Frédéric Mazenc
email: frderic.mazenc@l2s.centralesupelec.fr.

Michael Malisoff
email: malisoff@lsu.edu

Miroslav Krstic
email: krstic@ucsd.edu.

Vector Extensions of Halanay's Inequality

Keywords: Delay, stability, interval observer

We provide two extensions of Halanay's inequality, where the scalar function in the usual Halanay's inequality is replaced by a vector valued function, under a Metzler condition. We provide an easily checked necessary and sufficient condition for asymptotic convergence of the function to the zero vector in the time invariant case. For time-varying cases, we provide a sufficient condition for this convergence, which can be easily checked when the systems are periodic. We illustrate our results in cases that are beyond the scope of prior asymptotic stability results.

I. INTRODUCTION

Halanay's inequality is an efficient stability analysis tool, especially for systems with time-varying and poorly known delays, because for such cases, no general Lyapunov-Krasovskii functional construction is available in general. This celebrated inequality has the form

v(t) ≤ -av(t) + b sup ∈[t-τ,t] v( ) (1)
where a > 0, b ≥ 0 and τ ≥ 0 are constants and v : [-τ, +∞) → [0, +∞) is a scalar function of class C 1 . The usual Halanay's inequality result [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF] is the following: if a > b, then v(t) exponentially converges to zero as t → +∞. In addition to our works [START_REF] Mazenc | Stability analysis using new variant of Halanay's inequality[END_REF] and [START_REF] Mazenc | Stability analysis for timevarying systems with asynchronous sampling using contractivity approach[END_REF] that relax the requirement that the decay rate a is strictly larger than the gain b, several other extensions of this result are available in the literature e.g., in [START_REF] Di Ferdinando | Exponential inputto-state stability of globally Lipschitz time-delay systems under sampled-data noisy output feedback and actuation disturbances[END_REF], [START_REF] Pepe | On global exponential stability preservation under sampling for globally Lipschitz time-delay systems[END_REF], [START_REF] Selivanov | Distributed event-triggered control of transport-reaction systems[END_REF], and [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF]. Time-varying versions have been studied in [START_REF] Baker | Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag[END_REF] and [START_REF] Mazenc | Extension of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay[END_REF]. The fact that v in (1) is scalar valued is a limitation, because when one analyzes a system with delay, such a function may not be available, but functions v i : [-τ, +∞) → [0, +∞) of class C 1 and a Metzler matrix M , and a matrix P with all positive entries, such that

   v1 (t)
. . .

vn (t)    ≤ M    v 1 (t) . . . v n (t)   +P      sup l∈[t-τ,t] v 1 (l) . . . sup l∈[t-τ,t] v n (l)      (2)
for all t ≥ 0 may be available, where the inequality in (2) is componentwise. Using [START_REF] Baker | Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag[END_REF] to obtain a scalar valued function v satisfying the requirements of Halanay's theory does not seem to be possible in general. These remarks motivate this paper which continues our search for generalized or relaxed versions of Halanay's inequality, which we began in [START_REF] Mazenc | Stability analysis using new variant of Halanay's inequality[END_REF] and [START_REF] Mazenc | Stability analysis for timevarying systems with asynchronous sampling using contractivity approach[END_REF]. While [START_REF] Mazenc | Stability analysis using new variant of Halanay's inequality[END_REF] provided less restrictive versions of Halanay's inequality where the gain in the overshoot term can exceed the decay rate including applications to systems with scarce arbitrarily long sample intervals, and while [START_REF] Mazenc | Stability analysis for timevarying systems with asynchronous sampling using contractivity approach[END_REF] covered sampled cases that were beyond the scope of earlier Halanay's inequality formulations such as [START_REF] Mazenc | Stability analysis using new variant of Halanay's inequality[END_REF], here we pursue a very different direction, where the usual scalar decaying function in Halanay's inequality is replaced by a vector valued function. This provides an analog to vector Lyapunov function results that is applicable to stabilization problems that were beyond the scope of earlier Razumikhin function or diagonal stability based methods; see [START_REF] Matrosov | Vector Lyapunov function method: theory and application to complex industrial systems[END_REF], [START_REF] Ren | Vector-Lyapunov-function-based input-tostate stability of stochastic impulsive switched time-delay systems[END_REF], and [START_REF] Xu | Output feedback stabilization and estimation of the region of attraction for nonlinear systems: a vector control Lyapunov function perspective[END_REF] for vector Lyapunov functions and [START_REF] Dashkovskiy | An ISS small-gain theorem for general networks[END_REF] for input-to-state stability (or ISS) for interconnected systems via combinations of Lyapunov functions, or under small gain conditions that we do not require here.

We propose two extensions of Halanay's result in the case where vector Halanay's inequalities are satisfied. First, we consider a vector and time invariant version of this inequality. In Section II, we provide an easily checked necessary and sufficient conditions for the convergence of the v i 's to zero as time converges to +∞. Second, we propose sufficient conditions for this convergence, for a vector and time-varying version of Halanay's inequality, in Section III. We prove the results using ideas for Metzler matrices and cooperative systems, e.g., from [START_REF] Berman | Nonnegative Matrices in the Mathematical Sciences[END_REF] and [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. Then in Section IV, we provide three examples that illustrate how our results add value to the literature.

We use standard notation, which is simplified when no confusion would arise, where the dimensions of our Euclidean spaces are arbitrary unless otherwise noted. The standard Euclidean norm and induced matrix norm are denoted by | • |, and | • | ∞ is the usual sup norm. We define Ξ t by Ξ t (s) = Ξ(t + s) for all Ξ, s ≤ 0, and t ≥ 0 for which t+s is in the domain of Ξ, N = {1, 2, . . . , }, and • denotes the floor function. For matrices M ∈ R n×p and N ∈ R n×p with entries m i,j and n i,j in row i and column j respectively, we write M ≤ N when m i,j ≤ n i,j for all i ∈ {1, ..., n} an j ∈ {1, ..., p}, and similarly for < and for vectors. A matrix is called nonnegative (resp., positive) provided all of its entries are nonnegative (resp., positive). A matrix is called Metzler provided its off diagonal entries are nonnegative, and I is the identity matrix. A continuous linear system of the form Ξ(t) = L(t)Ξ t having a delay that is bounded above by a constant τ > 0 is called cooperative provided for each initial function satisfying Ξ(t) ≥ 0 for all t ∈ [-τ , 0], the corresponding solution satisfies Ξ(t) ≥ 0 for all t ≥ 0. We also use the n-fold product notation [0, +∞) n = [0, +∞) × . . . [0, +∞) and usual definitions and properties for state transition matrices (i.e., fundamental solutions) from [START_REF] Sontag | Mathematical Control Theory[END_REF]Appendix C.4].

II. TIME INVARIANT CASE A. Statement of Result and Remarks

Let M ∈ R n×n be a Metzler and Hurwitz matrix and P ∈ R n×n be a nonnegative matrix. Let τ > 0 be a constant and V : [-τ, +∞) → [0, +∞) n be C 1 and

V (t) = M V (t) + P S(V t ) (3)
hold for all t ≥ 0, where V = (v 1 ... v n ) and

S(V t ) = sup l∈[t-τ,t] V (l), (4)
and where

sup l∈[t-τ,t] V (l) = sup l∈[t-τ,t] v 1 (l) . . . sup l∈[t-τ,t] v n (l) , (5)
where v i is the ith component of V for each i, and similarly for vector valued functions W below. We prove:

Theorem 1. All C 1 solutions V : [-τ, +∞) → [0, +∞) n of (3) converge exponentially to the origin as t → +∞ if and only if M + P is Hurwitz.

Remark 1. One can prove that if M + P is Hurwitz, then a C 1 function V : [-τ, +∞) → [0, +∞) n such that V (t) ≤ M V (t) + P S(V t ) (6)
holds for all t ≥ 0 converges to 0 as t → +∞. This can be proved by the following variant of the usual comparison principle. Consider a function W such that

Ẇ (t) = M W (t) + P S(W t ) (7)
with S(W 0 ) > S(V 0 ), and suppose there were a t c > 0 such that W (t) > V (t) for all t ∈ [0, t c ) and such that there is a i ∈ {1, ..., n} such that v i (t c ) = w i (t c ). Then

V (t c ) ≤ e M tc V (0) + tc 0 e M (tc-) P S(V )d and W (t c ) = e M tc W (0) + tc 0 e M (tc-) P S(W )d , (8)
where we used the fact that the Metzler property of M implies that e M s ≥ 0 for all s ≥ 0 [START_REF] Mazenc | Design of continuousdiscrete observers for time-varying nonlinear systems[END_REF].

If we subtract the equality in (8) from the inequality in [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF] and recall that e M s ≥ 0 for all s ≥ 0 (so e M tc is nonzero and nonnegative) and P ≥ 0, we get the contradiction V (t c ) < W (t c ). Hence, W (t) > V (t) for all t ≥ 0, and Theorem 1 ensures that W (t) exponentially converges to 0 as t → +∞. Hence, since V is nonnegative valued, V (t) also exponentially converges to 0 as t → +∞.

Remark 2. We can use Theorem 1 to find novel sufficient conditions for the origin to be a globally exponential stable equilibrium on R n for systems of the form

Ẋ(t) = AX(t) + p i=1 B i X(t -τ i (t)) (9)
with multiple bounded delays τ i for any integer p ≥ 1, including cases where A is not required to be Metzler and the B i 's might not be nonnegative; see Section IV-B. We can rewrite (3) in the form

V (t) = M V (t) + P [v 1 (t -τ 1 (t)), . . . , v n (t -τ n (t))
] with time-varying delays τ i , which is reminiscent of but beyond the scope of [START_REF] Ngoc | A Perron-Frobenius theorem for a class of positive quasipolynomial matrices[END_REF]Theorem 4.1], because [START_REF] Ngoc | A Perron-Frobenius theorem for a class of positive quasipolynomial matrices[END_REF] was confined to constant delays. Thus, no extension of [START_REF] Ngoc | A Perron-Frobenius theorem for a class of positive quasipolynomial matrices[END_REF]Theorem 4.1] to (9) is possible. When M is not Metzler or P is not nonnegative, then the nonnegative orthant is not positively invariant for (3), and this motivates our conditions on M and P .

B. Proof of Theorem 1

First Part: Necessity. We prove that if M + P is not Hurwitz, then the asymptotic convergence condition of the theorem does not hold. To this end, notice that if M + P were not Hurwitz, then the system Ẋ(t) = (M + P )X(t) [START_REF] Matrosov | Vector Lyapunov function method: theory and application to complex industrial systems[END_REF] with X = (x 1 , . . . , x n ) is not exponentially stable. Consider a solution V : [-τ, +∞) → [0, +∞) n of (3) such that v i (r) = 2 for all i ∈ {1, ..., n} and r ∈ [-τ, 0] and the solution X of (10) with the initial condition X(0) = (1, ..., 1) . Then X(t) ≥ 0 for all t ≥ 0, because M + P is Metzler; see [START_REF] Mazenc | Design of continuousdiscrete observers for time-varying nonlinear systems[END_REF]Lemma 1]. We prove that

V (t) > X(t) (11)
for all t ≥ 0 by proceeding by contradiction.

