Frédéric Mazenc

Michael Malisoff

Local Observer for Parameters and for State Variables of Nonlinear Systems

We provide new observer designs to simultaneously identify parameters and states of systems whose nonlinearities have order two near the origin, which include cubic terms arising in the study of jump phenomena, process control, and bistable models of aerospace systems. This yields local exponential convergence of the state estimation error to zero, basin of attraction estimates, and fixed time parameter identification. We illustrate our result using Duffing's equation, whose cubic term puts it outside the scope of prior methods.

I. INTRODUCTION

We continue our search for observers for unknown states and unknown model parameters that we began in [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF], [START_REF]Finite-time observers for parameters and state variables of nonlinear systems[END_REF]. Whereas [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF], [START_REF]Finite-time observers for parameters and state variables of nonlinear systems[END_REF] were confined to systems where the state dependent nonlinearity satisfies a linear growth condition and provided global exponential stability, here we replace this linear growth condition by an order 2 condition near the origin (which covers cubic terms that are important in the study of jump phenomena, process control, and bistable models in aerospace engineering), but we instead prove local exponential stability for the state observer with basin of attraction estimates, and fixed time parameter identification, where by fixed time, we mean finite time convergence where the convergence time can be chosen to be independent of the initial state. This contrasts with our prior works (e.g., [START_REF] Ahmed | Finite time estimation for time-varying systems with delay in the measurements[END_REF], [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF], [START_REF]Reduced order finite time observers and output feedback for time-varying nonlinear systems[END_REF], [START_REF] Mazenc | Almost finite-time observers for a family of nonlinear continuous-time systems[END_REF], [START_REF]New finite-time and fast converging observers with a single delay[END_REF], [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF], [START_REF] Mazenc | Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence[END_REF]) for finite-time observers for nonlinear systems, which provided arbitrarily fast convergence of state observers for perturbed systems (and fixed time convergence when the uncertainties are zero and when the system was affine in the unmeasured state), but did not identify model parameters. These studies are motivated by the fact that observers are useful, e.g., to solve feedback control problems; see, e.g., [START_REF] Borri | Luenberger-like observers for nonlinear time-delay systems with application to the artificial pancreas: the attainment of good performance[END_REF], [START_REF] Buccella | Observerbased control of LCDC/DC resonant converter using extended describing functions[END_REF], [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF], [START_REF]A new approach to design interval observers for linear systems[END_REF], [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF], [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit Lyapunov function approach[END_REF], [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], [START_REF] Parikh | Framework for guaranteed convergence of image-based observers with intermittent measurements[END_REF], [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF].

As in [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF], [START_REF]Finite-time observers for parameters and state variables of nonlinear systems[END_REF], a key motivation is artificial neural network expansions that represent unknown time-varying functions as linear combinations of known basis functions whose constant weights must be identified. While adaptive control can estimate parameters, it is usually not amenable for fixed time parameter identification, and does not estimate states. We address these challenges when there is a linear output that can have measurement delays. By allowing nonlinearities that violate uniform global Lipschitzness conditions on the statedependent nonlinearity, we cover types of jump dynamics Key Words: Observers, nonlinear, delay. F. Mazenc is with Inria EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: frederic.mazenc@centralesupelec.fr).

M. Malisoff is with Department of Mathematics, 303 Lockett Hall, Louisiana State University, Baton Rouge, LA 70803-4918 USA (email: malisoff@lsu.edu). Supported by NSF Grant 2009659. that we study in Section IV below. Our results are novel and of interest even if no unknown parameters or measurement delays are present, because of the order 2 nonlinearities.

We use standard notation. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise, and | • | is the standard Euclidean and corresponding matrix norm. We set g t (s) = g(t + s) for functions g and all s ≤ 0 and t ≥ 0 such that t + s is in the domain of g, |f | J is the usual supremum over any interval J in the domain of functions f , |f | ∞ is the supremum of a function f , λ max (M ) (resp., λ min (M )) denote largest (resp., smallest) eigenvalues of square matrices M whose eigenvalues are all real, 0 is the zero matrix, and I denotes the identity matrix.

II. MAIN RESULT

A. Studied System and Statement of Main Result

Consider the system

     ẋ(t) = Ax(t) + B q i=1 ϵ i γ i (t) + u(t) + ∆(x(t), t) y(t) = Cx(t -d) (1)
whose state x is valued in R n , where the constants ϵ j in the parameter vector ϵ = [ϵ 1 , . . . , ϵ q ] ⊤ are unknown, the control u will be specified, the measurement y is real valued, the matrices A ∈ R n×n , B ∈ R n×m , and C ∈ R 1×n are known, γ(t) = [γ 1 (t), . . . , γ q (t)] is a matrix of known C 1 bounded functions where γ i : R → R m for each i, d ≥ 0 is a known constant, and ∆ is locally Lipschitz in x uniformly in t and has order 2 at the origin, where the order 2 condition is the requirement that there is a nondecreasing positive definite continuous function

L : [0, +∞) → [0, +∞) such that |∆(x, t)| ≤ L(|x|)|x| 2 (2)
for all x ∈ R n and t ≥ 0. The sum in (1) (which can be written as γ(t)ϵ) can represent an uncertainty that is parameterized by unknown weights ϵ j on known basis functions γ i . Condition (2) often holds, e.g., for systems with time invariant C 2 nonlinearities G(x) such that G(0) = 0, since we can use a Maclaurin expansion to write G(x) = G 1 x + G 2 (x) for a constant matrix G 1 such that the choice ∆(x, t) = G 2 (x) satisfies (2), and then we can add G 1 to A to get one constant coefficient of x, when the given system has the form ẋ(t) = Ax(t) + B[γ(t)ϵ + u(t)] + G(x(t)).

Later we specify u such that the closed loop system satisfies standard forward completeness conditions for all initial states x(0) in the basin of attraction of our locally exponentially stabilized system. Our first assumption is:

Assumption 1: The pair (A, B) is controllable, and the pair (A, C) is observable.

Assumption 1 implies that the matrix

E = 0 -h e A ⊤ s C ⊤ Ce As ds (3)
is invertible for each constant h > 0. We fix the constant h > 0 in what follows. In terms of the real valued functions

η i (t) = CE -1 t t-h e A ⊤ (s-t) C ⊤ C t-d s-d e A(s-m-d) Bγ i (m)dmds (4)
for i = 1, . . . , q and the function η(t) = [η 1 (t), . . . , η q (t)], our second assumption is the following, which agrees with our main invertibility assumption in our works [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF], [START_REF]Finite-time observers for parameters and state variables of nonlinear systems[END_REF] on systems without order 2 state dependent nonlinearities: Assumption 2: There are known nonnegative constants γ and τ j for j = 1, . . . , q such that τ 1 < τ 2 < . . . < τ q and such that the q × q matrix

W (t) = [η ⊤ (t -τ 1 ), . . . , η ⊤ (t -τ q )] (5)
is nonsingular for each t ≥ τ q + h + d, and such that 0

< |Bγ(W -1 ) ⊤ | [τq+h+d,+∞) ≤ γ.
Assumption 1 also provides matrices L and K such that

H = A + LC and M = A + BK (6)
are Hurwitz matrices. We fix L and K that satisfy the preceding requirements in the sequel, and corresponding H and M as defined by [START_REF] Mazenc | Almost finite-time observers for a family of nonlinear continuous-time systems[END_REF]. Then the Hurwitzness of H and M provides positive definite matrices Q 1 and Q 2 and constants c 1 > 0, c 2 > 0, and d > 0 such that the time derivative of

V (z t ) = V 0 (z(t)) + c 2 t t-2d t s |z(ℓ)| 2 dℓds (7) along all solutions z = (z 1 , z 2 ) : [0, +∞) → R 2n of the 2n-dimensional system    ż1 (t) = M z 1 (t) + LCz 2 (t) + BK[z 1 (t -d) -z 1 (t)] ż2 (t) = Hz 2 (t) (8) with constant initial functions satisfies d dt V (z t ) ≤ -c 1 V (z t ) for all t ≥ 0 and all d ∈ [0, d], where V 0 (z) = z ⊤ 1 Q 1 z 1 + z ⊤ 2 Q 2 z 2 .
We assume that the initial functions for all of our delayed systems are constant at the initial time t = 0.

