Konstantin Zimenko
email: kostyazimenko@gmail.com

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Artem Kremlev
email: kremlevartem@mail.ru

Homogeneous Observer Design for Linear MIMO Systems

Keywords: State estimation, nonlinear observer, homogeneity, robustness

The paper is devoted to the problem of state observation (particularly, in finite time) of linear MIMO systems. The presented nonlinear observer does not require system transformation to a canonical form and guarantees finite-time (asymptotic with a fixed-time attraction of any compact set containing the origin) stability of observation error equation if homogeneity degree is negative (positive). The proposed observer is robust in input-to-state sense with respect to disturbances and measurement noises. Performance of the observer is illustrated by a numerical example.

INTRODUCTION

The problem of nonlinear observer design has been widely investigated over the past decades (see, for example, [START_REF] Afri | State and parameter estimation: A nonlinear Luenberger observer approach[END_REF][START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF][START_REF] Andrieu | Highgain observers with updated high-gain and homogeneous correction terms[END_REF][START_REF] Ortega | On dynamic regressor extension and mixing parameter estimators: two Luenberger observers interpretations[END_REF][START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF][START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF][START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF]. It is worth noting that a finite/fixed-time convergence of observed states to the real ones is preferable specially for the cases when observers are used for systems with processes strongly restricted by time or for fault detection. To achieve finite-time stability of the observation error equation, sliding-mode differentiators [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF][START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF][START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] and homogeneity based observers [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit lyapunov function approach[END_REF][START_REF] Andrieu | Highgain observers with updated high-gain and homogeneous correction terms[END_REF][START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF]) may be used.

The present paper addresses the problem of homogeneous observer design for linear MIMO plants. Depending on the sign of degree of homogeneity the presented approach guarantees different types of observation: observation error equation is finite-time stable if degree of homogeneity is negative; it is asymptotically stable with a fixed-time attraction of any compact set containing the origin if degree of homogeneity is positive.

In comparison with existing approaches the presented approach has the following advantages:

• it is applicable for high-order systems;

• it does not require special canonical forms of the system and output matrices or block decomposition as in Lopez-Ramirez et al., 2018 (in some cases block This work was partially supported by the Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

decomposition can be accompanied by significant computational errors); • parameter tuning is based on solution of a system of linear matrix equations and inequalities.

Moreover, due to homogeneity property the observer is robust (in an input-to-state sense) in the presence of disturbances and measurement noises.

The paper is organized as follows. The problem statement is introduced in Section 2. Section 3 considers preliminaries used in the paper. The results on homogeneous observer design are presented in Section 4. Finally, considered example and conclusions are given in Sections 5 and 6, respectively.

Notation: R + = {x ∈ R : x > 0},

√

x T P x with x ∈ R n and P > 0);

A A = sup x∈R n Ax x
for A ∈ R n×n ; I n ∈ R n×n is the identity matrix; the minimal and maximal eigenvalues of a symmetric matrix P = P T are denoted by λ min (P ) and λ max (P ), respectively; R(λ) denotes the real part of the complex number λ; L ∞ (R p ) denotes the set of essentially bounded measurable functions f : R + → R p ; a continuous function σ : R + ∪ {0} → R + ∪ {0} belongs to class K if it is strictly increasing and σ (0) = 0. It belongs to class K ∞ if it is also radially unbounded; a continuous function

β : R + ∪ {0} × R + ∪ {0} → R + ∪ {0} belongs to class KL if β(•, r) ∈ K and β(r, •) is decreasing to zero for any fixed r ∈ R + .

PROBLEM STATEMENT

Let us consider the system in the form ẋ

(t) = Ax(t) + Bu(t) + d x (t), y(t) = Cx(t) + d y (t), (1)
where x ∈ R n is the state variable, y ∈ R k is the measured output, u : R → R s is the control input, d x , d y denote disturbances and measurement noises, respectively, A ∈ R n×n is the system matrix, B ∈ R n×s is the matrix of input gains and the matrix C ∈ R k×n is the output matrix which links the measured outputs to the state variables. The matrix pair (A, C) is assumed to be observable and rank(C) = k.

The main goal of the paper is to propose a homogeneitybased dynamic observer for the system (1) with constructive tuning rules. For non-perturbed case the observer must guarantee finite-time stability (asymptotic stability with a fixed-time attraction of any compact set containing the origin) of the error equation, depending on the degree of homogeneity. In the presence of ∞ -bounded disturbances and measurement noises the observer has to ensure robustness property in the sense of input-to-state stability.

PRELIMINARIES

Stability Notions

Consider the following system ẋ(t) = f (x(t)), x(0) = x 0 , t ≥ 0, (2) where x(t) ∈ R n is the state vector, f : R n → R n is a vector field, f (0) = 0.

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 ), (3)
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 N ⊂ R n is zero. Note that the system (2) may have non-unique solutions for a given x 0 ∈ R n and may admit both weak (a property holds for a solution) and strong stability (a property holds for all solutions) (see, for example [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF][START_REF] Roxin | On finite stability in control systems[END_REF]. This paper deals only with the strong stability properties, which ask for stable behavior of all solutions of the system (2).

Definition 1 [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 ( 2) is said to be globally finite-time stable if it is globally asymptotically stable and any solution x(t, x 0 ) of the system (2) reaches the equilibrium point at some finite time moment, i.e. x(t, x 0 ) = 0 ∀t ≥ T (x 0 ) and x(t, x 0 ) = 0 ∀t ∈ [0, T (x 0 )), x 0 = 0, where T : R n \ {0} → R + is the settling-time function.

Note that in the case the system (2) has non-unique solutions for some x 0 ∈ R n then T (x 0 ) should be considered as a set-valued map (since for the same x 0 ∈ R n there may be solutions that reach the equilibrium point at different time moments). However, for the sake of simplicity, further in such cases the designation T (x 0 ) will be understood as the greatest value of the settling time for a given x 0 ∈ R n .

Definition 2 [START_REF] Polyakov | Nonlinear feedback design for fixedtime stabilization of linear control systems[END_REF] A set M ⊂ R n is said to be globally finite-time attractive for (2) if any solution x(t, x 0 ) of ( 2) reaches M in some finite time moment t = T (x 0 ) and remains there ∀t ≥ T (x 0 ), T : R n → R + ∪ {0} is the settling-time function. It is fixed-time attractive if in addition the settling-time function T (x 0 ) is globally bounded by some number T max > 0. Theorem 1. [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]. Suppose there exists a positive definite C 1 function V defined on an open neighborhood of the origin D ⊂ R n and real numbers C > 0 and σ ≥ 0, such that the following condition is true for the system ( 2

) V (x) ≤ -CV σ (x), x ∈ D \ {0}.
Then depending on the value σ the origin is stable with different types of convergence:

• if σ = 1, the origin is exponentially stable;

• if 0 ≤ σ < 1, the origin is finite-time stable and

T (x 0 ) ≤ 1 C(1 -σ) V 1-σ 0 , where V 0 = V (x 0 ); • if σ > 1 the origin is asymptotically stable and, for every ε ∈ R + , the set B = {x ∈ D : V (x) <
ε} is fixed-time (independent on the initial values) attractive with

T max = 1 C(σ -1)ε σ-1 .
If D = R n and function V is radially unbounded, then the system (2) admits these properties globally.

Let us give definitions of input-to-state stability notions that are widely used for robustness analysis of nonlinear systems. Rewrite (2) for the perturbed case:

ẋ(t) = f (x(t), d(t)), x(0) = x 0 , t ≥ 0, (4) where x ∈ R n , d ∈ L ∞ (R p ) is a disturbance and f ∈ R n+p → R n
is a continuous or discontinuous vector field that satisfies Filippov conditions [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF].

Definition 3 [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] The system (2) is called

• Input-to-state stable (ISS), if there exist functions ζ ∈ KL and ϑ ∈ K such that for any d ∈ L ∞ (R p ) and any x 0 ∈ R n x(t, t 0 , d) ≤ ζ( x 0 , t) + ϑ( d [0,t) ), ∀t ≥ 0. • Integral Input-to-state stable (iISS), if there are some functions ϑ 1 , ϑ 2 ∈ K ∞ and ζ ∈ KL such that for any d ∈ L ∞ (R p
) and any x 0 ∈ R n the following estimate holds:

ϑ 1 ( x(t, t 0 , d) ) ≤ ζ( x 0 , t)+ t 0 ϑ 2 ( d(s) )ds, ∀t ≥ 0.

Generalized Homogeneity

The homogeneity is a property that specifies sort of symmetry of an object with respect to a group of transformations (dilation operation). [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]. The generator G d satisfies the following properties

d ds d(s) = G d d(s) = d(s)G d , d(s) = e G d s = +∞ i=0 s i G i d i! ,
where s ∈ R.

