Andrey Polyakov
email: drey.polyakov@inria.fr

Yuri Orlov

Harshal Oza

Sarah Spurgeon

Robust Finite-time Stabilization and Observation of a Planar System Revisited

The second order planar nonlinear affine control problem is studied. A homogeneous robust finite-time stabilizing control is developed for the most general case of matched and, the more challenging, mismatched nonlinear perturbations. A homogeneous observer is designed for the planar system. Explicit restrictions on the observer gains and nonlinearities are presented. The main contribution lies in the proposed combination of the explicit and implicit Lyapunov function methods as well as weighted homogeneity while providing finitetime stability analysis. Theoretical results are supported by numerical simulations.

I. INTRODUCTION

The problem of finite-time stabilization goes back to the classical results of optimal control design [START_REF] Fuller | Relay control systems optimized for various performance criteria[END_REF]. Indeed, the time-optimal bang-bang algorithm represented in state feedback form is perhaps the most famous example of a finitetime stabilizing control. The first theoretical investigations of finite-time stability and stabilization appear in [START_REF] Roxin | On finite stability in control systems[END_REF], where, in particular, the definition of finite-time stability as well as the corresponding Lyapunov theorems are developed for general dynamical systems. Subsequently, finite-time controllers are presented in [START_REF] Haimo | Finite time controllers[END_REF]. Currently this topic is popular in the context of sliding mode [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF], nonlinear [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Oza | Continuous uniform finite time stabilization of planar controllable systems[END_REF], [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], robust [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF], [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF] and optimal [START_REF] Chernousko | Control of nonlinear dynamical systems: methods and applications[END_REF], [START_REF] Polyakov | Suboptimal LMI-based solution of minimum time control problem[END_REF] control systems. The problem of finite-time stability also appears in mechanical systems with dry friction [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], [START_REF] Biemond | Bifurcations of equilibrium sets in mechanical systems with dry friction[END_REF].

Finite-time stability analysis is frequently related to the so-called homogeneity property of a dynamical system. Homogeneity theory has an established history going back to Euler and his homogeneous function theorem. Currently the so-called weighted homogeneity [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF] has been introduced in the paper of V.I. Zubov [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]. It is wellknown [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF] that asymptotically stable systems with negative homogeneity degree are finite-time stable. It is noteworthy that the problem of computing an upper bound on the finite settling time of homogeneous finite time systems without the help of finite time Lyapunov functions is an interesting problem. Recent results for the planar case can be found in [START_REF] Oza | Continuous uniform finite time stabilization of planar controllable systems[END_REF] which makes use of a general result provided in [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF].

Second order nonlinear models cover rather a large class of the physical (in particular, mechanical) and biological systems. This paper deals with finite-time control and estimation of planar systems. It revises results of [START_REF] Oza | Continuous uniform finite time stabilization of planar controllable systems[END_REF] for the case of both matched and mismatched uncertainties. The principal contribution of this analysis is two fold. Firstly, a combination of the explicit and (homogeneity-based) implicit Lyapunov function analysis is presented for asymptotic and finite-time stability, respectively. It allows the control parameters and system nonlinearities to be quantitatively explored as well settling time estimates to be specified. Secondly, the upper bound on the unmatched disturbances for the double integrator system covers a broader class of disturbances with a non-Lipschitz upper bound. The paper is organized as follows. Section II presents the problem formulation followed by Sections III and IV which outline the controller and the observer respectively. Numerical examples are given in Section V. Section VI concludes the paper.

II. PROBLEM STATEMENT

Consider the system

ẋ1 (t) = x 2 (t) + ω 1 (t, x 1 (t), x 2 (t)), ẋ2 (t) = u(t) + ω 2 (t, x 1 (t), x 2 (t)), t ∈ R, (1)
where x 1 , x 2 ∈ R are states, u is the control input, ω 1 : R 3 → R and ω 2 : R 3 → R are non-linear functions.

In this paper we deal with two problems:

• finite-time stabilization of the system (1) under the assumption that the nonlinear functions ω 1 and ω 2 are unknown but appropriately bounded; • finite-time observation of the system (1) with the output

y(t) = x 1 (t) (2)
under the assumption that the nonlinear functions ω 1 and ω 2 are known. Recent Lyapunov based results on finite-time control and observation for the perturbed double integrator can be found in [START_REF] Moreno | A Lyapunov approach to output feedback control using second-order sliding modes[END_REF], [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Orlov | Finite time stabilization of a perturbed double integrator part I: Continuous sliding mode-based output feedback synthesis[END_REF]. The most recent robust finite time stabilization result when ω 1 = 0 and |ω 2 | ≤ M |x 2 | α , for some a priori known constant M > 0, can be found in the reference [START_REF] Oza | On robustness of a class of homogeneous continuous finite time controllers[END_REF]. The results in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Oza | Continuous uniform finite time stabilization of planar controllable systems[END_REF] present finite time stabilization for the case of matched non-Lipscitz perturbations while the result in [START_REF] Hong | On an output feedback finitetime stabilization problem[END_REF] studies robustness to C 1 matched disturbances, i.e. ω 1 = 0 and ω 2 = 0. The paper [START_REF] Li | Continuous finite-time output regulation for disturbed systems under mismatching condition[END_REF] presents output feedback finite time stabilization for mismatched perturbations. It is noted in this reference that the output and not the whole state is finite time stabilized. In comparison, the proposed full state feedback based results in this paper are superior for the planar case as finite time stabilization of both the states x 1 (t) and x 2 (t) is achieved via the implicit Lyapunov function approach. Existing homogeneous1 controllers [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF], [START_REF] Orlov | Finite time stabilization of a perturbed double integrator part I: Continuous sliding mode-based output feedback synthesis[END_REF] and homogeneous observers [START_REF] Perruquetti | Homogeneous finite time observer for nonlinear systems with linearizable error dynamics[END_REF] are utilised in this paper. The main challenge is to achieve finite-time stabilization as well as observation of the system (1) in the presence of the perturbations ω 1 = 0 and ω 2 = 0.

III. FINITE-TIME CONTROLLER

Define the homogeneous feedback law [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF], [START_REF] Orlov | Finite time stabilization of a perturbed double integrator part I: Continuous sliding mode-based output feedback synthesis[END_REF] as follows

u(t)= -µ 1 |x 1 (t)| α 2-α sign[x 1 (t)]-µ 2 |x 2 (t)| α sign[x 2 (t)], (3)
where α ∈ [0, 1), µ 1 > 0, µ 2 > 0 are control parameters. It is well-known [4, Th. 2], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF]Th. 3.1], that in the unperturbed case with ω 1 = ω 2 = 0, the closed-loop system (1)-( 3) is homogeneous of negative degree. Hence, asymptotic stability of the origin implies finite-time stability, i.e. the origin will be reached after some finite instant of time. The theorem below characterizes (possibly uncertain) functions ω 1 and ω 2 , which do not destroy the finite-time stability property.

Theorem 1: If for some α ∈ [0, 1) and p ij ≥ 0, i, j = 1, 2 the nonlinear functions ω 1 and ω 2 are bounded as follows

|ω 1 (t, x 1 , x 2 )| ≤ p 11 |x 1 | 1 2-α + p 12 |x 2 |, |ω 2 (t, x 1 , x 2 )| ≤ p 21 |x 1 | α 2-α + p 22 |x 2 | α , (4)
for all (t, x 1 , x 2 ) ∈ R 3 and the control parameters µ 1 > 0, µ 2 > 0 satisfies the restrictions

2µ1-3p11-(µ2+p22)(2-α) 2ξ1+ξ3(α+1) > 2 3-α 3-α 2 µ 2-α 2 1 , 2µ1-3p11-(µ2+p22)(2-α) 2ξ1+ξ3(α+1) > 2+p11+2p21+α(µ2+p22) 2ξ2-ξ3(1-α) > 0, ξ 1 = (3-α)(µ1p11+α(µ1p12+p21)/(α+1)) 2 3-α 2 (µ1(2-α)) α-1 2 , ξ 2 = (3-α)(µ2-p22-(µ1p12+p21)/(α+1)) 2 3-α 2 ξ 3 = (µ 1 (2 -α)) α-1 2 ξ 1 , (5)
then the feedback (3) stabilizes the system (1) to the origin in finite time with the settling time estimate [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF]. Proofs of all the theorems are given in the Appendix, where settling-time estimates are also presented.

The nonlinear structure of the closed-loop system (1)-( 3) implies a nonlinear restriction (5) to the control gains µ 1 and µ 2 . Note that the system of nonlinear inequalities ( 5) is feasible for sufficiently small p 11 ≥ 0 and p 12 ≥ 0.

Consider some particular cases.

• If p 11 = p 12 = p 21 = 0 then the system (5) is feasible for any p 22 ≥ 0, since it is equivalent to µ 1 > (µ 2 + p 22 )(2 -α)/2 and µ 2 > p 22 . This coincides with the conditions derived in [START_REF] Oza | On robustness of a class of homogeneous continuous finite time controllers[END_REF].

• In the case of matched perturbations (p 11 = p 12 = 0), it can be shown that

µ 2 > p 22 + 2-α 2 p 21 , µ 1 > 2-α 2 (µ 2 +p 22 )+ αp21 (3-α) 1-α 2 (2-α) 1-α 2 (α+1) √ µ1 + 0.5 µ 2-α 2 1 , µ1-2-α 2 (µ2+p22) (µ1(2-α)) 0.5(1-α) +0.5(α+1) > αp21(1+p21+0.5α(µ2+p22)) (α+1)(µ2-p22-0.5p21(2-α)
) . The obtained system of inequalities is feasible with respect to µ 1 , µ 2 for any p 21 ≥ 0 and p 22 ≥ 0.

• In the sliding mode case (α = 0), the inequalities (4) become

|ω 1 (t, x 1 , x 2 )| ≤ p 11 |x 1 | 1 2 + p 12 |x 2 |, |ω 2 (t, x 1 , x 2 )| ≤ p 21 + p 22 ,
and the inequalities ( 5) can be rewritten as follows

µ 1 > µ 2 + 3/2 + 0.5/ √ 3 p 11 + p 22 + p 11 √ µ 1 √ 6 µ 2 > p 21 + p 22 + µ 1 (p 12 + p 11 ) µ1-µ2-3 2 p11-p22 µ1p11 1+2 3 2 √ µ1 > 1+ 1 2 p11+p21 µ2-µ1(p12+p11)-p22-p21 .
Evidently, the last system of inequalities is feasible for p 12 < 1 and sufficiently small p 11 < 1 -p 12 .

IV. FINITE-TIME OBSERVER Consider now the problem of finite-time observation of the system (1) with the measured output [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]. Consider the so-called homogeneous observer [START_REF] Perruquetti | Homogeneous finite time observer for nonlinear systems with linearizable error dynamics[END_REF] of the form

dx1(t) dt = -ν 1 |e 1 (t)| 1+β 2
sign[e 1 (t)]+x 2 (t)+ω 1 (t,y(t),x 2 (t))

dx2(t) dt = -ν 2 |e 1 (t)| β sign[e 1 (t)] + ω 2 (t, y(t), x2 (t)) (6
) where ν 1 > 0, ν 2 > 0, β ∈ [0, 1) and e 1 (t) = x1 (t) -y(t). Denote e 2 (t) = x2 (t)-x 2 (t). The observation error equation has the form

ė1 (t) = -ν 1 |e 1 (t)| 1+β 2 sign[e 1 (t)] + e 2 (t) + ∆ω 1 , ė2 (t) = -ν 2 |e 1 (t)| β sign[e 1 (t)] + ∆ω 2 , (7)
where

∆ω 1 = ω 1 (t, y(t), x2 (t)) -ω 1 (t, y(t), x 2 (t)) and ∆ω 2 = ω 2 (t, y(t), x2 (t)) -ω 2 (t, y(t), x 2 (t)).
Theorem 2: If for some β ∈ [0, 1), q 1 ≥ 0 and q 2 ≥ 0 the inequalities

|∆ω 1 | ≤ q 1 |e 2 | and |∆ω 2 | ≤ q 2 |e 2 | 2β 1+β (8)
hold and the observer parameters ν 1 > 0 and ν 2 > 0 satisfy the restrictions

1 2 -q1- q 2 β (1+β) √ ν 2 √ ν2 η2+ 1+3β 2(1+β) η3 > 2(1+β) 3+β 3+β 2(1+β) , 1 2 -q1- q 2 β (1+β) √ ν 2 √ ν2 η2+ 1+3β 2(1+β) η3 > √ ν2+ ν 2 1 2 √ ν2 + q2 1+β η1- 1-β 2(1+β) η3 > 0, η 1 = (3+β) 2(1+β) 3+β 2(1+β) ν 1 -2q1β 1+3β , η 2 = (3+β) 2(1+β)(2ν2) 1-β 2(1+β) q1(1+β) 1+3β + q2 ν2 , η 3 = (3+β) 2(1+β) 3+β 2(1+β) q1(1+β) 1+3β + q2 ν2 . (9)
then the system ( 7) is finite-time stable.

