition (3a).

Conditions (5b) and C3) guarantee that (3b) holds.

Indeed, if V 1 (χ) ≤ 1, then Q 2 (1, χ) = Q 1 (1, χ) ≤ Q 1 (V 1 (χ), χ) = 0 = Q 2 (V 2 (χ), χ) and, consequently, V 2 (χ) ≤ 1.
Let x t = χ be a solution of [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]. Consider the functions Vk (t

) := V k (x t ), Qk (V k , t) := Q k (V k , x t ) and d dt Vk (t) := d dt V k (x t ).
Clearly, Qk ( Vk (t), t) = 0 for all t ≥ 0 such that x t ̸ = 0. Then the implicit function theorem [4, p. 221] for Euclidean spaces and (5c), (5d

) imply that d dt Vk (t) = -Q′ t,k (V k , t)/ Q′ V,k (V k , t) ≤ -θ k V 1+µ k k (t)
. Thus, all steps of the proof of Theorem 1 can be repeated. 2 [START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF] with respect to ξ, we obtain:

C Proof of Proposition 2 P1)-P2) First, it is clear to see that φ ′ 1 (ξ) = -1 < 0 for all ξ ∈ [0, 1]. Differentiating
φ ′ i (ξ):=    φ i-1 (ξ) -1, ξ ∈ [0, 1], φ i-1 (ξ) -φ i-1 (ξ -1), ξ ∈ (1, i -1), -φ i-1 (ξ -1), ξ ∈ [i -1, i]. (C.1)
Obviously, using induction, one can prove that φ ′ i (ξ) < 0 on ξ ∈ (0, i) for i ≥ 2. Indeed, if φ ′ i-1 (ξ) < 0 on ξ ∈ (0, i-1), then φ i-1 (ξ) is strictly decreasing. Then taking into account that (10) implies φ i (0) = 1 and φ i (i) = 0, we finish the proof. P3) Property 4) from Proposition 2 in [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF] postulates that φi (hξ) + φi (h(i -ξ)) = 1, where functions φi (hξ) are such that φi (hξ) = φ i (ξ). Thus, φ i (ξ)+φ i (i-ξ) = 1. P4) Differentiating (C.1) with respect to ξ, we get:

φ ′′ i (ξ) :=    φ ′ i-1 (ξ), ξ ∈ [0, 1], φ ′ i-1 (ξ) -φ ′ i-1 (ξ -1), ξ ∈ (1, i -1), -φ ′ i-1 (ξ -1), ξ ∈ [i -1, i]. For i = 2 it is obvious that φ ′′ 2 (ξ) < 0 on ξ ∈ [0, 1) and φ ′′ 2 (ξ) > 0 on ξ ∈ (1, 2], since φ ′ 1 (ξ) = -1 < 0 for all ξ ∈ [0, 1]. Moreover, φ ′
2 (ξ) is strictly decreasing and strictly increasing on corresponding intervals. Applying property P1) for i > 2, it is sufficient to prove by using induction that function φ ′′ i (ξ) has the unique zero at ξ 0i = i/2. Indeed, φ ′′ i-1 ((i-1)/2) = 0 implies that φ ′ i-1 (ξ) is strictly decreasing on ξ ∈ (0, (i -1)/2) and strictly increasing on ξ ∈ ((i -1)/2, 1). Therefore, φ ′′ i (ξ) has the only one zero on ξ ∈ (0, i). Let us show that ξ 0i = i/2. Using (C.1), the condition

φ ′′ i (i/2) = φ ′ i-1 (i/2) - φ ′
i-1 (i/2 -1) = 0 can be equivalently rewritten as

2φ 1 (1/2) = 1 for i = 3, 2φ i-2 i-2 2 = φ i-2 i 2 + φ i-2 i-4 2
for i ≥ 4, since 3/2 ∈ [1, 2] and (3/2 -1) ∈ [0, 1] for i = 3, i/2 and (i/2 -1) ∈ [1, i -1] for i ≥ 4. Applying property P3), one can see that these relations hold and, therefore, ξ 0i = i/2 is the unique inflection point of φ i (ξ) for i > 2.

P5) It is obvious that ψ i (i) = 0. Then using the change of variable λ = i -λ and property P3), we obtain

ψ i (0) = i/2 0 φ i (λ)dλ + i i/2 φ i ( λ)d λ = i/2 0 φ i (λ)dλ + i/2 0 φ i (i -λ)dλ = i/2 0 [φ i (λ) + φ i (i -λ)]dλ = i/2.
P6) Function ψ i (ξ) could be rewritten as follows:

ψ i (ξ) = i 2 - ξ 0 φ i (λ)dλ, ξ ∈ [0, i/2], i ξ φ i (λ)dλ, ξ ∈ [i/2, i].
Taking into account property P4), integral terms can be estimated by the area of a trapezoid from below and a triangle from above, respectively:

ψ i (ξ) ≤ i/2 -ξ[1 + φ i (ξ)]/2, ξ ∈ [0, i/2], (i -ξ)φ i (ξ)/2, ξ ∈ [i/2, i]. Since i -ξ ≤ iφ i (ξ) for [0, i/2] and i -ξ ≤ i for all ξ ∈ [0, i], we conclude the proof. P7) Since function φi (ξ) := ψ -1 i (ξ)φ 2 i (ξ) is continuous on ξ ∈ [0, i), it is sufficient to prove that φi (i+0 -) < ∞.
Indeed, applying L'Hôpital's rule, we get φi (i + 0 -) = -2φ ′ i (i). From (C.1) it follows that φ ′ i (i) = φ i-1 (i -1) = 0. Therefore, ζ i = i 0 φi (ξ)dξ is well-defined and function φi (ξ) can be prolonged to ξ = i by defining φi (i) = 0. 2

D Proof of Theorem 3

Let us show that ILKFs [START_REF] Gu | Stability of Time-Delay Systems[END_REF] satisfy all conditions of Theorem 2.