Definition 5 (Polyakov et al., 2016a;[START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] The dilation d is said to be strictly monotone if ∃β such that d(s) A ≤ e βs for s ≤ 0.

Thus, monotonicity means that d(s) is a strong contraction for s < 0 (strong expansion for s > 0) and implies that for any x ∈ R\{0} there exists a unique pair (s 0 , x 0 ) ∈ R× S such that x = d(s 0 )x 0 . Theorem 2. [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]. If d is a dilation in R n , then

• the generator matrix G d is anti-Hurwitz, i.e. R(λ i (G d )) > 0, i = 1, ..., n and there exists a matrix P ∈ R n×n such that P G d + G T d P > 0, P > 0.

(5) • the dilation d is strictly monotone with respect to the weighted Euclidean norm x = √ x T P x for x ∈ R n and P satisfying (5):

e αs ≤ d(s) A ≤ e βs if s ≤ 0, e βs ≤ d(s) A ≤ e αs if s ≥ 0, (6)
where α = 1 2 λ max P

1 2 G d P -1 2 + P -1 2 G T d P 1 2 , β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G T d P 1 2 . Definition 6 (Polyakov et al., 2016b) A vector field f : R n → R n (a function g : R n → R) is said to be d- homogeneous of degree ν ∈ R if f (d(s)x) = e νs d(s)f (x), ∀x ∈ R n \ {0}, ∀s ∈ R. (resp. g(d(s)x) = e νs g(x), ∀x ∈ R n \ {0}, ∀s ∈ R.) (7)
A special case of homogeneous function is a homogeneous norm [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF] The following lemma gives the necessary and sufficient condition of d-homogeneity of linear systems. Lemma 3. [START_REF] Zimenko | Robust feedback stabilization of linear MIMO systems using generalized homogenization[END_REF]. Let G d ∈ R n×n be a generator of the dilation d(s) = e G d s , s ∈ R. Then the linear system ẋ

= Cx, x ∈ R n , C ∈ R n×n is d- homogeneous of degree ν ∈ R if and only if CG d -G d C = νC. (9)
The sign of homogeneity degree of stable systems determines the type of convergence. Theorem 4. [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF]). An asymptotically stable d-homogeneous system ẋ = f (x), f : R n → R n of degree ν ∈ R is uniformly finite-time stable if and only if ν < 0.

The homogeneity theory provides many other advantages to analysis and design of nonlinear control system. For instance, some results about ISS of homogeneous systems can be found in [START_REF] Bernuau | Robustness of homogeneous and homogeneizable differential inclusions[END_REF][START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF][START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF].

Let us develop the result of [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] about ISS stability for generalized homogeneity case.

Let

f (y, d) -f (y, 0) ≤ σ(|d|), σ(s) = cs max if s ≥ 1, cs min if s < 1, ∀y ∈ S, d ∈ R p
(10) be satisfied for some c ∈ R + , max ≥ min > 0. Then following [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] one may obtain:

Theorem 5. Define G d ∈ R n×n , R(λ i (G d )) > 0, i = 1, ..., n and G d ∈ R p×p , R(λ j (G d)) ≥ 0, j = 1, ..., p.
Let the vector field f (x, d) T 0 p T ∈ R n+p be homogeneous with the generators G d , G d and with degree η ≥ -min 1≤i≤n R(λ i (G d )), i.e. f (d(s)x, d(s)d) = exp(ηs)d(s) f (x, d) for all x ∈ R n , d ∈ R p and all s ∈ R. Assume that the system (4) is globally asymptotically stable for d = 0, then the system (4) is

ISS if min 1≤j≤p R(λ j (G d)) > 0 iISS if min 1≤j≤p R(λ j (G d)) = 0 and η ≤ 0.
For the case min 1≤j≤p R(λ j (G d)) = 0 a relaxed constraint min ≥ 0 has to be satisfied, which follows the continuity of f . In the case f is continuous with respect to both arguments, the condition (10) is satisfied. Note that only continuity with respect to the second argument is not enough. For example, the system

ẋ1 = -x 1 + x 1 arctan d 1/x 2 , x 2 = 0 0, x 2 = 0 , ẋ2 = -x 2
is continuous with respect to d, globally asymptotically stable for d = 0, homogeneous with G d = I 2 , G d = 1, and not ISS.

MAIN RESULTS

Let us consider the following observer

d dt x(t) = Ax(t) + Bu(t) -g F T (y(t) -C x(t)), (11)
where x(t) ∈ R n is the observer state vector and the function g F T : R k → R n is defined as

g F T (ς) = L 0 ς + | P ς| ρ- -ρ+ν d ln | P ς| 1 -ρ+ν L F T ς,
where ς ∈ R k , d is a dilation with the generator G d , and

P ∈ R k×k , L F T ∈ R n×k , L 0 ∈ R n×k , ν ≥ -1, , ρ ∈ R
are parameters of the observer, to be determined. Then the error equation in the disturbance-free case (d x = 0, d y = 0) has the following form (12) where e = x -x. Theorem 6. Let d x = 0, d y = 0 and

ė = A + L 0 C + | P Ce| ρ- -ρ+ν d ln | P Ce| 1 -ρ+ν L F T C e,
• the matrix L 0 ∈ R n×k is chosen such that the matrix A + L 0 C is nilpotent;
• there exists ν ≥ -1 such that the system of matrix equations

(A + L 0 C)H -H(A + L 0 C) = A + L 0 C, (13)
C(νH + (ρ -ν)I n ) = 0, (14) has a solution H ∈ R n×n , ρ ∈ R;
• for ξ > δ > 0 the system of matrix inequalities

P A + A T P + P L 0 C + C T L T 0 P + Y C + C T Y T + ξP P P -Z ≤ 0, (15)
P > 0, Z > 0, X > 0, ( 16)

δ 2 X Y T Y P ≥ 0 (17) P ≥ C T XC (18) Ξ T (λ)ZΞ(λ) ≤ 1 δ P, ∀λ ∈ [0, 1], (19)
be feasible for some

P, Z ∈ R n×n , symmetric X ∈ R k×k , Y ∈ R n×k and Ξ(λ) = exp((νH +ρI n ) ln λ)-I n .
Then the error equation ( 12) with

L F T = P -1 Y , P = X 1/2 , G d = νH + I n and > -0.5λ min (νP 1/2 HP -1/2 + νP -1/2 H T P 1/2 ) (20)
is globally asymptotically (for ν > 0) / exponentially (for ν = 0) / finite-time (for 0 > ν ≥ -1) stable.

Note that (20) guarantees strict monotonicity of the dilation d due to Theorem 2.

The nonlinear observer (11) guarantees different types of estimation error convergence depending on the sign of the parameter ν. According to Theorem 1 we have:

• for 0 > ν ≥ -1 the system ( 12) is finite-time stable and the settling time function is bounded as follows

T (e 0 ) ≤ 2α (ξ -δ)ν V -ν 0 ,
where α is defined in Theorem 2;

• for ν > 0 the system ( 12) is asymptotically stable and for any ε ∈ R + , the set B = {e ∈ R n : V (e) < ε is fixed-time attractive with

T (e 0 ) ≤ T max = 2α (ξ -δ)νε ν .
Note that in the case ν = 0 the presented observer becomes the classical Luenberger one.

In order to apply Theorem 6 in practice we need to solve the nonlinear matrix inequality (19) together with the linear ones ( 15)-( 18). Due to the smoothness of Ξ(λ) with respect to λ ∈ (0, 1], this can be done on a proper grid constructed over this interval. The following proposition provides sufficient feasibility condition of the inequality ( 19). Proposition 7. The parametric inequality (19) holds if there exists N > 0 such that

2(ρ + a)Z + H T Z + ZH Z (ρI n + H) ρI n + H T Z M ≥ 0, (21)
P > 0, M > 0, Z > 0, ( 22
)
q 2a i Ξ(q i )ZΞ(q i )+q 2a-1 i (q i -q i-1 )M ≤ q 2a i-1 1 δ P, i = 1, ..., N, (23)
where 0 = q 0 < q 1 < ... < q N = 1, a ∈ R + , H, P, M, Z ∈ R n×n .

The result of Proposition 7 allows to solve the parametrized system of matrix inequalities ( 15)-( 19) using the following algorithm with fixed ν, ξ, δ, a. Algorithm 1. Initialization: N = 1, q 0 = 0, q N = 1, Σ = {q 0 , q N }.

Loop: While the system of LMI ( 15)-( 18), ( 21)-( 23 Corollary 8. The system of matrix equations ( 13), ( 14) and matrix inequalities (15)-( 18), ( 21)-( 23) is feasible provided that |ν| is sufficiently close to zero.