The preceding constants and V can be found using the Lyapunov-Krasovskii methods from [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]; see the appendix below. We assume that d ∈ [0, d] and we use the constants

S = h sup CE -1 e A ⊤ s C ⊤ Ce A(s-m) : s ∈ [-h, 0], m ∈ [s, 0] and m 0 = min{λ min (Q 1 ), λ min (Q 2 )}, (9)
and the functions

µ 1 (s) = 2L s λmin(Q1) + s λmin(Q2) 1 m0 , Γ 1 (s) = 2|Q 2 | s λmin(Q2) µ 1 (s), Γ 2 (s) = 1
2 µ 1 (s), and

Γ 3 (s) = 4 s λmin(Q2) |Q 2 BγW -1 | [h+τq+2d,∞) S √ q ( 10
)
where L is from the order 2 condition (2) on the nonlinearity ∆. We also use the functionals

G ♯ (x t ) = [G(x t-τ1 ), . . . , G(x t-τq )], θ ♯ (y t , u t ) = [θ(y t-τ1 , u t-τ1 ), . . . , θ(y t-τq , u t-τq )],
and

Ω(t, y t , u t ) = γ(t)[θ ♯ (y t , u t )W -1 (t)] ⊤ , (11)
where

G(x t ) = -CE -1 t t-h e A ⊤ (s-t) C T t-d s-d e A(s-d-m) ∆ x (m)dmds and θ(y t , u t ) = y(t) -CE -1 t t-h e A ⊤ (s-t) C ⊤ y(s)ds -CE -1 t t-h e A ⊤ (s-t) C T t-d s-d e A(s-d-m) Bu(m)dmds (12)
and ∆ x (m) = ∆(x(m), m) and C T = C ⊤ C. We then use the dynamic controller

u(t) = K x(t), t ∈ [0, h + τ q + 2d) K x(t) -Ω(t, y t , u t ), t ≥ h + τ q + 2d (13)
where the estimator x of x has the dynamics

ẋ(t) = M x(t) + BK[x(t -d) -x(t)] + L[C x(t) -y(t)] (14)
with x(0) = 0. Letting v * > 0 be a constant such that

Γ 1 (v * ) + (h + τ q + 2d)Γ 2 (v * )Γ 3 (v * ) < c 1 ( 15
)
where c 1 is the decay rate on V given above, and setting

E(t) = sup{|Q 2 Bγ(t)λ| : λ ∈ E} (16)
for any bounded set E ⊆ R q that is known to contain ϵ, our last assumption is this smallness condition on E, which can be checked in practice because the right side of ( 17) is independent of ϵ and bounds on ϵ are typically known: Assumption 3: With the preceding choices, the inequality

|E| [0,h+τq+2d] < 1 2 v * λ min (Q 2 ) c 1 -Γ 1 (v * ) (17) is satisfied. Also, d ∈ [0, d].
Using the variable x(t) = x(t) -x(t -d) and constants

R * = max{λ max (Q 1 ), λ max (Q 2 )}(1 + 2d 2 c 2 )|c d | 2 , where c d = (1 + c d2 , c d2 ) and c d2 = d 0 |e (M +LC-BK)ℓ LC|dℓ (18)
and

∆ E = 4 λmin(Q2) |E| [0,h+τq +d] c1-Γ1(v * ) 2 , (19)
our main result is as follows (but see Remark 2.1 and Section III below for ways to express u in terms of a dynamical extension without distributed terms, and a way to express ϵ in terms of the estimator x instead of the state x): Theorem 1: Let Assumptions 1-3 hold. Consider (1) in closed loop with [START_REF]A new approach to design interval observers for linear systems[END_REF], where the dynamics of x is defined by [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF], with x(0) = 0. Then for each initial state x(0) in

S 0 = z ∈ R n : |z| 2 < 1 R * (v * -∆ E ) , (20)
the pair (x(t), x(t)) satisfies lim t→+∞ (x(t), x(t)) = 0, and

ϵ ⊤ = [θ ♯ (y t , u t ) + G ♯ (x t )]W -1 (t) (21)
holds for all t ≥ h + τ q + d.

Remark 2.1: The proof of our theorem allows us to construct positive constants r i for i = 1, 2 such that

|(x(t), x(t))| ≤ r 2 e -r1t (|x(0)| + (h + τ q + 2d)B * ) (22)
for all t ≥ τ q + h + 2d and initial states x(0) ∈ S 0 , where B * = sup{|Bγλ| ∞ : λ ∈ E} and E is the set we used in ( 16). This will be done by transforming the functional V from [START_REF]New finite-time and fast converging observers with a single delay[END_REF] into a new Lyapunov-Krasovskii functional V ♯ , with the choice (z 1 , z 2 ) = (x, x), and then building a uniform global exponential stability estimate for V ♯ that is valid for all t ≥ τ q +h+2d, and then applying a variation of parameters to the dynamics for (x, x) on [0, τ q +h+2d], after first showing that all solutions (x(t), x(t)) for all initial states x(0) ∈ S 0 stay in a bounded set in which the dynamics satisfy a linear growth condition. This will allow us to construct a nondecreasing function

L ♯ : [0, +∞) → [0, +∞) such that ϵ ⊤ W (t) = θ ♯ (y t , u t ) + G ♯ (x t ) + L ♯ J t (J t ) 2 (23)
for all t ≥ 2(h + τ q + d), where J t = e -r1t (|x(0)| + (h + τ q )B * ), and where G ♯ (x t ) is defined as in ( 11)-( 12) except with x in the preceding formulas replaced by x. The construction of L ♯ follows because our growth condition (2)

on ∆ gives |∆(x(ℓ), r)| = |∆(x(ℓ + d) -x(ℓ + d), r)| ≤ 2L(|x| [t-h-τq-2d,t+d] + |x| [t-h-τq-2d,t+d] )z(t) for all ℓ ∈ [t -h -τ q -2d, t], r ≥ 0, and t ≥ h + τ q + 2d, where z(t) = (|x| [t-h-τq-2d,t+d] + |x| [t-h-τq-2d,t+d] ) 2
Hence, ϵ is expressible in terms of the measured quantitites y t , xt , and u t at each time t, with an additional error term that exponentially converges to 0 as t → +∞ if W -1 is bounded. See also Section III for ways to express u and W in terms of solutions of ordinary differential equations.

B. Proof of Theorem 1

The proof has three parts. First, we prove [START_REF] Schieni | Control of bistable structures using a modified hybrid position feedback controller[END_REF], and we derive the dynamics for (x, x), whose linearization at 0 agrees with [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF] and so admits the Lyapunov functional [START_REF]New finite-time and fast converging observers with a single delay[END_REF]. In the second part, we show that lim t→+∞ V (x t , xt ) = 0 from all initial states x(0) ∈ S 0 , provided sup ℓ∈[0,h+τq+2d] V (x ℓ , xℓ ) < v * , where v * is from [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit Lyapunov function approach[END_REF]. In the last part, we prove that our choice of S 0 implies that

sup ℓ∈[0,h+τq+2d] V (x ℓ , xℓ ) < v * if x(0) ∈ S 0 .
First Part: Dynamics for (x, x). Let ρ(t) = Bu(t) + ∆(x(t), t). Then, for each time t ≥ h + d when the solution of ( 1) is defined, we can prove that

y(t) = CE -1 t t-h e A ⊤ (s-t) C ⊤ y(s)ds + η(t)ϵ +CE -1 t t-h e A ⊤ (s-t) C T t-d s-d e A(s-d-m) ρ(m)dmds. ( 24
)
where C T = C ⊤ C and η(t) = [η 1 (t), . . . , η q (t)] with each η i defined in (4). To see why (24) holds, first note that we can apply the method of variation of parameters to (1) to get

e A(s-t) x(t -d) = x(s -d) + t-d s-d e A(s-d-ℓ) B(γ(ℓ)ϵ +u(ℓ))dℓ + t-d s-d e A(s-d-ℓ) ∆(x(ℓ), ℓ)dℓ (25)
for all t ≥ s and s ≥ d. Then (24) follows by left multiplying (25) through by e A ⊤ (s-t) C T , then integrating the result over s ∈ [t -h, t], and then left multiplying the result by CE -1 . Thus, (24) and our choice of θ in [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] gives

ϵ ⊤ η ⊤ (t) = y(t) -CE -1 t t-h e A ⊤ (s-t) C ⊤ y(s)ds -CE -1 t t-h e A ⊤ (s-t) C T t-d s-d e A(s-d-m) ρ(m)dmds (26)
and so also

ϵ ⊤ η ⊤ (t) = θ(y t , u t ) -CE -1 t t-h e A ⊤ (s-t) C T t-d s-d e A(s-d-m) ∆(x(m), m
)dmds for all t ≥ h + d. Therefore, our choice of G in [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] and our formula for W in Assumption 2 give

ϵ ⊤ W (t) = [θ(y t-τ1 , u t-τ1 ) + G(x t-τ1 ), . . . , θ(y t-τq , u t-τq ) + G(x t-τq ) (27)
for all t ≥ h+τ q +d. Since Assumption 2 ensures that W (t) is nonsingular for each t ≥ τ q + h + d, (21) follows from our formulas for G ♯ and θ ♯ in [START_REF] Buccella | Observerbased control of LCDC/DC resonant converter using extended describing functions[END_REF]. Hence,

q i=1 ϵ i γ i (t) = γ(t)ϵ = γ(t)[θ ♯ (y t , u t )W -1 (t)] ⊤ +γ(t)[G ♯ (x t )W -1 (t)] ⊤ (28)
for all t ≥ h + τ q + d. Inserting (28) into the x dynamics (1) and recalling our formula for Ω from ( 11) now gives

