Frederic Mazenc

Michael Malisoff
email: malisoff@lsu.edu

Corina Barbalata
email: cbarbalata@lsu.edu

Fr Éd Éric Mazenc

Event-Triggered Prediction-Based Delay Compensation Approach

Keywords: Event-triggered, delay, prediction

We provide a new event-triggered delay compensation approach for linear systems with arbitrarily long constant input delays. Our prediction map is expressible as a solution of a discrete time system. Our method ensures input-to-state stability. We also provide an analog under measurement delays, where the prediction map is expressible as a solution of a continuous-discrete system. Significant novel features are our combined use of matrices of absolute values and our prediction based event triggers, instead of Euclidean norms, and the fact that the predictor dynamics always has the same dimension as that of the original system. Our marine robotic example illustrates an advantage of using our new methods.

I. INTRODUCTION

E VENT-triggered control provides the basis for consider- able significant ongoing research, owing to the need to take communication constraints into account; see [START_REF] Brunner | Robust event-triggered MPC with guaranteed asymptotic bound and average sampling rate[END_REF] and [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]. One important focus in event-triggered control theory is delay compensation. Chain predictors have been used to compensate for arbitrarily long input delays in event-triggered systems, using a dynamic extension whose dimension grows linearly with the length of the delay [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF]. Standard event triggers use standard Euclidean norms to measure deviations of the current state from a reference state.

By contrast, recent synergies of event triggering, chain predictors, and interval observers [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] were based on replacing the usual Euclidean norm by vectors of absolute values; see [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF]- [START_REF] Mazenc | Event-triggered control for systems with state delays using a positive systems approach[END_REF], and [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], which illustrate the benefits of this replacement. Although (as indicated, e.g., in [START_REF] Cacace | Stabilization of strictfeedback nonlinear systems with input delay using closed-loop predictors[END_REF]) chain predictors eliminate the need for the distributed terms arising in standard delay compensating predictors by enlarging the dimension of the dynamic extension to compensate for longer delays, it is beneficial to bypass the need for distributed terms without having the dimension of the dynamic extension increase for longer delays. This motivates this work, where we use a new predictive approach to compensate for arbitrarily long constant input delays, where the predictor can be expressed as a solution of a discrete time system. The predictor is a dynamic extension having the same dimension as the given system, regardless of the size of the input delay. We also provide an analog for output delays, whose predictor map is a solution of a continuous-discrete system. This contrasts with notable works such as [START_REF] Wei | Regulation of linear input delayed systems without delay knowledge[END_REF], whose pole conditions are not needed here.

We use standard notation, which we simplify when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. Our inequalities involving matrices of the same size are to be understood as being entrywise, i.e., if M = [m ij ] and N = [n ij ], then M ≤ N means m ij ≤ n ij for all i and j and similarly for <. Set Z 0 = {0, 1, 2, ...} and N = Z 0 \{0}. Given 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. We also set G + = [max{g ij , 0}] and G -= G + -G. For a matrix valued function G(r) = [g ij (r)] and an interval J in G's domain on which all of its entries g ij are bounded real valued functions, we set

sup r∈J |G(r)| = [sup r∈J |g ij (r)|],
i.e., the supremum is entrywise. A square matrix is called Metzler provided all of its off-diagonal entries are nonnegative. We let || • || denote the standard Euclidean norm of matrices, ||h|| S denote the sup norm in this norm for functions h over a subset S of the domain of h, 0 be the matrix whose entries are all zeros, and I denote the identity matrix. We use the standard definitions of input-to-state stability (or ISS, which we also use to abbreviate input-to-state stable) from [START_REF] Sontag | Input-to-state stability: basic concepts and results[END_REF].

II. SYSTEM WITH DELAY IN THE INPUT

A. Studied system

We consider the system

ẋ(t) = Ax(t) + Bu(t -τ ) + δ(t) (1)
with x valued in R n , the control u valued in R p , τ > 0 being a constant, constant matrices A ∈ R n×n and B ∈ R n×p , and each component of δ being piecewise continuous. We use two assumptions (but see Remark 1 for sufficient conditions under which Assumption 1 holds after a change of variables): Assumption 1: There is a matrix K ∈ R p×n such that the matrix H = A + BK is Hurwitz and Metzler.

Assumption 2: There is a known piecewise continuous function

δ : [0, +∞) → [0, +∞) n such that |δ(t)| ≤ δ(t) (2)
for all t ≥ 0. Under Assumption 1, there are a constant p > 0 and a vector V ∈ R n such that V > 0 and such that the inequality

V ⊤ H ≤ -pV ⊤ (3)
holds (e.g., by [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]Lemma 2.3,p.41]). We use the function

ω(s) = e As + s 0 e Aℓ dℓBK. (4)
We can then fix a small enough constant ν > 0 such that ω(s) is nonsingular for all s ∈ [0, ν] and such that the matrix

Γ 1 = sup r∈[0,ν] ω(r) -1 -I (5)
is such that the inequality

-pV ⊤ + V ⊤ |BK| Γ 1 < 0 (6)
is satisfied. The existence of such a constant ν is a consequence of the facts that p > 0, V > 0, and ω(0) = I. Remark 1: Assumption 1 can be satisfied for systems ẋ = A 0 x + B 0 u after a change of coordinates, when (A 0 , B 0 ) is a controllable pair. This is done by first finding a matrix K 0 such that A 0 +B 0 K 0 is Hurwitz with distinct real eigenvalues, then choosing a matrix P such that P -1 (A 0 + B 0 K 0 )P is diagonal, and then choosing A = P -1 A 0 P , B = P -1 B 0 , and K = K 0 P , to obtain an H that is Hurwitz and Metzler.

B. Main result

Our event-triggered control will use the matrices

λ = sup m∈[0,ν] ω(m) -1 and Γ 2 = νλe |A|ν |BK|e |A|τ (7)
with ν defined above and the simplifying notation

ξ i = e Aτ x(t i -τ ) + ti-τ ti-2τ e A(ti-τ -m) Bu(m)dm. (8)
Given any constant T ≥ ν + τ and K from the previous subsection, we propose the event triggered control

u(t) = 0 if t ∈ [-2τ, ν) u(t -τ ) = Kξ i if t ∈ [t i , t i+1 ) and i ≥ 1, (9)
where the sequence t i is defined by these three conditions:

            
(i) t 0 = 0, (ii) t 1 = ν + τ, and (iii) for each i ≥ 1, the value t i+1 is defined by

t i+1 = sup{b ∈ [t i , t i + T ) : |z i (s) -x(t i )| ≤ Γ 1 |z i (s)| + Γ 2 s s-ν-τ δ(ℓ)dℓ for all s ∈ [t i , b]} (10)
where z i is the solution of the initial value problem

żi (t) = Az i (t) + BKξ i , z i (t i ) = x(t i ) (11)
for each i ≥ 1 (so u(t -τ ) = 0 if t ∈ [-τ, t 1 )). We prove: Theorem 1: The system (1) with u and the sequence {t i } defined in ( 9)-( 11) is ISS with respect to δ. Also, t i+1 -t i ≥ ν for all i ≥ 0.

Remark 2: The control u and triggering times t i can be computed in the following recursive way. The formulas [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF] with i = 1 define the control on [0, t 2 ), where t 2 is defined as follows: Either t 2 is the supremum of times b ≥ t 1 for which

|z 1 (s) -x(t 1 )| ≤ Γ 1 |z 1 (s)| + Γ 2 s s-ν-τ δ(ℓ)dℓ (12)
holds for all s ∈ [t

T i ) holds for all s ∈ [t i , t i + T ] t i+1 = t i + T B i < t i + T t i+1 = B i This implies that t i+1 -t i ∈ [0, T ] for all i ≥ 1.
This differs significantly from standard triggers, which use Euclidean norms instead of the vectors of absolute values.

Remark 3: Our proof of Theorem 1 will show how condition (iii) is needed to allow cases where δ is not the zero function. It is used to compare solutions of the z i dynamics [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF] and the x dynamics. We use the values z i (s) in the triggering conditions instead of x(s) to eliminate the need to know future δ(t) values when determining future triggering times. Moreover, by using the values z i (s) of the dynamic extensions (11) instead of x(s) in the event trigger condition in (iii), we eliminate the need to continuously measure the state x(t) to determine future triggering times.

Remark 4: It is tempting to surmise from ( 8)-( 9) that the control u is not available in explicit form, because substituting ( 9) into (8) produces ξ i on both sides of (8). However, this is not the case, because for each i ≥ 1, we can write ξ i as the state of the discrete time system

ξ i = e Aτ x(t i -τ ) +e Ati i-1 j=J(i,τ ) tj+1 tj e -Am dmBKξ j + t J(i,τ )
ti-τ e -Am dmBKξ J(i,τ )-1 [START_REF] Sontag | Mathematical Control Theory[END_REF] with the initial state ξ 0 = e Aτ x(-τ ), where J(i, τ ) is the smallest integer j such that t j ≥ t i -τ (so J(i, τ ) ≤ i), and with the notational convention that the sum in ( 13) is not present if J(i, τ ) = i. The preceding allow us to write the control u from (9) in closed form. Also, by induction on i, ξ i does not depend on event times t j for any values j > i, and that it also does not depend on any values x(ℓ) at times ℓ ≥ t i -τ . Since u(t -τ ) = Kξ i is the constant control value that we use in (1) for all times t ∈ [t i , t i+1 ) for each i ≥ 1, this ensures the implementability of our control; see The proof has two parts. In the first part, we prove the lower bound condition inf i {t i+1 -t i } ≥ ν, to rule out Zeno's phenomenon (which would have allowed infinitely many triggering times on an interval of finite length). In the second part, we use interval observers to prove the ISS assertion.

1) Ruling out Zeno's phenomenon: Consider any i ≥ 1. Then substituting ( 8) into [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF] gives

żi (t) = Az i (t) + BK e Aτ x(t i -τ ) + ti-τ ti-2τ e A(ti-τ -m) Bu(m)dm (14)
for all t ≥ t i , and (1) and variation of parameters give

x(t) = e Aτ x(t -τ ) + t-τ t-2τ e A(t-ℓ-τ ) Bu(ℓ)dℓ + t t-τ e A(t-ℓ) δ(ℓ)dℓ (15)
for all t ≥ τ . As an immediate consequence,

żi (t) = Az i (t) + BK x(t i ) - ti ti-τ e A(ti-ℓ) δ(ℓ)dℓ (16) for all t ≥ t i . Since x(t i ) = z i (t i ), it follows that żi (t) = Az i (t) + BKz i (t i ) + δ ♯ (t) (17)
for all t ≥ t i . where

δ ♯ (t) = -BK ti ti-τ e A(ti-m) δ(m)dm. ( 18
)
By applying variation of parameters to (17) on the interval [t i , t] for any t ∈ [t i , t i + ν] with the choice (18), we get

z i (t) = ω(t -t i )z i (t i ) + t ti e A(t-ℓ) δ ♯ (ℓ)dℓ (19)
where ω was defined in [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. As an immediate consequence,

z i (t i ) = ω(t -t i ) -1 z i (t) -ω(t -t i ) -1 t ti e A(t-ℓ) δ ♯ (ℓ)dℓ (20) for all t ∈ [t i , t i + ν]. It follows that |z i (t) -z i (t i )| ≤ I -ω(t -t i ) -1 |z i (t)| + ω(t-t i ) -1 t ti e A(t-ℓ) δ ♯ (ℓ)dℓ ≤ Γ 1 |z i (t)| + ω(t-t i ) -1 t ti e A(t-ℓ) δ ♯ (ℓ)dℓ (21)
for all t ∈ [t i , t i + ν]. By our choice of δ ♯ , we deduce that

|z i (t) -z i (t i )| ≤ Γ 1 |z i (t)| +λ t ti e A(t-ℓ) BK ti ti-τ e A(ti-m) δ(m)dm dℓ (22)
and so also

|z i (t) -z i (t i )| ≤ Γ 1 |z i (t)| +νλe |A|ν |BK|e |A|τ t t-ν-τ |δ(m)| dm, (23)
where we used the fact that |e Ar | ≤ e |A||r| for all real values r (which follows from the Maclaurin series representation of the matrix exponential and subadditivity of the matrix norm) and the fact that

t-ν -τ ≤ t i -τ ≤ t i ≤ t for all t ∈ [t i , t i +ν].
Thus, since [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF] gives

x(t i ) = z i (t i ), (23) gives |z i (t) -x(t i )| ≤ Γ 1 |z i (t)| + Γ 2 t t-ν-τ |δ(m)| dm (24) for all t ∈ [t i , t i + ν]. We conclude that t i+1 -t i ≥ ν.
2) Stability analysis: We perform a stability analysis of (1) with u and t i defined by ( 9)- [START_REF] Mazenc | Event-triggered control for systems with state delays using a positive systems approach[END_REF]. We consider any i ≥ 1.