Note, that the system (12) is d-homogeneous of degree ν. Indeed, according to Lemma 3 we have that ė(t) = Ae(t)+ L 0 Ce(t) is d-homogeneous of degree ν, then:

A+L 0 C +| P Cd(s)e| ρ- -ρ+ν d ln | P Cd(s)e| 1 -ρ+ν L F T C d(s)e = (A + L 0 C) d(s)e+ exp(ρs-s)| P C(s)e| ρ- -ρ+ν d(s)d ln | P Ce| 1 -ρ+ν L F T C d(s)e = exp(ν)d(s) A+L 0 C +| P C(s)e| ρ- -ρ+ν d ln | P Ce| 1 -ρ+ν L F T C e.
Basing on homogeneity property a qualitative assessment of robustness can be presented for the observer (11). (24) Then, basing on homogeneity the following result may be obtained.

Corollary 9. Consider the perturbed error equation ( 24) and assume that all conditions of Theorem 6 are satisfied and additionally to (20) the parameter is chosen such that ≥ -0.5λ min (νH + νH T + νI n ).

(25) Then the system (24) is iISS. If the inequality ( 25) is satisfied strictly then the system (24) is ISS.

The quantitative analysis is out the scope of this paper and needs further research developments (for example, with the use of results [START_REF] Prasov | A nonlinear highgain observer for systems with measurement noise in a feedback control framework[END_REF][START_REF] Sanfelice | On the performance of high-gain observers with gain adaptation under measurement noise[END_REF][START_REF] Menard | Fixed-time observer with simple gains for uncertain systems[END_REF].

Remark The proposed observer is an extension of the results presented in [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit lyapunov function approach[END_REF] and based on the use of weighted homogeneity (the special case of generalized homogeneity). Indeed, consider the system in the block form

à =      0 A 12 0 • • • 0 0 0 A 23 • • • 0 • • • • • • • • • • • • • • • 0 0 • • • 0 A m-1 m 0 0 • • • 0 0      , C = [I k 0], ( 26
)
where m is an integer, A j-1 j ∈ R nj-1×nj , n j = rank(A j-1 j ), j = 2, . . . , m. Then, applying the presented approach for (26), one obtains an observer with the same structure as in the paper [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit lyapunov function approach[END_REF]. Therefore the main advantage of this paper is not in the better transients or robustness properties, but in the fact the presented approach does not require block decomposition which in some cases may be accompanied by computational errors. It can be viewed also as a nonlinear generalization of Luenberger observer, having non-asymptotic convergence rates for ν = 0 and coinciding with it for ν = 0 The result for the case of noisy measurements is shown in Fig. 2, where a band-limited white noise of power 10 -5 has been added to the output signal. The paper is devoted to homogeneous observer design for linear MIMO systems. The presented observer guarantees finite-time stability (asymptotic stability with a fixedtime attraction of any compact set containing the origin) of observation error equation if homogeneity degree is negative (positive). The key feature of the observer is in the fact that there is no requirements to transform the system to canonical form. The parameters tuning is based on solution of linear matrix equations and inequalities. The qualitative analysis of observers' robustness against bounded measurement noises and disturbances has been studied, namely, it is shown that the observer is iISS/ISS. Numerical example demonstrates effectiveness of the proposed control.

  : a continuous positive definite dhomogeneous function of degree 1. Define the canonical homogeneous norm • d : R n → R + as x d = e sx , where s x ∈ R such that d(-s x )x = 1. Note that d(s)x d = e s x d and d(ln x d )x = 1. (8) Define the unit sphere in the homogeneous norm by S = {x ∈ R n : x d = 1}.

  Consider the system (1) with nonzero d x : R + → L ∞ (R n ) and d y : R + → L ∞ (R k ). In this case the perturbed error equation takes the form ė = (A + L 0 C) e + L 0 d y + d x + | P (Ce+d y )| ρ--ρ+ν d ln | P (Ce+d y )| 1 -ρ+ν L F T (Ce+d y ).

.

  the system (1) in the disturbance-free case for n = 3, u(t) = 0, Define the observer in the form (11) with the parameter ν = -0.2, where the matrixL 0 ∈ R 2×3 is chosen A + L 0 C is nilpotent and the matrices P ∈ R 2×2 , L F T ∈ R 3×2 , G d ∈ R 3×3 are obtained from the inequalitiesThe results of simulation are shown in Fig.1with using the logarithmic scale in order to demonstrate fast (finite-time) convergence rate of the observer.

Fig. 1 .

 1 Fig. 1. Simulation plot of |e| for three different initial conditions x(0) = (0, 1, 2), x(0) = 10 3 (0, 1, 2), x(0) = 10 7 (0, 1, 2)

Fig. 2 .

 2 Fig. 2. Simulation plot of |e| for the observer with measurement band limited noise and initial conditions x(0) = 10 3 (0, 1, 2)

  The type of homogeneity, dealing with linear transformations, is called generalized homogeneity.Definition 4(Polyakov et al., 2016a,b) A map d : R → R n×n is called dilation in the space R n if it satisfies: • group property: d(0) = I n and d(t + s) = d(t)d(s) = d(s)d(t) for t, s ∈ R; • continuity property: d is a continuous map; • limit property: lim s→-∞ d(s)x = 0 and lim s→+∞ d(s)x = +∞ uniformly on the unit sphere S. The dilation d is a uniformly continuous group. Its generator is a matrix G d ∈ R n×n defined by G d = lim s→0

	d(s)-In
	s


Keywords: Fixed-time stability, Homogeneity, Nonlinear systems

Fixed-time stability of a class of nonlinear time-varying systems is investigated using the introduced notions of sub-and sup-homogeneity. These concepts allow the systems (even if they do not admit homogeneous approximations) to be analyzed using the homogeneity of their extensions. Then, finite-time and fixed-time stability properties can be recovered. The proposed stability conditions are not based on Lyapunov arguments. In the last section, an example illustrates the obtained results.

On fixed-time stability of a class of nonlinear time-varying systems

Youness Braidiz * , * * , Andrey Polyakov * * , * * * , Denis Efimov * * , * * * , and Wilfrid Perruquetti *

INTRODUCTION

In the last decades, analysis of asymptotic and exponential stability of dynamical systems were widely developed in the control community, e.g., see [START_REF] Hahn | Theory and application of Lyapunov's direct method[END_REF]; [START_REF] Yoshizawa | Stability theory by Lyapunov's second method[END_REF]; [START_REF] Hahn | Stability of motion[END_REF]; [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF] and references therein. However, in many applications the time is a critical constraint which must be taken into account. Therefore, there are lot of researchers who work on the so called finite-time stability (FTS) and fixed-time stability (FxTS) as [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF]; Bhat andBernstein (2000b, 2005a); [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control[END_REF]; Braidiz et al. (2019). These concepts require that the system's trajectories converge to an equilibrium state in finite-time, (which does not depend on initial states in the FxTS case) and they are kept there then after.

FxTS plays an important role in many engineering problems, such as robot control, fixed-time attitude tracking of spacecraft, networked control issues, and so on. The concept of FxTS was proposed by [START_REF] Polyakov | Nonlinear feedback design for fixedtime stabilization of linear control systems[END_REF]; [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] and he showed that this property can be investigated by using Lyapunov approach (necessary and sufficient Lyapunov characterizations of FxTS are obtained in [START_REF] Lopez-Ramirez | Conditions for fixed-time stability and stabilization of continuous autonomous systems[END_REF]). However, in many applications, to find a Lyapunov function to study FTS and FxTS is not obvious. Therefore, in this paper, we propose a new method, which is based on construction of set-valued extensions. These extensions will satisfy some kind of symmetry called sup-or sub-homogeneity. This new relaxed symmetry still preserves the scaling property of the solutions of a sup-and sub-homogeneous differential inclusion (DI).

Homogeneous extensions were used in Braidiz et al. (2019) to investigate FTS of some class of nonlinear time-varying systems, which do not have a homogeneous approximation. Due to the features of their construction, all results that have been obtained for homogeneous systems/DIs (see [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]; [START_REF] Moulay | Finite time stability of differential inclusions[END_REF]; [START_REF] Bernuau | Finite-time stability of continuous autonomous systems[END_REF]) can be used to analyze these homogeneous extensions. Then, a non homogeneous system should satisfy some conditions to guarantee the existence of their homogeneous extensions. Similarly, in this paper, we introduce the so-called sup-and subhomogeneous extensions, which will be used to study stability properties of some class of nonlinear systems.

The main contribution of this work is a method for establishment of the FxTS of some class of nonlinear systems which are uniformly globally asymptotically stable (GAS). The notion of sub-homogeneity which will be proposed in this work can also be used to study the FTS of uniformly GAS systems. Hence, this allows us to enlarge a lot the class of considered systems. Sup-homogeneity will be defined for DIs which will be given here as an extension of nonlinear systems. Particularly, we will prove that if a uniformly GAS system has a sup-homogeneous extension with positive degree of homogeneity, then it is nearly FxTS.