Let us assume there is a t e > 0 such that V (t) > X(t) for all t ∈ [0, t e ) and that there is i ∈ {1, ..., n} such that v i (t e ) = x i (t e ). By integrating, we obtain

V (t e ) = e M te V (0) + te 0 e M (te-) P S(V )d and X(t e ) = e M te X(0) + te 0 e M (te-) P X( )d . ( 12
)
As in Remark 1, it follows that V (t e ) > X(t e ) because M is Metzler. This yields a contradiction. Since X(t) does not converge to 0 as t → +∞ and is nonnegative valued, it follows from (11) that V (t) also does not exponentially converge to zero. This proves the necessity of Hurwitzness of M + P for the convergence condition in Theorem 1.

Second Part. Sufficiency. We show that if (3) is such that M + P is Hurwitz, then the convergence conclusion of Theorem 1 holds. To this end, we introduce the function W (t) = e -M t V (t) for all t ≥ 0. Then (3) gives

Ẇ (t) = e -M t P S(V t ). (13)
By integrating ( 13) on [t -h, t] for any t ≥ h + τ and h > 0, we obtain

e -M t V (t) = e M (h-t) V (t -h) + t t-h e -M P S(V )d (14)
which gives

V (t) = e M h V (t -h)+ t t-h e M (t-) P S(V )d . (15)
For all ≤ t, the matrix e M (t-) P is nonnegative, because P ≥ 0 and M is Metzler; see, e.g., [START_REF] Mazenc | Design of continuousdiscrete observers for time-varying nonlinear systems[END_REF]Lemma 1]. Hence,

V (t) ≤ e M h V (t-h)+ t t-h e M (t-) P d S h+τ (V t ) = e M h V (t-h)+M -1 e M h -I P S h+τ (V t ), (16)
where

S h+τ (V t ) = sup ∈[t-h-τ,t] V ( ). Thus, V (t) ≤ e M h + M -1 e M h P + R S h+τ (V t ) (17) with R = -M -1 P. (18)
By Lemma 1 in the appendix below, the matrix R is nonnegative and Schur stable. Hence we can find a real value µ > 0 such that R + µH is Schur stable and positive where H is the constant n×n matrix each of whose entries is 1 (because of the continuity of eigenvalues of a matrix as a function of the entries of the matrix). Also, since M is Hurwitz, it follows that lim h→+∞ |e M h | = 0. Hence, since R is Schur stable, there is a constant h > 0 such that for all h ≥ h , the matrix e M h + M -1 e M h P + R + µH is Schur stable and positive. Moreover, ( 17) is satisfied with e M h + M -1 e M h P + R replaced by e M h + M -1 e M h P + R + µH, since V is nonnegative valued. Then, by the nonnegativity and Schur property of e M h + M -1 e M h P + R + µH, it follows from the proof of [1, Lemma 1] that V (t) converges exponentially to zero as t → +∞.

III. TIME-VARYING CASE A. Studied Problem

Our main assumption throughout this section is: Assumption 1. The matrix valued functions M : R → R n×n and P : R → R n×n are bounded piecewise continuous functions satisfying the following properties:

(i) M (t) is Metzler for all t ∈ R, (ii) P (t) is nonnegative for all t ∈ R, and (iii) the system

Ẋ(t) = M (t)X(t) (19)
is uniformly globally exponentially stable on R n .

Let Φ be the state transition matrix of M . Note for later use that, for all s 1 ≤ s 2 , we have Φ(s 2 , s 1 ) ≥ 0 because M is Metzler (e.g. by [START_REF] Mazenc | Design of continuousdiscrete observers for time-varying nonlinear systems[END_REF]Lemma 1]). Also, there are constants c 1 > 0 and c 2 > 0 such that |Φ(t, s)r| ≤ c 1 e -c2(t-s) when t ≥ s ≥ 0 for all unit vectors r.

B. General Case

We introduce the following assumption: Assumption 2. There are a constant α > 0 and a Schur stable matrix R ≥ 0 such that

Φ(t, s) + t s Φ(t, )P ( )d ≤ R (20)
for all s ≥ 0 and t ≥ s + α.

Since M and P are bounded, it follows from the exponential stability condition on (19) that we can satisfy Assumption 2 when α > 0 is large enough and |P | is small enough. Note for later use that since R is nonnegative and Schur stable, [7, Lemma 2.7, p.79] implies that there are a vector U > 0 and a constant q ∈ (0, 1) such that

U R ≤ qU . ( 21
)
We prove:

Theorem 2. Let Assumptions 1-2 hold. Consider a constant τ > 0 and a vector valued function

V : [0, +∞) → [0, +∞) n of class C 1 such that, for all t ≥ 0, V (t) ≤ M (t)V (t) + P (t)S(V t ), (22)
where S(V t ) is defined in (4). Then lim t→+∞ V (t) = 0.

Proof. First part. First, let us establish that V (t) is bounded. By variation of parameters, we obtain

V (t) ≤ Φ(t, s)V (s) + t s Φ(t, m)P (m)S(V m )dm (23) when t ≥ s ≥ 0. It follows that V (t) ≤ Φ(t, s)+ t s Φ(t, m)P (m)dm sup ∈[s-τ,t] V ( ). (24)
Assumption 2 ensures that for all t ≥ s + α,

V (t) ≤ R sup ∈[s-τ,t] V ( ). (25)
Now, consider t ≥ 0 and t ≥ t + α. Then, from [START_REF] Sontag | Mathematical Control Theory[END_REF], it follows that for all m ∈ [t + α, t], the inequalities

V (m) ≤ R sup ∈[t -τ,m] V ( ) ≤ R sup ∈[t -τ,t] V ( ) (26)
are satisfied. It follows that

sup m∈[t +α,t] V (m) ≤ R sup ∈[t -τ,t] V ( ). (27)
Therefore

U sup m∈[t +α,t] V (m) ≤ qU sup ∈[t -τ,t] V ( ) ( 28
)
where U is the vector in [START_REF] Rantzer | Distributed control of positive systems[END_REF]. Using sup

∈[t -τ,t] V ( ) ≤ sup ∈[t -τ,t +α] V ( ) + sup ∈[t +α,t] V ( ) (29)
it follows from (28) that

U sup m∈[t +α,t] V (m) ≤ qU sup ∈[t -τ,t +α] V ( ) +qU sup ∈[t +α,t] V ( ). (30)
Since q ∈ (0, 1), this inequality is equivalent to:

U sup m∈[t +α,t] V (m) ≤ q 1-q U sup ∈[t -τ,t +α] V ( ). (31)
It follows that

U V (t) ≤ q 1-q U sup ∈[t -τ,t +α] V ( ). (32)
Hence, since U is a positive vector, V (t) is bounded.

Second part. Let us prove that lim t→+∞ V (t) = 0. Let us introduce the functions

N 1 i (a, b) = sup m∈[a+α,b] v i (m)
and

N 2 i (a, b) = sup m∈[a-τ,b] v i (m), (33)
for i = 1 to n, having the domains

E 1 = {(a, b) ∈ [0, +∞) 2 : b ≥ a + α} and E 2 = {(a, b) ∈ [0, +∞) 2 :
b ≥ a} respectively, where we continue using the notation from Section II-A. We have proved in the first part of the proof that the functions N j i are bounded. Moreover, they are continuous, nonincreasing in their first argument and nondecreasing in their second argument.

Hence, there are bounded functions

I j i (a) such that lim b→+∞ N j i (a, b) = I j i (a) (34)
for j ∈ {1, 2} and i ∈ {1, ...., n}. The functions I j i are nonincreasing and lower bounded by 0. It follows that there are contants I j i,∞ ≥ 0 such that

lim a→+∞ I j i (a) = I j i,∞ (35)
for all j ∈ {1, 2} and i ∈ {1, ...., n}. Now, recall that for any t ≥ 0 the inequality ( 27) is satisfied for all t ≥ t +α, which we rewrite as

   N 1 1 (t , t) . . . N 1 n (t , t)    ≤ R    N 2 1 (t , t) . . . N 2 n (t , t)    . ( 36
)
It follows that    I 1 1 (t ) . . .

I 1 n (t )    ≤ R    I 2 1 (t ) . . . I 2 n (t )    . ( 37
)
Since

lim a→+∞ sup m∈[a+α,+∞) v i (m) = I 1 i,∞ (38)
and

lim a→+∞ sup m∈[a-τ,+∞) v i (m) = I 2 i,∞ (39)
we deduce that I 1 i,∞ = I 2 i,∞ for all i ∈ {1, ...., n}. Thus,

V I ≤ RV I , (40)
with

V I = [I 1 1,∞ , . . . , I 1 n,∞ ]
, by (37). Hence, U V I ≤ qU V I . Since q ∈ (0, 1), it follows that I 1 i,∞ = 0 for all i ∈ {1, ..., n}. Thus lim a→+∞ sup m∈[a+α,+∞) v i (m) = 0 for each i, so lim t→+∞ V (t) = 0. This concludes the proof.

C. Periodic Case

Consider the particular case where M and P are both periodic of some period ω > 0; many systems are periodic. However, one cannot simply apply Floquet theory to reduce the periodic case to the constant coefficient case from Theorem 1 because (a) Floquet theory is usually nonconstructive and (b) the time-varying changes of coordinates from Floquet theory applied to positive systems do not necessarily yield a positive system. We propose a stability condition which can be more easily checked than Assumption 2 in this case. Let us introduce the function

ξ(t) = (I -Φ(t + ω, t)) -1 t
t-ω Φ(t, m)P (m)dm (41) where the existence of the inverse follows from the exponential stability of [START_REF] Ngoc | A Perron-Frobenius theorem for a class of positive quasipolynomial matrices[END_REF], because if there were a nonzero vector z ∈ R n and a t ≥ 0 such that Φ(t+ω, t)z = z, then the periodicity and semigroup properties and the global asymptotic stability of ( 19) would give the contradiction z = Φ(t + ω, t) k z = Φ(t + kω, t)z → 0 as k → +∞ with k ∈ N. We also use the condition: Condition 1. There is a positive Schur stable matrix B such that ξ(t) ≤ B (42)

for all t ∈ [0, ω].
We state and prove the following result:

Corollary 1. Let Assumption 1 hold, with M and P both periodic with some period ω > 0. Then, Condition 1 is satisfied if and only if Assumption 2 is satisfied.

Proof. First part. Assume that Assumption 2 is satisfied.

For notational convenience, we use the functions

ζ(t, s) = t s Φ(t, m)P (m)dm and Λ(t, s) = Φ(t, s) + ζ(t, s). ( 43
)
Then there are h ∈ N and a Schur stable matrix R ≥ 0 such that

Φ(t, s) + ζ(t, s) ≤ R (44)
for all s ≥ 0 and t ≥ s + hω.