From ( 8)-( 9) and ( 15), it follows that

u(t -τ ) = K x(t i ) - ti ti-τ e A(ti-ℓ) δ(ℓ)dℓ (25)
for all t ∈ [t i , t i+1 ). Hence, the system (1) is

ẋ(t) = Hx(t) + BK[x(t i ) -x(t)] + δ♯ (t) (26)
for all t ∈ [t i , t i+1 ), where δ♯ = δ ♯ + δ with δ ♯ as defined by (18) as before. Then we define a comparison system by

ẋ(t) = Hx(t) + (BK[x(t i ) -x(t)]) + + δ♯ (t) + ẋ(t) = Hx(t) -(BK[x(t i ) -x(t)]) --δ♯ (t) - (27)
for all t ∈ [t i , t i+1 ) and i ≥ 1.

Consider solutions of ( 26)-( 27) such that x(t 1 ) ≤ x(t 1 ) and x(t 1 ) ≤ x(t 1 ). Since H is Metzler, it follows that

x(t) ≤ x(t) ≤ x(t) (28)
holds for all t ≥ t 1 ; this follows by noting that X = x -x and X = x -x are both solutions of a dynamics of the form Ẋ(t) = M X(t) + G(t) for a Metzler M and a nonnegative valued piecewise continuous G(t) and by then applying [6, Lemma 1]. Similarly, 0 ≤ x(t) and 0 ≤ -x(t) for all t ≥ t 1 .

We deduce that x(t) -x(t) ≤ x(t) ≤ x(t) -x(t) and so also

|x(t)| ≤ x(t) -x(t) for all t ≥ ν + τ. ( 29
)
We next use the variable s and the function U in s(t) = x(t) -x(t) and U (s) = V ⊤ s. From (27), we get ṡ(t) = Hs(t) + |BK[x(t i ) -x(t)]| + δ♯ (t) for all t ∈ [t i , t i+1 ) and i ≥ 1. Hence, by (3), the time derivative of U satisfies

U (t) ≤ -pV ⊤ s(t)+V ⊤ |BK||x(t i )-x(t)|+V ⊤ δ♯ (t) (30)
for all t ∈ [t i , t i+1 ). Also, by the structures of the dynamics (1) and ( 11) and the fact that z i (t i ) = x(t i ), we have

||z i (t) -x(t)|| = t ti e (t-ℓ)A δ(ℓ)dℓ ≤ e ||A||T T ||δ|| [ti,t]
for all t ∈ [t i , t i+1 ), by applying variation of parameters to the dynamics for x -z i . It follows from ( 24), (30), ( 18), (29), and the fact that T ≥ τ that for all t ≥ t 1 , we have

U (t) ≤ -pV ⊤ s(t) + δ B (t) + V ⊤ |BK| Γ 1 |x(t)|+Γ 2 t t-ν-τ δ(ℓ)dℓ ≤ -pV ⊤ + V ⊤ |BK|Γ 1 s(t) +V ⊤ |BK| Γ 2 t t-ν-τ δ(ℓ)dℓ + δ B (t) (31)
where

δ B (t) = (||V ⊤ BK||e ||A||T T [2+||Γ 1 ||]+||V ||)||δ|| [0,t] ,
the 2 in the δ B formula came from our using the bound |x(t i )-

x(t)| ≤ |z i (t) -x(t i )| + |z i (t) -x(t)
| to bound the second right side term in (30), and the Γ 1 in the δ B formula came from the bound

Γ 1 |z i (t)| ≤ Γ 1 |x(t)| + Γ 1 |x(t) -z i (t)|.
Using (6), we can find a constant µ > 0 such that U (t) ≤ -µU (s(t))+β 1 ||δ|| [0,t] for all t ≥ t 1 , where (32)

β 1 = ||V || T |BK|Γ 2 +(2 + ||Γ 1 ||)||BK||e ||A||T T +1 . ( 33
)
It follows from integrating (32) that for all t ≥ t 1 , we have

U (s(t)) ≤ e -µ(t-t1) U (s(t 1 )) + t t1 e -µ(t-m) β 1 ||δ|| [0,m] dm ≤ e -µ(t-t1) U (s(t 1 )) + β1 µ ||δ|| [0,t] . (34)
Since V > 0, the last inequality allows us to conclude that there are constants β i > 0 for i = 2, 3 such that

||s(t)|| ≤ β 2 e -µ(t-t1) ||s(t 1 )|| + β 3 ||δ|| [0,t] (35

III. SYSTEM WITH A DELAY IN THE MEASUREMENTS

A. Studied system

We next consider the system

ẋ(t) = Ax(t) + Bu(t) + δ(t) y(t) = Cx(t -τ ) ( 37
)
where x is valued in R n , the known function u is valued in R p and will be specified, C ∈ R q×n is known, y is valued in R q , τ > 0 is a constant delay, and the unknown function δ has piecewise continuous components. In addition to Assumptions 1 and 2, we now assume: Assumption 3: The pair (A, C) is observable. Note that Assumption 3 implies that with the choice

C △ = Ce -Aτ , (38)
the pair (A, C △ ) is observable for all measurement delays τ > 0. We let ν, p, V , Γ 1 , ω and λ be as defined in Section II.

B. Preliminary step: finite time observer

The following finite time observer differs from works such as [START_REF] Mazenc | New finite-time and fast converging observers with a single delay[END_REF] and is more amenable to event-triggered control:

Lemma 1: With the preceding notation and under Assumptions 1-3, and in terms of the matrix an functions

E = 0 -τ e A ⊤ ℓ C ⊤ △ C △ e Aℓ dℓ, (39)
y £ (t) = E -1 t t-τ e A ⊤ (m-t) C ⊤ △ y(m) +C △ t m-τ e A(m-ℓ) Bu(ℓ)dℓ dm, (40) and δ £ (t) = -E -1 t t-τ e A ⊤ (m-t) C ⊤ △ C △ t m-τ e A(m-ℓ) δ(ℓ)dℓdm, (41)
we have

y £ (t) = x(t) + δ £ (t) (42)
for all t ≥ 2τ for all solutions of the system (37). □ Proof: The matrix (39) is invertible because (A, C △ ) is observable; see, e.g., [START_REF] Sontag | Mathematical Control Theory[END_REF]Section 3.5]. Next notice that

x(t -τ ) = e -Aτ x(t) - t t-τ e A(t-ℓ-τ ) Bu(ℓ)dℓ - t t-τ e A(t-ℓ-τ ) δ(ℓ)dℓ for all t ≥ τ, (43)
by applying variation of parameters to (37). Hence,

y(t) = Ce -Aτ x(t) -C t t-τ e A(t-ℓ-τ ) Bu(ℓ)dℓ -C t t-τ e A(t-ℓ-τ ) δ(ℓ)dℓ (44)
for all t ≥ τ . We next use the function

y ⋆ (t) = y(t) + C t t-τ e A(t-ℓ-τ ) Bu(ℓ)dℓ (45)
for all t ≥ τ . The function ( 45) is available for measurement, because y and its integral term are known. Moreover,

y ⋆ (t) = C △ x(t) + δ △ (t) (46)
with C △ defined in (38) and

δ △ (t) = -C t t-τ e A(t-ℓ-τ ) δ(ℓ)dℓ (47)
is available for all t ≥ τ . By applying variation of parameters to (37) on [m, t] and then substituting the result into the relation 46), we get

y ⋆ (m) -δ △ (m) = C △ x(m) from (
C △ e A(m-t) x(t) = y ⋆ (m) -δ △ (m) +C △ t m e A(m-ℓ) Bu(ℓ)dℓ +C △ t m e A(m-ℓ) δ(ℓ)dℓ (48) for all m ∈ [t -τ, t] and t ≥ τ . It follows that e A ⊤ (m-t) C ♯ e A(m-t) x(t) = e A ⊤ (m-t) C ⊤ △ y ⋆ (m) -e A ⊤ (m-t) C ⊤ △ δ △ (m) +e A ⊤ (m-t) C ♯ t m e A(m-ℓ) Bu(ℓ)dℓ +e A ⊤ (m-t) C ♯ t m e A(m-ℓ) δ(ℓ)dℓ, (49)
where

C ♯ = C ⊤ △ C △ .
By integrating both sides of (49) over all m ∈ [t -τ, t] for a fixed value of t and then left multiplying the result by E -1 , it follows that for all t ≥ 2τ , we get

x(t) = E -1 t t-τ e A ⊤ (m-t) C ⊤ △ y ⋆ (m)dm + E -1 t t-τ e A ⊤ (m-t) C ♯ t m e A(m-ℓ) Bu(ℓ)dℓdm -E -1 t t-τ e A ⊤ (m-t) C ⊤ △ δ △ (m)dm + E -1 t t-τ e A ⊤ (m-t) C ♯ t m e A(m-ℓ) δ(ℓ)dℓdm. ( 50
)
From the definition (45) of y ⋆ (m) and (38), we deduce that

x(t) = E -1 t t-τ e A ⊤ (m-t) C ⊤ △ y(m)dm +E -1 t t-τ e A ⊤ (m-t) C ⊤ △ C m m-τ e A(m-ℓ-τ ) Bu(ℓ)dℓdm +E -1 t t-τ e A ⊤ (m-t) C ♯ t m e A(m-ℓ) Bu(ℓ)dℓdm -E -1 t t-τ e A ⊤ (m-t) C ⊤ △ δ △ (m)dm +E -1 t t-τ e A ⊤ (m-t) C ♯ t m e A(m-ℓ) δ(ℓ)dℓdm = E -1 t t-τ e A ⊤ (m-t) C ⊤ △ y(m)dm +E -1 t t-τ e A ⊤ (m-t) C ♯ t m-τ e A(m-ℓ) Bu(ℓ)dℓdm -E -1 t t-τ e A ⊤ (m-t) C ⊤ △ δ △ (m)dm +E -1 t t-τ e A ⊤ (m-t) C ♯ t m e A(m-ℓ) δ(ℓ)dℓdm (51)
for all t ≥ 2τ , which we can combine with the formula (47) for δ ∆ to obtain the final decomposition (42).

C. Main result

In terms of the function y £ from (40) and the matrix

Γ 3 = λe |A|ν ντ |BK||E -1 |e |A ⊤ |τ |C ⊤ △ C △ |e 2τ |A| (52)
and the other notation from the previous sections, we define the control u by

u(t) = 0 if t ∈ [-2τ, t 1 ) u(t) = Ky £ (t i ) if t ∈ [t i , t i+1 ) and i ≥ 1, (53)
where the sequence of event trigger times t i are defined by

             (i) t 0 = 0, (ii) t 1 = ν + 2τ
, and (iii) for each i ≥ 1, t i+1 is defined by

t i+1 = sup{b ∈ [t i , t i + T ) : |z i (s) -x(t i )| ≤ Γ 1 |z i (s)| + Γ 3 s s-ν-2τ δ(ℓ)dℓ for all s ∈ [t i , b]} (54)
where z i is the solution of the initial value problem

żi (t) = Az i (t) + BKy £ (t i ), z i (t i ) = x(t i ) (55)
for each i ≥ 1, and where we now require our constant T to be such that T ≥ ν + 2τ . We prove: Theorem 2: Let Assumptions 1-3 hold. Then, the system (37) with u and t i defined in (53)-( 55) is ISS with respect to δ. Also, t i+1 -t i ≥ ν for all i ≥ 0.

