Frederic Mazenc

Michael Malisoff

Continuous-Discrete Sequential Observers under Sampling and Input Delays

Keywords: Delays, observers, sampling

We provide continuous-discrete sequential observers for linear systems with input delays, sampling in the controls and in the outputs, and uncertainties. We also include uncertainties and delays in the output. Our observer allows arbitrarily long constant delays in the input and in the output, and can be used to provide input-to-state stabilizing controllers under uncertain sampled outputs, sampling in the control, and arbitrarily long input and output delays. Since our control does not contain distributed terms, this paper is an extension of the recent prediction technique based on subsequential observers. This provides a useful alternative to existing methods that contained distributed terms. Our example illustrates the large class of systems that are amenable to our methods.

I. INTRODUCTION

The stabilization of linear systems under discrete sampling of an output is a challenging problem with important implications in engineering applications where the output is not available for continuous measurement [START_REF] Dinh | Dynamic neural network-based robust observers for uncertain nonlinear systems[END_REF], [START_REF] Parikh | A switched systems framework for guaranteed convergence of image-based observers with intermittent measurements[END_REF]. This motivated a large literature on delay compensation for systems with outputs and sampling, largely based on prediction or the reduction model approach. The reduction and prediction approaches are notable for their ability to compensate for arbitrarily long input delays, often using dynamic extensions that contain distributed terms; see, e.g., [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF], [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF], [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF].

The work [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF] applied the reduction model method to linear systems with piecewise constant inputs, but was confined to cases where there are continuous measurements available for the entire state. Several approaches exist to cope with cases where there are only discrete output measurements. For instance, if the sampling time intervals in the input and the output measurement all have the same length c > 0, and if the input delay is a multiple of c and the system is linear and time invariant, then [START_REF] Kazantzis | Control-relevant discretization of nonlinear systems with time-delay using Taylor-Lie series[END_REF] makes it possible to discretize the system so that one can apply a discrete time reduction model approach from [START_REF] Castillo-Toledo | Stability analysis for a class of sampled nonlinear systems with time-delay[END_REF], [START_REF] Gonzalez | Robustness of a discrete-time predictor-based controller for timevarying measurement delay[END_REF]. Two other alternative approaches entail (i) combining continuous observers with the reduction model approach and (ii) using continuous-discrete observers. Approach (ii) was used in [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF].

Continuous-discrete observers were used in the seminal works [START_REF] Deza | High gain estimation for nonlinear systems[END_REF] and [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF], and many more recent works such as [START_REF] Andrieu | Observer design for Lipschitz systems with discretetime measurements[END_REF], [START_REF] Goffaux | Improving continuous-discrete interval observers with application to microalgaebased bioprocess[END_REF], [START_REF] Hammouri | Constant gain observer design for continuous-discrete time uniformly observable systems[END_REF], [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF], [START_REF] Tellez-Anguiano | Nonlinear continuous-discrete observer application to distillation columns[END_REF]. Potential advantages of continuous-discrete observers are that they can be used in Mazenc is with EPI DISCO INRIA-Saclay, Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS, CentraleSupélec, Université Paris-Sud, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France, frederic.mazenc@l2s.centralesupelec.fr.

Malisoff is with the Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA, malisoff@lsu.edu.

Malisoff was supported by NSF Grants 1408295 and 1711299.

cases that are not amenable to continuous observers or static piecewise constant state feedback. The novelty of [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] was that it applied continuous-discrete observers to design globally asymptotically stabilizing output feedback under input delays. This contrasted with other works such as [START_REF] Wang | Piecewise-continuous observers for linear systems with sampled and delayed output[END_REF] that treated delays, discrete measurements, and sampling separately, with the exception of the nonlinear stabilization results [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zeroorder hold[END_REF], [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] for systems with inputs with zero-order hold and delay and with discrete delayed inputs. For the special case of linear time invariant systems, the feedbacks in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zeroorder hold[END_REF], [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] are expressed in terms of a sum of past values of the feedback. The work [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF] on reduction model methods for linear systems with delayed sampled inputs [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] does not apply to the problem we study in the present work, because [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF] requires differentiability of an operator whose analog in the case of impulsive systems would not satisfy the required differentiability conditions.

Here, we build on a series of recent works, which started with [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] and has been developed in [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] and [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF]. The fundamental new idea of this approach entails determining future values of the solutions of a system, using a simple dynamic extension where several pointwise delays are present, thus removing the need to introduce distributed delays in the expressions of the control laws or of the dynamic extensions. In the present work, for the first time, this strategy is adapted to the case where continuous-discrete observers are appropriate. Moreover, for the first time, two different delays are taken into account, one in the input and another in the output, under sampling, so we use ideas introduced in [START_REF] Cacace | Predictor-based control of linear systems with large and variable measurement delays[END_REF] to handle systems where the delays are only in the inputs. The proofs of our paper owe a great deal to [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF], [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF].

The strategy in [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] for stabilization of linear time invariant systems under sampling in both the input and the output was to use a dynamic extension that is similar to the one from [START_REF] Mondie | Finite spectrum assignment of unstable timedelay system with a safe implementation[END_REF]. However, works such as [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zeroorder hold[END_REF], [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF], [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] can lead to controls with distributed terms that may not always lend themselves to implementations. Therefore, the present work uses a nontrivial variant of our sequential predictor methods from [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF], [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] to provide sequential observers and stabilizing feedback controls for linear time invariant systems with sampling in the input, uncertainties in both the output measurements and the original plant, and sampling and uncertainties in the output, which can also ensure inputto-state stability [START_REF] Angeli | Input-to-state stability with respect to inputs and their derivatives[END_REF] robustness properties under arbitrarily long input and output delays without distributed terms.

In the next section, we state and discuss our main result, and then we prove our result in Section III. Our example in Section IV illustrates the utility of our theory in a detailed example, and we conclude in Section V with an overview of our contribution and our suggestions for future research.

II. STATEMENT OF MAIN RESULT

We study systems of the form

ẋ(t) = Ax(t) + Bu(g(t) -τ ) + δ 1 (t) y(t i ) = Cx(t i-r ) + δ 2 (t i ) , i ≥ r (1)
where t i = iν for a known constant ν > 0 for all integers i ≥ 0, the state x and the output y are valued in R nx and R ny respectively for any dimensions n x and n y , the constants τ > 0 and r ∈ N and the constant matrices A, B, and C are known, g is a right continuous nondecreasing piecewise continuous unbounded function that satisfies g(t) ≤ t for all t and so can be used to represent sampling in the control u, and the unknown δ i 's are assumed to be piecewise continuous and bounded and to possess finite right limits at each time t ≥ 0. We also assume that our initial functions are constant at time t 0 = 0. The definition of the sequence t i implies that

y(t i ) = Cx(t i -rν) + δ 2 (t i ).
Thus, y has the delay rν. We assume the following, where I is the identity matrix: Assumption 1: There is a matrix L ∈ R nx×ny such that

M = (I + LC)e νA (2)
is Schur stable, and τ /ν is a rational number.