D.1 Proof of conditions C1)-C3), (5a) and (5b)

The functionals Q k (V k , χ) defined by [START_REF] Gu | Stability of Time-Delay Systems[END_REF] are continuously Fréchet differentiable on R ⋆ + × W 1 ℏ , where ℏ = (n -1)h. Indeed, the following operators

DQ k,V (∆V k ) := -(∆V k /V k ) χ(0) ⊤ Λ -r k V k D k Λ -r k V k χ(0) - n-1 i=1 im k (i) 2Si V -2r k,i+2 +µ k k 0 -ih ψ i ( -τ h ) χ2 i+1 (τ )dτ , DQ k,χ (∆χ) := 2χ ⊤ (0)Λ -r k V k P Λ -r k V k ∆χ(0) + n-1 i=1 i Si V -2r k,i+2 +µ k k 0 -ih ψ i ( -τ h ) χi+1 (τ ) d∆χi+1(τ ) dτ dτ ,
where

∆V k ∈ R ⋆ + , ∆χ ∈ W 1 ℏ , D k := H r k P + P H r k and m k (i) := 2r k,i+2 -µ k , i = 1, n -1, are continuous partial Fréchet derivatives of function V k → Q k (V k , χ) and functional χ → Q k (V k , χ), respectively, for all V k ∈ R ⋆ + and χ ∈ W 1 ℏ .
Since P ≻ 0, then the following inequalities

λmin(P )∥χ(0)∥ 2 max{V 2-2µ k k ,V 2-2nµ k k } -1 ≤ Q k (V k , χ) ≤ λmax(P )∥χ(0)∥ 2 + n-1 i=1 i 2 4S i V -µ k k 0 -(n-1)h | χi+1(τ )| 2 dτ min{V 2-2µ k k ,V 2-2nµ k k } -1 hold for all V k ∈ R ⋆ + and χ ∈ W 1 ℏ .
Hence, it is easy to see that for any χ ∈ W 1 ℏ there exist

V k ∈ R ⋆ + such that Q k (V k , χ) = 0.
Taking into account (14b), introduce the functions q 1,k , q 2,k ∈ IK ∞ by the formulas

q 1,k (ρ 1,k , s) = 2s 2 max{ρ 2-2µ k 1,k ,ρ 2-2nµ k 1,k } -1, q 2,k (ρ 2,k , s) = max{ 1 ∥σ∥ , ρ -µ k 2,k max i=1,n-1 i 2 4S i }s 2 min{ρ 2-2µ k 2,k ,ρ 2-2nµ k 2,k } -1, where ρ 1,k , ρ 2,k , s ∈ R ⋆ + . The obtained estimates also guarantee that q 1,k (V k , ∥χ(0)∥) ≤ Q(V k , χ) ≤ q 2,k (V k , ∥χ∥ W ) for all V k ∈ R ⋆ + and χ ∈ W 1 ℏ . Moreover, condition (5b) obviously holds. One can see that m k (i) ≤ 2ω k and 0 ≺ D k ≼ 2ω k P due to (14a). Taking into account that by definition Q ′ V,k (V k , χ)∆V k = DQ k,V (∆V k ), we conclude that -2ω k ≤ V k Q ′ V,k (V k , χ) < 0, ∀(V k , χ) ∈ Ω k . (D.1) Therefore, the condition C3) of Theorem 2 hold. 2 D.2 Proof of conditions (5c) and (5d) If x t = x t (Φ, d)
is the solution of the system ( 6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF], then using property P5), we obtain

Q ′ t,k (V k , x t , d) = R 1,k + R 2,k + R 3,k , (D.2)
where

R 1,k := 2x ⊤ Λ -r k V k P Λ -r k V k f (x t , ẋt , d), R 2,k := V µ k k n-1 i=1 i 2 4Si V -2r k,i+2 k ẋ2 i+1 (t), R 3,k := - n-1 i=1 i 2hSi V -2r k,i+2 +µ k k t t-ih φ i ( t-s h ) ẋ2 i+1 (s)ds. Taking into account that Λ -r k V k A = V µ k k AΛ -r k V k , Λ -r k V k B = V -r k,n k B = V -1+µ k k
B and (11), R 1,k could be rewritten as follows:

R 1,k = 2V µ k k x ⊤ Λ -r k V k P (A + BK)Λ -r k V k x +BK(d h,k + d µ,k + d y,k ) + BV -1 k (d 1 + ax) , where disturbance terms d h,k := Λ -r k V k [0, δ 1 , . . . , δ n-1 ] ⊤ , d µ,k := V -1 k ⌈ỹ 1 ⌋ α1(∥ỹ∥) , . . . , ⌈ỹ n ⌋ αn(∥ỹ∥) ⊤ -Λ -r k V k ỹ and d y,k := Λ -r k V k [ d2,1 , d2,2 , . . . , d2
,n ] ⊤ represent finitedifference approximation error, nonlinear deviation of feedback and presence of the output perturbation, respectively.

Since c 2 , c 3 > 0, then R 2,k has the following estimate:

R 2,k ≤ V µ k k x ⊤ Λ -r k V k M -1 Λ -r k V k x + V µ k k (n-1) 2 4Sn-1 (V -1 k ẋn ) 2 . Note that V -1 k ẋn = Θz k with Θ := [Y, B ⊤ Ξ 12 ] and z k = x ⊤ Λ -r k V k P, d ⊤ h,k P √ ϱ , d ⊤ µ,k P √ ∥σ∥ , d ⊤ y,k P √ b0 , d1 V k , x ⊤ a ⊤ V k √ ε ⊤ .
Term R 3,k either can be upper-bounded by using P6):

R 3,k ≤ -2h -1 (1 -x ⊤ Λ -r k V k P Λ -r k V k x)/(n -1) (D.3) or by applying Lemma 1 with ϑ = φ i , ϕ = ẋi+1 , ϖ = 1 and noting that d ⊤ h,k C ⊤ Cd h,k = 0: R 3,k ≤ -V -µ k k h -2 d ⊤ h,k N -1 d h,k /(n -1). (D.4)
Adding and subtracting corresponding terms to (D.2) to construct a quadratic form with respect to the vector z k and matrix Ξ := Ξ11 Ξ12 * Ξ22 , we obtain