Consequently, for all t ≥ hω where h is any integer larger than h, the inequality Φ(t, t-hω)+ζ(t, t-hω) ≤ R holds. It follows from the semigroup property of Φ that

(t, h) + h-1 k=0 t-kω t-(k+1)ω Φ(t, m)P (m)dm ≤ R (45) with (t, h) = Φ(t, t -ω) h . This equality implies that H(h, t) ≤ R (46)
with H(h, t) = h-1 k=0 t t-ω Φ(t, t -kω)Φ(t -kω, mkω)P (m)dm, since (t, h) ≥ 0 and again using the semigroup and periodicity properties. Since the matrix M is periodic of period ω > 0, (46) is equivalent to

h-1 k=0 Φ(t, t -kω)ζ(t, t -ω) ≤ R. (47)
In terms of (41), (47) can be rewritten as

ξ(t) ≤ R + κ(t, h)ζ(t, t -ω), (48)
where

κ(t, h) = (I -Φ(t + ω, t)) -1 - h-1 k=0 Φ(t,
with j = t-s ω . We deduce that

Λ(t, s) ≤ j-1 k=0 t-kω t-(k+1)ω Φ(t, m)P (m)dm + ε. (53)
Recalling the periodicity of M and P and using the semigroup property of Φ, it follows that

Φ(t, l -kω) = Φ(t, t -kω)Φ(t -kω, l -kω) = Φ(t, t -kω)Φ(t, l) for all l ∈ [t -ω, t],
and so also

Λ(t, s) ≤ j-1 k=0 t t-ω Φ(t, l -kω)P (l)dl + ε = j-1 k=0 Φ(t, t -kω)ζ(t, t -ω) + ε. ≤ ξ(t) + ε, (54)
where the last inequality can be deduced from the definition (41) of ξ, (50), the fact that the partial sums in (54) form a nondecreasing sequence, the nonnegative valuedness of ζ, and the fact that the periodicity of M and the semigroup property of state transition matrices give Φ(t + ω, t) k = Φ(t, t -ω) k = Φ(t, t -kω) for all integers k ≥ 0. Hence, for all t ≥ s + α 1 , we have Λ(t, s) ≤ ξ(t) + ε ≤ B + ε. Hence, since B + ε is Schur stable, Assumption 2 is satisfied with R = B + ε.

IV. ILLUSTRATIVE EXAMPLES

A. First Illustration of Time Invariant Case

Consider the n-dimensional dynamics

             Ż1 (t) = -c 1 Z 1 (t) + Z 2 (t -τ 1 ) Ż2 (t) = -c 2 Z 2 (t) + Z 3 (t -τ 2 ) . . . Żn-1 (t) = -c n-1 Z n-1 (t) + Z n (t -τ n-1 ) Żn (t) = -c n Z n (t) + W (t) (55)
with the input W , which occurs in [3, Lemma 2] in the context of stabilization of linear strict-feedback systems with delayed integrators, where the constants c i and τ i are positive for i = 1, 2, . . . , n -1 (but the same reasoning applies if the delays τ i and τ ij in this subsection are bounded continuous time-varying functions, provided the upper bound τ in the following analysis is taken to be a positive constant). We assume that W takes the form

W (t) = d 1 Z 1 (t-τ n1 )+. . .+d n Z n (t-τ nn ) for constants τ ij ≥ 0 and d i > 0 for i = 1, 2, . . . , n.
For such a function W , the system (55) is cooperative; see [11, Lemma 1]. Thus, when proving asymptotic stability properties for (55), we can restrict our attention to its nonnegative valued solutions. Choosing τ > 0 such that τ > max{τ 1 , . . . , τ n-1 , τ n1 , . . . , τ nn }, it follows that all C 1 solutions Z : [-τ, +∞) → [0, +∞) n of (55) are solutions of (6) in the special case where M = diag{-c 1 , . . . , -c n } is a diagonal matrix having -c i as its ith main diagonal entry for i = 1, . . . , n and P is the n × n nonnegative matrix

P =        0 1 0 . . . 0 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 1 d 1 d 2 d 3 . . . d n-1 d n        . ( 56
)
Then we can find conditions on the c i 's and d i 's such that the corresponding matrix M + P is Hurwitz, to ensure that all C 1 solutions X : [-τ, +∞) → [0, +∞) n of (55) exponentially converge to the origin as t → +∞.

For instance, in the special case where n = 2, it follows from the quadratic formula that M +P is Hurwitz provided

c 1 + c 2 -d 2 > 0 and c 1 (c 2 -d 2 ) > d 1 .
For n = 3, the Hurwitzness condition on M + P is that all roots of the characteristic polynomial χ M +P (λ) =

λ 3 + (c 1 + c 2 + c 3 -d 3 )λ 2 + [c 2 (c 3 -d 3 ) + c 1 (c 3 + c 2 -d 3 ) -d 2 ]λ + c 1 [c 2 (c 3 -d 3 ) -d 2 ] -d 1 of M + P have negative real parts, which is equivalent to the requirements (c 1 + c 2 + c 3 -d 3 )[c 2 (c 3 -d 3 ) + c 1 (c 3 + c 2 -d 3 ) -d 2 ] > c 1 [c 2 (c 3 -d 3 )-d 2 ]-d 1 > 0 and c 1 +c 2 +c 3 -d 3 > 0 (
by the Routh-Hurwitz criterion for third order polynomials). The preceding conditions can be checked even if there is uncertainty in the positive values c i or in the nonnegative values d i , under suitable conditions on known intervals containing these unknown parameter values. Moreover, we can allow uncertainty in the positive delays τ i and τ ni (including continuous and time-varying delays), if we know a bound τ > 0 satisfying our conditions above. This illustrates how our work applies for delayed linear systems with uncertain coefficients and uncertain delays.

B. Second Illustration of Time Invariant Case

Consider the system

Ẋ(t) = AX(t) + p i=1 B i X(t -τ i (t)) (57)
for any integer p ≥ 1 with X valued in R n with constant matrices A and B i for i = 1, . . . , p where A is Hurwitz (but not necessarily Metzler), and where τ > 0 will denote a known bound on the piecewise continuous delays τ i : [0, +∞) → [0, τ ] for all i. We provide novel conditions that are independent of the delays τ i 's and that ensure that one can build an interval observer whose existence implies that (57) is globally exponentially stable to the origin; see, for instance, [START_REF] Mazenc | ISS interval observers for nonlinear systems transformed into triangular systems[END_REF] for the notion of interval observer. While more complicated than standard analysis for linear time invariant systems, our analysis is called for because of the mildness of the conditions on the delays and coefficient matrices, which puts this example outside the scope of existing results for linear time invariant systems.

One key ingredient in our interval observer design will be the proof of [START_REF] Mazenc | Interval observers for linear timeinvariant systems with disturbances[END_REF]Theorem 2] for Hurwitz matrices A, which constructs a C 1 function Q : [0, +∞) → R n×n with a bounded inverse and a constant Metzler matrix M such that

Q(t)Q(t) -1 + Q(t)AQ(t) -1 = M for all t ≥ 0.
To build the interval observer, first note that in terms of any Q that satisfies the requirements from the preceding paragraph, the new variable

Z(t) = Q(t)X(t) satisfies Ż(t) = M Z(t) + p i=1 L i (t)Z(t -τ i (t)) (58)
for all t ≥ τ , where L i (t) = Q(t)B i Q(t -τ i (t)) -1 for all i and t ≥ τ . Next, we introduce the dynamic extension

                       Ż(t) = M Z(t) + p i=1 L i (t) + Z(t -τ i (t)) - p i=1 L i (t) -Z(t -τ i (t)) Ż(t) = M Z(t) + p i=1 L i (t) + Z(t -τ i (t)) - p i=1 L i (t) -Z(t -τ i (t)) (59)
where C + = [max{c i,j , 0}] and

C -= C + -C for all matrices C = [c i,j ]. The change of coordinates Z ‡ (t) = -Z(t) yields                        Ż(t) = M Z(t) + p i=1 L i (t) + Z(t -τ i (t)) + p i=1 L i (t) -Z ‡ (t -τ i (t)) Ż ‡ (t) = M Z ‡ (t) + p i=1 L i (t) + Z ‡ (t -τ i (t)) + p i=1 L i (t) -Z(t -τ i (t)). ( 60
)
Since M is Metzler, it follows that (60) is cooperative; this follows by a variant of the argument from the appendix in [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functions and a positive systems approach[END_REF], which also explains why global exponential stability of (60) to the origin follows if all positive valued solutions of (60) exponentially converge to the origin as t → +∞. Thus, we focus on the positive solutions of (60). Let

Z(t) = Z(t) + Z ‡ (t). (61)
Then

Ż(t) = M Z(t) + p i=1 [L i (t) + +L i (t) -] Z(t-τ i (t)). (62) Setting S τ ( Zt ) = sup ∈[t-τ ,t] Z( ) gives Ż(t) ≤ M Z(t) + p i=1 [L i (t) + + L i (t) -] S τ ( Zt ) ≤ M Z(t) + LS τ ( Zt ) (63)
where the matrix L ≥ 0 is such that

p i=1 [L i (t) + + L i (t) -] ≤ L (64)
for all t ≥ 0. Hence, if M + L is Hurwitz, then we can apply Remark 1 to conclude that (62) is globally exponentially stable to the origin. Since Z and Z ‡ are nonnegative valued, it follows from (61) that (60) is also globally exponentially stable to the origin, so the origin of ( 59) is also a globally exponentially stable equilibrium. Since L i = L + i -L - i for each i, the reasoning we used in our proof of cooperativity of (60) shows cooperativity of the dynamics for (Z + , Z -) = (Z -Z, Z -Z). Hence, Z(t) ≥ Z(t) ≥ Z(t) for all t ≥ 0 if we choose any initial functions for Z and Z such that Z(t) ≥ Z(t) ≥ Z(t) for all t ∈ [-τ , 0]. Therefore, (59) provides an interval observer for (58) and all the solutions Z(t) exponentially converge to the origin as t → +∞. The inequality |X(t)| ≤ |Q(t) -1 ||Z(t)| for all t ≥ 0 and the boundedness of Q(t) -1 allow us to conclude that the X dynamics are globally exponentially stable to the origin. Remark 3. In many cases, we can take Q to be constant, notably when all eigenvalues of A are real, by picking Q such that QAQ -1 = M is the Jordan canonical form of A. Then our sufficient condition for (57) to be globally exponentially stable to the origin is that QAQ -1 + L is Hurwitz. When A is Metzler, we can take Q = I and then our sufficient condition is that A+ p i=1 [B + i +B - i ] is Hurwitz. See also [START_REF] Mazenc | Interval observers for linear timeinvariant systems with disturbances[END_REF]Section 4.3] for a Hurwitz matrix A that has a conjugate pair of complex (nonreal) eigenvalues and that calls for a nonconstant choice of the matrix Q.

