Delay-Dependent Positivity: Application to Interval Observers

New delay-dependent conditions of positivity for linear systems with time-varying delays are introduced. These conditions are applied to interval observer design for systems with time-varying delays in the state equations and in the measurements.

I. INTRODUCTION

An estimation in nonlinear delayed systems is rather complicated [START_REF] Sipahi | Stability and stabilization of systems with time delay limitations and opportunities[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF], as well as analysis of functional differential equations [START_REF] Richard | Time delay systems: an overview of some recent advances and open problems[END_REF]. Especially the observer synthesis is problematical for the cases when the model of a nonlinear delayed system contains parametric and/or signal uncertainties, or when the delay is time-varying or uncertain [START_REF] Briat | Design of LPV observers for LPV time-delay systems: an algebraic approach[END_REF], [START_REF] Califano | On the observer canonical form for nonlinear time-delay systems[END_REF], [START_REF] Zheng | On observation of time-delay systems with unknown inputs[END_REF]. An observer solution for these more complex situations is highly demanded in many applications. Interval or set-membership estimation is a promising framework to observation in uncertain systems [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], [START_REF] Jaulin | Nonlinear bounded-error state estimation of continuous time systems[END_REF], [START_REF] Kieffer | Guaranteed nonlinear state estimator for cooperative systems[END_REF], [START_REF] Moisan | Near optimal interval observers bundle for uncertain bio-reactors[END_REF], [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF], [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF], when all uncertainty is included in the corresponding intervals or polytopes, and as a result the set of admissible values (an interval) for the state is provided at each instant of time.

In this work an interval observer for time-delay systems is proposed. A peculiarity of an interval observer is that in addition to stability conditions, some restrictions on positivity of estimation error dynamics have to be imposed (in order to envelop the system solutions). The existing solutions in the field [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF], [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Efimov | Interval estimation for systems with time delays and algebraic constraints[END_REF] are based on the delay-independent conditions of positivity from [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF]. The first objective of this work is to introduce the delay-dependent positivity conditions, which are based on the theory of non-oscillatory solutions for functional differential equations [START_REF] Agarwal | Nonoscillation theory of functional differential equations with applications[END_REF]. Next, two designs of interval observers are given for linear systems with delayed measurements (with time-varying delays) for the case of observable and detectable systems.

The paper is organized as follows. Some preliminaries are given in Section 2. The delay-dependent positivity conditions are presented in Section 3. The interval observer design is performed for a class of linear time-delay systems (or a

II. PRELIMINARIES

In the rest of the paper, the following notation will be used: means that the matrix is negative (semi) definite.

• R is the Euclidean space (R + = {τ ∈ R : τ ≥ 0}), C n τ = C([-τ, 0], R n ) is the set of continuous maps from [-τ, 0] into R n for n ≥ 1; C n τ + = {y ∈ C n τ : y(s) ∈ R n + , s ∈ [-τ, 0]}; • x t is an element of C n τ associated with a map x t : R → R n by x t (s) = x(t + s) for all s ∈ [-τ, 0]; • |x| denotes the absolute value of x ∈ R, ||x|| is the Euclidean norm of a vector x ∈ R n , ||ϕ|| = sup t∈[-τ,0] |ϕ(t)| for ϕ ∈ C τ ; • for

A. General definition of positivity for time-delay systems

Consider a linear non-autonomous functional differential equation

ẋ(t) = m k=1 A k (t)x[h k (t)] + b(t), t ≥ t 0 ≥ 0, (1)
x(t) = φ(t) for t < t 0 , x(t 0 ) = x 0 , where x(t) ∈ R n is the state vector and

x 0 ∈ R n ; all elements of A k are from L ∞ for all 1 ≤ k ≤ m, m > 0; the signals h k : [t 0 , +∞) → R are Lebesgue measurable, h k (t) ≤ t and lim t→+∞ h k (t) = +∞ for all 1 ≤ k ≤ m; b ∈ L n ∞ is the input; φ : (-∞, t 0 ) → R n is a Borel measurable bounded function of initial conditions. Definition 1. For each s ≥ 0, the solution Υ(t, s) ∈ R n×n of the problem ∂Υ(t, s) ∂t = m k=1 A k (t)Υ[h k (t), s], Υ(t, s) = 0 for t < s, Υ(s, s) = I n ,
is called the fundamental (or the Cauchy) matrix of (1).

Lemma 1.

[17] (Lemma 9.1) For each x 0 ∈ R n and φ : (-∞, t 0 ) → R n there exists a unique solution of (1), and it can be presented in the form

x(t) = Υ(t, t 0 )x 0 + ˆt t0 Υ(t, s)b(s)ds + m k=1 ˆt t0 Υ(t, s)A k (s)φ[h k (s)]ds, where φ[h k (s)] = 0 if h k (s) > t 0 .
Now we are in position to give the definition of positivity, denote Ω a subset of Borel measurable bounded functions

φ : (-∞, t 0 ) → R n . Definition 2. The system (1) for x 0 ∈ R n + and b(t) ∈ R n + is called Ω-positive, if x(t) ≥ 0 for all t ≥ t 0 provided that φ ∈ Ω.
A direct consequence of this definition is that for Ω = {0} the system is {0}-positive (or we will write simply positive next) iff Υ(t, s) ∈ R n×n + for all t ≥ s.

B. Linear C n

τ + -positive systems with constant delays Consider a linear system with a constant delay (a variant of (1) for m = 2)

ẋ(t) = A 0 x(t) -A 1 x(t -τ ) + b(t), t ≥ t 0 ≥ 0, (2) x(θ) = φ(θ) for -τ ≤ θ ≤ 0, φ ∈ C n τ , where x(t) ∈ R n , x t ∈ C n τ is the state function, τ ∈ R + is the delay; b ∈ L n
∞ is the input; the constant matrices A 0 and A 1 have appropriate dimensions. The matrix A 0 is called Metzler if all its off-diagonal elements are nonnegative.

Lemma 2. [16] The system (2) is C n τ + -positive iff A 0 is Metzler, A 1 ≤ 0 and b(t) ∈ R n + for all t ∈ R. A C n τ + - positive system (2) is asymptotically stable for b(t) ≡ 0 for all τ ∈ R + iff there are p, q ∈ R n + (p > 0 and q > 0) such that p T [A 0 -A 1 ] + q T = 0.
Under conditions of the above lemma the system has bounded solutions for b ∈ L n ∞ with b(t) ∈ R n + for all t ∈ R. Note that for linear time-invariant systems the conditions of positive invariance of polyhedral sets have been similarly given in [START_REF] Dambrine | Feedback control of timedelay systems with bounded control and state[END_REF], as well as conditions of asymptotic stability in the nonlinear case have been considered in [START_REF] Dambrine | Stability analysis of time-delay systems[END_REF], [START_REF]Stability and stability domains analysis for nonlinear differential-difference equations[END_REF], [START_REF] Borne | Stability Theory at the end of the XXth Century[END_REF].

III. DELAY-DEPENDENT POSITIVITY

Consider a scalar time-varying linear system with timevarying delays [START_REF] Agarwal | Nonoscillation theory of functional differential equations with applications[END_REF]:

ẋ(t) = a 0 (t)x[g(t)] -a 1 (t)x[h(t)] + b(t), (3) x(θ) = 0 for θ < 0, (4)
where

a 0 ∈ L ∞ , a 1 ∈ L ∞ , b ∈ L ∞ , h(t)-t ∈ L ∞ , g(t)-t ∈ L ∞ and h(t) ≤ t, g(t)
≤ t for all t ≥ 0. For the system (3) the initial condition in ( 4) is, in general, not a continuous function (if x(0) = 0).

The following result proposes delay-independent positivity conditions following Definition 2. Lemma 3. [START_REF] Agarwal | Nonoscillation theory of functional differential equations with applications[END_REF] (Corollary 15.7) Let h(t) ≤ g(t) and 0 ≤ a 1 (t) ≤ a 0 (t) for all t ≥ 0. If x(0) ≥ 0 and b(t) ≥ 0 for all t ≥ 0, then the corresponding solution of (3),( 4) x(t) ≥ 0 for all t ≥ 0.

Recall that in this case positivity is guaranteed for "discontinuous" initial conditions and time-varying delays. The peculiarity of the condition 0 ≤ a 1 (t) ≤ a 0 (t) is that it may correspond to an unstable system (3). In order to overcome this issue, delay-dependent conditions can be introduced. Lemma 4. [START_REF] Agarwal | Nonoscillation theory of functional differential equations with applications[END_REF] (Corollary 15.9) Let h(t) ≤ g(t) and 0 ≤ 1 e a 0 (t) ≤ a 1 (t) for all t ≥ 0 with (e = exp(1))

sup t∈R+ ˆt h(t) [a 1 (ξ) - 1 e a 0 (ξ)]dξ < 1 e .
If x(0) ≥ 0 and b(t) ≥ 0 for all t ≥ 0, then the corresponding solution of (3), (4) x(t) ≥ 0 for all t ≥ 0.

These lemmas describe positivity conditions in the sense of Definition 2 for the system (3), (4), which is more complex than (2), but scalar.

Corollary 1.

[17] (Theorem 15.23) Consider the autonomous system (3) with a 0 (t) = a 0 , a 1 (t) = a 1 , h(t) = t -τ and g(t) = t-σ with 0 ≤ σ < τ . Let 0 ≤ a 0 ≤ ea 1 < a 0 +τ -1 . If x(0) ≥ 0 and b(t) ≥ 0 for all t ≥ 0, then the corresponding solution of (3), (4) x(t) ≥ 0 for all t ≥ 0.

