Tonametl Sanchez
email: tonametl.sanchez@ipicyt.edu.mx

Andrey Polyakov
email: drey.polyakov@inria.fr

Denis Efimov
email: de-nis.efimov@inria.fr

Lyapunov-based Consistent Discretization of Quasi-Continuous High Order Sliding Modes

In this chapter we propose an explicit discretization scheme for class of disturbed systems controlled by homogeneous quasi-continuous High Order Sliding Mode controllers which are equipped with a homogeneous Lyapunov function. Such a Lyapunov function is used to construct the discretization scheme that preserves important features from the original continuous-time system: asymptotic stability, finite-time convergence, and the Lyapunov function itself.

Introduction

. For the case of linear systems we can obtain exact dsicretized models, however, exact discretization of nonlinear systems is in general impossible due to the lack of explicit solutions. Hence, approximating discretization techniques must be used in the nonlinear setting. Nonetheless, for many nonlinear systems it is not a trivial task: first of all, standard discretization techniques usually impose some smoothness requirements on the system; and secondly, they do not preserve some relevant characteristics from the continuous-time system.

For the particular case of High Order Sliding Mode (HOSM) systems, which are non-smooth by nature, it is well known that standard discretization methods produce undesirable behaviors in the discrete-time approximation [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF][START_REF] Utkin | Variable Structure and Lyapunov Control, chap. Sliding mode control in discretetime and difference systems[END_REF][START_REF] Levant | Higher-Order Sliding Modes, differentiation and output-feedback control[END_REF][START_REF] Koch | Discrete-time equivalents of the super-twisting algorithm[END_REF][START_REF] Acary | Chattering-Free Digital Sliding-Mode Control With State Observer and Disturbance Rejection[END_REF][START_REF] Huber | Lyapunov Stability and Performance Analysis of the Implicit Discrete Sliding Mode Control[END_REF][START_REF] Efimov | Realization and Discretization of Asymptotically Stable Homogeneous Systems[END_REF][START_REF] Levant | On Fixed and Finite Time Stability in Sliding Mode Control[END_REF][START_REF] Sanchez | Homogeneous Discrete-Time Approximation[END_REF].

That is why several new strategies to discretize sliding-mode systems have been designed, e.g.: the implicit discretization of standard sliding modes [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF][START_REF] Utkin | Variable Structure and Lyapunov Control, chap. Sliding mode control in discretetime and difference systems[END_REF] and HOSM controllers [START_REF] Brogliato | The Implicit Discretization of the Supertwisting Sliding-Mode Control Algorithm[END_REF]; the discrete-time redesign of the robust exact differentiator of arbitrary order proposed in [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF]; the consistent implicit or semi-implicit discretization algorithms for finite-time and fixed-time stable systems developed in [START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF], which is based on an adequate transformation of the system; the digital implementation of sliding-mode controllers based on the discretization of differential inclusions by means of the implicit Euler method, see e.g. [START_REF] Acary | Chattering-Free Digital Sliding-Mode Control With State Observer and Disturbance Rejection[END_REF][START_REF] Huber | Lyapunov Stability and Performance Analysis of the Implicit Discrete Sliding Mode Control[END_REF].

In this chapter we propose a technique to discretize a class of systems controlled by quasi-continuous HOSM controllers whose origin is asymptotically stable. It is well known that one of the advantages of quasi-continuous HOSM is that the only discontinuity is at the origin [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF]. The proposed technique is based on the discretization procedure provided in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF][START_REF] Sanchez | A consistent discretisation method for stable homogeneous systems based on lyapunov function[END_REF] 1 for homogeneous systems without disturbances. On one hand we particularize the method for quasi-continuous HOSM, nonetheless, on the other hand we extend the method by allowing time-varying disturbances in the model. A relevant feature of the proposed method is that it takes advantage of the information provided by the Lyapunov function for the closed-loop system. As in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF] the discretization scheme has the following properties:

1. Lyapunov function preservation: the Lyapunov function of the continuous-time system is also a Lyapunov function for discrete-time system, guaranteeing this way that the origin of the obtained discrete-time system is Lyapunov stable; 2. Consistency: the origin of the discrete-time approximating system is finite-time stable (this means that the discretization is consistent in the sense described in [START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF]); 3. Independence of the discretization step: the properties of stability and consistency of the obtained discrete-time systems are not affected by the size of the discretization step.

Chapter organization: In Section 2 we state the problem to be solved and provide some definitions and preliminary results. In Section 3 we analyze the dynamics of the studied system by projecting it on a level set of its Lyapunov function. The discretization scheme, proposed in this chapter, is introduced and explained in Section 4. In Section 5, we present some examples of the proposed discretization method. In Section 6 some conclusions are stated.

Notation: The set of integer numbers is denoted by Z. R * + denotes the set R + \ {0}, analogously for the set Z. For a function V : R n → R + , which is continuous and positive definite, we denote the set S V = {x ∈ R n : V (x) = 1}. The class of functions η : R + → R + with η(0) = 0, which are strictly increasing and continuous, is denoted by K .

Problem statement and preliminaries

In this section we describe the class of systems to be studied in this chapter, we also give the statement of the problem to be solved, and we recall some important properties of homogeneous systems.

In this chapter we consider the following continuous-time system

ẋ1 (t) = x 2 (t) . . . ẋn-1 (t) = x n (t) ẋn (t) = d 1 (t) + d 2 (t)u(x(t)) (1)
where x(t) ∈ R n is the state, and u(x(t)) ∈ R is the control signal. The disturbances

d 1 (t), d 2 (t) ∈ R are piece-wise continuous functions such that |d 1 (t)| ≤ d1 , d 2 ≤ |d 2 (t)| ≤ d2 , (2)
for all t ∈ R for some known constants d1 , d 2 , d2 ∈ R * + . For (1), we consider the following sub-class of quasi-continuous controllers u(x) proposed in [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF],

u(x) = -k n σ n (x) σn (x) , (3)
where σ 1 (x) = x 1 , σ1 (x) = |x 1 |, and for i ∈ {2, . . . , n} we have that

r i = n + 1 -i , σ i (x) = ⌈x i ⌋ n r i + k n r i i-1 σ i-1 , σi (x) = |x i | n r i + k n r i i-1 σi-1 , (4)
where we used the notation ⌈x⌋ p := |x| p sign(x).