Remark 5: The reasoning from Remarks 1-3 also applies to Theorem 2, except with Γ 2 replaced by Γ 3 , and τ replaced by 2τ in the lower bound of integration in (iii) from [START_REF] Mazenc | Event-triggered control for systems with state delays using a positive systems approach[END_REF]. In particular, t i+1 -t i ∈ [0, T ] for all i ≥ 1.

Remark 6: By reasoning analogously to the input delay case in Remark 4, we can express the function y £ as a solution of a continuous-discrete system. To see how it can be done, notice that differentiating the formula for y £ from (40), and then substituting the formula u(t) = Ky £ (t i ) from (53), implies that each t ∈ [t i , t i+1 ) and i ≥ 1, we have

ẏ£ (t) = E -1 C ⊤ △ y(t) + C △ e At M 1 (t) -E -1 A ⊤ Ey £ (t) + BKy £ (t i ) -E -1 e -A ⊤ τ C ⊤ △ y(t -τ ) + C △ e A(t-τ ) M 2 (t) , (56)
where for k = 1, 2, we use the definitions and formulas

M k (t) = t t-kτ e -Aℓ Bu(ℓ)dℓ = t ti e -Aℓ dmBKy £ (t i ) + i-1 j=J(t-kτ ) tj+1 tj e -Am dmBKy £ (t j ) + t J(t-kτ ) t-kτ e -Am dmBKy £ (t J(t-kτ )-1 ) (57)
if t i ≥ t -kτ with J(t -kτ ) being the smallest j such that t j ≥ t -kτ and with the notational convention that the sum in (57) is not present if J(t -kτ ) = i, and

M k (t) = t t-kτ e -Aℓ dmBKy £ (t i ) (58)
if t i < t -kτ , and with the initial function

y £ (ℓ) = E -1 t1 ν+τ e A ⊤ (m-t1) C ⊤ △ y(m)dm (59)
for all ℓ ∈ [0, t 1 ]. The equations ( 56)-(59) show how ( 40) is expressible as a solution of a continuous-discrete system.

D. Proof of Theorem 2

The proof has two parts. First, we prove the lower bound condition inf i {t i+1 -t i } ≥ ν, to rule out Zeno's phenomenon. Then, we use interval observers to prove the ISS assertion.

1) Ruling out Zeno's phenomenon: To prove that Zeno's phenomenon does not occur, we fix an i ≥ 1, and we introduce the function δ ⋆ defined by δ ⋆ (t) = BKδ £ (t i ) for all t ≥ t i . Then from the decomposition of y £ from (42) and z i (t i ) = x(t i ), the dynamics in (55) satisfy żi (t) = Az i (t) + BKz i (t i ) + δ ⋆ (t) for all t ≥ t i . It follows that

z i (t) = ω(t -t i )z i (t i ) + t ti e A(t-ℓ) δ ⋆ (ℓ)dℓ (60)
for all t ∈ [t i , t i + ν], and so also

z i (t i ) = ω(t -t i ) -1 z i (t) -ω(t -t i ) -1 t ti e A(t-ℓ) δ ⋆ (ℓ)dℓ, (61)
by the reasoning that gave (20). We deduce that

|z i (t) -z i (t i )| ≤ ω(t -t i ) -1 -I |z i (t) + ω(t -t i ) -1 t ti e |A|(t-ℓ) |δ ⋆ (ℓ)|dℓ. (62)
Thus, for all t ∈ [t i , t i + ν], we have

|z i (t) -z i (t i )| ≤ Γ 1 |z i (t)| + λe |A|ν t ti |δ ⋆ (ℓ)|dℓ. (63)
Setting C ♯ = C ⊤ △ C △ as before, it follows from our formula (41) for δ £ that for all ℓ ∈ [t i , t], we have

|δ ⋆ (ℓ)| ≤ |BK| E -1 ti ti-τ e A ⊤ (m-ti) C ♯ ti m-τ e A(m-ℓ) δ(ℓ)dℓdm . (64)
Consequently, for all t ∈ [t i , t i + ν], we have

t ti |δ ⋆ (ℓ)|dℓ ≤ (t -t i )τ |BK||E -1 |e |A ⊤ |τ |C ♯ |e 2τ |A| ti ti-2τ |δ(ℓ)|dℓ ≤ ντ |BK||E -1 |e |A ⊤ |τ |C ♯ |e 2τ |A| ti ti-2τ |δ(ℓ)|dℓ. ( 65
)
Using the last inequality in (65) to bound the integral in (63), it follows that for t ∈ [t i , t i + ν], we have

|z i (t) -z i (t i )| ≤ Γ 1 |z i (t)| + Γ 3 t t-ν-2τ |δ(m)| dm. (66)
Hence, t i+1 -t i ≥ ν, so Zeno's phenomenon does not occur.

2) Stability: We have ẋ(t) = Ax(t) + BKy £ (t i ) + δ(t) for all t ∈ [t i , t i+1 ) and i ≥ 1. We deduce from (42) that

ẋ(t) = Hx(t) + BK(x(t i ) -x(t)) + BKδ £ (t i ) + δ(t) (67)
for all t ∈ [t i , t i+1 ). Then the remaining part of the proof is similar to the second part of the proof of Theorem 1.

IV. ILLUSTRATION

We revisit a dynamics for the control of the depth and pitch degrees-of-freedom (or DOF) of an autonomous underwater vehicle that we studied in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], e.g., the BlueROV2 vehicle, which is widely used in environmental surveys such as the study of corals. As in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], we assume that the vehicle has a Doppler Velocity Logger (or DVL) for estimating its velocity. The DVL commonly experiences bottom lock, making it impractical to continuously change the control values. Hence, we show how Theorems 1-2 apply, and so cover measurement delays which were beyond the scope of [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF] or other eventtriggered studies of the dynamics.

As noted in [12, Equation (9.28)], after linearization and assuming that the vehicle is neutrally buoyant, the linearized dynamics for the depth plane are given by

(m -X ẇ(t) ) ẇ(t) -(mx g + Z q ) q(t) -Z w w(t) -(mU (t) + z q )q(t) = Z γs u Z (mx g + M ẇ(t)) ẇ(t) + (I yy -M q ) q(t) -M w w(t) + (mx g U -M q )q(t) -M θ θ = M γs u M (68)
whose parameter values were experimentally computed and presented in [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF]. As in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], we assume that the surge nominal velocity is U = 0.1m/s. The states x = [w, q] ⊤ represent the depth and pitch velocity, and the controls u Z and u M are the force and moment required to produce motion of the vehicle. Using the parameter values and controller from [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF], the system (68) becomes ẋ(t) = A 0 x(t) + B 0 u with

A 0 = -0.
Choosing K 0 such that A 0 + B 0 K 0 has any two distinct negative eigenvalues (e.g., using the command FeedbackGains in the Mathematica program) and then applying a diagonalizing similarity transformation to obtain the matrices A = P -1 A 0 P , B = P -1 B 0 , and K = K 0 P as in Remark 1 above, we can satisfy the assumptions of Theorem 1 are satisfied. Also, when (A, C) is observable, the assumptions of Theorem 2 can be satisfied. Hence, Theorems 1-2 from the previous sections provide a useful alternative to results that could be obtained from [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], by only requiring a 2-dimensional piecewise continuous predictor using the values ξ i (which are needed to compensate for the input delays) regardless of the input delay length (instead of the chains of dynamical extensions that would be required when using [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF]), and also covering measurement delays which were beyond the scope of eventtriggered works like [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] and [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF]. In Figs. 2a and 2b, we show our MATLAB plots of the closed loop solutions for the preceding system obtained from applying Theorems 1 and 2, respectively. We chose τ = 0.1 s, ν = 0.4077, K 0 = [0.941, 0.637], P as in the preceding paragraph, and C = [1, 1]P (i.e., only the sum of the components of the states in the original variables is measured). The elements of Γ 1 were 0.001, and both elements of V were 1. We used p = 0.5, and a constant δ = [δ 1 , δ 2 ] ⊤ , with each δ i for i = 1, 2 found with the function rand(•) in MATLAB which draws values from the open interval (0, 1). Since our plots show desired convergence, they illustrate the value of our method, in the special case of the BlueROV2 with input or measurement delays.

V. CONCLUSION

We proposed new event-triggered controls for linear timeinvariant systems with a known arbitrarily long constant delays. Key novel features included (a) our alternative event triggers based on vectors of absolute values and (b) that our predictor maps were expressible as solutions of a discrete time system (under arbitrarily large constant input delays) or as solutions of continuous-discrete dynamic extensions (under measurement delays), with each of these predictor systems having its dimension equaling the dimension of the original systems that we render ISS and no distributed terms. This is a useful alternative to chain predictors whose dynamic extensions become arbitrarily large in dimension for bigger delays. Time-varying extensions are expected.

  Fig 1.

Fig. 1 :

 1 Fig. 1: Implementing Control from Theorem 1, Illustrating Control Values on Time Intervals and Feasibility of Method C. Proof of Theorem 1

Fig. 2 :

 2 Fig. 2: Simulation for (a) input and (b) measurement delays

  ) and the constant values B i = sup{b ∈ [t i , t i + T ) : (T i ) holds for all s ∈ [t i , b]}, our recursive algorithm is summarized in this table, which explains how to choose t i+1 after having chosen t i for each i ≥ 0 in the two cases: Case Choice (

[START_REF] Brunner | Robust event-triggered MPC with guaranteed asymptotic bound and average sampling rate[END_REF]

, b] if this supremum lies in [t 1 , t 1 +T ), or t 2 = t 1 +T otherwise. Then we use

[START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF]

to define ξ 2 to define the control values on [t 2 , t 3 ), where t 3 is found in the same way that we found t 2 except with the index i increased by 1.

Continuing inductively defines all triggering times t i and the control values for all times t ≥ 0. Using the test inequalities

|z i (s) -x(t i )| ≤ Γ 1 |z i (s)| + Γ 2 s s-ν-τ δ(ℓ)dℓ (T i

  ) for all t ≥ t 1 . From (29), we deduce that||x(t)|| ≤ β 2 e -µ(t-t1) ||x(t 1 ) -x(t 1 )|| + β 3 ||δ|| [0,t]Since we can assume that x(t 1 ) ≤ 2x(t 1 ) + and x(t 1 ) ≥ -2x(t 1 ) -, this gives||x(t)|| ≤ 2β 2 e -µ(t-t1) ||x(t 1 )|| + β 3 ||δ|| [0,t]

	(36)
	for all t ≥ t 1 . Also, (1) gives ||x(t)|| ≤ ||x(0)||e ||A||(2t1-t) +
	t 1 e ||A||t 1 ||δ|| [0,t] for all t ∈ [0, t 1 ], which we can add to (36)
	to get the desired ISS estimate.

Supported by US National Science Foundation Applied Mathematics Program Grant 2009659.

F.


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

Michael Malisoff
email: malisoff@lsu.edu

Feedback Stabilization and Robustness Analysis using Bounds on Fundamental Matrices

Keywords: Stabilization, sampling, positive system

We prove a new robust stabilization theorem for time-varying systems that have sampled measurements and time-varying disturbances, and we provide new observer designs, using novel bounds for fundamental matrices for systems with disturbances. Our main tools use properties of Metzler matrices and positive systems.

Introduction

This paper is devoted to the problem of constructing matrix valued interval observers for fundamental matrices of time-varying linear systems with unknown coefficient matrices, and to demonstrating the usefulness of these constructions for feedback stabilization under sampling and for observer design. Time-varying systems can be difficult to study because in general, no explicit formulas for fundamental solutions (i.e., state transition matrices) can be obtained and because they can be difficult to estimate when the vector fields defining the systems are uncertain. An added difficulty arises when one must find globally asymptotically stabilizing feedback controls when only sampled state measurements are available. While sampling problems can be expressed as delay compensation problems (such as those of [START_REF] Borri | Event-triggered control of nonlinear systems with time-varying state delays[END_REF] and [START_REF] Cacace | An enhanced observer for nonlinear systems with time-varying measurement delays[END_REF]) with time-varying sawtooth shaped delay functions, delay compensation methods generally require exact knowledge, or close approximations, of vector fields defining the given system, or can lead to bounds on the allowable delays (and so also bounds on the allowable sampling rates) that may be too small for applications. See [START_REF] Mattioni | Immersion and invariance stabilization of strict-feedback dynamics under sampling[END_REF], [START_REF] Mattioni | Feedforwarding under sampling[END_REF], and [START_REF] Tanasa | Backstepping control under multi-rate sampling[END_REF] for motivation for analyzing the effects of sampling in control systems.