Corollary 2. The system

ẋ(t) = A 0 x(t) -A 1 x(t -τ ) + b(t), t ≥ t 0 ≥ 0, x(θ) = 0 for -τ ≤ θ < 0, x(0) ∈ R n + , is positive if -A 1 is Metzler, A 0 ≥ 0, 0 ≤ (A 0 ) i,i ≤ e(A 1 ) i,i < (A 0 ) i,i + τ -1
for all i = 1, . . . , n, and b(t) ≥ 0 for all t ≥ 0.

From these corollaries it is easy to conclude that the delay-dependent case studied in Lemma 3 and Lemma 4 is crucially different from the delay-independent C n τ + -positivity conditions given first in Lemma 2, where in the scalar case the restriction a 1 ≤ 0 implies positivity of (2) and the condition a 0 < a 1 according to Lemma 2 ensures stability for any delay τ . These results do not contradict to Remark 3.1 of [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF], since x(θ) = 0 for -τ ≤ θ < 0 there. A graphical illustration of different delay-independent conditions (C n τ +positivity from Lemma 2 and positivity from Lemma 3) and delay-dependent ones (from Lemma 4, the stability conditions are also satisfied in this case) for the system (3) is given in Fig. 1 in the plane (a 0 , a 1 ). It is worth stressing that an extension of the positivity domain in Lemma 4 is also achieved due to restrictions imposed on initial conditions in [START_REF] Briat | Design of LPV observers for LPV time-delay systems: an algebraic approach[END_REF].

In order to use the results of Lemma 3 and Lemma 4 it is necessary to pass from discontinuous initial conditions in (4) to continuous ones usually studied [START_REF] Sipahi | Stability and stabilization of systems with time delay limitations and opportunities[END_REF]. In this section we will be interested in the case a 0 (t) = a 0 , a 1 (t) = a 1 , t-τ ≤ h(t) ≤ t-τ , g(t) = t, [START_REF] Califano | On the observer canonical form for nonlinear time-delay systems[END_REF] where τ ≥ τ ≥ 0 are minimum and maximum delays respectively. Then the following corollary of Lemma 4 can be formulated. Corollary 3. Let 0 ≤ a 0 ≤ ea 1 < a 0 + τ -1 . If x(0) ≥ 0 and b(t) ≥ 0 for all t ≥ 0, then the corresponding solution of (3), ( 5), (4) x(t) ≥ 0 for all t ≥ 0.

Remark 1. As mentioned in [START_REF] Liu | Delay-dependent methods and the first delay interval[END_REF], the first delay interval 0 ≤ t ≤ τ is important for delay-dependent conditions giving solution bounds (and not just stability conditions).

Now let us substitute the initial condition ( 4) with a continuous one:

x(θ) = φ(θ) for θ ∈ [-τ , 0], (6)
and consider the conditions providing delay-dependent C n τ +positivity for (3), ( 5), [START_REF] Zheng | On observation of time-delay systems with unknown inputs[END_REF].

Proposition 1. Let 0 ≤ a 0 ≤ ea 1 < a 0 + τ -1 . If x(0) ≥ 0, b(t) ≥ 0 for all t ≥ 0 and b(t) ≥ a 1 φ[h(t)] ∀t ∈ {0 ≤ t ≤ τ : h(t) < 0},
then the corresponding solution of (3), ( 5), (6) satisfies x(t) ≥ 0 for all t ≥ 0.

All proofs are excluded due to space limitations.

Let us show how these conditions can be used for the design of interval observers.

IV. INTERVAL OBSERVER DESIGN UNDER DELAYED

MEASUREMENTS

In this section a useful inequality for interval analysis and a statement of the problem are given. Next, a motivating benchmark example from [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF] is investigated, using the results of the previous section, in order to clarify the main idea. Finally, a delay-dependent approach for an interval observer design is presented.

A. Interval bounds

Given a matrix A ∈ R m×n define A + = max{0, A}, A -= A + -A and |A| = A + + A -. Let x ∈ R n be a vector variable, x ≤ x ≤ x for some x, x ∈ R n , and A ∈ R m×n be a constant matrix, then

A + x -A -x ≤ Ax ≤ A + x -A -x. ( 7
)
This claim follows from the equation Ax = (A + -A -)x, that for x ≤ x ≤ x gives the required estimates.

B. Problem statement

Consider a linear system with a time-varying delay:

ẋ(t) = A 0 x(t) + A 1 x[h(t)] + b(t), (8) y
(t) = Cx[h(t)] + v(t),
where x(t) ∈ R n , t -τ ≤ h(t) ≤ t -τ is a known timevarying delay (t -h(t) ∈ L ∞ ), 0 ≤ τ ≤ τ are minimum and maximum delays respectively, x 0 ∈ C τ ; y(t) ∈ R p is the system output available for measurements with the noise v ∈ L p ∞ ; b ∈ L n ∞ is the system input; the constant matrices A 0 , A 1 and C have appropriate dimensions. It is assumed that for given b and h the system has a unique solution defined at least locally. Assumption 1. For given input b(t), delay h(t) and initial condition x 0 ∈ C τ , the corresponding solution of (8) x(t) is bounded. In addition, there exist known functions

x 0 , x 0 ∈ C τ such that x 0 (θ) ≤ x 0 (θ) ≤ x 0 (θ) for all θ ∈ [-τ , 0].
Boundedness of the state x(t) is a usual assumption in the estimation theory [START_REF] Besançon | Nonlinear Observers and Applications[END_REF], [START_REF] Fossen | New Directions in Nonlinear Observer Design[END_REF]. The assumption about a known set [x 0 , x 0 ] for the initial conditions x 0 is standard for the interval or set-membership estimation theory [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], [START_REF] Jaulin | Nonlinear bounded-error state estimation of continuous time systems[END_REF], [START_REF] Kieffer | Guaranteed nonlinear state estimator for cooperative systems[END_REF], [START_REF] Moisan | Near optimal interval observers bundle for uncertain bio-reactors[END_REF]. We will assume that the values of matrices A 0 , A 1 and C are known, for the uncertain delay h(t) only the bounds τ , τ are given, the instant values of the signals b(t) and v(t) are unavailable.

Assumption 2. There exist known signals b, b ∈ L n ∞ and v, v ∈ L p ∞ such that b(t) ≤ b(t) ≤ b(t) and v(t) ≤ v(t) ≤ v(t) for all t ≥ 0.
Therefore, the uncertain inputs b(t), h(t) and v(t) in ( 8

) belong to known intervals [b(t), b(t)], [t -τ , t -τ ] and [v(t), v(t)] respectively for all t ≥ 0.
It is required to design an interval observer,

ξ(t) = F [ξ t , b(t), b(t), v(t), v(t), y(t)], ξ t ∈ C s τ , x(t) = G[ξ t , b(t), b(t), v(t), v(t), y(t)], x(t) = G[ξ t , b(t), b(t), v(t), v(t), y(t)]
such that x(t) ≤ x(t) ≤ x(t) for all t > 0 provided that x 0 ≤ x 0 ≤ x 0 , and x, x ∈ L n ∞ , s > 0. A similar problem has been studied in [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF] but for constant delays.

C. Motivating example

Consider a motivating example introduced in [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF], where the problem of a framer design (the definition of a framer can be found in [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF], roughly speaking it is an interval open-loop estimator independent of y(t)) has been posed for a scalar system

ẋ(t) = -x(t -τ ) (9)
with initial condition x 0 ∈ C τ . This system is globally asymptotically stable if τ < π 2 . It has been proven in [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF] that this system has no framer of the form

F (ξ t ) = F 1 ξ(t) + F 2 ξ(t -τ ), (10)
G(ξ t ) = H 1 ξ(t), G(ξ t ) = H 2 ξ(t),
where F i , H i (i = 1, 2) are matrices of appropriate dimensions.

Applying the result of Proposition 1 or Corollary 1, the system (9) has positive solutions for a discontinuous initial condition (4) with x(0) ≥ 0 if τ < 1 e . Actually in this case it has a non-oscillating solution which is asymptotically converging to zero (since 1 e < π 2 ), and which does not cross the zero level for all t ∈ R + . Further, using the result of Proposition 1, we can design a framer for (9) of a form similar to [START_REF] Moisan | Near optimal interval observers bundle for uncertain bio-reactors[END_REF]. Claim 1. For the system (9) with any initial condition x 0 ∈ C τ and τ < 1 e , the system

ẋ(t) = -x(t -τ ) + δ(||x 0 ||), ẋ(t) = -x(t -τ ) + δ(||x 0 ||), δ(s) = s if t ≤ τ, 0 otherwise
is a framer, i.e. x(t) ≤ x(t) ≤ x(t) for all t > 0, provided that x 0 ≤ x 0 ≤ x 0 , and x, x ∈ L n ∞ . Therefore, a framer of a form similar to (10) can be designed for [START_REF] Kieffer | Guaranteed nonlinear state estimator for cooperative systems[END_REF] with a restricted value of delay τ < 1 e (it differs from [START_REF] Moisan | Near optimal interval observers bundle for uncertain bio-reactors[END_REF] only on the interval [0, τ ]). The results of simulation for this framer are given in Section 5.

Let us extend this idea of interval observer design to a more generic system [START_REF] Jaulin | Nonlinear bounded-error state estimation of continuous time systems[END_REF].