For the closed-loop system (1), (3) it is also provided in [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF] the Lyapunov function V n : R n → R + given by the following construction for i ∈ {2, . . . , n},

V i (x) = V i-1 (x) +W i (x) , (5)
where

W i (x) = r i 2n |x i | 2n r i -⌈ν i-1 ⌋ 2n-r i r i x i + (1 -r i 2n )|ν i-1 | 2n r i , ν i = -k i ⌈σ i ⌋ n-i n , ν 1 = -k 1 ⌈x 1 ⌋ n-1 n , and V 1 (x) = 1 2 |x 1 | 2 .
Theorem 1 ( [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF]). Consider the closed-loop of (1) with (3). There exist large enough k i > 0 such that the origin of the system is globally finite-time stable and V n given by ( 5) is a Lyapunov function for the system.

A procedure to compute the gains k i is given in [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF]. Let us show three explicit examples of controllers (3) and their respective Lyapunov functions [START_REF] Brogliato | The Implicit Discretization of the Supertwisting Sliding-Mode Control Algorithm[END_REF].

Example 1. For n = 1 we have the controller u(x) = -k 1 sign(x 1 ) , and the Lyapunov function

V 1 (x) = 1 2 x 2 1 . Example 2.
For n = 2 we have the controller

u(x) = -k 2 ⌈x 2 ⌋ 2 + k 2 1 x 1 x 2 2 + k 2 1 |x 1 |
, and the Lyapunov function

V 2 (x) = 1 2 x 2 1 + 1 4 x 4 2 + k 3 1 ⌈x 1 ⌋ 3 2 x 2 + 3 4 k 4 1 x 2 1 .
Example 3. For n = 3 we have the controller

u(x) = -k 3 ⌈x 3 ⌋ 3 + k 3 2 σ 2 (x) |x 3 | 3 + k 3 2 |x 2 | 3 2 + k 3 2 1 |x 1 | , where σ 2 = ⌈x 2 ⌋ 3 2 + k 3 2
1 x 1 . For this case, the Lyapunov function is

V 3 (x) = 1 2 x 2 1 +W 2 (x) +W 3 (x) , where W 2 (x) = 1 3 |x 2 | 3 + k 2 1 ⌈x 1 ⌋ 4 3 x 2 + 2 3 k 3 1 x 2 1 , W 3 (x) = 1 6 x 6 3 + k 5 2 ⌈σ 2 ⌋ 5 3 x 3 + 5 6 k 6 2 σ 2 2 ,
In Section 5 these examples are resumed to illustrate the discretization procedure developed in Section 4.

In order to make the exposition clearer, let us rewrite the closed-loop system (1), (3) as follows

ẋ(t) = f x(t), d(t) , x(t) ∈ R n , d(t) ∈ R 2 , (6)
where

f i (x, d) = x i+1 for i ∈ {1, . . . , n -1}, and f n (x, d) = d 1 -d 2 k n σ n (x)
σn (x) . Note that f : R n+2 → R n is continuous except at x = 0. Let us also denote with D to the set of piece-wise continuous functions d : R → R 2 satisfying (2).

Since the input d is unknown, a usual procedure in sliding mode control to analyze ( 6) is to replace it by the differential inclusion [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF][START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]

ẋ1 (t) = x 2 (t) . . . ẋn-1 (t) = x n (t) ẋn (t) ∈ [-d1 , d1 ] -[d 2 , d2 ]k n σ n (x) σn (x) . (7)
Thus, the solutions of ( 7) are understood as the solutions of a differential inclusion

ẋ ∈ B(x) , x ∈ R n , (8)
associated with [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF], where the set-valued map B satisfies the following basic conditions [10, p. 77]: for all x ∈ R n the set B(x) is nonempty, compact and convex, and the set-valued function B is upper-semicontinuous. In this context, a (generalized) solution of ( 7) is defined as a function x : Γ ⊂ R + → R n which is absolutely continuous and satisfies [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF] for almost all t ∈ Γ [10, p. 50]. Moreover, the existence of solutions of the differential inclusion is guaranteed since B satisfies the basic conditions. Following [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], we refer to (8) as a Filippov differential inclusion, which is obtained by means of a kind of Filippov regularization of (7) [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. In general, the solutions of ( 7) or ( 8) are non-unique, however, observe that the right-hand side of (6) satisfies the conditions2 to guarantee uniqueness of solutions on R n \ {0}.

Homogeneity

Let us begin this section by recalling the definition of Weighted Homogeneity.

Definition 1 ([17, 26, 22]). Given a set of coordinates (x 1 , x 2 , . . . , x n ) for R n , ∆ r (ε)x denotes the family of dilations characterized by the square diagonal matrix ∆ r (ε) = diag(ε r 1 , . . . , ε r n ), where r = [r 1 , . . . , r n ] ⊤ , r i ∈ R * + , and ε ∈ R * + . The components of r are called the weights of the coordinates. Thus:

1. a function V : R n → R is r-homogeneous of degree m ∈ R if V (∆ r (ε)x) = ε m V (x) , ∀x ∈ R n , ∀ε ∈ R * + ; 2. a vector field f : R n → R n , is r-homogeneous of degree µ ∈ R if f (∆ r (ε)x) = ε µ ∆ r (ε) f (x) , ∀x ∈ R n , ∀ε ∈ R * + ; 3. a set-valued map x → B(x) ⊂ R n , is r-homogeneous of degree µ ∈ R if B (∆ r (ε)x) = ε µ ∆ r (ε)B(x) , ∀x ∈ R n , ∀ε ∈ R * + .
A differential inclusion ( 8) is said to be r-homogeneous of degree µ ∈ R if its vector-set field (or set-valued vector field) B is r-homogeneous of degree µ. Now, we recall some important features of r-homogeneous differential inclusions 3 .