Q ′ t,k (V k , x t , d) ≤ V µ k k z ⊤ k Ξ + Θ ⊤ (n-1) 2 4Sn-1 Θ z k +V µ k k x ⊤ Λ -r k V k M -1 -P ZP Λ -r k V k x + ϱR 3,k + V µ k k 2 n-1 (1 -x ⊤ Λ -r k V k P Λ -r k V k x) +(1 -ϱ) R 3,k + V µ k k 1 (n-1)ϱ d ⊤ h,k P d h,k +V µ k k 1 4(n-1) c 1 (V -1 k d 1 ) 2 + 2 ϵ (V -1 k ax) 2 -2 +V µ k 1 2(n-1) 1 b0 d ⊤ y,k P d y,k + 1 ∥σ∥ d ⊤ µ,k P d µ,k -2 -V µ k k 1 2(n-1) . (D.5)
Let us show that first six terms in (D.5) are nonpositive for all (V k , x t ) ∈ Ω k such that V 1 ∈ (max{v, w(∥d∥ ∞ )}, 1] and V 2 ∈ (max{1, w(∥d∥ ∞ )}, γ). Firstly, applying Schur complement to the first and the second terms, it is easy to see that they are not positive due to (14c).

Secondly, it follows from (D.3) that the third term in (D.5) is negative if ϱ/h > V µ k k , i.e. if v = γ-1 = h 1/ √ µ . Taking into account that (14c) implies N -1 ≽ ϱP , it is obvious that the fourth term is negative due to (D.4).

Thirdly, for all (V k , x t ) ∈ Ω k ILKFs ( 12) and (14b) imply (V -1 k ax) 2 ≤ (ϵ/2) max{V 2(r k,1 -1) k , V 2(r k,n -1) k }. Since r 1,1 > r 1,n > 1 and r 2,1 < r 2,n < 1, then it is clear that (V -1 k ax) 2 ≤ ϵ/2 for V 1 ≤ 1 and V 2 > 1. Moreover, one can see that c 1 (V -1 k d 1 ) 2 ≤ η 2 (V -1 k d) 2 ≤ 1 if w(s) ≥ ηs. Thus, the fifth term is also negative.

Finally, taking into account (11b), it can be proven that | d2,j | ≤ (2/h) j-1 ∥d 2 ∥ ∞ for j = 1, n. As a result, ∥d y,k ∥ 2 ≤ (2/h) 2n -1

(2/h) 2 -1 ∥d∥ 2 ∞ max{V -2r k,1 k , V -2r k,n k
} and d ⊤ y,k P d y,k ≤ b 0 if w(s) ≥ max{(ηs) 1/r1,1 , (ηs) 1/r2,1 }.

Then assuming that ∥d µ,k ∥ 2 ≤ ∥σ k ∥ 2 (see the proof in the next subsection), it is clear that the sixth term in (D.5) is negative due to (14c).

Since (D.1) implies that -1 ≤ V k

1 2ω k Q ′ V,k (V k , x t )
, one can conclude that conditions (5c) and (5d) are proven with v = γ-1 = h 1/ √ µ , w(s) = max{(ηs) 1/r1,1 , (ηs) 1/r2,1 } and θ -1 k = 4(n -1)ω k . Taking into account formulas of q 1,k and q 2,k , k = 1, 2 parameters γ, κ, T, v and function w ∈ K can be easily calculated using (4). 2 D.3 Proof of the estimate ∥d µ,k ∥ 2 ≤ ∥σ k ∥ 2

The disturbance term ∥d µ,k ∥ 2 can be rewritten as:

∥d µ,k ∥ 2 = n j=1 V -1 k |ỹ j | 1/r k,j -|V -r k,j k ỹj | +V -1 k |ỹ j | αj (∥ỹ∥) -|ỹ j | 1/r k,j 2 .
(D.6)

First, applying Lemma 1 with ϑ = φ i , ϕ = ẋi+1 , ϖ = ψ i /φ i to [START_REF] Gu | Stability of Time-Delay Systems[END_REF] for all (V k , x t ) ∈ Ω k such that V -µ k k ϱ/h > 1, we deduce that:

x ⊤ Λ -r k V k P Λ -r k V k x + n-1 i=1 i 2ϱζiSi V -r k,i+1 k δ i 2 ≤ 1.
Due to (14b) and upper bound on S i the following holds:

2V k -2r k,1 x 2 1 + 2 n-1 i=1 V k -2r k,i+1 (x 2 i+1 + δ 2 i ) ≤ 1.
Then Since V 1 ≤ 1 implies that ∥ỹ∥ ≤ b 1 , we deduce that α j (∥ỹ∥) = 1/r 1,j for V 1 ≤ 1. Therefore, the second term in (D.6) is zero and ∥d µ,1 ∥ 2 ≤ ∥σ 1 ∥ 2 .

∥Λ -r k V k ỹ∥ 2 ≤ 1 + 2∥d y,k ∥ 2 ≤ 1 + b 0 = b 2 1 . So it fol- lows that ∥ỹ∥ ≤ b 1 max{V r k,1 k , V
On the other hand, if |ỹ j | ≥ b 2 > b 1 for all j = 1, n, then ∥ỹ∥ ≥ b 2 and α j (∥ỹ∥) = 1/r 2,j . Thus, the second term in (D.6) for V 2 > 1 and |ỹ j | ≤ b 2 could be estimated as:

V -1
2 (|ỹ j | αj (∥ỹ∥) -|ỹ j | 1/r2,j ) ≤ max |ỹ|∈[0,b2] ||ỹ j | 1/r1,j -|ỹ j | 1/r2,j | = ḡ(b 1/r2,j 2 , r2,j r1,j ). Taking into account (D.7), one can finally conclude that ∥d µ,2 ∥ 2 ≤ ∥σ 2 ∥ 2 . 2

E Proof of Proposition 3

It is a well-known fact [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF] that the system (6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] with µ = 0 is exponentially ISS with a decay rate β ∈ (0, β 0 ), where β 0 > 0 is the decay rate of the corresponding state-feedback control, i.e. for all Φ ∈ W 1 ℏ and d ∈ D := {d ∈ L m ∞ : ∥d∥ ∞ < κ} there exists a constant c 0 > 0 and a function w ∈ K such that ∥x(t, Φ, d)∥ ≤ c 0 ∥Φ∥ W e -βt + w(∥d∥ ∞ ), ∀t ≥ 0.