To help us address these challenges, this paper presents a new construction of functions that are expressed in terms of fundamental solutions and which upper and lower bound the entries of the state transition matrix of a linear system whose bounded vector field is uncertain, where the in-Malisoff was supported by NSF Grants 1711299 and 2009659. A preliminary version [START_REF] Mazenc | Feedback stabilization with discrete measurements using bounds on fundamental matrices[END_REF] was accepted for presentation at the 2021 IEEE Conference on Decision and Control; see Section 1 for a comparison between this work and the conference version.

equalities are componentwise. Since the fundamental solutions in the bounds are those of time-varying systems with known vector fields (which can in turn be computed using a dynamical extension), this helps overcome the challenge of estimating the effects of uncertain vector fields. We apply this result to construct robust stabilizing controls for linear systems that have discrete measurements. Our control formula is the one in [9, Chapter 2, Section 2.9], which is based on the controllability Gramian and which is applied in [START_REF] Klamka | Controllability and Minimum Energy Control[END_REF] to systems whose state is available for continuous measurement. By contrast, here we apply the approach from [START_REF] Klamka | Controllability and Minimum Energy Control[END_REF] to systems whose state variables are only available for measurement at discrete instants. This makes it possible for us to cover situations in which the sup norm of an uncertainty in the dynamics or the sample rates are arbitrarily large, which we believe is a novel and significant contribution. Our proofs are based on comparison systems which share common features with interval observers (as defined, e.g., in [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF], [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], and [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]), but our approach is outside the scope of earlier interval observer methods that did not take sampling and unknown vector fields into account.

Our main strategy entails novel applications of properties of positive systems and Metzler matrices. Although positive systems have been used in aerospace engineering, mathematical biology, and other fields, we believe that our work is the first one to use them for sample data feedback stabilization when there are uncertain vector fields, and where the inter-sample intervals or sup norms of the uncertainty can be arbitrarily large. This paper improves on our preliminary conference version [START_REF] Mazenc | Feedback stabilization with discrete measurements using bounds on fundamental matrices[END_REF] by allowing timevarying coefficients and additional uncertainties in the vector fields defining the system that were not considered in [START_REF] Mazenc | Feedback stabilization with discrete measurements using bounds on fundamental matrices[END_REF] (in the B coefficient that multiplies the controls) and by applying our fundamental solution estimation approach to discrete time systems and to design new observers [START_REF] Borri | Luenberger-like observers for nonlinear time-delay systems with application to the artificial pancreas: The attainment of good performance[END_REF], using novel applications of the Gramian matrix. This contrasts with [START_REF] Mazenc | Feedback stabilization with discrete measurements using bounds on fundamental matrices[END_REF], which was confined to time-invariant coefficients and which did not include observer designs or discrete time systems or the uncertainty δ B below.

After providing the required definitions and notation, we provide our main feedback stabilization theorem in Section 3. In Section 4, we state our new estimation theorems, which are of independent interest because they help address challenges of estimating fundamental matrices for uncertain dynamics, and which we use to prove our main stabilization theorem in Section 5. In Section 6, we provide a discrete time analog of our stabilization theorem, whose proof uses discrete time versions of our theorems on estimating fundamental solutions for uncertain dynamics. In Section 7, we use our approach to provide new observer designs. We illustrate potential advantages of our approach in Section 8, and we close in Section 9 with a summary of our contributions and our suggestions for future research.

Definitions and Notation

We use the following definitions and notation thoughout this paper, where the dimensions of our systems are arbitrary unless we indicate otherwise. The notation will be simplified when no confusion could arise given the context, and all matrices in this paper are assumed to have only real entries. A square matrix is called Metzler provided all of its off-diagonal entries are nonnegative. A matrix is called nonnegative provided all of its entries are nonnegative. A square matrix is called Schur stable provided all of its eigenvalues have norm strictly less than one. For vectors We set G + = [max{g ij , 0}] and G -= G + -G. Notice for later use that if M 1 and M 2 are matrices of the same size and M 1 ≤ M 2 , then the ordering properties

V 1 = (v 1,1 .... v 1,n ) and V 2 = (v 2,1 .... v 2,n ) , we write V 1 < V 2 provided for all i ∈ {1, ..., n}, v 1,i < v 2,i ; and V 1 ≤ V 2 provided for all i ∈ {1, ..., n}, v 1,i ≤ v 2,
M + 1 ≤ M + 2 and M - 2 ≤ M - 1
and satisfied; the preceding two properties follow by separately considering the possible signs of the (i, j) entries of M + 1 , M + 2 , M - 1 , and M - 2 for all pairs (i, j). Let I be the identity matrix of any dimension, and 0 denote the matrix of any dimensions whose entries are all zero. Two square matrices M 1 and M 2 of the same size are called similar provided there is an invertible matrix P so that M 1 = P -1 M 2 P . For any continuous function F : [0, +∞) → R n×n , the fundamental (or state transition) matrix solution Φ F is defined to be the unique matrix valued function satisfying

∂Φ F ∂t (t, t 0 ) = F(t)Φ F (t, t 0 ), Φ F (t 0 , t 0 ) = I
(1) for all real t 0 and t. We use basic properties of input-to-state stability (or ISS, which we also use to mean input-tostate stable), e.g., from [START_REF] Sontag | Input to state stability: basic concepts and results[END_REF], and we set Z ≥0 = {0, 1, 2, . . .}.

Stabilization Theorem

We study systems of the form [START_REF] Bof | Lyapunov theory for discrete time systems[END_REF] with X valued in R n , the control U valued in R p , M : [0, +∞) → R n×n and B : [0, +∞) → R n×p being continuous matrix valued functions, and δ 1 : [0, +∞) → R n×n , δ B : [0, +∞) → R n×p , and δ 2 : [0, +∞) → R n being piecewise continuous locally bounded functions. The output measurement Y is defined by Y (t) = X(t i ) for all t ∈ [t i , t i+1 ), where t 0 = 0 and the t i 's are times when new state measurements become available, and we define σ : [0, +∞) → [0, +∞) by σ(t) = t i for all t ∈ [t i , t i+1 ). We assume the following conditions (but see Remark 2 for non-Metzler cases and a discussion of the class of time-varying systems that can be transformed into the form (2)):

Ẋ(t) = M (t)-δ 1 (t) X(t)+ B(t)+δ B (t) U (t)+δ 2 (t),
Assumption 1 There are known constant matrices δ ≥ 0 and δ B ≥ 0 such that for all t ≥ 0, we have 0 ≤ δ 1 (t) ≤ δ and |δ B (t)| ≤ δB .

(3) Also, the matrix M (t) is Metzler for each t ≥ 0, M and B both have some period p 0 > 0, and there are known positive constants η and ν such that η ≤ t i+1 -t i ≤ ν for all i ≥ 0 and such that with the choice

S = {(r, s) ∈ R 2 : r ∈ [0, p 0 ], η ≤ s -r ≤ ν}, (4)
the system ż(t) = M (t)z(t) + B(t)u(t) is controllable on [σ, τ ] for all (σ, τ ) ∈ S.
We then use the functions and matrices

ξ(r, s) = Φ M +δ (r,s)+Φ M -δ (r,s) 2 -Φ M (r, s), ξ(r, s) = Φ M -δ (r,s)-Φ M +δ (r,s) 2
, S(r) = B(r)B (r), and χ(r, s) = s r Φ M (r, )S( )Φ M (r, )d .

(

) 5
Then our controllability assumption from Assumption 1 is equivalent to the existence of the inverse χ -1 (r, s) for all (r, s) ∈ S, e.g., by [START_REF] Sontag | Mathematical Control Theory, Second Edition[END_REF]Theorem 5]. Then, for all (r, s) ∈ S, and with

J δ (r, s) = (s-r)e (|M |∞+|δ|)(s-r) δ B |B | ∞ e |M |∞(s-r) |χ -1 (r, s)|, (6)
the following matrices are well defined:

F c (r, s) = s r ξ(s, ) S( )Φ M (r, )χ -1 (r, s) - -ξ(s, ) S( )Φ M (r, )χ -1 (r, s) + d + ξ(s, r) + J δ (r, s) F c (r, s) = s r ξ(s, ) S( )Φ M (r, )χ -1 (r, s) - -ξ(s, ) S( )Φ M (r, )χ -1 (r, s) + d + ξ(s, r) -J δ (r, s) (7)
Using the preceding definitions, and the entrywise supre-

mum notation sup v∈S G(v) = [sup v∈S G ij (v)] for matrix valued functions G(v) = [G ij (v)]
, our stabilization theorem is:

Theorem 1 Let (2) satisfy Assumption 1, and choose p 0 , ν, η, and S to satisfy the requirements from Assumption 1. Then [START_REF] Bof | Lyapunov theory for discrete time systems[END_REF], in closed-loop with the control defined by

U (t) = -B (t)Φ M (t i , t)χ -1 (t i , t i+1 )X(t i ) (8)
for all ∈ [t i , t i+1 ) and i ≥ 0, is ISS with respect to δ 2 (t) if the matrix

Γ = max (r,s)∈S F + c (r, s) F - c (r, s) F - c (r, s) F + c (r, s) (9)
is Schur stable.

Before presenting our fundamental matrix estimation theorems (and then our proof of Theorem 1, in Section 5), we provide comments on the novelty and value of Theorem 1.

Remark 1 A key aspect of Theorem 1 is that one can check whether Γ is Schur stable because the matrices (7) are expressed in terms of known functions, while the state transition matrix of M -δ 1 (t) cannot be found because δ 1 is an unknown uncertainty. The periodicity in Assumption 1 makes it possible to express the maximum in the Γ formula over the compact set S (where the maximum exists by the continuity of the entries of the matrix in (9) as functions of (r, s) ∈ S), and then the eigenvalues of Γ can be computed numerically to check the Schur stability condition, e.g., using the command Eigenvalues in Mathematica. Moreover, in the important special case where M is constant, Φ M is a matrix exponential, so in that case we can express F+ c (r, s) and F - c (r, s) as functions of s -r. Then the maximum of the matrix in (9) can be computed over the interval [η, ν]. This includes the special case where the sample times are t i = iν for all i ∈ Z 0 , in which case we can take η = ν and S = {(r, r + ν) ∈ R 2 : r ∈ [0, p 0 ]}. Although fundamental solutions for the known matrices M and M ± δ are used in the formula for the control and in [START_REF] Klamka | Controllability and Minimum Energy Control[END_REF], they can be computed from solving matrix valued differential equations, e.g., using the method from [START_REF] Malisoff | Tracking and parameter identification for model reference adaptive control[END_REF]. In fact, as noted in [START_REF] Malisoff | Tracking and parameter identification for model reference adaptive control[END_REF], we can compute Φ A (t, s) for any piecewise continuous locally bounded square matrix valued function A by writing Φ A (t, s) = α A (t)β A (s) where α A and β A are the unique solutions of the matrix differential equations

αA (t) = A(t)α A (t) and βA (s) = -β A (s)A(s) ( 10
)
that satisfy α A (0) = β A (0) = I. When δ 2 (t)
is the zero function, our proof of Theorem 1 will imply that the origin of the closed-loop system from Theorem 1 is a globally exponentially stable equilibrium on R n . For methods to compute F c and F c when M and B are constant, see [1, p.34].