Theorem 2 ([31, 26, 3]). Let (8) be r-homogeneous of degree µ < 0 with B satisfying the basic conditions. If x = 0 is strongly globally asymptotically stable then 1. x = 0 is strongly globally finite-time stable; 2. for any positive integer p and any real m > p max{r 1 , . . . , r n } there exists a positive definite function V : R n → R + such that a. V is of class C ∞ for all x ̸ = 0 and of class C p for all x ∈ R n ; b. V is r-homogeneous of degree m; c. there exists a continuous positive definite function W : R n → R + such that it is r-homogeneous of degree m + µ, and

∂V (x) ∂ x b ≤ -W (x) , ∀x ∈ R n , ∀b ∈ B(x) . (9)
Remark 1. It is important to mention that ( 7) is an r-homogeneous differential inclusion of degree µ = -1 with weights r = [n, n -1, . . . , 1] ⊤ . Also note that (since (6) describes the closed-loop system (1), ( 3)), for any function d the vector field f is such that

f (∆ r (ε)x, d) = ε µ ∆ r (ε) f (x, d) , ∀x ∈ R n , ∀ε ∈ R * + . (10)
Now, if V is as in Theorem 2, then the derivative of V along ( 6) is given by

V = -W (x, d) , W (x, d) := - ∂V (x) ∂ x f (x, d) , (11)
where the function W : R n+m → R satisfies the following

W (∆ r (ε)x, d) = ε m+µ W (x, d) , ∀x ∈ R n , ∀d ∈ R 2 , ∀ε ∈ R * + ,
which is a direct consequence of the fact that ∂V (∆ r (ε)x) [START_REF] Sepulchre | Homogeneous Lyapunov functions and necessary conditions for stabilization[END_REF]Prop. 1]) and [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF].

∂ x = ε m ∂V (x) ∂ x ∆ -r (ε) (see, e.g.
Moreover, the solutions of ( 6) are in the set of solutions of the Filippov differential inclusion [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF]. Therefore, by Theorem 2, the derivative of V along the solutions of ( 6) satisfies V = -W (x, d) ≤ -W (x), with W as given in [START_REF] Efimov | Realization and Discretization of Asymptotically Stable Homogeneous Systems[END_REF]. In such a case, there exists α ∈ R * + such that [START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF][START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF] V ≤ -αV

m+µ m (x) . (12)
As stated in Theorem 2, W is r-homogeneous, hence, the constant parameter α in ( 12) can be computed as follows

α = inf x∈S V W (x) . (13)
We know from Theorem 2 that the degree of homogeneity of W is m + µ, which is strictly positive if the homogeneity degree of V is restricted to m > -µ. Observe that this is always the case for ( 5) since m = 2n and µ = -1.

The properties explained so far prove the following result (analogous to those in [START_REF] Haimo | Finite time controllers[END_REF][START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF][START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF] for unperturbed systems).

Lemma 1. Let (8) be a Filippov differential inclusion associated with [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF]. Also let (8) and V be as in Theorem 2. Then, in [START_REF] Crouzeix | Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons[END_REF], for all x(0) ∈ R n , for all t ∈ R + , and for any d ∈ D, the following holds (with α as given in [START_REF] Haimo | Finite time controllers[END_REF]

): V (x(t)) ≤ V (x(0),t) where V (x(0),t) =    V -µ m (x(0)) --µ m αt m -µ , t < m -µα V -µ m (x(0)) 0, t ≥ m -µα V -µ m (x(0)) . ( 14
)
From Lemma 1, we can see that the trajectories of (1) converge to the origin in finite-time. Moreover, the convergence time T (x(0)) to the origin, for the initial condition x(0), is bounded as follows

T (x(0)) ≤ m -α µ V -µ m (x(0)).

Problem statement

As already mentioned in the introduction, an exact discretization for a nonlinear system is (in general) not possible, this due to the lack of knowledge of the exact solution of the system. However, any suitable discretization scheme should preserve relevant properties of the solutions, e.g., the type of convergence of the trajectories to the origin in case of asymptotic stability. If we are able to extract some relevant information from a Lyapunov function, e.g., stability properties and convergence rates, just as it is done in Lemma 1, then such a Lyapunov function should be used to develop a discretization scheme. Hence, the problem to be solved in this chapter is: Develop a discretization scheme, for (6) (which describes the closed-loop system (1), ( 3)) such that: if the origin of (6) is finite-time stable, then the generated discrete-time approximating system preserves the finite-time stability property of the continuous-time system.

In this chapter we solve this problem by taking advantage of the information provided by the homogeneous Lyapunov function of the system. This is done by making a homogeneous projection of the dynamics of the system on a level set of the Lyapunov function. Thus, the evolution of the system's trajectories can be determined from the trajectories of the projected dynamics and an expansion computed by using the information of the decaying rate of the Lyapunov function along the solutions of the system (see Section 3 for more details).

We have to mention that, in the literature, there exist some discretization methods that also utilize the idea of projecting the trajectories of the system onto level sets of the Lyapunov functions. Unfortunately, although some of those methods are able to keep the same Lyapunov function for the discrete-time system, they cannot guarantee that the convergence rates are preserved as well, see [START_REF] Grimm | Geometric Integration Methods that Preserve Lyapunov Functions[END_REF] and the references therein. In general, another disadvantage of those methods is that the projection is not explicit (i.e., an algebraic equation must be solved to find the projection). In contrast, in the discretization method of this chapter, the projection onto the level set of the Lyapunov function is explicit, which represents a procedural advantage.

Projected dynamics

In this section we compute and analyze the projection of the dynamics of ( 6) onto a unitary level set of its Lyapunov function. The developments of this section constitute the fundamentals for the construction of the discretization method proposed in Section 4.