The system of inequalities ( 9) is feasible for sufficiently small q 1 ≥ 0 and q 2 ≥ 0. To estimate the upper bounds of the parameters q 1 and q 2 , consider particular cases.

• If q 1 = q 2 = 0, the inequalities (9) imply that the system (7) is finite time stable for any positive ν 1 > 0 and ν 2 > 0. • If q 2 = 0 then the system of inequalities ( 9) is feasible for any q 1 ∈ [0, 0.5). • For q 1 = 0 the system (9) is feasible ∀ q 2 ≥ 0.

V. NUMERICAL EXAMPLES

A. Finite-time stabilization of the uncertain planar system Consider the nonlinear system of the form

ẋ1 (t) = x1 4+4|x1| + x2|x2| 8 √ x 2 1 +x 2 2 + x 2 , ẋ2 (t) = u(t), (10)
where

x 1 , x 2 ∈ R. It is straightforward to see that the restriction (

B. Finite-time observation of the planar system

Consider the non-linear positive system

ẋ1 (t) = -2x 1 (t) + 1 -1 2 x 1 (t)x 2 (t), ẋ2 (t) = 1 4 -x 1 (t) x ρ 2 (t), (11)
where x 1 ≥ 0 and x 2 ≥ 0 are state variables, ρ ∈ [0, 1).

The paper [START_REF] Cattani | Qualitative analysis of second-order models ot tumor-immune system competition[END_REF] utilizes this system with ρ = 1 as the tumourimmune model for the so-called "steady control of tumour growth". In this case, the variable x 1 is related to the density of the lymphocyte population and x 2 is proportional to the density of the tumour cells. Due to physical restrictions, the variables are assumed to be bounded as follows 0 < x min 1 ≤ x 1 (t) ≤ x max 1 and 0 ≤ x 2 (t) ≤ x max 2 for all t ∈ R. It is assumed that the variable x 1 can only be measured and the problem is to reconstruct the state variable x 2 by means of the model [START_REF] Haimo | Finite time controllers[END_REF].

Introduce a state observer of the form

dx1(t) dt = 1-2x 1 (t)-x1(t) 2 x2 (t)-ν 1 (t)|e 1 (t)| 1+β 2 sign[e 1 (t)], dx2(t) dt = 1 4 xρ 2 (t) -x 1 (t)x ρ 2 (t) + ν2 (t)|e 1 (t)| β sign[e 1 (t)], where e 1 (t) = x1 (t) -x 1 (t), ν1 (t) = ν 1 x 1 (t) 2 and ν2 (t) = ν 2 x 1 (t) 2 ,
with ν 1 > 0 and ν 2 > 0. The change of time τ = t 0 2 x1(s) ds allows the error equation to be expressed in the form

ė1 (τ ) = e 2 (τ )-ν 1 |e 1 (τ )| 1+β 2 sign[e 1 (τ )], ė2 (τ ) = (4x1(τ )-1)(x ρ 2 (t)-x ρ 2 (t)) 2x1(τ ) -ν 2 |e 1 (τ )| β sign[e 1 (τ )],
where e 2 = x 2 (t) -x2 (t) Therefore, the obtained error equation is equivalent to [START_REF] Biemond | Bifurcations of equilibrium sets in mechanical systems with dry friction[END_REF] with ∆ω 1 = 0 and |∆ω 2 | ≤ q 2 • |e 2 | ρ , where q 2 = max

x1∈{x min 1 ,x max 1 } 2 -1 2x1
. This satisfies the conditions of Theorem 2 if β = ρ 2-ρ . Note that for the case ρ = 1, the finite-time homogeneous observer β ∈ [0, 1) can be constructed locally using the estimate of ∆ω 2 and a-priori boundedness of the state variables x 1 and x 2 .

VI. CONCLUSIONS Results on robust finite time stabilization are presented.

A combination of homogeneous Lyapunov functions and implicit Lyapunov functions are utilised to prove finite time stability of planar uncertain system. Both the problems of finite time stable controller design and finite time stable observer design are analysed in the presence of mismatched disturbances that admit a non-Lipschitz upper bound. This is superior to existing results in the area. An interesting open problem is to identify similar Lyapunov functions to extend these results to more general n-dimensional system.

VII. APPENDIX Below Young's inequality is used whereby:

|z 1 ||z 2 | ≤ |z1| r r + |z2| q q , 1 r + 1 q = 1, r, q > 1, (12
) where z 1 , z 2 are real numbers.

A. Proof of Theorem 1 I. Asymptotic Stability (Explicit Lyapunov Method)

Consider the Lyapunov function candidate

V 0 (x 1 , x 2 ) = cU 3-α 2 (x 1 , x 2 ) + µ 2-α 2 1 x 1 x 2 , ( 13
)
where c > 0 and x 1 (t) x 2 (t)

U (x 1 , x 2 ) = µ 1 2 -α 2 |x 1 | 2 2-α + 1 2 x 2 2 . ( 14
)
0 1 2 3 -4 -2 0 2 4 t x2(t) and x2(t)
x 2 (t) x2 (t) Fig. 2. Simulation results for finite-time observation Applying Young's inequality for q = 3 -α and r

= 3-α 2-α produces µ 2-α 2 1 |x 1 | 2 3-α |x 2 | 2 3-α ≤ µ 1 2 -α 3 -α |x 1 | 2 2-α + 1 3 -α x 2 2 .
Hence, for

c > 2 3 -α 3-α 2 (15)
it follows that V 0 (x 1 , x 2 ) > 0 outside the origin, i.e. V 0 is a positive definite function. Since

d dt U 3-α 2 (x 1 , x 2 ) = (3-α) µ1|x1| α 2-α sign[x1]ω1-µ2|x2| α+1 +x2ω2 U 1-α 2 2 ≤ (3-α) µ1p11|x1| α+1 2-α +(µ1p12+p21)|x1| α 2-α |x2|-(µ2-p22)|x2| α+1 U 1-α 2 2
holds true, applying Young's inequality for q = α + 1 and r = α+1 α produces

|x 1 | α 2-α |x 2 | ≤ α |x1| α+1 2-α α+1 + |x2| α+1 α+1 .
The following may now be derived

d dt U 3-α 2 (x 1 , x 2 ) ≤ - (3-α)(µ2-p22- µ 1 p 12 +p 21 α+1 ) 2 U 1-α 2 |x 2 | α+1 + (3-α)(µ1p11+α(µ1p12+p21)/(α+1)) 2 U 1-α 2 |x 1 | α+1 2-α ≤ -ξ 2 x 2 2 + ξ 1 |x 1 | 2 2-α + ξ 3 |x 1 | α+1 2-α |x 2 | 1-α ≤ -(ξ 2 -0.5ξ 3 (1 -α))x 2 2 + (ξ 1 + 0.5ξ 3 (α + 1))|x 1 | 2 2-α ,
where

ξ 1 = (3 -α)(µ 1 p 11 + α(µ 1 p 12 +p 21 )/(α+1)) 2 3-α 2 (µ 1 (2 -α)) α-1 2 , ξ 2 = (3-α) (µ 2 -p 22 -(µ 1 p 12 +p 21 )/(α+1)) 2 3-α 2 , ξ 3 = (µ 1 (2 -α)) α-1 2 ξ 1 ,
and Young's inequality is applied on the final step. On the other hand,

d dt (x 1 x 2 )=x 2 2 +x 2 ω 1 -µ 1 |x 1 | 2 2-α -µ 2 x 1 |x 2 | α sign[x 2 ]+x 1 ω 2 ≤ (1 + p 21 )x 2 2 + p 11 |x 1 | 1 2-α |x 2 | -(µ 1 -p 11 )|x 1 | 2 2-α + (µ 2 + p 22 )|x 1 ||x 2 | α Applying Young's inequality for q = 2 α and p = 2 2-α |x 1 ||x 2 | α ≤ 2 -α 2 |x 1 | 2 2-α + α 2 |x 2 | 2
and the inequality

|x 1 | 1 2-α |x 2 | ≤ |x 1 | 2 2-α 2 + x 2 2 2 produce V0 (x 1 , x 2 ) ≤ -µ 2-α 2 1 µ 1 -3p11 2 -(µ2+p22)(2-α) 2 -cξ 1 -cξ3(α+1) 2 |x 1 | 2 2-α -cξ 2 -cξ3(1-α) 2 -µ 2-α 2 1 1+ p11 2 +p 21 + α(µ2+p22) 2 x 2 2 .
Therefore, if the inequalities (15) and

µ1- 3p11 2 - (µ2+p22)(2-α) 2 ξ1+ ξ3(α+1) 2 > c µ 2-α 2 1 > 1+ p 11 2 +p21+ α(µ 2 +p 22 ) 2 ξ2- ξ 3 (1-α) 2 (16)
hold then the function ( 13) is a strict Lyapunov function for the closed-loop system (1). In the sliding mode case (α = 0) then the right-hand side of the closed-loop system is discontinuous. The Lyapunov function-based stability analysis can be provided in this case using the concept of Filippov solutions (see, for example, [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF]).

II. Finite-time Stability (Implicit Lyapunov Method)

The function

V 0 (x 1 , x 2 ) is r-homogeneous of degree 3-α 1-α
with the weights:

r 1 = 2-α 1-α and r 2 = 1 1-α . Indeed, V 0 (λ r1 x 1 , λ r2 x 2 ) = λ 3-α 1-α V 0 (x 1 , x 2 ), ∀x 1 , x 2 ∈ R and ∀λ > 0. Let us denote Ω = (z 1 , z 2 ) ∈ R 2 : V 0 (z 1 , z 2 ) = 1 and γ = inf (z1,z2)∈Ω l 1 |z 1 | 2 2-α + l 2 z 2 2 ,
where

l 1 = µ 2-α 2 1 µ 1 -3p11 2 -(µ2+p22)(2-α) 2 -cξ 1 -cξ3(α+1) 2 > 0, l 2 = cξ 2 -cξ3(1-α) 2 -µ 2-α 2 1 1+ p 11 2 +p 21 + α(µ 2 + p 22 ) 2 > 0.
In this case, V0 (

x 1 , x 2 ) ≤ -γ < 0 if (x 1 , x 2 ) ∈ Ω.
The finite-time Lyapunov function will be designed using the implicit Lyapunov function method:

Theorem [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF] If there exists a continuous function

Q : R + × R n → R (V, x) → Q(V, x)
that is C1) continuously differentiable in the domain R + × R n ; C2) for any x ∈ R n \{0} there exist V -∈ R + and V + ∈ R + :

Q(V -, x) < 0 < Q(V + , x); C3) for Ω = (V, x) ∈ R n+1 : Q(V, x) = 0 the limits lim x → 0 (V, x) ∈ Ω V = 0, lim V → 0 + (V, x) ∈ Ω x = 0, lim x → ∞ (V, x) ∈ Ω V = +∞; exist; C4) for ∀V ∈ R + and ∀x ∈ R n \{0} the inequality -∞ < ∂Q(V, x) ∂V < 0 holds; C5) ∀t ∈ R + , ∀V ∈ R + , ∀x ∈ R n \{0} : Q(V, x) = 0 ∂Q(V, x) ∂x f (x) ≤ cV 1-µ ∂Q(V, x)
∂V where c > 0 and 0 < µ ≤ 1 are some constants, then the equilibrium point x = 0 of the system

ẋ = f (x)
is globally finite time stable with the following settling time estimate

T (x 0 ) ≤ V µ (x 0 ) cµ ,
where V (x) : Q(V, x) = 0.