ẋ(t) = Ax(t) + ∆(x(t), t) + B [Ω(t, y t , u t ) +γ(t)[G ♯ (x t )W -1 (t)] ⊤ + u(t) . (29)
This motivates our choice

ẋ(t) = Ax(t) + B [Ω(t -d, y t-d , u t-d ) + u(t -d)] + L[C x(t) -y(t)] (30)
of the observer. Combining (29)-(30) gives the dynamics

ẋ(t) = H x(t) -Bγ(t -d)[G ♯ (x t-d )W -1 (t -d)] ⊤ -∆(x(t -d), t -d) (31)
for the error x(t) = x(t) -x(t -d) for all t ≥ h + τ q + 2d. Hence, substituting ( 13) into (30) gives

   ẋ(t) = M x(t) + LC x(t) + BK[x(t -d) -x(t)] ẋ(t) = H x(t) -Bγ(t -d)[G ♯ (x t-d )W -1 (t -d)] ⊤ -∆(x(t -d), t -d) (32)
when t ≥ h + τ q + 2d. The dynamics (32) have the linear approximation ( 8) with (z 1 , z 2 ) = (x, x) which admits the Lyapunov functional V from [START_REF]New finite-time and fast converging observers with a single delay[END_REF]. Hence, along all solutions of (32), our V from (7) satisfies

V (t) ≤ -c 1 V (x t , xt ) for all t ≥ h + τ q + 2d when ∆ is not present. Second Part: Proof that lim t→+∞ V (x t , xt ) = 0 if sup ℓ∈[0,h+τq+2d] V (x ℓ , xℓ ) < v * .
Using the first part of the proof and recalling the lower bounds

V 0 (x, x) ≥ max{λ min (Q 1 )|x| 2 , λ min (Q 2 )|x| 2 } and V 0 (x, x) ≥ min{λ min (Q 1 ), λ min (Q 2 )}|(x, x)| 2 ,
it follows from our choices of the Γ i 's in [START_REF] Borri | Luenberger-like observers for nonlinear time-delay systems with application to the artificial pancreas: the attainment of good performance[END_REF] that when ∆ is present,

V (t) ≤ -c 1 V (x t , xt ) + Γ 1 (V (x t , xt ))V (x t , xt ) + t t-h-τq-2d Γ 2 (V (x m , xm ))V (x m , xm )dmΓ ♯ 3 (t) (33)
along all solutions of (32) for all t ≥ h + τ q + 2d, where

Γ ♯ 3 (t) = Γ 3 (V (x t , xt ))
, and where (in terms of the m 0 we contradiction V (x t b , xt b ) < v * . This proves the theorem.

III. IMPLEMENTING THE CONTROL

We can express our control [START_REF]A new approach to design interval observers for linear systems[END_REF] in terms of dynamic extensions that eliminate integrals from the control expression. This makes our method more amenable to implementations. To see how, first note that for all t ≥ h, our formula for θ in [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] gives

θ(y t , u t ) = y(t) -CE -1 [L 1 (t) -e -Ah L 1 (t -h)] -CE -1 t t-h A(s, t) t-d s-d e A(t-d-m) Bu(m)dmds (49)
and so also

θ(y t , u t ) = y(t) -CE -1 [L 1 (t) -e -Ah L 1 (t -h)] -CE -1 t t-h A(s, t)[L 2 (t -d) -e A(t-s) L 2 (s -d)]ds (50)
hence

θ(y t , u t ) = y(t) -CE -1 [L 1 (t) -e -Ah L 1 (t -h)] -CL 2 (t -d)+CE -1 t t-h e A ⊤ (s-t) C T L 2 (s -d)]ds (51)
which gives

θ(y t , u t ) = y(t) -CE -1 [L 1 (t) -e -Ah L 1 (t -h)] -CL 2 (t -d) + CE -1 [L 3 (t) -e A ⊤ h L 3 (t -h)], (52)
where A(s, t) = e A ⊤ (s-t) C T e A(s-t) and C T = C ⊤ C as before, and where L 1 , L 2 , and L 3 are the states of

     L1 (t) = -A ⊤ L 1 (t) + C ⊤ y(t) L2 (t) = AL 2 (t) + Bu(t) L3 (t) = -A ⊤ L 3 (t) + C T L 2 (t -d) (53)
with the initial states L 1 (s) = L 2 (s) = L 3 (s) = 0 for all s ≤ 0, and where the (49), (50), and (52) followed from applying variations of parameters to the L 1 , L 2 , and L 3 dynamics, respectively. This allows us to express the control values as

u(t) = K x(t) -γ(t)[W -1 (t)] ⊤ [θ(y t-τ1 , u t-τ1 ), θ(y t-τ2 , u t-τ2 ), . . . , θ(y t-τq , u t-τq )] ⊤ (54)
for times t ≥ τ q + h + 2d in terms of the y, L 1 , L 2 , and L 3 values. This expression for u in terms of y, L 1 , L 2 , and L 3 can then be substituted into (53) to get a dynamics for L 1 , L 2 , and L 3 that can be solved to produce L 1 , L 2 , and L 3 values that allow us to write u without using integral terms. Also, although our functions η i from (4) (and therefore also W (t) from ( 5)) are expressed in terms of integrals, the method from [1, Section III] makes it possible to write each η i in terms of delayed dynamical extensions. This follows because the integral from the η i formula can be rewritten as

t t-h C ♯ (r -t) t-d r-d e A(t-d-m) Bγ i (m)dm dr = t t-h C ♯ (r-t) L 4i (t -d)-e (t-r)A L 4i (r-d) dr = EL 4i (t -d) - t t-h e -A ⊤ (t-r) C ⊤ CL 4i (r -d)dr = EL 4i (t -d) -(L 5i (t) -e -A ⊤ h L 5i (t -h)) (55)
for each i and t ≥ 0, where C ♯ (r) = e A ⊤ r C ⊤ Ce Ar , and where L 4i and L 5i are the states of the dynamic extension

L4i (t) = AL 4i (t) + Bγ i (t) L5i (t) = -A ⊤ L 5i (t) + C ⊤ CL 4i (t -d) (56)
with L 4i (s) = L 5i (s) = 0 for all s ≤ 0, where the first equality in (55) followed from applying variation of parameters to the L 4i dynamics, and the last equality in (55) followed by applying variation of parameters to the L 5i dynamics. This eliminates the need to compute integrals to check Assumption 2, which can then be checked by finding a positive lower bound on the absolute value of the determinant of W (t) over all t ≥ τ q + h + d.

IV. ILLUSTRATION Consider Duffing's equation, which contains a type of cubic nonlinearity that plays an important role in the study of jump phenomena, process control, and bistable models in aerospace engineering [START_REF] Pearson | Nonlinear model identification[END_REF], [START_REF] Schieni | Control of bistable structures using a modified hybrid position feedback controller[END_REF]. It is the second order dynamics z + c ż + αz

+ βz 3 = u + δ(t), (57)
where c, α, and β are real constants, and where the locally bounded piecewise continuous added uncertainty δ(t) represents unmodelled terms or errors due to the parameter uncertainties. The special case where u(t) = ω 1 cos(ω 2 t) for constants ω 1 and ω 2 and where δ is the zero function is called Duffing's oscillator, and can be viewed as a forced oscillator with a spring whose restoring force F = -αz -βz 3 when α > 0 [START_REF] Korsch | Chaos: A Program Collection for the PC[END_REF]. Assuming that the additive uncertainty δ on the control that we must identify has the form δ(t) = ϵ 1 γ 1 (t) + . . . ϵ q γ q (t), the system (57) can be written as [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF] with

A = 0 1 -α -c , B = 0 1 , ∆(x, t) = 0 -βx 3 1 , (58)
and C = [0, 1]. Then we can satisfy our requirements with L(s) = βs, for many choices of the parameters and basis functions, which allows us to implement our control using the strategy from the preceding section. For instance, with α = -1, β = 0.01, c = 3, d = 0, q = h = 1, and γ 1 = 1, and with K and L such that H and M both have the eigenvalues -2 and -2.02 (where K and L can be found, e.g., using the StateFeedbackGains command in Mathematica), then simple Mathematica calculations show that we can satisfy our requirements with the positive definite quadratic matrices

and c 1 = 0.438442. This differs significantly from prior treatments of (57), which did not make it possible to identify the state or unknown additive uncertainties on u.

