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

Michael Malisoff
email: malisoff@lsu.edu.

Gonzalo Robledo
email: grobledo@uchile.cl.

Stability and Robustness Analysis for a Multi-Species Chemostat Model with Uncertainties

Keywords: Bioreactors, nonlinear, stability, robustness

We prove stability and robustness results for chemostat models with one substrate, an arbitrary number of species, a constant dilution rate, and constant inputs of the species. Unlike all previous works, we prove input-tostate stability under uncertainties in important cases where the controls are the input nutrient concentration and the species inputs. Our assumptions ensure global asymptotic stability for an equilibrium, which can allow persistence of multiple species, when the uncertainties are zero. We allow arbitrarily large bounds on the uncertainties in the species dynamics, and equilibria that can be in the boundary of the state space.

I. INTRODUCTION

The chemostat is a mathematical model and a laboratory device that is used for the continuous culture of microorganisms. Since its introduction in [START_REF] Monod | La technique de culture continue, théorie et applications[END_REF] and [START_REF] Novick | Description of the chemostat[END_REF], it has been studied extensively, because of its vital role in ecology and microbiology as an ideal representation of microorganism growth, natural environments such as lakes, and wastewater treatment processes [START_REF] Dikshitulu | Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications[END_REF]. It is used in industrial applications that are of compelling engineering interest. See [START_REF] Smith | The Theory of the Chemostat. Dynamics of Microbial Competition[END_REF].

This has motivated our ongoing work (begun in [START_REF] Gouzé | Feedback control for nonmonotone competition models in the chemostat[END_REF], [START_REF] Grognard | Polytopic Lyapunov functions for persistence analysis of competing species[END_REF], [START_REF] Lobry | Persistence in ecological models of competition for a single resource[END_REF], [START_REF] Mazenc | Stabilization in a chemostat with sampled and delayed measurements[END_REF], [START_REF] Mazenc | Global output feedback stabilization of a chemostat with an arbitrary number of species[END_REF], [START_REF] Mazenc | Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement[END_REF], [START_REF] Mazenc | Stability and stabilization for models of chemostats with multiple limiting substrates[END_REF], [START_REF] Mazenc | Further results on stabilization of periodic trajectories for a chemostat with two species[END_REF], and [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF]) on methods to ensure desired asymptotic behaviors in chemostats, including the coexistence of multiple competing species, convergence to equilibria, or delay compensation. As noted in [START_REF] Smith | The Theory of the Chemostat. Dynamics of Microbial Competition[END_REF], the classical model of competition in the chemostat is

   ṡ(t) = D[s in -s(t)] - n i=1 Y -1 i µ i (s(t))x i (t) ẋi (t) = x i (t)µ i (s(t)) -Dx i (t), i = 1, . . . , n (1)
where n ≥ 2 microbial species (with concentrations x 1 , . . . , x n ) compete for a nutrient with concentration s.

The positive constants D and s in are the dilution rate and input nutrient concentration, respectively, and Y i is a positive yield constant related to the conversion of the substrate into biomass for each i. The µ i 's for i = 1, . . . , n are strictly increasing, satisfy µ i (0) = 0, and are assumed to be continuously differentiable; they describe the consumption of the nutrient by species i. The model assumes that the growth of species i is proportional to the nutrient consumption.

Well known results (e.g., [START_REF] Hardin | Competitive exclusion principle[END_REF] and [START_REF] Smith | The Theory of the Chemostat. Dynamics of Microbial Competition[END_REF]) imply that if the preceding conditions hold and 0 < µ

-1 n (D) < µ -1 n-1 (D) < . . . < µ -1 2 (D) < µ -1 1 (D) < s in , then the following com- petitive exclusion principle holds: lim t→+∞ s(t) = µ -1 n (D), lim t→+∞ x n (t) = Y n [s in -µ -1 n (D)],
and

lim t→+∞ x i (t) = 0 if 1 ≤ i ≤ n -1.
The preceding conditions imply that only the nth species persists, because it only needs the lowest nutrient concentration.

However, multiple species are often observed to persist in chemostats with one substrate, which motivated many works on ways to explain coexistence in chemostats. There are several approaches to explaining coexistence in bioreactor models, including models whose qualitative behavior and analytical treatment are considerably more complex than (1) and we refer to [START_REF] Gouzé | Feedback control for nonmonotone competition models in the chemostat[END_REF], [START_REF] Hsu | On a system of reaction-diffusion equations arising from competition in an unstirred chemostat[END_REF], [START_REF] Lenas | Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate[END_REF], [START_REF] Lobry | Persistence in ecological models of competition for a single resource[END_REF], [START_REF] Mazenc | Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement[END_REF], [START_REF] Mazenc | Further results on stabilization of periodic trajectories for a chemostat with two species[END_REF], [START_REF] Nie | Coexistence of an unstirred chemostat model with Beddington-De Angelis functional response and inhibitor[END_REF], and [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF] without claiming completeness. Nevertheless, we will focus our attention on the approach from [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF], which introduces a model with constant inputs x 0 i ≥ 0 of the ith competing species (for i = 1, . . . , n) described by the system

   ṡ(t) = D[s in -s(t)] - n i=1 Y -1 i µ i (s(t))x i (t) ẋi (t) = x i (t)µ i (s(t)) + D[x 0 i -x i (t)], 1 ≤ i ≤ n. (2)
In [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF], the authors obtained sufficient conditions ensuring the coexistence of multiple species. The work [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF] used polytopic Lyapunov functions, which were also used in [START_REF] Grognard | Polytopic Lyapunov functions for persistence analysis of competing species[END_REF]. However, it is well known that chemostats can contain uncertainties (e.g., unmodeled features, or uncertainties in the input concentrations, which are common in applications). Therefore, an even more accurate model than (2) is

       ṡ(t) = D[s in -s(t)] - n i=1 µ i (s(t))x i (t) + δ 0 (t) ẋi (t) = x i (t)µ i (s(t)) + D[x 0 i -x i (t)] + δ i (t), 1 ≤ i ≤ n, (3)
where the µ i 's are as before, and the unknown measurable essentially bounded functions δ i : [0, +∞) → [d i , di ] for i = 0, 1, . . . , n represent uncertainties and have known upper and lower bounds di and d i , respectively, and where we used a change of coordinates (based on a scaling of the x i 's and x 0 i 's) to remove the Y i 's. Two of our assumptions will be that d 0 > -Ds in and d i ≥ -Dx 0 i for i = 1, 2, . . . , n; see Section II for our assumptions. Therefore, all solutions of (3) with initial states (s(0), x(0)) in X = (0, +∞) n+1 remain in X for all t ≥ 0, so (3) has the state space X , and (3) will be the subject of this paper.

In the next section, we provide our theorem for (3), which uses s in and the x 0 i 's as controls. Our work is novel in its use of the model (3), which we believe has not been studied in the presence of nonzero uncertainties. Our new Lyapunov construction is the key ingredient for proving valuable inputto-state stability (ISS) robustness properties, which cannot be deduced from the polytopic Lyapunov functions from [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF]. Also, the equilibria that we stabilize are in the boundary of X when at least one x 0 i is 0, so the present paper covers a broad class of equilibria. Hence, this work provides a new theoretical result with valuable implications for future real time applications to bioprocess engineering.

II. ASSUMPTIONS, DEFINITIONS, AND MAIN RESULT

We will prove ISS properties for the dynamics for the error E(t) = (s(t) -s * , x(t) -x * ) with respect to the disturbance vector δ = (δ 0 , δ 1 , . . . , δ n ), for a large class of possible equilibrium points E * = (s * , x * ), where (s, x) is the state of (3), x * = (x 1 * , . . . , x n * ), and x = (x 1 , . . . , x n ). The ISS framework is used extensively in engineering; see [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] for ISS for systems without state constraints. To allow state constraints, we use a variant of the usual ISS property.

To explain this variant, we first need two definitions. Let K ∞ be the set of all continuous strictly increasing unbounded functions γ : [0, +∞) → [0, +∞) such that γ(0) = 0; and let KL be the set of all continuous functions

β : [0, +∞) × [0, +∞) → [0, +∞) such that (i) for each t ≥ 0, the function f (s) = β(s, t) is of class K ∞ and (ii)
for each s ≥ 0, the function g(t) = β(s, t) is nonincreasing and satisfies lim t→+∞ g(t) = 0. By ISS of a system of the form Ė(t) = F(E(t), δ(t)) with respect to a pair (D, S), we mean that there exist β ∈ KL and γ

∈ K ∞ such that |E(t)| ≤ β(|E(0)|, t) + γ(|δ| [0,t]
) holds for all t ≥ 0, all solutions E(t) of this system that have initial states E(0) ∈ S, and all measurable essentially bounded functions δ : [0, +∞) → D. Here and in the sequel, |•| is the Euclidean norm, and

| • | [0,t] (resp., | • | ∞ ) is the essential supremum over [0, t] for all t ≥ 0 (resp., over [0, +∞)). Assume: Assumption 1: The µ i 's in (3) have the Monod form µ i (s) = mis ai+s for i = 1, 2, . . . , n, (4)
where the m i 's and a i 's are known positive constants. Assumption 2: The constants s * > 0 and s in are such that

µ i (s * ) < D for i = 1, 2, . . . , n and
s in = s * + n i=1 µi(s * )x 0 i D-µi(s * ) , (5)
the x 0 i 's are nonnegative constants, and 0 < D < µ n (s in ). From Assumption 2, we obtain µ n (s * ) < µ n (s in ), so since µ n is strictly increasing, it follows that s * ∈ (0, s in ). We can always satisfy Assumption 2 for all constants D ∈ (0, m n ), by first fixing s * > 0 such that µ i (s * ) < D for all i, and then picking the x 0 i 's to be large enough so that D < µ n (s in ) and s in > 0, i.e., we view s in as a constant control. By the symmetry of the system (3) in its components x i , we can replace the condition D < µ n (s in ) by the condition that D < µ i (s in ) for any i, by renumbering the species. By Assumption 2, it follows that when the δ i 's in (3) are 0, the system (3) has the equilibrium E * = (s * , x * ), where

x i * = Dx 0 i D-µi(s * ) for i = 1, ..., n. (6)
We can allow many x i * 's through different choices of x 0 i 's, i.e., we also use the x 0 i 's as controls. Since the x 0 i 's are nonnegative, we have E * ∈ [0, +∞) n+1 ; and E * ∈ (0, +∞) n+1 when the x 0 i 's are all positive. Our assumptions on the unknown measurable essentially bounded functions δ i in (3) are as follows, where P = {i ∈ {1, 2, . . . , n} :

x 0 i > 0}: Assumption 3: We have δ(t) ∈ [d 0 , d0 ] × . . . × [d n , dn ]
for almost all t ≥ 0, where the known constants d i and di ≥ 0 are such that Ds in +d 0 > 0, d0 < 0.5Ds * , and Dx 0 i +d i > 0 for all i ∈ P, and d i = 0 for all i ∈ {1, 2, . . . , n} \ P.