Remark 2 For any system of the form

ẋ(t) = M 0 (t)x(t) + B(t)U (t) (11)
that is controllable on [σ, τ ] for all (σ, τ ) ∈ S with state space R n with bounded continuous matrix valued coefficients M 0 (t) and B(t), we can find a matrix ∆M ∈ R n×n such that for any continuous time-varying matrix valued function

∆ M (t) that satisfies |∆ M (t)| ≤ ∆M for all t ≥ 0, the system Ẋ(t) = (M 0 (t) + ∆ M (t))X(t) + B(t)U (t) (12)
is also controllable on these intervals. This follows from the continuity of fundamental solutions for systems as functions of the entries of the matrix defining the system and standard characterizations of controllability, e.g., from [START_REF] Sontag | Mathematical Control Theory, Second Edition[END_REF]Chapter 3]. In the preceding situation, we can write

M 0 (t)+ ∆ M (t) = M (t) -δ 1 (t), where M (t) = M 0 (t) + (∆ M (t)) + and δ 1 (t) = (∆ M (t)) -. If, in addition, M 0 (t)
is Metzler for all t ≥ 0, then this defines a class of time-varying systems (2) that satisfy the requirements of Assumption 1 with δ = ∆M , because 0

≤ (∆ M (t)) -≤ (∆ M (t)) + + (∆ M (t)) -= |∆ M (t)| ≤ ∆M for all t ≥ 0.
See, e.g., [START_REF] Sontag | Mathematical Control Theory, Second Edition[END_REF]Theorem 5] for easily checked sufficient conditions for controllability of ( 11), based on using the invertibility of the Gramian.

Moreover, we can often use a preliminary change of coordinates to ensure that the vector field in the drift term has the required Metzler property. For instance, for any controllable pair (A 0 , B 0 ) of constant matrices for which all eigenvalues of A 0 are real, we can transform a linear system of the form

ẋ(t) = A 0 x(t) + B 0 U 0 (t) into a new system ż(t) = M 0 z(t) + BU (t)
that is controllable and for which M 0 is Metzler. This is done by choosing M 0 = P a A 0 P -1 a , B = P a B 0 , and z = P a x, where P a is chosen so that M 0 is the Jordan canonical form of A 0 , which makes M 0 Metzler. See also Section 7.2 where systems satisfying the requirements of Assumption 1 are useful for observer design.

By using Theorem 3 from Section 4.2 below, we can generalize Theorem 1 to cases where the matrices M (t) are not Metzler, but where they are similar to Metzler matrices, by first using a change of coordinates to transform [START_REF] Bof | Lyapunov theory for discrete time systems[END_REF]. The control [START_REF] Insperger An | Stability chart for the delayed Mathieu equation[END_REF] is the control from [9, Chapter 2, Section 2.9]. One can find matrices δ * such that Γ is Schur stable when

0 ≤ δ ≤ δ * and |δ B | ≤ δ * , (13)
because ξ and ξ are continuous and equal zero when δ = 0. This facilitates verifying the assumptions of Theorem 1.

Estimation of Fundamental Matrices

This section provides our theorems on estimating fundamental matrices. In Section 5, we show how these estimation theorems can be used to prove our Theorem 1, and we use them for observer designs in Section 7.

Metzler Case

First, we address the problem of estimating fundamental solutions, when there is a known Metzler valued term in the vector field defining the dynamics; see Section 4.2 for a method for relaxing this Metzler requirement to cases where the corresponding matrix only needs to be similar to a Metzler matrix at each time t. Throughout this subsection, we assume that the given function ∆ satisfies:

Assumption 2 The function ∆ : [0, +∞) → R n×n is piecewise continuous, and there is a known constant nonnegative matrix ∆ ∈ R n×n such that 0 ≤ ∆(t) ≤ ∆ holds for all t ≥ 0.

In Appendix A2 below, we prove:

Theorem 2 Let M : [0, +∞) → R n×n be a matrix valued function such that M(t) is Metzler for all t ≥ 0, and ∆ be a function satisfying Assumption 2. Choose the functions

Φ M,∆ (r, s) = Φ M-∆ (r,s)+Φ M+∆ (r,s) 2 and Φ M,∆ (r, s) = Φ M (r, s) + Φ M-∆ (r,s)-Φ M+∆ (r,s) 2 . ( 14
)
Then the inequalities

Φ M,∆ (t, t 0 ) ≤ Φ M-∆ (t, t 0 ) ≤ Φ M,∆ (t, t 0 ) ( 15
)
hold for all t ≥ t 0 and t 0 ≥ 0.

Remark 3

We believe that the preceding result is new, even in the special case where M is constant, in which case the fundamental solutions in the formulas ( 14) have the form

Φ M±∆ (r, s) = e (M±∆)(r-s) . ( 16
)
For cases where M is time-varying, the Metzler requirements from Theorem 2 are satisfied if M has the form M(t) = M 0 +δ M (t), when the constant Metzler matrix M 0 has positive off-diagonal entries, and when the sup norm of δ M is small enough. Another significant special case is when ∆ and M are both constant, i.e., ∆(t) = ∆ c for all t ≥ 0, in which case Φ M-∆ (t, t 0 ) = e (M-∆c)(t-t0) .

Remark 4 A different approach to estimating fundmanental solutions of uncertain matrices is as follows. For any square matrices A and B having the same dimension, the inequality

e A+B -e A ≤ ||B||e sup ∈[0,1] ||A+ B|| (17)
is satisfied; this follows by applying the Mean Value Theorem to the function f ( ) = e A+ B on [0, 1]. Taking ∆ = ∆ c , it follows from [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF] that in the special case where M is constant and ∆ is a constant matrix ∆ c , we have

||Φ M-∆ (t, t 0 ) -Φ M (t, t 0 )|| ≤ (t -t 0 )||∆ c ||e (t-t0) sup ∈[-1,1] ||M+ ∆c|| ( 18
)
for all t ≥ t 0 ≥ 0. By contrast, in the special case where M and ∆ are constant matrices, [START_REF] Mazenc | Feedback stabilization with discrete measurements using bounds on fundamental matrices[END_REF] gives

(t -t 0 ) ≤ Φ M-∆ (t, t 0 ) -Φ M (t, t 0 ) ≤ (t -t 0 ) (19)
where

( ) = e (M-∆c ) -e M +e (M+∆c ) -e M 2 and
( ) = e (M-∆c ) -e M +e M -e (M+∆c ) 2 (20)
which gives [START_REF] Sontag | Mathematical Control Theory, Second Edition[END_REF], because of [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]. The main advantage of using Theorem 2 in this time invariant case (instead of using the estimates that can be obtained directly from ( 17)) is that it produces estimates for each entry of Φ M-∆ (t, t 0 ); see Section 8 for an example that illustrates this advantage of using our approach from Theorem 2.

More General Case

Although Theorem 2 only applies when M(t) is Metzler for each t, we can relax this Metzler requirement to cover a broader family of matrices that are similar to Metzler matrices at each time t. The price to pay for this generalization is that the upper and lower bounding functions have more complicated forms. In the special case where M is constant, this generalization is motivated in part by the fact that Jordan canonical forms of matrices with real eigenvalues are Metzler. Using the theorem from this subsection, we can also relax the Metzler condition from Assumption 1 so that we only require M to be similar to a Metzler matrix at each time t ≥ 0. In Appendix A3 below, we prove the following:

Theorem 3 Let A : [0, +∞) →
Φ £ (r, s) = (P -1 ) + 2 (Φ M -ς (r, s) + Φ M +ς (r, s)) P + -(P -1 ) -(Φ M (r, s) + F(r, s)) P + -(P -1 ) + (Φ M (r, s) + F(r, s)) P - + (P -1 ) - 2 (Φ M -ς (r, s) + Φ M +ς (r, s)) P - Φ * (r, s) = (P -1 ) + (Φ M (r, s) + F(r, s)) P + -(P -1 ) - 2 (Φ M -ς (r, s) + Φ M +ς (r, s)) P + -(P -1 ) + 2 (Φ M -ς (r, s) + Φ M +ς (r, s)) P - +(P -1 ) -(Φ M (r, s)+F(r, s))P - (22)
where F(r, s) = Φ M -ς (r,s)-Φ M +ς (r,s) 2 (23) and ς = P + ρ -P -ρ (P -1 ) + -P + ρ -P -ρ (P -1 ) -+ -P + ρ -P -ρ (P -1 ) + + P + ρ -P -ρ (P -1 ) - (24) and M (t) = F (t) + P + ρ -P -ρ (P -1 )

+ + P -ρ -P + ρ (P -1 ) -+ (25)
for all t ≥ 0. Then the inequalities

Φ * (t, t 0 ) ≤ Φ A+ρ (t, t 0 ) ≤ Φ £ (t, t 0 ) (26)
hold for all t ≥ t 0 and t 0 ≥ 0.

Proof of Theorem 1

This section uses Theorem 2 to prove Theorem 1. The proof consists of three main parts. In the first part, we derive the dynamics for the discrete time variables x k = X(t k ), which is a sample data system that corresponds to the closed loop system from Theorem 1. In the second part, we prove an ISS estimate for this discrete time system that is based on our Schur stability assumption on (9) from the theorem. In the last part, we use the ISS property for the discrete time system to obtain the required ISS property for the system from the conclusion of Theorem 1.

First

Step: Deriving the Discrete Time System. Let i ≥ 0 be an integer. By applying the method of variation of parameters to the system (2) on the interval [t i , t] (e.g., [18, Formula (C.26)]), with t ∈ [t i , t i+1 ], we get

X(t) = Φ M -δ1 (t, t i )X(t i ) + t ti Φ M -δ1 (t, )B ( )U ( )d + t ti Φ M -δ1 (t, )δ 2 ( )d , (27)
where B ( ) = B( ) + δ B ( ).

Then the semigroup property of fundamental solutions gives Φ M (t, t i )Φ M (t i , ) = Φ M (t, ) and therefore also

X(t) = t ti Φ M -δ1 (t, )δ 2 ( )d + Φ M (t, t i ) X(t i ) + t ti Φ M (t i , )B ( )U ( )d + [Φ M -δ1 (t, t i ) -Φ M (t, t i )]X(t i ) + t ti [Φ M -δ1 (t, ) -Φ M (t, )]B ( )U ( )d . (28)
Also, the feedback [START_REF] Insperger An | Stability chart for the delayed Mathieu equation[END_REF] gives

X(t) = t ti Φ M -δ1 (t, )δ 2 ( )d + D(t, t i ) + H(t, t i ) + [Φ M -δ1 (t, t i ) -Φ M (t, t i )]X(t i ) + t ti [Φ M (t, ) -Φ M -δ1 (t, )]S d X(t i ) = - t ti Φ D (t, )S d + Φ D (t, t i ) X(t i ) + t ti Φ M -δ1 (t, )δ 2 ( )d + D(t, t i ) +H(t, t i ), (29)
where D(t, t i ) is the quantity in curly braces in (28),

S = S( ) (Φ M (σ( ), )) χ -1 i , (30)
H(t, t i ) = - t ti Φ D (t, )δ B ( )B ( )Φ M (t i , )d χ -1 i X(t i ), (31)
and

Φ D (a, b) = Φ M -δ1 (a, b) -Φ M (a, b), (32)
and where χ i = χ(t i , t i+1 ). Moreover, our control [START_REF] Insperger An | Stability chart for the delayed Mathieu equation[END_REF] gives

D(t i+1 , t i ) = Φ M (t i+1 , t i ) X(t i ) + ti+1 ti Φ M (t i , )B( )U ( )d + ti+1 ti Φ M (t i , )δ B ( )U ( )d = - ti+1 ti Φ M (t i+1 , )δ B ( )Φ M (t i , )d χ -1 i X(t i ) (33)
where

δ B ( ) = δ B ( )B ( ).
Specializing (29) to the case where t = t i+1 therefore gives

X(t i+1 ) = W (i)X(t i ) + ti+1 ti Φ M -δ1 (t i+1 , )δ 2 ( )d , (34)
where

W (i) = Φ D (t i+1 , t i ) - ti+1 ti Φ D (t i+1 , )S d - ti+1 ti Φ M -δ1 (t i+1 , )δ B ( )Φ M (σ( ), )d χ -1 i , (35)
where the last term in (35) was obtained by adding (31) and (33). Equation (34) defines the desired closed loop discrete time system.

