Andrey Polyakov

Denis Efimov
email: denis.efimov@inria.fr

Xubin Ping
email: pingxubin@126.com

Consistent discretization of homogeneous finite/fixed-time controllers for LTI systems

Keywords: nonlinear control, sampled-time control, homogeneous systems

Introduction

By definition, the homogeneity is a dilation symmetry introduced by Leonhard Euler in 18th century as follows: f (λx) = λ ν f (x), ∀λ > 0, where the coordinate transformation x → λx is know today as a standard (or Euler) dilation. A weighted (generalized) dilation is studied since 1950s. An introduction to stability theory of weighted homogeneous Ordinary Differential Equations (ODEs) can be found in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogeneous righthand sides[END_REF]. Extensions of the homogeneity theory to various finite-dimensional and infinitedimensional dynamical models are proposed in [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous righthand sides[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. Homogeneous differential equations/inclusions form an important class of control system models [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF]. They appear as local approximations [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF] or set-valued extensions [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] of nonlinear systems and include models of process control [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], mechanical systems with frictions [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], fluid dynamics [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], etc. Stability and stabilization problems were studied for both standard [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Andreini | Global stabilizability of homogenenous vector fields of odd degree[END_REF] and weighted homogeneous [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Sepulchre | Homogeneous Lyapunov Functions and Necessary Conditions for Stabilization[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Nakamura | Homogeneous stabilization for input-affine homogeneous systems[END_REF] systems which are the most popular today [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF], [START_REF] Mercado-Uribe | Multiple-input multiple-output homogeneous integral control design using the implicit lyapunov function approach[END_REF]. A homogeneous model predictive control is introduced in [START_REF] Coron | Model predictive control, cost controllability, and homogeneity[END_REF].

An asymptotically stable homogeneous system is finitetime stable in the case of negative homogeneity degree and nearly fixed-time stable in the case of the positive homogeneity degree (see, e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]). However, the finite/fixed-time stability is a fragile property, since an arbitrary small measurement delay or an improper discretization of a finite-time or a fixed-time stable ODE may result in a chattering [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Levant | Chattering analysis[END_REF] or even in a finite-time blow up [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF]. Existing approaches to sampled-time implementation of nonlinear controllers (see, e.g. [START_REF] Nesić | Sufficient conditions for stabilization of sampled-data systems via discrete-time approximations[END_REF], [START_REF] Nesić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]) do not study these issues. The explicit discretization (sampled-time implementation) of a finite-time control yields a chattering even if the original control law is a continuous function of state [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], [START_REF] Josse | Robustness of sampled-data homogeneous systems[END_REF]. That is why the discretization issues are very important for practical implementation of finite/fixed-time control/estimation algorithms [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Kikuuwe | Proxy-based sliding mode control: A safer extension of pid position control[END_REF], [START_REF] Livne | Proper discretization of homogeneous differentiators[END_REF], [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF], [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF], [START_REF] Brogliato | Digital implementation of sliding-mode control via the implicit method: A tutorial[END_REF], [START_REF] Michel | A noise less-sensing semi-implicit discretization of a homogeneous differentiator : principle and application[END_REF], [START_REF] Hanan | Low-chattering discretization of sliding mode control[END_REF].

The concept of consistent discertization introduced in [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF] postulates that stability properties of a continuoustime system must be preserved in its discrete-time counterpart (approximation). Consistent discretizations for stable generalized homogeneous ODEs were developed in [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF], [START_REF] Sanchez | Lyapunovbased consistent discretisation of stable homogeneous systems[END_REF] based on Lyapunov function theory. Some schemes with state dependent discretization step were given in [START_REF] Efimov | Discretization of homogeneous systems using euler method with a state-dependent step[END_REF]. Being efficient for numerical simulations, the mentioned schemes do not allow a consistent discretization (sampled-time implementation) of finite-time controllers in the general case. To the best of authors' knowledge, such implementations are developed only for some particular first order (see, [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF]) and some particular second order systems (see, [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF], [START_REF] Brogliato | The implicit discretization of the super-twisting slidingmode control algorithm[END_REF]). This paper presents a consistent discretization Preprint submitted to Automatica for a homogeneous controller designed in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] for multidimensional linear plants. It is shown that the sampled-time implementation of the controller according to the developed scheme preserves the finite-time and nearly fixed-time stability property of the original closed-loop continuous-time system in the disturbancefree case. We also prove an Input-to-State Stability (ISS) of the obtained sampled-time control system with respect to bounded additive perturbations and measurement noises. Algorithms are developed for both single-input and multi-input models. Numerical simulations show an efficiency of this scheme and a complete rejection of the so-called numerical chattering [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF] caused by a sampled-time implementation of a continuous-time control algorithm. The preliminary version of this paper [START_REF] Polyakov | Consistent discretization of a homogeneous finite-time control for a double integrator[END_REF] considers only the case of a finite-time stabilizer for a planar (two-dimensional) system. Notation: N is the set of natural numbers including 0; R is the field of real numbers; R + = {α ∈ R : α ≥ 0}; 0 is the zero of a vector space; I n ∈ R n×n is the identity matrix; e i = (0, ..., 0, 1, 0, ..., 0) ∈ R n is the i-th element of the Euclidean basis; W 0 means positive definiteness of a matrix W = W ∈ R n×n and λ max (W ) is a maximal eigenvalue of W ; x =

√

x P x is the weighted Euclidean norm in R n with P 0 specified below in each case when P is not arbitrary; A = sup x =0 Ax x for A ∈ R n×n ; S = {x ∈ R n : x = 1}; K is the class of strictly increasing positive definite continuous functions R + → R + ; γ ∈ K is of the class K ∞ if γ(s) → +∞ as s → +∞; σ : R + ×R + → R + is of the class KL if the function s → σ(s, τ ) is of the class K for any fixed τ ∈ R + and the function τ → σ(s, τ ) is monotonically decreasing to zero for any fixed s ∈ R + ; L ∞ (R, R n ) is the space of the essentially bounded measurable functions q : R → R n and q L ∞ ((a,b),R n ) = ess sup t∈(a,b) q(t) ; ∞ is a space uniformly bounded sequences; diag{a 1 , ..., a n } ∈ R n×n is a diagonal matrix.

Problem Statement

Let us consider a linear control system

ẋ(t) = Ax(t) + Bu(t), t > 0, x(0) = x 0 ∈ R n , (1) where x(t) ∈ R n is the system state, u(t) ∈ R m is the control input, A ∈ R n×n , B ∈ R n×m are known matrices.
Definition 1 Let the system (1) with a feedback control u(t) = ũ(x(t)), ũ ∈ C(R n \{0}, R m ) be globally uniformly finite-time 1 (resp. nearly fixed-time) stable. A family of

1 A system ẋ = f (t, x), x(0) = x0 is globally uniformly • Lyapunov stable if ∃σ ∈ K∞: x(t, x0) ≤ σ( x0 ), ∀t ≥
0, ∀x0 ∈ R n and for any solution x(t, x0) of the system; • finite-time stable if it is globally uniformly Lyapunov stable and there exists a locally bounded function T : R n → R+ such that any trajectory of the system vanishes to zero functions ũh : R n → R m parameterized by a scalar h > 0 is said to be a consistent discretization of ũ if

• Consistency of Stability: the system (1) with

u(t) = ũh (x(t i )), t ∈ [t i , t i+1 ), t i = ih, i ∈ N (2)
is globally uniformly finite-time (resp., nearly fixedtime) stable for any h > 0;

• Control Approximation: ∀r 1 > 0, ∀r 2 > r 1 ,∃ ω r ∈ K : sup r1≤ x ≤r2 ũh (x) -ũ(x) ≤ ω r (h), ∀h > 0. (3)
The first condition of Definition 1 asks the sampled-time control system to preserve the stability property of the original system for any fixed sampling period h > 0. The second condition guarantees that the control ũh is, indeed, an approximation of ũ, i.e., ũh (x) → ũ(x) as h → 0 + uniformly on compacts from R n \{0}. The origin is excluded since a finite-time stabilizing feedback is always non-smooth or even discontinuous at zero. The aim of the paper is to propose a possible method of consistent discretization of finite/fixed-time controllers for LTI systems and to design a consistent discretization for one particular class of generalized homogeneous controllers studied in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF]. First, we design a universal discretization being a mixture of feedforward/feedback algorithms, which guarantees an exact tracking of the states of the original continuous-time closed-loop system at time instances t nk , k ∈ N. Next, we present the consistent (in the sense of the above definition) discretization scheme and study its robustness under the condition: Assumption 1 The pair {A, B} is controllable, the matrix A is nilpotent and m = 1.

Recall [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] that a linear system is generalized homogeneous of non-zero degree if and only if A is nilpotent. Finally, we generalize both schemes to multi-input systems, which can be decomposed into a cascade of singleinput subsystems satisfying Assumption 1.