We can prove the following result, where (s(t), x(t)) is the state of (3):

Theorem 1: If Assumptions 1-3 hold, then for all constants x > 0 and s ≥ s in , the dynamics for the error vector E = (s, x) -E * have the ISS property with respect to (D, S)

with the disturbance set D = [d 0 , d0 ] × . . . × [d n , dn ] and S = {E : E + E * ∈ (0, s] × (0, +∞) n-1 × (x, +∞)}.
Before discussing our proof, we make several remarks about the novelty and value of our theorem.

Remark 1: Our choice of S in Theorem 1 corresponds to the requirement that (s(0), x(0)) ∈ (0, s] × (0, +∞) n-1 × (x, +∞). However, since s ≥ s in and x > 0 are arbitrary, we conclude that when the δ i 's are zero, all solutions (s(t), x(t)) of ( 3) starting in X = (0, +∞) n+1 remain in X at all positive times and satisfy lim t→+∞ (s(t), x(t)) = (s * , x * ). This ensures persistence of the ith species for all i ∈ P (and lim t→+∞ x i (t) = 0 for all i ∈ {1, . . . , n} \ P).

Remark 2: We do not restrict the values of di ≥ 0 for i ≥ 1, so Theorem 1 ensures ISS under arbitrarily large sup norms on the δ i 's for i ≥ 1. A key ingredient in our proof is a Lyapunov-like function V and a function T 3 ∈ K ∞ such that V satisfies the usual Lyapunov positive definiteness and decay conditions along all solutions E(t) of the error dynamics for all t ≥ T 3 (|E(0)|). Using V instead of the polytopic Lyapunov functions from [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF] allows us to prove key ISS results that were beyond the scope of [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF].

III. SKETCH OF PROOF OF THEOREM 1 A. Preliminary State Bounds

Since (3) is forward complete on X = (0, +∞) n+1 , we can first fix any solution (s(t), x(t)) of (3) all of whose components are positive for all t ≥ 0 for which E(0) ∈ S. Set s = s + ( d0 /D) and

(s, x) = (s -s * , x -x * ). (7)
Then Assumption 3 implies that s(t) ≤ s for all t ≥ 0. We next produce functions T i ∈ K ∞ for i = 1, 2, 3, whose class K ∞ properties will be used later to build an ISS estimate that is valid for all times t ≥ 0, using three lemmas. Our first lemma is: Lemma 1: If Assumptions 1-3 hold, then there is a T 1 ∈ K ∞ such that s(t) ≤ s in + ( d0 /D) for all t ≥ T 1 (|s(0)|).

Proof: (Sketch.) If there is a t l ≥ 0 such that s(t l ) ≤ s in + ( d0 /D), then for all t > t l , we have s(t) ≤ s in + ( d0 /D). Next, consider the case where s(0) > s in +( d0 /D). Consider any t ≥ 0 such that min ∈[0,t] s( ) > s in +( d0 /D). For any such t, we also have max ∈[0,t] ṡ( ) < 0. Then since D < µ n (s in ), and since µ n is nondecreasing and Dx 0 n +d n ≥ 0 and s * < s in ≤ s( ) for all ∈ [0, t], we deduce from (3) that ẋn ( ) ≥ 0 and so also x n ( ) ≥ x n (0) ≥ x for all ∈ [0, t], and ṡ( ) ≤ -µ n (s( ))x n ( ) for all ∈ [0, t], so

tµ n (s in )x ≤ t 0 µ n (s in )x n ( ) d ≤ s(0) -s(t) ≤ s(0) -s * ≤ |s(0)|. (8)
C = 2 s in + n i=1 x 0 i + 1 D n i=0 di (9)
and fix any constant λ 1 > 1. We prove:

Lemma 2: If Assumptions 1-3 hold, then there is a T 2 ∈ K ∞ such that σ(t) ≤ λ 1 C holds for all t ≥ T 2 (|E(0)|).
Proof: (Sketch.) By (3), we have σ(t) ≤ 0.5CD -Dσ(t) for all t ≥ 0. It follows that σ(t) ≤ u(t) for all t ≥ 0, where u is the solution of the initial value problem

u(t) = CD 2 -1 2 Du(t), u(0) = σ(0). (10)
Since

λ 1 > 1, we deduce that σ(t) ≤ u(t) ≤ λ 1 C for all t ≥ 0 if σ(0) ∈ (0, λ 1 C].
Hence, there exists a function T 2 ∈ K ∞ such that σ(t) ≤ λ 1 C for all t ≥ T 2 (σ(0)) when σ(0) > λ 1 C. Therefore, the lemma will follow once we choose a function

T 2 ∈ K ∞ such that T 2 (|E(0)|) ≥ T 2 (σ(0))
holds for all solutions of (3) such that σ(0) > λ 1 C. To find T 2 , first note that our formulas for σ and s in give

σ(0) = s(0) + n i=1 xi (0) + s * + n i=1 Dx 0 i D-µi(s * ) ≤ |s(0)| + n i=1 |x i (0)| + s in + n i=1 x 0 i . (11)
We next consider two cases. Case 1:

If |E(0)| ≤ (s in + x 0 1 + . . . + x 0 n )/(n + 1)
, then the inequality in [START_REF] Mazenc | Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement[END_REF] gives σ(0) ≤ λ 1 C, so this case does not produce any restriction on the allowable values of

T 2 . Case 2: If |E(0)| > (s in + x 0 1 + . . . + x 0 n )/(n + 1
), then we use the fact that (11) gives

T 2 ((n + 1)|E(0)| + s in + x 0 1 + . . . + x 0 n ) ≥ T 2 (σ(0)). Setting a * = (s in + x 0 1 + . . . + x 0 n )/(n + 1)
, we can then take

T 2 ( ) = T 1 ( ) +    ( /a * )T 2 2(n + 1)a * , 0 ≤ ≤ a * T 2 (n + 1)( + a * ) , > a *
to satisfy our requirements. We next fix any constant λ 2 ∈ (0, 1) and set

s λ = λ 2 min        s * , Ds in + d 0 D + λ 1 C n i=1 mi ai        (12)
and x iλ = λ 2 min x i * , x 0 i + (d i /D) for all i and prove: Lemma 3: If Assumptions 1-3 hold, then there is a function T 3 ∈ K ∞ such that s(t) ≥ s λ and x i ≥ x iλ hold for all t ≥ T 3 (|E(0)|) and all i ∈ P.

Proof: (Sketch.) For all t ≥ T 2 (|E(0)|), Lemma 2 gives x i (t) ≤ λ 1 C for all i ∈ {1, 2, . . . , n}. Hence, for all t ≥ T 2 (|E(0)|), (3) and our formula (4) for the µ i 's give

ṡ(t) ≥ D(s in -s(t)) -λ 1 C n i=1 mis(t) ai + d 0 (13)
and

ẋi (t) ≥ -Dx i (t) + Dx 0 i + d i for all i ∈ P. ( 14
)
The right side of ( 13) is bounded below by

(Ds in + d 0 )(1 - λ 2 ) > 0 if t is such that s(t) ≤ s λ .
Also, for each i ∈ P, the right side of ( 14) is bounded below by

(1-λ 2 )(Dx 0 i +d i ) > 0 if t is such that x i (t) ≤ x iλ .
Hence, for each t 0 ≥ 0 such that s(t 0 ) ≥ s λ , we have s(t) ≥ s λ for all t ≥ t 0 ; and for each i ∈ P, and for each t 0 ≥ 0 such that x i (t 0 ) ≥ x iλ , we have x i (t) ≥ x iλ for all t ≥ t 0 . Therefore', it suffices to choose T 3 ∈ K ∞ such that: (i) If s(0) < s λ , then s(t) ≥ s λ for some t ∈ [0, T 3 (|E(0)|)] (which implies that s(t) ≥ s λ for all t ≥ T 3 (|E(0)|), by the preceding argument) and (ii) for each i ∈ P such that x i (0) < x iλ , we have x i (t) ≥ x iλ for some t ∈ [0, T 3 (|E(0)|)] (which implies that x i (t) ≥ x iλ for all t ≥ T 3 (|E(0)|), also by the preceding argument).

To find T 3 ∈ K ∞ , first note that if we pick any constant

T L > 1 1-λ2 max s λ Dsin+d 0 , max x iλ Dx 0 i +d i : i ∈ P
then the Fundamental Theorem of Calculus and the positiveness of s(t) and the x i (t)'s imply that: (A) If s(0) < s λ , then s( ) ≥ s λ for some ∈ [0, T L ], hence for all ≥ T L and (B) if i ∈ P is such that x i (0) < x iλ , then x i ( ) ≥ x iλ for some ∈ [0, T L ], hence for all ≥ T L . Conditions (A)-(B) follow because s( ) ≥ s(0) + (Ds in + d 0 )(1 -λ 2 ) for all such that max r∈[0, ] s(r) < s λ , and

x i ( ) ≥ x i (0) + (1 -λ 2 )(Dx 0 i + d i ) for all such that max r∈[0, ] x i (r) < x iλ , so ≤ T L . On the other hand, if |E(0)| ≤ T M where T M = (1 -λ 2 ) min{s * , min{x i * : i ∈ P}}, then s(0)-s * ≥ -|E(0)| ≥ (λ 2 -1)
s * ; and for all i ∈ P, we have

x i (0) -x i * ≥ -|E(0)| ≥ (λ 2 -1)x i * ; so s(0) ≥ λ 2 s * ≥ s λ and x i (0) ≥ λ 2 x i * ≥
x iλ for all i ∈ P, which imply that s(t) ≥ s λ and x i (t) ≥ x iλ hold for all t ≥ 0 and i ∈ P, by the previous paragraph. Hence,

T 3 (r) = T 2 (r) + (T L + T M ) r T M , 0 ≤ r ≤ T M T L + r, r > T M (15)
satisfies our requirements.