The preceding assumption agrees with [18, Assumption 1] except we added the condition that τ /ν is a rational number. Although [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] also allows nonperiodic sampling in the output, [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] uses distributed terms in the controls and so does not cover sequential observers that we present here. By the continuity of the eigenvalues of a matrix as functions of the entries of a matrix, it follows from Assumption 1 that for each sufficiently large integer d, the matrix

M n = e νA + LCe A(ν-τ d ) (3)
is also Schur stable. We prove the following, where | • | I denotes the supremum over any interval I, and the superscript + refers to the right limit; see Remark 1 for the definition of solutions for our continuous-discrete systems (4)-( 5). Theorem 1: Let Assumption 1 hold, and choose any positive integers d > max{3, τ /ν} and such that τ /ν = d/ and such that M n in (3) is Schur stable. Set p = d + r, and consider the families of continuous-discrete systems

ż1 (t) = Az 1 (t) + Bu g(t + τ d -rν) -τ ∀t ∈ J i z 1 (t + i ) = z 1 (t i ) + L Cz 1 t i -τ d -y(t i ) , i ≥ r (4)
and

żj (t) = Az j (t) + Bu g(t+ jτ d -rν) -τ ∀t ∈ J i z j (t + i ) = z j (t i ) + LCψ j t i -τ d , i ≥ r (5)
for j = 2, . . . , p, where J i = (t i , t i+1 ] for all i and we set

ψ 1 (t) = z 1 (t) -x t + τ d -rν and ψ j (t) = z j (t) -z j-1 t + τ d for j = 2, . . . , p (6)
for any choice of the piecewise continuous locally bounded function u. Set Ψ = (ψ 1 , . . . , ψ p ). Then the following conclusions hold: (a) We can find positive constants c 1 and c 2 such that

|Ψ(t)| ≤ c 1 e -c2t |Ψ(t + r )| + |δ| [0,t]
for all t ≥ t r (7) holds along all solutions of (1) and ( 4)-( 5). (b) The equality

z p (t) = x (t + τ ) + p s=1 ψ s t + (p-s)τ d (8)
holds for all t ∈ R along all solutions of (1) and ( 4)-( 5). (c) If, in addition, there is a matrix K such that

ẋ(t) = Ax(t) + BKx(g(t)) + δ a (t) (9)
is exponentially input-to-state stable with respect to δ a , then the (x, Ψ) dynamics in closed loop with u(t) = Kz p (t) are input-to-state stable with respect to δ = (δ 1 , δ 2 ). Remark 1: According to (4)-( 5), the z j subsystems are continuous time ones (whose dynamics are defined by the first equations in ( 4)-( 5) for all t > 0), save for the fact that their states are reset to the initial states z j (t + i ) at the times t i for all i ≥ r when computing z j ( ) for values on the intervals J i . Conclusion (a), in conjunction with conclusion (b), implies that z p provides an observer for the state x of (1), since conclusions (a)-(b) give

|z p (t-τ ) -x (t)| ≤ pc 1 e -c2(t-τ ) |Ψ(t + r )|+sup t≥0 |δ(t)|
for all t ≥ t r + τ . Then we can use conclusion (c) to render the (x, Ψ) system input-to-state stable with respect to δ, using our dynamic control that only depends on the perturbed sampled output. When (A, B) is a controllable pair, there are many g's such that ( 9) is input-to-state stable with respect to δ 1 , e.g., if we have sufficiently frequent sampling.

III. PROOF OF THEOREM 1

The proof has two parts. First we derive key relations that will enable us to prove the conclusions (a)-(c) of the theorem. In the second part, we use the relations we obtained from the first part of the proof to prove the conclusions (a)-(c).

First Part of Proof. From the definition of the sequence t i and the definition of y, we can rewrite (4) as

ż1 (t) = Az 1 (t) + Bu g(t + τ d -rν) -τ ∀t ∈ J i z 1 (t + i ) = z 1 (t i ) + LC z 1 t i -τ d -x(t i -rν) -Lδ 2 (t i ), i ≥ r. Since ẋ t + τ d -rν = Ax t + τ d -rν +Bu g t + τ d -rν -τ + δ 1 t + τ d -rν (10)
holds for all t ≥ -τ d + rν, we deduce that ψ 1 from our definition [START_REF] Chen | Impulsive observer-based stabilisation of uncertain linear systems[END_REF] satisfies

     ψ1 (t) = Aψ 1 (t) -δ 1 t + τ d -rν , ∀t ∈ J i ψ 1 (t + i ) = z 1 (t i ) + LC z 1 t i -τ d -x(t i -rν) -x t i + τ d -rν -Lδ 2 (t i ), i ≥ r. (11)
Hence, if i ≥ r, then we have

ψ 1 (t + i ) = ψ 1 (t i ) + LCψ 1 t i -τ d -Lδ 2 (t i ) . ( 12
)
Bearing in mind that t i -τ d ∈ (t i-1 , t i ) for all i (which follows from the fact that t i+1 -t i = ν > τ /d for all i), we can integrate the first equality in [START_REF] Gonzalez | Robustness of a discrete-time predictor-based controller for timevarying measurement delay[END_REF] along (t i-1 , t i ] and then on (t i-1 , t i -τ /d], and then substitute the resulting formulas for ψ 1 (t i ) and ψ 1 (t i -τ /d) into [START_REF] Hammouri | Constant gain observer design for continuous-discrete time uniformly observable systems[END_REF], to obtain

ψ 1 (t + i ) = e Aν ψ 1 (t + i-1 ) + LCe A(ν-τ d ) ψ 1 (t + i-1 ) +N 0 (δ, i) = M n ψ 1 (t + i-1 ) + N 0 (δ, i) , (13)
where

N 0 (δ, i) = - ti ti-1 M(δ, t i , )d -Lδ 2 (t i ) -LC ti-τ /d ti-1 M(δ, t i -τ /d, )d (14)
and M(δ, s, ) = e (s-)A δ 1 ( + τ /d -rν).