In order to apply the implicit Lyapunov function theorem to the system (1), define

Q(V, x) = V 0 V -r1 x 1 , V -r2 x 2 -1, (17)
where

x = (x 1 , x 2 ) T ∈ R 2 , r 1 = 2-α 1-α , r 2 = 1
1-α are homogeneity weights of the system (1). It can be easily checked that the conditions C1)-C4) hold. On the one hand,

∂Q ∂V = V 0 (x 1 , x 2 ) ∂ ∂V V -3-α 1-α =-3-α 1-α V0( x1 V r 1 , x2 V r 2 ) V =-3-α 1-α 1 V if Q(V, x) = 0.
On the other hand, for the system (1)

∂Q ∂x f (x) = V -3-α 1-α V0 (x 1 , x 2 ) ≤ -V -3-α 1-α (l 1 |x 1 | 2 2-α +l 2 x 2 2 ) = -V -1 l 1 V -r1 x 1 2 2-α +l 2 V -r2 x 2 2 ≤-γV -1 if Q(V, x) = 0.
Therefore, the condition C5) also holds for µ = 1 and c = γ(1-α)

3-α and the settling time admits the following estimate:

T (x) ≤ 3-α γ(1-α) V 1-α 3-α 0 (x 1 , x 2 ). (18)
III. Estimation of γ. If z 2 = 0 then V 0 (z 1 , z 2 ) = 1 can be rewritten as follows

c µ1(2-α) 2 |z1| 2 2-α z 2 2 + 1 2 3-α 2 + µ 1 |z1| 2 2-α z 2 2 2-α 2 sign[z 1 z 2 ] = |z 2 | α-3
Hence, denoting y 1 = |z1|

γ = inf y1≥0 l1y1+l2 c µ 1 (2-α) 2 y1+ 1 2 3-α 2 +µ 2-α 2 1 y 2-α 2 1 2 3-α
In the general case, the parameter γ can be calculated numerically. However, if l2 l1 ≤ 1 µ1(2-α) then the function under inf is monotone decreasing by y 1 ∈ [0, +∞) and

γ = 2l 1 µ 1 (2 -α)c 2
V 0 (e 1 , e 2 ) = cU 3+β 2(1+β) (e 1 , e 2 ) - 1 √ ν 2 e 1 e 2 , (19)
where c > 0 and

U (e 1 , e 2 ) = 1 β+1 |e 1 | β+1 + 1 2ν2 e 2 2 . (20)
Applying Young's inequality for r = 3+β 2 and q = 3+β 1+β it can be derived that

|x 1 | 2(1+β) 3+β |x2| √ ν2 2(1+β) 3+β ≤ 2 3+β |x 1 | 1+β + 1+β (3+β)ν2 x 2 2 .
Therefore, for c > 2(1+β) 3+β 3+β 2(1+β) [START_REF] Oza | Continuous uniform finite time stabilization of planar controllable systems[END_REF] the function V 0 is positive definite. Since

d dt U 3+β 2(1+β) (e 1 , e 2 ) = 3+β 2(1+β) U 1-β 2(1+β) U = (3+β)U 1-β 2(1+β) 2(1+β) -ν 1 |e 1 | 1+3β 2 +∆ω 1 |e 1 | β sign[e 1 ]+ e2∆ω2 ν2 ≤ (3+β)U 1-β 2(1+β) 2(1+β) -ν 1 |e 1 | 1+3β 2 +q 1 |e 1 | β |e 2 |+ q2|e2| 1+3β 1+β ν2 , then applying Young's inequality for r= 1+3β 2β , q= 1+3β 1+β |e 1 | β |e 2 | ≤ 2β 1+3β |e 1 | 1+3β 2 + 1+β 1+3β |e 2 | 1+3β 1+β it follows that d dt U 3+β 2(1+β) (e 1 , e 2 ) ≤ (3+β) 2(1+β) U 1-β 2(1+β) × -ν 1 -2q1β 1+3β |e 1 | 1+3β 2 + q1(1+β) 1+3β + q2 ν2 |e 2 | 1+3β 1+β ≤ -η 1 |e 1 | 1+β + η 2 |e 2 | 2 + η 3 |e 1 | 1-β 2 |e 2 | 1+3β 1+β ,
where

η 1 = (3+β) ν1- 2q1β 1+3β 2(1+β) 3+β 2(1+β) , η 3 = (3+β) q1(1+β) 1+3β + q 2 ν 2 2(1+β) 3+β 2(1+β) , η 2 = (3+β) 2(1+β)(2ν2) 1-β 2(1+β) q1(1+β) 1+3β + q2 ν2 ,
Applying again Young's inequality for r = V0 (e 1 , e 2 ) ≤ -

2(1+β) 1-β , q = 2(1+β) 1+3β it is obtained that |e 1 | 1-β 2 |e 2 | 1+3β 1+β ≤ 1-
1 2 -q 1 -q2β (1+β) √ ν2 √ ν 2 -c η 2 + 1+3β 2(1+β) η 3 e 2 2 -c η 1 -1-β 2(1+β) η 3 - √ ν 2 + ν 2 1 2 √ ν2 + q2 1+β |e 1 | 1+β .
Hence, the inequality ( 21) and the inequalities 1 2 -q1-

q 2 β (1+β) √ ν 2 √ ν2 η2+ 1+3β 2(1+β) η3 > c > √ ν2+ ν 2 1 2 √ ν2 + q2 1+β η1- 1-β 2(1+β) η3
guarantee that V 0 is a strict Lyapunov function for [START_REF] Biemond | Bifurcations of equilibrium sets in mechanical systems with dry friction[END_REF].

II. Finite-time Stability (Implicit Lyapunov Method) The function V 0 (x 1 , x 2 ) is r-homogeneous with the same weights as the original system : r 1 = 2 1-β and r 2 = 1+β 1-β . Namely, V 0 (λ r1 x 1 , λ r2 x 2 ) = λ 3+β 1-β V 0 (x 1 , x 2 ) for all x 1 ∈ R, x 2 ∈ R and λ > 0.

Denote Ω = (z 1 , z 2 ) ∈ R 2 : V 0 (z 1 , z 2 ) = 1 and γ = inf (z1,z2)∈Ω l 1 |z 1 | -c η 2 + 1+3β 2(1+β) η 3 > 0. In this case, V0 (x 1 , x 2 ) ≤ -γ < 0 if (x 1 , x 2 ) ∈ Ω.

The proof of finite-time stability can be provided by means of the implicit Lyapunov function method similarly to Theorem 1 and the settling time estimate has the form

T (x) ≤ 3 + β γ(1 -β) V 1-β 3+β 0 (x 1 , x 2 ). (22)

2 Fig. 1 . 4 .

 214 Fig. 1. Simulation results for finite-time stabilization

Fig. 2 1 = 0 . 1 , x max 1 = 1

 210111 presents simulation results for ρ = 0.5, x min and the parameters ν 1 = 10 and ν 2 = 400 satisfying (9).

and y 2 =

 2 |z 2 | 3-α allows the following derivation:

2 2 - 2 1 2 √ ν2 + q2 1+β > 0 and l 2 = 1 2

 2221+β1 α + l 2 z 2 2 > 0, where l 1 = c η 1 -1-β 2(1+β) η 3 -√ ν 2 + ν

  β 2(1+β) |e 1 | 1+β + 1+3β 2(1+β) e 2 2 ,

		3+β
	dU	2(1+β) (e1,e2) dt	≤ -η 1 -(1-β)η3 2(1+β) |e 1 | 1+β + η 2 + (1+3β)η3 2(1+β)	e 2 2 .
	On the other hand,
	d dt -e1e2 √ ν2 ≤ -1-q1 √ ν2 e 2 2 + ν1|e1|	1+β 2 |e2|+q2|e1||e2| √ ν2	2β 1+β	+	√	ν 2 |e 1 | 1+β
	and applying Young's inequality for r = 1 + β, q = 1+β β it
	is obtained that |e 1 | ν 2 -1/2	|e 2 |	2β 1+β	≤ |e1| 1+β 1+β + β 1+β e 2 2 ν -1 2 ,
	d dt -e1e2 √ ν2 ≤ -	1 2 -q1-√ ν2 (1+β) q 2 β √ ν 2	e 2 2 +	√	ν 2 +	ν 2 1 √ ν2 + q2 2 1+β |e 1 | 1+β
	Hence,

A function g : R n → R (resp. a vector field f : R n → R n ) is said to be r-homogeneous of degree m iff for all λ > 0 and for all x ∈ R n we have g(D(λ)x) = λ m g(x) (resp. f (D(λ)x) = λ m D(λ)f (x)), where D(λ) = diag{λ r i } n i=1 and r = (r 1 , . . . , rn) T ∈ R n + .


Andrey Polyakov

Laurentiu Hetel

On Finite-time Stabilization via Relay Feedback Control

The problem of finite-time stabilization of multiinput linear system by means of relay feedback is considered. A new control design procedure, which combines convex embedding technique with implicit Lyapunov function method, is developed. The issues of practical implementation of the obtained implicit relay feedback are discussed. Theoretical result is supported by numerical simulation.

I. INTRODUCTION

Theory of relay automatic control systems has a long outstanding history. Relay feedbacks appeared in the early technological developments of the 19th century. However, the first theoretical study of relay control methods was provided in 1950s [START_REF] Flugge-Lotz | Discontinuous automatic control[END_REF], [START_REF] Tsypkin | Relay Control Systems[END_REF]. The modern frequency domain approach to analysis of the relay systems can be found in [START_REF] Boiko | Discontinuous Control Systems: Frequency-Domain Analysis and Design[END_REF] When the sliding mode control methodology [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF] was invented, it suggested to utilize a proper fast relay switching strategy in order to maintain the motion of the control system on a prescribed surface in the state space. Indeed, the classical example of the sliding mode system has the form of relay feedback:

ẋ(t) = -sign[x(t)], t > 0, x(0) = x 0 ∈ R,
where the sign function is defined as follows: sign[ρ] = 1 if ρ > 0 and sign[ρ] = -1 if ρ < 0. Any trajectory of this system reaches the state x = 0 in a finite time and remains thereafter. In fact, finite-time stability frequently accompanies the relay and sliding mode feedback systems [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Principles of 2-sliding mode design[END_REF], [START_REF] Polyakov | Unified Lyapunov function for a finitetime stability analysis of relay second-order sliding mode control systems[END_REF]. The main application domains of sliding mode approach are electrical and electro-mechanical systems [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF].

The modern theoretical framework of hybrid dynamical systems [START_REF] Van Der Schaft | An introduction to hybrid dynamical systems[END_REF], [START_REF] Liberzon | Switching in Systems and Control[END_REF] includes the relay feedbacks as a particular case of switched affine systems [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF], [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF], [START_REF] Xu | Practical stability and stabilization of hybrid and switched systems[END_REF]. Recently, the ideas of convex embedding have been applied in order to design an exponentially stabilizing relay switching law based on the existence of a stabilizing static linear feedback [START_REF] Hetel | Variable structure control with generalized relays: A simple convex optimization approach[END_REF]. The present paper addresses the finite-time stabilization of linear multi-input system using relay control. The main goal of the article is to show how the convex embedding procedure can be used with the implicit Lyapunov function method [START_REF] Korobov | A solution of the problem of synthesis using a controllability function[END_REF], [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit lyapunov function approach[END_REF] in order to derive a finitetime stabilizing relay feedback.

The paper is organized as follows. The next section presents notations used in the paper. After that, the problem statement and basic assumptions are discussed. Some This research is supported by ANR ROCC-SYS (ANR-14-CE27-0008). A. Polyakov (andrey.polyakov@inria.fr) is with Non-A team-project, Inria Lille-Nord Europe, 40, avenue Halley, 59650 Villeneuve d'Ascq, France. L. Hetel (laurentiu.hetel@ec-lille.fr) and A. Polyakov are with CRIStAL UMR CNRS 9189, 59650 Villeneuve d'Ascq, France.

preliminary facts are considered in Section IV. Next, the main results are presented. Finally, numerical simulation example and concluding remarks are given. Some supporting constructions are provided in Appendix.