B. Representing the Error Dynamics

Let us introduce the functions and the constant

Γ(s) = D +
Then Γ(s(t)) ≥ Γ 0 for all t ≥ 0. By our formulas for s in and the x i * 's from ( 5) and ( 6), we have Ds in = Ds * + µ 1 (s * )x 1 * + . . . + µ n (s * )x n * and Dx 0 i = Dx i * -µ i (s * )x i * for all i ∈ {1, 2, . . . , n}. Hence, using (3) and the formulas for x i * from (6), and reorganizing terms gives:

               ṡ(t) = -Ds(t) + n i=1 [µ i (s * ) -µ i (s(t))]x i * - n i=1 µ i (s(t))x i (t) + δ 0 (t) ẋi (t) = [µ i (s(t)) -µ i (s * )]x i (t) + xi (t)µ i (s * ) -Dx i (t) + δ i (t), i = 1, ..., n. (17)
For all s > 0, we can use our formulas (4) for the µ i 's to check that Γ can be rewritten as

Γ(s) = D + n i=1 µi(s)-µi(s * ) s-s * x i * (18)
for all s = s * . Using the constants p i = D -µ i (s * ) (which are positive, by ( 5) in Assumption 2), we obtain

       ṡ(t) = -Γ(s(t))s(t)- n i=1 µ i (s(t))x i (t)+δ 0 (t) ẋi (t) = -p i xi (t) + c i x i (t)µ i (s(t)) s(t) s(t) + δ i (t), i = 1, ..., n , (19)
where c i = ai ai+s * for all i ∈ {1, ..., n}.

C. Construction of a Lyapunov-Like Functional

Let us define the C 1 function V by

V (E) = ν(s) + n i=1 1 ci Ψ i (x i ), where ν(s) = s -s * ln s+s * s * and Ψ i (x i ) = xi -x i * ln xi+xi * xi *
for all i ∈ P, and Ψ i (x i ) = x i for all i ∈ {1, . . . , n} \ P. By the chain rule, it follows that for all t ≥ T 3 (|E(0)|), the time derivative of V along the solutions of (19) satisfies

V (t) ≤ -N (E(t)) + N |δ| [0,t] ,
where

N (E(t)) = Γ 0 s2 (t) s(t) + n i=1 q i x2 i (t) xi(t) (20)
and where q i = p i /c i = (D -µ i (s * ))/c i and the formula

N = (n + 1) max s +s * s λ , max i ∈P 1 ci , max
Then Assumption 2 is satisfied with s in = 1.34412, and our formulas (6) for the x i * 's give

x 1 * = 1.29412 and x 2 * = 1.1. (22)
In this illustration, we will only use the uncertainty vectors δ(t) to model uncertainties in applying the constant input concentrations x 0 1 and x 0 2 (which may occur in applications, because it may be difficult to maintain the inputs x 0 i at constant levels), so we set δ 0 (t) = 0 and therefore can choose d 0 = d0 = 0, d 1 = -0.39, d 2 = -0.21, and any constants d1 ≥ 0 and d2 ≥ 0 to satisfy all of our assumptions.

We simulated (3) using the command NDSolve in Mathematica, with the preceding choices of the parameters, and the disturbance vector δ(t) = (δ 0 (t), δ 1 (t), δ 2 (t)) = (0, -0.1 sin(t), 0.1 cos(t)). We report our results in Fig. 1, with the initial state (s(0), x 1 (0), x 2 (0)) = (0.2, 0.1, 1), and then with the initial state (s(0), x 1 (0), x 2 (0)) = (1.3, 0.2, 0.1). The figure shows rapid convergence towards an oscillatory steady state, with a deviation from the equilibrium (s * , x 1 * , x 2 * ) = (0.5, 1.29412, 1.1) that can be explained by the presence of the uncertainties δ 1 and δ 2 , and therefore helps illustrate our theory.

V. CONCLUSIONS

We solved a key input-to-state stabilization problem for a chemostat model with one limiting substrate, an arbitrary number of competing species, a constant dilution rate, and uncertainties, using constant inputs of the species. In the special case where the uncertainties are zero, this implies that all solutions with initial states in (0, ∞) n+1 remain in (0, ∞) n+1 at all future times and asymptotically converge to an equilibrium, which corresponds to persistence of all species whose constant inputs are positive. This contrasts with the competitive exclusion principle, which does not consider the possibility of introducing positive constant inputs of the species. The uncertainties can represent unmodeled features that commonly occur in biotechnological applications, so our work has the potential to benefit the study of some robustness issues.

(

  ai+s * )(ai+s) , s(t) = s(t) -s * , xi (t) = x i (t) -x i * for i = 1,2, . . . , n, and Γ 0 = D + n i=1 aimixi * (ai+s * )(ai+s ) .

  2-3. Although (20) only holds for times t ≥ T 3 (|E(0)|), we can combine (20) with Gronwall's inequality to produce the final ISS estimate; see the appendix. IV. ILLUSTRATION Consider the system (3) with n = 2, D = 0.4, s * = 0.5, x 0 1 = 1, and x 0 2 = 0.55 and the growth functions µ 1 (s) = s 5+s and µ 2 (s) = s 2+s .

Fig. 1 .

 1 Fig. 1. Solution Components of (3) on Time Interval [0, 25]. Species x 1 (t) and x 2 (t) are Green and Red Curves, Respectively. Substrate s(t) is Blue Curve. Top Panel: Using Initial State (s(0), x 1 (0), x 2 (0)) = (0.2, 0.1, 1). Bottom Panel: Using Initial State (s(0), x 1 (0), x 2 (0)) = (1.3, 0.2, 0.1).

  Hence, t ≤ |s(0)|/(µ n (s in )x), so there is a t * ∈ [0, 2|s(0)|/(µ n (s in )x)] such that s(t * ) ≤ s in + ( d0 /D). Hence, we can choose T 1 (r) = 2r/(µ n (s in )x).Set σ(t) = s(t) + x 1 (t) + . . . + x n (t) for all t ≥ 0 and

Partially supported by the Regional Program MATH-AmSud STADE (Mazenc and Robledo) and by NSF Grant 1408295 (Malisoff).

Mazenc is with EPI DISCO Inria-Saclay,

APPENDIX: LAST PART OF PROOF OF THEOREM 1

To convert (20) into an ISS estimate for all t ≥ 0, first note that we can use our lemmas to find a γ ∈ K ∞ such that Γ 0 s2 (t)

and therefore also

along all trajectories of the E dynamics starting in our set S of initial states from the statement of our theorem, and for all t ≥ T 3 (|E(0)|). One method for finding γ is as follows. First, let O be the set of all points (s, x1 , . . .

for all i ∈ P, and (iii) x ∈ (0, λ 1 C] for all i ∈ {1, . . . , n} \ P.

Next, pick a function γ 0 ∈ K ∞ such that

Hence, we can use Lemma 2 to obtain

for all t ≥ T 3 (|E(0)|). Therefore, we can choose

Hence, combining (20) and (A.1) gives (A.2), so standard ISS arguments [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] provide β 0 ∈ KL and γ 0 ∈ K ∞ such that

holds for all t ≥ T 3 (|E(0)|). Then the structure of V provides functions β 1 ∈ KL and γ 1 ∈ K ∞ such that

To extend (A.6) to obtain an ISS estimate on [0, +∞), first note that the structure of the E dynamics [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF], combined with our bounds on the µ i 's and Γ and the Lipschitzness of the µ i 's, provide a constant L (that is independent of the choice of the solution) such that

holds for all ∈ [0, T 3 (|E(0)|)]; this can be done by rewriting x i (t) in [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF] in the form xi (t) + x i * . Integrating (A.7) over [0, t] for any t ∈ [0, T 3 (|E(0)|)] and applying the Fundamental Theorem of Calculus to E gives

We now apply Gronwall's inequality [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] to |E| to get

for all t ∈ [0, T 3 (|E(0)|)], by applying the triangle inequality to the terms in curly braces in (A.9). The ISS estimate now follows from adding the bounds (A.6) and (A.9), and using (A.


Jerome Weston
email: jwesto3@lsu.edu

Michael Malisoff
email: malisoff@lsu.edu

Frederic Mazenc
email: frederic.mazenc@lss.supelec.fr.

Sequential Predictors under Time-Varying Delays: Effects of Delayed State Observations in Dynamic Controller

Keywords: Delays, robustness, time-varying systems I

In a 2016 IEEE Conference on Decision and Control paper, our team designed sequential predictors for timevarying linear systems with time-varying delays, to prove global exponential stabilization properties using a feedback control that is computed in terms of the state of the last sequential predictor. This allowed feedback delays of arbitrarily large sup norm in the original system. Here we provide a significant generalization to more challenging cases with arbitrarily large feedback delay bounds, and where, in addition, current values of the plant state are not available to use in the sequential predictors. We illustrate our work in a pendulum example.

 [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]require that the current state x(t) of the original systems be available to use in the predictors when the control is computed. This can be a limitation when the sequential predictor must be computed on a computer that is far from the physical system, which can cause delays in the transmission of the state measurements from the physical system to the dynamic controller. This can occur for small robotic vehicles whose controls are computed remotely (as in our team's work [START_REF] Mukhopadhyay | Collaborative autonomous surveys in marine environments affected by oil spills[END_REF] on marine robots) and is modeled by replacing x(t) in the first sequential predictor by x(t -τ (t)) with measurement delays τ .

Therefore, this work studies the combined effects of (i) feedback delays h(t) in the original system and (ii) measurement delays τ (t) in the sequential predictors. As in our team's paper [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF], the new method in this work can allow arbitrarily long feedback delays h(t) in the original system, but [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] did not allow delays τ (t) in the state values in the sequential predictors, and in addition, we illustrate below how our new method can lead to a smaller number of required sequential predictors than were required by [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]. Hence, this work can benefit engineering systems that contain measurement delays, and reduce the computational burden relative to [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF].

II. MAIN RESULT

We study systems of the form

ẋ(t) = A(t)x(t) + B(t)u(t -h(t)), (1)
whose state x and control u are valued in R n and R , respectively, for any dimensions n and , where h : R → [0, +∞) is a known time-varying delay. The dynamics (1) will be interconnected with a dynamic controller whose right side uses delayed values x(t -τ (t)) of the state, where the delay τ : R → [0, +∞) may differ from h. We make these assumptions, which agree with those of [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] when τ = 0: Assumption 1: The nonnegative valued functions h and τ are C 1 and bounded from above by constants c h > 0 and c τ ≥ 0 respectively, their first derivatives ḣ and τ have finite lower bounds, ḣ and τ are bounded from above by constants l h ∈ (0, 1) and l τ ∈ (0, 1) respectively, and ḣ has a global Lipschitz constant n h > 0.

Assumption 2: The functions A and B in (1) are bounded and continuous on R, and there is a known bounded continuous function K : [0, +∞) → R ×n such that

ẋ(t) = [A(t) + B(t)K(t)]x(t) (2)
is uniformly globally exponentially stable on R n to 0.

