Jes Ús Mendoza-Avila

Denis Efimov
email: denis.efimov@inria.fr

Jaime A Moreno
email: jmorenop@ii.unam.mx

Leonid Fridman
email: lfridman@unam.mx

Analysis of Singular Perturbations for a Class of Interconnected Homogeneous Systems: Input-to-State Stability Approach

Keywords: Stability of Nonlinear Systems, Singular Perturbations, Input-to-State Stability, Homogeneity, Lyapunov Methods

In this work an interconnection of two singularly perturbed homogeneous systems of different degrees is considered. Under relaxed restrictions on the smoothness of the right-hand sides of the system, and some standard assumptions, the conditions of local or practical asymptotic stability of the interconnection are established by means of ISS properties and the Small-Gain Theorem. Moreover, the domains of stability and attractions are estimated. Finally, the results are illustrated through an example with a homogeneous system of negative degree.

INTRODUCTION

Homogeneous systems [START_REF] Bacciotti | Lyapunov functions and stability in control theory[END_REF][START_REF] Zubov | Methods of A.M. Lyapunov and their application[END_REF] constitute a subclass of nonlinear dynamics and admit interesting properties, e.g., scalability of solutions, robustness, different rates of convergence, etc. A very important feature of homogeneous controllers with a negative degree is that they are not Lipschitz, possessing an infinite gain near to the origin and providing finite-time convergence [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF][START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF][START_REF] Cruz-Zavala | Homogeneous high order sliding mode design: a Lyapunov approach[END_REF] Commonly, control systems are affected by parasitic dynamics. Such interactions can be seen as the interconnection of systems with different scales of time, which are represented by singularly perturbed models. For smooth singularly perturbed systems, methods of stability analysis are based on Klimushchev-Krasvoskii Theorem [START_REF] Klimushchev | Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms[END_REF] where asymptotic stability of the interconnection is concluded from uniform exponential stability of the slow and fast dynamics. Relaxing the last assumption, [START_REF] Saberi | Quadratic-type Lyapunov functions for singularly perturbed systems[END_REF] addresses asymptotic and exponential stability by means of quadratic-type Lyapunov functions (LF) and estimates the domain of attraction with the upper bound of the perturbation parameter. Moreover, the problems of controller design for singularly perturbed systems have been subject of a wide interest This work was supported in part by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898; by CONACyT (Consejo Nacional de Ciencia y Tecnología) project 241171, 282013 and CVU 711867; by PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) IN 115419 and IN 113617. (see the list of references in the book by [START_REF] Kokotovic | Singular perturbation methods in control: analysis and design[END_REF]). It is clear that these methods cannot be used for analysis of homogeneous systems in general case.

On the other hand, for systems affected by external inputs (e.g., exogenous disturbances, measurement noise, etc.), the concept of Input-to-State Stability (ISS) was introduced by [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF]. The connexion between Lyapunov stability and ISS has permitted the development of many stability concepts very useful for the analysis and design of nonlinear control systems (see the list of references in [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]). For instance, the so-called Small-Gain Theorem provides a sufficient condition to warranty the stability of interconnected systems [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF][START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF][START_REF] Dashkovskiy | Input-to-state stability of interconnected hybrid systems[END_REF]. In this context, [START_REF] Christofides | Singular perturbations and input-to-state stability[END_REF] studies ISS properties of smooth singularly perturbed systems providing a kind of "total stability" under standard assumptions.

To the best of our knowledge, existent works about stability analysis of singularly perturbed systems assume enough smoothness of the vector fields, which is a strong hypothesis for homogeneous systems of negative degree. The aim of this paper is to analyze the effect of singular perturbations on the stability of the interconnection of homogeneous systems by means of ISS properties and the Small-Gain Theorem. Unlike previous works, our analysis permits to relax smoothness requirements till assuming only continuity and at most continuous differentiability. Moreover, estimations of the region of attraction and ultimate bounds for the systems trajectories are provided.

The rest of the paper has the following structure. In Section 2 some useful definitions and preparatory results are presented.

The problem statements and the main results of the paper with the stability proof are given in Section 3 and 4, respectively. In Section 5, the obtained results are illustrated by an example, where a homogeneous system with a negative degree is considered. Finally, the conclusions are presented in Section 6.

Notation

• N and R are the sets of natural and real numbers, respectively. Moreover, R + represents the set of non-negative real numbers, i.e., R + = {x ∈ R : x ≥ 0}.

u : [0, ∞) → R m such that u L∞ = sup t≥0 |u(t)| < ∞.

PRELIMINARIES

Consider the system

ẋ = f (x, u), (1)
where x ∈ R n is the state vector and u ∈ R m is an input. In addition, f : R n+m → R n ensures forward existence and uniqueness of the system solutions at least locally in time and f (0, 0) = 0.

Weighted homogeneity

The presentation of this subsection follows by [START_REF] Zubov | Methods of A.M. Lyapunov and their application[END_REF][START_REF] Bacciotti | Lyapunov functions and stability in control theory[END_REF][START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. For real numbers r i > 0 (i = 1, . . . , n) called weights and λ > 0, one can define

• the vector of weights r = (r 1 , . . . , r n ) T , r max = max 1≤j≤n r j and r min = min 1≤j≤n r j ; • the dilation matrix function Λ r (λ) = diag(λ ri ) n i=1 , such that, for all x ∈ R n and for all λ > 0, Λ r (λ)x = (λ r1 x 1 , . . . , λ ri x i , . . . , λ rn x n )

T (along the paper the dilation matrix is represented by Λ r wherever λ can be omitted);

• the r-homogeneous norm of x ∈ R n is given by ||x|| r = n i=1 |x i | ρ r i 1 ρ
for ρ ≥ r max (it is not a norm in the usual sense, since it does not satisfied the triangle inequality);

• for s > 0, the sphere and the ball in the homogeneous norm are defined as S r (s) = {x ∈ R n : ||x|| r = s} and B r (s) = {x ∈ R n : ||x|| r ≤ s}, respectively. Definition 1. A function g : R n → R is r-homogeneous with a degree µ ∈ R if for all λ > 0 and all x ∈ R n : λ -µ g(Λ r (λ)x) = g(x).

A vector field f : R n → R n is r-homogeneous with a degree ν ≥ -r min if for all x ∈ R n and all λ > 0, f (Λ r (λ)x) = λ ν Λ r (λ)f (x). The system (1) (with u = 0) is r-homogeneous of degree ν if the vector filed f is r-homogeneous of degree ν.

Following [START_REF] Efimov | Homogeneous Lyapunov functions: From converse design to numerical implementation[END_REF], by its definition, || • || r is a rhomogeneous function of degree 1, and there exists σ, σ ∈ K ∞ such that

σ(||x|| r ) ≤ ||x|| ≤ σ(||x|| r ) ∀x ∈ R n , (2)
i.e., there is an equivalence between the norms || • || and || • || r . Moreover, for r max = 1 and ρ ≥ r max , || • || r is locally Lipschitz continuous.

Input-to-state stability

The next definitions and theorems were introduced by [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF][START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF][START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF]. Definition 2. The system (1) is said to be input-to-state practically stable (ISpS), if there exist a class KL function β, a class K function γ and a constant c ≥ 0, such that, for any u ∈ L ∞ and any

x 0 ∈ R n , the solution x(t) with initial condition x(0) = x 0 satisfies x(t) ≤ max{β( x 0 , t), γ( u(t) ), c} (3) for all t ≥ 0. The function γ is called nonlinear asymptotic gain. The system (1) is called ISS if c = 0. Definition 3. A smooth function V : R n → R is called the ISpS-LF for system (1) if there exist some c ≥ 0, α 1 , α 2 , α 3 ∈ K ∞ and χ ∈ K, such that, for all x ∈ R n and u ∈ R m α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), (4) and
x ≥ χ(max{ u , c}) =⇒ ∂V ∂x f (x, u) ≤ -α 3 ( x ). (5) hold. Moreover, if c = 0 then V is called a ISS-LF for system (1). Remark 4. The function γ(•) in (3) can be computed from the functions α 1 (•), α 2 (•) and χ(•). It is given by

γ(r) = α -1 1 • α 2 • χ(r).
(6) Theorem 5. The system (1) is ISS (ISpS) iff it admits an ISS (ISpS) LF.

Input-to-state stability of interconnected systems

Consider the system

ẋ = f (x, y), (7)
ẏ = g(x, y, u), (8) where x ∈ R n , y ∈ R m , u ∈ R p
and the origin x = y = u = 0 is an equilibrium point. The system ( 7)-( 8) can be seen as two interconnected subsystems where y is an input to the system (7) and x, u is an input to the system (8). Assume that both systems are ISpS w.r.t. their corresponding inputs. Therefore, from condition (3) we have

x(t) ≤ max{β 1 ( x 0 , t), γ 1 ( y(t) ), c 1 } y(t) ≤ max{β 2 ( y 0 , t), γ 2 ( x(t) ), γ 3 ( u(t) ), c 2 }
where x 0 ∈ R n and y 0 ∈ R m are the initial conditions for each variable, u ∈ L ∞ , β 1 , β 2 ∈ KL and γ 1 , γ 2 , γ 3 ∈ K are some functions from the indicated classes.

Sufficient conditions for ISpS stability of the interconnected system ( 7)-( 8) can be found in [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] as follows. Theorem 6. Let each subsystem ( 7) and (8) be ISpS. If there exists some c 0 ≥ 0 such that γ 1 (γ 2 (r)) < r, for all r > c 0 .

(9) then the interconnected system ( 7)-( 8) is ISpS. Furthermore, if c 0 = c 1 = c 2 = 0 then the system is ISS.

The inequality ( 9) is commonly refered as the small-gain condition. In particular, if the nominal system ( 7)-( 8) is ISS then its solutions with u = 0 are globally asymptotically stable.

Roughly speaking, the Small-Gain Theorem establishes that the interconnected system ( 7)-( 8) is ISS, if the composite function

γ 1 • γ 2 (•) is a simple contraction. 3. PROBLEM STATEMENT Consider the interconnected system ẋ = f (x, y), (10)
ẏ = g(x, y) = A(x)y + R(x), (11)
where x ∈ R n and y ∈ R m are the state variables, and > 0 is a small parameter, f ∈ C 0 , A ∈ C 1 with det(A(x)) = 0 for all x ∈ R n and R ∈ C 1 with R(0) = 0 and R(x) = 0 for all x ∈ R n \ {0}. Moreover, the system (10) is r-homogeneous with a degree ν, while the system (11) is r-homogeneous with a degree µ for the corresponding vector of weights r and r. Under the introduced restrictions on A and R, the equation g(x, h(x)) = 0 admits a solution h(x) = -A -1 (x)R(x), which is locally Lipschitz continuous.