Set z 0 (t) = x(t -rν). By our choices (6) of the ψ j 's,

żj (t) = Az j (t) + Bu (G(t, j) -τ ) ∀t ∈ J i z j (t + i ) = z j (t i ) + L Cz j t i -τ d -Cz j-1 (t i ) żj-1 (t) = Az j-1 (t) + Bu (G(t, j -1) -τ ) ∀t ∈ J i z j-1 (t + i ) = z j-1 (t i ) -L∆ j-1 (t i ) + L Cz j-1 t i -τ d -Cz j-2 (t i ) (15)
for all j ∈ {2, . . . , p} and integers i ≥ r, where G(t, s) = g t + sτ d -rν , and where

∆ k = δ 2 if k = 1 and ∆ k = 0 otherwise. Consequently, since τ /d < ν, ψj (t) = Aψ j (t) ∀t ∈ J i ψ j (t + i ) = ψ j (t i ) + LCψ j t i -τ d ( 16
)
holds for all integers j ∈ {2, . . . , p} and i ≥ r.

Integrating the first equation in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]

over t i -τ d , t i gives ψ j (t i ) = e Aτ /d z j t i -τ d -z j-1 t + i , (17)
by [START_REF] Chen | Impulsive observer-based stabilisation of uncertain linear systems[END_REF]. Bearing [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zeroorder hold[END_REF] in mind, we deduce that

ψ j (t i ) = e Aτ /d z j t i -τ d -z j-1 (t i ) -LC z j-1 t i -τ d -z j-2 (t i ) + L∆ j-1 (t i ) (18)
holds for all integers j ∈ {2, . . . , p} and i ≥ r. Hence,

ψ j (t i ) = e Aτ /d ψ j t i -τ d + L∆ j-1 (t i ) -e Aτ /d LCψ j-1 t i -τ d (19)
holds for all integers j ∈ {2, . . . , p} and i ≥ r, so by [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF],

     ψj (t) = Aψ j (t) , ∀t ∈ J i ψ j (t + i ) = e Aτ /d + LC ψ j t i -τ d -e Aτ /d LCψ j-1 t i -τ d + ∆ j-1 (t i ) (20)
for all integers j ∈ {2, . . . , p} and i ≥ r, where ∆ j-1 = e Aτ /d L∆ j-1 . By integrating the first equality of (20) over the interval

(t i-1 , t i -τ /d], it follows that      ψj (t) = Aψ j (t) , ∀t ∈ J i ψ j (t + i ) = e Aτ /d + LC e A(ν-τ d ) ψ j t + i-1 -e Aτ /d LCψ j-1 t i -τ d + ∆ j-1 (t i ) (21)
and therefore also

   ψj (t) = Aψ j (t) , ∀t ∈ J i ψ j (t + i ) = M n ψ j t + i-1 -e Aτ /d LCψ j-1 t i -τ d + ∆ j-1 (t i ) (22)
for all j ∈ {2, . . . , p} and i ≥ r, by the definition (3) of M n . Since the first lines of ( 11) and ( 22) (integrated along the intervals J i ) and the fact that τ /d < ν combine to give

ψ 1 t i -τ d = e A(ν-τ /d) ψ 1 (t + i-1 ) - ti-τ /d ti-1 e A(ti-τ /d-) δ 1 + τ d -rν d (23)
and

ψ j t i -τ d = e A(ν-τ /d) ψ j (t + i-1 ) (24)
for all integers i ≥ r and j ∈ {2, . . . , p}, we can combine [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] with the second equality of ( 22) to obtain the discrete time system

       ψ 1 (t + i ) = M n ψ 1 (t + i-1 ) + N 0 (δ, i) ψ j (t + i ) = M n ψ j (t + i-1 ) + N 1 ψ j-1 (t + i-1 ) + N 1 (δ, j, i) (25)
for all integers i ≥ r and integers j ∈ {2, . . . , p}, where

N 1 = -e Aτ /d LCe A(ν-τ /d) (26)
and

N 1 (δ, j, i) = δ 2 (t i ) +e Aτ /d LC ti-τ /d ti-1 e A(ti-τ /d-) δ 1 -τ d -rν d (27)
when j = 2 and N 1 (δ, j, i) = 0 if 3 ≤ j ≤ p. Second Part of Proof. We use the preceding calculations to prove conclusions (a)-(c) of the theorem. Since M n is Schur stable, it follows from the structure of the discrete time system (25) that the dynamics for the discrete time variable

D(i) = Ψ(t + i ) = (ψ 1 (t + i ), . . . , ψ p (t + i )) (28)
is exponentially input-to-state stable with respect to δ = (δ 1 , δ 2 ), by the following argument. We will use the following discrete time variant of the continuous time recursive Lyapunov function construction from [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF], [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] to derive positive constants c a and c b such that

|Ψ(t + i )| ≤ c a e -c b (ti-tr) |Ψ(t + r )| + |δ| [0,t] (29)
is satisfied for all integers i ≥ r. We begin by using standard results (e.g., [START_REF] Sontag | Mathematical Control Theory[END_REF]Lemma 5.7.19]) to find a positive definite matrix P such that

M n P M n -P = -I, (30)
and we set

V (x) = x P x. (31)
Then substituting in the formula from the right side of the second equation in [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] and then using (30) and Young's inequality and setting

∆V (j, i) = V (ψ j (t + i )) -V (ψ j (t + i-1 )) (32)
gives

∆V (j, i) = -ψ j (t + i-1 )ψ j (t + i-1 ) + N 1 ψ j-1 (t + i-1 ) + N 1 (δ, j, i) × P N 1 ψ j-1 (t + i-1 ) + N 1 (δ, j, i) +2(M n ψ j (t + i-1 )) × P N 1 ψ j-1 (t + i-1 ) + N 1 (δ, j, i) ≤ -|ψ j (t + i-1 )| 2 + |N 1 (δ, j, i)| 2 |P | + |ψ j-1 (t + i-1 )| 2 |N 1 | 2 |P | + 2{|N 1 ||P ||N 1 (δ, j, i)|}{|ψ j-1 (t + i-1 )|} +2(M n ψ j (t + i-1 )) × P N 1 ψ j-1 (t + i-1 ) + N 1 (δ, j, i) ≤ -1 2 |ψ j (t + i-1 )| 2 + N 2 |ψ j-1 (t + i-1 )| 2 + N 2 (δ, j, i) (33)
for j = 2, . . . , p and i ≥ r, where

N 2 = |N 1 | 2 |P | + 1 4 + 4|M n | 2 |P | 2 |N 1 | 2 and N 2 = |N 1 | 2 (|P | + 4|N 1 | 2 |P | 2 + 4|M n | 2 |P | 2
) and where the second inequality applied the inequality 2ab ≤ 1 4 a 2 + 4b 2 three times with suitable a and b (with a = |ψ j-1 (t + i-1 )|, and then twice with a = |ψ j (t + i-1 )|) to bound the terms in squared brackets in (33). We also have