We also assume throughout that the initial functions at time 0 are constant, e.g., x( ) is constant on (-∞, 0]. Assumption 1 can model many delays, e.g., using a denseness argument to closely approximate non-C 1 delays, including the approximation of discontinuous delays [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]. In terms of an integer m > 1 that we specify later, we set

Ω i (t) = t -i m h(t) and θ j (t) = Ω -1 m-j+1 (Ω m-j (t)) (3
) for all i ∈ {0, ..., m} and j ∈ {1, . . . , m}, and define

R 1 = θ1 and R i (t) = θi (t)R i-1 (θ i (t)), 2 ≤ i ≤ m. (4)
The preceding functions were also used in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF], and are used to define the coefficients in our predictors, and they exist by our upper bounds c h and l h ∈ (0, 1) from Assumption 1, which imply that the Ω i 's are strictly increasing and that their ranges are all of R, so the θ i 's are also C 1 and strictly increasing. The inverses in (3) can be computed numerically using standard programs, e.g., Mathematica [START_REF]The world's definitive system for modern technical computing[END_REF]. Setting

u c = n h c h (1-l h ) 2 + l h 1-l h , (5)
we then prove the following, where

2 1+ 1+ uc m m |A| ∞ 2 1+ uc m 1-lτ c τ + c h m(1-l h ) < 1, (6)
then we can construct positive constants µ 1 and µ 2 such that for all solutions x(t) of (1) in closed loop with the control

u(t) = K(Ω -1 m (t))z m (t), (7)
where z m is the last n components of the system

żi (t) = R i (t) A(G i (t))z i (t)+B(G i (t))u(Ω m-i (t)) + L i (t)E i (θ -1 i (φ(t))), 1 ≤ i ≤ m (8)
having the state space R nm , and with the choices

L i (t) = -I n -R i (t)A(G i (t)), φ(t) = t -τ (t), and
G i (t) = Ω -1 m (Ω m-i (t)), (9)
we have

|(x(t), E(t))| ≤ µ 1 |(x, E)| [-c h -cτ ,0] e -µ2t (10)
for all t ≥ 0, where

E(t) = (z 1 (t) -x(θ 1 (t)), z 2 (t) - z 1 (θ 2 (t)), . . . , z m (t) -z m-1 (θ m (t))).
Before proving Theorem 1, we provide several remarks on the novelty and value of the theorem.

III. REMARKS ON NOVELTY AND VALUE OF THEOREM 1

Remark 1: The z i dynamics in [START_REF] Ito | Construction of Lyapunov-Krasovskii functionals for networks of iISS retarded systems in small-gain formulation[END_REF] is called the ith sequential (sub)predictor for i = 1, 2, . . . , m and has state space R n , so [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] only depends on time and on the state of the last sequential predictor. Our system [START_REF] Ito | Construction of Lyapunov-Krasovskii functionals for networks of iISS retarded systems in small-gain formulation[END_REF] has the advantage that it does not depend on current x(t) values (because of the φ entering the E 1 (θ -1 1 (φ(t))) in the first sequential predictor in [START_REF] Ito | Construction of Lyapunov-Krasovskii functionals for networks of iISS retarded systems in small-gain formulation[END_REF]), so the current state x(t) is not needed when computing the z i 's at times t. Our system (8) also involves delays τ through the E i (θ -1 i (φ(t)))'s for i ≥ 2, since these delays are also needed for our global exponential stability proof. When τ = 0, we can satisfy (6) by choosing m large enough, so we allow arbitrarily large bounds c h on the delay h.

However, since there are no sequential predictors available to compensate for the delays τ (t) in ( 8), condition ( 6) is needed to put a restriction on the allowable c τ 's. By using the definition of e as a limit to obtain

e uc = lim m→∞ 1 + uc m m/uc uc , (11)
it follows that (6) holds if m is large enough and if

cτ √ 1-lτ ∈ 0, 1 2(1+e uc |A|∞) 2 . ( 12
)
Condition ( 12) is not too restrictive, since the τ 's from [START_REF] Ito | Construction of Lyapunov-Krasovskii functionals for networks of iISS retarded systems in small-gain formulation[END_REF] only arise from measurement delays, whereas the delay h(t) in (1) comes from both transmission delays from the mth sequential predictor to the physical plant, and from delays in the control actuation in [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF]. Hence, in practice, the feedback delay h would be much larger than τ in the sup norm. Remark 2: Theorem 1 is notable even in the special case where τ (t) is the zero function, since it can allow a smaller number m of sequential predictors than were allowed in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]. The work [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] required the delay τ in the x observations in the dynamic control [START_REF] Ito | Construction of Lyapunov-Krasovskii functionals for networks of iISS retarded systems in small-gain formulation[END_REF] to be the zero function and that the number m of sequential predictors satisfy

max 2, 4 b1 √ 2 + b 3 c h 1-l h < m, (13)
where, in terms of our notation from (5) and Assumption 1,

b 1 = 1+ 1 + uc m m |A| ∞ 1+ uc m m |A| ∞ , b 2 = 1 + 1+ uc m m |A| ∞ 2 , and b 3 = b 2 1+ uc m . (14)
In Section VI, we illustrate how (6) can lead to a significantly smaller number of required sequential predictors as compared with (13) when τ is the zero function. Our example also illustrates how we can allow nonzero τ 's. Remark 3: The estimate [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] implies that the (x, E) dynamics are uniformly globally exponentially stable to 0. Key features of Theorem 1 are (a) our allowing arbitrarily large bounds c h on h as well as measurement delays τ , (b) that our control [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] has no distributed terms, and (c) that only the bottom n components of ( 8) are needed to compute [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF].

IV. THREE LEMMAS TO PROVE THEOREM 1

Our proof of Theorem 1 will follow from three lemmas. The first lemma was shown in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF], and the second follows from simple calculations as were done in Part 1 of the proof of [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]Theorem 1] except with E i (θ -1 i (t)) in the dynamics for the error variable E in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] replaced by E i (θ -1 i (φ(t))) for each i, so we only prove our third lemma in this section.

Lemma 1: The θ i 's in (3) and the constant (5) satisfy

| θi ( ) -1| ≤ uc m and |θ -1 i ( ) -| ≤ c h 1-l h 1 m (15)
for all ∈ R and i ∈ {1, 2, . . . , m}.

Lemma 2:

The dynamics for the variable

E in Theorem 1 are            Ė1 (t) = R 1 (t)A(G 1 (t))E 1 (t) + L 1 (t)E 1 (θ -1 1 (φ(t))) Ėi (t) = R i (t)A(G i (t))E i (t)+L i (t)E i (θ -1 i (φ(t))) -θi (t)L i-1 (θ i (t)) ×E i-1 (θ -1 i-1 (φ(θ i (t)))), 2 ≤ i ≤ m. (16)
Also, the system (1) in closed loop with ( 7)-( 8) can be written as

ẋ(t) = (A(t) + B(t)K(t))x(t) +{B(t)[K(t)E 1 (θ 2 (...θ m (t -h(t)...))) +... + K(t)E m (t -h(t))]}, (17)
and

z m (t) = x(Ω -1 m (t)) + E 1 (θ 2 (...θ m (t)...)) + ... + E m (t) (18
) holds for all t ≥ 0. Finally, θ -1 i (t) ≤ t holds for all t ≥ 0 and all i ∈ {1, 2, . . . , m}.

Setting

c m = c τ + c h m(1-l h ) , (19)
and f t (s) = f (t + s) for any function f and t ≥ 0 and s ≤ 0 for which the equality holds, our final lemma is: Lemma 3: Let the assumptions of Theorem 1 hold, i ∈ {1, 2, . . . , m} be given, and λ > 1 be a constant such that λH < 1, where H denotes the left side of [START_REF] Cacace | Optimal control of linear systems with large and variable input delays[END_REF]. Then there is a constant c 0 > 0 such that the time derivative of

Q (s t ) = 1 2 |s(t)| 2 +(1 + |R i | ∞ |A| ∞ ) 4 λ 2 c m 1-lτ 1 + uc m I(s t ), where I(s t ) = t t-2c m t |s(r)| 2 drd (20)
along all solutions of

ṡ(t) = R i (t)A(G i (t))s(t) + L i (t)s(θ -1 i (φ(t))) (21)
satisfies

Q (t) ≤ -c 0 Q (s t ) (22)
for all t ≥ 0. Proof: Since θ -1 i is strictly increasing, as a composition of the strictly increasing functions Ω -1 m-i and Ω m-i+1 , it follows from Lemma 2 that θ -1 i (φ(t)) ≤ θ -1 i (t) ≤ t for all t ≥ 0. Also, our choices of the L i 's in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] and the Fundamental Theorem of Calculus let us rewrite [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] as

ṡ(t) = -s(t) + [I n + R i (t)A(G i (t))] D i (t), (23)
where

D i (t) = s(t) -s(θ -1 i (φ(t)))
, and where for all t ≥ 0,

|D i (t)| ≤ t θ -1 i (φ(t)) |R i ( )A(G i ( ))s( )|d + t θ -1 i (φ(t)) |L i ( )s(θ -1 i (φ( )))|d , (24)
by the constantness of the initial functions at time 0.

Then the second inequality in [START_REF] Merad | Isometric torque control for neuromuscular electrical stimulation with time-varying input delay[END_REF] in Lemma 1 (applied with = φ(t)) gives

0 ≤ t -θ -1 i (φ(t)) = τ (t) + φ(t) -θ -1 i (φ(t)) ≤ τ (t) + |φ(t) -θ -1 i (φ(t))| ≤ τ (t) + c h m(1-l h ) ≤ c m (25)
for all t ≥ 0. Therefore, for all t ≥ 0, we can now use Jensen's inequality (applied to the convex squaring function) to obtain 15) and ( 25) and the change of variables u

t θ -1 i (φ(t)) |s( )|d 2 ≤ c m t θ -1 i (φ(t)) |s( )| 2 d . (26) Also, (
= θ -1 i (φ( )) give t θ -1 i (φ(t)) |s(θ -1 i (φ( )))|d 2 ≤ t t-τ (t)- c h m(1-l h ) |s(θ -1 i (φ( )))|d 2 ≤ c m t φ(t)- c h m(1-l h ) |s(θ -1 i (φ( )))| 2 d ≤ c m θ -1 i (φ(t)) θ -1 i (φ(t)-c m ) |s(u)| 2 θi(u) 1-lτ du, (27)
where the last inequality followed by combining the bound

φ(φ(t) -(c h /(m(1 -l h )))) ≥ φ(t) -c m with the fact that θ -1 i
is strictly increasing and from the bounds τ (t) ≤ l τ < 1. We next combine the estimates ( 26)-( 27) with [START_REF] Smith | A controller to overcome dead time[END_REF] to get a suitable decay estimate on Q(t) = |s(t)| 2 /2 along solutions of ( 23), as follows. We first introduce the functions