If we consider a small parameter ≈ 0 then the trajectories y of the system (11) remain in a neighborhood of the solution ȳ = h(x) as it is predicted by the Tikhonov's theorem [START_REF] Vasil'eva | The Boundary Function Method for Singular Perturbed Problems[END_REF]. However, in most of the existence results smoothness (at least continuous differentiability) of the vector fields of the system (10)-( 11) is required.

In following section, two different problems will be addressed:

• Under what conditions the stability of the interconnected system ( 10)-( 11) can be ensured. • Moreover, is it possible to estimate the region of convergence and domain of attraction for the system solutions using their homogeneity?

MAIN RESULTS

The next theorem presents the main result of this paper and it establishes the conditions ensuring stability of the interconnected system (10)-( 11). First, let us introduce the hypotheses under which we propose a solution. Assumption 7. For the system (10)-(11):

A. The system ẋ = f (x, h(x)), (12) is GAS at the origin. B. The system ẏ = A(x)y (13) is GAS at the origin, uniformly w.r.t. x, and there exists

P = P > 0 and Q = Q > 0 such that A (x)P + P A(x) ≤ -Q (14) for all x ∈ R n .
Assumption 7 is standard for singular perturbation analysis [START_REF] Vasil'eva | The Boundary Function Method for Singular Perturbed Problems[END_REF][START_REF] Kokotovic | Singular perturbation methods in control: analysis and design[END_REF]. Nevertheless, in the most of existing results, a sufficient smoothness of the vector fields involved in the analysis is also required. However, as it will be shown below in our main result, such a restrictive requirement can be relaxed for the class of r-homogeneous. Theorem 8. Let the subsystems ( 10) and (11) be r-homogeneous with a degree ν and r-homogeneous with a degree µ, respectively. Moreover, consider functions f ∈ C 0 and A, R ∈ C 1 . If Assumption 7 is satisfied then there is 0 > 0 such that for all ∈ (0, 0 ] the interconnected system (10)-( 11) is

• globally asymptotically stable for µ = ν,

• locally asymptotically stable, for µ < ν,

• globally asymptotically practically stable, for µ > ν.

Proof. Following the ideas of the singular perturbation theory (see [START_REF] Vasil'eva | The Boundary Function Method for Singular Perturbed Problems[END_REF]; [START_REF] Kokotovic | Singular perturbation methods in control: analysis and design[END_REF]), let us define an "error" variable z = y -h(x), such that, the system (10)-( 11) in the new coordinates is given by

ẋ = f (x, z + h(x)), (15)
ż = 1 A(x)z -∂h(x) ∂x f (x, z + h(x)).
(16) The main advantage of this representation is that the system (15)-( 16) has an equilibrium point at the origin instead of the system (10)-( 11) where the trajectories y tend to h(x). Hence, by the homogeneity of the system (10)-( 11), the following expressions hold

f (Λ r x, Λ r y) = λ ν Λ r f (x, y), A(Λ r x)Λ r y + R(Λ r x) = λ µ Λ r A(x)y + λ µ Λ r R(x), h(Λ r x) = Λ r h(x).
First, let's analyze the stability of the subsystem (15). Since the system (12) is GAS and rhomogeneous of degree ν for z = 0, there exists a LF V (x) satisfying

V (Λ r x) = λ κ V (x), (17) a x x κ r ≤ V (x) ≤ āx x κ r , (18)
∂V (x)

∂x f (x, h(x)) ≤ -b x x ν+κ r , (19)
sup x r ≤1 ∂V (x) ∂x ≤ c x , (20)
for all x ∈ R n and for some a x , a x , b x , c x > 0. Using V (x) as an ISS-LF candidate for the system (15), its derivative is given by

V (x) = ∂V (x) ∂x f (x, z + h(x)) = ∂V (x) ∂x f (x, h(x)) + ∂V (x) ∂x (f (x, z + h(x)) -f (x, h(x))) , ≤ -bx x ν+κ r +cx x ν+κ r f (ξ, h(ξ) + Λ -1 r ( x r )z) -f (ξ, h(ξ)) ,
where the dilations Λ r ( x r ) and Λ r ( x r ) have been applied and x = Λ r ( x r )ξ for some ξ ∈ S r (1) . By the continuity on the unit sphere of f (x, z + h(x)), we have that for any b x , c x and 0 < θ < 1 there exist δ such that if

Λ -1 r ( x r )z r ≤ δ then f (ξ, h(ξ) + Λ -1 r ( x r )z) -f (ξ, h(ξ)) ≤ θb x c x ,
for all ξ r = 1. Therefore,

V (x) ≤ -b x x ν+κ r + θb x x ν+κ r , ≤ -(1 -θ)b x x ν+κ r , if x r ≥ δ -1 z r ,
where the properties 0 < θ < 1 and Λ -1 r ( x r )z r = x -1 r z r were used. Accordingly, these properties imply that the system (15) is ISS w.r.t. an input z. Moreover, from Definition 2, the solution x(t) of the system (15) is bounded by

x(t) r ≤ max{β 1 ( x 0 r , t), γ 1 ( sup τ ∈[0,t) z(τ ) r )} (21)
for all t ≥ 0, where β 1 is a KL function and, considering equations ( 6) and ( 18), γ 1 is a K function given by

γ 1 (s) = δ -1 āx a x s. (22)
Now, let's study the stability of the systems ( 16). Taking into account Assumption 7.B., there exists a LF W (z) = z P z, where P is a solution of equation ( 14), satisfying

W (Λ r z) = λ 2 W (z) (23) a z z 2 r ≤ W (z) ≤ āz z 2 r , (24)
∂W (z)

∂y A(x)z ≤ -b z z µ+2 r , ( 25
)
sup ζ r ≤1 ∂W (ζ) ∂ζ ≤ c z , (26)
for all z ∈ R m and for some a z , a z , b z , c z > 0. The function W (z) can be used as an ISpS-LF candidate for the system ( 16), such that

Ẇ (z) = 1 ∂W (z) ∂z A(x)z -∂W (z) ∂z ∂h(x) ∂x f (x, z + h(x)), ≤ bz z µ+2 r + λ 2+ν ∂W (ζ) ∂ζ ∂h(ξ) ∂ξ f (ξ, ζ + h(ξ)) .
where λ = max{ x r , z r } and the dilations Λ r (λ -1 ) and Λ r (λ -1 ) were applied, such that, ξ r = x r λ ≤ 1 and ζ r = z r λ ≤ 1. By arguments of continuity of functions h(x) and f (x, z + h(x)) on the unit ball,

sup ξ r ≤1, ζ r ≤1 ∂h(ξ) ∂ξ f (ξ, ζ + h(ξ)) ≤ b z η c z . ( 27
)
for some constant η. Thus,

Ẇ (z) ≤ -bz z µ+2 r + b z ηλ 2+ν ≤ -bz z µ+2 r + b z η x 2+ν r + z 2+ν r ≤ -(1 -θ) bz z µ+2 r , if z 2+µ r ≥ max{ η θ1 x 2+ν r , η θ2 z 2+ν r } ( 28
)
where θ1 + θ2 = θ and 0 < θ < 1. Thus, Ẇ (z) < 0, if

z r > η θ1 1 µ+2 x ν+2 µ+2 r , ( 29
)
or, on the other hand,

z µ-ν r > η θ2 . ( 30
)
where three different behaviors of system ( 16) can be observed:

• If µ = ν, the system ( 16) is ISS w.r.t. x for a sufficiently small . • If µ < ν, the system ( 16) is locally ISS w.r.t. x.

• If µ > ν, the system ( 16) is ISpS w.r.t. x.

Combining all these cases, from Definition 2 the trajectories of the system (15) are bounded by z(t) r ≤ max{β 2 ( z 0 r , t), γ 2 ( sup τ ∈[0,t)

x(τ ) r ), ρ} for all t ≥ 0, where β 2 is a KL function, ρ is a constant which can be estimated from (30), i.e., ρ = 0 for µ ≤ ν, and

ρ = āz a z η θ 1 µ-ν
for µ > ν, and considering equations ( 6), ( 29) and ( 24), γ 2 is a class K function given by

γ 2 (s) = āz a z η θ 1 µ+2 s ν+2 µ+2 .
(31)

Finally, the internal stability of the interconnection ( 15)-( 16) is investigated by using the Small-Gain Theorem. Likely, from equations ( 9), ( 22) and ( 31) we have

γ 1 (γ 2 (s) = δ -1 āxāz a x a z η θ 1 µ+2 s ν+2 µ+2 .
According to the small-gain condition (9), the stability of the interconnected system ( 15)-( 16) is insured for

δ -1 āxāz a x a z η θ 1 µ+2 < s µ-ν µ+2 .
Thus, depending on the homogeneity degrees of systems ( 15) and ( 16), we have three different cases:

• For µ = ν, the system ( 15)-( 16) is globally asymptotically stable if

< θ η δ a x a z āxāz µ+2 . (32) .
• For µ < ν, the system (15)-( 16) is locally asymptotically stable if

x r < δ -1 āxāz a x a z µ+2 µ-ν η θ 1 µ-ν , (33)
• For µ > ν, the system (15)-( 16) is globally asymptotically practically stable if

x r > δ -1 āxāz a x a z µ+2 µ-ν η θ 1 µ-ν , (34)
Since the changes of variables z = y-h(x) is a difeomorphism, the results (32),( 33) and ( 34) are preserved in the coordinates x, y for the system (10)-( 11) and this concludes the proof of Theorem 8.

In addition, the following corollary provides the estimations of the domains of attraction and the ultimate bounds for the trajectories of the system (10)-( 11) with initial condition x 0 = x(0) and y 0 = y(0). Corollary 9. Let the system (10)-( 11)satisfies all the requirements of Theorem 8.

• For µ = ν, lim t→∞ x(t) r = 0, lim t→∞ y(t) r = 0, for all x 0 ∈ R n and all y 0 ∈ R m . • For µ < ν, lim t→∞ x(t) r = 0, lim t→∞ y(t) r = 0, for all x 0 r < θ η a x a z δ āxāz µ+2 1 ν-µ , and y 0 -h(x 0 ) r < θ η a z āz µ+2 a x δ āx ν+2 1 ν-µ . • For µ > ν, lim t→∞ x(t) r ≤ η θ āxāz a x a z δ µ+2 1 µ-ν , lim t→∞ y(t) -h(x(t)) r ≤ η θ āz a z µ+2 āx a x δ ν+2 1 µ-ν for all x 0 ∈ R n and all y 0 ∈ R m .
Remember that by definition z = y -h(x) and h(0) = 0 hence the estimation for variable y(t) and its initial condition y 0 can be readily derived in the same way that (32), ( 33) and ( 34) in the proof of Theorem (8).

Note that for µ = ν the system (15)-( 16) always possesses some kind of stability (locally or practically), and by decreasing the value of it is possible to enlarge the domain of attraction for µ < ν or to decrease the size of the neighborhood of the origin that attracts all the solution for µ > ν.