D. Delay-dependent conditions for interval estimation

The equation ( 8) can be rewritten as follows:

ẋ(t) = A 0 x(t) + {A 1 -LC}x[h(t)] + Ly(t) + b(t) -Lv(t),
where L ∈ R n×p is an observer gain to be designed. Assumption 3. There exists an invertible matrix S ∈ R n×n such that S(A 1 -LC)S -1 = R 1 , where R 1 is a Metzler matrix and

R 1 = R † 1 + R 1 , R † 1 = diag[-r 1,1 , . . . , -r 1,n ], R 1 ≥ 0
with R † 1 is the diagonal matrix composed by all elements on the main diagonal of R 1 , r 1,i > 0 for all i = 1, . . . , n, and R 1 is formed by the rest elements of R 1 out of the main diagonal.

The conditions of existence of such matrices S and L can be found in [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF], in particular Assumption 3 is satisfied if the pair (A 1 , C) is observable. Under this assumption in the new coordinates z = Sx the system (8) takes the form:

ż(t) = R 0 z(t) + R 1 z[h(t)] + SLy(t) + β(t), (11)
where

R 0 = SA 0 S -1 and β(t) = S[b(t) -Lv(t)] is a new additive uncertain input, the initial condition z 0 = Sx 0 ∈ C τ and z 0 ≤ z 0 ≤ z 0 ,
where z 0 = S + x 0 -S -x 0 and z 0 = S + x 0 -S -x 0 are calculated using [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]. From Assumption 2 and the relations [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] we obtain that

β(t) ≤ β(t) ≤ β(t) ∀t ≥ 0, where β(t) = S + b(t) -S -b(t) -(SL) + v(t) + (SL) -v(t) and β(t) = S + b(t) -S -b(t) -(SL) + v(t) + (SL) -v(t).
Then the following interval observer can be proposed for the representation [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]:

ż(t) = R + 0 z(t) -R - 0 z(t) + R 1 z[h(t)] +SLy(t) + β(t) -δ, (12) ż
(t) = R + 0 z(t) -R - 0 z(t) + R 1 z[h(t)] +SLy(t) + β(t) + δ, δ = [δ 1 , . . . , δ n ] T , δ i = r 1,i ||z 0,i -z 0,i || if t ≤ τ 0 otherwise , i = 1, . . . n
with initial conditions z 0 , z 0 for the variables z(t), z(t) respectively. Finally interval estimates for the variable x(t) can also be obtained using [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]:

x(t) = (S -1 ) + z(t) -(S -1 ) -z(t), (13) x
(t) = (S -1 ) + z(t) -(S -1 ) -z(t).
For all i = 1, . . . , n denote r 0,i = (R + 0 ) i,i .

Theorem 1. Let assumptions 1-3 be satisfied and r 0,i ≤ er 1,i < r 0,i + τ -1 for all i = 1, . . . , n. Then for the system (8) and the interval observer ( 12), ( 13)

x(t) ≤ x(t) ≤ x(t) ∀t ≥ 0. (14)
∈ R 2n×2n , Σ ∈ R 2n×2n , Ξ ∈ R 2n×2n and Θ ∈ R 2n×2n such that the LMIs Ξ Θ Θ Ξ 0, (15)
    Φ T 0 P + P Φ0 + Σ -Ξ Θ P Φ1 + Ξ -Θ τ Φ T 0 Ξ Θ -Σ -Ξ Ξ -Θ 02n×2n Φ T 1 P + Ξ -Θ Ξ -Θ Θ + Θ T -2Ξ τ Φ T 1 Ξ τ ΞΦ0 02n×2n τ ΞΦ1 -Ξ     ≺ 0,
where

Φ 0 = R + 0 -R - 0 -R - 0 R + 0 , Φ 1 = R 1 0 n×n 0 n×n R 1 ,
are satisfied, then x, x ∈ L n ∞ . Let us illustrate this result by numerical simulations.

V. FRAMER FOR THE MOTIVATING EXAMPLE

The results of simulation of the framer from Claim 1 for the system (9) are shown in Fig. 2. For simulation the value τ = 0.1 < 1 e and it has been assumed that ||x 0 || ≤ 5. The red solid curves represent trajectories of (9) for different initial conditions, the blue and green dash-dot lines correspond to the the interval estimates x(t) and x(t) generated by the framer. As we can see, in this case the additional term δ "freezes" the estimates x(t), x(t) on the interval t ∈ [0, τ ], next they are converging to zero.

VI. CONCLUSION

In the paper, new delay-dependent positivity conditions for linear systems with time-varying delays have been proposed. These conditions are related with non-oscillatory behavior of solutions [START_REF] Agarwal | Nonoscillation theory of functional differential equations with applications[END_REF]. They nicely complement the existing delayindependent conditions of [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF] (see Fig. 1). Next, these new conditions have been employed to design interval observers for the systems with delayed measurements extending the theory of [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF], [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Efimov | Interval estimation for systems with time delays and algebraic constraints[END_REF]. The results have been applied for the benchmark system from [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF]. Extension of these results for the case of sampled-data measurements is a direction of future research.

Figure 1 .

 1 Figure 1. Different positivity conditions for (3)

Figure 2 .

 2 Figure 2. The results of simulation for the motivating example

  All authors except E. Fridman are with Non-A team @ INRIA -LNE, Parc Scientifique de la Haute Borne, 40 avenue Halley, 59650 Villeneuve d'Ascq, France and CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France.

	class of nonlinear systems in the output canonical form) with
	delayed measurements in Section 4. Example of numerical
	simulation is presented in Section 5.

E. Fridman is with School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. D. Efimov is with Department of Control Systems and Informatics, University ITMO, 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia. This work was supported in part by Région Nord-Pas de Calais, by the Government of Russian Federation (Grant 074-U01), the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031) and by Israel Science Foundation (grant No 754/10).

  a measurable and locally essentially bounded inputu : R + → R p the symbol ||u|| [t0,t1] denotes its L ∞ norm ||u|| [t0,t1] = ess sup{||u(t)||, t ∈ [t 0 , t 1 ]},or simply ||u|| if t 0 = 0 and t 1 = +∞, the set of all such inputs u ∈ R p with the property ||u|| <

	∞ will be
	denoted as L p ∞ ; • for a matrix A ∈ R n×n the vector of its eigenvalues is
	denoted as λ(A);
	• I n and 0 n×m denote the identity and zero matrices of
	dimensions n × n and n × m respectively;
	• a R b corresponds to an elementwise relation R (a and
	b are vectors or matrices): for example a < b (vectors)
	means ∀i : a i < b i ; for φ, ϕ ∈ C τ the relation
	φ R ϕ has to be understood elementwise for all domain
	of definition of the functions, i.e. φ(s) R ϕ(s) for all
	s ∈ [-τ, 0];
	• for a symmetric matrix Υ, the relation Υ ≺ 0 (Υ 0)


Emmanuel Bernuau

Denis Efimov

Emmanuel Moulay

Wilfrid Perruquetti

Homogeneous continuous finite-time observer for the triple integrator

 in the case of the triple integrator. In [1], convergence of the algorithm was only proved when the degree of homogeneity was sufficiently close to 0 without more tractable information. We show here that, in the case of the triple integrator, the observer presents global finite-time stability for any negative degree under constructive conditions on the gains. This is achieved with a homogeneous Lyapunov function design. Simulations of the proposed observer are also provided.

Introduction

Even though mechanical or electrical systems often have a second order model, problems where third order systems appear are also common and the simplest and canonical form of these systems is a triple integrator. Most of the current techniques for linear or nonlinear feedback stabilization and observation provide an asymptotic or exponential stability: the obtained closed-loop dynamics is locally Lipschitz and the system trajectories settle at the origin when the time is approaching infinity. Such a rate of convergence is not admissible in many applications, this is why the Finite-Time Stability (FTS) notion has been quickly developing during the last decades: solutions of a FTS system reach the equilibrium point in a finite time. For example, for x ∈ R and α ∈ (0, 1), the solutions of ẋ = -sign(x)|x| α starting from x 0 ∈ R at t 0 = 0 are

sign(x 0 )[|x 0 | 1-α -(1 -α)t] 1 1-α if 0 ≤ t ≤ |x 0 | 1-α 1-α 0 if t > |x 0 | 1-α 1-α .
Let us note that the right hand side of the above differential equation is not Lipschitz. In fact, finite-time convergence implies non-uniqueness of solutions (in backward time) which is not possible in the presence of Lipschitzcontinuous dynamics, where different maximal trajectories never cross. The problem of finite-time stability has been developed for continuous systems giving sufficient and necessary condition (see [START_REF] Haimo | Finite time controllers[END_REF][START_REF] Moulay | Finite time stability of non linear systems[END_REF]). In addition, necessary and sufficient conditions appear for discontinuous systems (see [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]). It was observed in many papers that FTS can be achieved if the system is locally asymptotically stable and homogeneous with negative degree [START_REF] Rosier | Etude de Quelques Problèmes de Stabilisation[END_REF]. This is why the homogeneity plays a central role in the FTS system design. The reader may found additional properties and results on homogeneity in [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF][START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF][START_REF] Hermes | Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF][START_REF] Orlov | Finite time stability of homogeneous switched systems[END_REF]. The homogeneity property was used many times to design FTS state controls [START_REF] Bhat | Continuous finite-time stabilization of the translational and rotational double integrator[END_REF][START_REF] Floquet | Higherorder sliding mode stabilization for a class of nonholonomic perturbed systems[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Qian | Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization[END_REF][START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], FTS observers [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF][START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF][START_REF] Menard | A global high-gain finite-time observer[END_REF], consensus protocols [START_REF] Zhang | Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements[END_REF] and FTS output feedback [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF][START_REF] Li | Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems[END_REF].

