Frédéric Mazenc

Sampled-Data Observers: Scarce Arbitrarly Large Sampling Intervals

Keywords: Observer, continuous-discrete, asynchronous sampling

. A scarcety condition on these intervals is exhibited. This result is established through a recent stability analysis technique called trajectory based approach.

I. INTRODUCTION

In many engineering applications, the measurements are available at discrete instants only. Moreover, digital implementation sometimes affects continuous-time observers. These facts are obstacles to the design or the use of continuous observers. This motivated many works on the design of observers using discrete outputs, as illustrated for instance by the papers [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF], [START_REF] Mazenc | Design of Continuous-Discrete Observers for Time-Varying Nonlinear Systems[END_REF] and [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] and the references therein. In the pioneering work [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF], an important approach was initiated for the design of observers for continuous-time nonlinear systems with discrete outputs. In this paper, a system of the type:

ẋ(t) = F(x(t)) , y(t) = Cx(t i ) , ∀t ∈ [t i , t i+1 ) , (1)
with x ∈ R n , y ∈ R q and where F is a nonlinear function was considered. A standard assumption on the sequence t i was imposed: the existence of a constant T > 0 such that for all i ∈ N, 0 ≤ t i+1 -t i ≤ T was assumed. Then the main result was the design of a new type of observer and the determination of a constant T > 0 ensuring that if T ≤ T , then the proposed observer converges exponentially to the solutions of [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF]. A key aspect of this result is that the size of all the sampling intervals must be smaller than a constant which depends on F and C, otherwise the convergence of the observer is not guaranteed. But in practice, due for instance to a temporary loss of measurements, the outputs are sometimes not available during long time intervals. Then the condition T ≤ T mentioned above may be violated. But this fact does not imply that the solutions of the observer of [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] do not converge to the solutions of the system (2): the intuition suggests that if, for many integers i, t i+1 -t i is sufficiently small and t i+1 -t i is large for a sufficiently small number of integers i, then the solutions of the observer can still converge to the solutions of the studied system.

The present work is build upon this intuitive idea. For systems with asynchronous sampling belonging to the subfamily of the one studied in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF], we show that the assumption T ≤ T can be relaxed and that an arbitrarily large value for T is allowed, provided that t i+1 -t i is sufficiently small for a sufficiently large number of integers i.

We prove the main result of our work via the trajectory based approach, which is a stability analysis for nonlinear time-varying systems introduced in [START_REF] Mazenc | Trajectory Based Approach for the Stability Analysis of Nonlinear Systems with Time Delays[END_REF], developed in [START_REF] Mazenc | Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory Based Approach[END_REF] and applied to the design of observers for continuous-time switched systems in [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF]. This approach makes it possible to establish the main result under assumptions which can be checked in practice.

The notation will be simplified whenever no confusion can arise form the context. The Euclidean norm is denoted by | • |. We denote by I the identity matrix of any dimension.

The paper is organized as follows. The main result is stated and proved in Section II. An illustrative example is given in Section III. Concluding remarks are drawn in Section IV.

II. MAIN RESULT

A. Studied system

We consider the system

ẋ(t) = Hx(t) + ϕ(Cx(t)) , y(t) = Cx(t i ) , ∀t ∈ [t i , t i+1 ) , (2)
with i ∈ N, x ∈ R n is the state, y ∈ R q is the output, C ∈ R q×n , C = 0 is a constant matrix, ϕ is a nonlinear function, H ∈ R n×n is a constant Hurwitz matrix and t i is an increasing sequence with t 0 = 0 and lim i→+∞ t i = +∞.

Let σ : [0, +∞) → [0, +∞) be the function defined by σ(t) = t i when t ∈ [t i , t i+1 ).

Remark.

The assumption that H in (2) is Hurwitz is not restrictive. Indeed, for any system ẋ(t) = Ax(t) + φ(Cx(t)) such that (A, C) is detectable, there is a matrix L so that the matrix A + LC is Hurwitz. Then the system ẋ(t) = Ax(t) + φ(Cx(t)) can be rewritten as ẋ(t) = Hx(t) + ϕ(Cx(t)) with H = A + LC and ϕ(y) = φ(y) -Ly and this system is of the type (2).

Remark. Since the matrix H is Hurwitz, there are constants c 1 > 0, p 1 > 0 and p 2 > 0 and a symmetric positive definite matrix P ∈ R n×n such that the LMIs:

P H + H P ≤ -2c 1 P (3)
and

p 1 I ≤ P ≤ p 2 I (4)
are satisfied.

We introduce three assumptions.

Assumption A1. There is a constant k ϕ > 0 such that for all y 1 ∈ R q and y 2 ∈ R q , the inequality

|ϕ(y 1 ) -ϕ(y 2 )| ≤ k ϕ |y 1 -y 2 | (5)
holds.

Assumption A2. There are 2 constants T > 0 and T > 0 such that for all i ∈ N, the inequality

T ≤ t i+1 -t i ≤ T (6)
is satisfied.

Let us introduce the constant

b 1 = k 2 ϕ |P ||CH| 2 c 1 p 1 (7)
and the function

b 2 (t) = b 1 2|C|k ϕ (t -σ(t)) e 2|C|kϕ(t-σ(t)) -1 . (8)
Assumption A3. There are constants g > 0, δ ∈ (0, 1) and t ≥ g such that

e -gc1 + t t-g e c1(r-t) b 2 (r)dr ≤ δ (9)
for all t ≥ t .

Remark. The condition ( 9) is not a restriction on the constant T in ( 6): it can be satisfied for an arbitrarily large constant T , as illustrated in Section III.

B. Observer

Let us introduce the dynamic extension:

   ω(t) = CHz(t) + Cϕ(ω(t)) , t ∈ [t i , t i+1 ) ω(t i ) = Cx(t i ) , i ∈ N , ż(t) = Hz(t) + ϕ(ω(t)) , t ≥ 0 (10) with i ∈ N, ω ∈ R q .
This dynamic extension was proposed in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF].

We are ready to state and prove the main result: Theorem 1. Assume that the system (2) satisfies Assumptions A1 to A3. Then the system (10) is an asymptotic observer for the system (2).

Proof. Since the system (2) satisfies Assumptions A1 to A3, the solutions of the system (2)-( 10) are well-defined and the system (2)-( 10) is forward complete. Now, let us introduce the variables:

e ω = ω -Cx , e x = z -x . (11)
Elementary claculations give:

           ėω (t) = CHz(t) + Cϕ(ω(t)) -CHx(t) -Cϕ(Cx(t)) , ∀t ∈ [t i , t i+1 ) , e ω (t i ) = 0 , ėx (t) = Hz(t) + ϕ(ω(t)) -Hx(t) -ϕ(Cx(t)) , ∀t ∈ [t i , t i+1 ) (12) for i ∈ N. Thus, we have            ėω (t) = CHe x (t) + Cϕ(ω(t)) -Cϕ(Cx(t)) , ∀t ∈ [t i , t i+1 ) , e ω (t i ) = 0 , ėx (t) = He x (t) + ϕ(ω(t)) -ϕ(Cx(t)) , ∀t ∈ [t i , t i+1 ) . (13)
Assumption A1 ensures that there is a function θ, bounded in norm by k ϕ , such that