Assumption 2 Let us assume that

A = A1 * ... * * 0 A2 ... * * ... ... ... ... ... 0 0 .... Am-1 * 0 0 .... 0 Am , B = B1 0 ... 0 0 0 B2 ... 0 0 ... ... ... ... ... 0 0 ... Bm-1 0 0 0 ... 0 Bm, , (4)
where

A i ∈ R ni×ni , B i ∈ R ni , n 1 + n 2 + .... + n m = n,
* is a (possibly) nonzero block. The pair {A i , B i } is controllable and the matrices A i are nilpotent, i = 1, ..., m.

in a finite time: x(t, x0) = 0, ∀t ≥ T (x0), ∀x0 ∈ R n ; • nearly fixed-time stable if it is globally uniformly Lyapunov stable and ∀r > 0, ∃Tr > 0:

x(t, x0) < r, ∀t ≥ Tr, ∀x0 ∈ R n .
If the pair {A, B} is controllable and rank(B) = m then there exists a coordinate transformation [START_REF] Luenberger | Canonical forms for linear multivariable systems[END_REF] to a canonical form similar to (4). Assumption 2 asks that {A, B} is controllable, A is nilpotent, rank(B) = m and the system admits a transformation to the block form (4).

3 Preliminaries: Homogeneous systems

Linear dilation and homogeneous norm

The so-called linear (geometric) dilation [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 6] in R n is given by

d(s) = e sG d = ∞ i=0 (sG d ) i i! , s ∈ R, (5)
where G d ∈ R n×n is an anti-Hurwitz matrix2 known as the generator of linear dilation. The latter guarantees that d satisfies the limit property, d(s)x → 0 as s → -∞ and d(s)x → +∞ as s → +∞, required for a group d to be a dilation in R n (see, e.g., [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF]). The linear dilation introduces an alternative norm topology in R n by means the so-called canonical homogeneous norm.

Definition 2 [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] The function • d : R n → R + given by x d = 0 for x = 0 and

x d = e sx , where s x ∈ R : d(-s x )x = 1, x = 0 (6)
is called the canonical homogeneous norm in R n , where d is a linear monotone dilation3 .

Notice that x = 1 (resp. x ≤ 1) is equivalent to

x d = 1 (resp. x d ≤ 1). For the uniform dilation d(s) = e s I n , s ∈ R we have • = • d .
Theorem 1 [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] If d is a monotone dilation and x = √

x P x with a symmetric matrix P ∈ R n×n satisfying P G d +G d P 0, P 0 then the canonical homogeneous norm • d is continuous on R n and smooth on R n \{0}:

∂ x d ∂x = x d x d (-ln x d )P d(-ln x d ) x d (-ln x d )P G d d(-ln x d )x , ∀x = 0; (7) Moreover, σ( x ) ≤ x d ≤ σ( x ), ∀x ∈ R n , with σ(r) = r 1/α if r≥1, r 1/β if r<1, σ(r) = r 1/β if r≥1, r 1/α if r<1, where α = 0.5λ max P 1/2 G d P -1/2 +P -1/2 G d P 1/2 > 0 and β = 0.5λ min P 1/2 G d P -1/2 +P -1/2 G d P 1/2 > 0.
Below the canonical homogeneous norm is utilized as a Lyapunov function for analysis and control design.

Remark 1 (On computation of • d ) Since the canonical homogeneous norm is defined implicitly, a computational algorithm is required for its practical implementation. Issues of numerical estimation of • d are studied in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] based on a bisection method. In [45, Chapter 9.2.4] a scheme for an approximation of • d by an explicit homogeneous function is presented.

Homogeneous continuous-time systems

Definition 3 [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF] A vector field f : R n → R n (resp. a function h : R n → R) is said to be d-homogeneous of degree

µ ∈ R if f (d(s)x) = e µs d(s)f (x) (resp. h(d(s)x) = e µs h(x)), for all x ∈ R n , s ∈ R.
If f is d-homogeneous of degree µ then solutions of ẋ = f (x) are symmetric [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF]: x(e -µs t, d(s)x 0 ) = d(s)x(t, x 0 ), where x(t, z) denotes a solution with x(0) = z. For instance [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF], the linear vector field

x → Ax, A ∈ R n×n is d-homogeneous of the degree µ = 0 ⇔ A is nilpotent ⇔ AG d = (µI n + G d )A.
The homogeneity degree specifies the convergence rate.

Theorem 2 [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Nakamura | Positive definiteness of generalized homogeneous functions[END_REF] Let f : R n → R n be dhomogeneous of a degree µ ∈ R. If the system ẋ = f (x) is asymptotically stable then it is globally uniformly finite-time (nearly fixed-time) stable for µ < 0 (µ > 0).

The homogeneous control systems are robust (ISS) with respect to a rather large class of perturbations [START_REF] Hong | H ∞ control, stabilization, and inputoutput stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

Homogeneous stabilization of linear plant

The following theorem merges results of [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF], [START_REF] Nekhoroshikh | Finite-time stabilization under state constraints[END_REF].

Theorem 3 Let a pair {A, B} be controllable. Then 1) any solution Y 0 ∈ R m×n , G 0 ∈ R n×n of the linear algebraic equation

AG 0 -G 0 A + BY 0 = A, G 0 B = 0 (8)
is such that the matrix G 0 -I n is invertible, the matrix

G d = I n + µG 0 is anti-Hurwitz for any µ ∈ [-1, 1/ñ],
where ñ is a minimal natural number such that rank[B, AB, ..., A ñ-1 B] = n, the matrix

A 0 = A + BY 0 (G 0 -I n ) -1 satisfies the identity A 0 G d = (G d + µI n )A 0 , G d B = B; (9)
2) the linear algebraic system

A 0 X +XA 0 +BY +Y B +ρ(G d X +XG d ) = 0, G d X + XG d 0, X = X 0 (10)
has a solution X ∈ R n×n , Y ∈ R m×n for any ρ > 0;

3) the canonical homogeneous norm • d induced by the weighted Euclidean norm x = √ x P x with P = X -1 is a Lyapunov function of the system (1) with

u = K 0 x + x 1+µ d Kd(-ln x d )x, (11)
K 0 = Y 0 (G 0 -I n ) -1 , K = Y X -1 , ( 12
)
where d is a dilation generated by G d ; moreover,

d dt x d = -ρ x 1+µ d , x = 0; (13)
4) the feedback law u given by [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] is continuously differentiable on R n \{0}, u is continuous at zero if µ > -1 and u is discontinuous at zero if µ = -1; 5) the system (1), ( 11) is d-homogeneous of degree µ.

Obviously, due to (13) the closed-loop system (1), [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] is uniformly finite-time stable if µ < 0 and it is nearly fixed-time stable if µ > 0. For µ = 0 the control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] becomes u = K 0 x + Kx. Such a control law (under some variations and/or simplifications) has been presented in the literature as a solution to a finite-time stabilization problem for linear plants [START_REF] Korobov | A solution of the synthesis problem using controlability function[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF].

Remark 2 Under Assumption 1, the equation (8) has a unique solution such that Y 0 = 0 (i.e., A 0 = A) and ∃J ∈ R n×n : J -1 G 0 J = -diag{n -1, ..., 1, 0}. This follows from the fact then the system (1), in this case, is equivalent to a controlled integrator chain.

A topological equivalence of any stable d-homogeneous system to a standard homogeneous one [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] allows an explicit representation of solution for (1), [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] to be derived.

Corollary 1 (Explicit representation of solutions)

Under conditions of Theorem 3 with µ = 0, a solution of the closed-loop system (1), ( 11) is unique and

x(t + τ ) = Q τ ( x(t) d )x(t), ( 14
)
where τ, t ≥ 0, Q τ (0) = 0 and for r > 0 one has

Q τ (r) = e G d ln r Q ln(1+µρτ r µ ) ρµ e -G d ln r if 1 r µ >-µρτ, 0 if 1 r µ ≤-µρτ, (15)
Q(ŝ) = e -ρG d ŝe (A+B(K0+K)+ρG d )ŝ , ŝ ≥ 0. ( 16
)
The proof of this corollary, as well as the proofs of other main and auxiliary results, are given in the Appendix. The matrix-valued function Q nh (•) can be easily computed, since elements of a matrix exponential e sM can always be represented as polynomial functions of s, e ρi , cos(ω i s) and sin(ω i s), where ρ i + iω i are eigenvalues of the matrix M ∈ R n×n . Moreover, if

A 0 = A + BK 0 , B, K, G d satisfy (10) then the matrix X 1/2 (A 0 + BK + ρG d )X -1/2 is skew-symmetric and e (A0+BK+ρG d )φ = X -1/2 R(φ)X 1/2 , ( 17
)
where R(φ) is a rotation matrix for any φ ∈ R, i.e.,

R(φ)R (φ) = R (φ)R(φ) = I n .
Corollary 2 (On cascade homogeneous control) Let Assumption 2 be fulfilled. Let G di ∈ R ni×ni , K i ∈ R 1×ni , P i ∈ R ni×ni and the control u i (x i ) with x i ∈ R ni be defined by Theorem 3 for the pairs {A i , B i } and some µ i ∈ [-1, 1/ñ i ], ρ i > 0 , respectively. Then the system (1) with the control u = (u 1 , ..., u m ) is globally uniformly finite-time stable if µ i < 0 (resp., nearly fixed-time stable if µ i > 0) for all i = 1, 2, ..., m.

Discretization of Homogeneous Control

Single-input case

Let us represent the system (1) with the sampled-time control

u(t) = u(t k ) for t ∈ [t k , t k+1
) in the form:

x k+1 = A h x k + B h u(t k ), k ∈ N, (18)
where

x k = x(t k ), t k = kh, A h = e hA and B h = h 0 e sA Bds.
The system (18) can be rewritten as follows:

x k+n = B h u(t k+n-1 )+...+A n-1 h B h u(t k )+A n h x k . ( 19
)
The controllability of the pair {A, B} implies the controllability of the pair {A h , B h } and the invertability of

W h = [B h , A h B h , ...., A n-1 h B h ] (20)
(see the formulas ( 45), [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] and Lemma 5 in Appendix).