C. Illustration of Corollary 1

We show that Corollary 1 makes it possible to prove exponential stability in cases where ( 6) is satisfied and some coefficients of P take large values at some instants, and without any restriction on the delay bound, which we believe puts this example outside the scope of previous results. Given p ∈ N, consider the system

         v1 (t) = -3v 1 (t) + (1 -1 4 cos(t))v 2 (t) + c * sin 2p (t) sup l∈[t-τ,t] v 2 (l) v2 (t) = -2v 2 (t) + 9 10 (1 -sin(t)) sup l∈[t-τ,t] v 1 (l) (65)
where τ and c * are positive constants, and v 1 and v 2 are nonnegative valued. Let us show that for any c * > 0, the origin of ( 65) is globally exponentially stable when c * < 0.12 1 + p.

With the notation of Section III, we can take ω = 2π,

M (t) = -3 0 0 -2 and
P (t) = 0 H(t) 9 10 (1 -sin(t)) 0 , (67)
where H(t) = 1 -1 4 cos(t) + c * sin 2p (t). Thus,

Φ(t, r) = e -3(t-r) 0 0 e -2(t-r) . (68)
Consequently,

(I -Φ(t + 2π, t)) -1 = 1 1-e -6π 0 0 1 1-e -4π
. (69) Also, since

Φ(t, )P ( ) = e -3(t-) 0 0 e -2(t-) 0 H( ) 9 10 (1 -sin( )) 0 = 0 e -3(t-) H( ) 9 10 e -2(t-) (1 -sin( )) 0 , (70)
the choice H * (t) =

ξ(t) = 0 θ 1 (t) + θ 2 (t) θ 3 (t) 0 , (73)
with the nonnegative valued functions

θ 1 (t) = 1 1-e -6π t t-2π e -3(t-) 1 -1 4 cos( ) d , (74) θ 2 (t) = c * 1-e -6π t t-2π e -3(t-) sin 2p ( )d , (75)
and By the integration by parts formula u( )v ( )d = u( )v( ) -u ( )v( )d with u = sin 2p-1 and v = -cos and the formula cos 2 = 1 -sin 2 , we solve for the second integral in (80) to conclude that for all p ∈ N, we have

θ 3 (t) = 9 10(1-e -4π ) t t-2π e -2(t-) (1 -sin( ))d . (

V. CONCLUSION

We proved extensions of the stability analysis technique based on Halanay's inequality, which is suitable for the analysis of interconnected systems. Key features included our allowing time-varying delays, and our novel use of positive systems and interval observers. This produced vector analogs of Halanay's inequality. Our results can be used to study time-varying systems with uncertain timevarying delays that were beyond the scope of the literature for linear time invariant systems. The ISS property with respect to additive disturbances can be proved. We hope to obtain extensions for PDEs and sampling, where instead of continuous time systems, we have continuous-discrete systems whose states are reset at the sample times.

APPENDIX

We used this lemma in our proof of Theorem 1: Lemma 1. Let the matrix M be Metzler and Hurwitz. Let P ≥ 0 be a matrix such that M + P is Hurwitz. Then the matrix -M -1 P is nonnegative and Schur stable.

Proof. By [21, Proposition 1], -M -1 ≥ 0. Hence, -M -1 P is nonnegative. Also, [21, Proposition 1] provides a vector V > 0 and a real number c > 0 such that (M + P )V ≤ -cP V . Since -M -1 ≥ 0, we deduce that -M -1 (M + P )V ≤ cM -1 P V , which is equivalent to -M -1 P V ≤ 1 1+c V . Since 1 1+c ∈ (0, 1), [21, Proposition 2] allows us to conclude.

π 2 0 2 0 2 0 2 0

 2222 sin 2p ( )d = 1 -1 2p π sin 2(p-1) ( )d . (81)Thus, since ln(1 -a) ≤ -a for all a ∈ (0, 1), we get π sin 2p ( )d = π 81) recursively to reduce the power of sin in the integrand to 0. sin 2p ( )d ≤ π 2 e -ln(1+p)

  Let us assume that Condition 1 is satisfied. Let ε > 0 be a matrix such that B+ε is Schur stable. Since (19) is uniformly exponentially stable, there is α 1 > 0 such that for all t ≥ s + α 1 , |Φ(t, s)| ≤ 1 2 ε and

	t-	t-s ω ω

t -kω). (49)

For each > 0, there is a h ∈ N with h ≥ h such that for all h ≥ h , we have |κ(t, h )ζ(t, t -ω)| ≤ for all t ≥ 0; this follows because of the geometric sum formula

+∞ k=0 Φ(t + ω, t) k = (I -Φ(t + ω, t)) -1

(50)

and boundedness of M and P . Hence, Condition 1 holds. Second part. s Φ(t, m)P (m)dm ≤ 1 2 ε, (51) by picking α 1 > ω such that sup ∈[-1,1] Φ(t, s + ω) is small enough. Thus, if t ≥ s + α 1 , then (43) gives Λ(t, s) ≤ t t-jω Φ(t, m)P (m)dm + ε

  Condition (87) holds if (66) is satisfied. Hence, Condition 1 is satisfied. Then Corollary 1 implies that Assumption 2 is satisfied, so Theorem 2 applies.

	0.853	0	√	1 1+p	π 2	.	(86)
	The matrix G is Schur stable if and only if the inequality
	0.413 + 4c * 1-e -6π	π/2 √ 1+p 0.853 < 1		(87)
	is satisfied.

Supported by US National Science Foundation Grants 1711299 (Malisoff) and 1711373 (Krstic).


Frederic Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

Corina Barbalata
email: cbarbalata@lsu.edu

Event-Triggered Control for Continuous-Time Linear Systems with a Delay in the Input

Keywords: Delay, event-trigger, stability, positive system

We provide an event-triggered control technique for a family of linear time-varying continuous-time systems with a constant known pointwise delay in the input. We adopt a subpredictor based prediction technique, and we provide sufficient conditions that ensure that Zeno behavior does not occur. At each time, only delayed measurements are needed to implement the control. Also, the delay can be an arbitrarily large constant. We prove an input-to-state stability property for the closed-loop system, using the theory of cooperative systems. We apply our method to a gyroscopic control problem for a curve tracking dynamics arising in marine robotics.

Introduction

Event-triggered control provides the basis for numerous significant recent developments in systems and controls; see, e.g., [START_REF] Astrom | Computer-Controlled Systems: Theory and Design, Second Edition[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF], and references therein. While recently developed computing methods can ease the burden of recalculating control values, the widespread use of shared wired (or shared wireless) networks strongly motivates taking communication, computation, and energy constraints into consideration when designing feedback controls. This motivated the development of event-triggered techniques to reduce the number of control value recomputation times needed to achieve stability properties (compared, e.g., with other sampled based feedback methods), to reduce the communication load in networked systems. When input delays are present, event triggered control can be complicated by the fact that traditional event-triggered controls were not designed to compensate for time delays, and so would not provide the performance guarantees that they would guarantee in the absence of input delays.

On the other hand, many systems have input delays, which frequently arise from transport and delays in sensors; see, e.g., [START_REF] Cacace | Output transformations and separation results for feedback linearizable delay systems[END_REF], [START_REF] Cacace | Delay identification for a class of nonlinear systems[END_REF], [START_REF] Cacace | Optimal control of linear systems with large and variable input delays[END_REF], [START_REF] Cacace | Stabilization of strict-feedback nonlinear systems with input delay using closedloop predictors[END_REF], [START_REF] Cacace | Output feedback control of linear systems with input, state and output delays by chains of predictors[END_REF], [START_REF] Cacace | Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment[END_REF], [START_REF] Cacace | Predictor-based control of linear systems with large and variable measurement delays[END_REF], and [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based approach[END_REF]. Even small input delays can have detrimental effects, so it can be essential to take them into account; see, e.g., [START_REF] Bhogaraju | Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties[END_REF] for a gridtied inverter dynamics that is destabilized by a delay of τ = 0.001. To cope with input delays, several stabilization techniques exist in the frequency and time domains, in-Supported by NSF Grants 1711299 and 2009659.

cluding the celebrated Smith predictor, stabilization that is robust to small delays (which computes bounds on allowable input delays), and the reduction model and prediction approaches from [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF] and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. See also the subpredictor approach [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] for known arbitrarily large delays. An advantage of subpredictors is that they only use pointwise delays, meaning that at each time t, the control uses state arguments from time t-τ for a constant delay τ > 0. However, we are not aware of results on event-triggered controls that ensure robustness to small known delays in the input. The reduction model approach was combined with eventtriggered control to stabilize systems with an input delay in [START_REF] Nozari | Event-triggered stabilization of nonlinear systems with time-varying sensing and actuation delay[END_REF]. See also [START_REF] Iwaki | Event-triggered PI control of time-delay systems with parametric uncertainties[END_REF] for a Smith predictor approach. To the best of our knowledge, the problem of designing eventtriggered controls under input delays and involving only pointwise delays has not been addressed in the literature. This motivates the present paper, which builds on recent studies of event-triggered control for systems with delay; see in particular [START_REF] Cacace | Output feedback control of linear systems with input, state and output delays by chains of predictors[END_REF], [START_REF] Li | Event-triggered impulsive control for nonlinear delay systems[END_REF], [START_REF] Wang | Event-triggered scheduling control for linear systems with quantization and time-delay[END_REF], and [START_REF] Zhu | Event-based leader-following consensus of multi-agent systems with input time delay[END_REF]. We use a subpredictor based prediction approach, which was developed in works such as [START_REF] Mazenc | Stabilization of nonlinear timevarying systems through a new prediction based approach[END_REF] and [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] in the absence of eventtriggering, to stabilize linear continuous time systems with a constant pointwise delay in the input and an additive uncertainty. We use an event-triggered control, which combines aspects of co-design of controls and triggering rules with emulation. We assume that the delay is constant and known, and that a bound for the additive uncertainty term is known. The stability analysis we perform is reminiscent of the one of the recent contribution [START_REF] Mazenc | Eventtriggered control using a positive systems approach[END_REF] which did not allow input delays. We rely on comparison systems whose stability is analyzed through tools of the theory of cooperative systems. Such tools have been used in notable works such as [START_REF] Aleksandrov | Absolute stability and Lyapunov-Krasovskii functionals for switched nonlinear systems with time-delay[END_REF], [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF], [START_REF] De Iuliis | Internally positive representations and stability analysis of linear differential systems with multiple time-varying delays[END_REF], and [START_REF] Ngoc | Stability of positive differential systems with delay[END_REF], but these earlier works did not consider event triggering. We prove that, for the control law we propose, the Zeno behavior is excluded, meaning each finite length time interval contains only finitely many event triggering times. The limitation of our approach resides mostly in the fact that we assume that the delay is known. To the best of our knowledge, no result for unknown input delay with a known upper bound has been established for event-triggered control systems. However, in addition to our allowing arbitrarily large constant delays, a significant novel feature is that only time lagged measurements are needed in the control or event-triggering formula. In Section 2, we present the systems under study. We also introduce the subpredictors. We state and prove our main result in Sections 3-4. Then Section 5 has an application to a marine robotic example. We conclude in Section 6 by summarizing our findings and our ideas for possible follow up research.