Let V be as in Theorem 2, and define the following change of variable

y = ∆ r (V -1 m (x))x , ∀x ∈ R n \ {0} . (15)
Observe that, according to [28, p. 159], [START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF] constitutes the homogeneous projection of the point x over the level set {x ∈ R n :

V (x) = 1}, thus, y ∈ S V for all x ∈ R n \{0}.
By taking the derivative of ( 15) along ( 6) we obtain

ẏ = ∆ r (V -1 m (x)) I -1 m V -1 (x)Gx ∂V (x) ∂ x f (x, d) , ( 16
)
where I is the n × n identity matrix, and G := diag(r 1 , . . . , r n ). Recall from Remark 1 that f in ( 6) satisfies [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF]. From [START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF] we obtain x = ∆ r (V 1 m (x))y, which can be substituted in [START_REF] Huber | Lyapunov Stability and Performance Analysis of the Implicit Discrete Sliding Mode Control[END_REF] to obtain

ẏ = ∆ r (V -1 m (x)) f (x, d) -1 m V -1 (x)∆ r (V -1 m (x))G∆ r (V 1 m (x))y ∂V (x) ∂ x f (x, d) , = V µ m (x) f (y, d) + 1 m W (x,d) V (x) Gy , = V µ m (x) f (y, d) + 1 m V m+µ m (x)W (y,d) V (x)
Gy , (where W is given by ( 11)), therefore,

ẏ = V µ m (x) f (y, d) + 1 m W (y, d)Gy . (17)
Equation ( 17) describes the dynamics (6) projected onto S V . However, note that we cannot recover the trajectories of (6) directly from the trajectories of ( 17) since ( 15) is not bijective. Here is where we can exploit the information provided by the Lyapunov function. Thus, we proceed to study the dynamics of V , i.e., its derivative along the trajectories of (6). Thus, considering [START_REF] Hong | On an Output Feedback Finite-Time Stabilisation Problem[END_REF], we obtain from ( 11) that

V = -W ∆ r (V 1 m (x))y, d = -V m+µ m (x)W (y, d) . (18)
Note that ( 17) and ( 18) still depend on x, thus, we introduce two auxiliary equations that will be useful for our purposes. Thus, we define a function v : R + → R + such that it is solution to the differential equation (cf. ( 18))

v(t) = -v m+µ m (t)W (z(t), d(t)) , (19)
where the function z : R + → R n is the solution to the following system (cf. ( 17))

ż(t) = v µ m (t) f (z(t), d(t)) + 1 m W (z(t), d(t))Gz(t) . ( 20
)
From the developments made up to this point, we are now ready to state the main results of this section. The first one of these results consists in verifying that the set S V is positively invariant with respect to the trajectories of (20).

Lemma 2. Consider (20) with z(0) ∈ S V , and any continuous function v : R + → R. If µ < 0 and v(t) ̸ = 0 for all t ∈ [0, T ) for some T ∈ R * + , then z(t) ∈ S V for all t ∈ [0, T ), and any d ∈ D.

Proof. To verify that S V is positively invariant, let us compute the derivative of V (z) along the trajectories of (20), thus

V = v µ m ∂V (z) ∂ z f (z, d) + 1 m W (z, d) ∂V (z) ∂ z Gz . Since ∂V (z) ∂ z f (z, d) = -W (z, d),
we can see that if W (z, d) = 0 then V (z(t)) = 0. Thus, without loss of generality, we assume that W (z, d) ̸ = 0. According to the equality4

∂V (z) ∂ z Gz = mV (z) , (21)
we obtain

V = v µ m W (z, d) -1 +V (z) .
Hence V (z(t)) = 0 if and only if V (z(t)) = 1 for all t ∈ [0, T ). □

The second main result of this section is with respect to a useful representation of the solutions of [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF].

v(t) =    v(0) -µ m --µ m Ŵ0 (t) m -µ , -µ m Ŵ0 (t) < v(0) -µ m 0, -µ m Ŵ0 (t) ≥ v(0) -µ m , (22)
where Ŵ0 (t) := t 0 W (z(τ), d(τ)) dτ, satisfies [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] with v(t) > 0 for all t in the interval [0,Θ d (v(0))), and v(t) → 0 as t → Θ d (v(0)).

Proof. This lemma is proven by direct integration of (19) to obtain [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. Nonetheless, let us provide some clarifying details. Since W (z, d) > 0 for all z ∈ S V and all d ∈ D, then v in ( 22) is strictly decreasing to zero, hence, for each v(0

) ∈ R * + there exists a maximal Θ d (v(0)) ∈ R * + such that Ŵ0 (t) < v(0) -µ
m for all t ∈ 0,Θ d (v(0)) . Note that, 0,Θ d (v(0)) is the interval of time such that the right-hand side of ( 20) is well-defined on it. □ Note that v : R + → R + , given by ( 22), is a continuous function, also note that v(t) ̸ = 0 for all t ∈ [0,Θ d (v(0)) and for initial conditions v(0) ∈ R * + . Therefore, ( 22) satisfies the hypothesis required in Lemma 2 for the function v.

The last results of this section (the following theorem and its corollary) constitute the fundamentals of the discretization technique that is developed in Section 4.

Theorem 3. Let (6) and V be as in Lemma

1. Define ζ = [v, z ⊤ ] ⊤ ∈ Z, where Z = R * + × S V . Consider (6) with x ∈ R n \ {0}, d ∈ D, and (19)
-( 20) with ζ ∈ Z. The solutions of (6) and the solutions of ( 19)-( 20) are equivalent with the homeomorphism Φ : R n \ {0} → Z given by

Φ(x) = V (x) ∆ r V -1 m (x) x . (23)
Proof. Since V is a continuous function of x, we can ensure that Φ is also a continuous function of x, moreover, it has a continuous inverse Φ -1 : Z → R n \ {0} given by

Φ -1 (ζ ) = ∆ r v 1 m z . ( 24
)
The remaining steps of the proof are straightforward, it is only needed to note that ζ (t) = Φ(x(t)) satisfies ( 19)-( 20) and x(t) = Φ -1 (ζ (t)) satisfies [START_REF] Crouzeix | Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons[END_REF]. □ Corollary 1. If v is a solution of (19) with initial condition v(0) = V (x(0)), and if z is a solution of (20) with initial condition z(0) = ∆ r v -1 m (0) x(0), for any x(0) ∈ R n \ {0}, then the function x : R + → R n given by

x(t) = ∆ r v 1 m (t) z(t), t < Θ d (v(0)), 0, t ≥ Θ d (v(0)), (25)
(with Θ d as given in Lemma 3) is solution of (6) for all t ∈ R + .