           ėx (t) = He x (t) +θ(x(t), ω(t))e ω (t) , ∀t ∈ [t i , t i+1 ) , ėω (t) = Cθ(x(t), ω(t))e ω (t) +CHe x (t) , ∀t ∈ [t i , t i+1 ) , e ω (t i ) = 0 . (14)
By integrating the e ω -subsystem of ( 14) and using the bound of θ, we deduce that there is a function ζ 1 (t, s), bounded by µ(t, s) = |CH|e |C|kϕ(t-s) , such that

e ω (t) = t ti ζ 1 (t, s)e x (s)ds , ∀t ∈ [t i , t i+1 ) . ( 15
)
This equality and the e x -subsystem in [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] give

ėx (t) = He x (t) + t ti ζ 2 (t, s)e x (s)ds , t ∈ [t i , t i+1 ) , (16) with ζ 2 (t, s) = θ(x(t), ω(t))ζ 1 (t, s).
Now, we analyze the stability properties of (16) by using the positive definite quadratic function:

V (e x ) = e x P e x . (17)
The LMI (3) ensures that the derivative of this function along the trajectories of (16) satisfies

V (t) ≤ -2c 1 V (e x (t
V (t) ≤ [-2c 1 + a(t -t i )]V (e x (t)) + 1 a t ti k 2 ϕ µ(t, s) 2 |P ||e x (s)| 2 ds ≤ [-2c 1 + a(t -t i )]V (e x (t)) + k 2 ϕ |P | ap1 t ti µ(t, s) 2 V (e x (s))ds = [-2c 1 + a(t -t i )]V (e x (t)) + k 2 ϕ |P ||CH| 2 ap1
t ti e 2|C|kϕ(t-s) V (e x (s))ds , (21) where the second inequality above is a consequence of (4). Let > 0 be any constant and let

a = c 1 t -t i + . Then V (t) ≤ -2c 1 + c1 t-ti+ (t -t i ) V (e x (t)) + t-ti+ c1 k 2 ϕ |P ||CH| 2 p1 t ti e 2|C|kϕ(t-s) V (e x (s))ds ≤ -c 1 V (e x (t)) + t-ti+ c1 k 2 ϕ |P ||CH| 2 p1
t ti e 2|C|kϕ(t-s) V (e x (s))ds .

(22) Since is arbitrarily small, it follows that

V (t) ≤ -c 1 V (e x (t)) +b 1 (t -t i ) t ti e 2|C|kϕ(t-s) V (e x (s))ds , (23)
where b 1 is the constant defined in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF]. Since t i+1 -t i ≤ T for all i ∈ N, we have

V (t) ≤ -c 1 V (e x (t)) +b 1 (t -σ(t)) t σ(t) e 2|C|kϕ(t-s) ds sup s∈[t-T ,t]
V (e x (s)) .

(

) Thus V (t) ≤ -c 1 V (e x (t)) + b 2 (t) sup s∈[t-T ,t] V (e x (s)) , ( 24
) 25
where b 2 is the function defined in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]. Now, one cannot apply Halanay's inequality [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF] or Razumikhin's theorem [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 1] to (25) because for some instants t ≥ 0, b 2 (t) may be larger than c 1 . But we can apply the trajectory based approach, initiated in [START_REF] Mazenc | Trajectory Based Approach for the Stability Analysis of Nonlinear Systems with Time Delays[END_REF]. By integrating (25) over [t -g, t], where g is the constant in Assumption A3, we obtain:

V (e x (t)) ≤ e -gc1 V (e x (t -g))

+ t t-g e c1(r-t) b 2 (r)dr sup s∈[t-T -g,t]
V (e x (s))

for all t ≥ T + g. It follows that

V (e x (t)) ≤ e -gc1 + t t-g e c1(r-t) b 2 (r)dr × sup s∈[t-T -g,t]
V (e x (s))

≤ δ sup s∈[t-T -g,t]
V (e x (s))

(27)
for all t ≥ T +g, where the last inequality is a consequence of Assumption A3. This inequality and [10, Lemma 1] allow us to conclude that e x (t) converges exponentially to the origin. Now, from ( 15) and ( 6), we deduce that e ω (t) converges exponentially to the origin. This concludes the proof.

Remark. One can apply the main result of [START_REF] Mazenc | Extension of Razumikhin's Theorem and Lyapunov-Krasovskii Functional Constructions for Time-Varying Systems with Delay[END_REF] to establish a result similar to the one of Theorem 1. But we have chosen to conclude via the trajectory based approach because it leads to Assumption A3 which is simpler than the assumption that we could derived from [START_REF] Mazenc | Extension of Razumikhin's Theorem and Lyapunov-Krasovskii Functional Constructions for Time-Varying Systems with Delay[END_REF].

III. ILLUSTRATION

We illustrate Theorem 1 by applying it to the pendulum model, studied for instance in [START_REF] Mazenc | Design of Continuous-Discrete Observers for Time-Varying Nonlinear Systems[END_REF].

Let T > 0 and T > T be two positive numbers. Let t i be the sequence periodic of k > 1, i.e. for all i, t i = t i+k , such that: t 1 -t 0 = T and for all i ∈ {1, ..., k -1}, t i+1 -t i = T . Now, consider the two dimensional system:

ẋ1 (t) = x 2 (t) ẋ2 (t) = sin(x 1 (t)) , (28)
with the output

y(t i ) = x 1 (t i ) . (29)
Let us check that Assumptions A1 and A2 are satisfied.

• The system (28) can be rewritten as

ẋ(t) = Hx(t) + ϕ(x 1 (t)) (30)
with

H = -2 1 -1 0 (31)
and

ϕ(x 1 ) = 2x 1 x 1 + sin(x 1 ) . ( 32
)
Then Assumption A1 is satisfied with

k ϕ = 2 √ 2.
• The definition of the sequence t i implies that Assumption A2 is satisfied. Now, with the notation of the previous section, we can take:

P = 1 -1 2 -1 2 1 , (33)
and

c 1 = 1 4 , p 1 = 9 20 , p 2 = 1. We have b 2 (t) = 200 3 √ 2(t -σ(t)) e 4 √ 2(t-σ(t)) -1 . (34)
Then for any constant g > 0, the function (36) Since the sequence t i is periodic of period k, the function t -σ(t) is periodic of period = (k -1)T + T .

γ(g, t) = e -gc1 + t t-g e c1(r-t) b 2 (r)dr (35)
(37)

Now, we choose g = . The fact that t -σ(t) is periodic of period implies that the function γ( , t) is identiquality equal to the constant (43) Using t 1 = T and t l+1 = t l + T for all l ∈ {1, ..., k -1} and the inequality e a -1 ≤ ae a for all a ≥ 0, we deduce that

Λ = e -4
ζ b = e
Λ ≤ e -(k-1)T +T 4 × 1 + 200 √ 2 3 4T 16 √ 2+1 - 16 277+32 √ 2 e (4 √ 2+ 1 4 )T +(16 -4T )e 1 4 T -4416+512 √ 2 277+32 √ 2 + 200 √ 2 3 e -(k-1)T +T 4 4 √ 2e kT 4 3 e T 4 -1 e ( 1 4 +4 √ 2)T T 3 ≤ e -(k-1)T +T 4 q a (T ) + q b (T ) (46) with q a (s) = 1 + 200 √ 2 3 ξ(s) (47) with ξ(s) = 4s 16 √ 2+1 - 16 277+32 √ 2 e (4 √ 2+ 1 4 )s +(16 -4s)e 1 4 s -4416+512 √ 2 277+32 √ 2 (48)
and

q b (s) = 1600e 1+16 √ 2 4 s 9 e s 4 -1 s 3 . (49)
Now, let T be any positive real number. Let us choose T > 0 such that

q b (T ) ≤ 1 4 . (50)
Next, one can choose an integer k such that

e -(k-1)T +T 4 q a (T ) ≤ 1 4 . (51)
These inqualities and (46) give:

Λ ≤ 1 2 , (52)
for all t ≥ 0. Thus assumption A3 is satisfied. We conclude that Theorem 1 applies. We conclude that, with the constants we have selected, the solutions of

      
ω(t) = -2z 1 (t) + z 2 (t) + 2ω(t) , t ∈ [t i , t i+1 ) ω(t i ) = x 1 (t i ) ż1 (t) = -2z 1 (t) + z 2 (t) + 2ω(t) ż2 (t) = -z 1 (t) + ω(t) + sin(ω(t))

(53) asymptotically converge to the solution of the system (28). Remark. In this example, T can take arbitrarily large values. This is not allowed by the stability conditions given in [START_REF] Mazenc | Design of Continuous-Discrete Observers for Time-Varying Nonlinear Systems[END_REF] or [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF].