Let parameters of a stabilizing homogeneous controller [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] be designed as in Theorem 3 for µ = 0. By Corollary 1, to track a trajectory of the continuous-time closedloop system (1), [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], the sampled-time control just has to fulfill the following identity

Q nh ( x k d )x k = B h u(t k+n-1 )+...+A n-1 B h u(t k )+A n h x k .
Indeed, if a sampled-time control is implemented as

u(t k+n-1 ) ... u(t k ) = W -1 h (Q nh ( x k d ) -A n h ) x k , (21)
then the discrete-time system ( 18), ( 21) tracks any trajectory of the continuous-time system (1), [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF]

at t = t kn , k ∈ N. In particular, if µ < 0 then for any x 0 ∈ R n there exists k * ∈ N such that x k = 0, ∀k ≥ k * .

Theorem 4

The system (1) with the sampled-time control ( 21) is globally uniformly finite-time stable if µ < 0 (nearly fixed-time stable if µ > 0).

Since u(t k+i-1 ) depends on x k = x(t k ) but not on x(t k+i-1 ), then the discretization (21) of the control (11) could be useful, for example, if the control sampling is n times faster than a measurement sampling. In other cases, the control ( 21) is a mixture of feedforward and feedback algorithms, where the state measurements x(t k+i-1 ) for i = 2, ..., n -1 are simply omitted during the control implementation. This could badly impact to a robustness and to a precision of the sampled-time controller. To avoid this drawback, let us consider the static feedback law

ũh (x k ) = Kh ( x k d )x k , (22)
Kh (

x k d ) = e n W -1 h (Q nh ( x k d ) -A n h ) , (23)
which is obtained from ( 21) selecting only u(t k ).

Proposition 1 (Approximation property) Let u be a homogeneous control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] designed by Theorem 3 under Assumption 1. Then ũh (x) → u(x) as h → 0 + uniformly on compacts from R n \{0}. This proposition, in particular, implies that for a sufficiently small h > 0 the system ( 18), ( 22) behaves similarly to the continuous-time system (1), ( 11) at least on small intervals of time. Let us denote

L h = B h e n W -1 h , F h = A h -L h A n h , h > 0, ( 24
)
M h ( x d )x = (F h + L h Q nh ( x d ))x, x ∈ R n , (25)
and rewrite the discrete-time system ( 18), ( 22) as follows

x k+1 = z h (x k ) := M h ( x k d )x k . ( 26
)
Lemma 1 (Homogeneity of discretization) The system (18), ( 22) is d-homogeneous as follows :

z h (d(s)x) = d(s)z e µs h (x) , (27) ũh (d(s
)x) = e s(1+µ) ũe µs h (x), (28)
for all s ∈ R, for all h > 0 and for all x ∈ R n

The dilation symmetry established by Lemma 1 guarantees that a global asymptotic stability of the discretetime system [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] for some h = ĥ > 0 is equivalent to the global asymptotic stability of this system for any h > 0. For simplicity, we select ĥ := (|µ|ρn) -1 .

(

) 29
As shown below, the key feature of the proposed control discretization is the nilpotence of the matrix F h , i.e., F n h = 0. Together with properties of Q nh ( x k d )x k , this allows the controller [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] to preserve stability properties of the original system in its sampled-time counterpart.

Lemma 2 Let u be a homogeneous control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] designed by Theorem 3 under Assumption 1. Then the closed-loop discrete-time system (18), ( 22) is 1) locally uniformly finite-time stable for µ < 0 and

x 0 d ≤ r -( ĥ/h) 1/µ ⇒ x k = 0, ∀k ≥ n,
where x d is the canonical homogeneous norm induced by the weighted Euclidean norm x = √

x P x with P = X -1 and r -> 0 is any number such that max i∈{1,...,n}

F i-1 ĥ d(ln r -) < 1;
2) globally practically4 finite-time stable for µ < 0 and the set

Ω -= x ∈ R n : x d ≤ r -( ĥ/h) 1/µ is in-
variant and finite-time stable for some r -≥ r -; 3) globally practically fixed-time stable for µ > 0:

x k d ≤ r + ( ĥ/h) 1/µ , ∀k ≥ n for all x 0 ∈ R n , where r + > max vi ≤1 n i=1 F i-1 ĥ L ĥv i d ;
4) locally asymptotically stable for µ > 0 and the set

Ω + = x ∈ R n : x d ≤ r + ( ĥ/h) 1/µ is an invariant attraction domain for some r + ∈ (0, r + ].
The latter lemma proves that the discretization ( 22) of the controller [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], indeed, preserves a stability property of the original system at least locally (close to zero and close to infinity). The discrete-time system with h = ĥ behaves similarly to the continuous-time system for x k d < r ± and x k d > r ± . Hence, if the set Ω± = {x : r ± < x d < r ± } does not contain an invariant set of the discrete-time system then the discretization is globally consistent. Lemma 2 does not provide explicit estimates for r -and r + . So, a criterion of the global consistency of the discretization ( 22) has be obtained in a different manner.

Let us consider a family of mappings Θ k : (0, +∞)×S → R n×n defined recursively as follows: Θ 0 (δ, v) = I n and

Θ k+1 (δ, v) = M δ ĥ( Θ k (δ, v)v d )Θ k (δ, v), k ∈ N, (30)
where δ ∈ R, v ∈ S, the parameter ĥ > 0 is defined by [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] and S = {x ∈ R n : x = 1} is the unit sphere.

Lemma 3 Any solution of the discrete-time system (18), [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] with h = ĥ and x 0 = 0 admits the representation

x k = d(ln x 0 d )Θ k ( x 0 µ d , v 0 ) v 0 , (31)
where

v 0 = d(-ln x 0 d )x 0 ∈ S.
The following theorem presents a necessary and sufficient condition of the consistency of the discretization [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] for the controller [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF].

Theorem 5 (Consistent discretization) Let u be a homogeneous control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] designed by Theorem 3 under Assumption 1 for µ = 0. Then ũh given by ( 22) is a consistent discretization of the control u if and only if there exists k * ≥ 1 such that

Θ k * (δ, v)v d < 1, ∀δ ∈ (0, r * ], ∀v ∈ S, (32)
where r * = (r -) µ if µ < 0 and r * = (r + ) µ if µ > 0. Therefore, the discrete-time system (18), ( 22) is uniformly finite-time(or nearly fixed-time) stable if and only if the canonical homogeneous norm • d is a kind of a homogeneous Lyapunov function. Indeed, [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF] and [START_REF] Livne | Proper discretization of homogeneous differentiators[END_REF] simply means that

x k * d < x 0 d , ∀x 0 : x 0 µ d ∈ (0, r * ).
Remark 3 (On feasibility of condition (32)) For k * = 1 the condition (32) is equivalent to the nonlinearly parameterized matrix inequality

M δ ĥ(1) X -1 M δ ĥ(1) < X -1 , ∀δ ∈ (0, r * ], (33)
which can be checked numerically on a sufficiently dense grid in (0, r * ] because of a continuous dependence of M δ ĥ(1) on the parameter δ. Denoting ∆(δ) = X -1 -M δ ĥ(1)X -1 M δ ĥ(1)) we conclude that the condition (33)

is fulfilled if λ min (∆(δ)) > 0 for all δ ∈ (0, r * ]. For ex- ample, for n = 2, µ = -1, ρ = 2,A = [ 0 1 0 0 ] , B = [ 0 1 ] Theorem 3 gives G d = diag{2, 1}, K = Y X -1 with X = x 11 X, x 11 > 0, X = 1 -ρ(1-µ) -ρ(1-µ) 7(2-µ) 2 ρ 2 (1-µ)/8 , Y = ρ 2 (2 -µ)(1 -µ)x 11 [ 8-7(2-µ) 8 -7ρ(2-µ)) 8
] . In this case, we have r * = 1 and Figure 1 depicts the evolution of the function δ → λ min (∆(δ)), which confirms that (33) is fulfilled. For k * > 1 a similar (but a bit more complicated) numerical procedure can be developed.

Robustness analysis

It is well known [START_REF] Hong | H ∞ control, stabilization, and inputoutput stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] that homogeneous systems are Input-to-State Stable (ISS) with respect to sufficiently large class of perturbations. Recall [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] that a system ẋ = f (t, x, q), t > t 0 [START_REF] Mercado-Uribe | Multiple-input multiple-output homogeneous integral control design using the implicit lyapunov function approach[END_REF] is practically ISS with respect to q ∈ L ∞ (R, R l ) if there exist, c > 0, ξ ∈ KL and γ ∈ K such that

x(t) ≤ c + ξ( x 0 , t -t 0 ) + γ( q L ∞ ((t0,t),R l ) ). ( 35
)
If c = 0 then the system is ISS. Local ISS restricts additionally the set of initial conditions x 0 ≤ r x and/or the maximal magnitude of the input q ≤ r q .

Let us consider the system

ẋ(t) = Ax(t) + Bu(t) + q(t), t > 0, u(t) = u(t k ), t ∈ [t k , t k+1 ), (36)
where u(t k ) is given by ( 21) and q ∈ L ∞ (R, R n ) is an exogenous input.