Discretization scheme

In this section we describe the proposed discretization scheme. The main idea of the method is a consequence of the developments presented in Section 3. Indeed, observe that (20) represents the dynamics of ( 6) but projected on a unit sphere S V , and that v (as given in Lemma 3) characterize the decay of the Lyapunov function V evaluated along the trajectories of (6). So, the main idea is to compute a numerical solution5 of ( 19)-( 20), and next, to define the numerical solution of the original system by using [START_REF] Mangasarian | Nonlinear Programming[END_REF].

Remark 2. Although, several different schemes can be used to obtain a numerical solution of ( 19)-( 20), we restrict ourselves in this chapter to the explicit (also known as forward) Euler method taking into account that v(t) is a nonegative variable and z(t) belongs to a manifold for all t ≥ 0.

To construct the discrete-time approximation of v, we see from Lemma 3 that for any h ∈ R + ,

v(t + h) =    v -µ m (t) --µ m Ŵ (t) m -µ , -µ m Ŵ (t) < v -µ m (t) 0, -µ m Ŵ (t) ≥ v -µ m (t) , (26)
where Ŵ (t) := t+h t W z(τ), d(τ) dτ. Note that [START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF] give us the exact value of v(t + h), but it requires the value of Ŵ (t). Hence, by defining a discrete-time approximation of Ŵ (t) we immediately obtain a discrete-time approximation of v. For example, if we use the forward Euler method with an integration step h, then we obtain the discrete-time approximation v k ∈ R to v(kh) given by

v k+1 =    v -µ m k --µh m W (z k , d k ) m -µ , -µh m W (z k , d k ) < v -µ m k 0, -µh m W (z k , d k ) ≥ v -µ m k , (27)
for all k ∈ Z + , where d k := d(kh), and z k ∈ R n is the discrete-time approximation to z(kh) given by

z k+1 = ∆ r V -1 m (z k+1 ) zk+1 , zk+1 = z k + hv µ m k f (z k , d k ) + 1 m W (z k , d k )Gz k , v k+1 > 0 z k , v k+1 = 0 . ( 28
)
Let us explain the main idea of [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. First, the term zk+1 can be regarded as an explicit Euler discretization of [START_REF] Lambert | Numerical methods for ordinary differential systems: The initial value problem[END_REF]; second, such a discretization is scaled by the factor ∆ r V -1 m (z k+1 ) . Note that such scaling is necessary since we have to ensure that z k ∈ S V for all k ∈ Z + . Also note that, the condition zk+1 ̸ = 0 is necessary to have [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] well defined, this is why we require the following assumption. Assumption 1. Consider (6) and V as in Theorem 3. For all z ∈ S V , all d ∈ D, and all τ ∈ R * + ,

z + τ f (z, d) + 1 m W (z, d)Gz ̸ = 0 .
In the following lemma we state some sufficient conditions that can be helpful in verifying Assumption 1.

Lemma 4. Assumption 1 holds in any of the following cases:

1. for all z ∈ S V such that ∂V (z) ∂ z z = 0 we have that z ⊤ F(z, d) ≥ 0, where F(z, d)

:= f (z, d) + 1 m W (z, d)Gz; 2. ∂V (z) ∂ z z ̸ = 0 for all z ∈ S V ; 3. the set {z ∈ R n : V (z) ≤ 1} is convex.
Proof. From Lemma 2 we know that, for all z ∈ S V and all d ∈ D, F(z, d) is tangent to S V . Hence, ∂V (z) ∂ z F(z, d) = 0 for all z ∈ S V . On the other hand, if there exist τ ∈ R * + , z ∈ S V and d ∈ D such that z + τF(z, d) = 0, then the vector F(z, d) is necessarily collinear to z but it has the opposite direction. Therefore, the following are necessary conditions to have z + τF(z, d) = 0: ∂V (z)

∂ z z = 0 and z ⊤ F(z, d) < 0. The analysis in the previous paragraph let us clearly see that a sufficient condition to guarantee that Assumption 1 holds is either ∂V (z) ∂ z z ̸ = 0 for all z ∈ S V (which is satisfied, for example, if the function z → ∂V (z) ∂ z z is positive definite), or z ⊤ F(z, d) ≥ 0 for all z such that ∂V (z)

∂ z z = 0. This proves the first two items of the lemma. The third item of the lemma is proven as follows. On one hand, if the set {z ∈ R n : V (z) ≤ 1} is convex, the fact that V is homogeneous guarantees that the sets {z ∈ R n : V (z) ≤ a} are also convex for all a ∈ R * + , hence, we have that the function V is quasi-convex (see, e.g. [START_REF] Boyd | Convex Optimization[END_REF]Section 3.4.1]). On the other hand, z = 0 is a global minimum of V since it is positive definite. From these reasoning we conclude that V is a pseudo-convex function (see, e.g. [6, Lemma 2.1]), therefore (by definition of pseudo-convexity), ∂V (z) ∂ z z > 0 for all z ∈ S V , see also [25, p. 40]. □

It is important to mention that in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF] it is wrongly stated that (in absence of disturbances) the first item in Lemma 4 is an equivalent condition to Assumption 1, however, this is only true for n = 2. Now, we can state the following theorem, which is the main result of this section. Theorem 4. Let (6) and V be as in Lemma 1. Suppose that Assumption 1 holds. Consider the discrete-time approximation of (6) given by

x k+1 = ψ(x k ) = ∆ r v 1 m k+1 z k+1 , x k ̸ = 0, 0, x k = 0, k ∈ Z + , (29)
where v k+1 and z k+1 are given by ( 27) and (28), respectively, with

v k = V (x k ), z k = ∆ r V -1 m (x k )
x k , and x 0 = x(0). Then V is a Lyapunov function for [START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF], and for all h ∈ R * + and all x(0) ∈ R n \ {0}, x k → 0 as k → ∞. Moreover, V (x k ) ≤ V (x 0 , kh) for all k ∈ Z + , with V given by [START_REF] Hairer | Solving Ordinary Differential Equations I[END_REF].