IV. CONCLUSION

In this paper, for a family of nonlinear systems, we have relaxed the assumption on the largest allowable sampling interval that is imposed in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] by establishing the convergence of the proposed observer to the studied system via the technique called trajectory based approach [START_REF] Mazenc | Trajectory Based Approach for the Stability Analysis of Nonlinear Systems with Time Delays[END_REF].

Our result can be extended to many other cases, which include time-varying systems with delay in the spirit of what is done in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] and [START_REF] Hann | Dynamical continuous high gain observer for sampled measurements systems[END_REF], interconnected sub-observers proposed in [START_REF] Mazenc | Stabilization and Robustness Analysis for Time-Varying Systems with Time-Varying Delays using a Sequential Subpredictor Approach[END_REF] and in [START_REF] Kahelras | Conception d'observateurs pour différentes classes de systèmes à retards non linéaires[END_REF] and ordinary differential equations in interconnection with Partial Differential Equations.

1 4 4 √ 2 (

 142 (r-t) (r -σ(r)) e r-σ(r)) -1 dr .

1 4 4 √ 2 ( 2 3 e -1 4 ζ 1 4 4 √ 2 (

 14224142 (r-) (r -σ(r)) e r-σ(r)) -1 dr .(38) Let us write Λ as the sum of three terms:Λ = e -4 + 200 √ (r-t l ) (r -t l ) e r-t l ) -1 dr .

  r ∈ [0, T ], e ( Since = (k -1)T + T , we deduce that

  Since |ζ 2 (t, s)| ≤ k ϕ µ(t, s) for all (t, s) ∈ R 2 , we obtain

	is satisfied.	Since the matrix P is symmetric and positive definite, for any a > 0 the inequality t ti 2e x (t) P ζ 2 (t, s)e x (s)ds ≤ a t e + 1 a t ti e x (s) ζ 2 (t, s) P ζ 2 (t, s)e x (s)ds (20)
		ti

)) + t ti 2e x (t) P ζ 2 (t, s)e x (s)ds . (18) x (t) P e x (t)ds + 1 a t ti e x (s) ζ 2 (t, s) P ζ 2 (t, s)e x (s)ds (19) holds. We deduce that the inequality V (t) ≤ [-2c 1 + a(t -t i )]V (e x (t))


J Langlois

H Mouchère

N Normand

C Viard-Gaudin

On The Fly Generated Data for Industrial Part Orientation Estimation with Deep Neural Networks

Keywords: Pose estimation, deep neural networks, convolutional neural networks, depth estimation, industrial parts, rendered data

In this paper a method is proposed for estimating the orientation of industrial parts. Classical 2d images of the part are used to train a deep neural network and infer the part pose by computing its quaternion representation. Another innovative point of this work is in the use of synthetic data which are generated on the fly during the network training from a textured cad model placed in a virtual scene. This way overcomes the difficulties to obtain pose ground truth images from real images. At the same time, using the cad model with several lighting conditions and material reflectances allows to anticipate challenging industrial situations. As a first step of the method, the part is separated from the background using a semantic segmentation network. Then a depth image of the part is produced employing an encoder-decoder network with skip connections. Finally, the input image is associated with its depth map to estimate the part orientation with a fully connected network using a geodesic loss function. This method can estimate the part pose from a rendered image with a 5 • accuracy while inferring from real images gives visually convincing results suitable for any pose refinement processes.

INTRODUCTION

Object pose estimation is a crucial functionality for various problems such as virtual reality or bin picking with robotic grasping. It remains a challenging task when the parts are poorly textured and have complex shapes. While several methods can rely on simple pattern detection for a pose estimation problem, the performances falls when the industrial environment is not light controlled or known a priori. Besides, any black and glossy plastic material can be difficult to handle because of the image low contrast and sometimes the light saliency on the part surface. Moreover, this task remains challenging when only classical RGB images are available. During the last years, convolutional neural networks (cnns) appeared relevant for the pose estimation problem. Their strong features learning abilities are essential to build a mapping from an object rgb image to its orientation in SO(3). In this paper, a neural-based orientation estimation technique is presented using only one rgb image of a scene containing an object of interest. Lots of samples are required to learn a rgb to SO(3) mapping and build a real image dataset can be time consuming. Moreover, an object inside a bin can be placed against other parts so that its orientation, can be any element of SO(3) which is difficult to obtain for a ground truth. To tackle this issue, the training dataset is composed with only rendered images. Using a data generation process enhances the dataset versatility by rendering random poses with various light conditions. When using an opengl shader in a screenless context, the image generation may be done on-the-fly during the training phase thus, no storage is required. The proposed pipeline for the pose estimation is divided in three distinct neural networks respectively segmentation, depth and orientation networks. The parts are assumed to be at a fixed distance from the camera lying on the same plane (eg. a conveyor) without any occlusions. The translations on both x and y can be easily obtained using each object pixel coordinates. This paper is divided as follows. First related works dealing with neural-based pose estimation are presented (Sec. 2). The image generation technique is then introduced (Sec. 3) followed by the segmentation technique (Sec. 4). The depth (Sec. 5) and orientation (Sec. 6) estimations are detailed before the experiments (Sec. 7) and conclusions (Sec. 8).