II. NOTATION

• R is the set of real numbers; R + = {x ∈ R : x > 0};

• x denotes the Euclidian norm of the vector x ∈ R n ;

• range(B) is the column space of the matrix B ∈ R n×m ;

• diag{λ 1 , ..., λ n } is a diagonal matrix with elements λ i ;

• the order relation P > 0(< 0, ≥ 0, ≤ 0) for P ∈ R n×n means that P is symmetric and positive (negative) definite (semidefinite); • if P > 0 then the matrix P 1/2 := B is such that B 2 = P ; • λ max (P ) and λ min (P ) denote maximum and minimum eigenvalues of the symmetric matrix P ∈ R n×n ; • I m ∈ R m×m is the identity matrix;

• a continuous function σ : R + → R + belongs to the class K if it is monotone increasing and σ(s) → +0 as s → +0; • co(U ) is the convex closure of the set U ⊂ R n ; int{U } denotes the interior of the set U .

III. PROBLEM STATEMENT

Let us consider a model of a control system described by the ordinary differential equation (ODE):

ẋ(t) = Ax(t) + Bu(t), t ∈ R + , (1)
where x ∈ R n is the state vector, u ∈ R m is the vector of control inputs, A ∈ R n×n is the system matrix, B ∈ R n×m is the matrix of control gains. It is assumed that the matrices A and B are known, rank(B) = m ≤ n and the pair (A, B) is controllable; the whole state vector x can be measured and utilized for control purpose. The control input u is assumed to be generalized relay, i.e. it can take values from a given discrete set:

u(t) ∈ U := {v 1 , v 2 , ..., v N } , v i ∈ R m , t ∈ R + , (2)
where N is a natural number. In addition, the assumption

0 ∈ int{co(U)} ⊂ R m (3)
is possessed in order to guarantee the existence of the stabilizing relay control (see, [START_REF] Hetel | Variable structure control with generalized relays: A simple convex optimization approach[END_REF] for the details). As we will see further, this configuration includes as a particular case the classical sliding control generated by sign functions. This control configuration may also be related to the simplex method in [START_REF] Bartolini | Simplex sliding mode control of multi-input systems with chattering reduction and mono-directional actuators[END_REF], [START_REF] Bajda | Vector method of design of sliding motion and simplex algorithms[END_REF] and to the stabilization of switched affine systems [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF]. Filippov theory of differential equations with discontinuous right-hand sides [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] is utilized below in order to take into account the discontinuity of the control law. The control aim is to stabilize the origin of the system (1) in a finite time and to specify the corresponding set of admissible initial conditions (i.e. the domain of finite-time attraction).

Following the ideas of [START_REF] Hetel | Variable structure control with generalized relays: A simple convex optimization approach[END_REF] the relay stabilizing control law can designed in two steps. Initially, some continuous finite-time stabilizing feedback should be selected. For this purpose the method of the Implicit Lyapunov Functions (ILF) is utilized [START_REF] Korobov | A solution of the problem of synthesis using a controllability function[END_REF], [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit lyapunov function approach[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF]. Next, a proper convex embedding procedure [START_REF] Hetel | Variable structure control with generalized relays: A simple convex optimization approach[END_REF] is applied in order to construct the relay switching law in the form

u(t) ∈ u r (x(t)) = argmin v∈U Γ T (t, x(t))v, (4)
where Γ : R n+1 → R m is a continuous (outside the origin) nonlinear function to be defined. The inclusion in (4) indicates that argmin is not unique in general case.

In particular, if m = 1 and

U = {-1, 1} then u r (x) = -sign[Γ(x)
] similarly to the sliding mode control [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], where

sign[ρ] =    1 if ρ > 0, -1 if ρ < 0, {-1, 1} if ρ = 0.
Note that in order to define the control input at the current state x(t) according to the formula (4) we just need to find the minimum of Γ T (x(t))v over finite set of values v ∈ U. This operation does not need applying any finite or infinite dimensional optimization procedure. We just need to calculate N scalar products Γ T (x(t))v i , i = 1, . . . , N and select the minimum.

IV. PRELIMINARIES

A. Finite-Time Stability

Let us consider the system of the form

ẋ(t) = f (x(t)), x(0) = x 0 , t ∈ R + , (5)
where x ∈ R n is the state vector, f : R n → R n is a nonlinear discontinuous (but locally measurable) vector field. Let Filippov procedure be applied for regularization of the discontinuous ODE, i.e by definition, an absolute continuous function x(•, x 0 ) is a solution to the Cauchy problem associated to (5) if x(0, x 0 ) = x 0 and almost everywhere it satisfies the differential inclusion

ẋ ∈ F (x) = δ>0 µ(N )=0 co f (x + B(δ)\N ), (6)
where µ(N ) = 0 means that a set N ⊂ R n has measure 0. Let the origin be an equilibrium point of the system (5), i.e. 0 ∈ F (0). Only strong uniform stability properties of the system [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] are studied in the this paper, so the corresponding words "strong uniform" will be omitted below for shortness and simplicity of the presentation.

Definition 1 ( [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]): The origin of system ( 5) is said to be finite-time stable if it is asymptotically stable and finite-time attractive, i.e. for any x 0 ∈ M\{0} there exists T (x 0 ) ∈ R + such that x(t, x 0 ) = 0 for all t ≥ T (x 0 ), where M is a neighborhood of the origin and T is called the settling-time function of the system [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]. If M = R n then the origin is globally finite-time stable.

B. Implicit Lyapunov Function Method

The next theorem is utilized below in order to design the feedback law.

Theorem 2: [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit lyapunov function approach[END_REF] If there exists a continuous function

Q : R + × R n → R that satisfies the conditions C1) Q is continuously differentiable in R + × R n \{0}; C2) for any x ∈ R n \{0} there exist V ∈ R + such that Q(V, x) = 0; C3) let Ω = {(V, x) ∈ R + × R n : Q(V, x) = 0} and lim x→0 (V,x)∈Ω V = 0 + , lim V →0 + (V,x)∈Ω x = 0, lim x →∞ (V,x)∈Ω V = +∞; C4) the inequality ∂Q(V,x) ∂V < 0 holds for all V ∈ R + and x ∈ R n \{0}; C5) there exist c ∈ R + and µ ∈ (0, 1] such that sup t∈R+,y∈K[f ](t,x) ∂Q(V, x) ∂x y ≤ cV 1-µ ∂Q(V, x) ∂V , (V, x) ∈ Ω;
then the origin of system ( 5) is globally finite time stable with the following settling time estimate:

T (x 0 ) ≤ V µ 0 cµ , where V 0 ∈ R + : Q(V 0 , x 0 ) = 0.
Theorem 2 provides the sufficient conditions of finite-time stability for implicit definition of Lyapunov function. The conditions C1)-C4) guarantee existence and uniqueness of a continuously differentiable (outside the origin) positive definite radially unbounded function V : R n → R + , which is implicitly defined by the equation Q(V, x) = 0. The implicit function theorem [START_REF] Courant | Introduction to calculus and analysis[END_REF] gives

∂V ∂x = - ∂Q ∂V -1 ∂Q ∂x .
Due to conditions C4), C5) the estimate

V (x) ≤ sup y∈F (x)
∂V ∂x y ≤ -cV 1-µ implies the finite-time stability of the origin of (5). Corollary 3: If the conditions C1)-C4) of Theorem 2 are fulfilled then the set

ε(V 0 ) = {z ∈ R n : Q(V 0 , z) ≤ 0} (7) is the V 0 -level set {s ∈ R n : V (s) ≤ V 0 } of the positive definite function V : R n → R + implicitly defined by the equation Q(V, s) = 0. Proof: Indeed, if s ∈ R n is such that V (s) = α, where α ∈ R + then Q(α, s) = 0, i.e. s ∈ ε(α, P ). The condition C4) implies that Q(V , s) < 0 (i.e s ∈ ε(V )) for any V > α and Q(V , s) > 0 (i.e. s / ∈ ε(V ))
for any V < α. Corollary 3 allows us to adapt Theorem 2 to local finitetime stability analysis possessing the condition C5) locally, i.e. 0 < V < V and x ∈ ε(V ) for some given V ∈ R + . In this case the level set ε(V ) specifies the finite-time attraction domain M (see Definition 1).

V. IMPLICIT RELAY FEEDBACK LAW A. Block Decomposition

Let us initially decompose the original multi-input system (1) to a block form [START_REF] Drakunov | Block control principle I[END_REF]. The block decomposition procedure studied in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF] is briefly discussed in Appendix. It constructs the non-singular coordinate transformation s = Θx [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF] reducing the original system (1) to the block form

ṡ(t) = Ãs + B(u + K lin s), (9)
where K lin ∈ R m×n is a rectangular matrix,

Ã=     0 A 12 ... 0 ... ... ... ... 0 ... ... A k-1 k 0 ... ... 0     , B=     0 ... 0 A k k+1     ∈ R n×n k , ( 10
)
A i i+1 ∈ R ni×ni+1 are matrices of full row rank, i = 1, 2, .., k, n 1 + ... + n k = n, n k = m and B 0 ∈ R m×m is a nonsingular matrix.

B. Relay Feedback Design

Introduce the ILF function

Q(V, s) := s T D r (V -1 )P D r (V -1 )s -1, (11)
where

s = (s 1 , ..., s k ) T , s i ∈ R ni , V ∈ R + , D r (λ) is the dilation matrix of the form D r (λ) =     λ r1 I n1 0 ... 0 0 λ r2 I n2 ... 0 ... ... ... ... 0 ... 0 λ r k I n k     , (12)
with

λ ∈ R + , r i = 1 + (k -i)µ, i = 1, 2, .., k, 0 < µ ≤ 1
and P ∈ R n×n is a symmetric positive definite matrix, i.e. P = P T > 0. Denote H µ := diag{r i I ni } k i=1 -the block diagonal matrix.

Theorem 4: Let µ ∈ (0, 1), X ∈ R n×n and Y ∈ R n k ×n satisfy the system of matrix inequalities:

ÃX + X ÃT + BY + Y T BT + H µ X + XH µ = 0, XH µ + H µ X > 0, X > 0. (13)
Consider V : R n → R + implicitly defined by the equation Q(V, s) = 0 and the function Q given by [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF] with P := X -1 . Then the control of the form (4) with

Γ T (x) = x T Θ T D r 1 V (Θx) P D r 1 V (Θx) B, (14)
locally stabilizes the origin of the system (1) in a finite time.

The settling-time function is bounded as follows

T (x 0 ) ≤ V µ 0 µ , ∀x 0 ∈ ε(V ) ( 15
)
where V 0 ∈ R + : Q(V 0 , Θx 0 ) = 0, the set ε(V ) is the finite-time attraction given by ( 7) with a positive V ∈ R + :

V = sup V ∈R+:ε(V )⊂C U V, C U = z ∈ R n : V 1-µ (Θz)YP -1 D r V -1 (Θz) Θz ∈ co(U) .
The system of matrix inequalities ( 13) can be easily solved using LMI toolbox of MATLAB or, for example, SeDuMi solver. The solution of (13) also can be constructed analytically using the proof (see, [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF]) of the next proposition.

Proposition 5 ([22]): The system of matrix inequalities ( 13) is feasible for any µ ∈ R + .

VI. PRACTICAL IMPLEMENTATION

In order to realize the control algorithm ( 4), ( 14) in practice we need to know V . In some cases the function V can be calculated analytically. See, for example [START_REF] Polyakov | A new homogeneous quasi-continuous second order sliding mode control[END_REF], where the analytical derivation has been provided for the case n = 2, m = 1. The function V can also be approximated numerically on a grid, which is constructed in the finitetime attraction domain ε(V ). Finally, the relay control law (4) can be applied by means of on-line estimation of V . The following corollary may be utilized for this purpose.

Corollary 6: If 1) the conditions of Theorem 4 hold; 2) {t i } +∞ i=0 is an arbitrary sequence of time instances such that 0 = t 0 < t 1 < t 2 < ... and lim i→+∞ t i = +∞; 3) the relay control u r has the form (4) with the sampled computation of the switching function Γ(t, x) = Γi (Θx) for t ∈ [t i , t i+1 ), where

ΓT i (s) = s T D r (V -1 i )P D r (V -1 i ) B, Q(V i , s(t i )) = 0.
Then the closed-loop system (1), ( 4) is globally asymptotically stable.

The corollary shows that if a the switching function is re-computed only at a countable number of time instants t i , then the asymptotic stability of the closed loop system is guaranteed. In practice, the estimation of the switching parameter V i can be obtained using the following algorithm [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF].