EXAMPLE: A HOMOGENEOUS SYSTEM OF NEGATIVE DEGREE

In [START_REF] Cruz-Zavala | Strict Lyapunov functions for homogeneous finitetime second-order systems[END_REF], a second order systems

ẋ1 = x 2 , ẋ2 = -k 1 r(x) 2r 2 -r 1 r 1 . ( 35
) where r(x) = k - r 1 r 2 2 x 2 r 1
r 2 + x 1 , and k 1 , k 2 are gains, was introduced. The parameters r 1 , r 2 ∈ R + satisfy 2r 2 > r 1 > r 2 , such that, the systems ( 35) is (r 1 , r 2 )-homogeneous with a degree ν = r 2 -r 1 < 0. Moreover, a strict LF

V (x) = 1 d x 1 -r 2 r 1 x 2 + r 1 k 2 |x 1 | r 1 + ( 2 + 1 ) 1 2 |x 2 | 2 + r 1 2r 2 k 1 |x 1 | 2r 2 r 1 2r 2 , (36)
where 1 , 2 ∈ R + , and

d 2 = 2 2 r2 r1k1 2r2( -r2) -r 2 r 2 with = 1
1+ 2 , was provided. Proposition 10. [START_REF] Cruz-Zavala | Strict Lyapunov functions for homogeneous finitetime second-order systems[END_REF]). Under 2r 2 > r 1 > r 2 > 0, the origin of ( 35) is globally finite-time stable for all k 1 , k 2 ∈ R + . Moreover, for each ≥ r 1 + r 2 , there exists a small enough such that (36) is a strict LF for (35).

In order to illustrate the results presented in Section 4, let put a parasitic dynamic in the system (35), such that,

ẋ1 = x 2 , ẋ2 = -k 1 y 2r 2 -r 1 r 1 , (37)
ẏ = -a(x)y + r(x), (38) where

a(x) = sin( x1 ρ 0 r 1 + x2 ρ 0 r 2 x r ) + 2 > 0, for all x 1 , x 2 ∈ R,
where ρ 0 > r 1 . Furthermore, the system (38) is r 1 -homogeneous with a degree µ = 0. Note that the function • q describing system (37) is just continuous for any 0 < q < 1, i.e., it belongs to the class C 0 and a(x), r(x) ∈ C 1 . Moreover, 1 < a(x) < 3. For = 0, the system (35) is recovered with

k1 = k 1 a(x) 2r 2 -r 1 r 1 .
It can be readily seen that the stability of the system ( 35) is kept despite of the term a(x) since it is bounded and positive for any x ∈ R 2 . On the other hand, by construction the function ( 36) is positive definite (see (Cruz-Zavala et al., 2018, Lemma 7)) and (r 1 , r 2 )-homogeneous with a degree hence it is also radially unbounded (Bhat and Bernstein, 2005, Lemma 4.1). Therefore, we can assume that (36) satisfies conditions ( 18)-( 20) for certain a x , āx , b x , c x ∈ R + .

Assuming a small parameter > 0, we can define the variable z = y -r(x) a(x) , such that, the system (37)-( 38) in the new coordinates is given by

ẋ1 = x 2 , ẋ2 = -k1 r(x) + a(x)z 2r 2 -r 1 r 1 , ( 39
) ż = - a(x) z + ∂ ∂x r(x) a(x) x 2 -k1 r(x) + a(x)z 2r 2 -r 1 r 1 (40)
To show that the system (39) is ISS w.r.t. z, we can use the LF (36) such that its derivative along the trajectories of the system (39) satisfies

V (x) ≤ -b x x ν+ r + k 1 ∂V (x) ∂x × r(x) 2r 2 -r 1 r 1 -r(x) + a(x)z 2r 2 -r 1 r 1
, where the property k1 ≤ k 1 was used. Taking advantage of the homogeneity of the system (39) and the LF V (x), and using the dilations Λ r ( x -1 r ) and Λ r (

x -1 r ), it is obtained V (x) ≤ -b x x ν+ r + k 1 c x x ν+ r × r(ξ) 2r 2 -r 1 r 1 -r(ξ) + a(ξ)Λ r ( x -1 r )z 2r 2 -r 1 r 1
, where ξ ∈ S r (1). By continuity arguments and taking into account that ξ r = 1, we can suppose that for any k

1 , b x , c x ∈ R + and 0 < θ 1 < 1 there exists δ 1 ∈ R + such that r(ξ) 2r 2 -r 1 r 1 -r(ξ) + a(ξ)Λ r ( x -1 r )z 2r 2 -r 1 r 1 ≤ b x θ 1 c x k 1 , for all z r x r ≤ δ 1 . Therefore, V (x) ≤ -b x x ν+ r + θ 1 b x x ν+ r , ≤ -(1 -θ 1 )b x x ν+ r
, ∀ x r ≥ δ 1 z r , where 0 < θ 1 < 1, hence we can say that the system (39) is ISS w.r.t z. By Definition 2, we can say that x(t) r ≤ max{β( x 0 r , t), γ 1 ( z(t) r )} (41) for all t ≥ 0, where β is a KL function and, considering equation ( 6) and since LF (36) satisfies (18), we have

γ 1 ( z(t) r ) = āx a x δ 1 z(t) r . (42)
On the other hand, the input-to state stability of the system (40) can be analyzed by the Lyapunov function W

(z) = 1 2 z 2 , such that Ẇ (z) ≤ -a(x)bz z 2 r +|z| ∂ ∂x r(x) a(x)
x 2 -k1 a(x) r(x) + a(x)z 2r 2 -r 1 r 1

.

For λ = max{ x r , z r }, we define the dilations Λ r (λ -1 ) and Λ r (λ -1 ) such that where 0 < θ 2 < 1. Accordingly, it can be concluded that

Ẇ (z) ≤ -(1 -θ 2 ) bz z 2 r , if z r ≥ max{ η θ3 -1 ν , η θ4 1 2 x ν+2 2 r }
where θ 3 + θ 4 = θ 2 and by its definition ν = r 2 -r 1 < 0 and 1 < a(x) < 3 for all x ∈ R 2 , thus system (40) is ISpS w.r.t

x. Following Definition 2, considering equation ( 6) and the LF W (z), it can be said that z(t) r ≤ max β( z 0 r , t), η θ2

1 2 x ν+2 2 r , η θ2 -1 ν , (43
) for all t ≥ 0, where β is a KL function. Finally, applying the small-gain theorem, the composition between functions (42) and ( 43) is given by

γ 1 (γ 2 (|x|)) = āx a x δ 1 η θ2 1 2 x ν+2 2 r , (44)
such that the small-gain condition fulfills for

x r > āx a x δ 1 -2 ν η θ2 -1 ν ,
with a small enough > 0, hence we can conclude that the interconnected system (37)-( 38) is ISpS.

CONCLUSION

This paper was devoted to study the stability of a class of rhomogeneous interconnected systems affected by a singular perturbation. According to the homogeneity degrees of the involved subsystems, sufficient conditions to warranty the stability (local or practical) of the inter-connexion were established. Moreover, estimations of the regions of attraction and ultimate bounds for the system trajectories were provided. Finally, the mentioned properties were illustrated by an example with an r-homogeneous system with a negative degree.

.

  where ς r ≤ 1 and ζ r ≤ 1. By homogeneity and continuity arguments, we can assume that there is a constant η such thatsup ς r ≤1, ζ r ≤1 ≤ -a(x)bz z 2 r + b z ηλ ν+2 , ≤ -(1 -θ 2 ) bz z 2 r -

  • | • | denotes the absolute value in R, • denotes the Euclidean norm in R n . • Expressions of the form | • | γ sign(•), γ ∈ R are written as • γ . • A function α : R + → R + belongs to the class K if it is continuous, strictly increasing and α(0) = 0. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is unbounded. A continuous function β : R + × R + → R + belongs to the class KL if, for each fixed t ∈ R + , β(•, t) ∈ K and, for each fixed s ∈ R + , β(s, •) is non-increasing and it tends to zero for t → ∞. • The space L m ∞ is defined as the set of piecewise continuous, bounded functions


Junfeng Zhang

Denis Efimov
email: denis.efimov@inria.fr

A Lyapunov-Razumikhin Condition of ISS for Switched Time-Delay Systems Under Average Dwell Time Commutation

Keywords: Switched nonlinear time-delay systems, Exponential stability, Razumikhin approach

A condition of ISS is proposed for nonlinear time-delay systems based on the Lyapunov-Razumikhin theory, which allows the rate of convergence to be evaluated. Then, this condition is used for ISS analysis of switched nonlinear time-delay systems with average dwell time switching. Finally, one example is given to verify the effectiveness of theoretical findings.

INTRODUCTION

Time delays are frequently used in technical systems to model the features of the information transmission and actuator constraints. Appearance of a time delay is usually related with a degradation of the quality of the transients [START_REF] Dugard | Stability and control of time-delay systems[END_REF][START_REF] Gu | Stability and control of time-delay systems[END_REF][START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], while a large delay can lead to instability of the systems. There are two main frameworks to deal with stability analysis of time-delay systems using Lyapunov-Razumikhin functions [START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF][START_REF] Myshkis | Razumikhin's method in the qualitative theory of process with delay[END_REF][START_REF] Mao | Razumikhin-type theorems on exponential stability of stochastic functional differential equations[END_REF] and Lyapunov-Krasovskii functionals [START_REF] Dugard | Stability and control of time-delay systems[END_REF][START_REF] Gu | Stability and control of time-delay systems[END_REF][START_REF] Hale | Introduction to functional differential equations[END_REF]. The latter approach is necessary and sufficient [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], and it can be used to establish the stability conditions [START_REF] Moon | Delay-dependent robust stabilization of uncertain statedelayed systems[END_REF][START_REF] Wu | Delaydependent criteria for robust stability of time-varying delay systems[END_REF]; whereas the former method is typically applied for analysis of delayindependent stability [START_REF] Fridman | On reachable sets for linear systems with delay and bounded peak inputs[END_REF][START_REF] Jiao | Adaptive feedback control of nonlinear time-delay systems: The LaSalle-Razumikhinbased approach[END_REF]. It should be pointed out that usually the asymptotic stability is considered when using Lyapunov-Razumikhin functions. These observations leave a space for improvements of the Lyapunov-Razumikhin approach.