V. CONCLUSION We provided a new observer design for systems whose state-dependent nonlinearities have order 2 at 0, which produced fixed time identification of model parameters and exponential convergence of the observation error for the state to 0. This is a significant departure from prior works that did not provide fixed time parameter convergence, or did not allow order 2 nonlinearities. Our method covers examples such as Duffing's equation that have cubic terms, which play important roles in the study of jump phenomena. We also allowed measurement delays. Due to the nonlinearity having order 2, this work provides a local result (instead of the global result from [START_REF] Mazenc | Almost finite-time observers for parameters and for state variables of nonlinear systems[END_REF], which was confined to systems where the state dependent nonlinearity is uniformly globally Lipschitz in the state uniformly in time), including estimates for basins of attraction. We aim to optimize the parameters in the observer to maximize the convergence rate of the observation error to zero and maximize the basin of attraction. APPENDIX: DERIVING V FROM [START_REF]New finite-time and fast converging observers with a single delay[END_REF] We derive the Lyapunov-Krasovskii functional [START_REF]New finite-time and fast converging observers with a single delay[END_REF]. We first solve the Riccati equations P 1 M + M ⊤ P 1 = -I and P 2 H + H ⊤ P 2 = -I for positive definite matrices P 1 and P 2 with the M and H in [START_REF] Mazenc | Almost finite-time observers for a family of nonlinear continuous-time systems[END_REF], and set V a (z 1 ) = z ⊤ 1 P 1 z 1 . Along solutions of ( 8 along all solutions of (8) for all t ≥ 0 when d ∈ (0, d], so we can take c 1 to be the minimum of the positive quantities in curly braces in (63).

Then 1 / 4 - 2 ( 1 + 2 - 1 4

 142121 of (8) for all t ≥ 0, whereM * = 2 √ 2|P 1 BK| max{|M | + |BK|, |LC|} and where c a = 16|P 1 BK| 2 (max{|M | + |BK|, |LC|}) 2 is assumed to be positive, by the relation |z 1 (ℓ)| + |z 2 (ℓ)| ≤ √ 2|z(ℓ)|, and where the last inequality in (61) followed from applying Young's inequality ab ≤ 1 4 a 2 + b 2 with a and b being the terms in curly braces in (61) and then using Jensen's inequality. We now pick any constant J ∈ (0, 1) and set J)d 2 c a > 0, and the formula[START_REF]New finite-time and fast converging observers with a single delay[END_REF] givesV (t) ≤ -1 4 -2(1 + J)d 2 c a |z(t)| Jdc a t t-d |z(ℓ)| 2 dℓ ≤ --2(1+J)d 2 camax{λmax(Q1),λmax(Q2) V 0 (z(t))

  ), Young's inequality gives2|z ⊤ 1 (t)P 1 LCz 2 (t)| ≤ 1 2 |z 1 (t)| 2 + 2|P 1 LC| 2 |z 2 (t)| 2 , hence Va ≤ -|z 1 (t)| 2 + 2|z ⊤ 1 (t)P 1 LCz 2 (t)| +2|z ⊤ 1 (t)P 1 BK|z 1 (t -d) -z 1 (t)| ≤ -1 2 |z 1 (t)| 2 + 2|P 1 LC| 2 |z 2 (t)| 2 +2|z ⊤ Hence, with the choices Q 1 = P 1 and Q 2 = (1+2|P 1 LC| 2 )P 2 , we can use the bound |z 1 (t)-z 1 (t-d)| ≤ t t-d | ż1 (ℓ)|dℓ and the structure of the z 1 subsystem of[START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF] to check that our functionV 0 (z) = z ⊤ 1 Q 1 z 1 + z ⊤ 2 Q 2 z 2 satisfies V0 (t) ≤ -1 2 |z| 2 + 2|z ⊤ 1 (t)P 1 BK|z 1 (t -d) -z 1 (t)| ≤ -1 2 |z| 2 + {|z 1 (t)|} M *

		t t-2d |z(ℓ)|dℓ
	≤ -1 4 |z| 2 + dc a	t t-2d |z(ℓ)| 2 dℓ

1 (t)P 1 BK|z 1 (t -d) -z 1 (t)| (60)

for all t ≥ 0.

defined in [START_REF] Mazenc | Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence[END_REF]) the term Γ 1 (V (x(t), x(t)))V (x(t), x(t)) in (33) came from the bound

λmin(Q2) L(V r (x(t), x(t))) V0(x(t),x(t)) m0 ,

and where the first inequality in (34) used [START_REF]Finite-time observers for parameters and state variables of nonlinear systems[END_REF] and the bound

) and where

V r (x(t), x(t)) = V0(x(t),x(t)) λmin(Q1) + V0(x(t),x(t)) λmin(Q2) , (35) and where the third right side term in (33) came from

where

using the formula x(t -d) = x(t) -x(t) and ( 2). Then, one can prove that if sup ℓ∈[0,h+τq+2d] V (x ℓ , xℓ ) < v * (with v * > 0 satisfying (15)), then for all t ≥ h + τ q + 2d, V (x t , xt ) < v ⋆ , as follows. If the assertion were not true, choose the smallest t a > h+τ q +2d such that V (x t a , xt a ) = v * , which exists by the continuity of V (x t , xt ) as a function of t. Then, since the Γ i 's are nondecreasing, [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit Lyapunov function approach[END_REF] implies that the right side of (33) at t = t a is bounded above by -c

Since the Γ i 's are nondecreasing, this and (33) give

for all t ≥ h + τ q + 2d. Choosing any constant p > 1 that is close enough to 1 so that

(which exists by ( 15)) implies that

along all solutions of (32) for all t ≥ τ q + h + 2d, where

where R * is the constant from [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF]. To this end, note that our choice (14) of our observer dynamics and our constant initial function 0 for x imply that for all t ∈ [0, d], we have

because our constantness condition on the initial functions gives x(t -d) = x(0) for all t ∈ [0, d], so we can apply the method of variation of parameters to (43) to get

where c d2 was defined in [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF]. Hence, for all t ∈ [0, d], our choice of V and our definitions of R * and c d from [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF] give

Hence, [START_REF] Pearson | Nonlinear model identification[END_REF] gives

We now use the preceding result to prove that sup ℓ∈[d,h+τq+2d] V (x ℓ , xℓ ) < v * when x(0) ∈ S 0 . First note that for values t ≥ d, the reasoning that led to (33) (except with γϵ not replaced by the right side of (28) as it was replaced in the second part of the proof, so on this interval, we do not have Ω in the u formula in (13)) gives

for all t ∈ [d, h + τ q + 2d]. Let t b be the supremum of all times t ∈ [d, h + τ q + 2d] such that V (x ℓ , xℓ ) < v * for all ℓ ∈ [d, t]. By the preceding argument and the continuity of V (x t , xt ) as a function of t, we have t b > d. Suppose, for the sake of obtaining a contradiction, that t b < h+τ q +2d. Then V (x t b , xt b ) = v * , again by the continuity of V (x t , xt ) as a function of t. Setting c ♭ 1 = c 1 -Γ 1 (v * ) (which is positive, because of ( 15)), we can then apply Young's inequality to the last term in (46) to check that for all t ∈ [d, t b ], we have


Michael Malisoff

Frédéric Mazenc

Corina Barbalata

Event-Triggered Control under Unknown Input and Unknown Measurement Delays using Interval Observers

Keywords:

We provide a new input-to-state stabilizing eventtriggered feedback design for linear systems with unknown input delays, unknown measurement delays, and unknown additive disturbances. Our trigger times are computed using only the matrices defining the system and time-lagged sampled state values. We use the theory of positive systems, interval observers, and a vector version of Halanay's inequality. We illustrate our method using a marine robotic model.

I. INTRODUCTION

This paper continues our development (begun in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], [START_REF] Mazenc | Event-triggered predictionbased delay compensation approach[END_REF], [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF], [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF]) of event-triggered control methods for continuous time systems using interval observers (as defined in [START_REF] Gouze | Interval observers for uncertain biological systems[END_REF], [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]), positive systems, and the use of matrices of absolute values instead of Euclidean norms. Although our prior eventtriggered works allowed input delays (as well as time-varying coefficients in linear systems and additive uncertainty [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF]), they assume that the input delays are known in advance.

Here we study more realistic systems having unknown delays in the input. We allow time-varying delays, and unknown disturbances. Although neither this work nor our prior eventtriggered controlled works assume that our systems to be controlled are positive, our analysis uses new synergies of the theory of positive systems, which has been shown in our prior works to lead to less triggering than traditional approaches, and the vector Halanay's inequality approach that was presented in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] in the absence of event triggering.

We prove input-to-state stability (or ISS), and we rule out Zeno's phenomenon. Allowing unknown input delays (as compared with our prior chain predictor based work [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] that required exact knowledge of the input delays to design the control and constantness of the delays), and trigger rules that only require sampled delayed state measurements with unknown measurement delays, make this work novel and significant. These innovations are made possible by our new event trigger rule and new Hurwitzness condition which have never appeared before, and which (as we illustrate below) let us achieve ISS under realistic bounds on the unknown time-varying input and measurement delays. This also contrasts with other notable event-triggered works for nonlinear systems, such as [START_REF] Borri | Event-triggered control of nonlinear systems with time-varying state delays[END_REF], [START_REF] Li | Observer-based event-triggered control for nonlinear systems with mixed delays and disturbances: The input-to-state stability[END_REF], [START_REF] Nozari | Event-triggered stabilization of nonlinear systems with time-varying sensing and actuation delay[END_REF] (where the delays are M. Malisoff is with Department of Mathematics, Louisiana State University (LSU), Baton Rouge, LA 70803, USA (e-mail: malisoff@lsu.edu).