RELATED WORKS

Deep learning approaches for pose estimation have been widely studied in the literature the past years for they provide useful pose dependent features from object images. One of the first work dealing with object pose estimation as a classification, 1 employs a simple k-nn algorithm working on features extracted from a convolutional neural network (cnn). The learning phase is done using synthetic object rgb-d images, obtained with an even camera placement on an encompassing sphere. The network is fed with image triplets composed of similar and dissimilar poses, to ensure that two close poses have a small distance in the feature space. The feature-based pose classification can also be done using a support-vector machine (svm). [START_REF] Schwarz | Rgb-d object recognition and pose estimation based on pre-trained convolutional neural network features[END_REF] However, the pose estimation acts as a classifier based on a subset discretization of the view space. To get a more detailed pose, a probabilist approach 3 employs a regression-classification random forest to define an energy function for pose estimation and an optimization scheme to minimize the error. Later, the energy function is replaced with a cnn as a probabilistic model to learn a comparison between the input image and a rendered object. [START_REF] Krull | Learning analysis-bysynthesis for 6d pose estimation in rgb-d images[END_REF] The method can also be improved using reinforcement learning. [START_REF] Krull | Poseagent: Budgetconstrained 6d object pose estimation via reinforcement learning[END_REF] The pose estimation problem can be treated as a regression problem in an end-to-end manner. A Siamese architecture among specific loss function terms, can enforce close poses to have similar features as well as handling occlusions. [START_REF] Doumanoglou | Siamese regression networks with efficient mid-level feature extraction for 3d object pose estimation[END_REF] Nonetheless, these works take profit of the image depth channel as the mapping from rgb to the pose space is complex. Yet, based on the GoogLeNet network, 7 PoseNet 8 computes a camera pose from rgb images suitable for outdoor scene images. To deal with objects, the single shot detector (ssd) framework 9 is adapted to compute a set of object orientations. The object pose among its translation is later classified. [START_REF] Kehl | Ssd-6d: making rgb-based 3d detection and pose estimation great again[END_REF] However, estimating the pose with an end-to-end method is not always the best solution. Several networks can also be used to generate region of interests then, infer the pose of multiple objects in the image. [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] The technique uses the camera intrinsic parameters to retro-project the object coordinates into the 3D scene. The iPose work [START_REF] Jafari | ipose: Instance-aware 6d pose estimation of partly occluded objects[END_REF] proposes to infer the object pixel-wise coordinates from and encoder-decoder network then, compute the pose with a ransac algorithm. Rather then inferring the coordinates, the network can provide the image optical flow. [START_REF] Dosovitskiy | Flownet: Learning optical flow with convolutional networks[END_REF] In DeepIM, [START_REF] Li | Deepim: Deep iterative matching for 6d pose estimation[END_REF] the features of the optical flow network are used to compute the object pose with dense layers reaching the state of the art in the linemod dataset. The training is done with real images from the dataset among rendered views with textures and light distributions extracted from the dataset. Alternatively, the pose can be computed with a pnp algorithm from the object 3D bounding box obtained with a vgg 15 network. [START_REF] Lepetit | Bb8: A scalable, accurate, robust to partial occlusion method for predicting the 3d poses of challenging objects without using depth[END_REF] These techniques can weather rely on the datasets images or complete the data with synthetic images using the same light distribution. Therefore, they are suitable for dataset challenges but might have difficulties to handle versatile industrial environment not know a priori.

IMAGE GENERATION

The cad model of the object is a triangular mesh composed with a set of surfaces {s} i and associated normals { n} i . Given an image size H × W , the object pose view I ∈ [0, 1] H×W is obtained with the model surfaces projections on the image plane, using a combination of a translation t ∈ R 3 , rotation R ∈ SO(3) and a camera intrinsic parameters matrix within an openglgl shader. The depth image is collected from the Opengl normalization of each surface z-coordinate. However, this normalization is first mapping [z n , z n ] to [-1, 1] with a non linear behavior (∝ 1 z ) than transforming it linearly to [0, 1]. Thus, a depth range set regarding to the object maximum radius r o and its z-translation t z as [t z -r o , t z + r o ], is enough to get a detailed depth map D ∈ [0, 1] H×W . Let (f X , f Y ) be the camera focal lengths in pixels and (c X , c Y ) the image center, the full projection (x, y, z) → (p x , p y , z) can be expressed as a tensor product using homogeneous coordinates:

    p x p y z 1     =     2f X 0 c x 0 0 -2f Y c y 0 0 0 -tz ro t 2 z -r 2 o ro 0 0 -1 0     R t 0 T 1     x y z 1     (1)
Each pixel is colored according to a Phong's shading technique controlled by 3 components respectively ambient L a , diffuse L d and specular L s , a material glowing factor and a light position (Fig. 2). The image background is set to be an alpha channel so that it directly represents the binary segmentation mask and can ease the insertion of future backgrounds. For each generated images, a Gaussian blur is added to smooth the part edges among random noise and gamma variations.

L s = 0.2 L s = 0.4 L s = 0.6 L s = 0.8 L d = 0.0 L d = 0.6

PARTS SEGMENTATION

The segmentation process is done using an encoder-decoder network with skip connections. All the layers are using ReLu activation except the last layer which uses a sigmoid activation. The training is done with a simple mse loss. Segmenting the full-sized image needs a heavy network with a long inference phase which is not suitable for a fast industrial process. On the other hand, using a low resolution segmentation leads to rough results often smoothed using crfs. In this work, the segmentation network is fed with small patches P I ∈ [0, 1] 128×128 taken close from the object location on the image. This means that the network is learning to segment an object which is entirely or partially seen in an image. As the networks needs to eliminate the background, some patches without object are also taken into account. The patches background are extracted from a conveyor textures set with random transformations (rotation, flip, noise...). The inference phase is done with a sliding window method where all inferred patches are summed then normalized. This gives a segmentation heat map of the scene that can be thresholded to get the objects binary mask. However, this method can not handle occlusions as it is assumed that all the parts are sequentially coming on a conveyor (not challenging if a proper supplier is used).

Various light conditions are applied to the object so that the network can absorb any light saliences (Fig. 3). The instance segmentation is done with a simple contour detection applied on the objects binary mask. Each object is then patched according to its bounding box before being resized to 128 × 128 using a bilinear interpolation. The patch background is set to -1 as the future depth network only uses the foreground information.

DEPTH PREDICTION

The normalized depth prediction is made with another encoder-decoder network with skip connection based on the one use for the segmentation. However, 2 fully connected layers (fc) are now inserted between the encoding and decoding phase enhancing the control of the latent space. The last fc layer output is then reshaped to a matrix to be fed into the first decoder deconvolution (transposed convolution) (Fig. 4). A dropout layer is also placed close to the input to anticipate any previous segmentation artifacts that may disturb the network during the inference. When trying to learn a depth patch P D from an image patch P I using a sigmoid activation and a Huber loss function (or mse), the inference appears to be noisy and the transposed convolutions sometimes give squared artifacts. To tackle this issue, rather than predicting a single channel, the network outputs two channels respectively P D and (1 -P D ) (so that the channel-wise sum is 1 for every pixel) using a softmax non-linearity. This gives smoother results as the softmax function tends to eliminate the output noises. The encoder-decoder shows a great light saliences robustness even with high contrasts. The loss L D is a cross-entropy function working a both output channels pixels (u, v) between a prediction P Di and its ground truth P Di :

L D = 1 N N i=1 u,v PDi log (P D i + ) (2)
Image patch Depth patch skip connections convolutions +pooling FC deconvolutions Figure 4. Depth estimation encoder-decoder network architecture using skip connections and dense layers within the latent space.

ORIENTATION ESTIMATION

The part orientation is given by a convolutional neural network using 3 convolutional layers and 3 fc layers. The network is fed with the previous depth patch P D that can be stacked with the image patch P I . However, if the encoder-decoder architecture infers correctly real images from a synthetic data based training, this is not necessary the case with a cnn. Using the image patch in the orientation network adds another virtual-real data bias in the pipeline and thus, it decreases the orientation estimation performances. The object orientation is expressed as a unitary quaternion q ∈ [-1, 1] 4 where ||q|| = 1. The quaternion is obtained with a tanh function before getting manually normalized. As q and -q represents the same orientation (obvious when using Rodrigues rotation formula), a L 2 loss is not a valid metric if the network is prediction the opposite of the ground truth.