Switched systems are a widely distributed subclass of hybrid dynamics. There are plenty of analysis and design results obtained for such a kind of systems [START_REF] El-Farra | Output feedback control of switched nonlinear systems using multiple Lyapunov functions[END_REF][START_REF] Zhao | Quadratic stability of a class of switched nonlinear systems[END_REF][START_REF] Zhao | Adaptive tracking control for a class of uncertain switched nonlinear systems[END_REF][START_REF] Fainshil | On the stability of positive linear switched systems under arbitrary switching laws[END_REF][START_REF] Zhai | Switched adaptive fuzzy tracking control for a class of switched nonlinear systems under arbitrary switching[END_REF][START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF][START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF]. Almost these results are derived under (average) dwell-time condition imposed on the commutation logic. A prerequisite to use the dwell-time approaches is that the exponential stability of each subsystem should be achieved [START_REF] Efimov | On input-to-output stability of switched nonlinear systems[END_REF]. Such a constraint imposes additional obstructions for utilization of Lyapunov-Razumikhin approach for analysis of switched time-delay systems. In [START_REF] Yan | Stability analysis of switched time delay systems[END_REF], the delay-independent asymptotic stability of switched linear time-delay systems was addressed using Lyapunov-Razumikhin criterion integrated with minimum dwell time. In [START_REF] Ren | Krasovskii and Razumikhin stability theorems for stochastic switched nonlinear time-delay systems[END_REF], for stochastic systems an average dwell-time switching was studied via the Lyapunov-Krasovskii approach and only asymptotic stability under a dwell-time switching was analyzed via Lyapunov-Razumikhin functionals. The work [START_REF] Jiang | Stability of nonlinear switched systems with delays[END_REF] considered the asymptotic stability of switched time-delay systems, where an average dwell-time switching law was integrated with small-gain conditions. Furthermore, the input-to-state stability (ISS) under presence of unstable subsystems was discussed in [START_REF] Jiang | Stability analysis for switched systems with ISS and unstable time-delayed subsystems[END_REF] (the authors used Lyapunov functions and small-gain conditions for stability analysis, but it is not in the framework of Lyapunov-Razumikhin approach).

Following the existing literature, this paper first presents a Lyapunov-Razumikhin criterion on ISS of nonlinear time-delay systems. It is verified that the corresponding systems without input exhibit exponential stability (in terms of Lyapunov-Razumikhin functions). Using this result, ISS of switched nonlinear time-delay systems is explored. For the systems without input, the exponential stability is reached and an average dwell time switching rule is provided. The remainder of the paper is organized as follows. Section 2 introduces the preliminaries, Section 3 presents an extension of ISS Lyapunov-Razumikhin conditions, the main results are formulated in Section 4, one example is provided in Section 5, and Section 6 concludes the paper.

Notation: Let (or + ), n (or n + ), n×m be the sets of (nonnegative) real numbers, n-dimensional (nonnegative) vectors and n × m matrices, respectively. The Euclidean norm |x| and ∞-norm of a vector x = (x 1 , . . . , x n ) T ∈ n are defined as |x| = (

D + V (x)v = lim sup h→0 + V (x+hv)-V (x) h
for any x ∈ n and v ∈ n .

PRELIMINARIES

Consider a class of nonlinear time-delay systems [START_REF] Gu | Stability and control of time-delay systems[END_REF]:

dx(t) dt = f (x t , u(t)), t ≥ 0, (1)
where

x(t) ∈ n and x t ∈ C [-d,0] is the state function, x t (s) = x(t + s), -d ≤ s ≤ 0 and d > 0 is a finite delay; u(t) ∈ m is the external input, u ∈ L m ∞ ; f : C [-d,0] × m → n is a continuous function, f (0, 0) = 0,
and such that existence and uniqueness of solutions in forward time for the system (1) is ensured. Denote such a unique solution satisfying the initial condition x 0 ∈ C [-d,0] with the input u ∈ L m ∞ by x(t, x 0 , u), and x t (s, x 0 , u) = x(t + s, x 0 , u) for -d ≤ s ≤ 0, which is defined on some interval [-d, T ). Definition 1. [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] The system (1) is called practical ISS, if there exist q ≥ 0, β ∈ KL and γ ∈ K such that |x(t, x 0 , u)| ≤ max{β( x 0 , t), γ(||u|| ∞ ), q} ∀t ≥ 0 for all x 0 ∈ C [-d,0] and all u ∈ L m ∞ . If q = 0 then (1) is called ISS.

There exist two methods evaluating ISS property of the system (1) based on a Lyapunov-Razumikhin function or a Lyapunov-Krasovskii functional. The former approach can be formulated as follows: Theorem 2. [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF] If there exists a locally Lipschitz continuous Lyapunov-Razumikhin function

V : n → + such that (i) for some α 1 , α 2 ∈ K ∞ , and all x ∈ n : α 1 (|x|) ≤ V (x) ≤ α 2 (|x|); (ii) for some α, γ x , γ u ∈ K and r ≥ 0, with γ x (s) < s for all s > 0: max{γ x max θ∈[-d,0] V (ϕ(θ)) , γ u (|u|), r} < V (ϕ(0)) ⇒ D + V (ϕ(0)) f (ϕ, u) ≤ -α(|ϕ(0)|) for all ϕ ∈ C [-d,0
] and all u ∈ m , then the system (1) is practically ISS, and it is just ISS if r = 0.

In the following, we introduce the notion of average dwell time switching. Definition 3. [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF] For a rightcontinuous signal σ : + → + and t 1 ≥ t 0 ≥ 0, let N σ (t 0 , t 1 ) denote the number of discontinuities of σ in the interval (t 0 , t 1 ). If N σ (t 0 , t 1 ) ≤ N 0 + t 1 -t 0 τ holds with τ > 0 and N 0 ≥ 1, then τ is called the average dwell time of the signal σ. If N 0 = 1 then τ is called the dwell time of σ.

ISS WITH GIVEN DECADE RATE

The drawback of Theorem 2 is that the form of the functions β and γ, which appear in Definition 1, is not evaluated. In order to use the dwell-time stability conditions [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF][START_REF] Efimov | On input-to-output stability of switched nonlinear systems[END_REF], we need to establish the shape of the function β, and for this purpose the following extension of Theorem 2 is developed: Lemma 4. If there exists a locally Lipschitz continuous Lyapunov-Razumikhin function

V : n → + such that (i) for some α 1 , α 2 ∈ K ∞ and all x ∈ n : α 1 (|x|) ≤ V (x) ≤ α 2 (|x|); (ii) for some γ < 1, α ≥ -ln γ d , γ u ∈ K and r ≥ 0: max{γ max θ∈[-d,0] V (ϕ(θ)) , γ u (|u|), r} < V (ϕ(0)) ⇒ D + V (ϕ(0)) f (ϕ, u) ≤ -α V (ϕ(0)) for all ϕ ∈ C [-d,0
] and all u ∈ m , then the system (1) is practically ISS, and for all

x 0 ∈ C [-d,0] , all u ∈ L m ∞ and all t ≥ 0: |x(t, x 0 , u)| ≤ α -1 1 • max{exp ln γ d t α 2 ( x 0 ) , γ u (||u|| ∞ ), r}.
Proof. Since all conditions of Theorem 2 are satisfied with α(s) = α α 1 (s) and γ x (s) = γ s, then the system (1) is practically ISS, and we have only to demonstrate that the proposed upper bound on |x(t, x 0 , u)| is valid.

For any x 0 ∈ C [-d,0] and any u ∈ L m ∞ , the corresponding solution x(t, x 0 , u) is defined for all t ≥ 0. Then the time domain of the solution existence

+ = T 1 ∪ T 2 ∪ T 3 , where max{γ max θ∈[-d,0] V (x(t + θ)) , γ u (|u(t)|), r} ≥ V (x(t)) for all t ∈ T 1 ∪ T 2 , with γ max θ∈[-d,0] V (x(t + θ)) > max{γ u (|u(t)|), r}
for all t ∈ T 1 and otherwise for all t ∈ T 2 :

γ max θ∈[-d,0] V (x(t + θ)) ≤ max{γ u (|u(t)|), r}. Consequently, max{γ max θ∈[-d,0] V (x(t + θ)) , γ u (|u(t)|), r} < V (x(t))
for all t ∈ T 3 . Since x(t) is a continuous function of time, and the same property is satisfied for V (x(t)), the sets T 1 and T 3 are concatenated by disjoint open intervals of the time, i.e. T i = Ji j=1 (t j i,s , t j i,e ) where J i > 0 is (possibly infinite) quantity of these intervals in T i , i = 1, 3.

By construction V (x(t)) ≤ max{γ u (||u|| ∞ ), r} for all t ∈ T 2 .
Next, for any j = 1, . . . , J 1 and the corresponding interval (t j 1,s , t j 1,e ) ⊂ T 1 , for all t ∈ (t j 1,s , t j 1,e ) there exists

θ t = min{ϑ ∈ [-d, 0] : V (x t (ϑ)) = max θ∈[-d,0] V (x t (θ))},
where we introduce the minimum over ϑ ∈ [-d, 0] to resolve the non-uniqueness issue. Note that the inequality θ t ≤ -ε xt is satisfied for some ε xt ∈ (0, d] dependent on x t , since the maximum is calculated under the restriction that

γ max θ∈[-d,0] V (x(t + θ)) ≥ V (x(t)) with γ < 1. Thus, V (x(t)) < γ V (x t (θ t )) = exp (ln γ ) V (x t (θ t )) ≤ exp -ln γ θ t d V (x t (θ t )) .
Recursively applying this estimate, i.e.,

V (x t (θ t )) < exp -ln γ θ t+θt d V (x t+θt (θ t+θt )) ,
we obtain

V (x(t)) ≤ exp -ln γ θ t + θ t+θt d V (x t+θt (θ t+θt )) ,
and by induction,

V (x(t)) ≤ exp ln γ d (t -t j 1,s ) max θ∈[-d,0] V x t j 1,s (θ) (2)
for all t ∈ (t j 1,s , t j ,1e ) (i.e., for t sufficiently close to t j 1,s it could be t + θ t < t j 1,s and max θ∈[-d,0] V x t j 1,s

(θ) has to be used).

Finally, for any j = 1, . . . , J 3 and the corresponding interval (t j 3,s , t j 3,e ) ⊂ T 3 , for all t ∈ (t j 3,s , t j 3,e ), we have

D + V (x(t)) f (x t , u(t)) ≤ -α V (x(t))
by the conditions of the lemma and, consequently,

V (x(t)) ≤ exp -α (t -t j 3,s ) V (x(t j 3,s )).
Since α ≥ -ln γ d we obtain that the estimate (2) is satisfied for all t ∈ (t j i,s , t j ,i,e ) with i = 1, 3. By definition V (x(0)) ≤ max θ∈[-d,0] V (x 0 (θ)), then combining the estimates derived for T 1 , T 2 and T 3 we get that the relation

V (x(t)) ≤ max{exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) , γ u (||u|| ∞ ), r}, (3)
for any x 0 ∈ C [-d,0] and any u ∈ L m ∞ , is satisfied for t = 0, and let t > 0 be a time instant such that (3) is true for all t ∈ [0, t ) and at t = t it is violated for the first time, i.e.