F. Mazenc is with Inria EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: frederic.mazenc@l2s.centralesupelec.fr).

C. Barbalata is with Department of Mechanical and Industrial Engineering, LSU, Baton Rouge, LA 70803, USA (email: cbarbalata@lsu.edu).

Supported by US National Science Foundation Grant 2009659.

known) and [START_REF] Zhang | Event-triggered control for nonlinear time-delay systems[END_REF], [START_REF] Zhang | Event-triggered stabilization of a class of nonlinear time-delay systems[END_REF] (where the measurement delays were only allowed to capture the effects of sampling). Since eventtriggered controls only change values when it is essential to change them [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF], they are useful for underwater robotics, owing to constraints in underwater communication. Hence, another contribution of this work is our application to a model of the BlueROV2 underwater vehicle, which is used to study corals, under uncertain input and measurement delays.

We use standard notation. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. We set

Z 0 = {0, 1, 2, ...}. For a matrix G = [g ij ] ∈ R r×s , we set |G| = [|g ij |]
, so the entries of |G| are the absolute values of the corresponding entries g ij of G. By G + , we denote the matrix whose entries are max{0, g ij } in the ith row and jth column for each i and j, and

G -= G + -G. Note for later use that |G| = G + + G -. For matrices D = [d ij ] and E = [e ij ] of the same size, we write D < E (resp., D ≤ E) provided d ij < e ij (resp., d ij ≤ e ij )
for all i and j, 0 denotes the zero matrix, and I denotes the identity matrix. A matrix M ∈ R n×n is called Metzler provided all of its off-diagonal elements are nonnegative. Given M ∈ R n×n , let D M denote the diagonal matrix such that all main diagonal entries of M -D M are equal to zero, R M = D M + (M -D M ) + , and N M = (M -D M ) -. For matrix valued functions M (r) = [m ij (r)], the supremum is entrywise, i.e., sup r∈S M (r) has the (i, j) entry sup r∈S m ij (r) for each subset S of the domain of M and all i and j when these suprema are finite. We also use the standard definitions of ISS (which we also use to denote input-to-state stable) and comparison functions KL and K ∞ , e.g., from [START_REF] Sontag | Input-to-state stability: basic concepts and results[END_REF]. We use ||•|| to denote the usual Euclidean norm and corresponding matrix norm, and || • || J denotes the supremum in this norm over any interval J.

II. STUDIED SYSTEM

We first consider the general class of linear systems

ẋ(t) = Ax(t) + Bu(t) + δ(t) (1)
with x valued in R n and the input u (which will contain an unknown nonzero time-varying delay ∆ in the sequel) valued in R m , and where A ∈ R n×n and B ∈ R n×m are known matrices, and δ is an unknown locally bounded piecewise continuous function (but see Section IV for local analogs for nonlinear systems). Our control values will have the form u(t) = Kx(t i -∆(t i )) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 for constant matrices K ∈ R m×n (which arises from zero-order hold and input delays on the controller side) where the t i 's admit positive constants T and T such that T ≤ t i+1 -t i ≤ T for all i ∈ Z 0 where t 0 = 0. The t i 's will be event trigger times, and we will add an assumption on K. We assume that the initial functions for (1) are constant. The t i 's will be found using sampled values of the time-lagged measurements y(t) = x(t -λ(t)) of the state of (1) at triggering times t i , whose delay λ is also unknown. Our first assumption is: Assumption 1: There are known constants ∆ ≥ 0 and λ ≥ 0 and a known function δ : R → [0, +∞) n such that

0 ≤ ∆(t) ≤ ∆, 0 ≤ λ(t) ≤ λ, and |δ(t)| ≤ δ(t) (2)
hold for all t ≥ 0.

Fixing A, B, and K as above, we use the function

Ω(r) = e Ar + r 0 e Aℓ dℓBK. (3)
Since Ω(0) = I, there is a constant ν > 0 such that the matrix Ω(r) is nonsingular for each r ∈ [0, ν]. Fixing ∆, λ, δ, and ν satisfying the preceding requirements and setting

Γ 1 = sup s∈[0,ν] I -Ω(s) -1 , Γ 2 = (Γ 1 + I) sup L∈[0,T ] L 0 e Aℓ BKdℓ , Γ 3 = sup ℓ∈[0,T ] |e Aℓ A|, and Γ 4 = sup ℓ∈[0,T ] |e Aℓ BK| (4)
for any constant T ≥ ν + ∆, we also assume:

Assumption 2: With the above notation and the choice H = A+BK, the matrix K ∈ R m×n is such that the matrix

M * = R H + N H + |BK|Γ 1 + max ∆, λ |BKA| +|(BK) 2 | + |BK|Γ 2 |A| + |BK| +λ|BK| (I + Γ 1 )(Γ 3 + Γ 4 ) + |A| + |BK| (5)
is Hurwitz.

Our novel Hurwitzness condition from Assumption 2 can always be satisfied after a change of coordinates if (A, B) is controllable and if ∆, λ, and ν are small enough. This is done by choosing K so that A + BK has distinct negative eigenvalues, then using a diagonalizing change of coordinates as in [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF], so A, B, and K and x are replaced by P -1 AP , P -1 B, KP , and P -1 x respectively for an invertible matrix P . After this coordinate change, H = R H + N H is diagonal and Hurwitz. Then, by continuity of eigenvalues as functions of the matrix entries, Assumption 2 holds when ∆, λ, and ν are small enough; see Section V for an illustration of how Assumption 2 can allow realistic delay values.

III. MAIN RESULT A. Control and Statement of Result

In terms of the control u that is defined by

u(t) = Kx(t i -∆(t i )) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 (6)
and the notation from the previous section, the closed-loop system and triggering times t i we consider are defined by:

ẋ(t) = Ax(t) + BKx(t i -∆(t i )) + δ(t) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , ḣi (t) = Ah i (t) + BKx(t i -λ(t i ))
for all t ≥ t i and i ∈ Z 0 , h i (t i ) = x(t i -λ(t i )) for all i ∈ Z 0 , and

t i+1 = sup {b ∈ [t i , t i + T ] : |h i (b) -h i (t i )| ≤ Γ 1 |h i (b)|} for all i ∈ Z 0 , (7)
where t 0 = 0. Then t 1 is the supremum of all b ≥ t 0 = 0 such that the inequality in the supremum in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] holds with i = 0 or t 1 = t 0 +T = T (whichever is smaller), and then the other trigger times t i for i ≥ 2 are inductively defined in the same way. Our supremum in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] ensures that t i+1 -t i ≤ T for all i ≥ 0, so there are infinitely many times t i . Also, only the discrete state measurements h i (t i ) are needed to find the t i 's, which are found using solutions h i (t) of the initial value problem for the h i dynamics from the second and third equalities in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], and T can be arbitrarily large. Hence, the triggering rule does not need to continuously monitor x(t) to determine future triggering times, which is a significantly novel feature. We prove the following, whose last conclusion rules out Zeno's phenomenon: Theorem 1: Let the system (1) satisfy Assumptions 1-2. Then (1) in closed loop with the control from ( 6)-( 7) is ISS to the origin on R n . Also, t i+1 -t i ≥ ν for all i ≥ 0.

Remark 1: Our theorem is new, even in the special case where there are no measurement delays, in which case we can choose λ = 0 and then Assumption 2 is the requirement that

R H + N H + |BK|Γ 1 + ∆ |BK|Γ 2 |A| + |BK| + |BKA| + |(BK) 2 |
is Hurwitz. The proof of Theorem 1 builds comparison functions in the final ISS estimate, namely, β ∈ KL and γ ∈ K ∞ such that all solutions x(t) of ( 7) for all initial states x(0) ∈ R n and δ's satisfy

||x(t)|| ≤ β(||x(0)||, t) + γ ||δ|| [0,t] (8)
for all t ≥ 0, with positive constants c a and c b such that these requirements hold with β(r, t) = c a re -c b t ; see also Remark 2 below. Hence, all solutions x(t) converge to 0 ∈ R n exponentially at rate c b when δ is zero, with an overshoot (given by the second right side term in ( 8)) for nonzero δ.

B. Proof of Theorem 1

1) Ruling Out Zeno's phenomenon: We prove that t i+1t i ≥ ν for all i ∈ Z 0 . We proceed by contradiction.