Define by T 0 the moment of time when the system (6), [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] with time delay h 0 and µ = 0 reaches the set A, i.e. T 0 = inf{t ≥ 0 : ∥x(t, Φ, d)∥ ≤ v + w(∥d∥ ∞ )}. Obviously, T 0 ≥ max{0, β -1 0 ln(c 0 ∥Φ∥ W /(v + w(κ)))} if v + w(∥d∥ ∞ ) ≥ w(∥d∥ ∞ ). Otherwise, the set A is unreachable. Therefore, it is easy to see that T ≤ T 0 if ∥Φ∥ W ≥ γ 0 := e β0 2+(n+1)µ 4(n-1)(1+(n+1)µ)µ (v + w(κ))/c 0 . Clearly, there is a small enough h 0 such that γ ≥ γ 0 . 2

  Denote by N and R the sets of natural and real numbers, respectively, R ⋆

2 3

 2 Input-to-state stability of neutral systems Consider a functional differential equation of neutral type with external disturbance:

Fig. 1 .

 1 Fig. 1. Plots of φi(ξ) for i = 1, 5

  ) are always feasible provided ϵ, µ, h, b 1 and b 2 are sufficiently small. Obviously, this is true for ϵ = µ = h = 0 and b 1 = b 2 = 1. Indeed, taking into account that in this case ∥σ∥ = ϱ = b 0 = 0 and r k,j = ω k = 1, one can see that LMIs (14) hold for some 0 ≺ X ≼ I n /2, Y , Z ≽ 0 and sufficiently large c l , S i . Clearly, LMIs (14) remain feasible for some positive nonzero ϵ, µ, h and 1 < b 1 < b 2 since r k,j , ω k , σ k,j , ρ and b 0 are continuous functions of µ, h, b 1 and b 2 .

5

 5 Numerical simulations Let n = 3 and ϵ = 0.05. Then LMIs (14) are feasible for µ = 0.01, h = 0.02, b 1 = 1.001 and b 2 = 1.1. Therefore, K = [-3.11, -5.95, -4.14], γ = 1.25 • 10 15 , v = 5 • 10 -18 , η = 5 • 10 6 and κ = 6 • 10 9 . For further comparison we set a = [1, 1, 1] • 0.125 such that aa ⊤ = 0.047 < ϵ. The numerical simulation of the closed-loop system (

Fig. 2 .

 2 Fig. 2. The norm of the solutions x(t, Φ, 0) (disturbance-free case)

Fig. 3 .

 3 Fig. 3. The norm of the solutions x(t, Φ, d) (disturbed case)

b 1 .

 1 Thus, applying Lemma 2, the first term in (D.6) can be bounded as follows:|V -r k,j k ỹj | 1/r k,j -|V -r k,j k ỹj | ≤ ḡ(b 1 , 1/r k,j ). (D.7)

Table 1 .

 1 Values of ζi for i = 1, 5.

	i	1	2	3	4	5
	ζi	2	2.1577	2.3282	2.4614	2.5680


Clément Boin

Xavier Guillaume

Gilles Grimaud

Tristan Groléat

Michaël Hauspie

One Year of DDoS Attacks Against a Cloud Provider: an Overview

Keywords: Network security, Distributed Denial of Service, Cloud provider

Distributed denial of service attacks represents one of the most important threats to cloud-providers. Over the years, volumetric DDoS attacks have become increasingly important and complex. Due to the rapid adaptation of attackers to the detection and mitigation methods designed to counter them, the industry needs to constantly monitor and analyse the attacks they face. In this paper, we present an overview of the attacks that were perpetrated against our infrastructure in 2021. Our motivation is to give an insight of the challenge that DDoS attacks still represent within a large European cloud provider

I. INTRODUCTION

For several years, distributed denial of service attacks have been one of the most important threats of the modern Internet. All Internet actors, as soon as they occupy positions that are either strategic or important to the public, are susceptible to DDoS attacks [START_REF] Zargar | A survey of defense mechanisms against distributed denial of service (ddos) flooding attacks[END_REF]. As we have seen during the Covid-19 pandemic, services that are essential to our lives today have been compromised by this type of attack. For example, the French insurer AXA was the victim of a DDoS attack in May 2021 on its information system [START_REF] Sharma | Insurer axa hit by ransomware after dropping support for ransom payments[END_REF]. In July of the same year, the website of the Ukrainian Ministry of Defense announced that it had suffered such an attack [START_REF] Rfe | Ukraine blames russian hackers for attack on navy website[END_REF]. The motivations of the attackers can be multiple. There are the attackers who launch a DDoS attack using an automated attack system that they rent from a third party for no other specific reason that they can. Attackers can also be motivated by political ideology or a desire to arm a competitor -either financially or by altering its brand image. Attackers can also be willing to perform attacks for personal reasons (e.g., disgruntled employee that was layed off) [START_REF] Mirkovic | A taxonomy of DDoS attack and DDoS defense mechanisms[END_REF], [START_REF] Nazario | DDoS attack evolution[END_REF].

Due to its position in the European cloud industry, OVHcloud and its customers may be subject to DDoS attacks. Therefore, OVHcloud needs to provide protection against DDoS attacks that is up to date against most recent attacks. Therefore, attacks performed against the infrastructure must be analysed. This paper present an overview of such analysis on the attacks performed in 2021.

This paper is organized as follows: Section II presents the different works that analyze the attacks on a cloud provider infrastructure. In Section III we give an insight on the methodology used to gather information on the attacks so that they can be analysed. Section IV presents the results of the analysis This work is also supported by CNRS IRCICA USR-3380 performed on last year attacks. Section V, concludes the paper and gives perspective on how the kind of analysis we performed could be used to ellaborate datasets that can be used to develop now DDoS detection and mitigation techniques.