V (x(t ))

> max{exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) , γ u (||u|| ∞ ), r}.
Obviously, exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) is a strictly decreasing function of time t ≥ 0, and always there is

T x0,u ≥ 0 such that exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) ≤ max{γ u (||u|| ∞ ), r} for all t ≥ T x0,u . Therefore, for t ∈ [0, T x0,u ) max{γ u (||u|| ∞ ), r} < exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) < V (x(t )) and V (x(t)) ≤ exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) for all t ∈ [max{0, t -d}, t ), then latter condition leads to V (x(t )) > γ max θ∈[-d,0] V (x(t + θ)). Hence, max{γ max θ∈[-d,0] V (x(t + θ)) , γ u (|u(t)|), r} < V (x(t ))
(4) and, consequently,

D + V (x(t )) f (x t , u(t )) ≤ -α V (x(t )) , (5)
which means that (3) cannot be violated due to α ≥ -ln γ d and the estimate (3) has to be also preserved at the instant

t . If t > T x0,u + d, then V (x(t)) ≤ max{γ u (||u|| ∞ ), r}, exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) ≤ max{γ u (||u|| ∞ ), r} for all t ∈ [t -d, t
), these facts imply that the relation (4) holds, then again ( 5) is valid and (3) cannot be violated at t . Finally, let

T x0,u ≤ t ≤ T x0,u + d, then exp ln γ d t max θ∈[-d,0] V (x 0 (θ)) < max{γ u (||u|| ∞ ), r} < V (x(t )) and V (x(t )) > γ max θ∈[-d,0] V (x(t + θ))
, which again leads to (4) and impossibility of violation of (3) at the instant t due to (5). The same arguments can be applied further for all t ≥ t , and the required upper estimate on the solutions of (1) follows.

Let α 1 (|x|) = a 1 |x| b and α 2 (|x|) = a 2 |x| b for some positive constants a 1 , a 2 , b in Lemma 4, then the exponential ISS of the system (1) is covered. The only additional restriction is imposed in Lemma 4 with respect to Theorem 2, which allows us to evaluate an exponential convergence rate, is α ≥ -ln γ τ (the chosen form of functions α(s) and γ x (s) is common in [START_REF] Gu | Stability and control of time-delay systems[END_REF]). Such a modification is not restrictive. Similar Lyapunov-Razumikhin conditions for exponential stability of nonlinear time-delay systems [START_REF] Myshkis | Razumikhin's method in the qualitative theory of process with delay[END_REF] and stochastic nonlinear timedelay systems [START_REF] Mao | Razumikhin-type theorems on exponential stability of stochastic functional differential equations[END_REF] were established, respectively. Remark 5. It is worth to highlight an important aspect of the Lyapunov-Razumikhin approach. Consider the formulation of Lemma 4. The rate of convergence, which is achievable in the system if the condition max{γ max

θ∈[-d,0] V (ϕ(θ)) , γ u (|u|), r} < V (ϕ(0))
is satisfied, is straightforward to derive under the Lyapunov-Razumikhin framework. It is given by the formulation of the method, since in such a case

D + V (ϕ(0)) f (ϕ, u) ≤ -α V (ϕ(0)
) . However, it is also necessary to substantiate that a similar rate of convergence is valid when this condition is not satisfied, i.e. when t ∈ T 1 ∪ T 2 , and max{γ max

θ∈[-d,0] V (ϕ(θ)) , γ u (|u|), r} ≥ V (ϕ(0)) .
This constitutes the core of the proof of Lemma 4.

For the system (1) with u(t) = 0 and r = 0, the exponential practical ISS presented in Lemma 4 is reduced to the exponential stability: Corollary 6. If there exists a locally Lipschitz continuous Lyapunov-Razumikhin function V : n → + such that (i) for some α 1 , α 2 ∈ K ∞ and all x ∈ n :

α 1 (|x|) ≤ V (x) ≤ α 2 (|x|);
(ii) for some γ > 1 and α ≥ ln γ d : max

θ∈[-d,0] V (ϕ(θ)) < γ V (ϕ(0)) ⇒ D + V (ϕ(0)) f (ϕ, 0) ≤ -α V (ϕ(0)) for all ϕ ∈ C [-d,0] , then for all x 0 ∈ C [-d,0
] and all t ≥ 0 the solutions of the system (1) with u(t) = 0 admit the estimate:

|x(t, x 0 , 0)| ≤ α -1 1 exp -ln γ d t α 2 ( x 0 ) .
In this work we will also be interested in the unstable case. We have to restrict ourselves by studying only forward complete systems (i.e., the systems as in (1) that for any x 0 ∈ C [-d,0] and any u ∈ L m ∞ admit solutions x(t, x 0 , u) defined for all t ≥ 0). A Razumikhin-type condition for forward completeness (1) is given below: Lemma 7. If there exists a locally Lipschitz continuous Lyapunov-Razumikhin function -d,0] and all u ∈ m , then solutions of the system (1) for all x 0 ∈ C [-d,0] , all u ∈ L m ∞ and all t ≥ 0 admit an upper estimate:

V : n → + such that (i) for some α 1 , α 2 ∈ K ∞ and all x ∈ n : α 1 (|x|) ≤ V (x) ≤ α 2 (|x|); (ii) for some α 00 > 0, γ u ∈ K, and r ≥ 0: max{ max θ∈[-d,0] V (ϕ(θ)), γ u (|u|), r} < V (ϕ(0)) ⇒ D + V (ϕ(0)) f (ϕ, u) ≤ α 00 V (ϕ(0)), for all ϕ ∈ C [
|x(t, x 0 , u)| ≤ α -1 1 •max{exp (α 00 t) α 2 ( x 0 ) , γ u (||u|| ∞ ), r}.
In Lemma 4, the Lyapunov-Razumikhin approach is presented to check ISS property for the system (1). Lemma 7 is a counterpart of Lemma 4, which provides an upper estimate on increasing rate of the states. Remark 8. Note that if the condition (ii) of Lemma 7 is replaced with max{ max

θ∈[-d,0] V (ϕ(θ)), γ u (|u|), r} > V (ϕ(0)) ⇒ D + V (ϕ(0)) f (ϕ, u) ≥ α 00 V (ϕ(0)), (6)
then we obtain a Razumikhin-type condition on instability of (1) with at least exponential growth of V (x(t)).

MAIN RESULTS

Consider the following switched time-delay systems:

dx(t)/dt = f σ(t) (x t , u(t)), t ≥ 0, (7)
where x(t) ∈ n and

x t ∈ C [-d,0] is the state, x t (s) = x(t+ s) for s ∈ [-d, 0] with a delay d > 0; x 0 ∈ C [-d,0] is the initial condition; u(t) ∈ m is the input, u ∈ L m ∞ ; σ(t) is a right-continuous switching signal taking values in a finite set S = {1, 2, . . . , M } for some integer M > 1; for each i ∈ S, f i : C [-d,0] × m → n is
a nonlinear function that guarantees the unique existence of solutions of (7) in forward time; f i (0, 0) = 0.

Our objective is to formulate Razumikhin-type conditions of ISS in (7) for a class of switching signals admitting an average dwell time property: Theorem 9. If there exist locally Lipschitz continuous Lyapunov-Razumikhin functions V i : n → + for all i ∈ S such that (i) for some α 1i , α 2i ∈ K ∞ and all x ∈ n :

α 1i (|x|) ≤ V i (x) ≤ α 2i (|x|); (ii) for some γ 1 ∈ K, γ 0 < 1, α 0 > -ln γ0 d , all ϕ ∈ C [-d,0] and all u ∈ m : max{γ 0 max θ∈[-d,0] V i (ϕ(θ)), γ 1 (|u|)} ≤ V i (ϕ(0)) ⇒ D + V i (ϕ(0))f i (ϕ, u) ≤ -α 0 V i (ϕ(0));
(iii) for some λ ≥ γ -1 0 , all j = i ∈ S and all x ∈ n : V i (x) ≤ λV j (x), (8) then the system ( 7) is exponentially ISS provided that the switching signal σ yields a dwell time bigger than d and the average dwell time τ > 1 -ln λ ln γ0 d.

Proof. Consider a time sequence 0

= t 0 ≤ t 1 ≤ • • • ≤ t m ≤ t < t m+1
, where t i , i = 1, . . . , m + 1 is the switching time instant, with σ(t i ) ∈ S being active in the interval [t i , t i+1 ). Since all conditions of Lemma 4 are satisfied for r = 0, then for all t ∈ [t m , t m+1 ):

V σ(tm) (x(t))
≤ max e ln γ 0

d (t-tm) max θ∈[-d,0] V σ(tm) (x(t m + θ)), γ 1 (||u|| ∞ ) .
Taking into account the relation ( 8) between V σ(tm) and V σ(tm-1) we obtain:

V σ(tm) (x(t)) ≤ max λe V σ(tm-1) (x(t m + θ)), γ 1 (||u|| ∞ ) .

In addition, max

s∈ [-d,0] V σ(tm) (x(t + s))

≤ max λe ln γ 0 d max{0,t-d-tm} × max θ∈[-d,0] V σ(tm-1) (x(t m + θ)), γ 1 (||u|| ∞ ) .
Iterative repeating these steps gives (9). By definition, m = N σ (t 0 , t) ≤ N 0 + t-t0 τ , note also that λe V σ(tm) (x(t)) ≤ max λe ≤ λ N0 e -ln γ0(N0-1) e -(t-ti) ≤ λ N0 e -ln γ0(N0-1) , hence,

ln γ 0 d (t-tm) max θ∈[-d,0] V σ(tm-1) (x(t m + θ)), γ 1 (||u|| ∞ ) ≤ max λ 3 e ln γ 0 d (t-tm+ m-1 i=m max{0,ti-d-ti-1}) max θ∈[-d,0] V σ(tm-3) (x(t m-2 + θ)), max{1, λe ln γ 0 d (t-tm) , λ 2 e ln γ 0 d (t-tm+max{0,tm-d-tm-1}) }γ 1 (||u|| ∞ ) ≤ • • • ≤ max λ m e ln γ 0 d (t-tm+ 2 i=m max{0,ti-d-ti-1}) max θ∈[-d,0] V σ(t0) (x(t 1 + θ)), max{1, λe
V σ(tm) (x(t)) ≤ λ N0 e -ln γ0(N0-1) × max e -(t-t0) max θ∈[-d,0] V σ(t0) (x(t 0 + θ)), γ 1 (||u|| ∞ ) . Noting that α 1 (|x|) ≤ V σ(tm) (x) and max θ∈[-d,0] V σ(t0) (x(t 0 + θ)) ≤ α 2 ( ϕ 0 ), where α 1 (s) = min 1≤i≤M {α 1i (s)} and α 2 (s) = max 1≤i≤M {α 2i (s)}, then |x(t)| ≤ α -1 1 λ N0 e -ln γ0(N0-1) × max e -(t-t0) α 2 ( x 0 ), γ 1 (||u|| ∞ ) ,
which implies the desired ISS property for (7). Remark 10. In [START_REF] Jiang | Stability of nonlinear switched systems with delays[END_REF][START_REF] Jiang | Stability analysis for switched systems with ISS and unstable time-delayed subsystems[END_REF], the asymptotic stability and ISS of switched time-delay systems were addressed by virtue of the average dwell-time switching method and some additional small-gain conditions. In Theorem 9, an additional exponential-kind estimate on the rate of convergence of solutions of the switched system is calculated.