Theorem 6

The system (36) is 1) locally ISS; 2) practically fixed-time stable if µ > 0; 3) ISS if µ > -β, where β > 0 is defined in Theorem 1.

The ISS can be established for consistent discretization of homogeneous controller. Let us consider the system

ẋ(t) = Ax(t) + Bu(t) + q p (t), t > 0, u(t) = ũh (x(t k ) + q m (t k )), t ∈ [t k , t k+1 ) ( 37
)
where ũh given by ( 22) is a discretization of a control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] designed by Theorem 3 under Assumption 1 and the exogenous input

q = (q p , q m ) ∈ L ∞ (R, R 2n ) is such that {q m (t k )} ∈ ∞ .
Here q p models an external perturbation and q m is a measurement noise.

Theorem 7

The system (37) is 1) locally ISS; 2) practically fixed-time stable if µ > 0; 3) practically ISS if µ > -β; 4) ISS if µ > -β and the unperturbed system (q = 0) is globally asymptotically stable.

Notice that q may contain an output of another system. In this case, a stability analysis of a cascade system can be based on ISS [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], [START_REF] Chaillet | Combining iiss and iss with respect to small inputs: the strong iiss 6 A system is forward complete if all its solutions are defined globally in the forward time. property[END_REF]. We follow this idea for the design of the consistent discretization in the multi-input case.

Multi-input case

Let the control u be designed using Corollary 2 under Assumption 2. Since A i is nilpotent then, as before, K 0 = 0 in Theorem 3 and in Corollary 1. Let Q nih (•) be defined by the formula (15

) for A i , B i , K i , G di , P i , µ i , i = 1, ..., m. Let us denote W h,i = [B h,i , A h,i B h,i , ...., A n-1 h,i B h,i ], (38)
A h,i = e hAi , B h,i = h 0 e τ Ai dτ B, and introduce the following discretization of the controllers u i :

ui(t k+n-1 ) ... ui(t k ) = W -1 h,i Q nih ( x i (t k ) d )-A n h,i x i (t k ). ( 39
)
Corollary 3 Under Assumption 2, the system (18) with the control (39) is globally uniformly finite-time stable if

µ i < 0 (nearly fixed-time stable if µ i > 0), ∀i = 1, ..., m.
Similarly to the single input case, a consistent discretization of the multi-input control system is designed as

u i (x k,i ) = Ki ( x k,i di )x k,i , (40)
Ki ( x k,i di ) = e ni W -1 h,i Q nh,i ( x k,i di )-A n h,i .
Corollary 4 Let a control law u be designed by Corollary 2 under Assumption 2. Then (40) is a consistent discretization of u provided that conditions of Theorem 5 are fulfilled for µ = µ i , ∀i = 1, ..., m and max µ i < 0 or min µ i > 0.

5 Numerical Examples

Single-input system

Let n = 3, A = 0 I2 0 0 , µ = -0.25, ρ = 2. By Theorem 3 we derive a finite-time homogeneous control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] with parameters K 0 = 0, G d = The numerical simulation has been done in MATLAB 5 . Simulation results of the system (1) with the consistently discretized control (2), ( 40) for x 0 = [1 -1 0] , h = 0.1 are presented on Fig. 2 (right). The system states converge to zero in a finite time in the sense that x k ≤ , ∀k ≥ k * = 24, where = 1.11 • 10 -16 is the machine zero. Since x k * -1 ≈ 10 -6 then t k * = 2.4 is the settling time estimate obtained by the numerical simulation, which coincides with the theoretical settling time

T (x 0 ) = x 0 -µ d /(-µρ) ≈ 2.
4526 up to an error of the order h. This property is observed in all simulations, for other initial conditions and other sampling periods, confirming very high precision of the approximation of the continuous time system by its sampled-time counterpart. The results for the explicit discretization

u(t) = ũ(x(t k )), t ∈ [t k , t k+1 ) ( 41
)
of the control [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] are depicted on Fig. 2 (left) for comparison reasons. The system (1) with the explicitly discretized homogeneous controller is not asymptotically stable and suffers of a chattering. For µ = 0.25, With the explicit discretization, the system simply blows up for larger initial conditions. Simulations of the consistently discretized control were made for initial conditions with various magnitudes up to x 0 ≈ 10 10 . They show the nearly fixed-time stability of the closedloop system as in Fig. 3 (right). ISS of both controllers (for µ = -0.25 and µ = 0.25) with respect to additive perturbations and measurement noises was also confirmed by simulations.

Multi-input system

For the multi-input case we consider the above singleinput system (n 1 = 3) with the finite-time control (µ 1 = -0.25, ρ 1 = 1) in the cascade with the second order system (n 2 = 2) with the finite-time control (µ 2 = -1, ρ 2 = 1) considered in Remark 3, i.e.,

A = A1 A12 0 A2 , B = B1 0 0 B2 , A 1 = 0 I2 0 0 , A 12 = 1 0 0 0 0 0 , A 2 = [ 0 1 0 0 ] , B 1 = 0 0 1 , B 2 = [ 0 1 ]
. The simulation results for the cascade system are shown on Fig. 4 for

x 1 (0) = [1 -1 0] , x 2 (0) = [1 0] .
They confirm finite-time stability of the closed-loop system with the consistently discretized control.

Conclusions

In the paper, two types of discretization (sampled-time implementation) schemes for a homogeneous control law are developed. Both preserves the finite/fixed-time stability properties of the original continuous-time closedloop system. The first scheme gives a mixture of feed- forward and feedback algorithms. It can be utilized if the control sampling is much faster than the sampling of measurements, or in the model-predictive framework. The second scheme (called consistent) provides a static feedback law, which always preserves the stability properties, at least, locally (close to zero and close to infinity). A necessary and sufficient condition of global consistency is presented. A particular sufficient condition is formalized in terms of a parametric LMI, by numerical simulations, is to be feasible in some cases. A development of numerical algorithms for control parameters tuning based on the obtained conditions of the consistency is an interesting problem for the future research.

Appendix

Auxiliary results

Lemma 4 [14, page 136] If

Z 1 Z 2 -Z 2 Z 1 = qZ 2 with q ∈ R and Z 1 , Z 2 ∈ R n×n then e Z1 e Z2 = e Z1+ q
1-e -q Z2 . Lemma 5 If Assumption 1 is fulfilled and d is a linear dilation defined in Theorem 3 then 1) for all s ∈ R the following identities hold: 2) the matrix W h given by (20) satisfies

W h =d * (ln h) 1 0 e τ A dτ [B e A B ... e (n-1)A B], (44)
W -1 h d * (ln h) n 0 e τ A dτ B = 1 ... 1 , (45)
where d * corresponds to d with µ = -1; • the parameter β defined in Theorem 1 satisfies β = 1 for any µ < 0.

Proof. 1) Under Assumption 1 of Theorem 3 the dilation d(s) = e sG d is uniquely defined such that G d satisfies (9) with A 0 = A. We derive Ad(s

) = A +∞ i=0 s i G i d i! = +∞ i=0 s i (µIn+G d ) i A i! = e µs d(s)A, d(s)B = +∞ i=0 s i G i d i! B = +∞ i=0 s i i! B, ∀s ∈ R. Hence, for all s ∈ R we have d(s)e τ A = d(s) n-1 i=0 h i A i i! = n-1 i=0 e -µis h i A i d(s) i!
= e e -µs τ A d(s) and d(s) h 0 e τ A dτ = h 0 e e -µs τ A d(s)dτ. and changing of the integration variable we derive [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF].

2) On the one hand, since e sA and h 0 e τ A dτ commute for any s ∈ R then taking into account A i h = e ihA we derive

W h = [B h A h B A 2 h B h ... A n-1 h B h ] =
e τ A dτ = 1 0 e τ A dτ n-1 i=0 e iA then W -1 h d * (ln h)• • n 0 e τ nA dτ B = [B e A B ... e (n-1)A B] -1 n-1 i=0 e iA B = n i e i .
3) For m = 1, under conditions of Theorem 3, we have

G d = U -1 diag{1 -µ(n -1), .
.., 1}U for some nonsingular matrix U ∈ R n×n . The latter means that β = 1 Lemma 6 The function V : R n → R + defined as

V (x) = σ -1 ( x d ),
x ∈ R n is positive definite radially unbounded and globally Lipschitz continuous with the Lipschitz constant 1, where σ ∈ K ∞ is as in Theorem 1.

Proof. The Lipschitz continuity for x ≥ 1 follows from [44, Proposition 1]. For x ≤ 1 this fact can be proven similarly (just replacing β with α). All other properties of V are obvious.

Lemma 7 For any C * > 0 and any ε ∈ (0, 1) there exist C > 0 and r > 0 such that

|u(x) -u ĥ(x)| ≤ C min{ x 1+2µ d , x 1+2µ }, (46)
for all x, x satisfying (1 -ε)

x d ≤ x d ≤ (1 + ε) x d , x µ d ≤ r, d(-ln x d )(x -x) ≤ C * x µ .
Proof. Using approximation of ũh (see, the proof of Proposition 1) we conclude ∃h 0 ∈ (0, 1), ∃C 0 > 0 such that sup y∈S |u(y) -ũh (y)| ≤ C 0 h, ∀ h < h 0 . Since u is locally Lipschitz continuous on R n \{0} then for any δ ∈ (0, 1) there exists C 1 such that |u(y)-u(z)| ≤ C 1 yz , ∀y ∈ S and ∀z : y -z ≤ δ. Using the Lipschitz condition and the dilation symmetry of u and ũh (see [START_REF] Korobov | A solution of the synthesis problem using controlability function[END_REF]) we derive we complete the proof.