In [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF], a generic class of homogeneous continuous observers for an nth integrator was developed. Their simple shape, combined with FTS and robustness properties granted by homogeneity were making them very promising candidates for observation. However, the proof of stability relied on a continuity argument and the result only held when the homogeneity degree was sufficiently close to 0, i.e., when the observer was almost linear. No constructive procedure was known, given a degree of homogeneity, to check whether the observer was stable or not, for n > 2.

In this paper, we shall give explicit conditions under which the result of [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF] holds in the case of the triple integrator. The paper is organized as follows. Section II is devoted to preliminaries. In section III we present the system of interest, the proposed observer and the corresponding error equation. In section IV, we introduce our candidate Lyapunov function and give conditions under which this function is positive definite. The derivative of this function is studied in section V, and the main results are compiled in theorems 1 and 2. Simulations are presented in section VI and finally a conclusion summarizes the paper.

Preliminaries

Through the paper the following notation will be used. For any real number α ≥ 0 and for all x ∈ R we define x α = sign(x)|x| α .

Proposition 1 (Young's inequality). For any x, y ∈ R, any p, q > 0 such that 1 p + 1 q = 1 and any ε > 0, the following inequality holds

|xy| ≤ ε |x| p p + 1 ε |y| q q .
This well known inequality will be used extensively throughout the paper.

Finite-time stabilization

Let us consider a continuous vector field f and the system

ẋ = f (x), x ∈ R n . (1)
Definition 1. [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] The origin of the system (1) is finitetime stable (FTS) iff there exists a neighborhood of the origin V such that:

1. For any x 0 ∈ V there exists t 0 ≥ 0 such that for any solution x(t) of (1) such that x(0) = x 0 we have x(t) = 0 for all t ≥ t 0 . The infimum T (x 0 ) of all such t 0 allows us to define the function T : V → R + , called the settling-time function of the system (1).

2. For any neighborhood of the origin U 1 ⊂ V , there exists a neighborhood of the origin U 2 such that for any x 0 ∈ U 2 and any solution x(t) of (1) such that x(0) = x 0 we have x(t) ∈ U 1 for all t ≥ 0.

Moreover, if the neighborhood V can be chosen to be R n , then the origin of the system (1) is said to be globally finite-time stable (GFTS).

Homogeneity

Let r = (r 1 , . . . , r n ) be a n-uplet of positive real numbers, thereafter called a generalized weight. Then Λ r x = (. . . , λ r i x i , . . . ) for any positive number λ represents a mapping x → Λ r x usually called a dilation (see [START_REF] Hermes | Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF]).

Definition 2. A function

h : R n → R is r-homogeneous of degree κ ∈ R if for all x ∈ R n and all λ > 0 we have h(Λ r x) = λ κ h(x).
Definition 3. A vector field f : R n → R n is rhomogeneous of degree κ if for all x ∈ R n and all λ > 0 we have f (Λ r x) = λ κ Λ r f (x), or equivalently, if the coordinate functions f i are r-homogeneous of degree κ + r i . When such a property holds, the corresponding system (1) is said to be r-homogeneous of degree κ.

Problem formulation

Consider the following system defined for z

= (z 1 , z 2 , z 3 ) T ∈ R 3        ż1 = z 2 ż2 = z 3 ż3 = u y = z 1 (2)
where z 1 , z 2 and z 3 are the states of the system, u is the input and y is the output. We follow [START_REF] Perruquetti | Finitetime observers: application to secure communication[END_REF] and define, for β ∈ ( 2 3 , 1), the following estimator

   ż1 = ẑ2 + l 1 y -ẑ1 β ż2 = ẑ3 + l 2 y -ẑ1 2β -1 ż3 = u + l 3 y -ẑ1 3β -2 . (3)
Consider now the estimation error x = zẑ. Its dynamics is given by

   ẋ1 = x 2 -l 1 x 1 β ẋ2 = x 3 -l 2 x 1 2β -1 ẋ3 = -l 3 x 1 3β -2 . ( 4
)
We intend to find explicit conditions on the gains l 1 , l 2 and l 3 and on the power β such that the origin is a GFTS equilibrium of the system (4). Remark 1. The right-hand side of system ( 4) is not Lipschitz continuous. Even though we could select β ≥ 1 to ensure Lipschitz continuity, we would lose the finitetime stability which is one of the desired properties of the algorithm. Therefore, we will stick to β < 1.

Lyapunov function design

Let us define, for ρ, δ > 0, the following candidate Lyapunov's function

V (x) = (l 2 -β ρl 3 ) |x 1 | 2β 2β -ρl 3 x 1 β x 2 + x 2 2 2 - l 1 l 2 x 2 x 3 β 2β -1 + δ 2β -1 2β |x 3 | 2β 2β -1 -x 1 x 3 . (5)
Using the Young's inequality, we find that V is definite positive if the following inequalities hold

l 2 -β ρl 3 -ε 1 ρl 3 β -ε 3 > 0 (6) 1 - ρl 3 ε 1 - l 1 ε 2 l 2 > 0 (7) δ (2β -1) - l 1 β l 2 ε 2 - 2β -1 ε 3 > 0 (8)
We claim that, under the two following conditions, we can find ε 1 , ε 2 and ε 3 such that ( 6), ( 7) and ( 8) hold

l 2 -β ρl 3 -β ρ 2 l 2 3 > 0 (9) δ > β l 2 1 l 2 -β 2 ρl 3 l 2 1 + 2 2β -1β ρl 1 l 2 l 3 + (2β -1)l 2 2 (2β -1)l 2 2 (l 2 -β ρl 3 -β ρ 2 l 2 3 ) . ( 10
)
Proposition 2. Under conditions ( 9) and ( 10), the function V defined by ( 5) is positive definite.

The proof is omitted due to space limitations. Let us remark that the conditions ( 9) and ( 10) do not really constrain the gains. Indeed, ρ and δ are free parameters appearing only in the Lyapunov function V and taking ρ small enough ensures condition (9) while taking δ large enough ensures condition [START_REF] Orlov | Finite time stability of homogeneous switched systems[END_REF].

Study of the derivative of V

Our goal in this section is to prove that an adequate choice of the parameters δ and ρ together with conditions on the gains l 1 , l 2 and l 3 can ensure that the derivative of V will be negative definite.

Let us start by mentioning that the function V is not differentiable on the plane x 1 = 0. It will not be a big deal because we will see that V is upper bounded by -CV (β +1)/(2β ) with C > 0 and because the only invariant subspace of the plane x 1 = 0 is {0}. However, to ensure the correctness of the following computations, we assume that x 1 = 0. A direct computation gives

V (x) = (ρl 3 l 2 + l 3 + l 1 β ρl 3 -l 1 l 2 )|x 1 | 3β -1 -β ρl 3 |x 1 | β -1 x 2 2 - l 1 l 2 |x 3 | 3β -1 2β -1 + ρl 2 β (l 1 -1) x 1 2β -1 x 2 + (l 1 -ρl 3 ) x 1 β x 3 + l 1 x 1 2β -1 x 3 β 2β -1 -δ l 3 x 1 3β -2 x 3 1 2β -1 + l 1 l 3 β l 2 (2β -1) x 1 3β -2 x 2 x 3 1-β 2β -1
.

From successive uses of the Young's inequality, we find that V (x) < 0 on {x 1 = 0} if there exist ε 1 , ε 2 , ε 3 > 0 such that the following conditions hold

l 1 > ρl 3 (11) l 1 [-(3β -1)l 2 + β ε 2 + ε 3 (2β -1)] + l 3 (3β -1) ρl 2 + 1 + l 1 β ρ + ρβ |l 1 -1|ε 1 2 -ρβ ε 2 + δ (3β -2) + l 1 l 2 β (5β -3) 2(2β -1) < 0 (12) ρ |l 1 -1| ε 1 -2 + l 1 l 2 (2β -1) < 0 (13) l 1 - 3β -1 l 2 + 2β -1 ε 2 + β ε 3 + l 3 - ρ(2β -1) ε 2 + δ + l 1 β (1 -β ) l 2 (2β -1) < 0. ( 14
)
There exists ε 1 > 0 such that inequation (13) holds if and only if

-2ρ + l 1 l 2 (2β -1) < 0 ( 15
)
and then ε 1 has to be chosen such that

ε 1 > |l 1 -1|l 2 ρ(2β -1) 2l 2 ρ(2β -1) -l 1 . (16)
Hence, there exist solutions (ε 1 , ε 2 , ε 3 ) of inequalities ( 12) -( 14) if and only if there exist solutions

(ε 2 , ε 3 ) of l 1 [-(3β -1)l 2 + β ε 2 + ε 3 (2β -1)] +l 3 (3β -1)(ρl 2 + 1 + l 1 β ρ + ρ 2 β |l 1 -1| 2 l 2 (2β -1) 4l 2 ρ(2β -1) -2l 1 ) -ρβ ε 2 + δ (3β -2) + l 1 l 2 β (5β -3) 2(2β -1) < 0, ( 17
)
together with ( 11), ( 14) and [START_REF] Qian | Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization[END_REF]. Now denoting

A 1 = -(3β -1)l 2 + β ε 2 + (2β -1)ε 3 A 3 = (3β -1)(ρl 2 + 1 + l 1 β ρ + ρ 2 β |l 1 -1| 2 l 2 (2β -1) 4l 2 ρ(2β -1) -2l 1 ) -ρβ ε 2 + δ (3β -2) + l 1 l 2 β (5β -3) 2(2β -1) B 1 = - 3β -1 l 2 + 2β -1 ε 2 + β ε 3 B 3 = - ρ(2β -1) ε 2 + δ + l 1 β (1 -β ) l 2 (2β -1)
we see that ( 17) and ( 14) can be written

A 1 l 1 + A 3 l 3 < 0, B 1 l 1 + B 3 l 3 < 0.
Hence, the negativity of V would be ensured for small enough l 3 if we were able to prove that the terms A 1 and B 1 are negative.