The SO(3) metric between two rotation matrices R 1 and R 2 , ||I -R T 1 R 2 ||, can be transformed to measure the difference between two quaternions q 1 and q 2 as 2 2 (1 -|q 1 .q 2 | 2 ) (where . is the inner product). The network loss is then defined as L O :

L O = 1 -(q 1 .q 2 ) 2 (3)
Two image patches of an object with different orientations and plane translations can however be similar. The orientation network is then also fed with the normalized pixel coordinates of the patch P C ∈ [-1, 1] 128×128 in the original image. This allows the network to know where the object is in the camera plane and distinguish the poses despite the images equivalences. Moreover, as the z-translation is known a priori, the coordinates retroprojection gives a good approximation of the object x and y-translations in the scene. A refinement technique such as icp, can be use to align the cad model within the image.

EXPERIMENTS 7.1 Protocol

As the part is supposed to stay on a conveyor, it can only shows balanced poses to the camera. However, in this work, no assumptions are made on the available part orientations. The rendered images are obtained using homogeneous poses on the object encompassing sphere given the two first rotation angles (φ, θ). The third angle ψ (camera plane rotation) is randomly taken from the [-π, π] interval. The x and y-translations are also randomly taken so that the part covers all the camera plane. This means that the networks are able to handle every part orientations without prior studies on the part geometry and inertia. The networks are trained for 1K epochs using adam optimizer for both segmentation and depth networks and Nesterov momentum for the orientation network with a 10 -4 learning rate. The weights giving the best results on a rendered validation dataset are kept.

To get a pose estimation error based on real images, an homography technique is employed using 4 known targets on a plane in an industrial environment (having various lights and contrasts). Using the camera intrinsic parameters, the homography matrix is computed then analyzed with a svd to obtain the rotation matrix R and translation t. The ground truth segmentation mask among the normalized depth are computed using the cad model via Opengl, placed according to the previous computed pose (Fig. 5). It is assumed that the homography matrix estimation is quite precise and suitable to establish a proper ground truth. The final dataset is composed with 100 real images covering a large portion of the object pose space (excluding z translations). A virtual images dataset is also build based on 500 random samples. The metric used for the segmentation evaluation is the mean intersection over union (miou) including 0.9 iou.

The orientation error can be expressed as a mean Euler angles error M E or based on the SO(3) geodesic distance between to rotation matrix R 1 and R 2 , ||log R T 1 R 2 ||, adapted to the quaternion space as M G with the predicted quaternion q and the truth q :

M G = 2 arccos(|q.q|)

The final pose estimation evaluation is done using several metrics including 2d-proj, 5cm5 • and add. [START_REF] Hinterstoisser | Model based training, detection and pose estimation of texture-less 3d objects in heavily cluttered scenes[END_REF] The 2d-proj metric evaluates the pixels distances, between the truth object pixels and the cad model projection (placed according to the estimated pose), that are inferior to 5px. The add metric quantifies the displacement, between the ground truth points cloud and the cad model (again placed according to the estimated pose), that are inferior to 10% of the object diameter. All inferences are made using the results of the previous network inferences so that each error can be seen as a cumulative error within the pipeline.

Results

CONCLUSION AND FUTURE WORKS

Figure 1 .

 1 Figure 1. Full pose estimation pipeline from left to right : the image of an object on a conveyor is first segmented then the local depth is inferred before estimating the orientation and computing the final pose in SE(3).

Figure 2 .

 2 Figure 2. Effects of the light diffuse L d and specular Ls components variation.

Figure 3 .

 3 Figure 3. Partial object segmentation inference on virtual image with different light parameters from flat appearance (left) to shiny material (right).

Figure 5 .

 5 Figure 5. Pose estimation via homography using a blob detection (top) for patch, mask and depth ground truth creation using opengl (bottom).

Figure 6 .

 6 Figure 6. Typical cases of good inferences (left) and badly orientated part (right).

Table 1 .

 1 Segmentation, depth, orientation and pose network performances for virtual images and real images.

	Dataset	Segmentation Depth Orientation x-y-Translations	Final
	Rendered images	0.9 iou miou	mse	M E M G	mse	2d-proj 5cm5 • add
	Real images	0.9 iou miou	mse	M E M G	mse	2d-proj 5cm5 • add
	ssss


Thach Ngoc Dinh
email: ngoc-thach.dinh@lecnam.net

Frederic Mazenc

Tarek Raïssi
email: tarek.raissi@cnam.fr

Finite-time guaranteed state estimation for discrete-time systems with disturbances

In this paper, we consider a family of discrete-time systems with input and output. A new technique, based on past values of the output which have been employed to construct some finite-time observers for continuous-time systems in the literature, is proposed for discrete-time case. The systems are affected by additive disturbances and disturbances in the output. Exact estimation or approximate estimation can be achieved depending on the absence or the presence of unknown but bounded uncertainties, respectively.

I. INTRODUCTION

The problem of designing state observers for dynamical systems, including automatic control systems, is a classical one and has rich history. For the purpose of feedback control to a system such as vehicles, chemical plants and biological systems, all of its state space variables must be available. However in general in engineering, some of the state variables are not available for measurements or measure of them is problematic (e.g. it is too expensive to measure all state variables). Thus, control designers have to face with the problem of estimating system state space variables based on a model and a limited set of possible input/output measurements. This can be done by building a dynamic extension, called an observer or an estimator. Roughly speaking, an observer is an auxiliary dynamic system coupled to the original system through the measured inputs and outputs. In many contributions, designs of observers are proposed. We can mention here traditional state estimators which are very popular, such as Luenberger observer [START_REF] Luenberger | An introduction to observers[END_REF]. Another cutting-edge technique of guaranteed state estimation is the interval observer based technique which was introduced two decades ago in [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]. Typically, interval observers give accurate component-wise estimations at any time instant when upper and lower bounds of the initial state are known, see for instance [START_REF] Raissi | Interval State Estimation for a Class of Nonlinear Systems[END_REF], [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF], [START_REF] Mazenc | Construction of Interval Observers for Continuous-time Systems with Discrete Measurements[END_REF] and the references therein. The usefulness of interval estimates is evident not only for feedback control purpose but also for monitoring purposes when large disturbances or uncertainties are present. Therefore, the topic is interesting from the mathematical point of view, and also has an important potential for use in a lot of industrial applications [START_REF] Alcaraz-Gonzalez | Robust Nonlinear Observers for Bioprocesses: Application to Wastewater Treatment[END_REF], [START_REF] Ifqir | Vehicle Lateral Dynamics Estimation using Switched Unknown Inputs Interval Observers: Experimental Validation[END_REF]. A promising concept, called fixed-time stability, is proposed in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. Using this concept, fixed-time controllers were proposed to ensure that frederic.mazenc@l2s.centralesupelec.fr some control performances are achieved regardless of the initial conditions of the system [START_REF] Defoort | Leader-follower fixed-time consensus for multiagent systems with unknown nonlinear inherent dynamics[END_REF]. Uniform robust exact differentiators were proposed in [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Lopez-Ramirez | Fixed-time output stabilization of a chain of integrators[END_REF], [START_REF] Rios | A hybrid observer for fixed-time state estimation of linear systems[END_REF] based on a Lyapunov analysis or homogeneity properties. A fixedtime observer, with linear matrix inequality algorithms for tuning the observer parameters, was introduced in [START_REF] Lopez-Ramirez | Finite-time and fixed-time observers design via implicit Lyapunov function[END_REF] for linear systems. Based on uniform robust exact differentiators, a uniformly convergent sliding mode observer was proposed in [START_REF] Mincarelli | Uniformly convergent sliding mode-based observation for switched linear systems[END_REF]. Although the settling time estimate does not depend on the initial conditions of the system in many works, it cannot be easily tuned and it is very over-estimated.