For simplicity, we only consider constant, pointwise, and known delays. We found these assumptions on the delay to be suitable for practice, because for instance, one can first apply delay estimation methods such as the notable work [START_REF] Chakraborty | Control of an input delayed uncertain nonlinear system with adaptive delay estimation[END_REF] or experiments for delay estimation. Also, in the surface marine robotic, power electronics, and other applications that we encountered, the delays are typically constant and pointwise; see, e.g., [START_REF] Bhogaraju | Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties[END_REF], and see [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] and [START_REF] Weston | Sequential predictors under timevarying feedback and measurement delays and sampling[END_REF] for sequential predictor methods under time-varying delays but no event triggering, which we conjecture can be extended to event triggering using the approach in the present work, because the present work uses the constantness of the delay primarily in the chain predictor construction.

m ij = sup{|g ij (t)| : t ∈ R}. We also set G + = [max{g ij , 0}] and G -= G + -G. A square matrix is called Metzler pro- vided all of its off diagonal entries are nonnegative. Given two matrices D = [d ij ] an E = [e ij ] of the same size, we write D < E (respectively, D ≤ E) provided d ij < e ij (re- spectively, d ij ≤ e ij )
for all i and j. Also, we write D E provided there is a pair (i, j) such that d ij > e ij . We use similar notation for vectors. Also, 0 denotes the zero matrix, and I is the identity matrix, of the size under consideration.

For each constant τ > 0, we let C([-τ, 0]) denote the set of all continuous functions φ : [-τ, 0] → R n for the dimension n under consideration, which will serve as the set of all initial functions for our closed loop systems having the input delay τ , where we assume that the initial times are t 0 = 0. A matrix M is called positive provided 0 < M . We set φ t (s) = φ(t + s) for all φ, t ≥ 0, and s ≤ 0 such that t + s is in the domain of φ. For a matrix M in R n×n , we let D M denote the diagonal matrix such that all of the diagonal entries of M -D M are equal to zero. We let

R M = D M + (M -D M ) + and N M = (M -D M ) -.
Hence, when M is Metzler, we have M = R M . We let || • || denote the standard Euclidean norm of matrices and vectors, and we let || • || ∞ (respectively, || • || I ) denote the sup norm of matrix valued functions in this norm over their domain (respectively, an interval I in their domain). We use standard notation and properties for input-to-state stability (or ISS, which we also use to abbreviate input-tostate stable). We use the usual definition and properties of fundamental solutions, e.g., from [START_REF] Sontag | Mathematical Control Theory[END_REF]Appendix C]. We use Φ M to denote the fundamental solution for a system of differential equations of the form ż(t) = M(t)z(t).

Studied system and subpredictors

In this part, we present the system we study. Also, we recall a result on prediction of future values of state variables, which can be found in the papers [START_REF] Mazenc | Stabilization of nonlinear timevarying systems through a new prediction based approach[END_REF] and [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF], for instance.

Studied system

We consider the system

ẋ(t) = A(t)x(t) + B(t)u(t -τ ) + δ(t) (1)
where x is valued in R n , u is valued in R p , A : R → R n×n and B : R → R n×p are given bounded continuous matrix valued functions, and the unknown function δ : [0, +∞) → R n is bounded and piecewise continuous and can represent control or model uncertainty. We assume that the constant delay τ ≥ 0 is known and that we know a bounded piecewise continuous function δ : [0, +∞) → R n such that

|δ(t)| ≤ δ(t) (2)
for all t ≥ 0. Also, u represents a control that will be specified by our theorem below.

Subpredictors

This section reviews basic ideas for chain predictors with event triggering, which were presented in [START_REF] Mazenc | Stabilization of nonlinear timevarying systems through a new prediction based approach[END_REF] and [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] and which we use to prove our main result in the next section. Choose a bounded continuous matrix valued function L : R → R n×n and an integer m > 0 such that with the choices of A and τ from (1), the origin of the system

Ẋ(t) = A(t)X(t) + L(t)X t - τ m + κ(t) (3)
is globally exponentially ISS on R n with respect to κ, where κ is an unknown measurable locally bounded function that represents uncertainty, i.e., there are constants p i > 0 for i = 1 to 3 such that for all t ≥ s and s ≥ 0, all solutions of (3) satisfy

||X(t)|| ≤ p 1 e -p2(t-s) sup r∈[s-2τ /m,s] ||X(r)|| + p 3 sup r∈[s,t] ||κ(r)||. (4)
Later, we specialize (3) to cases where X is a difference between subpredictor values evaluated at suitable times.

For instance, we can take

L(t) = -I -A(t) when m > 2 √ 2τ ||I + A|| ∞ max{||I + A|| ∞ , ||A|| ∞ }, (5)
by standard Lyapunov-Krasovskii functional arguments; see Appendix A below. We assume that m ≥ 2.

We introduce the subpredictors

                         ż1 (t) = A t + τ m z 1 (t) + B t + τ m u t -m-1 m τ + L t + τ m z 1 t -τ m -x(t) ż2 (t) = A t + 2τ m z 2 (t) + B t + 2τ m u t -m-2 m τ + L t + 2τ m z 2 t -τ m -z 1 (t) . . . żm (t) = A (t + τ ) z m (t) + B(t + τ )u (t) + L(t + τ ) z m t -τ m -z m-1 (t) (6)
with initial conditions in C([-τ, 0]), and the operators

e j (t) = z j (t) -z j-1 t + τ m (7)
for j = 1 to m for all t ≥ 0, where z 0 (t) = x(t).

Then we can use a telescoping sum argument to get

z m (t) = x(t + τ ) + ξ(t) (8)
for all t ≥ 0, where

ξ(t) = m j=1 e j t + (m-j)τ m , (9)
and then we can evaluate (8) at t -τ to remove the dependence of (6) on the current values x(t), as follows. Notice that by combining ( 1) and ( 6), we get

ė1 t + (m-1)τ m = A (t + τ ) e 1 t + (m-1)τ m +L (t + τ ) e 1 t + (m-2)τ m -δ (t + τ ) (10)
and

ėj t + (m-j)τ m = A(t + τ )e j t + (m-j)τ m + L(t + τ )e j t + (m-j-1)τ m -L(t + τ )e j-1 t + (m-j)τ m ( 11
)
for j = 2 to m for all t ≥ 0. Bearing in mind the properties of (3), it follows that we have a cascade of ISS systems ( 10)-( 11) having the state value components (e 1 , . . . , e m ).

According to (4), we deduce that there are two constants c 1 > 0 and c 3 > 0 such that for any initial condition of the system (1) and ( 6), we have

||ξ (t)|| ≤ c 1 e -p2(t-s) sup ∈[s-2τ /m,s] ||ξ( )|| + c 3 sup ∈[s,t+τ ] ||δ( )|| (12)
when t ≥ s ≥ 0.

Although the dynamics of the first subpredictor z 1 in (6) uses current x(t) values, our control will be computed from delayed values z m (t-τ ) of the last subpredictor z m , so only delayed values of x will be needed to compute our control; see (a) in Section 3 below. This makes our control practical for use where only time lagged values are available for use in the control. See also [START_REF] Weston | Sequential predictors under timevarying feedback and measurement delays and sampling[END_REF] for methods to compute a constant D * > 0 such that the conclusions of this subsection continue to hold if x(t) in the first subpredictor is replaced by a delayed version x(t -D) for any constant delay D ∈ [0, D * ], which can represent sensor delays. However, for simplicity, we do not consider these sensor delays in this work.

Main result

In this section, we introduce an event-triggered control and establish an ISS property for the corresponding closed-loop system. To this end, we first introduce a bounded continuous matrix valued function K : R → R p×n and the function

ω(t, s) = Φ A (t, s) + t s Φ A (t, )B( )K( )d . (13)
Since ω(s, s) = I for all real s, and since ω is a C 1 function such that sup{||(∂ω/∂t)(t, s)|| : |t -s| ≤ β} is bounded for each constant β > 0, we deduce that there is a constant ω L > 0 such that ω(t, s) is invertible for all (t, s) such that |t -s| ≤ ω L . Consequently, one can select constants ν > 0 and T > 0 such that T ≥ 2ν and such that ω(t, s) is invertible when |t -s| ≤ T . We deduce that the matrices

Γ = sup ω(t, s) -1 -I : 0 ≤ s ≤ t ≤ s + ν , (14)
λ = sup ω(t, s) -1 t s Φ A (t, )B( )K( )d : 0 ≤ s ≤ t ≤ s + ν} , and (15)
ω = sup ω(t, s) -1 : 0 ≤ s ≤ t ≤ s + ν (16
) are well defined, where the sup is computed entrywise; see Remark 1 for ways to compute the entries of the matrices in the suprema in ( 14), [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], and [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. To simplify the notation, we define a matrix valued function

H(t) = A(t) + B(t)K(t). (17)
Then, for each t ≥ 0, the matrix

M (t) = R H (t) + N H (t) + |B(t)K(t)|Γ (18)
is Metzler. We assume that the constant ν > 0 and matrix K specified above are such that the matrix (18) satisfies:

Assumption 1 There exist a positive vector V ∈ R n and a constant p > 0 such that V M (t) ≤ -pV for all t ≥ 0.

The motivation for Assumption 1 is that it can be viewed as a generalized exponential stability condition on the subset of all solutions of the system q(t) = M (t)q(t) that are nonnegative valued, which reduces to a robust decay condition that applies to the subset of all solutions of q(t) = H(t)q(t) that are nonnegative valued in the special case where H(t) is Metzler for each t, with |B(t)K(t)|Γ viewed as a perturbation term. Its role is analogous to the role played by the asymptotic stability condition on the nominal system in standard emulation-based event-triggered control with no delays. In our proof of our theorem, we use Assumption 1 to build a linear Lyapunov function In terms of the function ξ from (9), our event-triggered control and triggering times t i for (1) are defined by

V(ζ) = V ζ
(a) u(t -τ ) = K(t)z m (t i -τ ) if i ≥ 0 is such that t ∈ [t i , t i+1 ), (b) For each i ≥ 0 and each t ∈ [t i + τ, t i+1 + τ ), we have |x(t -τ ) -x(t i )| ≤ Γ|x(t -τ )| + λ|ξ(t i -τ )| + ω t-τ max{0,t-T -τ } e |A|∞(t-τ -) δ( )d , ( 19
) (c) |t -t i | ≤ T for each t ∈ [t i , t i+1
] and i ≥ 0, and (d) for each i, either t i+1 -t i = T , or for each > 0, there is a t c such that t c ∈ (t i+1 + τ, t i+1 + + τ ) and such that

|x(t c -τ ) -x(t i )| Γ|x(t c -τ )| + λ|ξ(t i -τ )| +ω tc-τ max{0,tc-T -τ } e |A|∞(tc-τ -) δ( )d ( 20
)
where t 0 = 0 < t 1 < t 2 < . . ., which uses the control value u(t -τ ) = K(t)z m (-τ ), until either time t 1 = T , or until some t 1 ∈ [0, T ) that is the infimum of the times when a violation ( 20) of ( 19) occurs if such a violation time occurs, and then we repeat the preceding process with t 0 = 0 and t 1 replaced by t 1 and t 2 respectively, and then argue inductively, to define the control, event triggering times t i , and closed loop system at all times t ≥ 0. Our justification for using t -τ and t c -τ instead of t and t c in ( 19)-( 20) is that it allows us to determine the event triggering times t i from measurements of x(t -τ ) and x(t i ), where t i in (b) and (d) can be viewed as the last event trigger time before t -τ or t c -τ . Hence, the three parts of our codesigned control and event-trigger (consisting of the subpredictors (6), the formula

u(t -τ ) = K(t)z m (t i -τ ) from (a),
t i+1 = inf {s ∈ [t i , t i + T ) : |x(s) -x(t i )| ≤ Γ|x(s)| +λ|ξ(t i -τ )| + ω s max{0,s-T } e |A|∞(s-) δ( )d , (21)
where the equivalence of the preceding condition and (d) above can be seen by noting that considering values of t such that t -τ ∈ [t i , t i + T ) is equivalent to considering values of s ∈ [t i , t i + T ). Later, we show that the constant ν > 0 we chose above is such that t i+1 -t i ≥ ν for all integers i ≥ 0, so the Zeno phenomenon does not occur.