P i (t) = (1+|R i | ∞ |A| ∞ )× t θ -1 i (φ(t)) |R i ( )||A(G i ( ))||s( )|d (28)
and

J i (t) = (1 + |R i | ∞ |A| ∞ )× t θ -1 i (φ(t)) |L i ( )||s(θ -1 i (φ( )))|d . (29)
We next use the facts that

|L i | ∞ ≤ 1 + |R i | ∞ |A|
Q(t) ≤ -|s(t)| 2 + |s(t)|P i (t) + |s(t)|J i (t) ≤ -1 2 |s(t)| 2 + α i1 t θ -1 i (φ(t)) |s( )| 2 d +α i2 θ -1 i (φ(t)) t-2c m |s(u)| 2 du (30)
along all solutions of [START_REF] Smith | A controller to overcome dead time[END_REF], where

α i1 = (1 + |R i | ∞ |A| ∞ ) 2 |R i | 2 ∞ |A| 2 ∞ c m and α i2 = (1 + |R i | ∞ |A| ∞ ) 4 c m 1-lτ 1 + uc m , (31)
and where we also used ( 15) and ( 24)-(25) and

θ -1 i φ(t) -c m ≥ θ -1 i t -c m -c τ ≥ t -2c m (32)
and ( 26)-( 27) to upper bound P 2 i and J 2 i , respectively. We next use the facts that for all t ≥ 0, we have

d dt I(s t ) = 2c m |s(t)| 2 - t t-2c m |s(r)| 2 dr (33) and I(s t ) ≤ 2c m t t-2c m |s(r)| 2 dr. (34)
Since α i1 ≤ α i2 , and since (20) can be written as

Q (s t ) = 1 2 |s(t)| 2 + α i2 λ 2 I(s t )
, it follows from combining the two integrals in (30) and then using (33) that the time derivative of Q along all solutions of (23) satisfies

Q (t) ≤ -1 2 + 2λ 2 c m α i2 |s(t)| 2 + (1-λ 2 )W i (t) where W i (t) = α i2 t t-2c m |s( )| 2 d . (35)
By [START_REF] Cacace | Optimal control of linear systems with large and variable input delays[END_REF] and our choice of λ > 1 and the fact that [START_REF] Merad | Isometric torque control for neuromuscular electrical stimulation with time-varying input delay[END_REF] gives |R i (t)| ≤ (1 + (u c /m)) m for all t ≥ 0, we conclude that the coefficients of |s(t)| 2 and W i (t) on the right side of (35) are negative. Hence, we can use (34) to check that

c 0 = min 1 -4λ 2 c m α i2 , λ 2 -1 2λ 2 c m (36)
satisfies our requirements. This proves Lemma 3.

V. PROOF OF THEOREM 1

We first study the dynamics for the transformed error vector

E (t) = E 1 (κ 1 (t)), . . . , E m (κ m (t)) on the set {t ≥ 0 : min{κ i (t) : 1 ≤ i ≤ m} ≥ 0}, where κ 1 (t) = t and κ i+1 = θ -1 i+1 • φ -1 • θ i • κ i for i = 1, .
. . , m, and where • means composition. Then

θ -1 i • φ • θ i+1 • κ i+1 = κ i for i = 1, 2, . . . , m. (37)
Note that Assumption 1 ensures that there are positive constants θ and θ such that θ ≤ κ i (t) ≤ θ hold for all i ∈ {1, 2, . . . , m} and t ∈ R (by noting that there are positive constants a and ā such that for all t ∈ R, we have a ≤ Ω i (t) ≤ ā and a ≤ (Ω -1 i ) (t) ≤ ā for all i and a ≤ (φ -1 ) (t) ≤ ā, and then reapplying the chain rule).

The decay estimate [START_REF] Sharma | Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation[END_REF] on Q , combined with Young's inequality ab ≤ 1 4 a 2 + b 2 for suitable a ≥ 0 and b ≥ 0 and our choice of the quadratic part of Q , implies that for all t ≥ 0 in the set S of all t ≥ 0 such that min{κ i (t) : 1 ≤ i ≤ m} ≥ 0 and along all solutions of the dynamics for

(E 1 (t), E 2 (t)) = (E 1 (κ 1 (t)), E 2 (κ 2 (t))), we have d dt Q (E 2,t ) ≤ -θc0 2 Q (E 2,t ) + 2 θ4 θc0 (| θ2 | ∞ |L 1 | ∞ ) 2 Q (E 1,t ) d dt Q (E 1,t ) ≤ -θc 0 Q (E 1,t ), (38)
where the first inequality used [START_REF] Sharma | Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation[END_REF], then Young's inequality, and then the fact that θ -1

1 • φ • θ 2 • κ 2 = κ 1 to get θ2 |E 2 (κ 2 (t))|| θ2 | ∞ |L 1 | ∞ |E 1 (θ -1 1 • φ • θ 2 • κ 2 (t))| ≤ θc0 4 |E 2 (κ 2 (t))| 2 + θ4 θc0 (| θ2 | ∞ |L 1 | ∞ ) 2 |E 1 (θ -1 1 • φ • θ 2 • κ 2 (t))| 2 ≤ θc0 2 Q (E 2,t ) + 2 θ4 θc0 (| θ2 | ∞ |L 1 | ∞ ) 2 Q (E 1,t
), by [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], where E i,t (s) = E i (t + s) for all s ≤ 0. Hence,

Q 2 (E 1,t , E 2,t ) = Q (E 2,t ) + 1+ 1 θc0 2 θ4 θc0 (| θ2 | ∞ |L 1 | ∞ ) 2 Q (E 1,t ) (39) satisfies Q 2 (t) ≤ -θc0 2 Q E 1,t + Q E 2,t (40)
for all t ∈ S and so is a Lyapunov-Krasovskii functional for the (E 1 (t), E 2 (t)) dynamics; we used the quantity in curly braces in (39) to cancel the effects of the nonnegative term in (d/dt)Q (E 2,t ) in (38). Arguing inductively (by adding one component of E at a time, and using (37), to cancel the effects of the nonnegative terms) provides positive constants ω i and η 1 such that

Q m (E t ) = m i=1 ω i Q (E i,t ) (41) satisfies Q m (E t ) ≤ -η 1 Q m (E t ) (42)
along all solutions of the E dynamics for all t ∈ S.

Assumption 2 provides a C 1 bounded function P such that V (t, x) = x P (t)x has a quadratic lower bound in |x| of the form p 0 |x| 2 for some constant p 0 > 0 and satisfies V ≤ -|x(t)| 2 along solutions of (2) (using [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Theorem 4.14]). Then along all solutions of (17), we have

V (t, x(t)) ≤ |P | ∞ |x(t)| 2 , so the triangle inequality ab ≤ 1 2 (a 2 +b 2 ) gives V ≤ -|x(t)| 2 + {|x(t)|}{2|P | ∞ |δ | ∞ } ≤ -1 2 |x(t)| 2 + 2|P | 2 ∞ |δ | 2 ∞ ≤ -1 2|P |∞ V (t, x(t)) + 2m (|P | ∞ |B| ∞ |K| ∞ ) 2 × |E 1 (θ 2 (...θ m (t -h(t)...)))| 2 + . . . + |E m (t -h(t))| 2 , ( 43
)
where δ is the quantity in curly braces in [START_REF] Mukhopadhyay | Collaborative autonomous surveys in marine environments affected by oil spills[END_REF], by applying the triangle to the terms in curly braces in (43), and using the Cauchy inequality for squaring a sum of m nonnegative terms to produce the 2m factor in (43).

Set

M i = κ -1 i • θ i+1 • • • • • θ m • Ω m if 1 ≤ i ≤ m -1 and M m = κ -1 m • Ω m .
Then the same reasoning we used to find θ and θ provides positive constants M and M such that M ≤ M i (t) ≤ M for all t ∈ R and i. Moreover,

|E i (θ i+1 • . . . • θ m (t -h(t)))| 2 = 2Q(E i (θ i+1 • . . . • θ m • Ω m (t))) = 2Q(E i (M i (t))) ≤ 2 ωi Q m E Mi(t) (44)
for all i ∈ {1, 2, . . . , m -1} (by ( 41)), and

|E m (t-h(t))| 2 = 2Q(E m (Ω m (t))) ≤ 2 ωm Q m E Mm(t)
. By the preceding upper bounds and (43), the function

Q + (E t ) = V (t, x(t)) + 1+ 2m(|P |∞|B|∞|K|∞) 2 Mη1 m i=1 2 ωi Q m E Mi(t) (45) admits a constant d 0 > 0 such that Q + (E t ) ≤ -d 0 Q + (E t ) (46)
holds along all solutions of the dynamics for

E (t) = x(t), E (M 1 (t)), . . . , E (M m (t)) (47)
for all t ≥ 0 such that

min{min{κ i (t), κ i • M i (t)} : 1 ≤ i ≤ m} ≥ 0. ( 48
)
Since the κ i 's and M i 's are strictly increasing and unbounded, we can find a constant a * > 0 such that (48) holds for all t ≥ a * . It follows from (46) that there are positive constants c 1 and c 2 such that

|E i (κ i • M i (t))| 2 ≤ Q + E t ≤ c 1 |(x, E )| 2
[a * -2c m ,a * ] e -c2t , 1 ≤ i ≤ m holds along all solutions of the dynamics for (47) for all t ≥ a * . Since the κ i •M i 's are strictly increasing and unbounded, we can then find a constant s > a * such that

|E i (t)| 2 ≤ c 1 |(x, E )| 2 [a * -2c m ,a * ] e -c2(κi•Mi) -1 (t) and |x(t)| 2 ≤ c1 p0 |(x, E )| 2 [a * -2c m ,a * ] e -c2t (49)
hold for all t ≥ s, e.g., any s > a * such that min{κ i (t),

κ i • M i (t), (κ i • M i ) -1 (t)} ≥ 0 for all i ∈ {1, 2, . . . , m} and t ≥ s; such a s exists because lim t→+∞ θ i (t) = lim t→+∞ θ -1 i (t) = lim t→+∞ Ω i (t) = lim t→+∞ Ω -1 i (t) = +∞ for all i ∈ {1, 2, . . . , m}. Since we can find positive constants h o and h 1 such that (κ i • M i ) -1 (t) ≥ h o t -h 1
for all i and all t ∈ R (by finding a positive lower bound on ((κ i • M i ) -1 ) (t) over all i and t and then applying the Mean Value Theorem to the (κ i • M i ) -1 's), it follows from (49) and the subadditivity of the square root that we can find positive constants ω 1 and ω 2 such that

|(x(t), E(t))| ≤ ω 1 |(x, E )| [a * -2c m ,a * ] e -ω2t (50)
holds along all solutions of the (x, E) systems for all t ≥ s.