In the present work, a new approach of finite-time guaranteed state estimations for estimating state variables is proposed where no appropriate knowledge of the initial conditions is known and the systems have no monotonicity property. The aim of this note is different from interval observers presented above, which always request an appropriate knowledge in terms of an upper and lower bound of the initial conditions and a direct or indirect notion of nonnegative and cooperative system. The approach is based on formulas incorporating past values of the input and the output of the studied plant. The technique has been developed in several contexts: in particular, some works are devoted to families of linear systems [START_REF] Engel | A Continuous-Time Observer Which Converges in Finite Time[END_REF], and others are devoted to nonlinear systems [START_REF] Sauvage | Design of a Nonlinear Finite-Time Converging Observer for a Class of Nonlinear Systems[END_REF], [START_REF] Liu | Finite time observer design for a class of nonlinear systems with unknown inputs[END_REF], [START_REF] Mazenc | Estimation of solutions of observable nonliear systems with disturbances[END_REF]. A common feature of these results is that they apply only to continuoustime systems. On the other hand, discrete-time systems are very important from a theoretical as well as an applied point of view and the problem of constructing observers or dynamic output feedbacks for them has been extensively studied [START_REF] Karafyllis | On the observer problem for discrete-time control systems[END_REF], [START_REF] Boutayeb | A reduced-order observer for nonlinear discrete-time systems[END_REF], [START_REF] Xiao | Nonlinear discretetime observer design with linearizable error dynamics[END_REF]. Furthermore, it is worth noting that discretization techniques transform continuous-time systems into discrete-time systems and systems with sampled data often lead to discrete-time system [START_REF] Astrom | Computer Controlled Systems. Theory and Design[END_REF]. These systems are usually affected by two types of bounded deterministic timevarying disturbances: in the dynamics and in the output. This fact motivates our work. Employing past values of the input and the output only, two goals can be achieved. In the absence of unknown uncertainties and after a finite time, the exact values of the solutions are given. Next when unknown disturbances are present and are upper and lower bounded by known constant vectors, after a finite time, the formulas we exhibit provide upper and lower bounds for each component of the solutions, as interval observers do.

The paper is organized as follows. Setups and objectives are introduced in Section II. Exact values of the solutions in the absence of uncertainties are given in Section III. Intervals for the solutions in the presence of uncertainties are constructed in Section IV. The main results are illustrated in Section V. Concluding remarks are given in Section VI. Notations, definitions. The set of natural numbers and real numbers are denoted by N and R, respectively. The Euclidean norm of a vector x ∈ R n is denoted by |x|. Inequalities must be understood component-wise, i.e., for

x a = [x a,1 , ..., x a,n ] ∈ R n and x b = [x b,1 , ..., x b,n ] ∈ R n , x a ≤ x b if and only if, for all i ∈ {1, ..., n}, x a,i ≤ x b,i . For a square matrix Q ∈ R n×n , the determinant of Q is denoted det(Q) and we define Q + , Q -∈ R n×n by Q + = max (Q, 0) and Q -= Q + -Q.

II. SETUPS AND OBJECTIVES

Consider the following discrete-time system:

x(k + 1) = Ax(k) + F (u(k), y(k)) + d(k) y(k) = Cx(k) + v(k) , k ∈ N.
(1) where

x(k) ∈ R n is the state, F is a nonlinear function, C ∈ R q×n , y(k) ∈ R q is the output, u(k) ∈ R p is an input, and d : N → R n , v : N → R q are
respectively disturbances in the dynamics and in the output.

We introduce the following assumptions : Assumption 1: The pair (A, C) is observable and A is invertible.

Assumption 2: There are known constant vectors

d ∈ R n , d ∈ R n and v ∈ R q , v ∈ R q such that for all k ∈ N, the inequalities d ≤ d(k) ≤ d (2) v ≤ v(k) ≤ v (3)
are satisfied.

Inspired by [START_REF] Mazenc | Estimation of solutions of observable nonliear systems with disturbances[END_REF], the observer designs proposed in this note achieve two objectives simultaneously:

• An exact estimation for the solutions in finite time when the functions d and v are known. • Design of two bounds for the solutions in finite time when the functions d and v are unknown but bounded by known values. Discussion of the Assumptions:

• Assumption 1 implies that there is a matrix L ∈ R n×q such that the matrix

H = A + LC ∈ R n×n (4)
admits a spectral radius smaller than the modulus of any eigenvalue of A. Then, we prove in Lemma 1 in Appendix that in this case, there is an integer h > 0 such that the matrix H -h -A -h is invertible. • The assumption that A is invertible is not restictive at all because when (A, C) is observable and A is not invertible, we can always decompose Ax + F (u, y) in an alternative way so that the new matrix A is invertible. • Assumption 2 is realistic and is frequently satisfied in practice. It can be relaxed by allowing the bounds to depend on time k but for the sake of the simplicity, we restrict ourselves to the case where they are constant.

III. EXACT ESTIMATION The results of this section provide with exact estimations of the solutions in finite time, but they can be applied only when the functions d and v are known. Let us state and prove the following result:

Theorem 1: Let the system (1) satisfy Assumptions 1. Let L ∈ R n×q and h ∈ N, h ≥ 1 be such that the matrix H -h -A -h is invertible. Then, for a given input u(k), any solution x(k) of the system (1) which exists over N satisfies, for all k ≥ h,

x(k) = -E h k-1 =k-h A k-h--1 F (u( ), y( )) + E h k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] -E h k-1 =k-h A k-h--1 d( ) + E h k-1 =k-h H k-h--1 Lv( ), (5)
with

E h = H -h -A -h -1 . (6)
Proof: From the definition of the output y and the definition of H, we deduce that the system (1) admits two representations:

x(k + 1) = Ax(k) + F (u(k), y(k)) + d(k), (7) x
(k + 1) = Hx(k) + F (u(k), y(k)) -Ly(k) + d(k) + Lv(k). (8)
By combining all equations of these two systems between two values m 1 ∈ N, m 2 ∈ N and m 1 ≥ m 2 , we obtain the equalities

x(m 1 ) = A m1-m2 x(m 2 ) + m1-1 =m2 A m1--1 [F (u( ), y( )) + d( )], (9)
x(m

1 ) = H m1-m2 x(m 2 ) + m1-1 =m2 H m1--1 [F (u( ), y( )) -Ly( ) + d( ) + Lv( )]. (10)
Now, consider a value k ≥ h. Then selecting m 2 = k -h and m 1 = k, the equalities ( 9)- [START_REF] Karafyllis | On the observer problem for discrete-time control systems[END_REF] give

x(k) = A h x(k -h) + k-1 =k-h A k--1 [F (u( ), y( )) + d( )], (11)
x(k) = H h x(k -h) + k-1 =k-h H k--1 [F (u( ), y( )) -Ly( ) + d( ) + Lv( )]. (12)
As an immediate consequence, we have

(H -h -A -h )x(k) = - k-1 =k-h A k-h--1 [F (u( ), y( )) + d( )] + k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( ) + d( ) + Lv( )].
Since H -h -A -h is invertible, we deduce that ( 5) is satisfied.

The formula (5) may contain many terms because h may be large and thus many values have to be stored. To overcome this drawback, we propose an alternative solution which is based on a dynamic extensions.