Assume that u(t) = 0, then Theorem 9 reduces to the exponential stability of the system (7): Corollary 11. If there exist locally Lipschitz continuous Lyapunov-Razumikhin functions V i : n → + for all i ∈ S such that (i) for some α 1i , α 2i ∈ K ∞ and all x ∈ n : 8) holds for some λ ≥ γ 0 , all j = i ∈ S and all x ∈ n , then solutions of the system (7) with u(t) = 0 admit an estimate |x(t)| ≤ α -1 1 λ N0 e ln γ0(N0-1) e -(t-t0) α 2 ( x 0 ) provided that the switching signal σ yields a dwell time bigger than d and the average dwell time with τ > 1 + ln λ ln γ0 d. Remark 12. In Theorem 9 and Corollary 12, it is required that all subsystems are ISS and exponentially stable, respectively. In Lemma 7, an upper bound estimate of the state of the system is provided. Combining Theorem 9, Corollary 12, and Lemma 7, the ISS and exponentially stability of the system (7) with part stable subsystems and part unstable (or, bounded) subsystems can be obtained.

α 1i (|x|) ≤ V i (x) ≤ α 2i (|x|); (ii) for some γ 0 > 1, α 0 > ln γ0 d , all ϕ ∈ C [-d,0] : max θ∈[-d,0] V i (ϕ(θ)) ≤ γ 0 V i (ϕ(0)) ⇒ D + V i (ϕ(0))f i (ϕ, 0) ≤ -α 0 V i (ϕ(0)); (iii) (

ILLUSTRATIVE EXAMPLE

This section provides one example to illustrate the obtained results. Consider the system (7) with two stable subsystems: ẋ1 (t) = -3x 1 (t) + 0 -0.5

x 1 (t + s)ds, ẋ2 (t) = 2x 1 (t) -3x 2 (t) + 0 -0.5

x 2 (t + s)ds, and ẋ1 (t) = -4x 1 (t) + 0 -0.5

x 1 (t + s)ds, ẋ2 (t) = x 1 (t) -4x 2 (t) + 0 -0.5

x 2 (t + s)ds.

Choose V 1 (x(t)) = 1 2 (x 2 1 (t) + x 2 2 (t)) and V 2 (x(t)) = From the above derivation, the conditions of Theorem 9 hold. Figs. 1 and 2 show the simulations of the states of the subsystems without switching and the state of the system with average dwell time switching.

ln γ 0 dd

 0 (t-tm) = λ N (tm,t) e -d-ti-1}) , and if t i > t i-1 + d (the dwell time is bigger than d), then (t-tm) = λ N (tm,t) e ln γ 0 d (t-tm-(N (tm,t)-1)d) , . . . , λ m-1 e ln γ 0 d (t-tm+ 3 i=m max{0,ti-d-ti-1}) = λ N (t1,t) e ln γ 0 d (t-t1-(Nσ(t1,t)-1)d) .

  -d-ti-1}) }γ 1 (||u|| ∞ ) ≤ max λ m e -d-ti-1}) }γ 1 (||u|| ∞ ) .

  imposed on the dwell time there is > 0 such that ln γ0 d (1 -d τ ) + ln λ τ ≤ -, then for all i = 1, . . . , m: λ N (ti,t) e ln γ 0 d (t-ti-(N (ti,t)-1)d)

Fig. 1 .

 1 Fig. 1. The simulations of the states of the subsystems

  = max 1≤i≤n |x i |, respectively. If it is not stated explicitly, |x| represents one of the norms introduced before for x ∈ n , or an absolute value for x ∈ . For a continuous function φ : [a, b] → n , a, b ∈ , its uniform norm is defined as φ = sup t∈[a,b] |φ(t)| 2 ; the space of such functions we will denote as C[a,b] . For a Lebesgue measurable function of time u : + → m define the norm ||u|| ∞ = ess sup t≥0 |u(t)| 2 and the space of u with ||u|| ∞ < +∞ we further denote as L m ∞ . A continuous function α : + → + belongs to class K if it is increasing and α(0) = 0, and α ∈ K ∞ if it belongs to the class K and is also unbounded. A continuous function β : + × + → + belongs to KL class if β(•, x) is a K class function and β(x, •) is decreasing to zero for any fixed x ∈ + . For a locally Lipschitz continuous function V : n → + , its upper directional Dini derivative is defined as:

	|x| ∞	n i=1 x 2 i )	1 2 and

This work was supported by the National Nature Science Foundation of China (No. 61873314), 111 project No. D17019 (China), the Government of Russian Federation (Grant 08-08), the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and the Natural Science Foundation of Zhejiang Province, China (No. LY20F030008).


Quentin Voortman
email: q.j.t.voortman@tue.nl

Denis Efimov

Alexander Pogromsky

Jean-Pierre Richard

Henk Nijmeijer

Event-triggered Data-efficient Observers of Perturbed Systems

Keywords: Observers, State estimation, Limited data rate, Discrete-time Systems, Perturbation

In this paper, an event-triggered, data-rate constrained observer for discrete-time linear systems with perturbations is presented. The system is connected to a remote location by a communication channel which can only transmit limited numbers of bits per time interval. The system is subject to perturbations in its state as well as errors in the output measurement. The objective is to reconstruct estimates of the state at the remote location, by sending messages over the communication channel. A new type of data-rate constrained observer which can be more efficient in terms of communication rate is presented. Relation between an admissible communication rate and the system parameters is evaluated. The observer's efficiency is illustrated by simulations.

INTRODUCTION

Since wireless technologies were invented in 1880 by Alexander Graham and Charles Tainter, much progress has been done on increasing the maximum communication rate as well as reduce the required energy to transmit messages. In modern days, wireless technologies are omnipresent in industrial applications. Many of these applications also involve dynamical systems and as such, there is an entire subfield of the dynamics and control area that is dedicated to studying interactions between dynamical systems and communication channels. The problems studied in this subfield all include one or several dynamical systems, possibly with sensors and actuators at locations remote from one another and all connected together via one or several data-rate constrained communication channels with or without packet losses on the channels. A few examples of such systems are: cooperative driving of wirelessly connect cars, synchronization or formation control of drones, distributed sensing via networks of connected sensors, etc... The presence of limited data-rates and/or packet losses in conjunction with sources of uncertainty implies that it is necessary to develop specific solutions for those particular problems. The sources of uncertainty include: noise, perturbations, parametric uncertainty, and uncertainties in the initial conditions.

The earliest works involving dynamical systems and data-rate constraints can be found in [START_REF] Wong | Systems with finite communication bandwidth constraints -Part I: State estimation problems[END_REF]. After that, a lot of attention was given to linear systems (see e.g. [START_REF] Elia | Stabilization of linear systems with limited information[END_REF] and references therein). Overviews of the results for linear systems can be found in [START_REF] Nair | Feedback Control Under Data Rate Constraints: An Overview[END_REF], [START_REF] Baillieul | Control and Communication Challenges in Networked Real-Time Systems[END_REF] and [START_REF] Andrievsky | Control and Estimation under Information Constraints: Toward a Unified Theory of Control, Computation and Communications[END_REF].

This paper was elaborated in the UCoCoS project which has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675080.

The nonlinear case has been studied extensively as well. The earliest results were obtained in De Persis (2003) and [START_REF] Baillieul | Data-rate requirements for nonlinear feedback control[END_REF]. More general results were then obtained in [START_REF] Nair | Topological feedback entropy and nonlinear stabilization[END_REF] and [START_REF] Liberzon | Stabilization of nonlinear systems with limited information feedback[END_REF] which generalized linear techniques to nonlinear systems. After that, different notions of entropy were used to provide bounds on the sufficient and/or necessary data-rates to observe/control nonlinear dynamical systems over data-rate constrained channels (see [START_REF] Kawan | Invariance Entropy for Deterministic Control Systems An Introduction[END_REF], [START_REF] Matveev | Estimation and Control over Communication Networks[END_REF], [START_REF] Kawan | State estimation , entropy and Lyapunov exponents[END_REF], [START_REF] Sibai | Optimal Data Rate for State Estimation of Switched Nonlinear Systems[END_REF], [START_REF] Liberzon | Entropy and minimal data rates for state estimation and model detection[END_REF], [START_REF] Matveev | Observation of nonlinear systems via finite capacity channels: Constructive data rate limits[END_REF], [START_REF] Sibai | State Estimation of Dynamical Systems with Unknown Inputs: Entropy and Bit Rates[END_REF], [START_REF] Voortman | Data-Rate Constrained Observers of Nonlinear Systems[END_REF] and [START_REF] Matveev | Observation of nonlinear systems via finite capacity channels, Part II: Restoration entropy and its estimates[END_REF]).

The recent inclusion of event-triggered control in the dynamics and control field (see e.g. [START_REF] Heemels | An Introduction to Event-triggered and Self-triggered Control[END_REF] or [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]) has also impacted the subfield of systems and control with communication channels. In [START_REF] Han | Stochastic event-triggered sensor schedule for remote state estimation[END_REF], [START_REF] Shi | Event-based state estimation of linear dynamic systems with unknown exogenous inputs[END_REF], [START_REF] Huang | Event-triggered state estimation with an energy harvesting sensor[END_REF][START_REF] Trimpe | Event-based state estimation: An emulationbased approach[END_REF] the authors consider different variants of the problem of state estimation for systems with Gaussian state perturbations and Gaussian output measurement errors. In [START_REF] Xia | Networked state estimation over a shared communication medium[END_REF] and [START_REF] Muehlebach | Distributed eventbased state estimation for networked systems: An LMI approach[END_REF] extensions for networked state estimation are presented.