The proof of Corollary 1

Denoting y = x d(x d )x, we derive y = x d d(ln x d )x = x d and conclude that the closed-loop system (1), ( 11) is topologically equivalent (homeomorphic on R n and diffeomorphic on R n \{0}, see [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]) to the standard homogeneous system:

ẏ = y µ (A 0 + BK + ρ(G d -I n ))y,
where the identities d(s)A 0 = e -µs A 0 d(s) and = e B, ∀s ∈ R are utilized on the last step. In this case, using [START_REF] Efimov | Discretization of homogeneous systems using euler method with a state-dependent step[END_REF] we conclude d dt y(t) = -ρ y(t) µ+1 and y(t+τ ) -µ = y(t) -µ +µρτ, for y(t) -µ +µρτ ≥ 0. Obviously, y(t + τ ) = 0 if y(t) -µ + µρτ ≤ 0. The latter corresponds to the negative homogeneity degree µ < 0 and the finite-time stability of the closed-loop system. Hence, denoting à = A 0 + BK + ρG d we obtain

y(t + τ ) = e ( Ã-ρIn)) τ 0 y(t+σ) µ dσ y(t) = e ( Ã-ρIn) 1
µρ ln(1+µρτ y(t) µ ) y(t).

Since y(t) = x(t) d then returning to the original coordinates we derive

x(t + τ ) = d(ln y(t + τ ) y(t+τ ) y(t+τ ) = Q τ ( x(t) d )
x(t) for all t ≥ 0 and all τ ≥ 0.

The proof of Corollary 2

Under Assumption 2, the model (1) is a system of interconnected subsystems with state vectors x i ∈ R ni and control inputs u i ∈ R. By the formula [START_REF] Chaillet | Combining iiss and iss with respect to small inputs: the strong iiss 6 A system is forward complete if all its solutions are defined globally in the forward time. property[END_REF], we derive

d dt x i di = -ρ i x i 1+µi di + x i di x i d i (-ln xi d i )Pidi(-ln xi d i ) m j=i+1 Aij xj x i d i (-ln xi d i )PiG d i di(-ln xi d i )xi
, where A ij are over diagonal blocks of the matrix A. By Theorem 3 the norm in R ni is defined as z = z P i z, z ∈ R ni and d i (-

x i di )x i = 1. Taking into account P i G di + G di P i 0 we derive x i d i (-ln x i di )P i G di d i (-ln x i di )x i ≥ β i > 0,
where β i = 0.5λ min (P

1/2 i G di P -1/2 + P -1/2 i G di P 1/2 ).
Applying the Cauchy-Schwartz inequality we obtain

x i d i (-ln xi d i )Pidi(-ln xi d i ) m j=i+1 Aij xj x i d i (-ln xi d i )PiG d i di(-ln xi d i )xi ≤ 1 βi d i (-ln x i di )Σ m j=i+1 A ij x j ≤ Σ m j=i+1 Aij xj βiσi( xi d i )
, where σ i = σ -1 and σ is given in Theorem 1 with α = α i = 0.5λ max (P

1/2 i G di P -1/2 + P -1/2 i G di P 1/2
) and β = β i . The estimate obtained for d dt x i di , the stability of the m-subsystem and the cascade structure imply the forward completeness of the whole system. [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] of the i-th subsystem with respect to the input m j=i+1 A ij x j provided that µ i > -β i . Since the m-th subsystem is globally uniformly asymptotically stable then using the cascade structure and the ISS property of each sysbsystem we conclude that the whole system is globally uniformly asymptotically stable as well. Moreover, the obtained estimate of d dt x i di implies that the finite-time, exponential and nearly fixed-time convergence rates are preserved as well dependently of the sign of µ i for all i = 1, ..., m. If µ i ≤ -β i , then the each subsystem with the zero input is finite-time stable. Taking into account forward completeness and the cascade structure we conclude that the whole system is finite-time stable too.

If V i (x i ) = σ i ( x i di ) then for x i di > 1 we have dVi dt = β i Vi• d dt xi d xi d ≤ -β i ρ i V 1+ µ i β i i + m j=i+1 A ij x j . For 0 < x i d < 1 we derive dVi dt = α i Vi• d dt xi d xi d ≤ -α i ρ i V 1+ µ i α i i + αi βi m j=i+1 A ij x j Since V i locally Lipschitz continuous on R ni \{0} then using the Clarke's gradient for x i di = 1 we have dVi dt ≤ -λ i (x i )α i ρ i V 1+ µ i α i i -(1 -λ i (x i ))β i ρ i V 1+ µ i β i i + λ i (x i ) αi βi + 1 -λ i (x i ) Σ m j=i+1 A ij x j with some λ(x i ) ∈ [0, 1]. The latter means that x i → V i (x i ) is an ISS Lyapunov function

The proof of Theorem 4

On the one hand, since x k+n = Q nh ( x k d )x k then, in the view of Corollary 1, the states of the discretetime system [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] with the control (21) coincides with the states of the original continuous-time system (1), [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] at time instances t kn , k = 0, 1, ..... In this case, if the discrete-time system is globally Lyapunov stable then the finite-time or nearly fixed-time stability property of the original continuous-time system is preserved.

On the one hand, in the view of Theorem 3 and Corollary 1, we have

Q τ ( x d )x -µ d = x -µ d + µρτ for all x ∈ R n : x -µ
d + µρτ ≥ 0 and ∀τ > 0, and Q τ ( x d )x = 0 otherwise. Hence, we conclude

Q τ ( x d )x d = ( x -µ d + µρτ ) -1/µ ≤ x d ≤ σ( x )
, where σ ∈ K ∞ is defined in Theorem 1. In this case, there exists ξ ∈ K ∞ such that |u(t k+j )| ≤ ξ( x k+j )), j = 0, 1, ..., n -1 and there exists σ 1 ∈ K ∞ such that x k+j+1 d ≤ σ 1 ( x k+j d ), j = 0, 1, ..., n -1. Consequently, there exists σ n ∈ K ∞ such that x k+n d ≤ σ n ( x k d ), ∀k ∈ N. On the other hand, by construction, we have

x k+n = Q nh ( x k d )x k , so, taking into account Q nh ( x k d )x k d ≤ x k d , we derive x k+n d ≤ min { x k d , σ n ( x k d )} and x k d ≤ min { x 0 d , σ n ( x 0 d } , ∀k ≥ 0.
Using Theorem 1 we substantiated the global Lyapunov stability of the system (18) (as well as the system (1)) with the sampled-time control [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF].

The of Proposition

First of all, notice that under Assumption 1 we have K 0 = 0 and A 0 = A (see Theorem 3). Let us denote s h = ln(1 + µρnhr µ ) 1/(ρµ) with r = x d and show Kh (r) → r 1+µ Kd(-ln r) as h → 0.

On the one hand, if d * (s) is defined by Lemma 5 then

d * (-ln h)e (A+BK+ρG d )s h = ∞ i=0 s i h d * (-ln h)(A+BK+ρG d ) i i! = ∞ i=0 s i h (A+BKd * (ln h)+ρhG d ) i d * (-ln h) h i i! = e s h (A+ρhG d ) h d * (-ln h) + n-1 i=1 s i h h i i! A i-1 BK + O(h). Indeed, for i = 2 we have (A + BKd * (ln h) + ρhG d ) 2 d * (-ln h) = (A + ρhG d ) 2 d * (-ln h) + ABK + O(h)
, and, by induction, we conclude (A+BKd * (ln

h)+ ρhG d ) i d(-ln h) = (A + ρhG d ) i d * (-ln h) + A i-1 BK + O(h). Since A is nilpotent then for i ≥ n + 1 we have A i-1 = 0 and (A + BKd * (ln h) + ρhG d ) i d(-ln h) = (A + ρhG d ) i d * (-ln h) + O(h).
On the other hand, since d(-s)Ad(s) = e µs A then d(-s)e τ A d(s) = e e sµ τ A for all s, τ ∈ R and [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. Since d * (s) commutes with d(τ ), ∀s, τ ∈ R then using the identities [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] and the estimate of d * (-ln h)e (A+BK+ρG d )s h we derive

Q nh (r) -A n h = d(ln r) Q(s h )d(-ln r) -e nhA = d(ln r)( Q(s h ) -e nhr µ A )d(-ln r) where Q(s h ) is given by
d * (-ln h) e (A+BK+ρG d )s h -e ρs h G d e nhr µ A = d * (-ln h)e (A+BK+ρG d )s h -e ρs h G d e nr µ A d * (-ln h) = n-1 i=0 s i h A i-1 BK h i i! + e s h (A+ρhG d ) h -e ρs h G d e nr µ A d * (-ln h)+O(h). If Z 1 = ρs h G d , Z 2 = nr µ A, q = -µρs h then the condition Z 1 Z 2 -Z 2 Z 1 = qZ 2 of Lemma 4 is ful- filled, so e Z1 e Z2 = e Z1+ q
1-e -q Z2 or, equivalently,

e ρs h G d e nr µ A = e ρs h G d + s h h A . Therefore, since s h = ln(1+µρnhr µ ) ρµ = µρnhr µ - (µρ2hr µ ) 2 2 +O(h 3 ) ρµ = hnr µ + O(h 2 ) then d * (-ln h) e (A+BK+ρG d )s h -e ρs h G d e nhr µ A = n-1 i=0 s i h A i-1 BK h i i! + O(h) = n-1 i=0 (nr µ ) i A i-1 BK i! + O(h) = nr µ 0 e τ A dτ BK + O(h). Since d(s) = e sG d then d * (-ln h)(Q nh (r) -A n h ) = d(ln r)e -ρs h G d nr µ 0 e τ A dτ BKd(-ln r) + O(h) = nr µ 0 d(ln r -ρs h )e τ A dτ BKd(-ln r) + O(h) = nr µ 0 e e -µ(ln r-ρs h ) τ A dτ d(ln r -ρs h )BKd(-ln r) + O(h) = nr µ 0 e 1+µρnhr µ r µ τ A dτ r (1+µρnhr µ ) 1/µ BKd(-ln r) + O(h) = nr µ 0 e 1 r µ τ A dτ BrKd(-ln r) + O(h) =
n 0 e τ A dτ Br 1+µ Kd(-ln r) + O(h). Therefore, using Lemma 5 and Remark 2 we conclude