Next note that the formula for the E 1 subsystem of ( 16) and the bounds θ

-1 i (φ(t)) ≤ φ(t) ≤ t provide a constant c a > 0 such that | Ė1 ( )| ≤ c a |E 1 | [-c h -cτ , ]
and so also

|E 1 ( )| ≤ |E 1 | [-c h -cτ ,0] + c a 0 |E 1 | [-c h -cτ ,s] ds (51)
for all ≥ 0. For all t ≥ 0, this gives

|E 1 | [-c h -cτ ,t] ≤ |E 1 | [-c h -cτ ,0] + c a t 0 |E 1 | [-c h -cτ ,s] ds.
Hence, Gronwall's inequality (applied to the function

|E 1 | [-c h -cτ ,t] of t) gives |E 1 (t)| ≤ |E 1 | [-c h -cτ ,t] ≤ |E| [-cτ -c h ,0] e cat
for all t ≥ 0. Since the E i subsystem in [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF]

depends affinely on E i-1 (θ -1 i-1 (φ(θ i (t)))) and since max{θ -1 i-1 (φ(θ i (t))) : 1 ≤ i ≤ m, t ∈ [0, s]} is finite, we can repeat this process m -1 times to find a constant c * > 0 such that |E(t)| ≤ c * |E| [-cτ -c h ,0] (52)
holds for all t ∈ [0, s * ], where s * ≥ s is any upper bound for the arguments of the E i 's in [START_REF] Mukhopadhyay | Collaborative autonomous surveys in marine environments affected by oil spills[END_REF] over all t ∈ [0, s]. Hence, ( 17) and (52) provide a constant c * * > 0 such that

|x(t)| ≤ c * * |(x, E)| [-c h -cτ ,0] (53)
for all t ∈ [0, s * ], again using Gronwall's inequality. Combining (52) and (53) provides a constant c n > 0 such that

|(x, E)(t)| ≤ c n |(x, E)| [-c h -cτ ,0] e s * -t (54) for all t ∈ [0, s * ]. Similar reasoning gives |E | [a * -2c m ,a * ] ≤ c * * |(x, E)| [-cτ -c h ,0] (55)
for a large enough constant c * * > 0. The final uniform globally exponential stability estimate (10) now follows by using ( 55) and (53) to upper bound the supremum on the right side of (50) and then adding the estimates (50) and (54). This completes the proof of Theorem 1.

VI. ILLUSTRATIONS

Theorem 1 applies to a broad class of time-varying linear systems for many choices of h, including cases where current values x(t) of the original plant state are not available to use in the sequential predictors at times t. For drift free systems (where A = 0), our conditions allow any constant measurement delay τ ∈ [0, 1/2), but we can also allow systems with drift. For example, we next illustrate Theorem 1 by revisiting a benchmark example from [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] where we show how Theorem 1 allows a smaller number of predictors than [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] when τ = 0. We also explain how Theorem 1 applies in cases where in addition to feedback delays, the measurement delays τ (t) are nonzero, which were beyond the scope of [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF].

The simple pendulum model from [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] is

ṙ1 (t) = r 2 (t) ṙ2 (t) = -g l sin(r 1 (t)) + 1 ml 2 v(t -h(t)) (56)
where h(t) is the time varying delay in the input v, l is the pendulum length in meters, m is the pendulum mass, and g = 9.8 m/s is the gravity constant. The objective in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] was to track a given C 1 reference trajectory (r 1,s (t), r 2,s (t)) that satisfies ṙ1,s (t) = r 2,s (t). Using the change of feedback

u(t -h(t)) = 1 σ 1 ml 2 v(t -h(t)) -ṙ2,s (t) -g l sin(r 1,s (t)) (57)
where σ > 0 is any constant, we obtain the tracking system

     ṙ1 (t) = σr 2 (t) ṙ2 (t) = g σl [sin(r 1,s (t)) -sin(r 1 (t)+r 1,s (t))] + u(t -h(t)). (58)
for the error variables r1 = r 1 -r 1,s (t) and r2 = (r 2r 2,s (t))/σ.

Our work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] showed that under the constant delay h = 1 and with the choices σ = 1 and r 1,s (t) = ωt where ω > 0 is a large enough constant, the linearization

ẋ1 (t) = σx 2 (t) ẋ2 (t) = -g σl cos(ωt)x 1 (t) + u(t -h) (59)
of (58) around 0, in closed loop with the distributed control

u(t) = -0.6x 1 (t) -0.4x 2 (t) - t t-1 0.6(t -r -1) + 0.4 u(r)dr, (60)
is globally exponentially stable to 0, and [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] provides a globally asymptotically stabilizing sequential predictor control for the original nonlinear dynamics (58) when h is constant. However, [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] did not apply under time-varying delays.

In [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF], we built sequential predictors for the linearized error dynamics (59) with time-varying delays h when σ = 1 and τ = 0, and the assumptions in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] were the same as Assumptions 1-2 above in the special case where τ = 0. For any constant ω > 0, the assumptions from [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] hold using

A(t) = 0 σ -g σl cos(ωt) 0 , B = 0 1 , and
K(t) = g σl cos(ωt) -σ -σ (61) which gives |A| ∞ = max g σl , σ . (62)
With the preceding choices, and with

h(t) = 1 + α sin(t), (63)
and σ = 1, condition ( 13) above (from [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]) requires that

m 4 -1+α 1-α |A|∞ √ 2 1+|A| ∞ 1+ uc m m 1+ uc m m + 1 + |A| ∞ 1 + uc m m 2 1 + uc m > 0. (64)
In the special case where α = 1/7 and σ = 1, and for any ω > 0 and any l > g, the smallest number m of sequential predictors for which (64) holds is m = 47. On the other hand, for the special case where the measurement delay τ is 0 and with the preceding choices of the parameters with σ = 1, our requirement ( 6) is satisfied with m = 17. Therefore, in the special case of (59) with σ = 1, our new method from Theorem 1 provided a 63% reduction in the number m of required sequential predictors as compared with [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF].

To see how we can also cover cases of nonzero measurement delays τ in the sequential predictors for (59), first note that in the special case where τ is constant and with the preceding choices of the parameters with l > g/σ 2 , our requirement [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] on the allowable measurement delays τ is τ ∈ 0, 1 2(1 + e uc σ) 2 , (65) since g/(σl) ≤ σ, which gives |A| ∞ = σ. Therefore, since u c is independent of A, we can allow any constant transmission delay τ ∈ [0, 0.5) by choosing the number m of sequential predictors to be large enough and σ > 0 small enough. For instance, with the choices τ = 0.15, h(t) = 1 + α sin(t) with α = 1/7, and σ = 0.1, the smallest integer m ≥ 2 such that our condition (6) on the number m of required sequential predictors is satisfied is m = 7.

VII. CONCLUSIONS We advanced the theory of sequential predictors for linear time-varying systems with time-varying delays by reducing the number of required sequential predictors as compared with the existing literature, which can reduce the computational burden from applying the method. Our result also incorporates measurement delays that model more realistic cases where the current state of the physical system may not be available for use in the sequential predictors. Such measurement delays had not been considered in previous treatments of sequential predictors for time-varying linear systems with time-varying delays. We hope to develop analogous results for PDE systems with feedback and measurement delays.

  ∞ and | θi (t)| ≤ 1 + u c /m both hold for all t ≥ 0 (by the first inequality in Lemma 1) and Young's Inequality ab ≤ 0.25a 2 + b 2 twice (with the choices a = |s(t)|, and with b = P i (t) and then with b = J i (t)) to obtain

Malisoff and Weston were supported by NSF Grant 1408295.


Frederic Mazenc
email: frederic.mazenc@lss.supelec.fr

Michael Malisoff
email: malisoff@lsu.edu

Laurent Burlion
email: lburlion@onera.fr.

Victor Gibert
email: victor.gibert@airbus.com.

Gibert

Bounded Backstepping through a Dynamic Extension with Delay

Keywords: Backstepping, delays, stabilization

We provide a bounded backstepping result that ensures global asymptotic convergence for a broad class of partially linear systems with an arbitrarily large number of integrators. We use one artificial delay, and we assume that the nonlinear subsystems satisfy a converging-input-convergingstate assumption. When the nonlinear subsystem is control affine with the state of the first integrator as the control, we provide sufficient conditions for our converging-inputconverging-state assumption to hold. Our example illustrates the novelty and utility of our main result.

I. INTRODUCTION

This work continues our team's search (begun in [START_REF] Mazenc | Backstepping with bounded feedbacks for time varying systems[END_REF], [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF], [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF], [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF], [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF], and [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF]) for new backstepping results that can help overcome the obstacles to using standard backstepping; see [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] for traditional backstepping. Classical backstepping synthesizes globally asymptotically stabilizing feedbacks, by recursively finding globally asymptotically stabilizing controls and Lyapunov functions for subsystems; see [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF] for more recent backstepping that includes nonlinearities and uncertainties, and see [START_REF] Bialy | Adaptive backstepping controller for a hypersonic air-breathing missile[END_REF], [START_REF] Dixon | Global adaptive partial state feedback tracking control of rigid-link flexiblejoint robots[END_REF], and [START_REF] Dupree | Adaptive Lyapunovbased control of a robot and mass-spring system undergoing an impact collision[END_REF] for applications of backstepping for adaptive, aerospace, and robotic systems. However, there are important cases that call for backstepping where existing backstepping methods do not apply, e.g., for general nonlinear subsystems having bounds on the allowable control magnitudes, which produce challenges that we help address here.

We focus on systems having the form 8 > > > < > > > :

ẋ(t) = F(t, x(t), z 1 (t)) żi (t) = z i+1 (t), i 2 {1, . . . , k 1} żk (t) = u(t) + k P j=1 v j z j (t) (1)
with a scalar control u and any number k 2 of integrators.

Here, x is valued in R n for any n, the v j 's are constant real numbers, and the nonlinear subsystem will enjoy a converging-input-converging-state condition that we define and discuss below. We write our closed loop controls as u(t), but they will be feedbacks that depend on t through their dependence on states of an enlarged system.

While we assume that the current state of the system can be measured, we also use a delay in the state values in our feedback since this artificial delay is needed to design our bounded control. Our work [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] used the converginginput-converging-state assumption and an artificial delay, but a significant improvement in this note in comparison with [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] is that here we allow an arbitrarily large number k of integrators, while [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF] only allowed one integrator. Our bounded backstepping work [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] also allowed an arbitrarily large number of integrators, but the present work produces a globally bounded control for (1) while the controls for the original systems in [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] were not bounded. In addition, while [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] required k artificial delays and did not use dynamic extensions, here we only require one artificial delay, which may make our feedback simpler to apply.