Lemma 1. For any β ∈ (2/3, 1) and any l 2 > 0 there exists a pair (ε 2 , ε 3 ) of positive reals such that

β ε 2 + (2β -1)ε 3 < l 2 (3β -1)
2β -1

ε 2 + β ε 3 < 3β -1 l 2 . ( 18
)
The proof is omitted due to space limitations. Finally we can state our main result.

Theorem 1. For any β ∈ (2/3, 1), any l 1 , l 2 > 0 and any ρ and δ such that

ρ > l 1 2(2β -1)l 2 (19) δ > β l 2 1 + (2β -1)l 2 (2β -1)l 2 2 ( 20
)
there exist l 3 > 0 such that V is positive definite and V (x) < 0 for all x ∈ {x 1 = 0}.

Proof. From all the discussion beforehand, it suffices to find positive reals l 3 , ε 2 and ε 3 such that the inequalities ( 9), ( 10), ( 11), ( 14), ( 15) and ( 17) hold.

Clearly, the condition ( 19) is just a rewritting of inequality [START_REF] Qian | Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization[END_REF]. Inequality ( 9) is equivalent to

l 3 < 1 2ρ 1 + 4l 2 β -1 . (21)
Condition [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF] ensures that inequality (10) has solutions. Inequality [START_REF] Orlov | Finite time stability of homogeneous switched systems[END_REF] can then be rewritten under the form

l 3 < κ(l 1 , l 2 , ρ, δ ). (22)
An explicit form of the function κ can be computed but is not presented here due to space limitations. Inequality ( 11) can be rewritten

l 3 < l 1 ρ . ( 23
)
Now, granted that we choose ε 2 and ε 3 for which [START_REF] Menard | A global high-gain finite-time observer[END_REF] holds, we see that A 3 > 0 and then inequalities ( 17) and ( 14) can be rewritten

l 3 < - A 1 A 3 l 1 (24) B 3 ≤ 0 or l 3 < - B 1 B 3 l 1 . (25)
Finally, by lemma 1, we get A 1 < 0 and B 1 < 0 and all inequalities ( 21) -( 25) hold if l 3 is chosen small enough, which concludes the proof.

Let us remark that, although this theorem seems to only state the existence of a gain l 3 > 0 such that the Lyapunov function V is positive definite and its derivative V is negative for x 1 = 0, it is actually constructive. To compute admissible values of l 3 > 0, it suffices to 1. select β ∈ (2/3, 1), l 1 , l 2 > 0, ρ and δ such that [START_REF] Zhang | Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements[END_REF] and ( 20) hold;

2. compute a couple (ε 2 , ε 3 ) such that (18) holds;

3. select l 3 such that all inequalities ( 21) -(25) hold.

Example 1. Let us select β = 0.75 and l 1 = l 2 = 1. Conditions ( 19) and ( 20) read ρ > 1 and δ > 2.5 so we choose ρ = 6 and δ = 3. Then we select ε 2 = 5 6 and ε 3 = The choice l 3 = 0.002 then ensures that V is positive definite and V < 0 for x 1 = 0.

We can now express the result in a more theoretical form.

Theorem 2. For any β ∈ (2/3, 1), for any l 1 > 0 and l 2 > 0, there exists l 3 > 0 such that the observer (3) recovers in finite time the state z of the system (2) for any initial state z 0 ∈ R 3 .

Proof. We need to prove that the origin is a GFTS equilibrium for the error equation ( 4). Let us select ρ, δ , ε 2 , ε 3 and l 3 such that inequalities [START_REF] Zhang | Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements[END_REF], [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF], [START_REF] Menard | A global high-gain finite-time observer[END_REF] and ( 21) -(25) hold.

Let us denote -a = sup{ V (x) : V (x) = 1, x 1 = 0}. Given that V < 0 on {V (x) = 1, x 1 = 0}, we have a ≥ 0. Moreover, {V (x) = 1} is compact: indeed, any continuous homogeneous function is proper [START_REF] Bhat | Continuous finite-time stabilization of the translational and rotational double integrator[END_REF] and a straightforward computation shows that V and V are (1, β , 2β -1)-homogeneous functions of degree 2β and 3β -1. Since 0 / ∈ {V (x) = 1}, we get a > 0. Classical manipulations on homogeneous functions then show that for any {x ∈ R 3 :

x 1 = 0} we have V (x) ≤ -aV (x) (3β -1)/(2β ) .
Now, given that no solution of (4), except x(t) = 0, stays on {x 1 = 0}, the set {V ≤ 1} is strictly positively invariant and therefore the origin is globally asymptotically stable for system (4) [START_REF] Bhat | Continuous finite-time stabilization of the translational and rotational double integrator[END_REF]. But system (4) is (1, β , 2β -1)-homogeneous of degree β -1 < 0, and therefore the origin is GFTS [START_REF] Bhat | Continuous finite-time stabilization of the translational and rotational double integrator[END_REF].

Sometimes the conditions ( 21) -(25) may lead to very small values of the gain l 3 . It is possible to rescale it with the following procedure. Proposition 3. Set β ∈ (2/3, 1). If the origin is GFTS for system (4) with a set of gains (l 1 , l 2 , l 3 ), then for any λ > 0 the origin is also GFTS for system (4) with the set of gains (λ l 1 , λ 2 l 2 , λ 3 l 3 ).

The proof is omitted due to space limitations. Remark that, for this new set of gains, corresponding values δ , ρ, ε 2 and ε 3 such that all the conditions (20) -(25) hold may not exist or be different, but the finite-time stability is still ensured.

Example 2 (Example 1 continued). We take again β = 0.75, ρ = 6, δ = 3, ε 2 = 5 6 and ε 3 = 6 5 . The set of gains (1, 1, 0.002) is valid, thus by proposition 3, taking λ = 10 3 , the set of gains (10 3 , 10 6 , 2.10 6 ) is also valid. However, inequalities [START_REF] Zhang | Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements[END_REF] and ( 15) read ρ > 10 -3 and ρ < 5.10 -4 and thus this set of gains could not have been found by the direct method exposed after theorem 1.

A local approach can also be used to extend our result to any linear system with an output of relative degree 3. and λ are real coefficients. Then, if the gains l i are selected according to the conditions presented before, the following observer leads the origin of the error equation to being locally finite-time stable:

ẋ = A x + Bu +   l 1 x 1 -x1 β l 2 x 1 -x1 2β -1 l 3 x 1 -x1 3β -2   .
Proof. Computing a local homogeneous approximation of the error equation leads to system (4) which is finitetime stable. Therefore the original system is locally finite-time stable for gains selected according to the conditions presented before. For details about local homogeneous approximations, see [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

Examples and Simulations

In this section, we would like to illustrate the applicability of the proposed algorithm in showing how to use it on concrete examples and demonstrate the efficiency of the proposed observer compared to a Luenberger observer.

Consider an electric motor acting on a rigid arm turning around a central point in the horizontal plane.

Let denote θ the angle of the arm and θ the angular velocity of the arm. The force generated by the motor is Ki where i is the intensity of the current and K > 0 a constant. We find

m θ = F -bθ u = L di dt + Ri + k θ
where u is the input tension, b > 0 is a coefficient of friction, m is the mass of the arm and L, R and k are positive coefficients of the motor. The output is supposed to be θ and is of relative degree 3. Following Corollary 1, we get a locally finite-time stable observer. Now, we would like to compare the proposed observer with a classical Luenberger observer. As in Example 1, we take β = 0.75, ρ = 6, δ = 3, ε 2 = 5 6 and ε 3 = 6 5 . The set of gains (1, 1, 0.002) is valid, thus by proposition 3, taking λ = 20, the set of gains [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF]400,[START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF]) is also valid. We will compare the estimation error given by ( 4) with the estimation error

   ẋ1 = x 2 -20x 1 ẋ2 = x 3 -400x 1 ẋ3 = -16x 1 . (26)
The results of simulations are presented in Fig. 1. The eigenvalues of the matrix corresponding to the linear system (26) are -9.980 + 17.309i, -9.980 -17.309i and -0.040, ensuring the asymptotic stability. However, the small value of the third eigenvalue does not allow the linear observer to recover quickly the signal. We can see that in about 1.2s the homogeneous observer has converged, while x 3 is still around -0.2 after 1.6s.

Conclusion

In this paper, we gave explicit conditions under which the observer (3) converges to the state of the system (2) so that the origin is a GFTS equilibrium of the corresponding error equation (4). We introduced a homogeneous Lyapunov function and gave explicit conditions under which this function is positive definite and its derivative is negative. Examples of use of these conditions were provided, as well as a method to derive other valid choices for the gains. Computer simulations demonstrated the efficiency of the proposed algorithm.

Future works include an optimization of the parameters ρ and δ to let the gain l 3 take values as large as possible. Also, we would like to go further and try to extend this method to n th -integrators.