∆V (1, i) = -ψ 1 (t + i-1 )ψ 1 (t + i-1 ) +2(M n ψ 1 (t + i-1 )) P N 0 (δ, i) +N 0 (δ, i) P N 0 (δ, i) ≤ -1 2 |ψ 1 (t + i-1 )| 2 + N 3 (δ, i) (34)
where

N 3 = (2|M n | 2 |P | 2 + |P |)|N 0 | 2
, and where we used

2(M n ψ 1 (t + i-1 )) P N 0 (δ, i) ≤ 1 2 |ψ 1 (t + i-1 )| 2 + 2|M n | 2 |P | 2 |N 0 (δ, i)| 2 . (35)
Applying (33) with the choices j = p and then j = p -1 and then adding the results gives

∆V (p, i) + (2N 2 + 1)∆V (p -1, i) ≤ -1 2 |ψ p-1 (t + i-1 )| 2 + |ψ p (t + i-1 )| 2 +2(N 2 + 1) N2 (δ, i) + (2N 2 + 1)N 2 |ψ p-2 (t + i-1 )| 2 , (36)
where N2 (δ, i) = max j N 2 (δ, j, i). By induction (by multiplying the decay estimate for ∆V (p -2, i) through by a large enough constant and then using the result to cancel the (2N 2 + 1)N 2 |ψ p-2 (t + i-1 )| 2 term in (36) and repeating this process), we obtain a nonnegative constant N 3 and constants ω i > 0 for which this exponentially input-to-state stable Lyapunov decay estimate holds along all solutions of the discrete time D dynamics from [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] for all i ≥ r:

p j=1 ω i ∆V (j, i) ≤ -ω 0 p j=1 ω i V (ψ j (t + i-1 ))+N 3 |δ| 2 [0,t] (37)
Then the desired exponentially stable input-to-state stability estimate in the D variable follows from well known discrete time input-to-state stability results [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. Conclusion (a) of the theorem follows by combining the exponential input-to-state stability estimate [START_REF] Wang | Piecewise-continuous observers for linear systems with sampled and delayed output[END_REF] with the fact that the first equalities in [START_REF] Gonzalez | Robustness of a discrete-time predictor-based controller for timevarying measurement delay[END_REF] and ( 21) provide a positive constant c such that

|Ψ(t)| ≤ c(|Ψ(t + i )| + |δ| [0,t] ) (38)
for all t ∈ J i and integers i ≥ 0, where (38) was used to convert the discrete time decay estimate [START_REF] Wang | Piecewise-continuous observers for linear systems with sampled and delayed output[END_REF] into a continuous time one in the variable Ψ(t).

To prove the remaining conclusions of the theorem, first note that by our definitions of the z j 's, we deduce that z 1 (t) = x t + τ d -rν + ψ 1 (t) and z j (t) = z j-1 t + τ d + ψ j (t) for all j ∈ {2, . . . , p} and t ≥ 0. It follows by induction on j ∈ {1, . . . , p} that

z p (t) = x t + pτ d -νr + p s=1 ψ s t + (p-s)τ d ( 39
)
for all t ∈ R. Next recall that our integer ∈ N was selected such that dν = τ , and that p -r = d. This gives

z p (t) = x t + pτ d -rτ d + p s=1 ψ s t + (p-s)τ d = x (t + τ ) + p s=1 ψ s t + (p-s)τ d , (40)
ẋ(t) = Ax(t) + BKz p (g(t) -τ ) + δ 1 (t) = Ax(t) + BKx(g(t)) + δ a (t) + δ 1 (t) (41)
for all t ≥ 0, where ) along all solutions of the closed loop system for all t ≥ 0.

δ a (t) = BK p s=1 ψ s g(t) -τ + (p-s)τ d , (42)

IV. ILLUSTRATION

Assumption 1 makes Theorem 1 applicable to a broad class of linear time invariant systems, leading to robust continuous-discrete observers that provide stabilizing delay compensating controllers without distributed terms under arbitrarily large constant input and output delays. To illustrate how Theorem 1 can be applied, consider the case where

A = 0 1 -1 0 , C = [ a b ], and ν = π 2 + 2kπ (43)
where a and b are constants and k is any nonnegative integer, for any choice of the matrix B. In this case, if we set L = [ L 1 L 2 ] for real constants L 1 and L 2 to be determined, then the matrix M from Assumption 1 takes the form

M = 1 + L 1 a L 1 b L 2 a 1 + L 2 b 0 1 -1 0 = -L 1 b 1 + L 1 a -1 -L 2 b L 2 a . (44)
Let us find conditions on the L i 's such that (44) is Schur stable, which will make it possible to satisfy Assumption 1 and apply Theorem 1. The matrix (44) has the eigenvalues

L 2 a -L 1 b ± (L 2 a -L 1 b) 2 -4(1 + L 1 a + L 2 b) 2 (45)
which will both be valued in (-1, 1) if the conditions

L 1 a + L 2 b = -1 L 1 b -L 2 a ∈ (-1, 1) (46
is exponentially input-to-state stable with respect to δ 1 , then Theorem 1 also provides an input-to-state stabilizing sampled output feedback control.

V. CONCLUSIONS

We developed new observer designs that are motivated by many control engineering applications where only sampled output values are available. We provided a new continuousdiscrete observer design for linear time invariant systems with input delays, uncertainties in the plant, sampling in the output observations, and additive uncertainties on the control and outputs. Unlike [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] and other works that use distributed terms, our sequential observer method produced controllers and stabilizing feedbacks that are free of distributed terms and which therefore may be better suited to implementations. We allow arbitrarily long constant input and output delays, and the input and output delays are allowed to differ. We hope to extend our work to aperiodic sampling in the output, and to use ideas from [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zeroorder hold[END_REF], [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] to cover nonlinear systems. We also hope to use ideas from [START_REF] Chen | Impulsive observer-based stabilisation of uncertain linear systems[END_REF] to cover cases where the matrices A and B in the original system can be uncertain.

  which gives the relation (8) from conclusion (b) of Theorem 1. Conclusion (c) follows from combining conclusions (a)-(b) and using the linear structure of the dynamics, by writing the closed loop system from conclusion (c) as

  ) are satisfied, and then the continuous-discrete observer is provided by Theorem 1. Viewing (46) as a linear system with unknowns L 1 and L 2 , we can solve (46) for values L i satisfying (46) if a and b are not both zero. If, in addition, the matrices B and K and the right continuous nondecreasing piecewise continuous unbounded function g are such that g(t) ≤ t for all t ≥ 0 and

	ẋ(t) =	0 1 -1 0

x(t) + BKx(g(t)) + δ 1 (t)


Frédéric Mazenc

Laurent Burlion

Victor Gibert

Stabilization with imprecise measurements: application to a vision based landing problem

Keywords: Stabilization, delay, backstepping, bounded control, vision-based control

The problem of stabilizing a nonlinear system approximated in a neighborhood of the origin by a saturated chain of integrators when the variables are not accurately measured is addressed. A recent backstepping approach with delay is adapted to the specificity of the studied system. The results are used to solve a control problem that arises in the context of vision based landing of a civil aircraft.