The paper is organized as follows. After introducing notations and definitions about asymptotic, finite-time, fixedtime stability and homogeneity in Section 2. We investigate the FTS and FxTS by using the introduced supand sub-homogeneous extensions (in Section 3). In the last section we present an example of a nonlinear model, (a form of mechanical system), to illustrate the obtained results of this paper.

PRELIMINARIES

Notations

• R + = {x ∈ R : x ≥ 0} , where R is the set of real numbers. | • | denotes the absolute value in R,

• denotes the Euclidean norm in R n and x α = |x| α sign(x) for x ∈ R and α ∈ R + .

• S = {x ∈ R n : x = 1} denotes the unit sphere in R n . B(x 0 , r) = {x ∈ R n : x -x 0 < r} denotes the open ball of radius r > 0 centered at a point x 0 , and B(x 0 , r) is the closure of B(x 0 , r). We define

B(K, r) = y ∈ R n : inf x∈K y -x < r for all com- pact K ⊂ R n . • M m,
∈ R + \ {0} and β(s, T ) = 0 for some 0 ≤ T < +∞) for each fixed s ∈ R + . • C(R + , R n ) denotes the space of continuous functions f : R + → R n . • For F : R n ⇒ R n , we denote by D(F ) the domain of definition of F. • coA denotes the closed convex hull of A ⊂ R n .

Stability properties and generalized homogeneity

In this subsection we give some definitions of homogeneity and different kinds of stability for a non-autonomous system:

ẋ(t) = f (t, x(t)), t ≥ t 0 ≥ 0 x(t 0 ) = x 0 ∈ R n , (1)
the vector field f : 1) possesses a unique solution in forward time for all initial conditions x 0 ∈ R n except the origin and t 0 ∈ R + , which we will denote by x(t, t 0 , x 0 )

R + × R n → R n , f (t, 0) = 0, ∀t ≥ 0 is considered such that (
Uniform asymptotic, finite/fixed-time stability:

The following definitions are inspired from [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF]; [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]; Bhat and Bernstein (2000a) and [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]. Definition 2.1. The origin of ( 1) is

• unifromly stable if ∀ε > 0, ∃δ = δ(ε) > 0, independent of t 0 , such that x 0 ∈ B(0, δ) =⇒ x(t, t 0 , x 0 ) ∈ B(0, ε), ∀t ≥ t 0 ≥ 0;
• uniformly attractive if ∃c > 0, independent on t 0 ≥ 0, such that ∀ε > 0, ∃ T (ε, c) > 0 and

x 0 ∈ B(0, c) =⇒ x(t, t 0 , x 0 ) ∈ B(0, ε), (2)
∀t ≥ t 0 + T (ε, c); • uniformly asymptotically stable if it is uniformly stable and uniformly attractive.

In addition, we say that (1) is uniformly GAS if it is uniformly stable, δ(ε) can be chosen to satisfy lim ε→+∞ δ(ε) = +∞ and for each pair of positive numbers ε and c, there is T (ε, c) such that the inequality (2) holds for all t ≥ t 0 + T (ε, c).

Definition 2.2. The origin of ( 1) is

• uniformly FTS, if it is uniformly stable and there exists a nonempty open neighborhood of the origin V ⊂ R n and a function T :

V → R + such that ∀t 0 ≥ 0, ∀x 0 ∈ V, x(t, t 0 , x 0 ) = 0, ∀t ≥ t 0 + T (x 0 ).
T is called a uniform settling-time function of the system (1);

• globally uniformly FTS, if V = R n ; • uniformly FxTS, if the settling-time function T is bounded for all x ∈ R n ; • uniformly nearly FxTS if for any r > 0, there exists 0 < T r < +∞ such that x(t, t 0 , x 0 ) ∈ B(0, r), ∀t ≥ t 0 + T r , ∀x 0 ∈ R n , ∀t 0 ≥ 0.
Now we define the stability properties for DIs. Let us consider the following DI

ẋ(t) ∈ F (x(t)), ∀t ≥ 0, x(0) = x 0 ∈ R n , (3)
where F : R n ⇒ R n . The next definition concerns the strong and weak stability properties of (3). We assume that the set-valued map F satisfies the classical conditions to guarantee the existence of solutions of (3) (see [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]). For each Ω ⊆ R n we let S(Ω) ⊂ C(R + , R n ) denote the set of solutions of (3), which satisfy x(0) ∈ Ω. If Ω is a singleton {ζ}, we will use the shorthand S(ζ). We denote S = S(R n ) as the set of all solutions of (3). Definition 2.3. The origin of the system (3) is weakly uniformly GAS (resp., weakly FTS/FxTS) if there exists a subset Ŝ of the set S of solutions of (3), such that all x(•) ∈ Ŝ are uniformly GAS (resp., FTS/FxTS). [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] Let a function F be upper semicontinuous on a compactum K and let for each x ∈ K the set F (x) be bounded. Then, the function F is bounded on K.

Definition 2.4. A set-valued map F is called upper semi- continuous at x ∈ D(F ) if ∀ε > 0, ∃δ > 0 such that ∀x ∈ B(x, δ), F (x) ⊂ B(F (x), ε). Lemma 2.1.
By the definition of weak GAS (resp., FTS) ∃β ∈ KL (resp., β ∈ GKL) such that x(t) ≤ β( x 0 , t), for all x 0 ∈ R n , x(•) ∈ Ŝβ (x 0 ). Notice that, in the case of the weak uniform GAS or FTS, the set Ŝ may depend on the function β:

Ŝ = Ŝβ .
In this case, for σ ∈ K ∞ and R > 0 we denote

C β σ (R) =    y ∈ C ([0, 1], R n ) ; y(1) = 0, y σ(t) 1 + σ(t) = x(t), t ≥ 0, x(•) ∈ Ŝβ (B(0, R))    .
The following lemma proves that the set Ŝβ of weakly uniformly stable solutions of ( 3) is homeomorphic to a compact in C([0, 1], R n ).

Lemma 2.2. If F is upper semicontinuous, Ŝβ is the set of all GAS solutions of (3) and R > 0. Then,

C β σ (R) is a com- pactum in (C([0, 1], R n ), • C ) with f C = sup x∈[0,1] f (x) .
Homogeneity: In control theory, homogeneity simplifies qualitative analysis of nonlinear dynamic systems. So that, it allows a local properties (e.g. local stability) to be extended globally using a symmetry of the solutions (homogeneity is a Lie symmetry). In order to define this concept, let us introduce the notion of dilation group. Definition 2.5. In the next section we will show the main results of this paper. A uniform GAS system will be considered and we will study its global fixed-time stability.

A map d : R → M n (R) is called a linear dilation in R n if it satisfies • Group property: d(0) = I n , d(t + s) = d(t)d(s), t, s ∈ R; • Limit property: lim

FINITE/FIXED-TIME STABILITY RESULTS

In this section, we consider the following system ẋ

(t) = H 1 (x(t))b 1 (t, x(t)) + H 2 (x(t))b 2 (t, x(t)); x(0) = x 0 ; x 0 , x(t) ∈ R n , ∀t ≥ t 0 ≥ 0. ( 4
)
with

H 1 : H 1 : R n → M n,m (resp., H 2 : R n → M n,m
) are continuous and d-homogeneous of degree ν 1 < 0, (resp., ν 2 > 0) (i.e., H 1 (d(s)x) = e ν1s d(s)H 1 (x) and

H 2 (d(s)x) = e ν2s d(s)H 2 (x)); H 2 : b 1 : R + × R n \ {0} → R m and b 2 : R + × R n \ {0} → R m are continuous, ∃M 1 , M 2 > 0 such that 0 < b 1 (t, x) ≤ M 1 and 0 < b 2 (t, x) ≤ M 2 , ∀x ∈ R n , ∀t ≥ 0; H 3 :
The origin of the system (4) is uniformly GAS.

Our goal is to prove that (4), under the assumptions H 1 , H 2 and H 3 , is FxTS. Therefore, we will study this system in two steps. The first step will analyze the FTS of (4) by using the notion of sub-homogeneous extension (Section 3.1). In the next step, we will show that (4) is nearly FxTS by using the notion of sup-homogeneous extensions (Section 3.2).

Sub-homogeneous extension

In this section we will investigate the FTS of the uniformly GAS system (4). To do so, we will introduce the notion of sub-homogeneous extensions. This concept will be used to prove that the system ( 4

) is FTS if ν 1 < 0. Definition 3.1. A set-valued map F : R n ⇒ R n is d-sub- homogeneous with degree of homogeneity ν ∈ R n , if for all x ∈ R n , F (d(τ )x) ⊆ e ντ d(τ )F (x), ∀τ ≤ 0,
where d is a generalized dilation and ν ∈ R.