Suppose that there were an i ∈ Z 0 such that t i+1 -t i < ν. By applying variation of parameters to the h i dynamics from [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], we deduce from our choice of

Ω in (3) that h i (t) = Ω(t -t i )h i (t i ) for all t ≥ t i . Since Ω(r) is invertible for each r ∈ [0, ν] and t i+1 -t i < ν, this provides a constant ε ∈ (0, ν + t i -t i+1 ) such that h i (t i ) = Ω(t -t i ) -1 h i (t) for all t ∈ [t i , t i+1 + ε), because 0 ≤ t i+1 + ε -t i < ν. Then |h i (t i ) -h i (t)| ≤ |I -Ω(t -t i ) -1 ||h i (t)| ≤ Γ 1 |h i (t)| for all t ∈ [t i , t i+1 + ε).
This contradicts the definition of t i+1 . Hence, t i+1 -t i ≥ ν holds for all i ∈ Z 0 .

2) Stability Analysis: The analysis has three parts. First, we derive a vector valued bound for |x(t) -x(t i )| at times t ∈ [t i , t i+1 ) for all i ∈ Z 0 . In the second part, we use the bound on |x(t) -x(t i )| from the first part to find a vector valued upper bound on ṡ(t) where s(t) = x(t) -x(t) is the difference between upper and lower vectors in an interval observer (x, x) for x. In the final part, we use the bound on ṡ(t) from the second part and a variant of the ISS vector Halanay's inequality from [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] to obtain our ISS estimate [START_REF] Mazenc | Event-triggered predictionbased delay compensation approach[END_REF].

First Part. Defining the function σ by σ(t) = t i when t ∈ [t i , t i+1 ) and i ≥ 0, and setting z(t) = Ax(t)+BKx(σ(t)-∆(t i )), we observe that our choice

H = A + BK gives ẋ(t) = Hx(t) + BK[x(t i ) -x(t)] +BK[x(t i -∆(t i )) -x(t i )] + δ(t) = Hx(t) + BK[x(t i ) -x(t)] -BK ti (ti-∆(ti)) + (z(ℓ) + δ(ℓ))dℓ + δ(t) (9)
for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , where we used the constantness of the initial functions for the x dynamics. Substituting our formula for z into [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF] gives 2 and R 3 = -BK. Next note that our choice of t i+1 from [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] gives

ẋ(t) = Hx(t) + BK[x(σ(t)) -x(t)] +R 1 σ(t) (σ(t)-∆(σ(t))) + x(ℓ)dℓ + R 2 σ(t) (σ(t)-∆(σ(t))) + x(σ(ℓ) -∆(σ(ℓ)))dℓ +R 3 σ(t) (σ(t)-∆(σ(t))) + δ(ℓ)dℓ + δ(t) (10) for all t ≥ 0 with R 1 = -BKA, R 2 = -(BK)
|h i (t) -h i (t i )| ≤ Γ 1 |h i (t)| (11)
for all t ∈ [t i , t i+1 ) and all i ∈ Z 0 . On the other hand, [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] and the Fundamental Theorem of Calculus give

ẋ(t)-ḣi (t) = A(x(t) -h i (t)) +BK[x(t i -∆(t i )) -x(t i -λ(t i ))] +δ(t) for all t ∈ [t i , t i+1 ) x(t i )-h i (t i ) = ti (ti-λ(ti)) + [Ax(ℓ) + δ(ℓ)] dℓ +BK ti (ti-λ(ti)) + x(σ(ℓ)-∆(σ(ℓ)))dℓ (12) for all i ∈ Z 0 . Since |z(t)| ≤ |Ax(t)| + |BKx(σ(t) - ∆(σ(t)
))|, we can use the second equality of (12) to get

|x(t i -∆(t i )) -x(t i -λ(t i ))| ≤ ti (ti-∆ ♯ ) + [|A||x(ℓ)| + |δ(ℓ)|] dℓ +|BK| ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ, (13)
|e (t-ti)A (x(t i ) -h i (t i ))| ≤ Γ 3 ti (ti-λ) + |x(ℓ)|dℓ +Γ 4 ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ + Γ 5 ti (ti-λ) + |δ(ℓ)|dℓ, (14)
and |x(t i ) -h i (t i )| ≤ ti (ti-λ) + [|A||x(ℓ)| + |δ(ℓ)|] dℓ +|BK| ti (ti-λ) + |x(σ ♭ (ℓ)))|dℓ (15) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , where the function σ ♭ is defined by σ ♭ (ℓ) = σ(ℓ) -∆(σ(ℓ)) for all ℓ ≥ 0, ∆ ♯ = max ∆, λ , Γ 3
and Γ 4 are defined in (4), and

Γ 5 = sup ℓ∈[0,T ] |e Aℓ |.
Letting G j (t i ) for j = 1, 2 denote the right sides of ( 13) and ( 14) respectively, and applying the method of variation of parameters to the dynamics for x -h i from ( 12) on [t i , t], it now follows from ( 12), [START_REF] Sontag | Input-to-state stability: basic concepts and results[END_REF], and (14) that

|x(t) -h i (t)| ≤ t ti e A(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) + G 2 (t i ), (16)
and so also 11) and ( 16) and then (17), and letting G 3 (t i ) denote the right side of (15), we can bound each of the three right side terms in |x(t) -

|h i (t)| ≤ |x(t)| + t ti e A(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) + G 2 (t i ) (17) for all t ∈ [t i , t i+1 ) and all i ∈ Z 0 , since |h i (t)| -|x(t)| ≤ |x(t) -h i (t)|. Using (
x(t i )| ≤ |x(t) - h i (t)| + |h i (t) -h i (t i )| + |h i (t i ) -x(t i )| (
which is obtained by applying the triangle inequality twice) separately to get

|x(t) -x(t i )| ≤ Γ 1 |h i (t)| + t ti e |A|(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) +G 2 (t i ) + G 3 (t i ) ≤ Γ 1 |x(t)| + Γ 1 t ti e |A|(t-ℓ) δ(ℓ)dℓ +Γ 2 G 1 (t i ) + t ti e |A|(t-ℓ) δ(ℓ)dℓ + (I + Γ 1 )G 2 (t i ) + G 3 (t i ) (18)
for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , so ( 14)- [START_REF] Zhang | Event-triggered stabilization of a class of nonlinear time-delay systems[END_REF] give

|x(t)-x(t i )| ≤ Γ 1 |x(t)| + Γ 2 |A| ti (ti-∆ ♯ ) + |x(ℓ)|dℓ +Γ 2 |BK| ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ +Γ 2 t (ti-∆ ♯ ) + δ(ℓ)dℓ +(Γ 1 + I) t ti e |A|(t-ℓ) δ(ℓ)dℓ +(Γ ♯ 3 + |A|) ti (ti-λ) + |x(ℓ)|dℓ +(Γ ♯ 4 +|BK|) ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ +(I + Γ ♯ 5 ) t (ti-λ) + δ(ℓ)dℓ, (19)
where Γ ♯ j = (I +Γ 1 )Γ j for j = 3, 4, 5. Setting Γ 6 = Γ 2 |BK| and Γ 7 = (Γ 1 + I)e T |A| + Γ 2 , this provides the bounds

|x(t) -x(t i )| ≤ Γ 1 |x(t)| + Γ 2 |A| ti (ti-∆ ♯ ) + |x(ℓ)|dℓ +Γ 6 ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ +Γ 7 t (ti-∆ ♯ ) + δ(ℓ)dℓ +(Γ ♯ 3 + |A|) ti (ti-λ) + |x(ℓ)|dℓ +(Γ ♯ 4 +|BK|) ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ +(I + Γ ♯ 5 ) t (ti-λ) + δ(ℓ)dℓ. ( 20
)
Second Part. We next introduce the interval observer

                                         ẋ(t) = R H x(t) -N H x(t) +(BK[x(σ(t)) -x(t)]) + + σ(t) (σ ♭ (t)) + (R 1 x(ℓ)) + dℓ + σ(t) (σ ♭ (t)) + (R 2 x(σ ♭ (ℓ))) + dℓ + σ(t) (σ ♭ (t)) + (R 3 δ(ℓ)) + dℓ + δ(t) + ẋ(t) = R H x(t) -N H x(t) -(BK[x(σ(t)) -x(t)]) - - σ(t) (σ ♭ (t)) + (R 1 x(ℓ)) -dℓ - σ(t) (σ ♭ (t)) + (R 2 x(σ ♭ (ℓ))) -dℓ - σ(t) (σ ♭ (t)) + (R 3 δ(ℓ)) -dℓ -δ(t) - (21)
which is defined for all t ≥ 0. Since the matrix

R H N H N H R H ∈ R 2n×2n (22)
is Metzler, formula [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF] for the x dynamics, (21), and the formula Hx(t) = R H x(t) -N H x(t) imply that if

x(0) ≤ x(0) ≤ x(0), x(0) ≤ 0, and 0 ≤ x(0), (23) then x(t) ≤ x(t) ≤ x(t), x(t) ≤ 0, and 0 ≤ x(t)

hold for all t ≥ 0, e.g., by applying [5, Lemma 1] to the dynamics for (x, -x), and the dynamics for (x -x, x -x). Now, let us consider initial conditions such that (23) are satisfied and let s(t) = x(t) -x(t). Then (21) gives