Kh (r) = e n W -1 n (Q nh (s h ) -A n h ) = e n W -1 n d * (ln h) n 0 e τ A dτ Br 1+µ Kd(-ln r) + O(h) = r 1+µ Kd(-ln r) + O(h 2 ),
and Kh (r) → r 1+µ Kd(-ln r) as h → 0 uniformly on r from compacts belonging to (0, +∞).

The proof of Lemma 1

Let us show that ( 27) holds. Indeed, on the one hand, since d(s)

x d = e s x d then Q ln(1+µρnhe -µs d(s)x µ d ) ρµ = Q ln(1+µρnh x µ d ) ρµ and ∀s ∈ R, ∀x ∈ R n we have Q nhe -µs ( d(s)x d ) = d(s)Q nh ( x d )d(-s).
On the other hand, using Lemma 5 we derive

W e -µs h = d * (-µs)W h , B e -µs h = e -µs-s d(s)B h , L e -µs h d(s) = d (s) L h , d (s) F h = F e -µs h d (s) ,
for all s ∈ R and for all h > 0, where the identities d * (τ ) = e τ In+τ G0 and d(s) = e sIn-sµG0 are utilized for the analysis L h in order to conclude that e -s-µs d * (µs)d(s) = I n , where G 0 is defined in Theorem 3. Hence, we derive [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF]. The identity (28) can be obtained in the same way.

The proof of Lemma 2

1) Local Finite-Time Stability for µ < 0. Let us show that the matrix F h is nilpotent. Notice that F h can be rewritten as follows

F h = I n -[0 0 ... 0 B h ]W -1 h A n-1 h A h = A 1-n h W h W -1 h A n-1 h -[0 0 ... 0 B h ]W -1 h A n-1 h A h = A 1-n h (W h -[0 0 ... 0 A n-1 h B h ])W -1 h A n h = A 1-n h [B h A h B h ... A n-2 h B h 0]W -1 h A n h Since W -1 h [A h B h ... A n-1 h B h 0] = [e 2 .
.. e n 0] then

F 2 h = A 1-n h [B h A h B h ... A n-2 h B h 0][e 2 e 3 ... e n 0]W -1 h A n h = A 1-n h [A h B h ... A n-2 h B h 0 0]W -1 h A n h .
Continuing the same considerations we derive F n h = 0. On the one hand, since for µ < 0 we have

Q( x d ) = 0 if x d ≤ (-µρnh) -1/µ then the closed-loop system becomes linear x k+1 = F h x k for x k d ≤ (-µρnh) -1/µ = ĥ/h 1/µ
. On the other hand, the inequality

F i h x 0 d ≤ ĥ/h 1/µ is equiva- lent to d 1 µ ln h ĥ F i h x 0 = F i ĥd 1 µ ln h ĥ x 0 ≤ 1,
and the inequality x 0 d ≤ r -ĥ/h

1/µ is equiva- lent to d(-ln r)d 1 µ ln h ĥ x 0 ≤ 1. Therefore, the inequality F i ĥd(ln r -) ≤ 1 F i 1 µ ln h ĥ x 0 ≤ d(-ln r -)d 1
µ ln h ĥ x 0 . Hence, we derive

x i d = F i h x 0 d ≤ ĥ/h 1/µ for i = 1, ..., n -1 provided that x 0 d ≤ r -ĥ/h 1/µ
. Taking into account the nilpotence of F h , the latter implies local Lyapunov stability of the closed-loop system and the finite-time convergence of solutions to zero.

2) Practical Finite-time Stability for µ < 0. The proof repeats the proof of Theorem 7, the case 1) for q = 0 and gives d x d dt ≤ -0.5ρ x 1+µ d for all x : x -µ d ≥ r. Using Lemma 1 we derive Ω -for h = ĥ with r -= r-1/µ , where r is defined in the proof of Theorem 7.

3) Practical Fixed-time Stability for µ > 0 Let us prove, now, the practical fixed-time stability. On the one hand, since, by Theorem 3 the canonical homogeneous norm is a Lyapunov function of the system satisfying

d dt x(t) d = -ρ x(t) 1+µ d , then x(t + nh) -µ d = x(t) -µ d + µρnh, and for µ > 0 we have x(t + nh) d < (µρnh) -1/µ = ĥ h 1/µ
independently of x(t). On the other hand, by Corollary 1, we have

x(t + nh) = Q nh ( x(t) d )x(t), so Q nh ( x d )x d ≤ ĥ h 1/µ
, ∀x ∈ R n . Since the right-hand side of the system can be represented as follows

z h (x) = F h x + L h Q nh ( x d )x, L h := B h e n W -1 h ,
then, for any x 0 ∈ R n the solution x k , k = 0, 1, 2, ... of the discrete-time system [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] satisfies

x 1 = F h x 0 + L h y 1 , x 2 = F 2 h x 0 + F h L h y 1 + L h y 2 , ... x n = F n h x 0 + F n-1 h L h y 1 + F n-2 h L h y 2 + ... + L h y n ,
where

y i = Q nh ( x i-1 d )x i-1 , i = 1, 2, .... Since the matrix F h is nilpotent then F n h = 0 and x k = F n-1 h L h y k-n+1 +F n-2 h L h y k-n+2 +...+L h y k , ∀k ≥ n. Since y i d ≤ ĥ h 1 µ ⇔ d 1 µ ln h ĥ y i d ≤ 1 ⇔ d 1 µ ln h ĥ y i ≤ 1 then d 1 µ ln h ĥ x k d = n i=1 d 1 µ ln h ĥ F i-1 h L h d - 1 µ ln h ĥ vi d ,
where v i = d 1 µ ln h ĥ y k+n-i . Taking into account F ĥ = d 1 µ ln h ĥ F h d -1 µ ln h ĥ , L ĥ = d 1 µ ln h ĥ L h d -1 µ ln h ĥ and v i ≤ 1 we derive d 1 µ ln h ĥ x k d < r+ , ∀k ≥ n.

4) Local Asymptotic

Stability for µ > 0. Since

d x d dt ≤ -0.5ρ x 1+µ
d for all x : x µ d ≤ r-1 then for µ > 0 the closed-loop system is locally asymptotically stable. Using Lemma 1 we derive Ω + with r + = r-1/µ .

The proof of Lemma 3

The symmetry proven by Lemma 1 yields M h ( x d )x = d(s)M e µs h ( d(-s)x d )d(-s)x for all x ∈ R n , for all s ∈ R and ∀h > 0. Hence, for any x 0 ∈ R n \{0} we have

x 1 = M h ( x 0 d )x 0 = d(ln x 0 d )Θ 1 ( x 0 µ d ĥ, v 0 )v 0 . Since x 1 d = x 0 d Θ 1 ( x 0 µ d , v 0 )v 0 d then x 2 = M h ( x 1 d )x 1 = M h ( x 1 d )d(ln x 0 d )Θ 1 ( x 1 d ĥ, v 0 )v 0 = d(ln x 0 d )M x0 µ d ĥ ( x 1 d / x 0 d ) Θ 1 ( x 0 µ d , v 0 ) v 0 = d(ln x 0 d )Θ 2 ( x 0 | µ d , v 0 ) v 0 .
Repeating the above considerations we derive [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF].

The proof of Theorem 5

The approximation property is proven by Proposition 1. Let us prove the consistency of stability properties. If the discrete-time system ( 26) is globally uniformly finite-time stable for some h > 0 then due to dilation symmetry (see Lemma 1) it is globally uniformly finitetime stable for any h > 0, in particular, for h = ĥ.

Necessity. Let us consider the case µ < 0. The uniformity of the finite-time stability and Lemma 2 guarantee that there exists k * ≥ 1 such that for any x 0 :

x 0 d ≥ r -we have x k * d < x 0 d . The latter means that x 0 d Θ k * ( x 0 µ , v 0 ) v 0 d < x 0 d and Θ k * (r µ , v 0 ) v 0 d < 1, ∀r ≥ r -, ∀v 0 ∈ S.
Denoting δ = r µ for µ < 0 we derive the inequality [START_REF] Livne | Proper discretization of homogeneous differentiators[END_REF].

The case µ > 0 can be treated similarly.