6 5

 6 and easily check that (18) holds. Then inequalities (21) -(24) read (numerical values have been truncated) l 3 ≤ 0.126 l 3 ≤ 0.145 l 3 ≤ 0.166 l 3 ≤ 0.002 while inequality (25) trivially holds because B 3 < 0.

Corollary 1 .c 1 c 2 c 3 

 13 Consider a linear system of dimension 3 such that the output y is of relative degree 3. The system reads ẋ = Ax + Bu with x = [y, ẏ, ÿ] T , A = and B = [0, 0, λ ] T where c 1 , c 2 , c 3

coming from the following observer of system ( 2

 2 + 20(y -ẑ1 ) ż2 = ẑ3 + 400(y -ẑ1 ) ż3 = u(y) + 16(y -ẑ1 ) .

Figure 1 .

 1 Figure 1. Comparison between homogeneous (top) and linear (bottom) estimation errors.

This work was supported in part by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031) and by the ANR funded project Qualiphe.


Denis Efimov

Romeo Ortega

Johannes Schiffer

ISS of multistable systems with delays: application to droop-controlled inverter-based microgrids Denis Efimov, Romeo Ortega and Johannes Schiffer

Abstract-Motivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov-Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given.

I. INTRODUCTION

The increasing penetration of renewable distributed generation (DG) units at the low and medium voltage levels has a strong impact on the power system structure [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Pravin | Smart operation of smart grid: Risk-limiting dispatch[END_REF], [START_REF] Fang | Smart grid -the new and improved power grid: a survey[END_REF]. This fact requires new control and operation strategies to ensure a reliable and efficient electrical power supply [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Green | Control of inverter-based micro-grids[END_REF]. An emerging concept to address these challenges is the microgrid [START_REF] Lasseter | Microgrids[END_REF], [14], [START_REF] Farhangi | The path of the smart grid[END_REF]. A microgrid is a locally controllable subset of a larger electrical network. It is composed of several DG units, storage devices and loads.

Typically, most DG units in an AC microgrid are connected to the network via AC inverters [START_REF] Green | Control of inverter-based micro-grids[END_REF]. Under ideal conditions, an inverter-based DG unit can be modeled as an ideal controllable voltage source [START_REF] Lopes | Defining control strategies for microgrids islanded operation[END_REF], [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF]. Furthermore, a popular control scheme to operate inverter-based DG units with the purpose to achieve frequency synchronization and power sharing in the network is droop control [START_REF] Chandorkar | Control of parallel connected inverters in standalone AC supply systems[END_REF], [START_REF] Guerrero | Advanced control architectures for intelligent microgrids -part I: Decentralized and hierarchical control[END_REF]. Conditions for stability in droop-controlled microgrids with inverters modeled as ideal controllable voltage sources have been derived, e.g., in [START_REF] Simpson-Porco | Synchronization and power sharing for droop-controlled inverters in islanded microgrids[END_REF], [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF], [START_REF] Münz | Voltage and angle stability reserve of power systems with renewable generation[END_REF].

In general, inverter-based microgrids operated with droop control have several equilibria [START_REF] Simpson-Porco | Synchronization and power sharing for droop-controlled inverters in islanded microgrids[END_REF], [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF]. Thus they are multistable systems. Stability analysis [START_REF] Angeli | Detection of multistability, bifurcations and hysteresis in a large class of biological positivefeedback systems[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF], [START_REF] Van Handel | Almost global stochastic stability[END_REF], [START_REF] Monzón | Local and global aspects of almost global stability[END_REF], [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF], [START_REF] Rajaram | Connection between almost everywhere stability of an ode and advection pde[END_REF], [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF], [START_REF] Rumyantsev | Stability and stabilization of motion with respect to part of variables[END_REF], [START_REF] Smale | Differentiable dynamical systems[END_REF] and robust stability analysis [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF], [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], [START_REF] Chaves | Bistable biological systems: A characterization through local compact input-to-state stability[END_REF], [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF] for this class of systems is rather complicated. Recently, the ISS theory [START_REF] Sontag | On the input-to-state stability property[END_REF] has been extended to multistable systems in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] (see also [START_REF] Christopher | Inputto-state stability, integral input-to-state stability, and unbounded level sets[END_REF] for discussion on ISS property with respect to an unbounded set).

Furthermore, in a practical setup, the droop control scheme is applied to an inverter by means of digital discrete time control. Besides clock drifts, see, e.g., [START_REF] Schiffer | Droop-controlled inverters are robust to clock drifts[END_REF], digital control usually introduces time delays [START_REF] Kukrer | Discrete-time current control of voltage-fed threephase PWM inverters[END_REF], [START_REF] Maksimovic | Small-signal discrete-time modeling of digitally controlled PWM converters[END_REF], [START_REF] Nussbaumer | Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system[END_REF]. According to [START_REF] Nussbaumer | Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system[END_REF], the main reasons for this are 1) sampling of control variables, 2) calculation time of the digital controller and 3) generation of the pulse-width-modulation. We refer the reader to, e.g., [START_REF] Nussbaumer | Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system[END_REF] for further details. To the best of the authors' knowledge this fact has yet not been considered in previous analysis of droop-controlled microgrids.

Motivated by the abovementioned phenomenon, the main contribution of the present paper is to extend the recently derived ISS framework for multistable systems [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] to multistable systems with delay. In particular, sufficient conditions for ISS of multistable systems in the presence of delays are given in terms of a Lyapunov-Razumikhin function. It is also shown that ISS multistable systems are robust with respect to feedback delays. This result is illustrated via the example of a nonlinear pendulum. Next, based on the established results, we provide a condition for asymptotic phase-locking in a microgrid composed of two droop-controlled inverters with delay. The analysis is conducted for a simplified inverter model derived under the assumptions of constant voltage amplitudes and ideal clocks, as well as negligible dynamics of the internal inverter filter and controllers. In that scenario, the delay merely affects the phase angle of the inverter output voltage. The stability results are illustrated by simulations.

II. PRELIMINARIES

For an n-dimensional C 2 connected and orientable Riemannian manifold M without a boundary, let the map f (x, d) : M × R m → T x M be of class C 1 , and consider a nonlinear system of the following form:

ẋ(t) = f (x(t), d(t)), (1)
where the state x ∈ M and d(t) ∈ R m (the input d(•) is a locally essentially bounded and measurable signal) for t ≥ 0. We denote by X(t, x 0 ; d) the uniquely defined solution of (1) at time t fulfilling X(0, x 0 ; d) = x 0 . Together with (1) we will analyze its unperturbed version:

ẋ(t) = f (x(t), 0). (2)
A set S ⊂ M is invariant for the unperturbed system (2) if X(t, x; 0) ∈ S for all t ∈ R and for all x ∈ S. Define the distance from a point x ∈ M to the set S ⊂ M as |x| S = min a∈S δ(x, a), where the symbol δ(x 1 , x 2 ) denotes the Riemannian distance between x 1 and x 2 in M , |x| = |x| {0} for x ∈ M or a usual euclidean norm of a vector x ∈ R n . For a signal d : R → R m the essential supremum norm is defined as d ∞ = ess sup t≥0 |d(t)|.

A. Decomposable sets

Let Λ ⊂ M be a compact invariant set for (2).

Definition 1. [22]

A decomposition of Λ is a finite and disjoint family of compact invariant sets Λ 1 , . . . , Λ k such that

Λ = k i=1 Λ i .
For an invariant set Λ, its attracting and repulsing subsets are defined as follows:

W s (Λ) = {x ∈ M : |X(t, x, 0)| Λ → 0 as t → +∞}, W u (Λ) = {x ∈ M : |X(t, x, 0)| Λ → 0 as t → -∞}. Define a relation on W ⊂ M and D ⊂ M by W ≺ D if W s (W) ∩ W u (D) = . Definition 2. [22] Let Λ 1 , . . . , Λ k be a decomposition of Λ, then 1. An r-cycle (r ≥ 2) is an ordered r-tuple of distinct indices i 1 , . . . , i r such that Λ i1 ≺ . . . ≺ Λ ir ≺ Λ i1 . 2. A 1-cycle is an index i such that [W u (Λ i ) ∩ W s (Λ i )] - Λ i = .
3. A filtration ordering is a numbering of the Λ i so that

Λ i ≺ Λ j ⇒ i ≤ j.
As we can conclude from Definition 2, existence of an rcycle with r ≥ 2 is equivalent to existence of a heteroclinic cycle for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. Furthermore, existence of a 1-cycle implies existence of a homoclinic cycle for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. Definition 3. The set W is called decomposable if it admits a finite decomposition without cycles, W = k i=1 W i , for some non-empty disjoint compact sets W i , which form a filtration ordering of W, as detailed in definitions 1 and 2.

B. Robustness notions

The following robustness notions for systems represented by [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF] have been introduced in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF]. Definition 4. We say that the system (1) has the practical asymptotic gain (pAG) property if there exist η ∈ K ∞ and a non-negative real q such that for all x ∈ M and all measurable essentially bounded inputs d(•) the solutions are defined for all t ≥ 0 and the following holds:

lim sup t→+∞ |X(t, x; d)| W ≤ η( d ∞ ) + q.
If q = 0, then we say that the asymptotic gain (AG) property holds.