We will consider the following example of sub-homogeneous extension:

F (x) = co s≤0 t≥0 e -ν1s d(-s)f (t, d(s)x) , ( 5
)
where f is defined in (1). Such a set-valued map F is closed, convex and d-sub-homogeneous by construction, and for

f (t, x) = H 1 (x)b 1 (t, x) + H 2 (x)b 2 (t, x), if ν 2 > 0 > ν 1 , F is bounded for every fixed x ∈ R n . Indeed, e -ν1s d(-s)f (t, d(s)x) = H 1 (x)b 1 (t, d(s)x) + e (ν2-ν1)s H 2 (x)b 2 (t, d(s)x), because (ν 2 -ν 1 ) > 0 and lim s→-∞ e (ν2-ν1)s = 0, we get lim s→-∞ sup x∈K H 1 (x)b 1 (t, d(s)x) + e (ν2-ν1)s H 2 (x)b 2 (t, d(s)x) ≤ M 1 sup x∈K H 1 (x) Mm,n ,
where K is a compact set in R n \ {0}. This implies that for every fixed x ∈ R n the set F (x) is compact. Therefore, the system (4) has a sub-homogeneous extension. Hence, the solutions of the system (4) are in the set of solutions of the DI (3) with F is sub-homogeneous. The following proposition ensures symmetry of solutions of the subhomogeneous differential inclusion (3). Proposition 3.1. [START_REF] Braidiz | On finite-time stability of homogeneous systems with multiplicative bounded function[END_REF] Let F be a d-subhomogeneous set-valued map of degree ν 1 ∈ R. If for x 0 ∈ R n , the trajectory x(•, x 0 ) is a solution of (3) starting at x 0 , then for all s ≥ 0, the absolutely continuous curve t -→ d(s)x(e ν1s t, x 0 ) is a trajectory of (3) starting at d(s)x 0 , (i.e., if x(•, x 0 ) ∈ S(x 0 ), then d(s)x(e ν1s •, x 0 ) ∈ S(d(s)x 0 ) for all s ≥ 0.

The proof of Proposition 3.1 is based on d-sub-homogeneity of the set valued map F (see Proposition 3.1 in [START_REF] Braidiz | On finite-time stability of homogeneous systems with multiplicative bounded function[END_REF]). Remark 3.1. Due to the continuity of the right side of (4), the solutions x(•, t 0 , x 0 ) exist, then the set S(x 0 ) is nonempty. All the solutions of the system

ẋ(t) = e -ν1s d(-s)f (t, d(s)x(t)), x(0) = x 0 s ≤ 0 and t ≥ 0, with f (t, x) = H 1 (x)b 1 (t, x) + H 2 (x)b 2 (t, x) are in S(x 0 ), ∀x 0 ∈ R n .
In the sequel of this paper, we will use Lemma 2.2. Therefore, to do so, the set-valued map F should be upper semicontinuous:

Lemma 3.1. If the vector field f (t, x) = H 1 (x)b 1 (t, x) + H 2 (x)b 2 (t, x
) is continuous and satisfies the assumptions H 1 , H 2 , then the set-valued map F given by (5) is upper semicontinuous.

Proof. The proof of this lemma will be based on proving that:

∃ h(x) = q(x) i=1 λ i (x)g i (x) such that h(x) ∈ F (x) =⇒ inf h(x)∈F (x) h(x) -h(x) ≤ ε. (6)
with

h(x) = q(x) i=1 λ i (x)g i (x), g i (x) = e -ν1si(x) d(-s i (x))f (t i (x), d(s i (x))x), {λ i } i∈N ⊂ C(R n , R + ), q : R n → N, {t i } i∈N ⊂ C(R n , R + ), {s i } i∈N ⊂ C(R n , (-∞, 0]) and q(x) i=1 λ i (x) = 1, ∀x ∈ R n .
Due to the continuity of f, one gets

h(x) = q(x) i=1 λ i (x)g i (x) ∈ B(F (x), ε), (H 1 , H 2 imply that the set valued F (x) is compact by construction) which means that F (x) ⊂ B(F (x), ε).
Then, we deduce that F is upper semicontinuous. Now, after showing that the set valued map F is upper semicontinuous. The results of Lemma 2.2 will be used to prove the FTS of the system (4). Theorem 3.1. If the system (4) satisfies the assumptions H 1 , H 2 and H 3 , then it is FTS.

Proof. Let Ŝβ be the set of solutions of the DIs (3), which are uniformly GAS. Let R > 0,

x 0 ∈ B 2R , x(•, x 0 ) ∈ Ŝβ (x 0 ), B R = {x ∈ R n : d(-ln(R))
x ≤ 1} and let us define the time τ (x 0 , B R ) for uniformly GAS solutions of (3) starting from x 0 , to reach and stay in B R by:

τ (x 0 , B R ) = sup x(•,x0)∈ Ŝβ (x0) inf {T > 0 : x(t, x 0 ) ∈ B R , ∀t ≥ T } .
Using Lemma 2.2, one can prove that τ is finite for every x 0 ∈ B R . The fact that ν 1 < 0 implies the weak FTS of the sub-homogeneous extension (3). Since the solutions of (4) are in the set of solutions of (3). ( 4) is finite-time stable.

Theorem 3.1 generalizes to time-varying systems the results that have been proven in [START_REF] Bernuau | Finite-time stability of continuous autonomous systems[END_REF].

Sup-homogeneous extension

In this subsection we consider (3) with F is given by

F (x) = co s≥0 t≥0 e -ν2s d(-s)f (t, d(s)x) , (7)
with f (t, x) = H 1 (x)b 1 (t, x) + H 2 (x)b 2 (t, x) and from the construction of F , we can see that for s = 0 and for t ≥ 0 we have f (t, x) ∈ F (x), for every fixed x ∈ R n and t ∈ R + . Let us now start by the following definition of sup-homogeneous set-valued map.

Definition 3.2. We say that F is d-sup-homogeneous with degree of sup-homogeneity ν 2 , if for all x ∈ R n , F (d(τ )x) ⊆ e ν2τ d(τ )F (x), ∀τ ≥ 0, where d is a generalized dilation and ν 2 ∈ R.

The extension (7), with ν 2 > 0 > ν 1 , is compact. Indeed, this happens because lim s→+∞ e (ν1-ν2)s = 0. The solutions to the DI (3), when F is d-sup-homogeneous, satisfy some sort of symmetry. The following proposition show that. Proposition 3.2. If F is d-sup-homogeneous set-valued map of degree ν 2 ∈ R, then for all x(•, x 0 ) ∈ S(x 0 ), x 0 ∈ R n , the absolutely continuous curve t -→ d(s)x(e ν2s t, x 0 ), s ≤ 0 is a trajectory of the system (3),( 7) starting at d(s)x 0 (i.e., if x(•, x 0 ) ∈ S(x 0 ), then d(s)x(e ν2s •, x 0 ) ∈ S(d(s)x 0 ) for all s ≤ 0.

Proof of Proposition 3.2: The proof is similar to the one of Proposition 3.1.

Lemma 3.2. If f (t, x) = H 1 (x)b 1 (t, x) + H 2 (x)b 2 (t, x
) is continuous and satisfies the assumptions H 1 , H 2 , then the set-valued map F given by ( 7) is upper semicontinuous.

Proof. To prove Lemma 3.2 we use the same method as in the proof of Lemma 3.1.

Let us now give the following theorem which shows the nearly FxTS of (4). Theorem 3.2. If the system (4) satisfies the assumptions H 1 , H 2 and H 3 , then the system (4) is nearly FxTS.

Proof. Let Ŝβ (x 0 ) be the set of solutions of the differential inclusion (3), starting from x 0 ∈ R n , which are uniformly GAS. Let x(•, x 0 ) ∈ Ŝβ (x 0 ), B R = {x ∈ R n : d(ln(R))x ≤ 1} for a given R > 0 and let us define τ (x 0 , B R ) = sup

x(•)∈ Ŝβ (x0)
inf{T > 0 : x(t, x 0 ) ∈ B R , ∀t ≥ T }, as in the proof of Theorem 3.1. By the definition of τ (x 0 , B R ) and the fact ν 2 > 0, one deduces that the ball B R is FxTS. Which means that all the solutions in Ŝβ (x 0 ) get in B R in fixed-time, and stay there.

In particular, the solutions of (4) are converging in fixedtime to the ball B R .

Combining Theorems 3.2 and 3.1, we deduce the FxTS of (4): Theorem 3.3. The system (4), under the assumptions H 1 , H 2 and H 3 , is FxTS.

Proof. The system (4) satisfies the assumptions H 1 , H 2 and H 3 , then 1) Theorem 3.2 implies that the system (4) is nearly FxTS. 2) Theorem 3.1 shows that the system (4) is FTS for all x 0 ∈ B R .