Our works [START_REF] Mazenc | Backstepping with bounded feedbacks for time varying systems[END_REF], [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF], [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF], and [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] did not use converginginput-converging-state conditions or artificial delays. Moreover, our work differs from [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] (which was based on a forwarding method for one integrator case), [START_REF] Mazenc | Backstepping with bounded feedbacks for time varying systems[END_REF] (which was limited to one integrator), [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF] (which gives unbounded controls), and [START_REF] Tsinias | Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation[END_REF] and [START_REF] Tsinias | Backstepping design for time-varying systems and application to partial-static feedback and asymptotic tracking[END_REF] (which employ Lie derivatives without satisfying input constraints). Hence, our novel combination of converging-input-converging-state conditions, artificial delays, and bounded controllers for (1) in this work is useful.

We use standard notation and definitions. We omit arguments of functions when they are clear, and the dimensions of our Euclidean spaces are arbitrary. We use |•| to denote the usual Euclidean norm and the induced matrix norm, and | | 1 (resp., | | I ) is the essential supremum (resp., supremum over any interval I) for bounded measurable functions; . Set N = {1, 2, . . .}. For a fixed constant T > 0, let C in denote the set of all continuous functions : [ T, 0] ! R a , which we call the set of all initial functions. We define ⌅ t 2 C in and ⌅t 2 C in by ⌅ t (s) = ⌅(t + s) and ⌅t (s) = ⌅ 0 (t + s) for all choices of ⌅, s  0, and t 0 for which the equalities are defined. We assume for simplicity that the initial times for our solutions are t 0 = 0 and that the initial functions are constant at time 0 (so the states are constant on [ T, 0], where T will be the artificial delay). We denote by f (i) (t) the ith derivative of a function f : [0, +1) ! R with f (0) (t) = f (t), and r : R ! [ r, r] is the standard saturation that is defined for all constants r > 0 by r (s) = s for all s 2 [ r, r] and r (s) = rsign(s) otherwise.

II. MAIN RESULT

To state our theorem, we require the following two lemmas, the first of which follows from elementary calculations that we omit: Lemma 1: Let q > 0 and T > 0 be given constants, and µ 0 : [ T, +1) ! R be any continuous function, and set

⇣(t) = R t t T e q(` t) Q(t, `, `+ T )µ 0 (`)d`, and ⌦ j (t) = ⇣ (j 1) (t) and µ i (t) = 1 T R t t T e q(` t) (t `)i 1 (i 1)! µ 0 (`)d`( 2
)
for all j 2 {1, ..., k + 1} and i 2 N, where Q(t, a, b) = (t a) k 1 (t b) k 1 for all a 2 R and b 2 R and k 2 N with k 2. Then there are constants c i,j (q, T ) 2 R for all i 2 {1, ..., 2k 1} and all j 2 {1, ..., k}, and constants

g i (q, T ) 2 R for all i 2 { 1, 0, ..., 2k 1}, such that ⌦ j (t) = 2k 1 P i=1 c i,j (q, T )µ i (t) and ⌦ k+1 (t) = 2k 1 P j=0 g j (q, T )µ j (t) + g 1 (q, T )µ 0 (t T ) (3)
hold for all t 0. ⇤ Lemma 1 follows by first using the Binomial Theorem to write Q(t, `, `+ T ) as a linear combination of the terms (t `)r with k 1  r  2k 2. In the next lemma (which can be deduced from [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF] on linear systems), we say that a linear system is not exponentially unstable provided its poles are all in the closed left-half plane:

Lemma 2: Let k 2 be an integer and v = (v 1 , . . . , v k ) be any vector of k real constants such that 8 <

:

żi (t) = z i+1 (t), i 2 {1, . . . , k 1} żk (t) = u + k P i=1 v i z i (4)
is not exponentially unstable when u = 0. Then there is a bounded locally Lipschitz function # such that (4), in closed loop with u = #(Z) where Z = (z 1 , . . . , z k ) > , is globally asymptotically and locally exponentially stable to 0. ⇤ In addition to the notation from the preceding two lemmas, our theorem will also use the nonzero constant

b T = R 0 T e q``k 1 (`+ T ) k 1 d`(5)
and the following assumption:

Assumption 1: (i) The function F in (1) is continuous in t and globally Lipschitz in (x, z 1 ) and satisfies F(t, 0, 0) = 0 for all t 0. (ii) There is a globally Lipschitz bounded function ! : R n ! [ !, !] having some constant bound ! > 0 such that !(0) = 0, and constants T > 0 and q > 0, such that for each continuous function : [0, +1) ! R that exponentially converges to 0, the following is true: All solutions ⇠ : [0, +1) ! R n of the system

⇠(t) = F ✓ t, ⇠(t), Z t t T e q(` t) Q(t,`,`+T )!(⇠(`)) bT d`+ (t) ◆ (6) satisfy lim t!+1 ⇠(t) = 0, where Q(t, a, b) = (t a) k 1 (t b) k 1 and k 2 is from (1).
⇤ We refer to part (ii) of Assumption 1 as our converginginput-converging-state assumption; see Section III for sufficient conditions. The system (6) differs from the nonlinear subsystem of (1) because the third argument of F in (1) has been replaced by the sum of (t) and an integral. In terms of the Jordan matrix

J 2k 1 = 2 6 6 6 6 6 6 4 q 1 0 . . . 0 0 q 1 . . . 0 . . . . . . . . . . . . . . . . . . . . . q 1 0 . . . . . . 0 q 3 7 7 7 7 7 7 5 2 R (2k 1)⇥(2k 1)
, our main result is as follows, where the forward completeness of the closed loop system follows from the boundedness of ! and the control and the global Lipschitzness of F and !: Theorem 1: Let the constants q > 0, k 2, and T > 0 and the functions F and ! be such that Assumption 1 holds, where k 2 N. Let # and v satisfy the requirements from Lemma 2. Consider the augmented (x, Z, Y ) system, consisting of (1) and the 2k 1 dimensional system

Ẏ (t) = J 2k 1 Y (t) + e 2k 1 T !(x(t)) bT (7)
where e 2k 1 = (0, 0, . . . , 1) > 2 R 2k 1 is the (2k 1)-st standard basis vector, in closed loop with the control

u(Z(t), Y t , x t ) = ā (M(Y t )) + g 0 (q, T ) !(x(t)) bT + g 1 (q, T ) !(x(t T )) bT + #(Z ? (t)) (8)
with the saturation level

ā = k P j=1 v j C j (q, T ) G(q, T ) e |J 2k 1 |T ! bT , (9)
where

M(Y t ) = G(q, T ) k P j=1 v j C j (q, T ) ! (Y t ) (10)
and

(Y t ) = Y (t) e T J 2k 1 Y (t T ) and Z ? (t) = 2 6 4 z 1 (t) C 1 (q, T ) (Y t ) . . . z k (t) C k (q, T ) (Y t ) 3 7 5 , (11)
and G(q, T ) = [g 2k 1 (q, T ) ...... g 1 (q, T )] and

C j (q, T ) = [c 2k 1,j (q, T ) ...... c 1,j (q, T )], 1  j  k (12
) and where the constants c i,j and g i satisfy the requirements from Lemma 1 for the function µ 0 (t) = !(x(t))/b

T . Then all maximal solutions (x, Z, Y )(t) of the augmented (x, Z, Y ) system satisfy lim t!+1 (x, Z, Y )(t) = 0. ⇤ We next provide sufficient conditions for our converginginput-converging-state condition from Assumption 1 to hold, and then we prove Theorem 1 in Section IV.

III. CHECKING ASSUMPTION 1

We provide sufficient conditions for our converging-inputconverging-state conditions on (6) to hold, based on Lyapunov functions, using

q(t) = F(t, q(t), !(q(t))), ( 13
)
where F is from (1), and using the standard definitions of positive definiteness, properness, and K 1 [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. In the next assumption, V t and V

x are the partial derivative with respect to t and the gradient with respect to x, respectively, and uniform global Lipschitzness in x means that the global Lipschitz constants can be chosen independently of t:

Assumption 2: There exist functions f : [0, +1) ⇥ R n ! R n and g : [0, +1) ⇥ R n ! R n that are uniformly globally Lipschitz in x and continuous on [0, +1) ⇥ R n , such that

F(t, x, p) = f (t, x) + g(t, x)p (14)
holds for all t 0, x 2 R n , and p 2 R. Also, there exist a C 1 uniformly proper and positive definite function V : [0, +1) ⇥ R n ! [0, +1); a uniformly continuous positive definite function W : R n ! [0, +1); positive constants r 0 , r 1 , and r 3 ; and a constant r 2 0 such that for all (t, x) 2 [0, +1) ⇥ R n , we have

V t (t, x) + V x (t, x) f (t, x) + g(t, x)!(x)  W (x), |V x (t, x)g(t, x)|  r 0 p W (x), |!(x)|  r 1 p W (x), |f (t, x)|  r 2 p W (x), and |g(t, x)|  r 3 , (15)
where ! : R n ! R is bounded and satisfies !(0) = 0, and there is a global Lipschitz constant C > 0 for ! on R n . ⇤ See [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] for conditions under which Assumption 2 holds. Proposition 1: If Assumption 2 holds, then for all constants T > 0 such that

4(T r 0 C) 2 " 2r 2 2 + 5 2 ✓ r 1 r 3 T 2k 1 |b T | ◆ 2 # < 1, (16)
and for all integers k 2, Assumption 1 is satisfied. ⇤ Proof: (Sketch.) The proof has a similar structure to the proof of the sufficient conditions from [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF], so we only sketch the proof. Fix any continuous function : [0, +1) ! R that exponentially converges to 0. By (5), we have

Z t t T e q(` t) Q(t, `, `+ T ) b T d`= 1 (17)
for all t T . Hence, along all solutions x(t) of ( 6), we get

V (t)  W (x(t)) + r 0 p W (x(t)) ⇥ sup `2[t T,t] |!(x(`)) !(x(t))| + | (t)| ! ( 18
)
for all t T and

| ẋ(t)|  r 2 p W (x(t))+ r 3 ✓ T 2(k 1) |b T | Z t t T |!(x(`))|d`+ | (t)| ◆ (19)
along all solutions of (6). It follows from the Young and Jensen's inequalities that along all solutions of (6), we have

| ẋ(t)| 2  2r 2 2 W (x(t))+ 2r 2 3 ✓ 5| (t)| 2 + 5 4 T 4(k 1) T r 2 1 b 2 T Z t t T W (x(`))d`◆ . (20)
Hence, Jensen's and Young's inequalities also give

V (t)  1 2 W (x(t)) + r 2 0 ✓ | (t)| 2 + C 2 sup `2[t T,t] |x(`) x(t)| 2 ◆  1 2 W (x(t)) + r 2 0 ✓ | (t)| 2 + C 2 T R t t T | ẋ(`)| 2 d`◆  1 2 W (x(t)) + r 2 0 ✓ | (t)| 2 + C 2 T  2r 2 2 R t t T W (x(`))d+ 10r 2 3 T | | 2 [t T,t] + 5 2 r 2 3 T 4(k 1) T 2 b 2 T r 2 1 R t t 2T W (x(`))d` ◆  1 2 W (x(t)) + N 1 R t t 2T W (x(`))d+ N 2 | | 2 [t T,t] (21)
along all solutions of ( 6) for all t T , where

N 1 = T (r 0 C) 2 ✓ 2r 2 2 + 5 2 ⇣ r1r3T 2k 1 |bT | ⌘ 2 ◆ ( 22
)
and N 2 = 10(r 0 r 3 T C) 2 + r 2 0 . Then our condition ( 16)

provides a constant > 1 such that 2T N 1 < 1/2, so V 1 (t, x t ) = V (t, x(t))+ N 1 R t t 2T R t s W (x(`))d`ds (23)
satisfies the following along all solutions of (6):

V1  1 2 2T N 1 W (x(t)) + N 2 | | 2 [t T,t] (24)
for all t 2T . The result now follows from our choice of and applying Barbalat's Lemma to W (x(t)).