II. RELATED WORK

Every year, organisations and companies that deal with DDoS attacks -either by being targets of such attacks or by being a provider of protection solution -publish reports on the development of DDoS over the past year.

By studying the following reports [START_REF]Ddos attack trends for q1 2021[END_REF]- [START_REF]Ddos attacks in q3 2021[END_REF], we could get a insight of important characteristics of DDoS attacks, from these organisations point of view. These characteristics are mainly geographical distribution, used attack vectors and time and throughput characteristics.

We can draw the following conclusions. The two leading countries in terms of the number of attacks, whether directed at infrastructure based in these countries or emanating from infrastructure hosted in these countries, are the United States and China. The most observed attack vectors are UDP and TCP, and according to the reports, they follow each other very closely, so they represent most of the attacks. Akamai report states that attacks using more than one attack vector is an increasingly common phenomenon. Over 90% of attacks are less than 4 hours long and most attacks are small, using about 50,000 packets per second to be carried out. Finally, almost all attacks are operated from Botnets.

Although interessting, these reports are from actors whose activities differ from the ones of OVHcloud. Being a mainly european actor, the geographical aspect of the analysis may not yield the same results, should it be conducted on our data. Moreover, the type of service provided is also quite different as one of the main activity of OVHcloud is to provide baremetal environnement as opposed to specific services. The specific services are the responsability of OVHcloud clients. This makes the attack surface different from a SaaS provider and may show different results on attack vectors.

The next section will present the methodology used to gather information needed to produce reports on attacks characteristics. Those reports will then be analysed and we will draw conclusion with regards to the attacks characteristics specific to OVHcloud.

III. METHODOLOGY

The statistics presented in this paper are taken from the archives of our DDoS attack detection system. As part of the fight against DDoS attacks, we have set up collection systems at various locations in our infrastructure, which provide realtime network traffic statistics to our detection agents. The data collected are mainly ip flows information similar to NetFlow [START_REF] Systems | RFC 3954 -cisco systems NetFlow services export version 9[END_REF]. If certain properties of the NetFlow analyzer match characteristics of a traffic that is suspicious, we generate alerts. These alerts are effective for as long as the traffic is considered suspicious by our heuristics. It is these alerts that we find in our archiving database. An alert is composed of the IP address of the victim, a start date, an end date, the vector used to carry out the attack and the number of packets and bits per second that make up the alert. As indicated in the literature, an attack can be composed of several vectors, in which case there will be several alerts for the same IP address at a similar time interval.

The main attack vectors we have identified are TCP and its different variations (TCP-SYN, TCP-ACK, etc), ICMP, UDP, IP, NTP, DNS, Fragmentation and Chargen. From our point of view, an attack is a set of malicious network traffic to an IP address of one of our users that can use one or more vectors. So, whether an attack is launched by an attacker using a botnet network hypothetically composed of 500 machines or whether an attack is launched by two separate attackers with 250 machines each and who may or may not have been working together, is for us one and the same attack. Therefore, we had to apply additional processing to the data in the database where the alerts are stored to be able to represent an attack as we have just described it.

To construct the attacks, we used the following methodology: each alert is grouped according to the IP address of the victim. For each group, if two alerts have a start and end date that overlap, it means that they are part of the same attack, so they must be merged. Two cases are possible. First, the two alerts use different attack vectors. In this case, the number of packets and bits received must be calculated for the total duration of each attack and then the number of packets and bits received must be divided by the new duration of this attack, which corresponds to the minimum start date and the maximum end date of the two attacks. For example, in 1, there are three alerts that have the same destination IP.

The first one is an alert that lasts five seconds, the BPS and PPS of this alert is ten, the attack vector used is ICMP, the second one also lasts five seconds, for the same number of PPS and BPS but with the UDP vector, the last one lasts ten seconds, still with a number of PPS and BPS equal to ten, it uses the TCP vector. These three alerts form a single attack, a multi-vector attack (ICMP, TCP and UDP) with a total duration of ten seconds, a PPS and BPS equal to twenty as shown in Figure 2.

The second case is when our detection system updates the number of packets and bits seen per second, in which case both alerts will have the same attack vectors. In this case, we need to count the number of packets and bits on the first potential part where the attacks do not overlap, then take the maximum number of packets and bits for the period where the attacks overlap and then the number for the last potential part where the attacks do not overlap. In 2021, our DDoS detection systems issued more than three million attacks. Figure 3 represents the number of attacks by month during the year. The month in which the fewer attacks are observed is July, with only a bit over 200,000 attacks, while March and April had over 400,000 attacks each. The median number of DDoS attacks per month observed over the year was just under 264,000 attacks per month. In [START_REF]Ddos attack trends for q3 2021[END_REF], the authors claim to have observed the same order of magnitude of attacks per month on its DDoS attack protection service. Compared to the figures published by OVHcloud in 2017 on its blog, the most productive month in terms of attacks generated just over 75,000 attacks. An important difference is that in 2017 we accounted for attacks to be detected and mitigated, whereas today we consider an attack as soon as it is detected, it may not be sent to our mitigation systems, for multiple reasons. These attacks were directed against over 650,000 of our IPs out of a total of 4 million IPs allocated to our customers. Figure 4 shows the number of distinct IP addresses that have been the victim of at least one attack per month. A slight discrepancy can be observed between Figure 3 and Figure 4, although the trend is the same, and it can be explained by the fact, that some IP addresses are attacked more than one time as shown in Figure 5. On this figure, we can see that for 10% of our customers, DDoS attacks represent a daily threat, as they are targeted several times a month. It is for these customers that our detection and mitigation systems are most important, as DDoS attacks can generally impact their business activities. We can see between the months of March and June, an increase in attack activity is observed.

The month with the highest number of IP address with at least one attack is April with more than 215,000 IPs were attacked while August was the quietest month with 35,000 attacked IPs recorded. The most affected months are March to May. During this period of the year, the public exposure of OVHcloud has strongly increased. This attracted more attackers and explained the net increase in the number of attacks over this period. We also notice that the quietest months in terms of number of attacks correspond to the month where the professional activity is lower because of the holiday periods in Europe. It may be less interesting for the attackers to target services that are little used during these periods.