In terms of the constants in [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF], the delay τ > 0 in (1), T and p chosen as above, and any constant c * > 0 such that

2p 2 (c * + 1)e p2(T +τ ) (c 1 + 1) < e c * p2 -1, (22)
which can be satisfied when c * > 0 is large enough, and using [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] and the constants

b = ||V |BK| ∞ (λ + I)||, v = min i V i , T e = max{c * , T }, T † = c * + T, p † = ||V ||+(1+b)c 2 * v , p * = min p, 1 2c * , and c e = ||V |BK|∞ωe |A∞ |T ||T +||V ||+c3(1+b)(c * +1) vp , (23)
where V i is the ith component of V from Assumption 1, the main result of this paper is as follows:

Theorem 1 Let Assumption 1 hold. Then the solutions of ( 1) and ( 6), with ξ defined by ( 9) and with the eventtriggered control defined by (a)-(d), satisfy

||x(t)|| ≤ p † e -p (t-r) sup ∈[r-T † ,r+τ ] ||x( )|| + sup ∈[r-T † -τ,r] ||z m ( )|| +c e sup ∈[r-Te,t+τ ] ||δ( )|| (24)
for all r ≥ T † + τ and t ≥ r.

Remark 1 Due to the p † e -p (t-r) in ( 24) and the positiveness of p , x(t) asymptotically approaches 0 when δ is zero, because in that case, we can take δ = 0. Also, ||x(t)|| approaches the ball of radius c e d * asymptotically when the uncertainty is bounded by a constant d * in the norm || • ||.

Hence, uncertainties with smaller sup norms ensure convergence of the states to arbitrarily small prescribed values.

In the special case where A, B, and K are all constant, we get ω(t, s) = Ω(t -s), where

Ω(r) = e Ar + r 0 e (r-)A d BK ( 25
)
is the corresponding function that was used in the undelayed case in [START_REF] Mazenc | Eventtriggered control using a positive systems approach[END_REF]. Also, when A, B, and K all have some period p > 0, then ω(t + p, s + p) = ω(t, s) for all s ≥ 0 and t ≥ s, so our invertibility requirement for ω can be checked by checking that ω( Remark 2 Our control algorithm (a)-(d) is valid in the special case where τ = 0, in which case ξ = 0. When τ = 0 and A, B, and K are constant, our proof of the theorem will show that we can remove condition (c) and then not consider the possibility t i+1 -t i = T in (d), and then (a)-(d) is the same as the event-triggered algorithm from [START_REF] Mazenc | Eventtriggered control using a positive systems approach[END_REF]Theorem 1] in this special undelayed case.

Remark 3

The design of the sequence t i ensures that for all i ≥ 0, we have t i+1 -t i ∈ [0, T ]. This constraint on this sequence is not very restrictive because, in general, ω(t, s) is invertible for all t and s satisfying |t -s| ≤ ω L for a large value of ω L , which implies that T can be a large constant.

Remark 4 Assumption 1 is a stabilizability condition. To evaluate how restrictive it is, let us consider the case where the pair (A, B) is a constant controllable pair. In this case, we can select a constant matrix K such that the corresponding matrix H = A + BK is Hurwitz and such that all of its eigenvalues are real numbers. It follows that one can find a matrix P ∈ R n×n such that P HP -1 is Hurwitz and Metzler, by choosing P such that P HP -1 is the Jordan canonical form of H. Hence, after a preliminary change of coordinates, we obtain a system for which there is a matrix K such that H is Hurwitz and Metzler. Then the matrix M defined in [START_REF] Iwaki | Event-triggered PI control of time-delay systems with parametric uncertainties[END_REF] satisfies M = H + |BK|Γ. Since Γ can be rendered arbitrarily small by an appropriate choice of ν > 0, we deduce that, in this case, Assumption 1 is satisfied for some constants ν (again by [16, Lemma 2.3, p.41]).

Remark 5 In (a)-(d), the function δ is present. Thus Theorem 1 does not apply when no upper bound for |δ| is known.

Proof of Theorem 1

The proof has four parts. In the first part, we prove that, for the inter-sampling intervals [t i , t i+1 ) arising from using the control (a)-(d), the Zeno phenomenon does not occur. In the second part, we show that the function V(ζ) = V ζ satisfies an ISS Lyapunov function decay condition along the dynamics for ζ(t) = x(t) -x(t), where (x, x) is the vector of states for a framer that we will construct for the state variable x(t), and where V > 0 is from Assumption 1. In the third part, we prove an ISS estimate for the functional η(ξ t ) = sup

∈[t-T -τ,t] ||ξ( )|| (27)
with δ viewed as the disturbance. In the final part, we combine the ISS estimate for [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] with the ISS decay estimate on V ζ to obtain the desired ISS estimate in the state of the closed loop system from the theorem.

First part: Inter-sampling intervals. We prove that the Zeno phenomenon does not occur, by proving that t i+1t i ≥ ν for all i where ν > 0 was chosen in Section 3.

The solutions of the closed-loop system satisfy the following for all integers i ≥ 0: According to the equality (8),

                                 ẋ(t) = A(t)x(t) + B(t)K(t)z m (t i -τ ) + δ(t) for all t ∈ [t i , t i+1 ); |x(t -τ ) -x(t i )| ≤ Γ|x(t -τ )| + λ|ξ(t i -τ )| +ω t-τ max{0,t-T -τ } e |A|∞(t
ẋ( ) = A( )x( )+B( )K( )x(t i ) + B( )K( )ξ(t i -τ ) + δ( ) (29)
for all ∈ [t i , t i+1 ). Here and in the sequel, all inequalities and equalities are for each integer i ≥ 0 unless otherwise indicated. To show that t i+1 ≥ t i + ν, we can first apply variation of parameters to (29) on [t i , t -τ ] for any t ∈ [t i + τ, t i + τ + T ] with the initial state x(t i ) to obtain

x(t -τ ) = Φ A (t -τ, t i )x(t i ) + t-τ ti Φ (t, )d [x(t i ) + ξ(t i -τ )] + t-τ ti Φ A (t -τ, )δ( )d = ω(t -τ, t i )x(t i ) + t-τ ti Φ (t, )d ξ(t i -τ ) + t-τ ti Φ A (t -τ, )δ( )d (30)
where Φ (t, ) = Φ A (t -τ, )B( )K( ) and ω was defined in [START_REF] Chakraborty | Control of an input delayed uncertain nonlinear system with adaptive delay estimation[END_REF].

Since ω(r, s) is invertible when |s -r| ≤ T , we deduce that

x(t i ) -x(t -τ ) = ω(t -τ, t i ) -1 -I x(t -τ ) -ω(t -τ, t i ) -1 t-τ ti Φ (t, )d ξ(t i -τ ) -ω(t -τ, t i ) -1 t-τ ti Φ A (t -τ, )δ( )d . (31)
Thus, for all t ∈ [t i + τ, t i + ν + τ ], we have |t -τ -t i | ≤ ν and t -τ ≥ t i , so our choices ( 14)-( 15) of Γ and λ give [START_REF] Ngoc | Stability of positive differential systems with delay[END_REF] and so also

|x(t i ) -x(t -τ )| ≤ Γ|x(t -τ )| + ω(t -τ, t i ) -1 t-τ ti Φ (t, )d |ξ(t i -τ )| + ω(t -τ, t i ) -1 t-τ ti Φ A (t -τ, t i )δ( )d
|x(t i ) -x(t -τ )| ≤ Γ|x(t -τ )| + λ |ξ(t i -τ )| + ω(t -τ, t i ) -1 t-τ ti Φ A (t -τ, )δ( )d . ( 33
)
Bearing in mind (2), we deduce that

|x(t i ) -x(t -τ )| ≤ Γ|x(t -τ )| + λ |ξ(t i -τ )| + ω(t -τ, t i ) -1 t-τ ti e |A|∞(t-τ -) δ( )d and |t -τ -t i | ≤ T (34) for all t ∈ [t i +τ, t i +ν +τ ]. Since ω(t -τ, t i ) -1 ≤ ω for all t ∈ [t i + τ, t i + ν + τ ],
we conclude that the inequality [START_REF] Khalil | Nonlinear Systems. Third Edition[END_REF] in (b) holds for all t ∈ [t i + τ, t i + τ + ν], so t i+1 ≥ t i + ν. This implies that the Zeno phenomenon does not occur.

Second part: ISS Lyapunov function decay estimate for

V(ζ) = V ζ.
To analyze the stability of the closed loop system (28), we use the approach from [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] and [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functions and a positive systems approach[END_REF]. Hence, our stability analysis relies on framers, which are used as comparison systems. For all integers i ≥ 0, we can evaluate (29) at = t to write (29) as ẋ(t) = H(t)x(t) + µ(t) + δ(t), (35) where µ is the function defined by 36) for all t ∈ [t i , t i+1 ) and where H = A + BK is the matrix valued function that we defined in [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF].