Definition 5. We say that the system (1) has the limit property (LIM) with respect to W if there exists µ ∈ K ∞ such that for all x ∈ M and all measurable essentially bounded inputs d(•) the solutions are defined for all t ≥ 0 and the following holds:

inf t≥0 |X(t, x; d)| W ≤ µ( d ∞ ).
Definition 6. We say that the system (1) has the practical global stability (pGS) property with respect to W if there exist β ∈ K ∞ and q ≥ 0 such that for all x ∈ M and measurable essentially bounded inputs d(•) the following holds for all t ≥ 0:

|X(t, x; d)| W ≤ q + β(max{|x| W , d ∞ }).
It has been shown in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] that to characterize pAG property in terms of Lyapunov functions the following notion is appropriate.

Definition 7. We say that a C 1 function V : M → R is a practical ISS-Lyapunov function for (1) if there exists K ∞ functions α 1 , [α 2 ]
, α 3 and γ, and scalar q ≥ 0 [and c ≥ 0] such that

α 1 (|x| W ) ≤ V (x) ≤ [α 2 (|x| W + c)],
the function V is constant on each W i and the following dissipation holds:

DV (x)f (x, d) ≤ -α 3 (|x| W ) + γ(|d|) + q.
If the latter inequality holds for q = 0, then V is said to be an ISS-Lyapunov function.

Notice that α 2 and c are in square brackets as their existence follows (without any additional assumptions) by standard continuity arguments.

The main result of [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] connecting these robust stability properties is stated below, it extends the results of [START_REF] Sontag | On characterizations of input-to-state stability with respect to compact sets[END_REF], [START_REF] Sontag | New characterizations of input-to-state stability[END_REF] obtained for connected sets.

Theorem 1. Consider a nonlinear system as in (1) and let a compact invariant set containing all αand ω-limit sets of (2) W be decomposable (in the sense of Definition 3). Then the following facts are equivalent.

1. The system admits an ISS Lyapunov function; 2. The system enjoys the AG property; 3. The system admits a practical ISS Lyapunov function; 4. The system enjoys the pAG property; 5. The system enjoys the LIM property and the pGS.

A system in (1), for which this list of equivalent properties is satisfied, is called ISS with respect to the set W [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF].

III. MULTISTABLE SYSTEMS

ẋ(t) = F (x t , d t ), x 0 ∈ C τ , (3)
where the map F : C τ × D → T x M is of class C 1 (we will denote a set of continuous functions ξ : [-τ, 0] → M by C τ ), x(t) ∈ M is the state, x t ∈ C τ and d t ∈ D for all t ≥ 0. We denote by X(t, x 0 ; d) the uniquely defined solution of (3) at time t fulfilling X(θ, x 0 ; d) = x 0 (θ) for all θ ∈ [-τ, 0];

X x0,d t (θ) = X(t + θ, x 0 ; d) for θ ∈ [-τ, 0]
. Define as in [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF]

|x t | = max θ∈[-τ,0] |x(t + θ)|, ||x|| t0 = sup t≥t0 |x t | = sup t≥t0-τ |x(t)|.
Again, together with (3), we will analyze its unperturbed version:

ẋ(t) = F (x t , 0). (4)
A set S ⊂ C τ is invariant for the unperturbed system (4) if X x0,0 t ∈ S for all t ∈ R + and for all x 0 ∈ S. Define the distance from a function ξ ∈ C τ to a set S ⊂ C τ as ||ξ|| S = min α∈S |ξ -α|.

Let W ⊂ M be a set, denote by W a subset of W = {ξ ∈ C τ : ξ(t) ∈ W ∀t ∈ [-τ, 0]} such that if ζ ∈ W then ζ = X ξ,0 τ for ξ ∈ W.
For stability analysis in time-delay systems it is required to define a distance to invariant sets in two spaces: in R n with respect to the set W and in C τ with respect to corresponding invariant set W (functions from C τ taking values in W and solutions of (3)). The following stability notions for (3) are considered in this work. Definition 8. The system (3) has the pAG property with respect to the set W if there exist η ∈ K ∞ and a non-negative real q such that for all x 0 ∈ C τ and all bounded piecewise continuous inputs d(•) the solutions are defined for all t ≥ 0 and the following holds:

lim sup t→+∞ |X(t, x 0 ; d)| W ≤ η( d t 0 ) + q.
If q = 0, then we say that the AG property holds. This property can be equivalently stated as

lim sup t→+∞ ||X x0,d t || W ≤ η( d t 0 ) + q
and it implies that (a subset of) W is invariant for (4) if q = 0. Definition 9. The system (3) has the pGS property with respect to the set W if there exist β ∈ K ∞ and q ≥ 0 such that for all x 0 ∈ C τ and all bounded piecewise continuous inputs d(•) the following holds for all t ≥ 0:

|X(t, x 0 ; d)| W ≤ q + β(max{||x 0 || W , d t 0 }).
To characterize pAG and pGS properties for a time-delay system (3) the Lyapunov-Razumikhin approach is used in this work. Given a continuous function

x : [-τ, +∞) → M with a C 1 function U : M → R denote U (t) = U (x(t)), if x(t) = X(t, x 0 ; d)
D + U (t) = lim sup h→0 + U (t + h) -U (t) h . Definition 10. A C 1 function U : M → R is a practical ISS-Lyapunov-Razumikhin (ISS-LR) function for (3) if there exist K ∞ functions α 1 , [α 2 ]
, α 4 , γ and γ U , γ U (s) < s for all s > 0, and scalar q ≥ 0 [and c ≥ 0] such that

α 1 (|x| W ) ≤ U (x) ≤ [α 2 (|x| W + c)], U (t) ≥ max{γ U (|U t |), γ(|d t |), q} ⇒ D + U (t) ≤ -α 4 [U (t)].
If the latter inequality holds for q = 0, then U is said to be an ISS-LR function.

The following result can be stated connecting pAG, pGS properties and existence of an ISS-LR function.

Theorem 2. Consider the system (3). Suppose there exists an ISS-LR function U : M → R as in Definition 10. Then the system (3) admits the pAG property from Definition 8 with η(s) = α -1 1 • γ(s) and pGS property from Definition 9. Proof. The proof mainly follows the ideas of [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF].

IV. ISS OF MULTISTABLE SYSTEMS WITH DELAYED

PERTURBATIONS

In this section we consider the robustness of the system (1) with respect to a disturbance d, which is dependent on a delayed state. The analysis is conducted under the assumption that the system (1) is ISS with respect to a set W.

A. Robustness analysis

If ( 1) is ISS with respect to the set W, then by Theorem 1 there exists an ISS Lyapunov function V as in Definition 7. From the inequalities α

3 [0.5α -1 2 • V (x)] ≤ α 3 (0.5[|x| W + c]) ≤ α 3 (|x| W ) + α 3 (c) we obtain DV (x)f (x, d) ≤ -α 4 [V (x)] + γ(|d|) + q,
where α 4 (s) = α 3 [0.5α -1 2 (s)] and q = q + α 3 (c). Assume that the input d has two terms d 1 and d 2 , and d 2 is a function of x t ∈ C τ for some τ > 0, i.e.:

d = d 1 + d 2 , d 2 = g(x t ), ( 5
)
where g is a continuous function, |g(x t )| ≤ υ(|V t |) + υ 0 for υ ∈ K ∞ and υ 0 ≥ 0. Denote further for simplicity of notation d = d 1 , then the system ( 1) is transformed to (3) with

F (x t , d t ) = f (x(t), d + g(x t )), and
D + V (t) ≤ -α 4 (V (t)) + γ(2υ(|V t |) + 2υ 0 ) + γ(2|d t |) + q.
This estimate can be rewritten as follows:

V (t) ≥ max{γ V (|V t |), γ(|d t |), q} ⇒ D + V (t) ≤ -0.5α 4 (V (t)), γV (s) = α -1 4 [6γ(4υ(s))], γ(s) = α -1 4 [6γ(2s)], q = α -1 4 [6q + 6γ(4υ 0 )].
It is straightforward to see that if γV (s) < s for all s > 0, then V is an ISS-LR function for (1) with ( 5), and by Theorem 2 this system possesses pAG and pGS properties.

B. Illustration for a nonlinear pendulum

Consider a nonlinear pendulum:

ẋ1 = x 2 , ẋ2 = -Ω 2 sin(x 1 ) -κx 2 + d, (6)
where the state x = [x 1 , x 2 ] takes values on the cylinder M := S × R, d(t) ∈ R is an exogenous disturbance, and Ω, κ are constant positive parameters. The unperturbed system (6) admits a Hamiltonian H(x) = 0.5x 2 2 + Ω 2 (1 -cos(x 1 )) and Ḣ = x 2 d -κx 2 2 . The unperturbed system (6) has two equilibria [0, 0] and [π, 0] (the former is attractive and the latter one is a saddle-point). Thus, W = {[0, 0] ∪ [π, 0]} is a compact set containing all αand ω-limit sets of ( 6) for d = 0. In addition, it is straightforward to check that W is decomposable in the sense of Definition 3.

Lemma 1. The system ( 6) is ISS with respect to the set W.