Figure 6 shows the number of DDoS attacks by time of day. We can see that the attacks are unevenly distributed throughout the day, with a peak of attacks occurring in the early evening between 5pm and 11pm. This distribution implies being able to detect as well as mitigate many attacks in a short period of time. The time of day when the attacks occur is not insignificant, it is generally when people go home and use many online services: video games, streaming platforms, etc. It is during these periods that the impact of an attack on our infrastructure would be more important.

Indeed, the slightest congestion would have a significant impact on the quality of service for all customers. Regarding the attack vectors, as shown in Figure 7 more than 45% of the attacks are syn-flood, as opposed to the figures published by [START_REF]Ddos attack trends for q1 2021[END_REF] this attack vector represents 54% of their attacks. TCP-based attacks represent most of the attacks observed at OVHcloud. If we add up the percentages of the different attack vectors known to be used over TCP, we get 55% of the attacks observed while [START_REF]Ddos attacks in q1 2021[END_REF] reported more than 75%. UDP represents around 20% of the total attacks, other vectors such as IP or ICMP share the rest.

What is interesting to note is that the most used attack vectors are also those that often rely on IP spoofing, which allows the attacks to stay anonymous. This technique also allows the implementation of amplification and reflection attacks. This technique is widely used by booters, DDoS-asa-Service providers [START_REF]Ddos attack trends for q1 2021[END_REF], [START_REF]Ddos attacks in q1 2021[END_REF]. The first conclusion is that online gaming services are the most targeted, with Minecraft game servers in the lead. For several years now, we have been observing attacks between game servers, who are engaged in real cyberwar. For example, the Hypixel community server, one of the largest Minecraft server, with over 24M total unique logins to date and a world record 216,000 concurrent players has undergone a campaign of multi-vector attacks with several attacks within this campaign peaking at over 600Gbps [START_REF]Ddos attack trends for 2021 q2[END_REF]. The motivation for attacking game servers seems to be mainly financial. Indeed, the biggest game servers generate revenues, which motivates "administrators" to attack each other to degrade the quality of service at their competitor's, and to convince players to join their servers.

In second place, we find e-commerce platforms. Surprisingly, the targeted platforms are very heterogeneous in size. They range from very large merchants to online shops with moderate traffic. It is rarely a question of rivality, but rather of extortion. The scheme is quite simple: after an initial attack, the attacker sends an e-mail to his target asking for money him to stop the attacks. For other services, it is more difficult to draw conclusions about the services targeted and the possible motivations of attackers. Startups, public administrations or news websites, everyone seems to be targeted by DDoS attacks, for different reasons: rivality between competitors, disputes between a service and customers, a desire for censorship in the case of the media, many motives are possible [START_REF] Nazario | DDoS attack evolution[END_REF].

For the statistics related to packets per second and bandwidth, we can see on figures 8 and 9 that the attacks of the first two months of the year the maximums in terms of pps and bps are correlated. Whereas the months of March and April, which as figure 3 terms of number of attacks, are only very representative in the figures. We can therefore deduce that despite the large number of attacks, these attacks were very low in pps and bps. On the other hand, we can see that larger attacks took place during the period from June to September, since although the number of attacks observed during these months is less important, these are the months with the highest number of attacks in pps and bps of the year.

V. CONCLUSION

In this paper, we present an overview of the attacks carried against the OVHcloud infrastructure in 2021. This overview can be used to give an example to the scientific community of the challenge that DDoS attacks still represent within a large European cloud provider. The activity of DDoS attacks is globally linked to the specificity of the targets, but also to that of their host. The propagation of attack vectors also seems to be correlated with the sector of activity of the target of these attacks. A service mainly based on HTTP will be targeted by attacks using TCP whereas a video game service is more likely to be attacked via UDP. The figures presented cannot be generalized to all cloud providers as variations in the heuristics used to detect a DDoS attack or even in the business model can for example change from one cloud provider to another. The statistics presented show what it looks like at a cloud provider, and we believe that they may premise the production of datasets that are more in line with what we observe. Therefore, we plan to produce a more accurate comparison between the datasets that are available to the scientific community and the network traffic that we observe in our infrastructure.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Alerts with the same attack vector

Fig. 3 .JFig. 4 .

 34 Fig. 3. Monthly observed attacks in 2021

JFig. 5 .

 5 Fig. 5. Frequency of attacks per month and per IP addres

Fig. 6 .

 6 Fig. 6. Occurrence of attacks by day of the week and time (UTC) By taking a closer look at the IPs targeted by DDoS attacks and the kind of infrastructure, we can draw the following conclusions about the attacked services at OVHcloud. The gaming and e-commerce related infrastructures are most attacked, the same as in our previous analysis in 2017. In fact, if we look

Fig. 8 .Fig. 9 .

 89 Fig. 8. Average attacks over the months, expressed in packets per second (pps) and the largest attacks per month in terms of pps

  Fig. 7. Attack vectors used in attacks targeting OVH IP addresses in 2021at the UDP attack vector that is most widely used for video game protocols, we see a slight increase in the latter during the holiday periods in Europe. Even though games such as Minecraft are mainly based on TCP, the fact that the UDP attack vector increases during certain periods shows that DDoS attacks are often linked to the activity of the target.

	Vectors percentage	0% 20% 40% 60% 80% 100%	January	February	March	April	May	June Months July	August	September	October	November	December	chargen dns fragment icmp ip_null ntp other tcp_ack tcp_fin tcp_null tcp_psh tcp_rst tcp_syn udp


Bruno Neckel Wesling
email: bruno.neckel-wesling@u-bordeaux.fr

Marina Deng

Chhandak Mukherjee

Magali De Matos

Abhishek Kumar

Guilhem Larrieu

Jens Trommer

Thomas Mikolajick

Cristell Maneux

Extraction of small-signal equivalent circuit for de-embedding of 3D vertical nanowire transistor

In this paper, we present an improved methodology to extract the small-signal electrical equivalent circuit of the parasitic elements using RF test structures for a 3D vertical nanowire transistor technology. The methodology is based on the extraction of the distributed parasitic elements from an open structure for which on-wafer S-parameter measurements were carried out up to 40 GHz. The electrical equivalent circuit of the passive device was then used for de-embedding of the transistor S-parameters for extraction of intrinsic smallsignal parameters such as the gate capacitances.