Algorithm 7:

INITIALIZATION: V 0 = 1; a = V min ; b = 1; STEP : If s T i D r (b -1 )P D r (b -1 )s i > 1 then a = b;b = 2b; elseif s T i D r (a -1 )P D r (a -1 )s i < 1 then b = a; a = max{ a 2 , V min }; else c = a+b 2 ; If s T i D r (c -1 )P D r (c -1 )s i < 1 then b = c; else a = max{V min , c}; endif; endif; V i = b; If s i ∈ R n is
some given vector and STEP of the presented algorithm is applied recurrently many times to the same s i then Algorithm 7 realizes:

1) a localization of the unique positive root of the equation

Q(V, s i ) = 0, i.e. V i ∈ [a, b];
2) improvement of the obtained localization by means of the bisection method, i.e. (b -a) → 0.

Such an application of Algorithm 7 allows us to calculate V i with rather high precision but it requests a high computational capability of a control device. If the computational power is very restricted, then STEP of Algorithm 7 may be realized just once at each sampled instant of time. Indeed, in the proof of Corollary 6 we show that the ellipsoid ε(V i ) is an invariant set of the closed-loop system (1), (4) with Γ(x) = Γi (Θx). If the root of the equation Q(V, s i ) = 0 is localized, Algorithm 7 always selects the upper estimate of V i providing that s(t i ) ∈ ε(V i ), i.e. V i do not increase in time.

The parameter V min defines lower admissible value of V . In practice, this parameter cannot be selected arbitrary small due to finite numerical precision of digital devices and measurement errors, which may imply s(t i ) / ∈ ε(V i ). Therefore, the real-life realization of the relay control may provide the only practical stabilization of the system with the attractive set ε(V min ).

VII. ACADEMIC EXAMPLE

Let us consider the system (1) with

A =   0 1 -0.5 1 -0.3 0.9 0.5 0 0.7   , B =   0 0 0.5 0 0 1   and U = {v 1 , v 2 , v 3 }, v 1 = 0 √ 3 3 , v 2 = 0.5 - √ 3 6
, v 3 = -0.5 -

√ 3 6
.

The considered system already has the required block form The relay control (4) with Γ of the form ( 14), P = X -1 is applied using the algorithm 7 with V min = 10 -3 . The explicit Euler discretization with the step size 10 -3 is utilized for the numerical simulation. The trajectories are depicted on Fig. 1.

The upper estimate of the settling time [START_REF] Hetel | Variable structure control with generalized relays: A simple convex optimization approach[END_REF] gives 1.4734. The simulation confirms this.

The numerical simulation (see Fig. 2) shows fast control switchings approving the expectable fact that the finitetime stabilization by means of the relay control implies appearance of sliding mode in the closed-loop system.

VIII. CONCLUSION

The paper presents relay feedback control algorithm for stabilization of linear multi-input system provided nonasymptotic transitions. The control design procedure combines the ILF method and convex embedding technique. This approach allows us to provide simple procedure for implicit switching surface design using LMIs. The algorithm of practical implementation of the obtained implicit relay feedback is also presented and justified. The robustness analysis of the proposed control scheme is considered as the subject for future research.

IX. APPENDIX

A. Transformation to Block Form

Let us denote by rown(W ) the number of rows of a matrix W and by null(W ) the matrix that has the columns defining an orthonormal basis of the null space of a matrix W .

Let the orthogonal matrices T i be defined by the following simple algorithm:

Initialization : A 0 = A, B 0 = B, T 0 = I n , k = 0. Loop: While rank(B k ) < rown(A k ) do A k+1 = B ⊥ k A k B ⊥ k T , B k+1 = B ⊥ k A k Bk , T k+1 = B ⊥ k Bk , k = k + 1, where B ⊥ k = null(B T k ) T , Bk = null B ⊥ k T .
In the paper [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] it was proven that the orthogonal matrix

G= T k 0 0 I w k T k-1 0 0 I w k-1 ... T 2 0 0 I w2 T 1 ,
where w i := n -rown(T i )

provides

GAG T =       A 11 A 12 0 ... 0 A 21 A 22 A 23 ... 0 ... ... ... ... ... A k-1 1 A k-1 2 ... A k-1 k-1 A k-1 k A k1 A k2 ... A kk-1 A kk       , GB = 0 0 ... 0 A T k k+1 T ,
where A k k+1 = B0 B 0 , A ij ∈ R ni×nj , n i := rank(B k-i ), i, j = 1, 2, ..., k and rank(A i i+1 ) = n i .

Recall that the B has full column rank (rank(B) = m). Consequently, A k k+1 is square and nonsingular. Since rank(A i i+1 ) = n i = rown(A i i+1 ) then A i i+1 A T i i+1 is invertible and A + i i+1 = A T i i+1 (A i i+1 A T i i+1 ) -1 is the right inverse matrix of A i i+1 . Introduce the linear coordinate transformation s = Φy, s = (s 1 , ..., s k ) T , s i ∈ R ni , y = (y 1 , ..., y k ) T , y i ∈ R ni by the formulas:

s i = y i + ϕ i , i = 1, 2, ..., k, ϕ 1 = 0, ϕ i+1 = A + i i+1 i j=1 A ij y j + i r=1 ∂ϕi ∂yr r+1 j=1 A rj y j . (17)
The presented coordinate transformation is linear and nonsingular. The inverse transformation y = Φ -1 s is defined as follows: (please see [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] for more details).

y i = s i + ψ i , i =

7 .

 7 The solution to the LMI (13) was obtained for µ = 0.999:

Fig. 1 .Fig. 2 .

 12 Fig. 1. Evolution of the system states of the closed-loop system

1 , 2 ,A

 12 ..., k, ψ 1 = 0, ij (s j + ψ j )   .For example, if k = 3 then the matrix Φ has the formΦ= 21 +A + 12 A 2 11 ) A + 23 (A 22 +A + 12 A 11 A 12 ) I n3   .In general case, the transformation Φ can be calculated numerically. Applying the transformation s = Θx with Θ = ΦG to the system (1) we obtain the system ... ... ... 0 ... ... A k-1 k Ãk1 ... ... Ãkk     s + Bu, which is equivalent to (9) with K lin = B + 0 Ãk1 ... ... Ãkk


Konstantin Zimenko
email: kostyazimenko@gmail.com

Andrey Polyakov

Denis Efimov

Stabilization of Chain of Integrators with Arbitrary Order in Finite-time*

. The theoretical results are supported by numerical simulations.

I. INTRODUCTION

Frequently, strict quality requirements are imposed for design, operation and control of complex technical processes. One such requirement is completion of all transients in a finite time. In particular, this issue is relevant in control design for a variety of robotic and mechatronic devices, safety evaluation, aerospace applications, vehicles control systems, etc. Therefore the problems of finite-time control and finite-time observation are intensively studied for many years (see, for example, [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]- [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF] and [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF]- [START_REF] Shen | Global finite-time observers for lipschitz nonlinear systems[END_REF] correspondingly).

Solutions of the problem of control design for chain of integrators are quite sought-after. This is motivated by the fact that control scheme for chain of integrators can be easily extended to larger classes of systems (see, for example, [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF]). Moreover, in many applications the nominal models have the form of chain of integrators, for instance, mechanical planar systems. There is a number of research results on the finite-time stabilization or regulation of a chain of integrators (see, for example, [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Bernuau | Finite-time output stabilization of the double integrator[END_REF]). However, most of them do not find an application in practice because of constructive and computational complexity. For instance, the methods mainly do not suggest constructive algorithms of parameter tuning and ensure only existence of suitable control parameters.

One of effective methods for the finite-time stability analysis is the method of Implicit Lyapunov Function (ILF) [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF], [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], [START_REF] Korobov | A general approach to synthesis problem[END_REF]. This method allows to check all stability conditions directly using implicitly defined Lyapunov function which is a solution of some algebraic equation. Note that for Villeneuve d'Ascq (e-mail: denis.efimov@inria.fr, andrey.polyakov@ inria.fr). They are also with CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cite Scientifique, 59651 Villeneuve-d'Ascq, France. such a stability analysis there is no need to solve this equation. Another issue that can be useful in analysis of finitetime stability is property of homogeneity. For instance, if an asymptotically stable system is homogeneous of negative degree, then it is finite-time stable.

The paper [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] is related to development of method for finite-time control of multiple integrators of arbitrary order together with implicit Lyapunov function of closed-loop system. Conditions of finite-time stability are formulated in the form of LMI that essentially simplifies the process of tuning of control parameters. However, the practical realization of the ILF control algorithm requires development of special computational procedure for on-line computation of the ILF value at the current state (the control law depends on ILF value explicitly).

The present paper addresses the problem of a control design for the finite-time stabilization of a chain of integrators with arbitrary order. The paper represents an extension and simplification of results in [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF]. The developed finitetime control law does not depend on implicitly defined Lyapunov function that allows to avoid any additional online procedures. The result is obtained by using the method of Implicit Lyapunov Function with applying properties of homogeneous systems. As in [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] parameter tuning is presented in LMI form.

The paper is organized as follows. Notation used in the paper is given in the next section. Problem statement is considered in Section III. Some background notions about finitetime stability, ILF method and homogeneity are introduced in Section IV. Control design algorithm and some remarks about proposed control scheme are presented in Section V. Section VI contains simulation results for effectiveness demonstration of proposed control. Finally, concluding remarks and possible directions for future research are presented in Section VII.

II. NOTATION

Through the paper the following notation will be used:

• R + = {x ∈ R : x > 0}, where R is the set of real number; • • is the Euclidian norm in R n , i.e. x = x 2 1 + • • • + x 2 n for x ∈ R n ;
• the positive (negative) definiteness (semi-definiteness) of a symmetric matrix P = P T ∈ R n×n is denoted by P > 0 (P < 0, P ≥ 0, P ≤ 0); • for a symmetric matrix P = P T the minimal and maximal eigenvalues are denoted by λ min (P ) and λ max (P ), respectively;

• a continuous function σ : R + ∪{0} → R + ∪{0} belongs to the class K if σ(0) = 0 and the function is strictly increasing; • diag {λ i } n i=1 is the diagonal matrix with the elements λ i on the main diagonal.

III. PROBLEM STATEMENT Consider the control system

ẋ = Ax + bu, (1)
where x ∈ R n is the state vector, u ∈ R is the control input,

A =        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 0 0 • • • 0        and b =        0 0 . . . 0 1       
that describes a chain of integrators. The aim of the paper is to develop the finite-time stabilizing control algorithm to the system (1), which satisfies two restrictions:

• it has the form of the explicit full-state feedback and its application does not require additional on-line computations; • it admits the simple computational scheme (e.g. LMIbased) scheme for parameters tuning.

IV. PRELIMINARIES A. Finite-time Stability

Consider the system of the form

ẋ f (t, x), x(0) = x 0 , (2)
where x ∈ R n is the state vector, f ∈ R + × R n → R n is a nonlinear continuous vector field, which can be discontinuous with respect to the state variable. In this case the solutions x(t, x 0 ) of the system (2) are understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF].

According to [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] an absolutely continuous function x(t, x 0 ) is called a solution of the Cauchy problem associated to (2) if x(0, x 0 ) = x 0 and for almost all t > 0 it satisfies the following differential inclusion

ẋ ∈ K[f ](t, x) = co ε>0 µ(N )=0 f (t, B(x, ε)\N ),
where co(M ) defines the convex closure of the set M , B(x, ε) is the ball with the center at x ∈ R n and the radius ε, the equality µ(N ) = 0 means that the measure of the set x ∈ R n is zero.

Assume that the origin is an equilibrium of the system (2) and it has uniqueness of solutions in forward time.

Definition 1 ([1], [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]): The origin of system ( 2) is said to be globally finite-time stable if:

• Finite-time attractivity: there exists so-called settling time function T : R n \{0} → R + such that lim t→T (x0) x(t, x 0 ) = 0 for all x 0 ∈ R n \{0}.

• Lyapunov stability: there exists a function δ ∈ K such that x(t, x 0 ) ≤ δ( x 0 ) for all x 0 ∈ R n . Notice that [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]:

• finite-time stability is obtained through an "infinite eigenvalue assignation" for the system at the origin; • the settling time function T (x 0 ) evaluates the time for a solution to reach the equilibrium; • the right hand side of the ordinary differential equation cannot be locally Lipschitz at the origin.