ṡ(t) = (R H + N H )s(t) + |BK[x(σ(t)) -x(t)]| + σ(t) (σ ♭ (t)) + |R 1 x(ℓ))|dℓ + σ(t) (σ ♭ (t)) + |R 2 x(σ ♭ (ℓ))|dℓ + σ(t) (σ ♭ (t)) + |R 3 δ(ℓ)|dℓ + |δ(t)| ≤ (R H + N H )s(t) + |BK[x(σ(t)) -x(t)]| +|R 1 | σ(t) (σ(t)-∆) + |x(ℓ)|dℓ +|R 2 | σ(t) (σ(t)-∆) + |x(σ ♭ (ℓ))|dℓ + σ(t) (σ(t)-∆) + |R 3 δ(ℓ)|dℓ + |δ(t)| (25)
for all t ≥ 0. By using (20) to bound the second term in the upper bounds in (25), we can use (25) to obtain

ṡ(t) ≤ (R H + N H )s(t) + |BK|Γ 1 |x(t)| +|BK|Γ 2 |A| σ(t) (σ(t)-∆ ♯ ) + |x(ℓ)|dℓ +|BK|Γ 6 σ(t) (σ(t)-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ + σ(t) (σ(t)-∆) + |R 3 δ(ℓ)|dℓ + |δ(t)| +|BK|Γ 7 t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ +|R 1 | σ(t) (σ(t)-∆) + |x(ℓ)|dℓ +|R 2 | σ(t) (σ(t)-∆) + |x(σ ♭ (ℓ))|dℓ +|BK|(Γ ♯ 3 + |A|) σ(t) (σ(t)-λ) + |x(ℓ)|dℓ +Γ * σ(t) (σ(t)-λ) + |x(σ ♭ (ℓ))|dℓ +|BK|(I + Γ ♯ 5 ) t (σ(t)-λ) + δ(ℓ)dℓ, (26)
where Γ * = |BK|(Γ ♯ 4 + |BK|). Since (24) gives x(t) -x(t) ≤ x(t) ≤ x(t) -x(t) and so also |x(t)| ≤ s(t) for all t ≥ 0, we can upper bound each right side term in (26) that involves x using s to get

ṡ(t) ≤ (R H + N H + |BK|Γ 1 )s(t) +|BK|Γ 2 |A| σ(t) (σ(t)-∆ ♯ ) + s(ℓ)dℓ +|BK|Γ 6 σ(t) (σ(t)-∆ ♯ ) + s(σ ♭ (ℓ))dℓ + δ(t) + |R 1 | σ(t) (σ(t)-∆) + s(ℓ)dℓ +|R 2 | σ(t) (σ(t)-∆ ♯ ) + s(σ ♭ (ℓ))dℓ + (|BK|Γ 7 + |R 3 |) t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ +|BK|(Γ ♯ 3 +|A|) σ(t) (σ(t)-λ) + |s(ℓ)|dℓ + Γ * σ(t) (σ(t)-λ) + |s(σ ♭ (ℓ))|dℓ + |BK|(I + Γ ♯ 5 ) t (σ(t)-λ) + δ(ℓ)dℓ (27)
for all t ≥ 0, by also assuming that the s dynamics has constant initial functions. Collecting terms in (27) gives

ṡ(t) ≤ (R H +N H +|BK|Γ 1 )s(t) + Γ 8 sup ℓ∈[(t-2T -2∆ ♯ ) + ,t] s(ℓ) + (|BK|(Γ ♯ 5 +Γ 7 +I)+|R 3 |) t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ + δ(t) (28)
for all t ≥ 0, with

Γ 8 = ∆ ♯ |BK|[Γ 2 |A| + Γ 6 ] + ∆ ♯ (|R 1 | + |R 2 |) + λ|BK|(Γ ♯ 3 + Γ ♯ 4 + |A| + |BK|). Third Part.
We apply [6, Theorem 1] to the function w(t) = s(t) and τ = 2(T + ∆ ♯ ), after extending s to the domain [-τ, +∞) by defining s to be constant on [-τ, 0], so we use a piecewise C 1 function w instead of assuming that w is C 1 as was assumed in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF]; this is valid because the proof of [6, Theorem 1] remains valid if the C 1 requirement on w in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] is relaxed to only requiring it to be piecewise C 1 . This ensures that s satisfies an ISS property when

R H + N H + |BK|Γ 1 + Γ 8 (
β 0 ∈ KL and γ 0 ∈ K ∞ such that ||s(t)|| ≤ β 0 (||s|| [0,h] , t) + γ 0 (||δ|| [0,t] ) (30)
for all t ≥ 0 for all solutions of the s(t) dynamics from (25).

The remaining part of the proof converts the estimate (30) to an ISS estimate for the state variable x(t), as follows.

Note that [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] provides the constant c1 = max{1,

||A|| + ||BK||} such that || ẋ(t)|| ≤ c1 (||x|| [t-T -∆,t] + ||δ|| [0,t]
) for all t ≥ 0. Hence, the fundamental theorem of calculus gives

||x(t)|| ≤ ||x(0)||+c 1 t 0 ||x|| [ℓ-T -∆,ℓ] dℓ+c 1 t||δ|| [0,t] (31)
for all t ≥ 0. Recalling our assumption that the initial functions are constant, it follows from (31) that the function

S(t) = ||x|| [t-T -∆,t] satisfies S(t) ≤ ||x(0)|| + c1 t 0 S(ℓ)dℓ + c1 t||δ|| [0,t] (32
||s|| [0,h] ≤ c3 (3 + 2h)||x(0)||+||δ|| [0,h] . (33)
Since |x(t)| ≤ s(t) holds for all t ≥ 0, and since the values x(t) do not depend on δ(ℓ) values for times ℓ ≥ t, we can substitute (33) into (30) and then use the fact that β 0 (a + b, t) ≤ β 0 (2a, t) + β 0 (2b, 0) holds for all a ≥ 0, b ≥ 0 and t ≥ 0, to satisfy the ISS requirement with β(r, t) = β 0 (2(3 + 2h)c 3 r, t) and γ(s) = γ 0 (s) + β 0 (2c 3 s, 0). Remark 2: In terms of diagonal matrices D whose main diagonal entries are all positive, matrices P > 0, constants h > 0 and τ > 0, and a constant c 0 ∈ (0, 1) and U 0 ∈ [1, +∞) n such that M 0 U 0 = c 0 U 0 where M 0 = e -Dh + 0 -h e Dℓ dℓP , the β 0 from [6, Theorem 1] and [6, Remark 1] is β 0 (r, t) = n||U 0 ||e (t-h) ln(c 0 )/(τ +h) r. The existence of c 0 and U 0 follows because M 0 > 0 is Schur stable [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF]. This and the last part of the proof of Theorem 1 ensure the exponential stability condition from Remark 1. We next use the preceding β 0 to obtain local analogs for nonlinear systems.

IV. LOCAL RESULT FOR NONLINEAR SYSTEMS

We provide local exponential stability analogs for

ẋ(t) = Ax(t) + Bu(t) + Φ(x(t)) (34)
where Φ satisfies the following (but similar arguments allow additive uncertainty δ(t) in (34) with suitable bounds on δ):

Assumption 3: Each component Φ i of Φ is continuous and satisfies lim z→0 Φ i (z)/||z|| = 0 for i = 1, . . . , n.
The preceding assumption is very general, because we can rewrite any

C 1 nonlinear function G : R n → R n that satisfies G(0) = 0 in the form G(x) = G 0 x + Φ(x)
, where Φ is the remainder term in the first order Maclaurin approximation of G (which will be of second order at the origin and so satisfy Assumption 3), and then combine the constant matrices A and G 0 to get one constant coefficient of x(t) when the given system has the form ẋ(t) = Ax(t) + Bu(t) + G(x(t)) for such a function G. Assumption 3 covers cases where each component Φ i grows quadratically near the origin, and we assume that (34) satisfies the standard uniqueness of solutions property, as in Section II. We continue our other notation from the preceding sections, where ∆ ♯ = max ∆, λ and Γ 5 = sup ℓ∈[0,T ] |e Aℓ |, and we use the following matrix:

B * = I + T +∆ ♯ |BK| (1+Γ 1 )(Γ 5 +e T |A| )+Γ 2 +2I (35)
Also, like in Theorem 1, our closed loop system will use ( 6), whose trigger times t i are defined by

t 0 = 0, ḣi (t) = Ah i (t) + BKx(t i -λ(t i )) for all t ≥ t i and i ∈ Z 0 , h i (t i ) = x(t i -λ(t i )) for all i ∈ Z 0 , and t i+1 = sup {b ∈ [t i , t i + T ] : |h i (b) -h i (t i )| ≤ Γ 1 |h i (b)|} for all i ∈ Z 0 , (36)
where T and ν satisfy the requirements from Section II. With H = A + BK where K satisfies Assumption 2, and letting D c denote the diagonal matrix whose main diagonal entries are all c and B c denote the closed 2-norm ball of radius c centered at 0 for any constant c > 0, and 1 ∈ R n×n be the matrix of all 1's, we prove the following, where the existence of the ϵ such that M * +ϵ1 is Hurwitz follows from continuity of eigenvalues of a matrix as functions of the matrix entries:

d 3 = d 2 2||BK|| + ∆(||BKA|| + ||(BK) 2 || +d 0 ||BK||) + d 0 max{1, h}e ||R H +N H ||h . (37)
Then, we can find positive constants c * and r * such that for each initial state x(0) in