µ(t) = B(t)K(t)[x(t i ) -x(t)] + B(t)K(t)ξ(t i -τ ) (
We introduce the framer

ẋ(t) = R H (t)x(t) -N H (t)x(t) + µ(t) + + δ(t) + ẋ(t) = -N H (t)x(t) + R H (t)x(t) -µ(t) --δ(t) - (37)
with initial conditions x(r) = x + (r) and x(r) = -x -(r) (38) with r ≥ T + τ . Let x ‡ (t) = -x(t). This produces the following system:

ẋ(t) = R H (t)x(t) + N H (t)x ‡ (t) + µ(t) + + δ(t) + ẋ ‡ (t) = N H (t)x(t) + R H (t)x ‡ (t) + µ(t) -+ δ(t) - (39)
Then since µ(t) + + δ(t) + ≥ 0 and µ(t) -+ δ(t) -≥ 0 for all t ≥ 0, and since the matrix

R H (t) N H (t) N H (t) R H (t) (40)
is Metzler for each choice of t, the system (39) is cooperative; see, e.g., [24, Lemma 1]. It follows that x(t) ≥ 0 and x ‡ (t) ≥ 0 (41) hold for all t ≥ r and all solutions of (39). Hence, the corresponding solutions of (37) satisfy x(t) ≥ 0 and x(t) ≤ 0 (42) for all t ≥ r. Similarly, since H = R H -N H , and since µ = µ + -µ -and δ = δ + -δ -, one can prove that x(t) ≤ x(t) ≤ x(t) (43) for all t ≥ r, by using ( 35) and [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF] to show the positivity of the dynamics for (x -x, x -x).

Let us introduce ζ(t) = x(t) -x(t). Then the inequalities (42) ensure that ζ(t) ≥ 0 for all t ≥ r. Moreover, for all t ≥ r, our formula [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF]

|µ(t)| ≤ |B(t)K(t)|Γ|x(t)| + |BK| ∞ (λ|ξ(t i -τ )| +ω t max{0,t-T } e |A|∞(t-) δ( )d + |BK| ∞ |ξ(t i -τ )| (45)
for all t ≥ τ with t ∈ [t i , t i+1 ). Thus, for all t ≥ r, we have

|µ(t)| ≤ |B(t)K(t)|Γ|x(t)| +|BK| ∞ ω t t-T e |A|∞(t-) δ( )d +|BK| ∞ (λ + I)|ξ(t i -τ )|. (46)
This inequality and (44) give

ζ(t) ≤ (R H (t) + N H (t))ζ(t) + |B(t)K(t)|Γ|x(t)| +|BK| ∞ ω t t-T e |A|∞(t-) δ( )d + |BK| ∞ (λ + I)|ξ(t i -τ )| + δ(t). ( 47
)
According to (42)-( 43), we have x(t)-x(t) ≤ x(t) ≤ x(t)x(t) and so also

|x(t)| ≤ ζ(t) (48)
for all t ≥ r, which we use to upper bound the right side of (47) to get

ζ(t) ≤ M (t)ζ(t) + |BK| ∞ (λ + I)|ξ(t i -τ )| +|BK| ∞ ω t t-T e |A|∞(t-) δ( )d + δ(t) ( 49
)
where M is the matrix defined in ( 18) for all t ≥ r.

We next use the linear Lyapunov function

V(ζ) = V ζ, ( 50
)
where V is the vector present in Assumption 1 as before.

According to Assumption 1 and the inequality (49), and also using the nonnegative valuedness of ζ, the time derivative of V(ζ) along the trajectories of (49) satisfies

V(t) ≤ -pV(ζ(t)) + V |BK| ∞ (λ + I)|ξ(t i -τ )| + ∆ 1 (t) (51)
for all t ≥ r, where

∆ 1 (t) = V |BK| ∞ ωe |A|∞T t t-T δ( )d + V δ(t). (52) Consequently V(t) ≤ -pV(ζ(t)) + b sup a∈[t-T -τ,t-τ ] ||ξ(a)|| + ∆ 1 (t), (53)
where b = ||V |BK| ∞ (λ + I)||, which is the desired ISS Lyapunov function decay estimation on V.

Third part. ISS estimate for [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF]. From the inequality [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF], it follows that the functional η we defined in [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] satisfies

η(ξ t ) ≤ c 1 e p2(T +τ ) e p2(s-t) sup ∈[s-2τ m ,s] ||ξ( )|| + c 3 sup ∈[s,t+τ ] ||δ( )|| (54)
for all s ≥ 0 and t ≥ s + T + τ . Now, consider the case where t ∈ [s, s+T +τ ] and s ≥ T +τ .

Then

η(ξ t ) ≤ sup l∈[s-T -τ,t] ||ξ (l)|| ≤ sup l∈[s-T -τ,s] ||ξ (l)|| + sup l∈[s,t] ||ξ (l)|| ≤ sup l∈[s,t]   c 1 e p2(s-l) sup ∈[s-2τ m ,s] ||ξ( )|| +c 3 sup ∈[s,l+τ ] ||δ( )|| + sup l∈[s-T -τ,s] ||ξ (l)|| (55)
where the last inequality is a consequence of [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF]. Since m ≥ 2, it follows that

η(ξ t ) ≤ η(ξ s ) + sup l∈[s,t]   c 1 sup ∈[s-2τ m ,s] ||ξ( )|| +c 3 sup ∈[s,l+τ ] ||δ( )|| ≤ η(ξ s ) + c 1 sup ∈[s-2τ m ,s] ||ξ( )|| + c 3 sup ∈[s,t+τ ] ||δ( )|| ≤ (c 1 + 1)η(ξ s ) + c 3 sup ∈[s,t+τ ] ||δ( )|| ≤ (c 1 + 1)e p2(T +τ ) e p2(s-t) η(ξ s ) + c 3 sup ∈[s,t+τ ] ||δ( )||, (56)
where the last inequality is a consequence of the fact that t -s ≤ T + τ . We deduce that η(ξ t ) ≤ c 4 e p2(s-t) η(ξ s ) + c 3 sup

∈[s,t+τ ] ||δ( )|| (57)
for all s ≥ T + τ and t ≥ s with c 4 = (c 1 + 1)e p2(T +τ ) . Condition (57) is our required ISS estimate for η.

Fourth part: ISS property for closed loop system. This part entails combining (53) and (57) into the ISS estimate as required by the theorem. It uses the functional

F(ζ(t), ξ t ) = V(ζ(t)) + (1 + b)W(ξ t ), ( 58
)
where b is the constant defined in [START_REF] Malisoff | Adaptive control for planar curve tracking under controller uncertainty[END_REF], and where

W(ξ t ) = t t-c * t a η(ξ l )dlda (59)
with c * > 0 being any constant satisfying our condition [START_REF] Malisoff | Adaptive planar curve tracking control and robustness analysis under state constraints and unknown curvature[END_REF]. For details, see Appendix B below.

Illustration

We illustrate our theorem, using a benchmark twodimensional (i.e., planar) curve tracking dynamical system from [START_REF] Malisoff | Adaptive planar curve tracking control and robustness analysis under state constraints and unknown curvature[END_REF], which we studied in our work [START_REF] Mazenc | Eventtriggered control for time-varying systems using a positive systems approach[END_REF] that was confined to unelayed cases. While simple (insofar that its only control is a steering control), it will illustrate the value of our approach for compensating any positive constant delay while reducing the computational burden as compared to continuous time controls that are not event-triggered.

The dynamics are

ḋ(t) = -sin(φ(t)) φ(t) = κ cos(φ(t)) 1+κd(t) + u(t -τ ) (60)
which describes curve tracking by a unit speed marine robot having a gyroscopic steering control u with a constant delay τ > 0, where d is the distance between the marine robot and the closest point (i.e., its projection) on the curve being tracked (assuming as in [START_REF] Malisoff | Adaptive planar curve tracking control and robustness analysis under state constraints and unknown curvature[END_REF][START_REF] Malisoff | Adaptive control for planar curve tracking under controller uncertainty[END_REF] that the closest point is unique at each time), φ denotes the heading angle (measuring the difference between the angles of the tangent lines of the path of the marine robot and of the path being traced by the closest point on the curve that is being tracked at each time t), and κ denotes the curvature at each time. We assume for simplicity that the curvature is a positive constant. Linearization of the system (60) around a trajectory (d r (t), φ r (t)) gives

ḋa (t) = a 12 (t)φ a (t) φa (t) = a 21 (t)d a (t) + a 22 (t)φ a (t) + u(t -τ ), (61)
where a 12 (t) = -cos(φ r (t)), a 21 (t) = -κ 2 cos(φr(t)) (1+κdr(t)) 2 , and a 22 (t) = -κ sin(φr(t)) 1+κdr(t) .

(62)

We consider the reference trajectory (d r , φ r ), where

d r (t) = 1 κ 1 + t∆ *
is that it provides the speed at which the robot is traveling towards the curve that is being tracked (if ∆ * < 0) or away from the curve being tracked (if ∆ * > 0). This contrasts with the undelayed case that was studied in [START_REF] Mazenc | Eventtriggered control for time-varying systems using a positive systems approach[END_REF], where the corresponding constant ∆ * was required to be nonpositive. Therefore, for this reference trajectory, (1 + ∆ * )/κ is the limit of the distance between the robot and the projection point on the curve being tracked as t → +∞. This gives

a 12 (t) = A 1 (t) -1, a 21 (t) = -κ 2 4 + A 2 (t)
, and a 22 (t) = A 3 (t), where

A 1 (t) = 1-1- ∆ 2 * κ 2 (1+t) 4 1/2 , A 2 (t) = κ 2   1 4 - 1- ∆ 2 * κ 2 (1+t) 4 (2+ t∆ * 1+t ) 2   ,
and

A 3 (t) = ∆ * (1+t) 2 1 2+ t∆ * 1+t . ( 65
)
Let us choose κ = 2. This produces the dynamics

ḋa (t) = (A 1 (t) -1)φ a (t) φa (t) = -(1 -A 2 (t))d a (t) + A 3 (t)φ a + u(t -τ ). ( 66
)
The change of coordinates (x 1 (t), x 2 (t)) = (d a (t), d a (t)φ a (t)) produces the dynamics ẋ(t) = A(t)x(t) + Bu(t -τ ), where

A(t) = A 1 (t) -1 1 -A 1 (t) A 4 (t) 1-A 1 (t)+A 3 (t) , (67)
B = [0, -1] , and

A 4 = A 1 -A 2 -A 3 .
Using Mathematica calculations, it follows that our requirements of Assumption 1 are satisfied with V = [1, 2] , δ = 0, K = [0, 1.75], each entry of Γ being 0.1, and p = 0.01 when the parameter in the reference trajectory (63) is ∆ * = 0.15. We obtained the preceding V , p, K, and Γ values by first finding constant choices of V , p, K and Γ for which Assumption 1 is satisfied when ∆ * = 0 (and with a constant H that is both Metzler and Hurwitz), and by then choosing ∆ * > 0 to be a small enough constant so that our requirements are still satisfied with these choices of V , p, K, and Γ. Then the requirement on ν is satisfied with ν = 0.026, so our theorem applies when the number m ≥ 2 of subpredictors satisfies m > 2 √ 2τ ||I + A|| ∞ max{||I + A|| ∞ , ||A|| ∞ }, which in this case gives the requirement m > 14.9701τ , meaning the number of subpredictors should be at least approximately 15 times the value τ of the input delay.