B. Homogeneity

Homogeneity [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF]- [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] is an intrinsic property whereby objects such as functions or vector fields remain consistent with respect to some scaling operation called a dilation.

For fixed r i ∈ R + , i ∈ {1, ..., n} called weights and λ > 0 one can define:

• the vector of weights r = (r 1 , ..., r n ) T ;

• the dilation matrix

D(λ) = diag{λ ri } n i=1 and correspondingly D(λ)x = (λ r1 x 1 , ..., λ rn x n ) T for x = (x 1 , ..., x n ) T ∈ R n . Definition 2 [17]: A function g : R n → R (vector field f : R n → R n ) is said to be r-homogeneous of degree m iff g(D(λ)x) = λ m g(x) (f (D(λ)x) = λ m D(λ)f (x)) for all λ > 0 and x ∈ R n .
Let us introduce the homogeneous unit sphere

S r = {x ∈ R n : x r = 1}, (3)
where x r represents the homogeneous norm of x ∈ R n defined by

x r = n i=1 |x i | ρ r i 1 ρ (4)
for ρ ≥ max i r i . Obviously, each homogeneous norm is r-homogeneous of degree 1 and positive definite. For any x ∈ R n there are y ∈ S r and λ = x r such that x = D(λ)y [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF].

Theorem 1 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] Let f be a r-homogeneous continuous vector field on R n such that the system (2) is a locally asymptotically stable. Then it is globally asymptotically stable and there exists a continuously differentiable r-homogeneous Lyapunov function V for [START_REF] Roxin | On finite stability in control systems[END_REF].

Note that in accordance with [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] there exist constants c 1 , c 2 > 0 such that

c 1 x m r ≤ V (x) ≤ c 2 x m r , (5)
where m is degree of r-homogeneous Lyapunov function.

Theorem 2 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] Let f : R n → R n be defined on R n and be a continuous r-homogeneous vector field with a negative degree. If the origin of the system (2) is locally attractive then it is globally finite time stable.

C. Implicit Lyapunov Function Method

The theorem given below presents the ILF method for finite-time stability analysis.

Theorem 3 [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] If there exists a continuous function

Q : R + × R n → R (V, x) → Q(V, x) such that C1) Q(V, x) is continuously differentiable for ∀x ∈ R n \{0} and ∀V ∈ R + ; C2) for any x ∈ R n \{0} there exist V -∈ R + and V + ∈ R + : Q(V -, x) < 0 < Q(V + , x); (6)
C3) for Ω = {(V, x) ∈ R n+1 : Q(V, x) = 0} lim x→0 (V,x)∈Ω V = 0 + , lim V →0 + (V,x)∈Ω x =0, lim x →∞ (V,x)∈Ω V= +∞; C4) the inequality -∞ < ∂Q(V, x) ∂V < 0 holds for ∀V ∈ R + and ∀x ∈ R n \{0}; C5) the inequality ∂Q(V, x) ∂x f (x) ≤ ηV 1-µ ∂Q(V, x) ∂V holds ∀(V, x)
∈ Ω, where 0 < µ ≤ 1 and η > 0 are some constants.

Then the origin of the system (2) is globally finite-time stable with the following settling time estimate

T (x 0 ) ≤ V µ 0 ηµ .

V. MAIN RESULT

Introduce the ILF function

Q(V, x) := x T D(V -1 )P D(V -1 )x -1, (7)
where P = P T ∈ R n×n is a symmetric positive definite matrix, D(λ) is the dilation matrix of the form

D(λ) = diag{λ 1+(n-i)µ } for 0 < µ ≤ 1. Let us denote the diagonal matrices H µ = diag {-1 -(n -i)µ} n i=1
and

H(λ) = diag λ (n+1-i)µ n i=1 . Theorem 4
If the following statements hold: 1) the system of matrix inequalities

       AX + XA T + by + y T b T + αX + βI n ≤ 0, XH µ + H µ X < 0, X > 0, βI n ≥ γX, γ y y T X ≥ 0, X I n -H(c) I n -H(c) βI n ≥ 0 (8) is feasible for some µ ∈ (0, 1], α, β, γ, c ∈ R + : α > β, X ∈ R n×n , y ∈ R 1×n ;
2) there exists a c u that satisfies one of the two following inequalities:

c 1 c ≥ c u ≥ c 2 (9) or c 1 ≥ c u ≥ c 2 c , (10)
where the coefficients c 1 and c 2 correspond to the inequality [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF].

Then the control of the form

u(x) = (c u x r ) 1-µ kD((c u x r ) -1 )x, (11)
where k = yX -1 stabilizes the origin of the system (1) in a finite time.

Let us make some remarks about presented control scheme:

• For practical implementation of the control [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] feasibility of inequality ( 9) or ( 10) is required. Therefore, it is necessary to estimate the coefficients c 1 and c 2 .

In this case in accordance with [START_REF] Shen | Global finite-time observers for lipschitz nonlinear systems[END_REF] one can conclude that the Lyapunov function

V (x) is r-homogeneous of degree 1 as Q(V, D(λ)x) = Q(λ -1 V, x), i.e. V (D(λ)x) = λV (x).
Then rewrite ( 5)

c 1 x r ≤ V (x) ≤ c 2 x r . (12)
The inequality ( 12) can be rewritten in the following form

c 1 D(V -1 )x r ≤ 1 ≤ c 2 D(V -1 )x r Denote z = (z 1 , . . . , z n ) T := D(V -1 )x. Hence c ρ 1 n i=1 |z i | ρ/ri ≤ 1 ≤ c ρ 2 n i=1 |z i | ρ/ri .
From the other side, x T D(V -1 )P D(V -1 )x = 1 implies

1 λ max (P ) ≤ n i=1 z 2 i ≤ 1 λ min (P )
.

If P ≥ I n (or X ≤ I n that coincides with the fourth inequality of (8) for β = γ), then z i ≤ 1 and hence |z i | ≤ 1. For ρ = 1 + (n -1)µ using inequality of arithmetic and quadratic means one can obtain

n i=1 |z i | ρ/ri ≤ n i=1 |z i | ≤ √ n z and c 1 ≤ 1 2ρ √ n (13)
On the other hand, if we choose µ ∈ 0, 1 n-1 then

ρ ri ≤ 2 and |z i | ≤ 1 implies |z i | 2 ≤ |z i | ρ r i . Therefore, n i=1 |z i | 2 ≤ n i=1 |z i | ρ/ri and c 2 ≥ ρ λ max (P ). (14)
Thus, one can rewrite the system (8)

       AX + XA T + by + y T b T + αX + βI n ≤ 0, XH µ + H µ X < 0, I n ≥ X > 0, β y y T X ≥ 0, X I n -H(c) I n -H(c) βI n ≥ 0 (15)
and inequalities ( 9), (10) take the forms

2ρ √ n c ≥ c u ≥ ρ λ max (P ) (16)
and

2ρ √ n ≥ c u ≥ ρ λ max (P ) c , (17)
correspondingly for µ ∈ 0, 1 n-1 , ρ = 1 + (n -1)µ. • The advantage of proposed control scheme is based on simple tuning of control parameters presented in LMI.

For any fixed α, µ ∈ R + the first three inequalities of the system ( 8) are feasible at least for sufficiently small µ ∈ (0, 1] [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF]. Moreover, the last three inequalities of the system (8) are defined by the inequality

β 2 P ≥ I n -H V cu x r k T k I n -H V cu x r
, which also becomes feasible for sufficiently small µ, since I n -H V cu x r → 0 for µ → 0. Thus, one can always find such µ for which the system (1) is finite-time stable with control in the form [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF].

• If c u satisfies the inequality (9) then the control ( 11) is bounded with |u(x)| ≤ (c 2 x r ) 1-µ √ γ. Indeed,

1 γ u 2 (x) = 1 γ (c u x r ) 2-2µ
x T D((c u x r ) -1 )k T kD((c u x r ) -1 )x ≤ -β) .

1 γ V 2-2µ x T D(V -1 )k T kD(V -1 )x ≤ V 2-2µ = V 2-2µ x T D(V -1 )P D(V -1 )x ⇒ 1 γ k T k
• The system (1) is r-homogeneous of degree -µ with the vector of weights r = (1+(n-1)µ, 1+(n-2)µ..., 1) T . Indeed, since homogeneous norm is r-homogeneous of degree 1, i.e. D(λ)x r = λ x r :

u(D(λ)x) = (c u D(λ)x r ) 1-µ kD((c u D(λ)x r ) -1 )D(λ)x = λ 1-µ x r kD(λ -1 (c u x r ) -1 )D(λ)x = λ 1-µ u(x)
and AD(λ)x + Bu(D(λ)x) = λ -µ D(λ)(Ax + Bu(x)). • In comparison with the results of the work [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] the control [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] does not require any additional computational power for on-line computation of the ILF value at the current state. This issue significantly simplifies the practical realization of the developed finite-time control scheme.

VI. NUMERICAL EXAMPLE

Consider the system (1) for n = 2. Define the finite-time control u in the form [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] with the parameter µ = 0.5, where the matrix P ∈ R 2×2 and the vector k ∈ R 1×2 are obtained from the system of matrix inequalities [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF] The corresponding simulation results for control algorithm presented in [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] with the same values of P , k, µ, α and β are shown in the Fig. 3, Fig. 4. Fig. 3. Transients of the system states for the finite-time control presented in [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] Fig. 4. The control signal u with its bounds for the finite-time control presented in [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] VII. CONCLUSIONS The paper presents control algorithm for finite-time stabilization of chain of integrators with arbitrary order. The control design procedure is based on application of the ILF method and some properties of homogeneous systems. Tuning of control parameters is presented in the form of linear matrix inequalities. Numerical example demonstrates the effectiveness of proposed control.

This work is an extension of the paper [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] and in comparison with it obtained control u does not require online procedure for computation of the ILF value at the current state. This issue allows to simplify the practical realization of the control scheme.

There are many possible directions for further research. For instance, studying robustness with respect to delays or extension of the present result for MIMO and nonlinear systems.

  ≤ P coincides with the matrix inequality γ y y T X ≥ 0 in LMI form. As V ≤ c 2 x r then u 2 (x) ≤ γ(c 2 x r ) 2-2µ . • In order to estimate the settling time function the following matrix inequality XH µ + H µ X ≥ -νX can be added to (8) for some ν > 0. Then the settling time function estimate has the form T (x 0 )

  with parameters c = 0.02, α = 0.1, ρ = 1 + (n -1)µ = 1.5 and γ = β = 3.8605 * 10 -4 : = -0.2245 -0.2515 .As γ = β in accordance with (13) and (14) we can select c 1 = 0.7937, c 2 = 38.9465 and coefficient c u due to (16) should satisfy the following expression 39.6850 ≥ c u ≥ 38.9465. The simulation results for c u = 39 and x 1 (0) = 3, x 2 (0) = 1 are presented in the Fig. 1, Fig. 2. The system states are shown in the Fig. 1. Fig. 2 presents the control signal u with its bounds.

Fig. 1 .

 1 Fig. 1. Transients of the system states for the finite-time control

Andrey Polyakov, Denis Efimov are with Non-A INRIA -LNE, Parc Scientifique de la Haute Borne 40, avenue Halley Bat.A, Park Plaza 59650

*This work was supported by the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031) and by Government of Russian Federation, Grant 074-U01.


Bernard Brogliato
email: bernard.brogliato@inria.fr.

Andrey Polyakov
email: andrey.polyakov@inria.fr.

Globally Stable Implicit Euler Time-Discretization of a Nonlinear Single-Input Sliding-Mode Control System

Keywords: Implicit Euler discretization, sliding mode, fixed-time stability, hyper-exponential convergence

In this note we study the effect of an implicit Euler time-discretization method on the stability of the discretization of a globally fixed-time stable, scalar differential inclusion representing a simple nonlinear system with a set-valued signum controller. The controller nonlinearity is a cubic term and it is shown that the fullyimplicit method preserves the global Lyapunov stability property of the continuous-time system, contrarily the explicit discretization which does not. It allows to obtain finite-time convergence to the origin when the plant is undisturbed, while the cubic term provides the hyperexponential convergence rate.