These two facts imply that the state of the system (4) converge to the origin at fixed-time (as we see in Fig. 1)

T ≤ τ x 2 ν 2 τ (R) 2 ν 2 -1
, B R + 2 ν 1 τ (R) 1-2 ν 1 . Remark 3.2. The results of this paper can be used to stabilize in fixed-time nonlinear systems, which have the form ẋ = H(x, u)b(t, x, u), with u is a control (e.g., the From this we have DU (z), f (t, z) ≤ 0, ∀z ∈ R 2 \ {0} and all t ≥ 0. Using LaSalle's theorem, one gets that the origin of the system (8) is uniformly GAS. The degree of homogeneity of H is negative and the degree of homogeneity of H is positive, then by using Theorem 3.3 the system (8) is FxTS. The Figure 2 shows the solution of the system (8) with initial condition (2, 4) which reaches the origin in fixed-time (less than 50).

CONCLUSION

FxTS of some class of nonlinear time-varying systems is investigated. The notions of sup-and sub-homogeneity were introduced. These concepts are used to reach the FxTS results of this paper. Specifically, we presented some sufficient conditions on the vector fields to guarantee the FxTS of nonlinear dynamical systems. As we showed, these properties allow systems which do not admit homogneous approximations to be analyzed. Example from Section 4 is considered to illustrate the obtained results.

  )x = +∞ uniformly on the unit sphere.Definition 2.6.The dilation d is monotone in R n , if d(s) Mn ≤ 1, ∀s ≤ 0.Now we will define the canonical homogeneous norm. Definition 2.7. If the dilation d is monotone, the continuous function• d : x ∈ R n → x d = e sx ∈ R+ , where s x ∈ R : d(-s x )x = 1, x d → 0 as x → 0 and d(s)x d = e s x d > 0, is called the canonical homogeneous norm. Definition 2.8. A vector field f : R n → R n (resp., a function h : R n → R) is said to be d-homogeneous of degree ν ∈ R (resp., µ ∈ R) iff for all s ∈ R and all x ∈ R n we have e -νs d(-s)f (d(s)x) = f (x), (resp., e -µs h(d(s)x) = h(x)).

Fig. 2 .

 2 Fig.2. The solutions of the system (4) with α = 1, β 1 = 2, β 2 = 1.5 and the initial condition (x 0 , y 0 ) = (2, 4).

  This work was partially supported by ANR 15 CE23 0007 (Project Finite4SoS), by the Government of Russian Federation (Grant 08-08) and the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

  n is the set of all m×n-matrices over the field of real numbers, and it forms a vector space. When m = n we write M n instead of M n,n . • Mm,n denotes the matrix norm induced by • , i.e. A Mm,n = sup The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. A continuous function β : R + ×R + → R + belongs to the class KL (resp., GKL) if β(•, t) ∈ K ∞ for each fixed t ∈ R + and lim

	Ax .
	x∈S
	• A continuous function α : R + → R + belongs to
	the class K if α(0) = 0 and the function is strictly
	increasing. t→+∞	β(s, t) = 0 (resp., if for each fixed

t ∈ (0, T ), β is a strictly decreasing function of its second argument t ∈ R + for any fixed first argument s

2 -β 2 2 -sin t z |y| 3 .

dynamic model of quadrotor ( ẋ = A(x)x + Bu)) with H is homogneoeus and b is bounded.

ACADEMIC EXAMPLE

Consider the following nonlinear system

where z = x y ∈ R 2 and β 1 , β 2 > 0. We can rewrite the system (8) in the following form ż

The functions b and b are bounded and continuous on every z ∈ R 2 \ {0}, for all t ≥ 0. The functions H and H are continuous and d-homogeneous of the degree of homogeneity respectively ν = -1 3 and ν = 2 3 provided that the dilation d is defined as d(s) = diag{e s , e 2 3 s }, which corresponds to a weighted homogeneity. We use the following Lyapunov function U (x, y) = 3 4 α|x| 4 3 + 1 2 y 2 , we obtain


Mariana Ballesteros
email: mballesteros@ctrl.cinvestav.mx

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Isaac Chairez

Alexander Poznyak

Adaptive Discontinuous Control for Homogeneous Systems Approximated by Neural Networks

Keywords: Homogeneous systems, Neural Networks, Adaptive Control, Discontinuous Control, System Identification

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The design of automatic controllers aimed to solve the trajectory tracking problem for systems described with uncertain models and subjected to external perturbations remains as a significant area in systems theory. In the recent literature, there are numerous works related with the tracking control problem for systems with unknown elements. One remarkable approach is the application of robust techniques [START_REF] Harashima | Tracking control of robot manipulators using Sliding Mode[END_REF][START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF][START_REF] Alazki | A class of robust bounded controllers tracking a nonlinear discrete-time stochastic system: Attractive ellipsoid technique application[END_REF]. The results obtained with robust controllers are reliable. However, a regular robust controller has the disadvantage of being designed considering only the worst possible scenario for the uncertain system. Therefore, it is insensitive to minor system changes providing the same energy in the control signal which could not be suitable in many real plants [START_REF] Doyle | Synthesis of robust controllers and filters[END_REF][START_REF] Utkin | Sliding mode control in electro-mechanical systems[END_REF].

On the other hand, compensating controllers, which use an approximation of the unknown dynamics have solved the trajectory tracking problems with a bounded tracking error for a wide variety of nonlinear systems. This error can be characterized by the quality of the approximation [START_REF] Join | Control of an uncertain three-tank system via on-line parameter indentification and fault detection[END_REF][START_REF] Castañeda | Robust trajectory tracking of a delta robot through adaptive active disturbance rejection control[END_REF]?). The approximation of system dynamics in control theory is known as non-parametric system identification. The structure of the identified systems is used to design the adaptive controllers. There are numerous algorithms ⋆ This work was partially supported by the Government of Russian Federation (Grant 08-08) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031). Mariana Ballesteros is sponsored by Mexico scholarship from the National Council for Science and Technology, (Scholar reference: 550803) for system identification [START_REF] Jagannathan | Identification of nonlinear dynamical systems using multilayered neural networks[END_REF][START_REF] Ljung | System Identification Theory for the User[END_REF][START_REF] Haber | Nonlinear System Identification Input-Output Modeling Approach[END_REF][START_REF] Ljung | Some aspects on Nonlinear System Identification[END_REF] considering an extensive class of systems with uncertainties. In such cases, the design of the identifier depends on the available data entering and coming out from the system. In this work, the considered systems are known as homogeneous systems [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]?).

The homogeneous dynamic systems are represented by ordinary differential equations, whose vector fields posses a kind of symmetry under a specific nonlinear transformation. In other words, a dilation over the arguments results in a scaling over the vector fields. This characteristic allows to extend the local attributes (in a compact, i.e. the unit sphere) globally.

A homogeneous representation of nonlinear systems can be more useful than a linear approximation. For example, a linearized system may not be conclusive on the stability of the original system, meanwhile the homogeneous approximation could give a stability conclusion [START_REF] Rosier | Homogeneous Lyapunov Function for Homogeneous Continuous Vector Field[END_REF]. In addition, the stability of homogeneous systems has been studied and the type of stability of the system is related with the degree of homogeneity (or if some part of the approximation is not homogeneous, the stability of the equilibrium) [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. Indeed, there exist techniques to obtain an homogeneous Lyapunov function [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF][START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF] for homogeneous systems.

Homogeneous approximations of known nonlinear vector fields have been studied in [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Kawski | Nilpotent Lie algebras of vector fields[END_REF], among others. However, studies in the design of non-parametric iden-tifiers or observers for homogeneous systems with uncertainties are scarce in the literature. An approach for realizing the system identification is the use of the approximation theory, which is based on the selection of a suitable function basis to estimate the uncertainty. The design of identifiers based on Artificial Neural Networks (ANN) is an example of this method [START_REF] Nelles | Nonlinear system identification: from classical approaches to neural networks and fuzzy models[END_REF]. ANN structures are known for their approximation capability of multivariate nonlinear continuous functions [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF][START_REF] Park | Universal approximation using radial-basis-function networks[END_REF][START_REF] Haykin | Neural networks: a comprehensive foundation[END_REF]. Based on this property, ANN structures have been used in control theory to approximate nonlinear dynamics of ordinary differential equations, partial differential equations and for the design of identifiers and observers. Another advantage of the ANN structures is that they can be used to approximate multi-input and multi-output systems [START_REF] Lewis | Neural network control of robot manipulators and non-linear systems[END_REF][START_REF] Poznyak | Differential neural networks for robust nonlinear control: identification, state estimation and trajectory tracking[END_REF][START_REF] Chairez | Adaptive neural network nonparametric identifier with normalized learning laws[END_REF]. This work presents the design of an adaptive controller to solve the trajectory tracking problem for uncertain homogeneous systems based on a homogeneous representation implementing differential ANNs. The main contributions and novelties of this work are:

• The design of an adaptive control for the tracking trajectory problem of homogeneous uncertain systems. • The design of an adaptive discontinuous controller that uses a compensating scheme with an ANN homogeneous identifier.