In the top panel of Fig. 1, we plot MATLAB simulations we obtained for the (d a , φ a ) dynamics in closed loop with the event-triggered control from our theorem, using the preceding values with τ = 0.1 with m = 2 subpredictors and constant initial functions. Since it shows fast convergence of the error states to zero, it helps to verify our theorem in the special case of the curve tracking dynamics (60). On the other hand, a key point in our method is its ability to compensate for arbitrarily long delays τ > 0. We illustrate this point in the bottom two panels of Fig. 1, which show our MATLAB simulations using the same parameters as the first simulation, except with τ = 1.

and ν = 0.2125, respectively, and keep the rest of the dynamics the same, then simple Mathematica calculations show that the requirements of our theorem are again satisfied for any delay τ when the number of subpredictors m satisfies m > 14.6989τ . This illustrates the possibility of changing the reference trajectory to obtain larger values of the guaranteed lower bound ν on the inter-sampling intervals t i+1 -t i , while not increasing the required lower bound on the ratio m/τ of the number of subpredictors to the delay.

Conclusion

We proposed a new event-triggered control for timevarying linear continuous time systems with a delay in the input and additive uncertainty terms. It uses a chain predictor, which is a dynamic extension of the original system that involves copies of the original system running on different scales and additional stabilizing terms, and yields ISS with respect to the uncertainty for an arbitrarily large constant delay. This contrasts with traditional exact predictor approaches, which would have instead produced distributed terms in the control that would have made numerical implementations more difficult.

The work was strongly motivated by the ubiquity of input delays, and the benefits of event-triggered control as a way to eliminate the need to continuously (or too frequently) change the control value. Instead of standard Lyapunov function approaches, we used framers and positive systems.

A key feature of our approach is that neither the control nor the trigger mechanism require current state measurements. Extensions to systems with a time-varying delay are expected, as well as extensions to output feedback de-signs. The case where the delay is poorly known, applications to networked systems [START_REF] Zhu | Sub-predictors for network-based control under uncertain large delays[END_REF], and comparisons of our approach with alternative approaches that could be based on quadratic Lyapunov functions, will be subjects of further studies too. where v = min i V i is the smallest entry of V as defined in [START_REF] Malisoff | Adaptive control for planar curve tracking under controller uncertainty[END_REF]. Using (48) and upper bounding the formula (59) using the formula (B.9), we deduce that This allows us to conclude.

  We use standard definitions and notation, which we simplify when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. Given a matrix G = [g ij ] ∈ R r×s , we set |G| = [|g ij |], i.e., the entries of |G| are the absolute values of the corresponding entries of G. When G is time varying with bounded entries, we set |G| ∞ = [m ij ], where

  by writing Φ A (t, m) = α A (t)β A (m) for all m and t ≥ m, where α A and β A are the unique solutions of αA (t) = A(t)α A (t) and βA (m) = -β A (m)A(m) (26) that satisfy α A (0) = β A (0) = I, and by then choosing Γ = sup{|I -ω -1 (t, s)| : s ∈ [0, p], t ∈ [s, s + ν]}. Similar reasoning allows us to compute λ and ω from (15)-(16).

  ts max{0,ts-T } e |A|∞(ts-) δ( )d .

1+t

  and φ r (t) = -arcsin ∆ * κ(1+t) 2 . (63)The physical meaning of the constant ∆ * ∈ (-min{1, κ}, min{1, κ})

  2 and m = 21 subpredictors used in the middle panel; and with τ = 3 and m = 45 used in the bottom panel. Each panel shows satisfactory control performance, and therefore helps to illustrate the value of our theorem. If we replace (63) by the reference trajectory d r (t) = 1 κ (1 -0.1 sin(t)) and φ r (t) = arcsin cos(t) 10κ ,(68)and if we replace V , p, Γ, and ν by V = [1, 3.7] , p = 0.001,

Fig. 1 .

 1 Fig. 1. Curve Tracking with K = [0, 1.75] and Delay τ = 0.1 (Top Panel), τ = 1.2 (Middle Panel), and τ = 3 (Bottom Panel)

  * η(ξ s )ds + c 3 sup ∈[t-c * ,t+τ ] ||δ( )|| (B.5) for all t ≥ c * +T +τ . Combining (B.5) and (B.1), we obtainẆ(t) ≤ -η(ξ t ) -t t-c * η(ξ l )dl + (c * + 1) c4p2 e p 2 c * -1 t t-c * η(ξ s )ds + c 3 (c * + 1)sup∈[t-c * ,t+τ ] ||δ( )|| (B.6)for all t ≥ c * + T + τ . Since our condition[START_REF] Malisoff | Adaptive planar curve tracking control and robustness analysis under state constraints and unknown curvature[END_REF] on c * and our choice c 4 = (c 1 + 1)e p2(T +τ ) imply that(c * + 1) c4p2 e p 2 c * -1 ≤ * η(ξ l )dl + c 3 (c * + 1)sup∈[t-c * ,t+τ ] ||δ( )||. (B.8)It follows from the boundW(ξ t ) ≤ c * t t-c * η(ξ l )dl (B.9) that Ẇ(t) ≤ -η(ξ t ) -1 2c * W(ξ t ) +c 3 (c * + 1)sup∈[t-c * ,t+τ ] ||δ( )|| (B.10)for all t ≥ c * + T + τ . Recalling our formula[START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] for η, and using (53) and (B.10), we then haveḞ(t) ≤ -pV(ζ(t)) + b sup a∈[t-T -τ,t-τ ] ||ξ(a)|| + ∆ 1 (t) -(1 + b)η(ξ t ) -1+b 2c * W(ξ t ) +c 3 (1 + b)(c * + 1)sup∈[t-c * ,t+τ ] ||δ( )|| ≤ -p V(ζ(t)) + 1+b 2c * p W(ξ t ) -η(ξ t ) + ∆ 1 (t) + c 3 (1 + b)(c * + 1)sup∈[t-c * ,t+τ ] ||δ( )|| ≤ -p F(ζ(t), ξ t ) + ∆ 2 (t) (B.11)for all t ≥ max{r, c * + T + τ }, where∆ 2 (t) = ∆ 1 (t)+c 3 (1+b)(c * +1) sup ∈[t-c * ,t+τ ] ||δ( )|| (B.12)and p = min{p, 1 2c * } is the constant from[START_REF] Malisoff | Adaptive control for planar curve tracking under controller uncertainty[END_REF]. Applying variation of parameters to (B.11) then givesF(ζ(t), ξ t ) ≤ e -p (t-s) F(ζ(s), ξ s ) + t s e -p (t-) ∆ 2 ( )d (B.13) for all t ≥ s ≥ max{r, c * + T + τ }. Since V(ζ) = V ζ ≥ v(ζ 1 + . . . ζ n ) ≥ v||ζ|| for all ζ ∈ [0, +∞) n , itfollows that v||ζ(t)|| + (1 + b)W(ξ t ) ≤ e -p (t-s) ||V ||||x(s) -x(s)|| +(1 + b)e -p (t-s) W(ξ s ) + t s e -p (t-) ∆ 2 ( )d , (B.14)

18 )

 18 ||x(t)|| ≤ ||V ||v e -p (t-s) ||x(s) -x(s)|| + (1+b)c * e -p (t-s) p (t-) ∆ 2 ( )d≤ ||V || v e -p (t-s) ||x(s) -x(s)|| + (1+b)c 2 * v e -p (t-s) sup ∈[s-c * -T -τ,s] p (t-) ∆ 2 ( )d (B.15) when t ≥ s ≥ max{r, c * + T + τ }. According to (8), we have ξ(t) = z m (t) -x(t + τ ), so we deduce that ||x(t)|| ≤ ||V || v e -p (t-s) ||x(s) -x(s)|| + (1+b)c 2 * v e -p (t-s) sup ∈[s-c * -T -τ,s] p (t-) ∆ 2 ( )d . (B.16)Consequently, when r ≥ c * + T + τ , our requirements[START_REF] Weston | Sequential predictors under timevarying feedback and measurement delays and sampling[END_REF] on the initial conditions give||x(t)|| ≤ ||V || v e -p (t-r) ||x + (r) + x -(r)|| + (1+b)c 2 * v e -p (t-r) sup ∈[r-c * -T -τ,r] p (t-) ∆ 2 ( )d ≤ e -p (t-r) c sup ∈[r-c * -T,r+τ ]It follows from our choice of c e in (23), and from our choices of ∆ 1 and ∆ 2 in (52) and (B.12), that with the choicec 5 = ||V || + c 3 (1 + b)(c * + 1), we have ||x(t)|| ≤ e -p (t-r) c sup ∈[r-c * -T,r+τ ] ||x( )|| +c ‡ sup ∈[r-c * -T -τ,r] ||z m ( )|| + 1 vp ||V |BK| ∞ ωe |A|∞T || t r-T δ( )d+c 5 sup ∈[r-c * ,t+τ ] ||δ( )|| ≤ e -p (t-r) c sup ∈[r-c * -T,r+τ ] ||x( )|| +c ‡ sup ∈[r-c * -T -τ,r] ||z m ( )|| + c e sup ∈[r-max{c * ,T },t+τ ] ||δ( )||. (B.19)

  to establish asymptotic stability conditions for a positive system whose state ζ is a difference between upper and lower estimates from an interval observer. Assumption 1 holds if M is a constant Hurwitz and Metzler matrix (e.g., by [16, Lemma 2.3, p.41]), and so also if M has the form M 0 + ∆ M (t) with M 0 being a constant Hurwitz and Metzler matrix when the sup norm of ∆ M is small enough; see Remark 4 for general conditions under which Assumption 1 is satisfied.

  -τ -) δ( )d for all t ∈ [t i + τ, t i+1 + τ ); and either t i+1 -t i = T, or |t i+1 -t i | < T and for each > 0 there exists a t s ∈ (t i+1 , t i+1 + ) such that |x(t s ) -x(t i )| Γ|x(t s )| + λ|ξ(t i -τ )| +ω

Appendix A: Lower bound on m

We show that (3) is globally exponentially stable to 0 on R n with the choice L(t) = -I -A(t) when m > 2 √ 2τ ||L|| ∞ max{||L|| ∞ , ||A|| ∞ } and τ > 0. To this end, we rewrite (3) as Ẋ(t) = -X(t) + (A(t) + I)(X(t) -X(t -τ /m)) + κ(t), which we can use to check that the time derivative of 2 , and the last inequality in (A.1) followed from two applications of Holder's inequality rs ≤ 1 4 r 2 + s 2 (first with r and s being the terms in curly braces in (A.1), and then applied with r = ||X(t)|| and s = ĉ1/2 ||κ|| ∞ ) followed by Jensen's inequality. Choosing any constant > 0 such that m > 2

along all solutions of (3) satisfies

for all t ≥ 0, where

is positive because of our lower bound on m. By applying variation of parameters to (A.3), then lower bounding the left side V (X t ) of the result by 1 2 |X(t)| 2 , then multiplying through by 2, and then finally taking square roots of both sides, we obtain the desired exponential ISS estimate