We focus on developing an event-triggered observer for dynamical systems with bounded state perturbations and bounded output measurement errors. This observer transmits estimates of the state to a remote location by sending messages over a data-rate constrained communication channel. It is designed in such a way that the time between two consecutive messages can be chosen. The main contribution of the paper is providing, in function of the chosen time interval between two consecutive communications, an upper bound on the resulting observation error as well as the required minimum data-rate to implement the observer. Moreover, since the observer functions on an event-triggered basis, it can be much more efficient in terms of required rate than the theoretical worst-case required data-rate, as is proven through simulations. This paper is structured as follows. In Section 2, we define the structure of the problem which we intend on solving. In Section 3, we provide a solution to the previously exposed problem in the form of a communication procedure. In Section 4, we prove two results. Firstly a proposition that gives an upper bound on the observation error, secondly a theorem which provides sufficient conditions on the minimum channel rate to implement the procedure of Section 3. Section 5 shows, through simulations, how this communication procedure can sometimes require bit-rates much lower than the theoretical maximum of the theorem in Section 4.

PROBLEM STATEMENT

We consider linear discrete-time systems of the following form

x[k + 1] = Ax[k] + d[k], y[k] = Cx[k] + w[k],
where

x[k] ∈ R n is the state, A ∈ R n×n , y[k] ∈ R m is the output, C ∈ R m×n , d[k] ∈ R n is
[k] verify d[k] 2 ≤ δ , w[k] 2 ≤ ω, ∀k ≥ 0,
where δ is the maximum state perturbation and ω is the maximum measurement error, both of which are known constants. In this paper, the notation • 2 refers to the Euclidean norm in R n . We use the notation σ i (M) to refer to the i-th singular value of the matrix M, where the singular values are ranked in non-increasing order (σ

1 (M) ≥ • • • ≥ σ n (M)).
The system is equipped with a smart sensor (a sensor admitting some computational capacities, which allows it to perform additional computations on the measured data) and it is connected to a remote location via a data-rate constrained communication channel which can only send messages that are of finite size. The objective is to provide estimates x[k] of x[k] at the remote location by sending messages over this communication channel. The sensor and the remote location have an initial estimate x[0] which verifies

x[0] -x[0] 2 ≤ ε 0 , (1)
where ε 0 is a user-specified parameter corresponding to the error of initial conditions. The constraint on the channel datarates is such that for any time interval between two consecutive transmission ∆k, the channel can only transmit a maximum number of bits b + (∆k).

In order to generate the estimates, messages m[k j ], where k j are the transmission times, are sent. Four ingredients interact with these messages: a sampler S , a coder C , an alphabet A , and a decoder D. The four devices together form a communication protocol. The following constants/parameters are known by all devices: the system matrices A and C, the maximum state perturbation δ , the maximum measurement error ω, the discretization error ε (which is induced by coding/decoding operation and for brevity of exposition in this note the constant ε 0 = ε), and the initial estimate x[0]. At the system side, the sampler S generates the instants of transmission in the following way

k j+1 = S (k j , y[0], y[1], . . . , y[k j ], m[k 1 ], . . . , m[k j ]), (2) k 0 = 0, m[0] = / 0.
The coder then generates the messages in the following way

m[k j ] = C ( x[0], y[0], . . . , y[k j ], m[k 1 ], . . . , m[k j-1 ]),
(3) ∀k j : j > 0. At each communication instant, the list of different possible messages is encoded into a finite-sized alphabet (the finite-sizedness being due to the data-rate constraints). The alphabet A determines the last index of the messages l j in the following way

l j = A ( x[0], m[k 1 ], . . . , m[k j-1 ]), ∀k j : j > 0.
(4) The restriction on the choice of messages is then m[k j ] ⊂ {1, . . . , l j }, ∀k j : j > 0. After encoding the messages into sequences of bits, the number of transmitted bits should not exceed the maximum number of bits that can be sent during the communication interval. This implies the following constraint on the alphabet length.

log 2 l j ≤ b + (k j+1k j ) ∀k j : j > 0.

(5) At the remote location, the decoder D receives the messages and interprets them to generate an estimate of the state x[k] in the following way

x

[k] = D( x[0], m[k 1 ], . . . , m[k j ]), ∀k ∈ {k j , . . . , k j+1 -1}, ∀ j ≥ 0.
Because of the perturbation, measurement error and finite data-rate, it is impossible to provide exact estimates at the remote location. Instead, the design of the communication protocol should ensure that the estimation error x[k]x[k] 2 is bounded and proportional to the different sources of perturbations/noise that affect the system. The upper bound on the estimation error is decomposed as follows

x[k] -x[k] 2 ≤ γ ε ε + γ δ δ + γ ω ω, ∀k ≥ 0, (6)
where γ ε , γ δ , γ ω ∈ R + are constants which indicate the contribution of the different errors/perturbation in the total error. The first objective of the paper is to design a data-rate constrained observer and communication scheme which on average performs better in terms of transmitted data if the perturbations are not the worst-case perturbations every time. The second objective of the present paper is to investigate the relationship between the time interval between subsequent communications ∆k j := k j+1k j , the maximum number of bits per time interval b + (•), and the error constants γ ε , γ δ , γ ω for the proposed communication scheme.

DESIGNING THE AGENTS

In this section, we introduce the different agents of the communication protocol. The main mechanism can be described as follows: at the sensor side, a local observer transforms the output into estimates of the state x [k]. A copy of the decoder is also simulated by the computational capacity of the sensor such that it is known what estimate x[k] the decoder currently has. This copy of the remote estimate which is provided by the smart sensor will be denoted xc [k]. Starting at the initial estimate x[0] and in the absence of messages, the decoder simply updates the estimate by iterating the matrix A. When the distance between x[k] and xc [k] = x[k] becomes larger than the prescribed maximum error (including a margin for the observation error ē

[k] := x[k] -x[k]
), the sampler decides to communicate and the coder sends a message to the decoder to provide a new estimate x[k]. Figure 1 depicts how the different elements interact. Below, each of these algorithms are presented in details.

The Local Observer

The local observer equation is

x[k + 1] = A x[k] -L(y[k] -C x[k]), (7)
L,γ 1 δ + σ 1 (L)ω -γ 1 s.t. σ 1 (A + LC) < 1 -γ 1 , γ 1 ≥ 0 (9)
This choice of nonlinear program implies that the worst case error is bounded, as proven by the following proposition. Proposition 1. Assume that there exists an L such that the inequality σ 1 (A + LC) < 1 has a solution and ε is chosen small enough, then choosing L as in (9) leads to the following inequality

ē[k] 2 ≤ δ + σ 1 (L)ω 1 -σ 1 (A + LC) , ∀k ≥ 0. Proof: We have that (8) implies that ē[k + 1] 2 ≤ σ 1 (A + LC) ē[k] 2 + δ + σ 1 (L)ω. (10) By choosing ε 0 ≤ δ + σ 1 (L)ω, we have ē[0] 2 ≤ δ + σ 1 (L)ω. Inequality (10) applied several times implies that ē[k] 2 ≤ k ∑ i=0 σ 1 (A + LC)(δ + σ 1 (L)ω),
which, due to the sum of a geometric series and because L is chosen such that σ 1 (A + LC) < 1 implies that

ē[k] 2 ≤ 1 -σ 1 (A + LC) k 1 -σ 1 (A + LC) (δ + σ 1 (L)ω) ≤ δ + σ 1 (L)ω 1 -σ 1 (A + LC)
.

For brevity, we define the notation η := δ +σ 1 (L)ω 1-σ 1 (A+LC) . The following proposition provides an alternative formulation of the previous nonlinear program in the form of an LMI program. Proposition 2. If the following LMI, is feasible, then (9) has a L = arg min

γ i , -γ 1 + δ 2 γ 2 + ω 2 γ 3 -γ 4 s.t.    -I n + γ 4 0 n×n 0 n×m A +C L 0 n×n -γ 2 I n 0 n×m I n 0 m×n 0 m×n -γ 3 I m L A + LC I n L -I n    0, -I n + γ 1 I n A +C L A + LC I n ≺ 0, γ i ≥ 0, ( 11
)
where I n is the n × n identity matrix and 0 n×m is the n × m zero matrix.

The proof of this proposition will be presented in the full version of this paper.

Remark 1. It is possible to use another Lyapunov function such as V ( ē) = ē P ē instead of V ( ē) = ē ē in the above developments for more generality. In such a case in order to keep tightness in the inequalities used in the present work, it is necessary to define the singular values differently for the previous analysis. One way to achieve these tight bounds is to replace σ 1 (M), with the square root of the P-generalized eigenvalues of the matrix M PM (see e.g. [START_REF] Pogromsky | Estimation of the topological entropy via the direct Lyapunov method[END_REF] for more details about P-generalized eigenvalues). For brevity, the mathematical details of the usage of V ( ē) are not explored in this paper.

The Protocol Description

We now present the communication procedure, which we will further reference as Procedure 1. It is composed of a sampler, alphabet, coder and decoder as described below. For this particular communication procedure, a minimum time interval between communications is going to be employed. This quantity, denoted as ∆k is known by all agents. It is a user-specified parameter which is to be chosen finite and it directly influences the upper bound on the estimation error. How exactly one might choose ∆k and how it influences the error will be discussed further in this paper.

To properly describe the communication instants, we will need several quantities. The indexes j of the communication instants are inherently known by all agents. The quantity j refers to the index of the last instant of communication (initially, j = 0). This quantity is always known by the sampler (because it knows how many communication instants it defined), the coder (because it knows how many messages it sent), as well as the decoder (because it knows how many messages it received). Finally, the sampler and coder interact to update the knowledge of the estimate x[k] at the decoder side. Procedure 1. The Sampler: At each time instant k ≥ k j + ∆k, the sampler verifies whether the following condition is satisfied

x[k] -A xc [k -1] 2 ≤δ ∆k-1 ∑ f =1 σ 1 (A) f -1 + σ 1 (A) ∆k-1 ε. (12)
If the condition is verified, the sampler sets xc

[k] = A xc [k -1].
If the condition is not verified, a message must be sent to provide a new estimate. The sampler thus sets j = j = j + 1 and k j = k ( j increases by one and j = j).

The Alphabet: If k = k j, the alphabet agent builds a covering of the set I j , where I j is defined as

I j := x ∈ R n | x -A xc [k j -1] 2 ≤ δ ∆k ∑ f =1 σ 1 (A) f -1 +σ 1 (A) ∆k ε + (σ 1 (A) ∆k + 1)η , (13)
with balls of size ε. The balls in the covering are numbered from 1 till l j, where l j is the length of the alphabet.

The Coder: At the communication instants, the coder function finds the index of the ball in the covering made by the alphabet whose center is the closest to x[k j ] and sends this index over the communication channel. The coder also updates the local estimate xc [k] by setting it to be equal to the center of the chosen ball.

The Decoder: If the decoder receives no message, it assigns

x[k] = A x[k -1], otherwise it uses the center of the ball whose index it received as the new estimate.