Sufficiency. Let us denote r * = (r -) µ for µ < 0 and r * = (r + ) µ for µ > 0. Let us consider the candidate Lyapunov function V : R n → R + defined as follows

V (x) = d(-µ -1 ln r * )x p d if x µ d ≤ r * , d(-µ -1 ln r * )x if x µ d ≥ r * , (47)
where p = β if µ < 0 and p = α if µ > 0, where α, β are given by Theorem 1. By construction, V is positive definite, radially unbounded and globally Lipschitz continuous with the Lipschtiz constant L = d(-µ -1 ln r * ) due to Lemma 6.

a) Let us show that V is a Lyapunov function for Lemma 2). This means that there exist γ * ∈ (0, 1) such that V (x k ) ≤ γ * V (x 0 ), x 0 µ > r * . Without loss of generality we may assume that k * ≥ n (otherwise we just take nk * instead of k * in all above considerations). d) Therefore, for any r * ∈ (0, r * ) and for any finite r * > r * there exists γ ∈ (0, 1) such that

x 0 µ d ∈ [r * , r * ]. Since (δ, v) → Θ k * (δ, v)v d is a con- tinuous function on the compact [r * , r * ] × S then using (32) we derive γ = max δ∈[r * ,r * ] Θ k * (δ, v)v d < 1 and for x 0 µ ∈ [r * , r * ] we have x k * ≤ γ 0 d . Since x k * d ≤ γ x 0 d ⇔ d(s)x k * d ≤ γ d(s)x 0 d , ∀s ∈ R then V (x k * ) ≤ γ p V (x 0 ) for x 0 µ d ∈ [r * , r * ]. b) Let us show that V is a Lyapunov function for x 0 µ d ≤ r * . Since d x d dt ≤ -0.5ρ x 1+µ d for all x : x µ d ≤ r * (see the proof of Lemma 2, case 2), then x 1 d ≤ γ * ( x 0 µ d ) x 0 d , ∀x 0 : x 0 µ d ≤ r * , or equiv- alently, V (x 1 ) ≤ γ p * ( x 0 µ d )V (x 0 ), ∀x 0 : x 0 µ d ≤ r * , where γ * (s) = (1 + 0.5ρµs) -1/µ with s > 0. c) Let us show that V is a Lyapunov function for x 0 µ d ≥ r * . If x 0 µ d > r * then for all k ≥ n we have x k = 0 if µ < 0 and d(-µ -1 ln r * )x k < 1 if µ > 0 (see the proof of
V (x k * ) ≤ γV (x 0 ), ∀x 0 : r * ≤ x 0 µ ≤ r * .
Taking into account local finite-time (resp., asymptotic) stability and practical finite-time (resp., fixed-time) stability proven by Lemma 2 for µ < 0 (resp., µ > 0) we complete the proof.

The proof of Theorem 6

In a discrete time, the system (36) can be rewritten as

x q k+n = Q nh ( x q k d )x q k + n-1 i=0 A n-i h qk+i
where qk = h 0 e A(h-τ ) q p (t k + τ )dτ is the sampled-time realization of the external perturbation, so {q k } ∈ ∞ for any h > 0. Let V be defined as in Lemma 6. Since

Q nh ( x d )x -µ d = x -µ d + µρnh for all x ∈ R n : x -µ d +µρnh ≥ 0 then x q k+n -qk -µ d = x q k -µ d + µρnh, where qk = n-1 i=0 A n-i h qk+i .
Moreover, since qk is uniformly bounded, then for µ > 0 it guarantees a practical fixed-time stability. In this case, we derive

V (x q k+n )-V (x q k )=V (x q k+n )-V (x q k+n -q k )+V (x q k+n -q k )-V (x q k ) ≤ qk +V (x q k+n -q k )-V (x q k )= qk +σ -1 ( x q k -q k d )-V (x q k ) = qk +W (V (x q k )),
where W (V ) = σ -1 (σ(V ) -µ +µρnh) -1/µ -V. For µ > -β we have W ∈ K ∞ and V is an ISS Lyapunov function [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF].

The proof of Theorem 7

In a discrete time, the system (37) can be rewritten as

x q k+1 = A h x q k + B h Kh ( x q k + qk )(x q k + qk ) + qk , (48)
where {q k }, {q k } ∈ ∞ for any h > 0, qk = q m (t k ) is the sampled-time realization of the measurement noise and qk = h 0 e A(h-τ ) q p (t k +τ )dτ is the sampled-time realization of the external perturbation. Denote q k = (q k , q k ). Due to the dilation symmetry proven by Lemma 1 it is sufficient to analyze ISS of (48) for h = ĥ.

1) Let us prove local ISS and practical ISS of [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF]. If x = e Aht x q k + B ht ũĥ (x q k + qk ) + qht , h t = t -t k and qht = ht 0 e A(ht-τ ) q p (t k + τ )dτ then x corresponds to a solution of the system (37) for t ∈ [t k , t k + ĥ). Let us denote q 1 k = d(ln x q k d )q k , q 2 k = d(ln x q k d )q ht . a) Let us show that x q k d is close to x d for a sufficiently large x q k -µ d and sufficiently small q i k , i = 1, 2. Using dilation symmetry (see [START_REF] Korobov | A solution of the synthesis problem using controlability function[END_REF]) we derive

x = d(ln x q k d ) e Ah t x q k µ d v k +B h t x q k µ d ũĥ x q k µ (v k +q 1 k )+q 2 k ,
with v k = d(ln x q k d )x q k ∈ S, Since e Ahts → I n as s → 0 and B hts → 0 as s → 0 then for any ε ∈ (0, 1) there exist r ε > 0 and δ ε > 0 such that

e Aht x q k µ d v k +B ht x q k µ d ũĥ (v k + q 1 k ) + q 2 k d -1 ≤ ε for x q k -µ d > r ε and q i k ≤ δ ε , i = 1, 2. Hence, we have (1 -ε) x q k d ≤ x d ≤ (1 + ε) x q k d and ∃ C0 , C1 > 0 : d(-ln x d )q k = d -ln x d x q k d q 1 k ≤ C0 q 1 k and d(-ln x d )x q k < C1 for x q k -µ d > r ε and q i k ≤ δ ε .
b) Let us show that u h (x q k + qk ) is close to u(x) for a sufficiently large x q k -µ d and sufficiently small q i k . Using Lemma 5 and the identity x q k = e -Aht xe -Aht B ht ũĥ (x q k + qk ) -e -Aht qht we derive

d(-ln x d )x-d(-ln x d )(x q k +q k )= d(-ln x d )(In-e -Ah t )x+d(-ln x d )e -Ah t B h t ũĥ (x q k +q k ) +d(-ln x d )e -Ah t qh t +d(-ln x d )q k = -d(-ln x d ) h t 0 Ae Aτ dτ x+e -Ah t x µ d d(-ln x d )B h t ũĥ (x q k +q k ) +e -Ah t x µ d d(-ln x d )q h t +d(-ln x d )q k = - h t x µ d 0 Ae Aτ dτ d(-ln x d )x +e -Ah t x µ d • B h t x µ d • ũ x µ d ĥ (d(-ln( x d / x q k d ))(v k +q 1 k )) +e -Ah t x µ d d(-ln( x d / x q k d ))q 2 k +d(-ln x d )q k .
Since ∃ C2 > 0 :

ht x µ d 0 Ae Aτ dτ ≤ C2 x µ d and ∃ C3 > 0 : B h x µ d = ht x µ d 0 e Aτ dτ ≤ C3 µ d then for any C4 > 0 there exist C 3 > 0 such that d(-ln x d )x -d(-ln x d )(x q k + qk ) ≤ C 3 x µ d for x -µ d ≥ rε = ( 1+ε 1-ε ) |µ| r ε , q i k ≤ δ ε and d(-ln x d )q k ≤ C4 x µ d .
In this case, by Lemma 7 there exists C > 0 and r > 0 such that ũĥ (

x q k + qk ) -u(x) ≤ C x 1+2µ . for x -µ d ≥ max{r ε , r -1 }, all t ∈ [t i , t i+1 ) and q i k ≤ δ ε and d(-ln x d )q k ≤ C4 x µ d .
c) Adding and subtracting u(x) we derive

d x d dt = x d x d (-ln x d )P d(-ln x d )(Ax+B ũĥ (x k +q k )+qp) x d (-ln x d )P G d d(-ln x d )x = x d (-ln x d )P (B(ũ ĥ(x k +q k )-u(x))+ x d d(-ln x d )qp) x d (-ln x d )P G d d(-ln x d )x -ρ x 1+µ d ≤β B(ũ ĥ(x k +q k )-u(x)) +β x d d(-ln x d )qp -ρ x 1+µ d ≤ x 1+µ d (β B C x µ d -ρ)+β x d d(-ln x d )qp that for x -µ d ≥ r = max{r ε , r -1 }, t ∈ [t i , t i+1 ), q i k ≤ δ ε and d(-ln x d )q k ≤ C4 x µ d .
d) Let us show that • d is an ISS Lyapunov function (close to zero for µ > 0 and close to infinity for -β < µ < 0). Let σ(s) = s µ σ -1 (s), where σ ∈ K ∞ is given in Theorem 1 and s ≥ 0. For q p ≤ 0.25β -1 ρσ( x d ) we derive β x d d(ln x d )q p ≤ 0.25ρ x 1+µ d . Notice that q m ≤ δ ε σ -1 ( x d ) implies q 1 k ≤ δ ε , q m ≤ δ 0 σ( x d ) with δ 0 = C-1 for x -µ d ≥ r, q p ≤ δσ( x d ), q m ≤ δ 0 σ( x d ) and q m ≤ δ ε σ( x d ), where δ = min{0.25β -1 ρ, δ }. The function σ belongs to the class K ∞ if µ > -β. Therefore, the system (37) is practically ISS if -β < µ < 0 and locally ISS if µ > 0 even when the conditions of Theorem 5 do not hold. e) Let us show that the system (37) is practically fixedtime stable and practically ISS for µ > 0. Since F n ĥ = 0 (see the proof of Lemma 2) then

x q 1 = F ĥx q 0 + L ĥy 1 + q1 , q1 = q0 -L ĥA n ĥ q0 , x q 2 = F 2 ĥ x q 0 +F ĥL ĥy 1 +L ĥy 2 + q1 , q2 = F ĥq 1 + q1 -L ĥA n ĥ q1 , ... ...