Second

Step: Stability Analysis for Discrete Time System (34). Theorem 2 ensures that for all a ≥ b, we have

ξ(a, b) ≤ Φ D (a, b) ≤ ξ(a, b), (36)
where ξ and ξ are defined in [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF]. We deduce that, for any integer i ≥ 0, we have

ξ(t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i + ≤ Φ D (t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i + ≤ ξ(t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i + (37)
and

ξ(t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i - ≤ Φ D (t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i - ≤ ξ(t i+1 , ) S( )(Φ M (σ( ), )) χ -1 i - (38)
for all ∈ (t i , t i+1 ). By multiplying (38) through by -1 and then adding the result to (37), we get

ξ(t i+1 , )(S ) + -ξ(t i+1 , )(S ) - ≤ Φ D (t i+1 , )S ≤ ξ(t i+1 , )(S ) + -ξ(t i+1 , )(S ) - (39)
for all ∈ (t i , t i+1 ), where we used the fact that S = (S ) + -(S ) -.

Also, by noting that

|Φ Q (t, s)| ≤ e |Q |∞|t-
|Φ M -δ1 (t i+1 , )δ B ( )Φ M (σ( ), )χ -1 i | ≤ e (|M |∞+|δ1|∞)(ti+1-ti) δ B |B | ∞ e |M |∞(ti+1-ti) |χ -1 i | (41)
for all ∈ [t i , t i+1 ]. We deduce from (36) that

F c (t i , t i+1 ) ≤ W (i) ≤ F c (t i , t i+1 ), (42)
where (41) was used to bound the last term in (35) and to obtain the expression for the function J δ that we used in (6) and our formulas [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] for F c and F c .

In order to prove our ISS property for (34), we first use our bounds (42) to study the system

Z(t i+1 ) = W (i)Z(t i ) ( 43
)
that corresponds to the δ 2 = 0 case in (34). By rewriting (43) as Z(t i+1 ) = (W (i) + -W (i) -)Z(t i ), we conclude that for any solution Z(t i ) of (43), the function

Ξ(t i ) = [(Z(t i )) , (-Z(t i )) ] is a solution of ζ(t i+1 ) = W (i) + ζ(t i ) + W (i) -ψ(t i ) ψ(t i+1 ) = W (i) -ζ(t i ) + W (i) + ψ(t i ). (44)
Moreover, (44) is a positive system (which can be checked by induction, using the fact that W (i) + and W (i) -are nonnegative matrices for each index i).

Next, let [(ζ p (t i )) , (ψ p (t i )) ] and [(ζ n (t i )) , (ψ n (t i )) ]
denote the solutions of (44) with initial conditions

ζ p (t 0 ) ψ p (t 0 ) = Z(t 0 ) + Z(t 0 ) - (45)
and

ζ n (t 0 ) ψ n (t 0 ) = Z(t 0 ) - Z(t 0 ) + (46)
respectively. By the positivity of the system (44), these two solutions are nonnegative valued. Then, noticing that

Ξ(t 0 ) = Z(t 0 ) + -Z(t 0 ) - Z(t 0 ) --Z(t 0 ) + = ζ p (t 0 ) ψ p (t 0 ) - ζ n (t 0 ) ψ n (t 0 ) , (47)
we deduce from the uniqueness of solutions property for the discrete time system (44) that

Ξ(t i ) = ζ p (t i ) ψ p (t i ) - ζ n (t i ) ψ n (t i ) (48)
for all i ≥ 0.

Next consider the system

ζ(t i+1 ) = F + c (t i , t i+1 )ζ(t i ) + F - c (t i , t i+1 )ψ(t i ) ψ(t i+1 ) = F - c (t i , t i+1 )ζ(t i ) + F + c (t i , t i+1 )ψ(t i ), (49)
and let [(ζ p (t i )) , (ψ p (t i )) ] and [(ζ n (t i )) , (ψ n (t i )) ] denote the solutions of (49) satisfying

ζ p (t 0 ) ψ p (t 0 ) = Z(t 0 ) + Z(t 0 ) - and ζ n (t 0 ) ψ n (t 0 ) = Z(t 0 ) - Z(t 0 ) + (50)
respectively. By (42) and the order preserving properties from Section 2, we have

W (i) + ≤ F+ c (t i , t i+1 ) and W (i) -≤ F - c (t i , t i+1
), (51) so by the comparison principle, we have

0 ≤ ζ p (t i ) ψ p (t i ) ≤ ζ p (t i ) ψ p (t i ) and 0 ≤ ζ n (t i ) ψ n (t i ) ≤ ζ n (t i ) ψ n (t i ) (52)
for all i ≥ 0; this follows by noting that the initial conditions in the right sides of (45)-( 46) agree with those of the right sides of (52), and that (51) implies that the states with the bars are growing faster. As an immediate consequence,

-Γ i ζ n (t 0 ) ψ n (t 0 ) ≤ - ζ n (t i ) ψ n (t i ) ≤ Ξ(t i ) ≤ ζ p (t i ) ψ p (t i ) ≤ Γ i ζ p (t 0 ) ψ p (t 0 ) (53)
for all i ≥ 0, where Γ is the Schur stable matrix from [START_REF] Klamka | Controllability and Minimum Energy Control[END_REF], and where we used (48), (52), and the periodicity assumption in Assumption 1 (which gives Fc (r + kp 0 , s + kp 0 ) = Fc (r, s) and F c (r + kp 0 , s + kp 0 ) = F c (r, s) for all integers k ≥ 0 and all r ≥ 0 and s ≥ r). Consequently,

-Γ i Z(t 0 ) - Z(t 0 ) + ≤ Z(t i ) -Z(t i ) ≤ Γ i Z(t 0 ) + Z(t 0 ) - , (54)
by our choice of Ξ. Thus,

-[I, 0]Γ i Z(t 0 ) - Z(t 0 ) + ≤ Z(t i ) ≤ [I, 0]Γ i Z(t 0 ) + Z(t 0 ) - . (55)
By our assumption that Γ is Schur stable, this implies exponential stability of (43), hence the desired ISS property of (34), by the boundedness of δ 1 (which can be shown, e.g., by showing that the strict Lyapunov function for (43) that can be constructed from [2, Theorem 5.13] is an ISS Lyapunov function for (34)).

Third

Step: ISS Conclusion of Theorem 1. We can find a constant c > 0 such that all solutions of the closed loop system from the statement of Theorem We can combine (56) with the ISS conclusion from the second step to obtain the ISS property of the closed loop system from the theorem. This proves Theorem 1.

Remark 5

The last part of the preceding proof shows how bigger values of δ and δB can lead to bigger c values, and therefore to larger bounds on the right side of the final ISS estimate. Since the formula for c also depends on the upper bound ν on the sample intervals, this can also provide guidance on how to choose sample times that lead to a suitable upper bound ν on the sample periods, by choosing ν so that c lies in a suitable range to ensure satisfactory performance in terms of a small enough upper bound on the right side of the bound for the state.

6 Discrete Time Analogs

Stability Theorem

Using discrete time analogs of Theorems 1 and 2, it is possible to prove asymptotic stability conditions for systems of the form

X i+1 = (M -∆ i )X i (
57) with M ≥ 0, and with the matrices ∆ i admitting a matrix ∆ ≥ 0 such that 0

≤ ∆ i ≤ ∆ for all i. Although M -∆ ≤ M -∆ i ≤ M (58)
holds for all i, the inequalities (58) cannot be used to study the stability of (57) when some entries of M -∆ are negative. On the other hand, we have the following theorem, whose proof in Appendix A4 uses discrete time analogs of our continuous time theorems from the preceding sections:

Theorem 4 Let S j = (M +∆) j +(M -∆) j 2 and S j = M j + (M -∆) j -(M +∆) j 2 (59)
for all i ≥ 0. Assume that lim j→+∞ S j = 0 and lim j→+∞ S j = 0. (60)

Then the system (57) is globally asymptotically stable to the origin on R n .

Illustration

To illustrate the preceding theorem let

M = 0 1 2 1 2 0 and ∆ i = δ 1,i 0 0 δ 2,i , (61)
and assume that there is a constant δ > 0 such that 0 ≤ δ k,i ≤ δ for k = 1, 2 and all i ≥ 0. Then Lemma A.1 from Appendix A1 below with n = 1 gives

M + ∆ = P -1 1 2 + δ 0 0 δ -1 2 P, (62)
where ∆ = δI,

P = 1 1 -1 1
, and (63)

M -∆ = P -1 1 2 -δ 0 0 -δ -1 2 P. (64)
Hence, with the choices

δ j = ( 1 2 +δ) j 2 and μ j = (δ-1 2 ) j 2 , ( 65
)
the functions from (59) are

S j = P -1 δ j +(-1) j μ j 0 0 μ j +(-1) j δ j P (66)
and

S j = M j + P -1 (-1) j μ j -δ j 0 0 (-1) j δ j -μ j P = P -1 ( 1 2 ) j +(-1) j μ j -δ j 0 0 -1 2 j +(-1) j δ j -μ j P, (67)
by Lemma A.1 from Appendix A1 below with α = 0 and β = 1/2. Hence, if δ < 1/2, then (60) will hold, which will ensure global asymptotic stability of the system.

Observer Designs

We next illustrate the usefulness of our continuous time transition matrix estimation and Gramian approaches, in the context of new observer designs. First we provide a finite time observer, based on our Gramian approach but not using our estimation methods. This finite time observer has the advantage of being a fixed time one, meaning, the convergence of the observer to the true state value occurs at a time that is independent of the initial state. Then we provide a new asymptotic observer for more general systems, using a convergence proof that is based on both transition matrix estimation and the Gramian. The proof of the convergence property for this asymptotic observer makes essential use of the arguments from the proof of Theorem 1 and the result from Theorem 2. Therefore, through its uses of the Gramian in our first observer design (which is an essential ingredient from Theorem 1) and the results from Theorems 1-2, this section is strongly connected to, and helps illustrate the value of, the earlier sections.

For simplicity, we assume in this section that the coefficient matrices are constant and that the sample intervals t i+1 -t i are of a constant positive length ν, but analogous arguments produce results for unevenly spaced sample times t i as in the previous sections and for time-varying coefficient matrices, by replacing matrix exponentials by transition matrices in the relevant places. In both of our observer designs, we will use the matrix valued functions

G(S) = ν 0 e S C Ce S d -1 (68)
for suitable functions S, where the existence of the inverse will follow from the observability of (S, C) for matrices S that we specify later (again by [18, Theorem 5, p.109]).

Finite Time Observer

We consider the system

ẋ(t) = Ax(t) + f (y(t), u(t)) y(t) = Cx(t) (69)
where x is valued in R n , f is locally Lipschitz, u is valued in R p , and y is valued in R q . We assume that this system is forward complete, and we introduce a sequence t i = iν with ν > 0 being a given constant. See, e.g., [START_REF] Insperger An | Stability chart for the delayed Mathieu equation[END_REF], [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF], and [START_REF] Mazenc | Reduced-order fast converging observers for systems with discrete measurements and measurement error[END_REF] (which did not use Gramian approaches and also did not use estimators for transition matrices) for the importance of systems with nonlinearities of the form we have in (69) for modeling the vibration of elastic membranes, pendulums, and single-link robotic manipulators. Our objective is to construct a finite time observer for this system, under this standard assumption:

Assumption 3 The pair (A, C) is observable.
We also use the matrix valued function R : R → R n×q defined by R( ) = e A G(A)e A C (70) for all ∈ R, where G was defined in (68).

Let us propose the observer that is defined by

ẋ(t) = Ax(t) + f (y(t), u(t)) +R(t -t i ) y(t) -Ce A(t-ti) x(t i ) -C ẑ(t) ż(t) = Aẑ(t) + f (y(t), u(t)), ẑ(t i ) = 0 (71)
for all t ∈ [t i , t i+1 ) and i ≥ 0, where the solutions of the x dynamics are defined by solving the first equation of (71) on [t 0 , t 1 ) = [0, ν) with the initial state x(0) = 0, then solving it on [t 1 , t 2 ) with the initial state x(t 1 ) = x(t - 1 ), and then repeating this process on subsequent intervals [t i , t i+1 ) for i ≥ 2. We prove: Theorem 5 Let Assumption 3 hold. Then for each solution x(t) of (69), the corresponding solution x(t) of the observer (71) is such that x(t) = x(t) for all t ≥ ν.