Introduction

In the context of emerging technologies for new computing paradigms such as neural networks, vertical nanowire field effect transistors (VNWFET) are a promising technology owing to their naturally 3D architectures [START_REF] Maneux | Modelling of vertical and ferroelectric junctionless technology for efficient 3D neural network compute cube dedicated to embedded artificial intelligence[END_REF]. For the circuit design using such a 3D emerging technology, it is necessary to build new and accurate compact models to capture the static and dynamic behavior of the device [START_REF] Mukherjee | Compact modeling of 3D vertical junctionless gate-all-around silicon nanowire transistors towards 3D logic design[END_REF]. However, on-wafer test structures of the devices contain parasitic elements introduced by the RF pads and interconnects that are necessary to probe the devices under test (DUT). The standard procedure to characterize the parasitic effects of a device at high frequencies can be divided into two parts. Firstly, the measurement system needs to be calibrated so that the reference plane is at the probe tips. This can be done using standard calibration techniques such as SOLT (Short-Open-Load-Thru). At this point, all the parasitic elements associated with the pads and interconnects are still included in the measurements. In the second step, S-parameters of the open and short structures are measured which are then used to remove the parasitic elements external to the DUT, through a subtraction process known as de-embedding [START_REF] Koolen | An improved de-embedding technique for on-wafer high-frequency characterization[END_REF].

Although the process of parasitic extraction is widely used in technologies targeted for high frequency applications, this process can also be applied for technologies such as the VNWFET that are mainly designed for computing applications to improve the modeling accuracy, which normally just relies on DC measurements [START_REF] Maneux | Modelling of vertical and ferroelectric junctionless technology for efficient 3D neural network compute cube dedicated to embedded artificial intelligence[END_REF]. Moreover, an accurate modeling of the parasitic elements allows the designer to optimize the 3D layout of the logic circuits and interconnects, thereby improving performance metrics such as compactness and energydelay product of the technology.

In this paper, we present an elaborate methodology for extracting the extrinsic interconnect parasitic network associated with the unconventional vertical and inherently 3D test structures of the VNWFETs. In the rest of the paper, results from the RF measurements and electro-magnetic (EM) simulations are used for the extraction of parasitic elements of the electrical equivalent circuit of an open test structure. Finally, we show the smallsignal capacitances of a VNWFET extracted using two different de-embedding processes.

Methodology

For de-embedding, open structures were fabricated using the same process and on the same wafer as the VNWFETs [START_REF] Larrieu | Sub-15nm gate-all-around field effect transistors on vertical silicon nanowires[END_REF]. The open device consists of all interconnects except for the vertical nanowires. Fig. 1(a) shows the open structure containing the RF pads, where the absence of the active device can be observed. For the demonstration of the de-embedding process, a VNWFET consisting of 81 nanowires in parallel with a diameter of 17 nm each was selected. A drawing of a full transistor can be seen on Fig 1(c).

On-wafer S-parameter measurements

The on-wafer S-parameters were measured using a Vector Network Analyzer (VNA) from Rohde & Schwarz up to 40 GHz, and a source monitor unit, Agilent 4142B, connected via a bias-tee. The DUTs were contacted using Ground-Signal-Ground (GSG) probes with a pitch of 100 μm following a SOLT calibration. A second test bench that allows S-parameter measurements up to 110 GHz, Agilent® E8361 PNA, was used for the open device to ascertain the credibility of the measurements.

Electromagnetic simulation

The 3D view of the open device constructed for EM simulations is shown in Fig. 1(b). The ADS-Momentum EM simulator was used for the computation of the S-parameters of the provided layout. To eliminate manual errors, the layout was imported into ADS using the same GDSII file that was used for the fabrication process. The electrical properties of the material layers were set based on the fabrication process [START_REF] Larrieu | Sub-15nm gate-all-around field effect transistors on vertical silicon nanowires[END_REF]. A cross-section of the materials stack is shown in Fig. 1(c). In Fig. 3, a comparison between the measured and the EM simulated S-parameters of the open device is shown. The EM simulation results exhibit very good agreement with the measured data up to 40 GHz. Considering that an open structure can be represented by a π-network consisting of three capacitors, the values of these capacitances were extracted from the Y parameters, which were obtained using the standard S to Y parameter transformation method. The capacitances are presented in Fig. 4 for both measurements and EM simulation. The capacitances obtained from the second measurement setup that reaffirmed the measurement results are presented as C1_MEAS_110G, C2_MEAS_110G and C12_MEAS_110G respectively. The capacitance C1 extracted from the EM simulation showed good agreement with the measurements beyond 10 GHz. Below 10 GHz, discrepancies can be observed that might be related to the limitation of EM simulation in reproducing all the details of the fabricated structures. This indicated that the measurement of the open device below 10 GHz cannot be modeled as a simple π-network. A more complete and lumped model was then investigated taking into account the EM simulation of different sections of the complete open structure.

Distributed equivalent circuit of the transistor interconnects

To construct the small-signal equivalent circuit (SSEC) of the open device, EM simulations of different sections of the complete structure were performed to estimate their individual contributions. The pad was considered as an LC network in series with a contact resistance and a π-network was used to model the coupling capacitances between the gate, drain, and the source accesses. The pad capacitance and impedance are estimated from the EM simulations of the virtual OPEN_PAD and SHORT_PAD structures presented in Fig. 5 (a) and (b), respectively. The coupling capacitances are obtained from the OPEN_DUT shown in Fig. 5 (c). A figure of the substrated used on ADS for the EM simulation is shown on Fig. 5 (d). The extracted elements are then combined to construct the distributed circuit 'model1'.