x q n = F n-1 ĥ L ĥy 1 +...+L ĥyn + qn, qn = F ĥ qn-1 + qn-1 -L ĥA n ĥ qn-1 ,

where y i+1 = Q n ĥ( x q i + qi d )(x q i + qi ), i = 0, ..., n -1. Since for µ > 0 we have y k+i ≤ 1 (see the proof of Lemma 2, case 3) and qi ≤ C max j=0,...,i-1 q j for some C > 0, then the system (37) with µ > 0 is practically fixed-time stable and practically ISS. f) Let us show that the system (37) is locally ISS for µ ≥ -1. For x q k + qk d ≤ r -we have x q k+1 = F ĥx q k + qk -L ĥA n ĥ qk , where F h is a Schur stable nilpotent matrix. Since asymptotically stable linear system is ISS w. r. t. additive perturbations then (37) is locally ISS.

2) Let us show that the system (37) is ISS for µ > -β provided that the unperturbed system is globally asymptotically stable. Our goal is to show ISS of x q (p+1)k * = Ξ(x q pk * , q pk * , ..., q (p+1)k * -1 ), p = 0, 1, ... (49) which describes evolution of [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] with the discrete step k * . The latter would imply ISS of [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF]. Notice that the local and practical ISS of (37) guarantees the local and practical ISS of the system [START_REF] Polyakov | Consistent discretization of a homogeneous finite-time control for a double integrator[END_REF]. Let k * ≥ 1, V , r * , r * , γ ∈ (0, 1), L = d(-µ -1 ln r * ) be defined as in the proof of Theorem 5. Let x k ∈ R n denote a solution of the non-perturbed system with x 0 = x q 0 . a) Let us show there exists ω k * ∈ K ∞ :

x q k * -x k * ≤ ω k * (max{ q 0 , ..., q k * -1 }) for r * ≤ x 0 µ d ≤ r * . Since the system (37) is practically ISS then for any σ ∈ K ∞ there exists a compact set Ω ⊂ R n such that x q i ∈ Ω for all i ∈ 0, ..., k * provided that r * ≤ x 0 µ d ≤ r * and max j=0,...,k * -1 q j ≤ σ( x 0 d ).

Denoting q0 = 0 and qk = F h qk-1 + qk +L h (Q n ĥ( x k-1 + qk-1 + qk d )(x k-1 + qk-1 + qk ) -Q n ĥ( x k-1 d )x k-1 ) for k ≥ 1 we derive x q k = x k + qk . Since the function x → Q τ ( x d )x is continuous on R n then by Heine-Cantor Theorem it is uniformly continuous on Ω and there exists

ω 0 ∈ K ∞ such that Q τ ( z 1 d )z 1 -Q τ ( z 2 d )z 2 ≤ ω 0 ( z 1 -z 2 )
for all z 1 , z 2 ∈ Ω and q1 ≤ q0 + L h ω 0 ( q0 ).

Repeating the above consideration, on k * -th step, we derive that ∃ω k * ∈ K ∞ : qk * ≤ ω k * ( max j=0,...,k * -1 q j ) for r * ≤ x 0 µ d ≤ r * and max j=0,...,k * -1 q j ≤ σ( x 0 d ).

b) Since V (x q k * ) = V (x q k * ) -V (x k ) + V (x k ) ≤ L qk * + γV (x 0 ) ≤ γ+1 2 V (x q 0 ) for r * ≤ x 0 µ d ≤ r * and qk * ≤ 1-γ 2L V (x 0 ) then V (x q k * ) ≤ γ+1 2 V (x q 0 ) for r * ≤ x 0 µ d ≤ r * and max j=0,...,k * -1 q j ≤ min σ( x 0 d ), ω -1 k * 1-γ 2L V (x 0 ) . Taking into account the local and practical ISS proven above, the latter guarantees global ISS of (49) by [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF].

The proof of Corollaries 3 and 4

Denote δ i (t) = m j=i+1 A ij x i (t), where i = 1, ..., m is a number of subsystem in the system (1), ( 2), [START_REF] Nekhoroshikh | Finite-time stabilization under state constraints[END_REF] and the matrices A ij are defined in the proof of Corollary 2. By Theorem 4, each subsystem with δ i = 0 is finitetime (for µ i < 0) or nearly fixed-time (for µ i > 0) stable. Moreover, it is forward complete 6 if δ i is uniformly bounded. The case µ i < 0. Since the m-th subsystem is finite-time stable then ∃T m > 0 such that δ m-1 (t) = 0 for all t ≥ T m . Considering subsequently the systems m -1, m -2,...,1 we conclude that the system (18), ( 39) is finite-time stable. The case µ i > 0. Since the m-th subsystem is fixed-time stable then d m-1 is uniformly bounded and (m -1)-th subsystem practically fixed-time stable (see Theorem but the property guarantees its global uniform asymptotic stability [START_REF] Sontag | Changing supply functions in inputistate stable systems[END_REF]. Using the cascade structure of the system we complete the proof of Corollary 3. The proof of Corollary 4 is literally the same but it uses Theorems 5 and 7 instead of Theorem 4 and 6, respectively.

  This work was supported by the National Natural Science Foundation of China under Grant 62050410352.

Fig. 1 .

 1 Fig. 1. The minimal eigenvalue of the matrix ∆(δ) for µ = -1/2, ρ = 2

  10.0750 -25.1875 6.4250 -25.1875 75.2507, K = [-34.2751 -25.175 -7.5 ].

Fig. 2 .,K

 2 Fig.2. The simulation results for the system (1) with explicitly (left) and consistently (right) discretized finite-time control[START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] for n = 3, m = 1, µ = -0.25, ρ = 2, h = 0.1.5 https://gitlab.inria.fr/polyakov/hcs-toolbox-for-matlab

Fig. 3 .

 3 Fig.3. The simulation results for the system (1) with explicitly (left) and consistently (right) discretized fixed-time control[START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] for n = 3, m = 1, µ = 0.25, ρ = 2, h = 0.2.

Fig. 4 .

 4 Fig. 4. The simulation results for the cascade system (1) with consistently discretized finite-time controllers with n1 = 3, n2 = 2, µ1 = -0.25, ρ1 = 1, µ2 = -1, ρ2 = -1, h = 0.05.

  Ad(s)=e µs d(s)A, d(s)B=e s B, d(s)e τ A =e e -µs τ A d(s), (42) d(s) h 0 e τ A dτ = e µs e -µs h 0 e τ A dτ d(s);[START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF]

h 0 e0 e sA ds = 1 0

 01 sA ds • •[B; e hA B e 2hA B ... e (n-1)hA B]. On the other hand, the homogeneity identities (42) imply hB = d * (ln h)B and e ihA d * (ln h) = d * (ln h)e iA , ∀i ∈ N, ∀h > 0. Hence, we have W h = 1 h h 0 e sA ds d * (ln h)[B, e A B, e 2A B, ..., e (n-1)A B]. Using the identity (43) we derive d * (-ln h) 1 h h e sA ds d * (-ln h), so the formula (44) holds. Finally, since n 0 e τ nA dτ =

  |u(x)-ũ ĥ(x)|≤|u(x)-u(x)|+|u(x)-ũĥ (x)|=x 1+µ d |u(d(-ln x d )x)-u(d(-ln x d )x)|+ x 1+µ d |u(d(-ln x d )x)-u x µ d ĥ(d(-ln x d )x)|≤ C1 x 1+µ d d(ln x d )x-d(-ln x d )x + x 1+µ d C0 x µ d ĥ= C1 x 1+µ d d(ln x d )x-d(-ln x d )x +C2 x 1+2µ d provided that d(ln x d )x -d(ln x d )x ≤ δ and x µd ≤ h 0 / ĥ, where C 2 = C 0 ĥ. Taking r ≤

d

  and d(ln x d )q k ≤ x µ d for x -µ ≥ r . For a sufficiently small δ > 0 the inequalitiesx q k -µ d ≥ r ε (or, equivalently, x -µ d ≥ rε ) and q p ≤ δ σ( x d ) imply q 2 k ≤ δ ε . Selecting r = max{r , 0.5ρ(β B C) -1 } we derive d x d dt ≤ -0.25ρ x 1+µ d

A matrix G d ∈ R n×n is anti-Hurwitz if -G d is Hurwitz.

A dilation in R n is monotone if for any x ∈ R n \{0} the function s → d(s)x , s ∈ R is strictly increasing.

Practical finite-time and fixed-time stability is introduced using the same definitions by replacing a norm (distance to 0) with a distance to a set being a neighborhood of zero.