• The extension of differential ANN with homogeneous functions to approximate homogeneous uncertain systems. • A two structure controller, the first related with the compensation and the second to ensure the tracking error convergence.

This work has the following outline: Section 2 describes the dynamic system under study and provides the problem statement. Section 3 contains the justification of the ANN approximation property for the class of systems in this manuscript. In Section 4, the identification of the uncertain system based on ANN is presented. Section 5 is devoted to the adaptive control design to solve the trajectory tracking problem based on the ANN identification. Section 6 studies the performance of the suggested control scheme by means of numerical simulations. Finally, Section 7 closes the paper with some conclusions and observations for future work related with the presented approach.

Notation: The symbol R + = {p ∈ R : p ≥ 0}, with R as the set of real numbers and R n is the vector space of n real elements.

For

M ∈ R r×n , vec(M) = m ⊤ 1 , . . . , m ⊤ n ⊤ .
The Kronecker product is described by ⊗. The identity matrix of dimension n is depicted by I n . The Euclidean norm is denoted as • and for the matrix space is considered the Frobenius norm

M F = tr{M ⊤ M}, where tr{H} = n ∑ i=1 H ii is the trace of a matrix H ∈ R n×n , H ii is the element of the i-th column and the i-th row, diag {a i } i=1,...,m is a diagonal matrix of dimension m, a i ∈ R is the i -th element of the diagonal, [p i ] i=1,...,m is a matrix with m columns p i ∈ R n .

PROBLEM STATEMENT

The class of systems with uncertain model considered in this work, is described by the following ordinary differential equation:

ż(t) = f 0 (z(t)) + m ∑ i=1 f i (z(t))u i , z(0) = z 0 , t ∈ R + , (1)
where z ∈ R n is the states vector, u i ∈ R are the control inputs, m is the number of inputs, m ≤ n, z 0 ∈ R n is the initial condition and f i : R n → R n , i = 0, ..., m are unknown vector fields.

The following assumptions are considered in the design of the adaptive control. Assumption 1. The unknown vector fields are homogeneous of degree k i ∈ R, i.e.

f i (εz) = ε k i f i (z), ∀ε > 0, ∀z ∈ R n , i = 0, . . . , m.
Assumption 2. The vector fields are continuous in the unit sphere:

S = {z ∈ R n : z = 1} . ( 2
)
Assumption 3. All the states are on-line measured and bounded.

The main goal of the control design is to obtain a control input, such that, the trajectory tracking error

e = z -z * , (3) is bounded, i.e. lim sup t→∞ e(t) ≤ σ(ψ + ) ≤ +∞, (4)
where z * ∈ R n is the vector of the reference trajectories, ψ + defines the approximation quality of the unknown vector fields f i , i = 0, . . . , m, and the function σ : R + → R + is a class-K function (See [START_REF] Khalil | Nonlinear systems[END_REF]). This goal considers the universal approximation property of ANN structures [START_REF] Hornik | Approximation capabilities of multilayer feedforward networks[END_REF].

The following assumption is considered for the references trajectories.

Assumption 4. The dynamics of the reference trajectories are given by:

d dt z * = ϕ(z * ,t).
(5) The vector field ϕ : R n × R + → R n is bounded and continuous.

The design of ϕ(z * ,t) can be realized using diverse trajectory planning methods.

NEURAL NETWORK APPROXIMATION OF HOMOGENEOUS FUNCTIONS

The approximation property of ANN is based in the weighted superposition of nonlinear functions such as polynomials, radial basis functions and sigmoidal functions [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF][START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF][START_REF] Funahashi | On the approximate realization of continuous mappings by neural networks[END_REF].

The following Theorem guarantees the approximate realization of continuous mappings using bounded and monotone increasing differentiable functions. Theorem 1. [START_REF] Funahashi | On the approximate realization of continuous mappings by neural networks[END_REF] Let φ(z) be a constant, bounded and monotone increasing continuous function. Let K be a compact subset (bounded closed subset) of R n and g(z 1 , . . . , z n ) be a real valued continuous function on K. Then, for an arbitrary ψ ∈ R + , there exists and integer N and real constants c i , θ i and ω i j , i = 1, . . . , N and j = 1, . . . , n, such that,

g(z 1 , . . . , z n ) = N ∑ i=1 c i φ n ∑ j=1 ω i j z j -θ i , (6)
satisfies max

z∈K |g(z 1 , . . . , z n ) -g(z 1 , . . . , z n )| < ψ.
Notice that Theorem 1 is formulated for multi-layer ANN structures with a static structure. This property has been used for the approximation of dynamic systems. Remark 1. In this work, the ANN structures are designed linear in the parameters [START_REF] Lewis | Neural network control of robot manipulators and non-linear systems[END_REF]. In [START_REF] Igelnik | Stochastic choice of basis functions in adaptive function approximation and the functional-link net[END_REF] and [START_REF] Lewis | Neural network control of robot manipulators and non-linear systems[END_REF], it is stated that the parameters w i j , and θ i in (6) can be selected randomly using a uniform distribution and the universal approximation property holds by finding only the parameters c i .

Theorem 1 states the approximation property of ANN for the estimation of continuous functions on a compact, the next Corollary extends the approximation property for the systems represented by (1). Then, the proposed ANN structure (8) provides a global approximation of the vector fields f i . Corollary 2. [START_REF] Ballesteros | Differential neural network identification for homogeneous dynamical systems[END_REF] Consider that (1) satisfies the assumptions 1-3. Then, for any ψ i ∈ R + and for any Hurwitz matrix A ∈ R n×n , there exist

N i ∈ R and W * i ∈ R n×N i , i = 0, 1, .., m such that: d(z, u) ≤ ψ 0 z k 0 + m ∑ i=1 ψ i z k i |u i |, ∀z ∈ R n , ∀u ∈ R m , ∀t ∈ R + , (7)
where d(z, u) := ż -F(z, u) and the approximate function F(z, u) is given by:

F(z, u) = z k 0 A z z + W * 0 φ 0 z z + m ∑ i=1 z k i W * i φ i z z u i . (8)
The elements of the vector functions φ i : R n → R N i , i = 0, . . . , m, are proposed as sigmoidal activation functions, that is:

(φ i (z)) j = 1 + e -(β ⊤ i j z+α i j ) -1 , (9)
where α i j ∈ R + and β i j ∈ R n are properly selected parameters with i = 0, . . . , m and j = 1, . . . , N i .

The proof of Corollary 2 was presented in [START_REF] Ballesteros | Differential neural network identification for homogeneous dynamical systems[END_REF]. In view of Remark 1, in the design of the identifier, the parameters α i j and β i j , are selected randomly with a uniform distribution. Therefore, the ANN structures are linear in the parameters and, in consequence, the identification is solved by finding the parameters W * i .

NEURAL NETWORK REPRESENTATION OF HOMOGENEOUS DYNAMIC SYSTEMS

The solution for the identification of (1) considers the representation (8). Notice that the ANN structures in (8) can be represented as follows

W * i φ i z z = Φ i z z w * i , (10)
where

Φ i z z = I n ⊗ φ ⊤ i z z
and w * i = vec (W * i ) ⊤ . Therefore, the identification section consists on finding the parameters w * i by some adaptive law, such that, the identification error ∆ = zẑ converges to zero or is small enough (depending on the quality of the ANN approximation), where ẑ ∈ R n represents the state vector of the following identifier:

d dt ẑ = z k 0 A ẑ z + m ∑ i=0 z k i Φ i z z w i u i + z k 0 Ω i K i Ω ⊤ i ∆ , (11)
where i = 0, 1 . . . , m, u 0 = 1, A ∈ R n×n is a Hurwitz matrix, and with the adaptive laws given by

d dt w i = -z k 0 K i Ω ⊤ i ∆, (12)
where K i ∈ R nN i ×nN i are positive definite matrices and the auxiliary variables

Ω i : R + → R n×nN i satisfy d dt Ω i = z k 0 -1 AΩ i -z k i u i Φ i z z . (13)
The following Theorem sums up the result on the convergence of the identification error. Theorem 3. [START_REF] Ballesteros | Differential neural network identification for homogeneous dynamical systems[END_REF] Let assumptions 1-3 be satisfied and the control input u be designed to be a uniformly bounded function, i.e. |u i (t)| ≤ U, U > 0, ∀t ∈ R + . Consider that (1) can be represented in the form ( 8) with an estimation error given in ( 7) and consider the identifier (11) with the adjustment laws ( 12) and the auxiliary variable adjusted by ( 13). If K i ∈ R nN i ×nN i , i = 0, . . . , m, are positive definite matrices and the control inputs u i in (1) are such that ∃z -> 0, ∃z + > 0, z -< z(t) < z + < +∞ ∀t ∈ R + , and the following persistent exitation (PE) condition holds for all t ∈ R + and some ϑ W > 0 and ℓ W > 0:

t+ℓ W t G ⊤ (s)G(s)ds ≥ ϑ W I n ∑ m i=0 N i , (14)
where the matrix

G ∈ R n×n ∑ m i=0 N i is given by G = [Ω 0 , Ω 1 , . . . , Ω m ].
Then, there exist two class-K functions σ 1 and σ 2 such that:

lim sup t→∞ ∆(t) ≤ σ 1 ψ + , ( 15
) lim sup t→∞ w i (t) -w * i ≤ σ 2 ε + , (16)
where ψ + = max i=0,...,m {ψ i } and ψ i are given by (7).