Proof. By applying variation of paramters to the dynamics in (69), we deduce that

y(t) = Ce A(t-ti) x(t i ) + C t ti e A(t-) f (y( ), u( ))d . (72) Since ẑ(t i ) = 0, it follows that y(t) = Ce A(t-ti) x(t i ) + C ẑ(t), so ẋ(t) = Ax(t) + f (y(t), u(t)) +R(t -t i )Ce A(t-ti) [x(t i ) -x(t i )]. (73)
Here and in the sequel, all equalities and inequalities are for all t ∈ [t i , t i+1 ) and i ≥ 0, unless otherwise indicated. Let x(t) = x(t) -x(t). Then

ẋ(t) = Ax(t) -R(t -t i )Ce A(t-ti) x(t i ). (74)
By integrating this system, we deduce that

x(t) = e A(t-ti) - t ti e A(t-) R( -t i )Ce A( -ti) d x(t i ). (75)
Thus, in particular, the equality

x(t i+1 ) = e Aν - ti+ν ti e A(ti+1-) R( -t i )Ce A( -ti) d x(t i ) (76)
holds. Consequently,

x(t i+1 ) = e Aν I - ν 0 e -A R( )Ce A d x(t i ) = e Aν I - ν 0 e -A e A G(A)e A C Ce A d x(t i ), (77)
where the last inequality is a consequence of the definition (70) of R. We deduce that

x(t i+1 ) = e Aν I - ν 0 G(A)e A C Ce A d x(t i ) = 0 (78)
Thus, in particular, x(ν) = 0. From (74), we deduce that x(t) = 0 for all t ≥ ν. This concludes the proof.

Observer for Time-Varying Systems

We next consider the more general class of systems

ξ(t) = [A -∆(t)]ξ(t) + f (Y (t), u(t)) Y (t) = Cξ(t) (79)
where ξ is valued in R n , u is valued in R p , and Y is valued in R q . We assume that f is locally Lipschitz, that this system is forward complete, and that Assumption 3 is satisfied, where A, ∆, and f are assumed to be known. Before presenting our observer design, we introduce the following dynamic extension whose purpose is to remove the term f (Y (t), u(t)) in (79):

ξ (t) = [A -∆(t)]ξ (t) + f (Y (t), u(t)) +L[Y (t) -Cξ (t)]. (80)
The term L[Y (t)-Cξ (t)] is here only to introduce a degree of freedom. Then, with the choice,

x(t) = ξ (t) -ξ(t), we obtain ẋ(t) = [M -∆(t)]x(t) y(t) = Cx(t) (81)
with M = A -LC. In terms of our notation (68), we assume the following, where the existence of the inverse in (83) follows from the observability of (A, C) as before:

Assumption 4 The pair (A, C) is observable, the origin of (81) is globally uniformly asymptotically stable, and M is Metzler. Also, there is a matrix ∆ such that 0 ≤ ∆(t) ≤ ∆ (82) for all t ≥ 0. Finally, with the choices

β(s) = e (M -∆)s +e (M +∆)s -2e M s 2 , β(s) = e (M -∆)s -e (M +∆)s 2 , H( ) = e M G(M )e M C , (83)
and

G c (r) = β(r) - r 0 β(ν -) H( )Ce M + -β(ν -) H( )Ce M -d G c (r) = β(r) - r 0 β(ν -) H( )Ce M + -β(ν -) H( )Ce M -d , (84) the matrix κ = G c (ν) + G c (ν) - G c (ν) -G c (ν) + (85)
is Schur stable.

Our condition on (81) in Assumption 4 holds if ∆ is sufficiently small and M is Hurwitz and Metzler. In this case, bounds ∆ on the allowable functions ∆(t) can be found using quadratic Lyapunov functions. For instance, we can solve the Riccati equation P * M + M P * = -I for a positive definite matrix P * , and then the time derivative of the quadratic Lyapunov function v(x) = x P * x along solutions of ẋ(t) = [M -∆(t)]x(t) satisfies v ≤ -||x(t)|| 2 -2x (t)P * ∆(t)x(t), and then can choose ∆ so that all of its entries are strictly less than 1/(2n||P * ||), because that will ensure that ||P * ||||∆(t)|| < 1 2 when ∆ satisfies (82). Here the n in the denominator of the upper bound on the entries of ∆ arises from the need to convert a condition on Euclidean 2-norms of ∆(t) into a condition of the form (82) on the entries of ∆(t). We define our observer by

ẋ(t) = [M -∆(t)]x(t) +H(t -t i )[y(t) -Ce M (t-ti) x(t i )] (86)
for all t ∈ [t i , t i+1 ) and i ≥ 0, where H is from (83), and the solutions of (86) are defined analogously to those of the x dynamics in (71). We then have the following theorem: Theorem 6 Let Assumption 4 hold. Then for each solution x(t) of (81) and the corresponding solution x(t) of the observer (86), we have lim t→+∞ (x(t) -x(t)) = 0.

Proof. Since x(t) = Φ M -∆ (t, t i )x(t i ) for all t ≥ t i , we have

ẋ(t) = [M -∆(t)]x(t) +H(t -t i )[CΦ M -∆ (t, t i )x(t i ) -Ce M (t-ti) x(t i )]. (87)
Here and in the sequel, all equalities and inequalities are for all t ∈ [t i , t i+1 ) and i ≥ 0, unless otherwise indicated. Thus

ẋ(t) = [M -∆(t)]x(t) +H(t-t i )Ce M (t-ti) [x(t i ) -x(t i )] +H(t-t i )C Φ M -∆ (t, t i )-e M (t-ti) x(t i ). ( 88
)
As an immediate consequence of (81) and (88), the variable x(t) = x(t) -x(t) satisfies the following:

ẋ(t) = [M -∆(t)]x(t) -H(t -t i )Ce M (t-ti) x(t i ) -H(t -t i )C Φ M -∆ (t, t i ) -e M (t-ti) x(t i ).
(89) Applying variation of parameters to (89) gives

x(t) = Φ M -∆ (t, t i )x(t i ) + t ti Φ M -∆ (t, ) -H( -t i )Ce M ( -ti) x(t i ) -H( -t i )CE( , t i )x(t i )] d = [Φ M -∆ (t, t i ) - t ti Φ M -∆ (t, )H( -t i )Ce M ( -ti) d x(t i ) - t ti Φ M -∆ (t, )H( -t i )CE( , t i )d x(t i ) (90)
and so also

x(t) = [Φ M (t, t i ) - t ti Φ M (t, )H( -t i )Ce M ( -ti) d x(t i ) + [E(t, t i ) - t ti E(t, )H( -t i )Ce M ( -ti) d x(t i ) - t ti Φ M -∆ (t, )H( -t i )CE( , t i )d x(t i ), (91)
where E(r, s) = Φ M -∆ (r, s) -e M (r-s) for all r ≥ 0 and s ≥ 0. Thus, in particular,

x(t i+1 ) = Φ M -∆ (t i+1 , t i ) -e M ν - ti+1 ti E(t i+1 , )H( -t i )Ce M ( -ti) d x(t i ) - ti+1 ti Φ M -∆ (t i+1 , )H( -t i )CE( , t i )d x(t i ) +e M ν I - ti+1 ti e M (ti-) H( -t i )Ce M ( -ti) d x(t i ). (92)
From the definition of H in (83), it follows that

x(t i+1 ) = e M ν {I - ti+1 ti G(M )e M ( -ti) C Ce M ( -ti) d x(t i ) + Φ M -∆ (t i+1 , t i ) -e M ν - ti+1 ti E(t i+1 , )H( -t i )Ce M ( -ti) d x(t i ) - ti+1 ti Φ M -∆ (t i+1 , )H( -t i )CE( , t i )d x(t i ). ( 93
)
From the definition of G in (68), we obtain x(t i+1 ) = G(t i )x(t i ) + H(t i )x(t i ) (94) with the choices

G(t i ) = E(t i+1 , t i ) - ti+1 ti E(t i+1 , )H( -t i )Ce M ( -ti) d H(t i ) = - ti+1 ti Φ M -∆ (t i+1 , )H( -t i )CE( , t i )d , (95)
since the quantity in curly braces in (93) is zero.

We next study the discrete time system ς(t i+1 ) = G(t i )ς(t i ).

(96) In terms of our notation from Assumption 4, we then have

β(t i+1 -) ≤ E(t i+1 , ) ≤ β(t i+1 -), (97)
by Theorem 2. This gives

β(t i+1 -) H( -t i )Ce M ( -ti) + ≤ E(t i+1 , ) H( -t i )Ce M ( -ti) + ≤ β(t i+1 -) H( -t i )Ce M ( -ti) + (98) and β(t i+1 -) H( -t i )Ce M ( -ti) - ≤ E(t i+1 , ) H( -t i )Ce M ( -ti) - ≤ β(t i+1 -) H( -t i )Ce M ( -ti) -. (99) Therefore β(t i+1 -) H( -t i )Ce M ( -ti) + -β(t i+1 -) H( -t i )Ce M ( -ti) - ≤ E(t i+1 , )H( -t i )Ce M ( -ti) ≤ β(t i+1 -) H( -t i )Ce M ( -ti) + -β(t i+1 -) H( -t i )Ce M ( -ti) -. (100)
Hence, with our choice of G from (95), and with our choices of G c and G c from Assumption 4, we have

G c (ν) ≤ G(t i ) ≤ G c (ν).
(101) Therefore, the last part of our proof of Theorem 1 implies that the discrete time system (96) is exponentially stable if the matrix κ we defined in (85) is Schur stable. Since the origin of (81) is globally uniformly asymptotically stable, we also deduce from the proof of Theorem 1 that the solutions of (94) exponentially converge to the origin, which gives the desired result, by the reasoning from the third step of the proof of Theorem 1.

Application

In the important special case where δ 2 and δ B in (2) are 0, the proof of Theorem 1 implies that its closed loop system with the control ( 8) is uniformly globally exponentially stable to 0 on R n when (9) is Schur stable. On the other hand, we show in this section how to use [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF] to get an alternative sufficient condition that ensures this exponential stability. We show that our Schur stability requirement from Theorem 1 is less restrictive than the condition that would come from only using [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF], by proving that the Schur condition is satisfied under larger bounds δ on δ 1 than the largest bound that we could allow if we instead only used [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]. This will illustrate an advantage of using our Schur stability condition. Although we assume in this section that δB = 0 and δ 2 = 0 and that t i = iν for all i ≥ 0 and that δ 1 is a constant matrix δ, similar reasoning applies for time varying δ 1 's, as well as for ISS cases with nonzero δ 2 's.

Consider the dynamics (2) with

M = B = I ∈ R 2×2 and δ = 0 δ * δ * 0 ∈ R 2×2 (102)
for a constant δ * ∈ (0, 2). This gives S = I, and the function χ from (5) is

χ(r, s) = s-r 0 e -2 d I = 1 2 (1 -e -2(s-r) )I. ( 103
and the bounding functions from [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] are

F c (r, s) = e L * (cosh(δ * L * ) -1) I + 2 1-e -2L * L * 0 e L * -2 sinh(δ * (L * -))d 0 1 1 0 and F c (r, s) = -e L * sinh(δ * L * ) 0 1 1 0 - 2 1-e -2L * L * 0 e L * -2 [cosh(δ * (L * -)) -1]d I, (107)
where L * = s -r, because in this case, (Se -M χ -1 ) -is the zero matrix. Since cosh(s) ≥ 1 and sinh(s) ≥ 0 for all s ≥ 0, we conclude that in this example, we have F+ c = Fc and F - c = -F c at all pairs (r, s) where L * > 0. Then with the preceding F+ c and F - c , the closed loop system from Theorem 1 will be globally exponentially stable to 0 on R n provided that the Γ from ( 9) is Schur stable.