S 0 = x 0 ∈ B x : max d 2 ||x 0 ||, β 0 [3 + 2h]d 3 ||x 0 ||, h < x ,
the solution x : [0, +∞) → R n of (34), in closed loop with ( 6) and (36), satisfies ||x(t)|| ≤ c * e -r * t ||x(0)|| for all t ≥ 0. Proof: The proof has two parts. In the first part, we find constants c * > 0 and r * > 0 such that the local exponential stability estimate holds for all t ≥ 0 and all solutions x : [0, +∞) → R n of the closed loop system such that

||x|| [0,+∞) ≤ x (38)
holds. In the second part, we show that all solutions x(t) of the closed loop system such that x(0) ∈ S 0 satisfy (38). Second Part. We show that (38) holds along all solutions x(t) of the closed loop system such that x(0) ∈ S 0 . First, we show that ||x|| [0,h] ≤ x. To this end, note that the first part of the proof gives ||x|| [0,t] ≤ d 2 ||x(0)|| for all initial states x(0) and t ∈ [0, h] for which the corresponding solution x(t) of the closed loop system satisfies ||x|| [0,t] ≤ x. Now we argue by contradiction. If x(0) ∈ S 0 were such that x(t) satisfied ||x|| [0,h] > x, then choose the largest t * ∈ [0, h) such that ||x|| [0,t * ] ≤ x. By the continuity of ||x|| [0,t] as a function of t, this gives ||x|| [0,t * ] = x. On the other hand, the preceding argument and our choice of S 0 give ||x|| [0,t * ] ≤ d 2 ||x(0)|| < x, which is a contradiction. Hence, ||x|| [0,h] ≤ x.

Therefore, we can apply the first part of the proof of this theorem to get ||x(t)|| ≤ β(||x(0)||, t) for all initial states x(0) ∈ B x and all t ≥ 0 for which ||x|| [0,t] ≤ x, where β(r, t) = β 0 ([3 + 2h]d 3 r, t) is from the proof of Theorem 1 except with the c i 's replaced by the d i 's and where we can use 3 + 2h instead of 2 [3 + 2h] in the β formula because δ = 0. Also, our choices of β and S 0 give β(||x 0 ||, h) = β 0 [3 + 2h]d 3 ||x 0 ||, h < x for all x 0 ∈ S 0 .

We next use the preceding observations to check that ||x|| [h,+∞) ≤ x. We again argue by contradiction. If x(0) ∈ S 0 were such that the corresponding solution x(t) satisfied ||x|| [h,+∞) > x, then choose the largest t * * ≥ h such that ||x|| [h,t * * ] ≤ x. Again using the continuity of ||x|| [0,t] , this gives ||x|| [h,t * * ] = x. On the other hand, the preceding paragraph implies that ||x|| [h,t * * ] ≤ β(||x(0)||, h) < x, which is a contradiction. Hence, (38) holds for all solutions of the closed loop system having initial states x(0) ∈ S 0 .

V. ILLUSTRATION Consider a dynamics for the depth and pitch degreesof-freedom of an autonomous underwater vehicle, e.g., the BlueROV2 which is used for environmental surveys such as the study of corals. The vehicle has a Doppler Velocity Logger (or DVL) that estimates its velocity. When close to the sea floor, the DVL experiences bottom lock, which makes it impractical to continuously change the control values. Hence, we use Theorem 1 to design a control system for the depth plane, under input and measurement delays, where unlike in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], the delays are time-varying and unknown. This is an innovative significant departure from event-triggered works like [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], which did not allow measurement delays, and where the chain predictors required exact knowledge of a constant input delay. Following [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF], we linearize the dynamics to get (1) with the diagonal matrix A = diag{-0.387, -1.8} and B = [0.038, 1.5] ⊤ . To choose K, we searched for a K that yielded the largest allowable ∆ and λ subject to the constraint that Assumption 2 holds with T = 1.

This produced K = [-0.0085, -0.054], which enabled us to satisfy Assumptions 1-2 with the lower bound ν = 0.16 on the intersample times, the bound λ = 2.86 on λ, and the bound ∆ = 0.64 on ∆. To validate the efficacy of our method, we used MATLAB to simulate the closed loop system from Theorem 1 with the preceding parameter values, and with random delays that took their values on the allowable intervals λ(t) ∈ [0, 1] and ∆(t) ∈ [0, 0.64]. We report the results in Fig. 1, where (w, q) represent the depth and pitch components of the state respectively, which have subscripts 1 and 2 to indicate two initial state vectors (0.2, 0.6) and (-0.15, -0.35) in our two simulations in each panel. In Fig. 1(b), the uncertainty had the constant components δ i (t) = 0.1 for i = 1, 2. Since our simulations exhibit ISS under realistic delays, they illustrate our result.

VI. CONCLUSION

Our new global input-to-state stability theorem for eventtriggered linear systems allowed uncertain input delays and uncertain measurement delays, and used new synergies of interval observers, positive systems, and an ISS vector Halanay's inequality approach. Our trigger times are computed from time-lagged sampled state measurements with unknown measurement delays. We also provided a basin of attraction estimate for a local analog that allows nonlinearities. We aim to develop methods to tune the control parameters to maximize the convergence rate, minimize overshoots, and maximize the basin of attraction in the nonlinear case. We also aim to use marine robotic experiments in our LSU lab to further compare the performance of our event-triggered controls with other possible solutions, such as adjustable time-delay processes based on time-delay tools that solve the event-triggered control problem for systems with delays.

Theorem 2 :

 2 Let H, K, M * , and Φ satisfy Assumptions 2-3, where ∆ and λ are constants such that 0 ≤ ∆(t) ≤ ∆ and 0 ≤ λ(t) ≤ λ hold for all t ≥ 0, and let d max be the largest main diagonal element of H. Let d 0 , ϵ, and x be positive values such that ||Φ(x)|| ≤ d 0 ||x|| for all x ∈ B x , d 0 B * 1 ≤ ϵ1, and M * + ϵ1 is Hurwitz. Choose any constants h > 0 and d > max{0, d max }, and then choose the function β 0 from Remark 2 with P = M * + ϵ1 + D d , D = D d , and τ = 2(T + ∆ ♯ ). Choose the constants d 1 = ||A|| + ||BK|| + d 0 , d 2 = e d1h , and

First Part.

  We indicate the changes that are needed in the proof of Theorem 1. The argument up through (26) remains the same, except with δ(t) and δ(t) replaced by Φ(x(t)) and |Φ(x(t))|, respectively. With these replacements, we collect the terms involving Φ and use the bounds B * |Φ(x)| ≤ d 0 B * 1|x| ≤ ϵ1|x| (which follow from our assumptions, since ||x|| ≤ |x 1 |+. . .+|x n |), and then the bound |x(t)| ≤ s(t) in the result to replace the terms involving |x|'s by terms that use s, to get (28) except with Γ 8 replaced by Γ 8 + ϵ1 and δ replaced by 0, if (38) holds. Since ||Φ(x)|| ≤ d 0 ||x|| for all x ∈ B x , we can then use the reasoning from the third part of the proof of Theorem 1 except with c i replaced by d i for i = 1, 2, 3 and δ = 0, to get the local exponential stability estimate for those solutions x(t) such that (38) holds.

Fig. 1 :

 1 Fig. 1: Simulations (a) without and (b) with uncertainty

  ) for all t ≥ 0. Hence, we can apply Gronwall's inequality to S to check that the constant c2 = max{1, c1h}e c1h is such that ||x(t)|| ≤ S(t) ≤ c2 (||x(0)||+||δ|| [0,t] ) for all t ∈ [0, h].This allows us to use the s dynamics from the equality in (25) and to then apply Gronwall's inequality to the function ||s(t)|| to get the constant c3 = max{1, h}(2c2 ||BK|| + c 2 (||R 1 || + ||R 2 ||)∆ + ∆||R 3 || + 1)e ||R H +N H ||h such that

	||s(t)|| ≤ c3 (||s(0)|| + ||x(0)|| + ||δ|| [0,t] ) for all t ∈ [0, h].
	Since we can assume that x(0) ≤ (1 + h)|x(0)| and x(0) ≥

-(1 + h)|x(0)|, it follows that we can then use the formula s = x -x to get ||s(0)|| ≤ 2(1 + h)||x(0)||, and so also ||s(t)|| ≤ c3 (3 + 2h)||x(0)|| + ||δ|| [0,t] for all t ∈ [0, h]. Hence,