The alphabet is based on the following idea. As was previously mentioned, in the absence of messages, new estimates are obtained at the decoder side by iterating

A (i.e. x[k + 1] = A x[k] and xc [k + 1] = A xc [k]
). After receiving a message, the state of the system x[k] is contained in a ball of a certain radius whose center is the estimate x[k]. In the absence of any messages, this "ball" of uncertainty is gradually deformed into a larger/small uncertain set. The uncertain set evolves due to two factors: first of all, the unknown state perturbation d[k], d[k + 1], . . . increases its radius by δ at each time step, secondly, the uncertainty set is stretched/compressed by action of the linear mapping A (the deformations are proportional to the singular values of A). Given that the communication intervals are chosen to be finite, this uncertain set remains of finite size in between communications. It can thus be covered by a finite number of balls of size ε > 0. The balls in the covering can be indexed from 1 till l max < ∞. In order to produce such a covering, the only information needed are the initial ball and the different upper bounds on the uncertainties/errors, which implies that both the coder as well as the decoder can build the set. In order to transmit a new estimate, one can simply send the index of one of the balls whose center then serves as a new estimate with a precision that will depend on ε. The cost of communicating in that fashion is dependent on how many balls of size ε are required to cover the uncertain set.

In practice, the previously described set is contained within the set I j . The alphabet relies on the assumption that the estimate x[k] will lie within the set I j when the communications occur. This assumption guarantees that x[k]x[k] 2 ≤ ε after each communication instant, which makes the procedure repeatable. The following lemma proves this assumption. Lemma 1. For the set I j as defined in (13) and estimates x[k] as generated by the observer (7) the following holds: x[k j ] ∈ I j , ∀ j ≥ 1.

Proof: We first consider the case

k j = k j-1 + ∆k. Starting from k = k j-1 + 1, the error ê[k] := x[k] -A x[k -1] follows particular dynamics. Indeed, we have ê[k + 1] = x[k + 1] -A x[k] = Ax[k] + d[k] -A x[k] = A ê[k] + d[k], (14)
∀k ∈ {k j-1 , . . . , k j -1}. Because the estimates that was trans-

mitted at k j-1 is of precision ε, we have ê[k j-1 ] 2 ≤ x[k j-1 ] -x[k j-1 ] 2 + ē[k j-1 ] 2 ≤ ε + η. This implies that we have ê[k j ] 2 ≤ σ 1 (A) ∆k (ε + η) + ∆k ∑ f =1 σ 1 (A) f -1 δ .
By using Proposition 1, we can decompose the distance between

x[k j ] and A x[k j -1] x[k j ] -A x[k j -1] 2 ≤ ê[k j ] 2 + ē[k j ] 2 ≤ σ 1 (A) ∆k (ε + η) + ∆k ∑ f =1 σ 1 (A) f -1 δ + η which implies that x[k j ] ∈ I j for k j = k j-1 + ∆k. For k j > k j-1 + ∆k, (12) was necessarily verified at k = k j -1, which implies that x[k j -1] -A x[k j -2] 2 ≤ δ ∆k-1 ∑ j=1 σ 1 (A) j-1 + σ 1 (A) ∆k-1 ε.
Because of the error dynamics of ê[k], we have that

x[k j ] -A x[k j -1] 2 ≤ ê[k j ] 2 + ē[k j ] 2 ≤ δ ∆k ∑ j=1 σ 1 (A) j-1 + σ 1 (A) ∆k ε + η, which implies that x[k j ] ∈ I j for k j > k j-1 + ∆k.
We finish the current section with several remarks on the different features of the observer.

Remark 2.

• The estimates of the state at the sensor side xc [k] are either updated by the sampler, if no message is sent, or by the coder, if a message is sent.

• The coordinates of the centers of the balls used in the covering are always relative to the previous estimates. By communicating in a relative fashion, it is possible to keep the size of the messages limited even if the system is unstable. Note that since xc [k] = x[k], both agents can build this set according to its definition (13). • The alphabet procedure is easy from a computational point of view since it consists of covering a set which always has the same shape (the shape depends on which norm is used in the definition of I j ) except the whole set is shifted by a certain vector from the origin. Moreover, since this set is centered around the previous estimate, both the coder and decoder can build a covering for it and thus have access to the alphabet.

RATE AND ERRORS

With the observer and its agents fully introduced, it is time to determine what minimum number of bits per time interval is sufficient to implement the observer. This quantity is closely related to the observation error. The first result we present provides closed-form expressions for the constants that indicate the proportionality of the different errors/perturbations in the total error. Proposition 3. The observer described in Procedure 1 ensures that (6) holds with

γ ε = max{σ 1 (A) ∆k-1 , 1}, (15)
γ δ = max{σ 1 (A) ∆k-1 , 1} 1 -σ 1 (A + LC) + ∆k-1 ∑ f =1 σ 1 (A) f -1 , (16)
γ ω = σ 1 (L) max{σ 1 (A) ∆k-1 , 1} 1 -σ 1 (A + LC) . ( 17
)
Proof: The proof is separated in three different cases, we first consider k = k j , j ∈ {0, 1, . . . }. At the instants when the messages are sent, and because the estimates are the centers of ball of size ε, we have

x[k j ] -x[k j ] 2 ≤ x[k j ] -x[k j ] 2 + x[k j ] -x[k j ] 2 , which, using Proposition 1, becomes x[k j ] -x[k j ] 2 ≤ ε + η.
Evidently, since σ 1 (A) ≥ 0, the previous equation is upper bounded by

x[k j ] -x[k j ] 2 ≤ max{σ 1 (A) ∆k-1 , 1}ε + max{σ 1 (A) ∆k-1 , 1}η.
which implies that ( 15)-( 17) hold for k = k j . For k ∈ {k j + 1, k j + 1, . . . , k j + ∆k -1}, we consider the dynamics of e

[k] = x[k] -x[k].
In between communication instants, we have e and d[k] 2 ≤ δ , we have by induction that

[k + 1] = x[k + 1] -x[k + 1] = Ax[k] + d[k] -A x[k] = Ae[k] + d[k]. Since e[k j ] 2 ≤ ε + η,
e[k] 2 ≤σ 1 (A) ∆k-1 (ε + η) + δ ∆k-1 ∑ f =1 σ 1 (A) f -1 ,
for k ∈ {k j + 1, k j + 1, . . . , k j + ∆k -1}. Evidently, the righthand side of the previous equation is again upper bounded by

e[k] 2 ≤ max{σ 1 (A) ∆k-1 , 1}ε + max{σ 1 (A) ∆k-1 , 1}η + δ ∆k-1 ∑ f =1 σ 1 (A) f -1 .
which implies that ( 15)-( 17) hold for k ∈ {k j +1, k j +1, . . . , k j + ∆k -1}. For k ∈ {k j + ∆k, . . . , k j+1 -1}, the sampler verifies whether ( 12) is satisfied. Since the instants k we consider in this case are not instants of communication, the condition ( 12) is necessarily verified, which implies that

x[k] -x[k] 2 ≤ x[k] -x[k] 2 + x[k] -x[k] 2 ≤η + δ ∆k-1 ∑ j=1 σ 1 (A) j-1 + σ 1 (A) ∆k-1 ε.
The proof is completed by upper bounding the above expression as previously.

The main result of this sections aims to provide a lower bound on b + (•) for designed communication scheme. In the following theorem, the notation • refers to the ceiling function.

Theorem 1. The observer described in Procedure 1 with ∆k ≥ 2 is implementable on any channel with b + (∆k) ≥

log 2 √ n σ 1 (A) ∆k ε + δ ∑ ∆k f =1 σ 1 (A) f -1 + (σ 1 (A) ∆k + 1)η ε n (18)
Proof: In order to implement Procedure 1, (5) should be verified for all j. The size of the alphabet is equal to the number of balls of radius ε required to cover I j . Since the radius of the set is

σ 1 (A) ∆k ε + δ ∆k ∑ f =1 σ 1 (A) f -1 + (σ 1 (A) ∆k + 1)η,
it can evidently be covered by no more than

σ 1 (A) ∆k + δ ∑ ∆k f =1 σ 1 (A) f -1 + (σ 1 (A) ∆k + 1)η ε n hypercubes of radius ε √ n .
These hypercubes are themselves contained in a sphere of radius ε. In order to verify (5), it thus suffices that b

+ (∆k) ≥ n log 2 √ n σ 1 (A) ∆k ε + δ ∑ ∆k f =1 σ 1 (A) f -1 + (σ 1 (A) ∆k + 1)η ε
for the observer to be implementable.

Remark 3. Without any state perturbations or measurement errors, the inequality of Theorem 1 becomes

b + (∆k) ≥ n log 2 √ nσ 1 (A) ∆k .
This right-hand side could be improved to ∑ i∈{1,...,n} max{0, log 2 ( √ nσ 1 (A) ∆k )} by more accurately defining the sets I j in (13). Since the objective of this paper is to deal with the case with perturbations, such considerations are left for further research.

SIMULATIONS

In this section we provide simulations of the data-rate constrained observer. We will also demonstrate, through simulations, that the fact that we only communicate if ( 12) is verified, greatly reduces the average communication rate in some cases.

For this purpose, we consider the following system The results in terms of average number of communications jmean and number of transmitted bits N bits per communication can be seen in Table 1. As can be seen from the table, the communication scheme transmits messages much more rarely than every ∆k timesteps (e.g. with ∆k = 2, the scheme could communicate up to 50 times every run, but only communicates 4.63678 times on average). Increasing the time interval between subsequent communications greatly reduces the total number of messages sent, which is due to the fact that a larger error between estimate and state is tolerated. In terms of rate rather than number of bits, for ∆k = 20, the observers sends an average of 0.63829 • 14/100 = 0.0893606 bits/timestep. Overall, these simulations prove the effectiveness of the event-triggeredness of the communication scheme. In this paper we presented an event-triggered, data-rate constrained observer for discrete-time linear systems with perturbations. After posing the problem statement, the design of the agents that form the communication scheme was explained.

x[k + 1] = -0.3913 0.3149 0.6140 0.9115 A x[k] + d[k], y[k] = [-0.6071 -0.8284] C x[k] + w[k] (19)
We then provided a theorem that upper bounded the minimum bit-rate required to implement this communication protocol.

The protocol was tested via simulations, which confirmed that the time interval between subsequent communications is much longer than the minimum allowable time interval. Such an observation implies that the communication scheme is very efficient at reducing the average number of communications. This work will be adapted for the observation of mobile robots as well as for more general nonlinear systems.

Fig. 1 .

 1 Fig. 1. Structure of setup. where L ∈ R n×m is a gain matrix. Note that the dynamics of ē[k] are thus ē[k + 1] = (A + LC)e[k] + d[k] + Lw[k]. (8) The observer uses x[0] = x[0] as an initial point, which implies that ē[0] 2 ≤ ε 0 . In order to minimize the worst case local observation error and to keep the local observation error bounded, L is computed as the solution of L = arg min

  Fig. 2. Evolution of the different errors for one particular simulation.