I. INTRODUCTION

The difficult problem of stabilizing systems with bounded control has a long history. To solve it, several techniques of control design have been developed for several families of systems. Semi-global state and output feedback stabilization results have been obtained via linear control laws acting inside a saturation function, as done for instance in the contribution [START_REF] Saberi | Control of linear systems with saturating actuators[END_REF]. Crucial regional stability results for families of linear and nonlinear systems, which rely in particular on LMI techniques, are presented in the research monograph [START_REF] Tarbouriech | Queinnec, Stability and stabilization of linear systems with saturating actuators[END_REF]. The techniques called the forwarding and the bounded backstepping provide with globally asymptotically stabilizing control laws for some families of nonlinear systems. Bounded backstepping has been developed in particular in the contributions [START_REF] Freeman | Integrators backstepping for bounded controls and control rates[END_REF] and [START_REF] Mazenc | Backstepping with Bounded Feedbacks[END_REF] and forwarding in many contributions, notably in [START_REF] Teel | Feedback stabilization: Nonlinear solutions to inherently nonlinear problems[END_REF], [START_REF] Mazenc | Adding an Integration and Global Asymptotic Stabilization of Feedforward Systems[END_REF] and [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF].

Recently, a significantly different backstepping design has been proposed in the papers [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF] and [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF]. It relies on the introduction in the expression of the control laws of artificial pointwise delays which circumvent the problem of determining Lie derivatives of the fictitious control laws used in the classical approach. Thus, it makes it possible to relax the smoothness requirement which is imposed on the fictitious control in all the previous contributions on backstepping. Moreover, for many systems of feedback or feedforward form, it can be adapted to the problem of determining stabilizing bounded feedbacks and leads to analytic expressions that are simpler than those of the feedbacks constructed in [START_REF] Freeman | Integrators backstepping for bounded controls and control rates[END_REF] and [START_REF] Mazenc | Backstepping with Bounded Feedbacks[END_REF]. Finally, by contrast with other techniques, global stability results are obtained via this approach.

Mazenc is with Inria, University Paris-Saclay, L2S (CNRS) Cen-traleSupélec, Gif-sur-Yvette, France. frederic.mazenc (at) l2s.centralesupelec.fr The advantages of the approach of [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF] led us in the present work to adapt it to a specific control design problem motivated by an engineering application. Indeed, we here interest us to the emergency landing of a civil aircraft : the runway being unequipped, the aircraft must rely an embedded video camera which, from the control point of view, poses the problem of controlling a perspective dynamical system [START_REF] Karagiannis | A new solution to the problem of range identification in perspective vision systems[END_REF]. In the present paper, we focus on the lateral motion and develop a new control law for a specific three dimensional feedforward system. Due to vision, the key difficulty presented by the model, is that only imprecise measurements of the two first state variables are available. As a consequence, we cannot directly apply results providing with semi-global or regional stability mentioned above, nor the design proposed in [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF] or its extension [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF]. Thus, we propose a new theoretical control design technique, which extends the theory presented in [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF] and [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF]. This theoretical result is new but owes a great deal to the forwarding theory [START_REF] Teel | Feedback stabilization: Nonlinear solutions to inherently nonlinear problems[END_REF], [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF]. The control laws we construct are bounded and result in a globally asymptotically stable closed-loop system. An important part of our work is devoted to the assessment of the control design through numerical simulations based on real data.

The paper is organized as follows. The problem is presented in Section II. Technical lemmas are stated in Section III. The main result is in Section IV. The main result is illustrated in the specific context of vision based landing of a civil aircraft in Section V. Concluding remarks in Section VI end the paper.

A. Notation

The notation will be simplified whenever no confusion can arise from the context. The Euclidean norm in R a , and the induced norm of matrices, are denoted by | • |. Given any constant T > 0, we let C in denote the set of all continuous functions φ : [-T, 0] → R a , which we call the set of all initial functions. We define Ξ t ∈ C in by Ξ t (s) = Ξ(t + s) for all choices of Ξ, s ≤ 0, and t ≥ 0 for which the equality is defined.

Given L > 0, sat L denotes the classical symmetric saturation function i. e. sat L (x) = max(-L, min(L, x)), ∀x ∈ R.

(

II. PROBLEM STATEMENT We consider the system

   ẋ1 = sat L 1 (x 2 ) ẋ2 = sat L 2 (x 3 ) ẋ3 = sat L 3 (u), (2)
where

(x 1 , x 2 , x 3 ) ∈ R 3 , u ∈ R is the input and L i > 0, i = 1, 2, 3.
We assume that the outputs are:

y 1 (t) = η(t)x 1 (t) + δ 1 (t), y 2 (t) = x 2 (t) + δ 2 (t), y 3 (t) = x 3 (t), (3)
where δ 1 , δ 2 and η are unknown but piecewise-continuous functions for which there are constants η > 0, η > η, δ 1 ≥ 0 and δ 2 ≥ 0 such that

η(t) ∈ [η, η] , |δ i (t)| ≤ δ i , i = 1, 2 (4)
for all t ≥ 0. The objective of this paper is the global robust stabilization of the origin of the system (2) through a bounded feedback using only the measurements y i in [START_REF] Gibert | Nonlinear observers in vision system: Application to civil aircraft landing[END_REF].

The fact that η(t) is present in y 1 is a serious difficulty. It implies that classical approaches do not apply. Although the system (2) is a feedforward system (see for instance [START_REF] Mazenc | Adding an Integration and Global Asymptotic Stabilization of Feedforward Systems[END_REF] for the definition of feedforward system), the forwarding approach (notably the results of [START_REF] Mazenc | Adding an Integration and Global Asymptotic Stabilization of Feedforward Systems[END_REF] and [START_REF] Teel | Feedback stabilization: Nonlinear solutions to inherently nonlinear problems[END_REF]) does not apply, except under an additional assumption pertaining to the size of η -η. But for practical reasons, we cannot impose an assumption of this type.

Since ( 2) is not in feedback form, classical backstepping results (see instance [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]) do not apply. Since the function η is unknown, not necessarily differentiable and with a possibly unbounded first derivative, bounded backstepping results in [START_REF] Mazenc | Backstepping with Bounded Feedbacks[END_REF], [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF] do not apply. Other approaches (see for instance [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF]) do not apply too. Finally, let us observe that one cannot directly apply the result of [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF], which does not take into account the case where there are uncertainties on the measurements. Thus, to the best of our knowledge, the problem we solve is open.