Proof. Developing ideas of [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], the result follows from Theorem 1 considering a Lyapunov function candidate

V (x) = H(x) + κ (1 -cos(x 1 )) + x 2 sin(x 1 ), which admits derivative V = -[κ -cos(x 1 )]x 2 2 -Ω sin 2 (x 1 ) + sin(x 1 )d + x 2 d (7) ≤ -0.5[κ -]x 2 2 -0.5 Ω 2 sin 2 (x 1 ) +0.5[ Ω -2 + 1 κ - ]d 2 provided that 0 < < κ.
Now consider a time-delay modification of (6):

ẋ1 (t) = x 2 (t), ẋ2 (t) = -Ω 2 sin[x 1 (t -τ )] -κx 2 (t) + d(t), (8)
where τ > 0 is a fixed delay. The unperturbed system (8) with d(t) = 0 has the same equilibria as (6), i.e. [0, 0] and [π, 0]. The system (8) can be represented as follows:

ẋ1 (t) = x 2 (t), ẋ2 (t) = -Ω 2 sin[x 1 (t)] -κx 2 (t) +d(t) + Ω 2 {sin[x 1 (t)] -sin[x 1 (t -τ )]}.
By the Mean value theorem

| sin[x 1 (t)] -sin[x 1 (t -τ )]| = | cos[x 1 (φ)]x 2 (φ)τ | ≤ |x 2 (φ)|τ
for some φ ∈ [t -τ, t]. Thus, the system (8) can be analyzed as a perturbed nonlinear pendulum with part of the input d dependent on the delay. By taking the estimate derived for V in [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] we obtain for µ = Ω -2 + 1 κ-:

D + V (t) ≤ -0.5[κ -]x 2 2 -0.5 Ω 2 sin 2 (x 1 ) +µΩ 4 x 2 2 (φ)τ 2 + µd 2 . It is straightforward to check that V (x) ≤ 0.5[1 + ]x 2 2 + 0.5 sin 2 (x 1 ) + 2[Ω 2 + κ ], x 2 2 ≤ 2 1 - V (x) + 1 - for 0 < < min{1, κ}, then for ρ = min{ κ- 1+ , Ω 2 } D + V (t) ≤ -ρ{V (t) -2[Ω 2 + κ ]} +µΩ 4 x 2 2 (φ)τ 2 + µd 2 ≤ -ρ{V (t) -2[Ω 2 + κ ]} + µΩ 4 1 - τ 2 [2V (φ) + ] + µd 2 .
Therefore,

V (t) ≥ 6 ρ max{2 µΩ 4 1 - τ 2 |V t |, 2ρ[Ω 2 + κ ] + µΩ 4 1 - τ 2 , µd 2 } ⇒ D + V (t) ≤ -0.5ρV (t)
and V is an ISS-LR function for [START_REF] Fang | Smart grid -the new and improved power grid: a survey[END_REF] provided that

12 ρ µΩ 4 1 - τ 2 < 1. (9)
The inequality ( 9) is a delay-dependent stability condition for [START_REF] Fang | Smart grid -the new and improved power grid: a survey[END_REF], which is always satisfied for a sufficiently small delay τ . If we assume that max{0, κ-Ω 2 1+Ω 2 } < < min{1, κ}, then min{ κ- 1+ , Ω 2 } = κ- 1+ and the condition (9) can be rewritten as follows:

τ 2 < 1 12Ω 2 1 - 1 + 1 (κ -) + Ω 2 .
Since the functions 1- 1+ and 1 (κ-)+Ω 2 are decreasing for ∈ (max{0, κ-Ω 2 1+Ω 2 }, min{1, κ}), selecting = max{0, κ-Ω 2 1+Ω 2 } + ε for a sufficiently small ε > 0 optimizes the value of the admissible delay τ to

τ * = 1 2Ω 1 - 1 + 1/3 (κ -) + Ω 2 ,
i.e. for any τ < τ * the system (8) admits V as an ISS-LR function.

V. APPLICATION TO A MICROGRID COMPOSED OF TWO DROOP-CONTROLLED INVERTERS WITH DELAY

By following [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF], under the assumption of constant voltage amplitudes, a lossless droop-controlled microgrid formed by two inverters with delay can be modeled as:

θ(t) = ω 1 (t) -ω 2 (t), (10)
τ P1 ω1 (t) = -ω 1 (t) -k P1 a 12 sin[θ(t -τ d1 )] + c 1 + d 1 (t), τ P2 ω2 (t) = -ω 2 (t) + k P2 a 12 sin[θ(t -τ d2 )] + c 2 + d 2 (t),
where θ(t) ∈ [0, 2π) is the phase difference in inverters, ω 1 (t), ω 2 (t) ∈ R are time-varying frequencies of the inverters; τ d1 > 0 and τ d2 > 0 are delays caused by the digital controls required to implement the droop controls; τ P1 > 0, τ P2 > 0, k P1 > 0, k P2 > 0, a 12 > 0, c 1 and c 2 = -k P 2 k P 1 c 1 are constant parameters, the disturbances d 1 (t) and d 2 (t) represent additional model uncertainties. We say that a solution of ( 10) is phase-locked if θ(t) = θ 0 is constant ∀t ∈ R + for some θ 0 ∈ [0, 2π) [START_REF] Franci | Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback[END_REF]. If this property holds asymptotically, i.e., for t → +∞, we speak about an asymptotic phase-locking.

For brevity of presentation, we impose the following restrictions on the values of parameters.

Assumption 1. τ P1 = τ P2 = τ P > 0 and τ d1 = τ d2 = τ > 0.

Under this assumption, define the new coordinates:

x 1 = θ, x 2 = ω 1 -ω 2 , x 3 = k P2 k P1 ω 1 -ω 2 .
Then the system (10) can be rewritten as follows:

ẋ1 (t) = x 2 (t), (11) τ
P ẋ2 (t) = -x 2 (t) -[k P1 + k P2 ]a 12 sin[x 1 (t -τ )] +[1 + k P2 k P1 ]c 1 + d 1 -d 2 , (12)
τ P ẋ3 (t) = -x 3 (t) + k P2 k P1 d 1 -d 2 . (13)
Thus, the system (10) is decomposed into two independent subsystems: (11), ( 12) and ( 13). The variable x 3 converges asymptotically to zero with the time constant τ P if d 1 = d 2 = 0. Hence, asymptotically the frequencies ω 1 and ω 2 are locked. The dynamics [START_REF] Green | Control of inverter-based micro-grids[END_REF], [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF] have the form of (8) for

d = [1 + k P 2 k P 1 ]c 1 + d 1 -d 2
and, as it has been established above, have pAG and pGS properties from definitions 8 and 9 respectively if condition (9) is satisfied, which for [START_REF] Green | Control of inverter-based micro-grids[END_REF], [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF] takes the form:

τ 2 < min τ -1 P - 1+ , [k P 1 +k P 2 ]a12 τ P 12 [k P 1 +k P 2 ] 2 a 2 12 τ 2 P (1-) [k P 1 +k P 2 ]a12 + 1 1-τ P (14)
for 0 < < min{1, τ -1 P }. Therefore, for a sufficiently small delay τ the inverters may demonstrate a phase-locking behavior. According to [START_REF] Nussbaumer | Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system[END_REF], a good estimate of the overall delay introduced by the digital control is τ = 1.75T S 1 , where T S = 1/f S and f S ∈ R >0 is the switching frequency of the inverter. Since usually f S ∈ [START_REF] Chandorkar | Control of parallel connected inverters in standalone AC supply systems[END_REF][START_REF] Monzón | Local and global aspects of almost global stability[END_REF] kHz [START_REF] Green | Control of inverter-based micro-grids[END_REF], τ is reasonably small in most practical applications. Hence, we expect condition (14) to be satisfied for most practical choices of parameters τ P , k P1 and k P2 .

The analysis is illustrated in a simulation example with the following set of parameters for the system (10): τ P = 1, k P1 = 10, k P2 = 20, a 12 = 0.1, c 1 = 0.2 and τ = 0.05. Condition ( 14) is satisfied for = 0.5 min{1, τ -1 P }. The simulation results are shown in Fig. 1. The solid lines represent the state (θ, ω 1 , ω 2 ) T trajectories for the case d 1 (t) = d 2 (t) = 0, and the dashed lines correspond to d 1 (t) = 0.8 sin(t), d 2 (t) = 0.9 sin(2t). The phase-locking phenomenon is observed in these simulation results.

VI. CONCLUSIONS

Sufficient conditions for ISS of multistable systems with delay have been derived. The conditions have been established using Lyapunov Razumikhin functions. The potential of the presented approach has been illustrated by providing several new robustness properties for a nonlinear pendulum with delay. Furthermore, it has been shown that phaselocking in a lossless droop-controlled microgrid formed by two inverters with delays can be analyzed based on the pendulum model. By exploiting this fact, a delay-dependent condition for ISS of such a microgrid has been presented. Future work will consider an extension of the analysis to more complex inverter models with delays and, e.g., timevarying voltages or internal filter and controllers.

  WITH DELAYS Let τ > 0, for a function d : [-τ, +∞) → R m and t ≥ 0 denote a function d t (•) : [-τ, 0] → R m defined by d t (θ) = d(t + θ) for θ ∈ [-τ, 0]. Denote by D a set of bounded and piecewise continuous functions d t (•) : [-τ, 0] → R m . Consider a functional differential equation on an n-dimensional C 2 connected and orientable Riemannian manifold M without a boundary:

  is a solution to (3) for some piecewise continuous d : [-τ, +∞) → R m and initial condition x 0 ∈ C τ , then the upper right-hand side derivative of U along this solution is

Figure 1 .

 1 Figure 1. Simulation results for the system (10). The solid lines show the state trajectories for the case d 1 (t) = d 2 (t) = 0. The dashed lines correspond to the case d 1 (t) = 0.8 sin(t), d 2 (t) = 0.9 sin(2t).

The overall delay reduces to τ = 1.5T S if no moving average function for the measurement is used[START_REF] Nussbaumer | Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system[END_REF].

This work was supported in part by Région Nord-Pas de Calais, by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).