The proof of the theorem is completely analogous to the proof of Theorem 2 in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF]. It is clear from Theorem 4 that the solutions of ( 29) reach the origin in a finite number of steps. Remark 3. Let us underline the main features of the proposed discretization scheme described in Theorem 4:

1. the discretization method is consistent, this means that the stability properties and the convergence rate from the solutions of the continuous-time system are preserved; 2. the Lyapunov function is preserved, i.e., the Lyapunov function from the continuous-time system is a Lyapunov function for the discrete-time approximating system as well; 3. the discretization method is explicit since the right-hand side of ( 29) does not depend on x k+1 but only on x k .

Numerical convergence to the solutions

In this section we verify that the solutions of the discrete-time approximating system converge to the solutions of the continuous-time system. The methodology is the same as that given in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF]. First, consider the solution x : [0, a] → R n of ( 6) for some fixed time a ∈ R * + . Second, suppose that the discretization step h is given by h = a/N, for some N ∈ Z * + , thus it is clear that h → 0 as N → ∞.

Let {x k } N k=0 be the sequence generated by means of some discretization of ( 6). Consider the step-function (associated to such a discretization method) defined as t → x(t) := x k , for all t ∈ [kh, (k + 1)h). An essential requirement for any discretization technique is that its associated step-function converges uniformly on [0, a] to the solution x as the step size h tends to zero (or equivalently, as N tends to infinity). The standard procedure to verify such a convergence property consists in confirming that the global truncation error6 tends to zero as the step h tends to zero. Such a confirmation can be achieved by verifying the existence of a function η ∈ K such that the local truncation error7 E(t + h) := x(t + h)x k+1 satisfies the following (see, e.g. [START_REF] Walter | Proof of Peano's existence theorem without using the notion of the definite integral[END_REF] and [14, pp. 37 and 159])

|E(t + h)| ≤ hη(h) . (30)
Thus, in this section we demonstrate the convergence property of the discretization scheme proposed in Section 4, by means of the verification of a local truncation error estimate given by 30. It is important to see that taking into account Theorem 3, we only need to demonstrate that v k and z k converge to v and z, respectively. Thus, we only have to analyze the local truncation error estimates of v k and z k as stated in the following theorem.

Theorem 5. Assume that the hypotheses of Theorem 4 hold with v(t) = v k and z(t) = z k for some t ∈ R + . Assume also that min{V (x(t + h)), v k+1 } ≥ b for some b ∈ R * + . Then, there exist functions η v , η z ∈ K such that in [START_REF] Nesić | Perspectives in Robust Control, chap. Sampled-Data Control of Nonlinear Systems: an Overview of Recent Results[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]:

|E v (t + h)| := |v(t + h) -v k+1 | ≤ h η v (h) , (31)
|E z (t + h)| := |z(t + h) -z k+1 | ≤ h η z (h) . ( 32
)
The proof of Theorem 5 is given below, but first we state the following lemma, which is used for the proof of the theorem. We use the following notation: the i-th element of the vector zk is denoted as zi k . Lemma 5. Consider [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. Given H ∈ R * + , under the assumptions of Theorem 5, there exist constants γ i , γ i ∈ R * + such that γ i ≤ |z i k+1 | ≤ γi for all h ≤ H and all i ∈ {1, . . . , n}.

Proof. On one hand, we have that Assumption 1 guarantees the existence of the constants γ i . On the other hand, from [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] we can see that for every i = 1, . . . , n,

|z i k+1 | ≤ ζ i + hv µ m k fi + 1 m ᾱr i ζ i , where D = [-d1 , d1 ] × [d 2 , d2 ], fi = sup z∈S V d∈ D | f i (z, d)| , ζ i = sup z∈S V |z i | , ᾱ = sup z∈S V d∈ D |W (z, d)| .
Now, since v k is decreasing and µ < 0, the hypotheses of the lemma ensure that

γ i = ζ i + Hb µ m fi + 1 m ᾱr i ζ i .

□

Proof of Theorem 5

First we analyze E v given by [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. From ( 26) and [START_REF] Nesić | Perspectives in Robust Control, chap. Sampled-Data Control of Nonlinear Systems: an Overview of Recent Results[END_REF] we have that (denoting

E v = E v (t + h)) |E v | ≤ v(t) 1 --µ m v µ m (t) Ŵ (t) m -µ -1 --µ m v µ m (t)W (z k , d k )h m -µ ,
where Ŵ (t) := t+h t W (z(τ), d(τ)) dτ. Note that we have used the hypothesis v(t) = v k . Now, we rewrite these inequalities as follows

|E v | ≤ hv(t) 1 --µ m v µ m (t) Ŵ (t) m -µ -1 --µ m v µ m (t)W (z k , d k )h m -µ h .
Assume that d is continuous at t. Recall that W is continuous for all z ∈ R n \ {0} and all d ∈ R 2 , moreover, z(t) = z k and d(t) = d k . Hence, it is clear (e.g. by using the L'Hôpital's rule) that

lim h→0 1 h 1 --µ m v µ m (t) Ŵ (t) m -µ -1 --µ m v µ m (t)W (z k , d k )h m -µ = 0 .
Observe that W (z, d) is positive and bounded for all z ∈ S V and all d ∈ D, therefore, there exists a function ηv ∈ K (which does not depend on z k ∈ S V ) such that

1 h 1 --µ m v µ m (t) Ŵ (t) m -µ -1 --µ m v µ m (t)W (z k , d k )h m -µ ≤ ηv (h) .
Thus, the result of the theorem is obtained by taking η v (h) = v(t) ηv (h). Now, we analyze the error E z , which is given by [START_REF] Sanchez | Homogeneous Discrete-Time Approximation[END_REF]. Observe from ( 28) that