The capacitance C1_MODEL1 in Fig. 6, extracted from 'model1', does not match well with the measurement data. This means that not all parasitic effects were accounted for, and a more detailed circuit is needed [START_REF] Lee | Equivalent Circuit Model for Si Avalanche Photodetectors Fabricated in Standard CMOS Process[END_REF]. Analyzing the device layout and the fabrication process one can identify that substrate parasitic effects might be present due to (i) a low silicon substrate resistivity [START_REF] Larrieu | Sub-15nm gate-all-around field effect transistors on vertical silicon nanowires[END_REF] and (ii) the absence of a ground shield metal underneath the signal pads [START_REF] Han | Impact of pad de-embedding on the extraction of interconnect parameters[END_REF]. The substrate parasitic elements introduce a lossy path, Cpad2-Rsub-Cpad3, shown in Fig. 7 along with a fringe capacitance Cpad1 [START_REF] Aktas | Pad de-embedding in RF CMOS[END_REF]. The final SSEC is presented in Fig. 7, denoted as 'model2'. The values of the components in the distributed circuit were a mix of values estimated mathematically from the physical dimensions and material properties of the device as well as from EM simulations. Finally, the capacitance C1_MODEL2 extracted from the SSEC 'model2', presented in Fig. 6, shows an excellent agreement with the measurement.

Transistor de-embedding

Two different de-embedding methods were used in this work. The first one is a simple open de-embedding process where the Y parameters of the open device are subtracted from those of the transistor, referred to as OPEN_deemb. The second approach is based on the methodology described in [START_REF] Deng | Benefits and validation of 4-dummies de-embedding method for characterization of SiGe HBT in G-band[END_REF], referred to as the accessmodeling based de-embedding (AM_deemb) henceforth. For this, ABCD matrices were used to remove the gate and drain parasitic access contributions, for which the matrix [Atransistor_meas] of the complete transistor measurement was multiplied by the inverse of MATRIX Agate and MATRIX Adrain, obtained from the equivalent circuit in Fig. 8,

[𝐴 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟1 ] = [𝐴 𝑔𝑎𝑡𝑒 ] -1 × [𝐴 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟_𝑚𝑒𝑎𝑠 ] × [𝐴 𝑑𝑟𝑎𝑖𝑛 ] -1 (1)
The Y-parameters of the CGD_access (Fig. 8) were then subtracted from the Y parameters obtained from the ABCD-to-Y transformation of matrix [𝐴 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟1 ] in (1) as,

[𝑌 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟2 ] = [𝑌 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟1 ] -[𝑌 𝐶 𝐺𝐷_𝑎𝑐𝑐𝑒𝑠𝑠 ] (2)
Y-to-Z parameter transformation is then performed on [𝑌 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟2 ] and subsequently the impedance LS_access (Fig. 8) was removed by subtracting its Z-parameters:

[𝑍 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟_𝑑𝑒𝑒𝑚𝑏 ] = [𝑍 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟2 ] -[𝑍 𝐿 𝑆_𝑎𝑐𝑐𝑒𝑠𝑠 ] (3)
The final transistor S parameters were obtained from the transformation of the [𝑍 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟_𝑑𝑒𝑒𝑚𝑏 ] parameters obtained from eq. ( 3).

De-embedding results

The gate and drain capacitances, CGS and CDS, obtained from the de-embedded Y-parameters of the transistor at a VGS of -1 V and a VDS of -0.1 V are shown in Fig. 9. The data was truncated at 20 GHz since the VNWFET has no RF gain at higher frequencies and the measurement noise becomes dominant. The CGS obtained from the AM_deemb has a mean value of 37 fF compared with 48 fF for the OPEN_deemb. The CDS shows a flat value between 5 and 12 GHz for the AM_deemb, while an unphysical negative value was obtained from the OPEN_deemb. The CDS frequency increase above around 12 GHz is a indicator of the limitation in the de-embedding method, even though the new proposed method provides more physical results compared with the conventional method.

Conclusions

In this work, the small-signal equivalent circuit of the VNWFET interconnects was achieved by using the measurements from the open structure and predictive EM simulation. Adding the substrate parasitic network to the distributed SSEC is crucial for higher accuracy of de-embedding since the capacitance of the open device is more than the intrinsic capacitance of the transistor. The extraction of capacitances could be achieved up to 20 GHz, demonstrating a promising efficiency of our proposed method. In the future scope of this work, the extracted values of the gate capacitances will be verified from 3D TCAD simulation of the transistors.

Figure 1 :

 1 Figure 1: (a) Open structure for S-parameters measurements, (b) a 3-D view of the open structure created using ADS MOMENTUM and (c) the drawing of the VNWFETs showing the nanowires.

Figure 2 :

 2 Figure 2: Drawing of the cross-section of the transistor presenting the stack of materials.

Figure 3 :

 3 Figure 3: Comparison of S-parameters from measurements and electromagnetic simulation for the open test structure.

Figure 4 :

 4 Figure 4: Open structure capacitances extracted from measurements and EM simulation up to 20 GHz. (a) C1, (b) C2 and (c) C12. The inset in (a) shows the equivalent π-network.

Figure 5 :

 5 Figure 5: Virtual structures for the EM simulation, a) OPEN_PAD, b) SHORT_PAD, c) OPEN_DUT and (d) the substrate definition on ADS for EM simulation.

Figure 6 :

 6 Figure 6: Capacitances as a function of frequency extracted from measurements and electrical simulation up to 20 GHz. (a) C1, (b) C2 and (c) C12.

Figure 7 :

 7 Figure 7: 3D view of the distributed model.

Figure 8 :

 8 Figure 8: Electrical equivalent circuit representing the parasitic elements obtained from the open test structure that is used for the de-embedding of VNWFET.

Figure 9 :

 9 Figure 9: Small signal capacitances as a function of frequency for the de-embedded transistor at a VGS = -1V and VDS= -0.1V.

Acknowledgments

This work was supported by the project FVLLMONTI funded by the European Union's Horizon 2020 research and innovation program under grant agreement N•101016776, by LEGO project (Grant ANR-18-CE24-0005-01) and by the LAAS-CNRS micro and nanotechnologies platform, a member of the Renatech French national network.