III. TECHNICAL PRELIMINARY RESULTS

A. Stability of a system with a distributed delay

The following lemma will have a fundamental role when we establish the main result of our paper.

Lemma 1:

Consider the system ξ = α(t, ξ t ) + d(t), (5)
where

ξ ∈ R, d : [0, +∞) → R is a piecewise continuous function such that |d(t)| ≤ d for all t ≥ 0, α(t, ξ t ) = -s t t-τ e q( -t) -τ e r(m-) φ (η(m)ξ (m))dmd , (6)
where the constant s is defined by:

s = qr (1 -e -qτ ) (1 -e -rτ ) , (7)
with τ > 0 q > 0, r > 0 and where φ : R → R is (i) odd, (ii) non-decreasing, (iii) globally Lipschitz, with Lipschitz constant φ c > 0 and such that there are positive constants φ s and φ such that

lim s→+∞ φ (s) = φ , φ s s ≤ φ (s) , ∀s ∈ 0, φ 2 . ( 8
) Moreover, 8τφ c η(φ + d) + 4d ≤ φ (9)
and τp < c [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF] with

p = φ 2 c η 2 and c = inf a∈E φ (η|a|) 3|a| (11)
where

E = {b ∈ R : 4τφ c η(φ + d) + 2d ≥ φ (η|b|)}. ( 12
)
Then there is t ≥ 4τ such that for all t ≥ t ,

|ξ (t)| ≤ (2φ c ητ + 1)d 2c(c -τp) . (13)
Remark. 1. The system ( 5) cannot be Input-to-State-Stable with respect to d because α is bounded. 2. The inequality [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] ensures that there is

k > 0 such that E = [-k, k]. 3.
The relations in [START_REF] Mazenc | New Control Design for Bounded Backstepping under Input Delay[END_REF] ensure that c > 0.

Proof. First, notice that finite escape time phenomenon does not occur for (5) because the function φ is bounded. Second, consider the positive definite quadratic function

V (ξ ) = 1 2 ξ 2 (14)
and let us define for later use the function

W (ξ ) = ξ φ (ηξ ). (15)
Simple calculations give

V (t) = -sξ (t) t t-τ e q( -t) -τ e r(m-) φ (η(m)ξ (t))dmd +sξ (t) t t-τ e q( -t) -τ e r(m-) φ (η(m)ξ (t)) -φ (η(m)ξ (m)) dmd + ξ (t)d(t) ≤ -sξ (t) t t-τ e q( -t) -τ e r(m-) φ (η(m)ξ (t))dmd +sφ c |ξ (t)| t t-τ e q( -t) -τ e r(m-) |η(m)| ×|ξ (t) -ξ (m)|dmd + ξ (t)d(t), (16)
where the last inequality is a consequence of the fact that φ is Lipschitz with Lipschitz constant φ c . From the other properties of φ and the fact that, for all t ≥ 0,

s t t-τ e q( -t) -τ e r(m-) dmd = 1, we deduce successively that V (t) ≤ -sξ (t) t t-τ e q( -t) -τ e r(m-) φ (ηξ (t))dmd +sφ c |ξ (t)| t t-τ e q( -t) -τ e r(m-) η ×|ξ (t) -ξ (m)|dmd + ξ (t)d(t) = -W (ξ (t)) + sφ c η|ξ (t)| t t-τ e q( -t) -τ e r(m-) ×|ξ (t) -ξ (m)|dmd + ξ (t)d(t). ( 17
)
Now, observe that for all m ∈ R,t ∈ R such that 0 ≤ m ≤ t, the inequality

|ξ (m) -ξ (t)| ≤ t m | ξ (s)|ds (18)
is satisfied. It follows that for all t ≥ 2τ,

V (t) ≤ -W (ξ (t)) + sφ c η|ξ (t)| t t-τ e q( -t) -τ e r(m-) × t m | ξ (s)|dsdmd + ξ (t)d(t) ≤ -W (ξ (t)) + φ c η|ξ (t)| t t-2τ | ξ (s)|ds + ξ (t)d(t), (19)
where the last inequality is a consequence of the fact that

t m | ξ (s)|ds ≤ t t-2τ | ξ (s)|ds for all m ∈ [t -2τ,t]. Now, observe that | ξ (t)| ≤ s t t-τ e q( -t) -τ e r(m-) φ (η|ξ (m)|)dmd +|d(t)| ≤ φ + d.
(20) Combining the last inequality with (19), we obtain

V (t) ≤ -W (ξ (t)) + [2τφ c η(φ + d) + d]|ξ (t)| = -1 2 W (ξ (t)) + 2τφ c η(φ + d) + d -1 2 φ (η|ξ (t)|) |ξ (t)|. (21) Therefore when 4τφ c η(φ + d) + 2d ≤ φ (η|ξ (t)|) (22) then V (t) ≤ - 1 2 W (ξ (t)). ( 23
)
Since W is positive definite, we deduce there is t 0 ≥ 4τ such that for all t ≥ t 0 , |ξ (t)| ∈ E, where E is the set defined in [START_REF] Oudin | Low Speed Protections for a Commercial Airliner: a Practical Approach[END_REF]. Then we deduce from (19) and the definition of c in (11) that

V (t) ≤ -3cξ (t) 2 + φ c η|ξ (t)| t t-2τ | ξ (s)|ds + ξ (t)d(t) (24) and | ξ (t)| ≤ φ c ηs t t-τ e q( -t) -τ e r(m-) |ξ (m)|dmd + |d(t)| (25) for all t ≥ t 0 . We deduce that V (t) ≤ -3cξ (t) 2 + φ c η|ξ (t)| t t-2τ φ c ηs s s-τ e q( -s) × -τ e r(m-) |ξ (m)|dmd + |d(s)| ds +ξ (t)d(t) ≤ -3cξ (t) 2 + ps t t-2τ s s-τ e q( -s) × -τ e r(m-) |ξ (t)||ξ (m)|dmd ds + Ω(t)|ξ (t)| (26)
with p defined in [START_REF] Mazenc | Adding an Integration and Global Asymptotic Stabilization of Feedforward Systems[END_REF] and