IV. PROOF OF THEOREM 1

Theorem 1 will follow from three more lemmas, which we state next. The following follows from [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite-Dimensional Systems[END_REF]Appendix A.3

(Y t ) = Y (t) e T J 2k 1 Y (t T ) for all solutions Y of Ẏ (t) = J 2k 1 Y (t) + e 2k 1 T µ 0 (t), (26)
are such that for all t T , we have

⌫ 2k 1 (t) = (Y t ) and | (Y t )|  e |J 2k 1 |T μ, (27)
where ⌫ 2k 1 (t) = (µ 2k 1 (t), . . . , µ 1 (t)) > for all t T . ⇤ Proof: By integrating (26), we deduce that

(Y t ) = Y (t) e T J 2k 1 Y (t T ) = %(t), (28)
where

%(t) = R t t T e J 2k 1 (t `) e 2k 1
T µ 0 (`)d`(29) for all t T . On the other hand, using (25), we obtain

%(t) = 1 T Z t t T e q(t `)2 6 6 6 6 4 (t `)2(k 1)
(2(k 1))!

. . .

t 1 3 7 7 7 7 5 µ 0 (`)d`= ⌫ 2k 1 (t), (30)
which proves the first conclusion of the lemma. The second conclusion of the lemma follows from (28)-(29). Lemma 5: Let µ 0 : [ T, +1) ! [ μ, μ] be any continuous function having a bound μ, and let the constants v i and the function # satisfy the requirements from Lemma 2. Consider the linear system 8 > < > :

żi (t) = z i+1 (t), i 2 {1, . . . , k 1} żk (t) = u(t) + k X j=1 v j z j (t) (31)
in closed loop with the control

u(Z(t), Y t , x t ) = ā (M(Y t )) + g 0 (q, T )µ 0 (t) + g 1 (q, T )µ 0 (t T ) + #(Z ? (t)) (32)
with the saturation level ā for ā defined by

ā = k P j=1 v j C j (q, T ) G(q, T ) e |J 2k 1 |T μ (33)
and where Y satisfies (26) and M, , Z ⇤ , G, and the C j 's and g j 's are defined as in Theorem 1. Then the dynamics for the vector Z(t) = (z 1 (t), ..., zk (t)) are globally asymptotically and locally exponentially stable to the origin, where zi (t) = z i (t) ⌦ i (t) for i = 1, 2, . . . , k and the ⌦ i 's are defined in (2) in Lemma 1.

⇤ Proof: The fact that ⌦i = ⌦ i+1 for all i 2 {1, 2, . . . , k} and the structure of the dynamics (31) allow us to conclude that the dynamics for the functions zi

(t) = z i (t) ⌦ i (t) are 8 > > < > > : żi (t) = zi+1 (t), i 2 {1, . . . , k 1} żk (t) = u(t) ⌦ k+1 (t) + k P j=1 v j [z j (t) + ⌦ j (t)] . (34)
Using our conclusion from Lemma 4 that

⌫ 2k 1 (t) = (Y t ) (35)
where ⌫ 2k 1 (t) = (µ 2k 1 (t), . . . , µ 1 (t)) > as before, it follows from (3) that żk (t) = u(t) G(q, T )⌫ 2k 1 (t) g 0 (q, T )µ 0 (t)

g 1 (q, T )µ 0 (t T ) + k P j=1 v j zj (t) + k P j=1 v j C j (q, T )⌫ 2k 1 . (36)
Hence (35) and our choice

Ẏ (t) = J 2k 1 Y (t) + e 2k 1 T µ 0 (t) (37)
of our dynamic extension imply that for all t T , we have 8 > > > > < > > > > :

żi (t) = zi+1 (t), i 2 {1, . . . , k 1} żk (t) = u(t) + k X j=1 v j zj (t) g 0 (q, T )µ 0 (t) g 1 (q, T )µ 0 (t T ) + ḡ (Y t ) (38)
where

ḡ = k X j=1 v j C j (q, T ) G(q, T ). (39)
Next note that since Lemma 1 gives

⌦ j = C j (q, T )⌫ 2k 1 for 1  j  k, (40)
it follows that for all i 2 {1, . . . , k}, we have

zi (t) = z i (t) ⌦ i (t) = z i (t) C i (q, T )⌫ 2k 1 (t
P i=1 v i zi (43)
so the lemma follows from our choice of # in Lemma 2. We now combine the preceding lemmas to prove Theorem 1. First notice that the closed loop system is as follows:

8 > > > > > > < > > > > > > : ẋ(t) = F(t, x(t), z 1 (t)) żi (t) = z i+1 (t), i 2 {1, . . . , k 1} żk (t) = u(Z(t), Y t , x t ) + k X j=1 v j z j (t) Ẏ (t) = J 2k 1 Y (t) + e 2k 1 T !(x(t)) bT (44)
which is forward complete by our global Lipschitzness assumptions. Using the preceding lemma, we deduce that

lim t!+1 |z i (t) ⌦ i (t)| = 0 (45)
for all i = 1 to k, and z 1 ⌦ 1 exponentially converges to 0.

Next notice that ⇣ = ⌦ 1 in Lemma 1, and therefore that

ẋ(t) = F(t, x(t), ⇣(t) + z 1 (t) ⌦ 1 (t)) (46)
when we choose the bounded function µ 0 (t) = !(x(t))/b T to obtain

⇣(t) = R t t T e q(` t) Q 1 (t, `, `+ T ) !(x(`)) bT d`. (47)
Hence, we can use Assumption 1 with

= z 1 ⌦ 1 to conclude that lim t!+1 |x(t)| = 0 (48)
and therefore that for all i 2 {1, 2, . . . , k}, we have

lim t!+1 ⌦ i (t) = 0, (49)
since !(0) = 0 and ! is continuous at 0. Therefore, Theorem 1 follows by combining (45) with ( 48)-(49).

V. ILLUSTRATION

We apply our method to the three-dimensional dynamics 8 > < > :

ẋ(t) = |x(t)| 1+|x(t)| + z 1 (t) ż1 (t) = z 2 (t) ż2 (t) = u(t) (50)
which is outside the scope of classical backstepping, as the right side of ẋ(t) is not differentiable. With our earlier notation, we choose k = 2, n = 1, q = 1, and

!(x) = |x| 1+|x| 2 x 1+|x| . (51)
We find a constant T > 0 such that Assumption 1 holds with

F(t, x, z 1 ) = |x| 1+|x| + z 1 . (52)
Since ( 51)-( 52) are globally Lipschitz functions in the states and F is affine in z 1 and ! is bounded, it suffices to produce constants r i for i = 0, 1, 2, 3 and functions V and W that satisfy Assumption 2 with f (t, x) = x/(1 + |x|) and g(t, x) = 1, and then we choose T such that (16) holds.

To this end, we check that Assumption 2 holds for and we can use Mathematica [START_REF] Mathematica | The world's definitive system for modern technical computing[END_REF] to check that the right side of (56) is 0.912536 for T = 0.11. Hence, Assumption 1 holds with T = 0.11, so the desired dynamic controller is provided by Theorem 1.

V (t, x) =

VI. CONCLUSIONS

Our new bounded backstepping technique applies to a large class of partially linear systems with arbitrarily large numbers of integrators, when the nonlinear subsystems satisfy a converging-input-converging-state assumption. For many cases, our Lyapunov functions provide sufficient conditions for our converging-input-converging-state assumption to hold. Our theorem provides bounded controllers for the original system. We plan to combine our methods with the time delay methods in [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] to also allow measurement delays.

R x 0 1 , and 1  r 3 .By separately considering points where x 2 [ 1 , 1 ] or x 6 2 3 p 2 , r 2 = 1 , and r 3 = 1 . 2 = 4 (3T ) 2  2 + 5 2 ⇣

 113211232213124222 (`)d`and W (x) = 2 1 (x)x 1+|x| .(53)We havef (t, x) + g(t, x)!(x) = 2x 1+|x|(54) so our conditions on the r i 's from (15) will hold if| 1 (x)|  r 0 [ 1, 1],we can readily check that Assumption 2 is satisfied using C = 3, r 0 = 1, r 1 = Moreover, for the case where k = 2 and q = 1, our constant b T from (5) that is used in our condition (16) on T in Proposition 1 is b T = 2 T e T (2 + T ), so our requirement (16) on T > 0 is that1 > 4(T r 0 C) 2 3T 3 p 2|2 T e T (2+T )| ⌘ 2(56)

  Let µ 0 : [ T, +1) ! [ μ, μ] be any continuous function having a bound μ. Then the functions µ

	e J 2k 1 t = e qt	2 6 6 6 6 6 6 6 4	1 t 0 1 . . . . . . . . .	t 2 2 t ... . . . . . . . . . . . . 1	t 2(k 1) (2(k 1))! t 2k 3 (2k 3)! . . . t	3 7 7 7 7 7 5 7 7	(25)
			0 . . . . . .	0	1
	holds for all t 2 R and integers k 2. Lemma 4: i ⇤
	from (2) in Lemma 1, and the functions

]: Lemma 3: For the Jordan matrix J 2k 1 , the equality

Malisoff was supported by NSF Grant 1408295.