Theorem 2: Let the system (1) satisfy the Assumptions 1, let L and h be defined as in Theorem 1 and let u be a given input. Consider the dynamic extensions

x(k + 1) = Ax(k) + F (u(k), y(k)) + d(k) (13)
and

x * (k + 1) = Hx * (k) + F (u(k), y(k)) -Ly(k) + d(k) + Lv(k). ( 14
)
Consider a solution x(k) of ( 1) defined over N. Then, for all k ≥ h,

x(k) = E h H -h x * (k) -x * (k -h) -A -h x(k) + x(k -h) . (15
) Remark 1: Notice that ( 14) is a classical observer for the system (1) when disturbances are known.

Proof: Consider a solution (x(k), x * (k)) of ( 13)-( 14) associated with the solution x(k) defined over N. Then, arguing as we did in the proof of Theorem 1, we have

x(m 1 ) = A m1-m2 x(m 2 ) + m1-1 =m2 A m1--1 [F (u( ), y( )) + d( )], (16) x * (m 1 ) = H m1-m2 x * (m 2 ) + m1-1 =m2 H m1--1 × × [F (u( ), y( )) -Ly( ) + d( ) + Lv( )]. (17) It follows that for all k ≥ h, k-1 =k-h A k-h--1 [F (u( ), y( )) + d( )] = A -h x(k) -x(k -h), (18)
k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( ) + d( ) + Lv( )] = H -h x * (k) -x * (k -h). (19)
Combining ( 5), ( 18) and ( 19), we obtain, for all k ≥ h,

E -1 h x(k) = H -h x * (k) -x * (k -h) -A -h x(k) + x(k -h). (20)
This allows us to conclude.

IV. APPROXIMATE ESTIMATION

Theorems 1 and 2 give in finite time the exact value of any solution x(k) of (1). However, these estimations cannot be used when the disturbances d and v are unknown. The second objective of the present note is to overcome this limitation by assuming only that the bounds d, d of d and v, v of v are known.

In this section, we consider the case where Assumptions 1 and 2 are satisfied and the matrix L is selected as described in [START_REF] Boutayeb | A reduced-order observer for nonlinear discrete-time systems[END_REF]. Next, we define the following matrices

F h = -E h A -(h+1) , G h = E h H -(h+1) . ( 21
)
and the vectors

d L = F h h =1 A + d -F h h =1 A - d, (22)
d S = F h h =1 A + d -F h h =1 A - d, (23)
v L = G h h =1 H L + v -G h h =1 H L - v, ( 24
) v S = G h h =1 H L + v -G h h =1 H L - v. (25)
We are ready to state and prove the following result: Theorem 3: Let the system (1) satisfy Assumptions 1 and 2 and let L and h ∈ N, h ≥ 1 be such that E h given in ( 6) is well-defined. Let u be a given input and consider a solution of the system (1) defined over N. Then, for all integer k ≥ h, the inequalities

x(k) ≤ x(k) ≤ x(k). (26)
with

x(k) = -E h k-1 =k-h A k-h--1 F (u( ), y( )) + E h k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] + d L + v L , (27)
x(k) = -E h k-1 =k-h A k-h--1 F (u( ), y( )) + E h k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] + d S + v S , (28) hold.
Proof: From (5), we have

x(k) = -E h k-1 =k-h A k-h--1 F (u( ), y( )) + E h k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] -E h A -(h+1) k-1 =k-h A k-d( ) + E h H -(h+1) k-1 =k-h H k-Lv( ). (29)
From ( 29) and the definition of F h and G h in ( 21), it follows that, for all k ≥ h,

x(k) = -E h k-1 =k-h A k-h--1 F (u( ), y( )) + E h k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] + F h h =1 A d(k -) + G h h =1 H Lv(k -). ( 30
)
From Assumption 2, we deduce that, for all k ≥ h,

F h h =1 A + d -F h h =1 A - d ≤ F h h =1 A d(k -) ≤ F h h =1 A + d -F h h =1 A - d, G h h =1 H L + v -G h h =1 H L - v ≤ G h h =1 H Lv(k -) ≤ G h h =1 H L + v -G h h =1 H L - v.
It follows that

d S ≤ F h h =1 A d(k -) ≤ d L , (31)
v S ≤ G h h =1 H Lv(k -) ≤ v L , (32)
with

d L = F h h =1 A + d -F h h =1 A - d, (33)
d S = F h h =1 A + d -F h h =1 A - d, (34)
and

v L = G h h =1 H L + v -G h h =1 H L - v, ( 35
) v S = G h h =1 H L + v -G h h =1 H L - v. ( 36
)
From ( 30), ( 31) and (32), we can conclude. The motivations of Theorem 2 also motivate the following result:

Theorem 4: Let the system (1) satisfy the conditions in Theorem 3. Let u(k) be a given input and consider a solution of (1) defined over N. Let us introduce several dynamic extensions:

z a (k + 1) = Az a (k) + F (u(k), y(k)), (37)
z h (k + 1) = Hz h (k) + F (u(k), y(k)) -Ly(k), (38)
Then, for all k ≥ h, the inequalities

Υ(Z k ) ≤ x(k) ≤ Υ(Z k ), (39)
with Z = (z a , z h ) and the bounds Υ, Υ are an estimated interval for the system (1) given by

Υ(Z k ) = E h [z a (k -h) -A -h z a (k) + H -h z h (k) -z h (k -h)] + d L + v L , (40)
Υ(Z k ) = E h [z a (k -h) -A -h z a (k) + H -h z h (k) -z h (k -h)] + d S + v S , (41)
where d L , d S , v L , v S are the vectors defined in ( 22), ( 23), ( 24), (25), are satisfied. Proof: For a solution x(k) of (1) defined over N, we have

z a (k) = A h z a (k -h) + k-1 =k-h A k--1 F (u( ), y( )), (42) z h (k) = H h z h (k -h) + k-1 =k-h H k--1 [F (u( ), y( )) -Ly( )]. (43)
These equalities rewrite as

- k-1 =k-h A k-h--1 F (u( ), y( )) = z a (k -h) -A -h z a (k), (44)
k-1 =k-h H k-h--1 [F (u( ), y( )) -Ly( )] = H -h z h (k) -z h (k -h). (45)
Theorem 3 ensures that the inequalites (26) hold. Then from these inequalities and the equalities (42) and (43), we deduce that the inequalities (39) are satisfied.

V. ILLUSTRATIVE EXAMPLE In this section, consider the system

x 1 (k + 1) = 5 4 x 1 (k) + x 2 (k) + 1 4 u 1 (k) + 1 9 sin(k), (46a)
x 2 (k + 1) = - 3 8 x 1 (k) + 1 8 u 2 (k) + 1 9 sin(k), (46b)
y(k) = x 1 (k) + 1 9 sin(k 2 ). (46c)
System ( 46) is of the form (1) with C = [1 0],

A = 5 4 1 -3 8 0 , F (u(k), y(k)) = 1 4 u 1 (k)

Now to determine an analytical expression of the matrix

H -h -A -h , let us observe that R 1 AR -1 1 = S 1 with R 1 = 1 4 3 1 2 and S 1 = 3 4 0 0 1 2 . ( 47
)
and

S 2 = R 2 HR -1 2 with R 2 = 1 4 1 8 and S 2 = 1 4 0 0 1 8 . (48)
Next, for an integer h > 0, we have

H -h -A -h = H h -1 -A h -1 = R -1 2 S -h 2 R 2 -R -1 1 S -h 1 R 1 = α 11 α 12 α 21 α 22 ,
with

α 11 = 2 1 4 -h - 1 8 -h -3 3 4 -h + 2 1 2 -h , α 12 = 8 1 4 -h -8 1 8 -h -4 3 4 -h + 4 1 2 -h , α 21 = 1 4 1 8 -h - 1 4 1 4 -h + 3 2 3 4 -h - 3 2 1 2 -h , α 22 = 2 1 8 -h - 1 4 -h + 2 3 4 -h -3 1 2 -h . The matrix H -h -A -h is invertible for all h ≥ 2 because det(H -h -A -h ) = 1 32 -h -6 1 16 -h -6 3 16 -h + 5 1 8 -h + 5 3 32 -h + 3 8 -h = 8 3 h 2 h -1 × × 5 × 2 h -5 × 3 h + 6 h -1 ,
which is strictly positive when h ≥ 2.