CONTROL DESIGN

The main goal of the input design for u = [u 1 , . . . , u m ] ⊤ is to guarantee that the tracking error is small enough (4) and bounded. The input design considers the differential ANN identifier, therefore, the following assumption is needed. Assumption 5. The matrix Ξ ∈ R n×m , defined by

Ξ = Φ i z z w i i=1,...,m
, is full-rank by columns (m) uniformly with respect to time. Remark 2. Assumption 5 implies a sufficient condition to justify the controllability of the ANN identifier. This is a reasonable assumption if the closed-loop controller is developed based on the ANN approximation. The demonstration of such fact implies the introduction of a penalty function associated to the controlled Lyapunov function which led to the learning laws in ( 12) and ( 13) [START_REF] Ballesteros | Differential neural network identification for homogeneous dynamical systems[END_REF]. For more details on how introduce weights restrictions in the identifier structure, please see [START_REF] Escudero | Adaptive linearization for nonlinear systems using continuous Neural Networks[END_REF].

The control solution to solve the tracking control by means of the ANN identification has two main elements, a compensating element u 1 and a second structure to guarantee the tracking convergence u 2 based on the theory of discontinuous controllers.

u = Π -1 (t)D -1 (t) (u 1 + u 2 ) , (17)
where

Π(t) = diag z k i i=1,...,m , D(t) = H(t)Ξ(t). The matrix H : R + → R m×n satisfies Ḣ(t) = H(t)Λ, H(0) = [H 0 , 0 m,n-m ],
(18) where 0 m,n-m is the matrix with m rows and nm columns with zero entries, Λ ∈ R n×n , Λ = Λ ⊤ , Λ > 0 and H 0 ∈ R m×m is a diagonal positive definite matrix. The control elements are described as follows

u 1 = -z k 0 H(t) m ∑ i=1 Ω i K i Ω ⊤ i ∆ + Φ 0 z z w 0 -z -k 0 ϕ(z * ,t) + A ẑ z -Ḣ(t)δ, (19)
where δ ∈ R n , δ = ẑz * , and

u 2 = -λ s Sign(s), (20)
with λ s > 0. Sign(s) = [sign(s i )] ⊤ i=1,...,m , s i is the i-th element of the manifold defined by s = Hδ, (21) and the function sign(ν) :=

1 if ν > 0 [-1, 1] if ν = 0 -1 if ν < 0. ( 22
)
The following Theorem presents the result of the tracking error convergence with the proposed controller. Theorem 4. Consider that assumptions 1-4 and 5 are fulfilled and consider that (1) is supplied with the input (17) using the ANN identifier structure (11). Then, the tracking error ( 3) is ultimately bounded by (4).

Proof. The dynamic of ( 21) satisfies ṡ = Ḣ(t)δ + H(t) δ.

(23) Taking into account the definition of δ, (5) and (11), equation ( 23) is such that

ṡ = z k 0 H(t) A ẑ z + Φ 0 z z w 0 + m ∑ i=1 Ω i K i Ω ⊤ i ∆ + Ḣ(t)δ -H(t)ϕ(z * ,t) + D(t)Π(t)u(t). (24)
Therefore, the dynamic (24) in closed loop with ( 17) is ṡ = -λ s Sign(s).

(25) Consider the following Lyapunov function candidate

V = s 2 , the time derivative is V = -2λ s s ⊤ Sign(s). (26) Then, the following inequality is valid V ≤ -2λ s √ mV . (27)
This result confirms that s

(t) = 0, ∀t ≥ (λ s √ m) -1 s(0) .
Considering the dynamic of the manifold (23) and the condition ṡ = 0, the dynamics of δ is

H(t) δ = -Ḣ(t)δ.
(28) By the substitution of ( 18) on (28), and considering that H is row full-rank, then, δ = -Λδ.

(29) Therefore, the origin is an asymptotically exponentially stable equilibrium point for δ lim sup

t→∞ δ(t) = 0. ( 30
)
Consider the following representation of (3) e = zẑ + ẑz * .

(31) The norm of the tracking error, considering (31), satisfies the following inequality:

e ≤ ∆ + δ , ∀t ∈ R + .

(32) Finally, considering Theorem 3, the bound (4) holds, with σ = σ 1 . This completes the proof.

NUMERICAL RESULTS

In this section, an academic example is given to show the performance of the designed controller.

ż1 = z k 0 2 + (z 1 + z 2 ) k 1 , ż2 = (z 2 -z 1 ) k 0 + z k 2 1 u, ( 33
)
where k 0 = 0.2, k 1 = 0.1 and k 2 = 0.2 and the initial condition z(0) = [2 3] ⊤ . The numerical simulations for the controller were made in Simulink Matlab ® applying Dormand-Prince as integration method with a variable integration step.

The ANN was designed with a structure of three activation functions for each vector field.

A = 0 1 -7.2 -4.3 . ( 34
)
The initial conditions of the weights are:

w 0 (0) = [1, 2, 1, 1, 3, 2] ⊤ , w 1 (0) = [1, 3, 1, 5, 3, 2] ⊤ , w 2 (0) = [1, 4, 1, 2, 1, 6] ⊤ . (35)
The initial condition of the identifier was ẑ(0) = [7 5]. The gain matrices of the identifier were K 0 = I 2 ⊗ K0 , and

K 1 = K 2 = I 2 ⊗ K1 , with K0 = 1 0 0 12 , K1 = 12 -1 -1 5 . ( 36
)
The sliding surface was developed with a gain matrix H(t) having an initial condition of H(0) = [181]. The parameters for the control were selected as

Λ = 8 -1 -1 6 , and λ s = 2. ( 37
)
These parameter were obtained using a recurrent approximation method.

Figure 1 compares the trajectories evolution of the first state of the homogeneous system, the reference and the proposed DNN identifier. Notice that either the identification and the tracking trajectory problems are solved within the first second of simulation. This figure also shows the evolution of the first state controlled with a proportional-integral-derivative PID controller with proportional gain of 20, derivative gain of 5 and integral gain of 1. Such variable does not convergence to the actual reference trajectory, as the proposed controller does.

Figure 2 presents a similar comparison between the second state of the homogeneous system, the corresponding reference and the proposed DNN identifier. In this case, the identification and the tracking trajectory problems are solved within the first second of simulation. However, larger oscillations appear at the beginning of the simulation evaluation. Notice that this figure also shows the evolution of the second state controlled with the aforementioned PID controller. Once more, the state obtained with the PID controller does not attain the reference state. The comparison of PID and the proposed NN based controller demonstrates the benefits of introducing the DNN compensation. Even if the PID controller may seem too simple for the class of systems considered in this study, it offers a comparative reference to highlight the benefits of the suggested adaptive control structure.

The realization of the control action u 1 + u 2 appears in Figure 3. This evolution proves that tracking the states with high precision depends on increasing the control amplitude. This behavior is a consequence of the compensation based on the DNN identifier. Notice that the sliding surface implies the increment of the control amplitude. This control action exhibits the high frequency of the sliding mode realization (undesired chattering effect). This is still an opportunity area which could use an adaptive gain in the sliding control design. Figure 4 depicts the filtered version (with a first order low-pass linear filter and a time constant of 0.1 seconds) of the sliding part of the controller. The crossing of this oscillation on the zero value confirms the resolution of δ = 0. The filtered sliding variable evolution (Figure 4) confirms the movement around the origin of the tracking error. This filtered information defines the resolution of the sliding mode base controller.

CONCLUSIONS

The design of an adaptive controller for tracking trajectory of homogeneous uncertain systems, based on differential ANN was developed in this work. The convergence of the tracking error is analyzed by means of the Lyapunov theory. The result on the stability of the tracking error proved that the bound depends on the quality of the approximation. The numerical evaluation of the suggested controller was realized over a two-states system. The suggested mixed controller using the compensation as well as the sliding resolution provides the tracking of the proposed reference trajectories. As a future extension of this work, it is plotted to extend the approximation for the case of weighted homogeneity and to use this property in the controller design to obtain a defined kind of convergence.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Comparison between the target trajectory z *1 , the state of the identifier ẑ1 and the state z 1 of the system.

Fig. 5 .

 5 Fig. 3. Evolution of the control signal