E z (t + h) = z(t + h) -∆ r V -1 m (z k+1 ) zk+1 ,
which can be rewritten as

E z (t + h) = z(t + h) -zk+1 + I -∆ r V -1 m (z k+1 ) zk+1 .
Hence, for i = 1, . . . , n, we have that

E z i (t + h) = z i (t + h) -zi k+1 + 1 -V -r i m (z k+1 ) zi k+1 . (33)
Since z(t + h) ∈ S V , the term 1 -V -r i m in (33) can be rewritten as follows

1 -V -r i m (z k+1 ) = 1 V r i m (z k+1 ) V r i m (z k+1 ) -1 = 1 V r i m (z k+1 ) V r i m (z k+1 ) -V r i m (z(t + h)) .
From Lemma 5, and for any H ∈ R * + , we can ensure the existence of constants

a 1 , a 2 ∈ R * + such that a 1 ≤ V (z k+1 ) ≤ a 2 for all h ≤ H. Hence, 8 V r i m (z k+1 ) -V r i m (z(t + h)) ≤ L i V (z k+1 ) -V (z(t + h)) ≤ L i L v zk+1 -z(t + h) , for some constants L i , L v ∈ R * + . Thus, we obtain 1 -V -r i m (z k+1 ) ≤ c i zk+1 -z(t + h) with c i := a -r i m 1 L i L v .
Hence, we find a bound for (33) as follows 8 Since [a 1 , a 2 ] ⊂ R is compact and a 1 > 0, the function g

: [a 1 , a 2 ] ⊂ R * + → R given by g(V ) = V r i m
is Lipschitz continuous. Also, z k and z belong to a compact subset of R n on which V is Lipschitz continuous.

|E z i (t + h)| ≤ |z i (t + h) -zi k+1 | + c i zk+1 -z(t + h) |z i k+1 | , ≤ |z k+1 -z(t + h)| + c i zk+1 -z(t + h) |z i k+1 | , ≤ (1 + c i |z i k+1 |) zk+1 -z(t + h) , ≤ ci zk+1 -z(t + h) , ci := 1 + c i γ i , (34)
with γ i as given in Lemma 5. To analyze the term zk+1z(t + h) define F(x, d) := f (x, d) + 1 m W (x, d)Gx. Thus, from ( 20) and [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] we have that z(t + h) = z(t) + k F(z k , d k ), respectively. By the Taylor's theorem, there exists a function h → R(t, h) such that z(t

+ h) = z(t) + v µ m (t)F(z(t), d(t))h + R(t, h), and 1 h R(t, h) → 0 as h → 0. Since z k = z(t), d(t) = d k , and v k = v(t), zk+1 -z(t + h) = z k + hv µ m k F(z k , d k ) -z(t) -v µ m (t)F(z(t), d(t))h -R(t, h) , = |R(t, h)| = h 1 h R(t, h) , lim h→0 1 h R(t, h) = 0 . (35)
Therefore, from [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF] and [START_REF] Sepulchre | Homogeneous Lyapunov functions and necessary conditions for stabilization[END_REF] we conclude that there exist c ∈ R * + and

η z ∈ K such that |E z (t + h)| ≤ hη z (h) , η z (h) ≥ c 1 h R(t, h) .

□ 5 Examples

In this section, firstly we resume the examples given in Section 2 to illustrate the discretization scheme proposed in Section 4. Secondly, in Example 7 we show a possible application of the proposed discretization scheme to construct a discretetime implementation of the controller 3. The disturbances to be used in the examples are given by

d 1 (t) = A 1 sin(ω 1 t) , d 2 (t) = 1 -A 2 cos(ω 2 t) , (36)
where d1 = A 1 , d 2 = 1 -A 2 and d2 = 1 + A 2 , with the parameters A 1 = 1, ω 1 = π, A 2 = 1/5, and ω 2 = 10π.

Example 4. For n = 1 consider (1) with the controller and the Lyapunov function given in Example 1. Observe that ∂V 1 (x) ∂ x x = x 2 1 . Hence, Lemma 4 guarantees that Assumption 1 holds. For the simulation, we use the gain k 1 = 3 and the initial condition x(0) = 5. Fig. 1 shows the behavior of the discretization scheme (29) with a step of h = 0.1. It is clear that the state of the system converges exactly to the origin in finite-time despite the disturbance. Additional details of this example are given in [START_REF] Sanchez | Lyapunov-based consistent discretization of stable homogeneous systems[END_REF], where it is even compared with the implicit discretization schemes from [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF] and [START_REF] Acary | Chattering-Free Digital Sliding-Mode Control With State Observer and Disturbance Rejection[END_REF].

∂V 2 (x) ∂ x x = 1 + 3 2 k 4 1 x 2 1 + 5 2 k 3 1 ⌈x 1 ⌋ 3 2 x 2 + x 4 2 .
By applying the Young's inequality to the term ⌈x 1 ⌋ 3 2 x 2 it can be verified that the function given by x →

∂V 2 (x) ∂ x x is positive definite if k 1 < 8 1 4 3 5 5 8 1 3 -4 - 1 4
.

Under this condition, Lemma 4 guarantees that Assumption 1 holds. For the simulation we use the initial conditions x 1 (0) = 2, x 2 (0) = 2 and the gains

k 1 = 1, k 2 = 4.
The discretization step is again set to h = 0.1. In Fig. 2 it can be seen the states of the discrete-time approximation [START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF] preserving the finite-time converge feature from the continuous-time model. Example 6. Now, for the case n = 3, consider (1) with the controller and the Lyapunov function given in Example 3. To simulate the discretization scheme (29) we use the initial conditions x 1 (0) = 2, x 2 (0) = 2, x 3 (0) = 2 and the gains k 1 = 0.6, k 2 = 1.7, k 3 = 1200. The discretization step is set to h = 0.001. Fig. 3 shows the states of the system converging exactly to the origin in finite-time. Since, for this example, Assumption 1 is not easily verifiable by means of Lemma 4, we confirm along the simulation that zk ̸ = 0 for k ≥ 0. This can be corroborated in Fig. 4, which shows the norm of zk . Example 7. In this example we show a possible application of the proposed discretization scheme to construct a discrete implementation of the controller (3).