Ω(t) = φ c η t t-2τ |d(s)|ds + |d(t)|. ( 27
)
Using

|ξ (t)||ξ (m)| ≤ 1 2 ξ (t) 2 + 1 2 ξ (m) 2 , we obtain V (t) ≤ -3c + τφ 2 c η 2 ξ (t) 2 + 1 2 ps t t-2τ s s-τ e q( -s) × -τ e r(m-) ξ (m) 2 dmd ds + Ω(t)|ξ (t)| ≤ -2cξ (t) 2 + ps 2 t t-2τ s s-τ e q( -s) × -τ e r(m-) ξ (m) 2 dmd ds + Ω(t)|ξ (t)|, (28)
where the last inequality is a consequence of [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF]. Then, using the inequality Ω(t)|ξ (t)| ≤ cξ (t) 2 + 1 4c Ω(t) 2 , we obtain

V (t) ≤ -cξ (t) 2 + ps 2 t t-2τ s s-τ e q( -s) × -τ e r(m-) ξ (m) 2 dmd ds + 1 4c Ω(t) 2 . ( 29
)
Since Ω(t) ≤ (2φ c ητ + 1)d, we deduce that

V (t) ≤ -2cV (ξ (t)) + 2τp sup a∈[t-4τ,t]
V (ξ (a))

+ (2φ c ητ+1) 2 d 2 4c . (30)
Then [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF] allows us to conclude.

B. Robust stabilization of a system

In this section, we establish a result of robust stabilization for the system

λ1 = sat U 1 (λ 2 ) λ2 = sat U 2 (ρ), (31)
with (λ 1 , λ 2 ) ∈ R 2 , U 1 > 0 and U 2 > 0 and where ρ is the input. This result draws its inspiration from [START_REF] Teel | Feedback stabilization: Nonlinear solutions to inherently nonlinear problems[END_REF] and [START_REF] Mazenc | Global Asymptotic Stabilization for Chains of Integrators with a Delay in the Input[END_REF]. Let q, r, v and w be positive constants satisfying

v < min U 2 2 , qU 1 , w ≤ v 3 . ( 32
)
Lemma 2: Let us consider the system (31) with

ρ(t) = -qλ 2 (t) + sat v (-rqλ 1 (t) -rλ 2 (t) + w(t)), ( 33
)
where w is a function such that, for all t ≥ 0,

|w(t)| ≤ w. ( 34
) Let λ 3 = λ 2 +qλ 1 .
Then there is t * ≥ 0 such that for all t ≥ t * ,

λ1 = -qλ 1 + λ 3 λ3 = -rλ 3 + w. ( 35
)
Proof. To ease the proof, let us introduce the simplifying notation

v(t) = sat v (-rqλ 1 (t) -rλ 2 (t) + w(t)). (36)
Then |v(t)| ≤ v for all t ≥ 0 and

λ2 = sat U 2 (-qλ 2 (t) + v(t)) . (37)
We deduce that for any ε > 0, there is

t ε ≥ 0 such that for all t ≥ t ε , |λ 2 (t)| ≤ (1 + ε) v q . ( 38
)
Consequently,

|-qλ 2 (t) + v(t)| ≤ (2 + ε)v. ( 39
)
From the first inequality in (32), it follows that there is ε p > 0 such that (2 + ε p )v < U 2 and (1 + ε p ) v q < U 1 . Then it follows from (38) and (39) that, for all t ≥ t ε p ,

|λ 2 (t)| < U 1 (40) and |-qλ 2 (t) + v(t)| < U 2 . (41)
As an immediate consequence, we have that, for all t ≥ t ε p ,

λ1 = λ 2 λ2 = -qλ 2 + sat v (-rqλ 1 -rλ 2 + w). (42)
Now, using λ 3 = λ 2 + qλ 1 , we have, for all t ≥ t ε p ,

   λ1 = -qλ 1 + λ 3 λ3 = -qλ 2 + sat v (-rqλ 1 -rλ 2 + w) + q(-qλ 1 + λ 3 ) = sat v (-rqλ 1 -rλ 2 + w). (43) Finally, λ1 = -qλ 1 + λ 3 λ3 = sat v (-rλ 3 + w). ( 44
)
Since |w(t)| ≤ w for all t ≥ 0, we deduce that for all δ > 0, there is T δ ≥ t ε p such that, for all t ≥ T δ ,

|λ 3 (t)| ≤ (1 + δ )w r . (45)
As an immediate consequence,

| -rλ 3 (t) + w(t)| ≤ r (1 + δ )w r + w = (2 + δ )w ≤ (2 + δ ) v 3 , (46)
where the last inequality is a consequence of the second inequality in (32). Choosing δ = 1 2 , we obtain

| -rλ 3 (t) + w(t)| ≤ v, (47)
for all t ≥ T1 2 . It follows that for all t ≥ T1 2 , (35) is satisfied.

IV. MAIN RESULT

We are ready to state and prove the main result of the paper.

Theorem 1: Let φ and τ be respectively a function and a constant chosen as in Lemma 1 with d = 2φ c δ 1 + 2δ 2 . Consider system (2) in closed loop with

u(Y t ) = -qy 3 (t) + sat v (-rqy 2 (t) -ry 3 (t) + ς (y 1t )) (48) with Y = (y 1 , y 2 , y 3 ), ς (y 1t ) = -sφ (y 1 (t)) + s (e -rτ + e -qτ ) φ (y 1 (t -τ))
-se -(q+r)τ φ (y 1 (t -2τ)), (49) with s defined in [START_REF] Mazenc | Backstepping with Bounded Feedbacks[END_REF] and where r > 0, q > 0, v and φ are constants such that

v < min L 3 2 , qL 2 , v ≤ 2rL 1 (50) and 1 + e -rτ + e -qτ + e -(q+r)τ φ ≤ v 6s . (51)
Then, when

δ 2 ≤ v 6qr , (52)
there is t ≥ 0 such that for all t ≥ t , X(t) = (x 1 (t), x 2 (t), x 3 (t))

|X(t)| ≤ ψ(δ 1 , δ 2 ) (53) with ψ(a, b) = x2 1 (a, b) + 10 9 qb + φ c η x1 (a, b) 2 +q 2 (1 + q) 2 10b 9 + q(1+e -qτ )φc 1-e -qτ x1 (a, b) 2 , x1 (a, b) = √ 2(2φ c ητ+1)(φ c a+b) √ c(c-τp)
.

(54) Remark. The key feature of u(Y t ) defined in (48) is that it depends only on the measurements and has no distributed term. Also, it is worth noticing that the functions φ and η do not need to be of class C 1 , which is important from an applied point of view.

Proof. The proof is omitted.

V. ILLUSTRATION A. Scenario and numerical data

φ i o j o x b y b Δ ψ Δ X Δ Y V Fig. 1.