Next, observe that in this case, the functions (35) are

W (i) = - ti+1 ti Φ D (t i+1 , )e -+σ( ) d 2 1-e -2ν I + Φ D (t i+1 , t i ), (108)
where Φ D (t, ) = Φ M -δ1 (t, ) -Φ M (t, ) as before. Then, in this particular case where δ 1 = δ and δ B = 0, we obtain

W (i) = e (ti+1-ti)(I-δ) -e (ti+1-ti)I -2 1-e -2ν
ti+1 ti e (ti+1-)(I-δ) -e (ti+1-)I e ti-d (109) and therefore also

W (i) = e ν(I-δ) -e νI - 2
1-e -2ν ν 0 e (ν-)(I-δ) -e (ν-)I e -d I.

(110) Therefore, we deduce from [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF] that (111)

||W (i)|| ≤ ν ν 0 (ν -)|| δ||e (ν-)(1+|| δ||) e -d +ν|| δ||e ν(1+|| δ||) = ν || δ|| ν 0 (ν -)e ν(1+|| δ||)-(2+|| δ||) d +ν|| δ||e ν(1+|| δ||) = ν || δ||e ν(1+|| δ||) ν ν 0 e -(2+|| δ||) d - ν 0 e -(2+|| δ||) d + ν|| δ||e ν(1+|| δ||)
Hence, by using ( 17), we conclude that the closed loop system from Theorem 1 satisfies the desired exponential stability property if the right side of (111) is in (0, 1), by the second part of our proof of Theorem 1. This is more restrictive than our Schur stability condition from Theorem 1, because for instance, if δ * = 0.36 and ν = 0.72 in the preceding example, then simple Mathematica calculations imply that the eigenvalues of the matrix Γ in (9) are {0.969621, -0.767802, 0.246164, -0.170381} (which means that the Schur stability requirement is satisfied) but that the right side of (111) is 1.0879. If instead we have δ * = 1 and ν = 0.36, then Γ has eigenvalues {0.95009, -0.688578, 0.292117, -0.178081} but the right side of (111) is 1.11283. It follows that Theorem 1 gives a better stability condition than the one that we would have obtained by only applying [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF].

In Figs. 1-2, we graphically illustrate the preceding result, using plots that we generated using the Mathematica program. In Fig. 1, we show the spectral radius (i.e., the largest of the absolute values of the four eigenvalues) of the matrix Γ from Theorem 1, as a function of the disturbance parameter δ * and sample rate ν, for the choices from the preceding example. It illustrates a range of δ * and ν values for which the eigenvalues of Γ remain in (-1, 1) to satisfy our Schur stability requirement from Theorem 1. In Fig. 2, we plot the norms of closed loop solutions obtained from applying Theorem 1 to the previous example with δ * = 0.36 and ν = 0.72. Fig. 2 shows rapid convergence of the solutions to zero, and it illustrates the effects of the sampling in our control (which produces cusps in the plots of the norms at the sample times), and so also illustrates Theorem 1.

Conclusions

We provided new results on feedback stabilization for dynamics with sampling and uncertainty. We allowed additive uncertainty on the right side of the systems, in addition to uncertainty in both vector fields defining our timevarying linear systems. Our methods combined positive systems approaches with new matrix inequalities that estimate fundamental solutions for time-varying linear systems that contain uncertain coefficient matrices. Our matrix inequality estimations are of independent interest, because of the difficulty in estimating fundamental matrix solutions for time-vayring linear systems that contain uncertainties in their coefficients. Our example illustrated the potential advantages of our new stabilization approach. In our future work, we aim to extend our work to nonlinear input delayed systems having right sides that include Metzler matrices.

Appendix A1: Key Lemmas

We first provide two lemmas that we used in our proofs of our transition matrix estimation theorems. The first one will use the fact that for the matrix (A.

2)

The following then follows from simple calculations:

Lemma A.1 For all matrices α ∈ R n×n and β ∈ R n×n ,

  i . We use analogous notation for matrices. Let || • || denote the standard Euclidean 2-norm of vectors and matrices. We also use the notation |M | ∞ = [||m ij || ∞ ] when M = [m ij ] is a bounded matrix valued function, where ||f || ∞ for a function f is the usual sup norm, and which we denote by |M | when M is a constant matrix, so |M | = [|m ij |].

  s| and |Φ Q (t, s)| ≤ e |Q|∞|t-s| (40) hold for any bounded piecewise continuous matrix valued function Q : [0, +∞) → R n×n and all s ≥ 0 and t ≥ 0 (e.g., because of the Peano-Baker formula from [18, Appendix C]), we obtain the bound

2 e 2 e 2 e 2 .

 2222 ) Also, since e (M ± δ) = e M e ± δ , we havee (M +δ) = e (1+δ * ) +e (1-δ * ) (1+δ * ) -e (1-δ * ) 2 e (1+δ * ) -e (1-δ * ) 2 e (1+δ * ) +e (1-δ * ) 2(104)ande (M -δ) = e (1-δ * ) +e (1+δ * ) (1-δ * ) -e (1+δ * ) 2 e (1-δ * ) -e (1+δ * ) (1-δ * ) +e (1+δ * ) (105)Consequently, ξ and ξ from (5) are ξ(r, s) = e r-s (cosh(δ * (r -s)) -1)I and ξ(r, s) = -e r-s sinh(δ * (r -s))

2 + 2 +

 22 and so also||W (i)|| = ν || δ||e ν(1+|| δ||) ν 1-e -ν(2+|| δ||) 2+|| δ|| + e -ν(2+|| δ||) ν 2+|| δ|| + -1+e -ν(2+|| δ||) (2+|| δ||) ν|| δ||e ν(1+|| δ||) = ν || δ||e ν(1+|| δ||) ν 2+|| δ|| + -1+e -ν(2+|| δ||) (2+|| δ||) ν|| δ||e ν(1+|| δ||) ,where ν = 2 1-e -2ν . This provides the condition ||W (i)|| ≤ 2δ * e ν(1+δ * ) (1-e -2ν )(2+δ * ) ν + e -ν(2+δ * ) -1 2+δ * + νδ * e ν(1+δ * ) .

Fig. 1 .Fig. 2 .

 12 Fig. 1. Spectral Radius Σ(Γ) of Γ as Function of Disturbance Parameter δ * and Sample Rate ν, Illustrating Range of δ * and ν Values for which our Schur Condition on Γ is Satisfied

  R n×n be a matrix valued function that admits an invertible matrix P ∈ R n×n and a matrix valued function F : [0, +∞) → R n×n such that F (t) is Metzler for each t ≥ 0 and such that P A(t)P -1 = F (t) for each t ≥ 0. Let the piecewise continuous function ρ : [0, +∞) → R n×n and the matrices ρ ∈ R n×n and ρ ∈ R n×n

	be such that
	ρ ≤ ρ(t) ≤ ρ	(21)
	for all t ≥ 0. Choose the functions

holds for all matrices γ ∈ R n×n and ρ ∈ R n×n .

We also use the following consequence of the uniqueness of solutions property for (1) with F = R, which is shown by checking that H(t, t 0 ) = LΦ A (t, t 0 )L -1 satisfies H(t 0 , t 0 ) = I and ∂ ∂t H(t, t 0 ) = RH(t, t 0 ) for all t ≥ 0 and t 0 ≥ 0: Lemma A.2 Let A : [0, +∞) → R n×n . Let L ∈ R n×n be an invertible matrix and let R = LAL -1 . Then

for all t 0 ∈ [0, +∞) and t ∈ [0, +∞).

Appendix A2: Proof of Theorem 2

We consider M(t) and ∆(t) such that M(t) is Metzler for all t ≥ 0 and such that there is a ∆ satisfying the requirements of Theorem 2. Let

,

and

Then Ω(t) ≤ Ω(t) ≤ Ω(t). Here and in the sequel, inequalities and equalities should be understood to hold for all t ≥ 0 unless otherwise indicated. It follows from the comparison lemma that

According to Lemma A.1,

and

Then, from (A.8) and Lemma A.2, we have

Hence, Lemma A.1 also gives

where

We straightforwardly deduce from considering the upper left and then the upper right submatrices in (A.12) that

are satisfied. Adding the corresponding left, middle, and right sides of the inequalities in (A. [START_REF] Mazenc | Design of continuous-discrete observers for time-varying nonlinear systems[END_REF] gives the desired conclusion of the theorem.

Appendix A3: Proof of Theorem 3

From (21), it follows that P + ρ ≤ P + ρ ≤ P + ρ and -P -ρ ≤ -P -ρ ≤ -P -ρ. We deduce from the formula P ρ = (P ρ) + -(P ρ) -that P + ρ-P -ρ ≤ P ρ ≤ P + ρ-P -ρ is satisfied. Therefore, we get P + ρ -P -ρ (P -1 ) + ≤ P ρ(P -1 ) + ≤ P + ρ -P -ρ (P -1 ) + (A.15)

and -P + ρ -P -ρ (P -1 ) - ≤ -P ρ(P -1 ) -≤ -P + ρ -P -ρ (P -1 ) -.

(A.16)

By adding the inequalities in (A.15)-(A.16), we obtain

where ς and ς are defined as follows:

ς = P + ρ -P -ρ (P -1 ) + -P + ρ -P -ρ (P -1 ) -and ς = P + ρ -P -ρ (P -1 ) + -P + ρ -P -ρ (P -1 ) -.

Then, since the formula (25) gives M = F + ς+ , we get

where M is from (25) and ∆ = ς + -P ρP -1 . Then M (t) is Metzler for all t ≥ 0, and (A.17) gives 0 ≤ ∆ ≤ ς + -ς.

Next, Theorem 2 ensures that

for all t ≥ t 0 ≥ 0. According to Lemma A.2, these inequalities in combination with (A.18) give µ(t, t 0 ) ≤ P Φ A+ρ (t, t 0 )P -1 = Φ M -∆ (t, t 0 ) ≤ µ(t, t 0 ) for all t ≥ t 0 ≥ 0, where

Consequently, (P -1 ) + µ(t, t 0 ) ≤ (P -1 ) + P Φ A+ρ (t, t 0 )P -1 ≤ (P -1 ) + µ(t, t 0 ) (A.21) and -(P -1 ) -µ(t, t 0 ) ≤ -(P -1 ) -P Φ A+ρ (t, t 0 )P -1 ≤ -(P -1 ) -µ(t, t 0 ). (A.22) Adding (A.21)-(A.22) and recalling that R = P -1 , we get

From these inequalities, we deduce that

and This appendix provides a proof of Theorem 4. The proof is based on discrete time analogs of our continuous time transition matrix estimation theorems that are of independent interest, and which we prove first. Consider matrices M i ∈ R n×n , ∆ i ∈ R n×n , and ∆i ∈ R n×n such that M i ≥ 0 and 0 ≤ ∆ i ≤ ∆ i (A.25) for all i ∈ Z ≥0 . Let us find an upper and a lower bound for

which is the state transition matrix for X i+1 = (M i -∆ i )X i , meaning X j+i = E i,j X i when j ≥ i with j ≥ 1. We use the matrices

and κ j,i = Ω j-1+i .....Ω i+1 Ω i ,and κ j,i = Ω j-1+i .....Ω i+1 Ω i . According to Lemma A.1 from Appendix A1,

holds for all i and j, where P is the (2n) × (2n) block matrix with each block being I ∈ R n×n , D i,j = (M j-1+i + ∆ j-1+i ).....(M i+1 + ∆ i+1 )(M i + ∆ i ) and E i,j defined in (A.26) and

where

and

(A.30)

Notice that since ∆ i ≥ 0 for all i, the inequalities κ j,i ≤ κ j,i ≤ Ω j-1+i .....Ω i+1 Ω i (A.31) are satisfied with

As an immediate consequence of (A.28), (A.29), and (A.31), the inequalities

holds. We deduce from Lemma A.1 that

Di,j +Ei,j 2 We now use the preceding observations to prove Theorem 4. To this end, notice that the preceding analysis and the choices (59) of S j and Sj give S j ≤ (M -∆ j-1+i )......(M -∆ i ) ≤ S j (A.36) for all j ≥ 1. Hence, the theorem follows by the order preservation property of limits.