Consider (1) for n = 2. Assume that the state x is measured at instants t k = kh, k ∈ Z + , h ∈ R * + , and the control signal must be constant for the interval I k = [kh, (k + 1)h), i.e. u(t) = u k for all t ∈ I k . It is well known that the standard discretization of u(x(t)) given by

u k = u(x(t k )) , (37)
generates numerical chattering as it can be seen in Fig. 5, which shows the behavior of the states of the system for d 1 = 0 and d 2 = 1, with the controller (3) discretized as in [START_REF] Walter | Proof of Peano's existence theorem without using the notion of the definite integral[END_REF]. For the simulation the initial conditions are x 1 (0) = 1, x 2 (0) = 1 and the gains k 1 = 1, k 2 = 4. The step for the control discretization is h = 0.01. The continuous-time dynamics is approximated by means of the explicit Euler discretization with a step of h s = 1 × 10 -5 . Fig. 5 States of (1) in closed loop with the discretization of (3) given by (37) (undisturbed case). Now, as it is done with implicit discretization techniques for sliding mode controllers (see, e.g. [START_REF] Drakunov | On Discrete-Time Sliding Modes[END_REF][START_REF] Acary | Chattering-Free Digital Sliding-Mode Control With State Observer and Disturbance Rejection[END_REF]) we consider the controller discretization u k = u(x k+1 ) .

(38)

We use the proposed discretization method to compute x k+1 . First, observe the function u : R n → R given by ( 3) is r-homogeneous of zero degree. Hence, if x = ∆ r (ε)z, then u(x) = u(z) for all ε ∈ R * + . Thus, by considering (29), we can replace (38) with u k = u(z k+1 ) .

To compute z k+1 we use [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] and v k+1 given by [START_REF] Nesić | Perspectives in Robust Control, chap. Sampled-Data Control of Nonlinear Systems: an Overview of Recent Results[END_REF] with the data

v k = V (x(t k )) , z k = ∆ r v -1 m k x(t k ) .
Now, note that both v k+1 and z k+1 given by ( 27) and ( 28), respectively, depend on the disturbance d, and it is generally unknown. Thus, we compute v k+1 and z k+1 by assuming the nominal case, i.e. with the disturbance d such that d 1 = 0 and d 2 = 1. Finally, observe that u is discontinuous at zero, hence we have to assign the value of u(z k+1 ) for the case z k+1 = 0. Since lim ε↓0 u(∆ r (ε)z) = u(z) for all z ̸ = 0, we set u(z k+1 ) = u(z k ) if z k+1 = 0. In Fig. 6 we can see the states of the system with the controller (3) discretized as in (39). The disturbance signals are set as before, i.e. d 1 ≡ 0 and d 2 ≡ 1. The initial conditions are x 1 (0) = 1, x 2 (0) = 1 and the gains k 1 = 1, k 2 = 4. It can be seen that the numerical chattering has been considerably reduced with the proposed discretization scheme. Fig. 6 States of (1) in closed loop with the discretization of (3) given by (39) (undisturbed case). Now we repeat the simulations by considering the disturbances given in [START_REF] Utkin | Variable Structure and Lyapunov Control, chap. Sliding mode control in discretetime and difference systems[END_REF]. In Fig. 7 and Fig. 8 we can appreciate the states of the system and the control signals with the two different methods. It is clear that the proposed Lyapunov-based discretization helps to reduce the numerical chattering effect in the state signals. It is also noticeable that the accuracy in the second state is improved with the Lyapunovbased method, but it is not for the first state.

Conclusion

We have provided in this chapter a discretization scheme for a class of systems controlled by a family of quasi-continuous HOSM controllers. Two of the most relevant properties of the method are that it preserves from the continuous-time system the finite-time convergence to the origin and the Lyapunov function. Another interesting feature of the technique is that both the discretization and the projection procedures are explicit. Finally, we have shown an example of the application of the proposed discretization-scheme to design a discrete-time implementation of a quasi-continuous controller that helps to reduce the numerical chattering effect. Fig. 8 States of (1) in closed loop with the discretization of (3) given by (39) (disturbed case).

Lemma 3 .

 3 Let (6) and V be as in Lemma 1. For any initial condition v(0) ∈ R * + , and any d ∈ D, there exists Θ d (v(0)) ∈ R * + such that the function v : R + → R + given by

m

  (τ)F(z(τ), d(τ)) dτ and zk+1 = z k + hv µ m

Fig. 1 1 . 5 .

 115 Fig. 1 Discrete-time approximation of (1), (3) for n = 1.

Fig. 2

 2 Fig. 2 Discrete-time approximation of (1), (3) for n = 2.

Fig. 3

 3 Fig. 3 Discrete-time approximation of (1), (3) for n = 3.

Fig. 4

 4 Fig.4Norm of zk in[START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] for the discrete-time approximation of (1), (3) for n = 3.

Fig. 7

 7 Fig.7States of (1) in closed loop with the discretization of (3) given by (37) (disturbed case).

Under the assumption of d ∈ D, it can be seen that the right-hand side of (6) is locally Lipschitz in x on R n \ {0}.

Following[START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF], we use the term strong stability (which involves all the solutions) in Theorem 2 to contrast with the term of weak stability, which claims properties of some solutions[START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF].

This equation is known as the Euler's theorem for weighted homogeneous functions, see, e.g. [2, Proposition

5.4].

In this chapter, we mean by numerical solution a sequence {z k } k∈Z+ such that z 0 = z(0), and for some h ∈ R * + , z k approximates z(kh).

The global truncation error can be understood as the accumulation of the errors generated at each step in a given compact interval [0, a], see, e.g.[START_REF] Lambert | Numerical methods for ordinary differential systems: The initial value problem[END_REF] or[14, p. 159].

The local truncation error is the one-step error computed by assuming that E(t) = 0, i.e. x(t) = x k .

Acknowledgements authors acknowledge the support of: the project ANR DIGITSLID (ANR 18-CE40-0008); CONACYT CVU-371652.