I. INTRODUCTION

Sliding mode (SM) method is the oldest approach to robust control design introduced almost 50 years ago (see, for example, [START_REF] Utkin | Variable structure systems with sliding modes[END_REF] and references therein). It utilizes "relay" (discontinuous) feedback in order to force a closed-loop system to slide on a given surface in a state space.

The main theoretical advantage of sliding mode algorithms is their insensitivity to the so-called matched disturbances and uncertainties, see [START_REF] Edwards | Sliding Mode Control: Theory And Applications[END_REF], [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF]. In practice, the discontinuous feedback application yields the so-called chattering phenomenon [START_REF] Boiko | Analysis of chattering in systems with second-order sliding modes[END_REF], [START_REF] Levant | Chattering analysis[END_REF] expressed in high-frequency destructive oscillations of the closedloop system. Several approaches have been proposed in order to overcome this drawback. The classical way of chattering reduction is the "linearization" of the relay feedback law close to a sliding (switching) surface. The high-order SM control principles were introduced in [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] as a desirable "chattering-free" alternative of the convectional technique. However, an improper implementation of SM control laws in a digital devices anyway implies tangible destructive chattering. Recently, the method of implicit discretization has been developed [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF] for effective digital realization of sliding mode algorithms. It has been tested on an experimental setup [START_REF] Wang | Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Toward input and output chattering suppression[END_REF], [START_REF] Huber | Discretetime twisting controller without numerical chattering: Analysis and experimental results with an implicit method[END_REF], providing drastic reduction of both input and output chattering for first and second order sliding mode algorithms.

One more feature of sliding mode algorithm is finite-time reaching of the sliding surface by trajectories of the closed-loop system. Finite-time control is the subject of intensive research (see, for example, [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Orlov | Discontinous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions[END_REF]) motivated by control problems, which need terminations of all transition processes in a finite time. Finite-time analysis of different sliding mode systems has been done in [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF], [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF]. Finitetime stable system with a bounded settling time on a whole attraction domain has been discovered in [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF]. In [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] this property was called fixed-time stability and utilized for sliding mode control design with fixed-time reaching of the sliding surface and the origin of the closed-loop system, providing a possible theoretical background of fast control system. Application of this idea to the first order sliding mode control design is presented in [START_REF] Cruz-Zavala | Uniform sliding mode controllers and uniform sliding surfaces[END_REF].

The present paper aims at extending the implicit discretization method introduced in [1], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF] to the nonlinear case. It shows that contrary to the explicit discretization (which is shown in [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF] to lack of global stability), the implicit method preserves the continuous-time closed-loop properties like global finite-time Lyapunov stability. Section II presents the problem statement and the basic assumptions. Then the discretization scheme is studied in the ideal undisturbed case in section III. The case when a perturbation acts on the system is treated in section IV.

Mathematical Preliminaries

: Let K ∈ IR n be a non empty convex closed set. The normal cone to K at x ∈ K is defined as N K (x) = {z ∈ IR n |z T (s -x) 0 for all s ∈ K}, while N K (x) = ∅ if x ∈ K. In particular let K = [a, b], a < b. Then N [a,b] (x) = {0} if a < x < b, IR + if x = b, IR -if x = a. The sign set-valued function is sgn(x) = 1 if x > 0, -1 if x < 0,

II. PROBLEM STATEMENT

A. Plant and control design

Let us consider a nonlinear single-input control system ż(t) = f (z(t), q(t)) + g(z(t), p(t))w + r(t), t > 0,

(1) where z(t) ∈ IR n is the state vector, f (•) and g(•) are vector fields, w(•) ∈ IR is the control input and q, p, r : IR n+2 → IR n are single-valued vector functions representing uncertainties and noise. Let us follow the sliding mode control methodology that recommends, initially, to select an appropriate sliding surface x(z) = 0 in the state space, where x : IR n → IR is a smooth function. Let us assume that after a possible suitable pre-feedback action w(z, v), the system (1) admits a surface with the sliding variable equation of the form

ẋ(t) = v(t) + d(t, z) (2)
where x(t) = x(z(t)) is the sliding variable, v(•) the new input, and the uncertain function d(t, z) is an equivalent disturbance, which is assumed to satisfy |d(z, t)| δ < 1 for all t and z1 . Next, let us design the fixed-time sliding mode control

v(t) = -x 3 (t) -sgn(x(t)). (3)
The closed-loop system (2), (3) is globally fixedtime stable, with the Lyapunov function

V (x) = x 2 . Indeed, d dt (V • x)(t) = ∂V ∂x ẋ(t) = -x(t)sgn(x(t)) - x(t) 4 + x(t)d(t) -(1 -δ)|x(t)| -x(t) 4 -(1 - δ)V 1/2 (x(t)) -V 2 (x(t))
, that is negative definite and V (x(t)) = 0 for all t > 2/(1 -δ) + 1 (see [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] for details).

In [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF] it has been shown that digital implementation of the control law (3) is difficult due to instability of the explicit Euler discretization of the closed-loop system (2) (3). This paper presents and alternative way for the realization of the fixed-time algorithm, which is based on the analysis of the implicit Euler discretization. The obtained (so-called implicit) control algorithm can be easily realized in a digital device.

B. Discretization of fixed-time sliding mode algorithm

In view of the above, let us consider the following sliding mode dynamics (without disturbance):

   ẋ(t) = v(t) v(t) ∈ -x(t) 3 -sgn(x(t)) x(0) = x 0 (4)
where x(t) ∈ IR. This system is well-posed in the sense of Fillipov, and enjoys the uniqueness of solutions property for any x 0 . In the following we shall work with the Euler discretization of (4), namely:

x k+1 = x k + hv k v k ∈ -x 3 i -sgn(x j ) (5)
where h > 0 is the step size,

t k = hk, x k ∆ = x(t k ).
From a general point of view, several discretization schemes may be designed:

• i = k + 1 and j = k + 1 (fully-implicit discretization) • i = k and j = k + 1, or i = k + 1 and j = k (half-implicit discretizations) • i = k and j = k (fully-explicit discretization)
In this note, we focus on the fully-implicit method and show that the results in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF] can be extended to the considered nonlinear case. Let us note that the fully-explicit discretization has already been proven to be unstable in [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF]. Equilibria of both continuous-time and discrete-time systems are given as the solutions of the generalized equation:

0 = v * v * ∈ -x 3 * -sgn(x * ) (6)
One infers that both (4) and ( 5) have the origin x * = 0 as a unique fixed point, with input v * = 0.

III. ANALYSIS OF THE FULLY-IMPLICIT

DISCRETIZATION

We first focus on the case without disturbance, and we prove that the implicit method preserves the continuous-time stability properties, while an explicit discretization does not as proved in [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF]. Let us denote the set-valued part of the control input v k as u k+1 , so that u k+1 ∈ -sgn(x k+1 ). Thus the discrete-time system reads as:

x k+1 = x k -hx 3 k+1 + hu k+1 u k+1 ∈ -sgn(x k+1 ) (7)
From ( 7) one obtains the following two generalized equations with unknown x k+1 and u k+1 , respectively:

(1 + hx 2 k+1 )x k+1 -x k + hζ k+1 = 0 ζ k+1 ∈ sgn(x k+1 ) (8)
and

ξ k+1 + hξ 3 k+1 -x k -hu k+1 = 0 ξ k+1 ∈ -N [-1,1] (u k+1 ) (9)
where N [-1,1] (u k+1 ) is the normal cone to the interval [-1, 1] computed at u k+1 . One has (see the Mathematical Preliminaries in section I):

x k+1 ∈ -N [-1,1] (u k+1 ) ⇔ u k+1 ∈ -sgn(x k+1
). Thus, in a sense, the two generalized equations in ( 8) and ( 9) are dual one to each other. We adopt the notation ξ k+1 instead of x k+1 in (9) to emphasize the fact that for this generalized equation, x k+1 is a dummy variable. The same holds for

ζ k+1 in (8) instead of u k+1 . If (x * k+1 , u * k+1
) is a solution ( 8) and ( 9), then the control v k in ( 5) with the fully-implicit method is defined as follows

v k = -(x * k+1 ) 3 -u * k+1 . (10)
Three comments arise at this stage:

• As the next two propositions show, such a control input at time t k is nonanticipative (i.e. it is causal), because both x * k+1 and u * k+1 are functions of x k and h, but not of the future state x k+1 .

• The ability to solve generalized equations as in ( 8) and ( 9) is at the core of the proposed implicit discretization method. For analytical purpose the solutions should be characterized. For implementation purpose, it may be sufficient that they are numerically solved with a specific solver. • A complete study should start with the exact ZOH discretization of the plant (1), then the application of a discrete-time pre-feedback to obtain the sliding-mode dynamics with its equivalent disturbance in [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], and finally the analysis of the closed-loop system when a Euler discretization of the set-valued nonlinear robust controller in (3) is applied. However the Euler discretization of the controller is a control design choice. This is quite in agreement with the results in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Wang | Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Toward input and output chattering suppression[END_REF], [START_REF] Huber | Discretetime twisting controller without numerical chattering: Analysis and experimental results with an implicit method[END_REF] which deal with similar generalized equations to calculate the controller, with no nonlinear term however.

Proposition 1: The generalized equation ( 8) has a unique solution x * k+1 for any h > 0 and any

x k . If x k > h then x * k+1 > 0, if x k < -h then x * k+1 < 0, and if |x k | h then x * k+1 = 0. Proof: • 1) let x k > h. Suppose first that x k+1 > 0. We obtain the nonlinear equation (1+hx 2 k+1 )x k+1 = x k -h > 0.
A simple graphical reasoning proves that this 3rd order equation has a unique solution x * k+1 > 0. Suppose now that x k+1 < 0, then we get (1 + hx 2 k+1 )x k+1 = x k + h > 0, however this nonlinear equation has no negative solution, thus one infers that x k+1 is necessarily non negative, a contradiction. Let now x k+1 = 0, thus we obtain 0 ∈ -x k + hζ k+1 and ζ k+1 ∈ [-1, 1]. Since x k > h this is not possible. We infer that the first solution satisfying x * k+1 > 0 is the unique solution.

• 2) The case x k < -h may be analysed similarly and we conclude that the solution is the unique x * k+1 < 0 solution of the nonlinear equation

(1 + hx 2 k+1 )x k+1 = x k + h < 0. • 3) Let |x k | < h. Suppose that x k+1 > 0, this yields (1 + hx 2 k+1 )x k+1 = x k -h < 0 which is impossible. Similalry if x k+1 < 0 we obtain (1 + hx 2 k+1 )x k+1 = x k + h > 0 which is impossible. Let now x * k+1 = 0, then ζ * k+1 = -x k
h solves the generalized equation ( 8) and is thus the unique solution.

• 4) Let now x k = h. Thus the generalized equation

is 0 = (1 + hx 2 k+1 )x k+1 -h + hζ k+1 , ζ k+1 ∈ sgn(x k+1 ). Let us choose x k+1 = 0, so that ζ k+1 ∈ [-1, 1]
. We obtain 0 = -h+hζ k+1 so that ζ k+1 = 1 is a solution. Take now x k+1 > 0, then we get 0 = (1 + hx 2 k+1 )x k+1 , a contradiction. If we choose x k+1 < 0 we get (1 + hx 2 k+1 )x k+1 = 2h, a contradiction. We infer that x * k+1 = 0 and ζ * k+1 = 1 is the unique solution. Proposition 1 yields in particular that |x k | h implies x k+1 = 0: if the state reaches the ball of radius h centered at the origin, then the next states stay at the origin. Most importantly the control input has to be calculated uniquely at each time step. This is the object of the next proposition. Proposition 2: The generalized equation ( 9) has a unique solution u * k+1 for any h > 0 and any

x k . If x k h then u * k+1 = -1, if x k -h then u * k+1 = 1, and if |x k | < h then u * k+1 = -x k h ∈ [-1, 1]. Moreover x *
k+1 is a function of x k and h only.

Proof: The proof is led in a similar way as the proof of Proposition 1, step by step in a constructive way.