Notations used in the alignment part of the glide phase

To illustrate our results, we consider the lateral dynamics of an Airbus airliner which must align with a runway using a body fixed monocular camera. This problem is a challenge of strong relevance in the case the runway is unequipped or in the case of GPS loss. More precisely, the position, size and heading of the runway are unknown, so that the relative position (∆ X , ∆ Y ) and heading ∆ ψ of the aircraft with respect to it are unmeasured (Note that these quantities are depicted on Figure 1) .

Airbus airliners safety is enhanced thanks to some flight envelope protections laws which can be automatically or fully automated [START_REF] Traverse | Airbus fly-by-wire: A total approach to dependability[END_REF]. For instance a protection, called high angle-of-attack prevents the aircraft from stalling [START_REF] Oudin | Low Speed Protections for a Commercial Airliner: a Practical Approach[END_REF]. Other protection laws are provided during lateral maneuvers: for instance, they can be designed

• either to limit the relative heading ∆ ψ between the aircraft and the runway (which is required to keep the runway in the field of view of the camera), • or to limit the aircraft roll angle ϕ and its time derivative φ (which is required to protect the aircraft from exceeding its operating limits). All these protection laws directly act at the control surface deflection level. In case of control saturations or multiple active flight control laws, the aircraft response should be as fast as possible while respecting the highest priority constraint.

As discussed in [4], a key step for the successful transfer to the aeronautics sector of the developed control methods is their demonstration on standardized industrial V&V processes. The control design methods are validated by using more and more complex aircraft models. Once the tuning process is satisfactory, the methods are finally applied to the nonlinear and most representative aircraft model.

The starting point of this research was thus a very simplified lateral guidance model provided by Airbus:

   ∆Y = V sat L ψ (∆ ψ ) ∆ψ = g V sat L ϕ (ϕ) φ = sat L u (u lat ), (55)
where u lat is the input, where V = 70m.s -1 is constant all along the final approach, g = 9.81m.s -2 and ϕ (resp. u lat ) is the aircraft roll angle (resp. the guidance/outer loop control action). After a simple change of coordinates, it is obvious that such a model belongs to the class of system [START_REF] Freeman | Integrators backstepping for bounded controls and control rates[END_REF]. Clearly, this model does not have any physical interpretation nonetheless it is currently used by Airbus for its lateral guidance design studies: indeed, even if this model is very simplified, it allows the user to preliminary tune a guidance law while keeping in mind that the full model possesses some protection laws. As a result, using such a model should provide a first quantitative feedback of the quality of the guidance design.

In our context and using the extraction method proposed in [START_REF] Burlion | Vision based anti-windup design with application to the landing of an Airliner[END_REF], we also assume that the (possibly noisy) quantities ∆ Y ∆ X and ∆ ψ can be derived by image processing from the body fixed monocular camera at each instant. The unknown term ∆ X = η -1 is the longitudinal distance between the aircraft center of gravity and the runway touchdown point (as shown on Figure 1). In this application, we focus on the glide phase just before switching to the final flare phase. In this phase, the term ∆ X (t = 0) belongs to [4000m, 6000m] and evolves at a constant speed V.

Moreover, it is assumed that ϕ can be perfectly measured by using the aircraft IMU 1 .

It is easy to see that system (55) together with the measured outputs can be transformed into system (2)-(3)

1 Inertial Measurement Unit by applying the change of coordinates x 1 = ∆ Y , x 2 = V ∆ ψ , x 3 = gϕ while changing the saturation limits accordingly and by multiplying the measured outputs by some known constants. The saturations limits are as follows:

L 1 = 25m.s -1 , L 2 = 7m.s -2 , L 3 = 2m.s -3 . (56)
Note that they correspond to L ψ ∼ = 20deg, L ϕ ∼ = 40deg and L u ∼ = 12deg/s.

B. Controller gains tuning

We first propose to use the measure ∆ Y ∆ X and to multiply it by ∆X = ∆X (t = 0) -V t where ∆X (t = 0) ∈ [4500m, 6000m] is a tuning parameter (another possibility would be to design a vision based estimation of ∆ X following for instance the work of [START_REF] Karagiannis | A new solution to the problem of range identification in perspective vision systems[END_REF], [START_REF] Gibert | Nonlinear observers in vision system: Application to civil aircraft landing[END_REF]). We decided to choose the minimum value ∆X (t = 0) = 4500 to obtain η = 1.

(57)

We thus used the following quantity in our control law:

y 1 = η(t)∆ Y = 4500 -V t ∆ X ∆ Y (58)
the following parameters were chosen for the unconstrained control law in order to satisfy some industrial specifications:

q = 0.3 , r = 0.59. (59)

C. Numerical results

Our design is applied on the aircraft benchmark which is a non-trivial application. Clearly this example goes beyond what is proved in the paper since simulations stop when ∆ X = 400m which corresponds to the beginning of the flare phase: this is a finite time problem and global stabilization results are lost. Interestingly, our control method succeeds to considerably lower the lateral deviation ∆ Y for a set of initial conditions δ X (0) ∈ [4500m, 6000m], ∆ Y (0) ∈ [-30m, 30m], ∆ ψ ∈ [-3deg, 3deg] as shown on Figure 2. As shown on Figure 3, the method is still efficient when we the vision based outputs are noisy (we here chose δ 1 (t) = 4.10 -3 ∆X (t) and δ 2 = 0.3deg).

D. Further discussion

As shown on Figure 4, the numerical results are even better when multiplying φ by 2. In this case, the conditions of our results are violated although the control law u lat is still far from its limitations. As a consequence, it may be possible to improve the control law performance. This issue is postponed to future studies. We adapted a recent backstepping strategy to the problem of stabilizing a class of nonlinear systems when the output measurements are imprecise. Our design was illustrated on the lateral dynamics of a vision based landing aircraft problem. Ongoing work is devoted to extend our design on a more representative and complex aircraft model. Such a design will finally be validated in simulation on an industrial platform including image processing algorithms on synthetic images.

Also, for φ and

  τ, we choose φ (s) = 0.15tanh(0.4s) (60) and τ = 2s in order to satisfy inequality (9) when d = 0. (Let us observe that we thus have φ = 0.15 and φ c = 0.06). The conditions of our main result are also satisfied choosing: v = 1.02. (61)

Fig. 2 .Fig. 3 .

 23 Fig. 2. Evolution of ∆ Y (t) and u lat(t) (below)

  Burlion is with the Department of Information Processing and Systems, ONERA -The French Aerospace Lab, Toulouse, France. Burlion was partly supported by ANR VISIOLAND project ANR-13-CORD-0012 dedicated to VISIOn based aircraft LANDing techniques. laurent.burlion (at) onera.fr

Gibert is with Airbus Operations, S.A.S, Toulouse, France.

victor.gibert (at) airbus.com