• 1) let x k > h, and take u k+1 = -1 so that ξ k+1 ∈ IR + . Then we obtain 0 = ξ k+1 +hξ 3 k+1 -

x k + h ⇒ ξ k+1 + hξ 3 k+1 = x k -h > 0.
This equation has a unique solution ξ * k+1 > 0 as may be checked graphically since the function x → x + hx 3 is strictly increasing. Now take u k+1 = 1. One obtains ξ k+1 + hξ 3 k+1 = h + x k while ξ k+1 ∈ IR -and h + x k > 0: this is impossible. Now take |u k+1 | < 1, from which it follows that ξ k+1 = 0. Then 0 = h -x k which is impossible. Hence there is a unique solution u * k+1 = -1.

• 2) The case x k < -h may be analysed similarly and the unique solution is found to be u * k+1 = 1.

• 3) Take now |x k | < h. Let u k+1 = -x k h , hence one obtains 0 = ξ k+1 + hξ 3 k+1 , with ξ k+1 ∈ -N [-1,1] (u k+1 ). Since |u k+1 | = |x k |
h < 1 one has ξ k+1 = 0 and we conclude that u k+1 = -x k h is a solution. Take now u k+1 = 1. We obtain ξ k+1 + hξ 3 k+1 = x k + h with ξ k+1 ∈ IR -. Since x k +h > 0 this is impossible. The same reasoning may be done for u k+1 = -1, and one infers that

u * k+1 = -x k h is the only solution. • 4) Finally consider x k = h. Then 0 = ξ k+1 + hξ 3 k+1 -h -hu k+1 . Consider u k+1 = -x k h = -1. Then ξ k+1 ∈ IR + and 0 = ξ k+1 (1 + hξ 2 k+1
) which has the unique solution ξ k+1 = 0. Consider now u k+1 = 1, hence the generalized equation is

ξ k+1 + hξ 3 k+1 = 2h with ξ k+1 ∈ IR -: this is impossible. Finally let |u k+1 | < 1 ⇒ ξ k+1 = 0.
The generalized equation becomes h = hu k+1 , a contradiction. Thus u * k+1 = -1. • 5) A similar reasoning holds for x k = -h to infer u * k+1 = 1. • The last point follows from the fact that in cases 1) and 2), x * k+1 is obtained as the root of a third order polynomial whose solutions are in [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] [START_REF] Levant | Chattering analysis[END_REF] below.

It is noteworthy that the framework of the fullyimplicit method is that of difference inclusions, since the signum function may take values in the interior of [-1, 1]. However, similarly to what happens in continuous-time (within Filippov's mathematical framework), the input u k+1 , which may be viewed as a selection of the set-valued right-hand side of the system, is uniquely defined as a function of the state. For a real x, x is the integer n such that n x < n + 1.

Proposition 3: Let h > 0. The discrete-time system (5) with the fully-implicit method, has a globally Lyapunov finite-time stable equilibrium x * = 0 and the equilibrium is reached after a finite number of steps N . Moreover x k = 0 for all k N + 1 and u k+1 = 0 for all k N + 2, while N • 1) Let x k > h, then from Proposition 2 u k+1 = -1 so from (5) we get:

x k+1 = x k -h -hx 3 k+1 . From Proposition 1 one has x k+1 > 0, conse- quently x k+1 < x k -h ⇒ |x k+1 | < |x k | -h. • 2) Let x k < -h, then from Proposition 2 u k+1 =
1 so from (5) we get:

x k+1 = x k + h -hx 3 k+1 . From Proposition 1 one has x k+1 < 0, conse- quently x k+1 > x k + h ⇒ |x k+1 | < |x k | -h. • 3) Let |x k |
h, from Proposition 1 one has x k+1 = 0 while from Proposition 2 u k+1 = -x k h . So one infers that x k+n = 0 for all n 1 and u k+1 = 0 for all n 2. Let us now prove that V (x k ) = |x k | is a Lyapunov function for [START_REF] Boiko | Analysis of chattering in systems with second-order sliding modes[END_REF] with the fully-implicit method, for any h > 0 and any x 0 . Outside the ball |x k | h, it strictly decreases as V (x k+1 ) = V (x k ) -h and h > 0. Suppose that the initial condition is x(t 0 ) = x 0 for some finite x 0 =. Then V (x k ) = |x 0 | -kh, from which it follows that there exists a finite integer N such that V (x N ) = |x N | = 0. From item 3) the result follows. For the last part of the proposition, suppose that x 0 > h. Let us show that N x0 h for any x 0 ∈ IR. One has x k < x 0 -kh for any k 1. We are seeking the least integer

N such that x 0 -kh h ⇔ k + 1 x0 h . Thus for all k + 1 x0 h + 1 one obtains x k+1 = 0. Similar calculations hold for the case x 0 < -h.
Remark 1: The Lyapunov function for the continuous-time case shown in section II, is

V (x) = x 2 . It is not too difficult to show that V (x k ) = x 2
k is also a Lyapunov function for the discrete-time case. Consider for instance case 1) in the proof of Proposition 3. One obtains

V (x k+1 ) -V (x k ) = (x k+1 -x k )(x k+1 + x k ) = -h(1 + x 3 k+1 )(x k+1 + x k )
, which is negative because both x k > 0 and x k+1 > 0 in this case. The other cases are treated similarly.

Remark 2: Proposition 3 proves that after a finite number of steps, the discrete-time state and the discrete-time input have no numerical chattering, i.e. no spurious oscillations around the sliding surface and no bang-bang behaviour of the input occur due solely to the discretization process, contrarily to what happens with an explicit discretization as shown analytically in [START_REF] Galias | Complex discretization behaviours of a simple sliding-mode control system[END_REF], [START_REF] Galias | Euler's discretization of single input sliding-mode control systems[END_REF], [START_REF] Galias | Analysis of zero-order holder discretization of two-dimensional sliding-mode control systems[END_REF] and experimentally in [START_REF] Wang | Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Toward input and output chattering suppression[END_REF], [START_REF] Huber | Discretetime twisting controller without numerical chattering: Analysis and experimental results with an implicit method[END_REF]. This is why it is worth analysing the disturbancefree case. Notice that the presented calculation of N does not incorporate the decrease due to the term -x 3 k+1 , and is therefore an overestimation of the real number of steps needed to attain the origin. In fact one may deduce a more accurate estimation for N . Let [•] + be the projector to IR + , i.e. [x] + = x for x > 0, [x] + = 0 for x 0.

Corollary 1: The fully-implicit method implies that x k attains the equilibrium after a finite number of steps N for any h > 0 and for any bounded initial condition x 0 , with:

N 2+ log 3 [ln( √ h|x0|)] + ln(2+h 3/2 ) +
for h 1 [START_REF] Huber | Discretetime twisting controller without numerical chattering: Analysis and experimental results with an implicit method[END_REF] and

N min{1,|x0|} h + [ln min{1/ √ h,|x0|}] + ln(1+2h) + log 3 [ln( √ h|x0|)] + ln(2+h 3/2 ) + for h < 1. (12)
Proof: In order to take into account the effect of the cubic term, at least for large initial conditions, let us define ϕ k := √ hx k , and suppose that ϕ 0 > 1 (i.e. x 0 > 1/ √ h). Multiplying both sides of the equality hx The proven corollary implies that, in addition to finite-time stability, the fully implicit system has hyper exponential convergence rate. In contrast to the continuous-time case, we are not able to prove that the settling time T = hN is globally bounded. However, the property of fast convergence is preserved in the discrete-time system. Moreover if |x 0 | 1 then the presented estimate coincides with the estimation given by Proposition 3, and if |x 0 | > 1 then N < |x0| h . Remark 3: In practice, the fully-implicit method may be implemented at time t k = kh by solving the generalized equations ( 8) and (9) with a suitable numerical solver, as implemented for instance in the open-source SICONOS software 2 . For the particular case of this paper one notices that if x k > h then x * k+1 is the unique root of

3 k+1 + x k+1 = x k -h by √ h we derive ϕ 3 k+1 + ϕ k+1 = ϕ k -h 3/2 .
x 3 k+1 + 1 h x k+1 -x k -h h = 0.
2 http://siconos.gforge.inria.fr/ Equivalently:

x * k+1 = 2 -1 3 x k -h h + x k -h h 2 + 4 1 27h 3 1 2 1 3 + 2 -1 3 x k -h h -x k -h h 2 + 4 1 27h 3 1 2 1 3 (13)
and similarly for x k < -h:

x * k+1 = 2 -1 3 x k +h h + x k +h h 2 + 4 1 27h 3 1 2 1 3 + 2 -1 3 x k +h h -x k +h h 2 + 4 1 27h 3 1 2 1 3 (14)
If |x k | h then x * k+1 = 0. One way to interpret the above results, is that they allow us to analyze such a switching difference equation. Having the explicit representation of x * k+1 the discrete control law (10) can be easily calculated using Proposition 2.

IV. THE CASE WITH A DISTURBANCE

Let us nevertheless consider now that a disturbance d(z, t) with |d(z, t)| δ < 1 acts on the system in (4): ẋ(t) = u -x(t) 3 + d(z, t). The continuous-time system is still globally finite-time Lyapunov stable. Similar results as in Propositions 1, 2, 3 and Corollary 1 can be obtained. However the system's behaviour near the origin is slightly different. The plant discretetime dynamics is given by:

x k+1 = x k + hu k+1 + hd k (15)
where d k = d(x k , t k ). Since d(x, t) is unknown, the controller is calculated from a nominal system as follows:

xk+1 = x k + hu k+1 u k+1 ∈ -x 3 k+1 -sgn(x k+1 ) (16)
where xk+1 replaces ξ k+1 . The same generalized equations as in ( 8) and ( 9) can be constructed from ( 16) which is the counterpart of [START_REF] Edwards | Sliding Mode Control: Theory And Applications[END_REF]. Mimicking Propositions 1 2 and 3 we get: Proposition 4: The generalized equation ( 16) has a unique solution x * k+1 for any x k and h > 0: if x k > h then x * k+1 < 0 (and u * k+1 = -1), if x k < -h then x * k+1 > 0 (and u * k+1 = 1), and if |x k | h then x * k+1 = 0 (and u * k+1 = -x k h ). Moreover the surface xk = 0 is attained after a finite number of steps.

The next corollary states what happens on the "sliding surface" xk = 0 in terms of the set-valued control input and disturbance compensation. Proof: From ( 16) it follows that xk = 0 implies u k+1 = -x k h . Suppose that the nominal system satisfies xk+1 = 0, then obviously, the discretized plant state satisfies x k+1 = hd k . Thus we infer that x k = hd k-1 . The result follows.

Remark 4: The presence of a cubic term in the control input, implies that v(•) takes large values during the transient period if the initial conditions are large. This certainly limits the practical usefulness of such controllers. The goal of this article is however merely to prove that the implicit discretization method introduced in [1], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF] and successfully experimentally tested in [START_REF] Wang | Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Toward input and output chattering suppression[END_REF], [START_REF] Huber | Discretetime twisting controller without numerical chattering: Analysis and experimental results with an implicit method[END_REF], also applies in a nonlinear setting, and has the very nice feature to preserve the continuous-time closed-loop properties like global finite-time Lyapunov stability. This is in big contrast with the widely used explicit discretization which does not share such properties.

V. CONCLUSIONS

In this note we have shown that the implicit timediscretization method introduced for linear slidingmode dynamics in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], extends nicely to the scalar, nonlinear case. It preserves the continuous-time closed-loop system properties like the global finitetime Lyapunov stability of the unique equilibrium point. Moreover it compensates the disturbances by a factor h, and suppresses the input and ouput numerical chattering. This papers treats only the case of a single-input first order sliding mode control system. It demonstrates the successful applicability of the implicit control design method for fixed-time algorithms. The extension of this concept to the case of the second order sliding mode systems is an important subject for future research.

  and sgn(0) = [-1, 1]. It follows from convex analysis that for any reals x and y: x ∈ N [-1,1] (y) ⇔ y ∈ sgn(x).

• 5 )

 5 The last case x k = -h yields the unique solution x * k+1 = 0 and ζ * k+1 = -1.

  x 0 ∈ IR. Proof: let us analyse the quantities |x k+1 | -|x k | when x k > h, x k < -h and |x k | h.

Corollary 2 :

 2 Let xk = 0 for all k n and some n 0. Then u k = -hd k-1 for all k n + 1.

Quite often in practice the equivalent disturbance d(t, z) in (2) is the result of not only parameter uncertainty and measurement noise, but also of the error made by the discretization of the pre-feedback.

Granted by the French National Research Agency project ChaSlim 2011-BS03-007-01.