Then for all h ≥ 2,

E h = H -h -A -h -1 = 11 12 21 22
,

with 11 = 2 1 8 -h -1 4 -h + 2 3 4 -h -3 1 2 -h 8 3 h (2 h -1) (5 × 2 h -5 × 3 h + 6 h -1) , 12 = -8 1 4 -h + 8 1 8 -h + 4 3 4 -h -4 1 2 -h 8 3 h (2 h -1) (5 × 2 h -5 × 3 h + 6 h -1) , 21 = -1 4 1 8 -h + 1 4 1 4 -h -3 2 3 4 -h + 3 2 1 2 -h 8 3 h (2 h -1) (5 × 2 h -5 × 3 h + 6 h -1) , 22 = 2 1 4 -h -1 8 -h -3 3 4 -h + 2 1 2 -h 8 3 h (2 h -1) (5 × 2 h -5 × 3 h + 6 h -1) .
We apply Theorem 2 and select the initial values x 1 (0) = 2.3, x 2 (0) = 1 and the input u 1 = 1 and u 2 = 2. Then, for different values of the delay h = 2, 5, 10, we implement the exact estimation of the state x given by [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]. The simulation result is plotted in Figs. 1-2-3.

Finally, we implement the dynamic extensions x a and x h given by ( 37)-(38), and the upper and lower bounds given by ( 40

VI. CONCLUSION

We have proposed a new technique of estimation of the solutions for a family of discrete-time nonlinear systems with disturbances in the dynamics and disturbances in the output. The key idea relies on the use of past values of the input and the output of the studied system. No information on the bound of the initial conditions was needed in our development and we provided exact values of the solutions in the absence of disturbances and a lower and upper bounds when the disturbances are present after a finite time which can be tuned. Extension to more general families of nonlinear discrete-time systems and output feedback stabilization of systems can be considered for future works.

APPENDIX

Lemma 1: Let (A, C) be an observable pair. Then there is a matrix L such that the matrix H = A + LC such that there is an integer h * such that for all h ∈ N, h ≥ h * , the matrix H -h -A -h is invertible.

Proof: Since the pair (A, C) is observable, there is a matrix L so that the spectral radius of H is strictly less than 1 and smaller than the smallest norm of the eigenvalues of A. As an immediate consequence, lim r→+∞ |H r ||A -r | = 0. It follows that, there is an integer h * such that, for all h ∈ N, h ≥ h * , |H h ||A -h | < 1. Consequently, it does not exist a vector V = 0 such that (H h A -h -I)V = 0. Therefore H h A -h -I is invertible. Note that the matrix H -h -A -h is invertible if and only if H h A -h -I is invertible. This concludes the proof.

  Notice that the pair (A, C) is observable. The choice L =

Fig. 1 .

 1 Fig. 1. Real state and exact estimation for h = 2.

Fig. 2 .

 2 Fig. 2. Real state and exact estimation for h = 5.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Real state and exact estimation for h = 10.

Fig. 5 .

 5 Fig. 5. Finite time interval estimation with upper and lower bounds for h = 4.


Laurent Burlion

Michael Malisoff

Frédéric Mazenc

Stabilization and Robustness Analysis for a Chain of Saturating Integrators Arising in the Visual Landing of Aircraft

Keywords: stabilization, vision-based control, backstepping, aircraft

We study a chain of saturating integrators with imprecise output measurements. Using a recent backstepping approach that leads to pointwise delays in the control and a dynamic extension, we provide an input-to-state stability result using a bounded control of arbitrarily small amplitude. We apply the result to a problem in the visual landing of aircraft.

I. INTRODUCTION

This paper continues our search for more effective feedback stabilization methods for cases where only imprecise output measurements are available for use in the control. This led to our novel backstepping approach in [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] and [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF] where pointwise delays are present in the feedback even if current output values are available, and then our work [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF], [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF] that uses the preceding backstepping approach to solve a feedback control problem for a chain of saturated integrators with imprecise output measurements using an unbounded control. In the present work, we use our backstepping approach to solve a stabilization problem for a chain of saturating integrators with imprecise measurements using dynamic output feedback controls of arbitrarily small amplitude; see Section II for more on the potential advantages of this work as compared with the method in [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF], [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF].

The present work is therefore a new development in a long history of research on stabilization under bounded controls. Earlier results on bounded feedback controls include the semi-global state and output feedback stabilization results [START_REF] Zhu | Linear backstepping output feedback control for uncertain linear systems[END_REF], which involve linear control laws inside saturations. Crucial regional [START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF] stability results for some linear and nonlinear systems (using LMI methods [START_REF] Chesi | LMI-based fixed order output feedback synthesis for two-dimensional mixed continuous-discrete-time systems[END_REF], [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF]) were presented in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Other earlier bounded backstepping and forwarding methods lead to globally asymptotically stabilizing controls for some nonlinear systems; see [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] for bounded backstepping, and see [START_REF] Mazenc | Adding an integration and global asymptotic stabilization of feedforward systems[END_REF] and [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF] for forwarding methods.

The backstepping designs proposed in [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] and [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF] are significantly different from prior backstepping methods, because they circumvent the problem of determining Lie derivatives of the fictitious controls by introducing artificial delays in the control (which are called artificial because they are present even if current state values are available for measurement). The artificial delays approach relaxes the smoothness requirement on the fictitious control that was present in previous backstepping approaches.

The advantages of [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] carry over to the present work, which adapts ideas from [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] to a control problem for a chain of saturating integrators for dynamics with outputs that occurs in the vision based [START_REF] Karagiannis | A new solution to the problem of range identification in perspective vision systems[END_REF] landing of aircraft. Since only imprecise measurements of the two first state variables are available in this application, we cannot apply the regional or semi-global results mentioned above, nor [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] or extensions such as [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF]. This motivates our new control, which is inspired by the forwarding theory from [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF]. Our controls in the present work ensure input-to-state stability with a saturated input. Moreover, for any positive constant ū, we can ensure that our control is bounded by ū.

The notation will be simplified whenever no confusion would arise given the context. Given any constant T > 0, C in denotes the set of all continuous functions φ : [-T, 0] → R n , which we call the set of all initial functions. We define Ξ t ∈ C in by Ξ t (s) = Ξ(t + s) for all Ξ, s ≤ 0, and t ≥ 0 for which the equality is defined. The Euclidean norm is denoted by | • |, and | • | S (resp., | • | ∞ ) denotes the corresponding supremum over any set S (resp., essential supremum). For each constant L > 0, we use the usual saturation function sat L (x) = max{-L, min{L, x}}, and ∂ will denote a boundary. We also use the standard definitions of input-to-state stability and class K L and K ∞ functions, as defined in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Chapter 4], and M denotes the set of all functions of the form γ + c where γ ∈ K ∞ and c ≥ 0 is a constant.

II. DYNAMICS

We consider the system

   ẋ1 = sat L 1 (x 2 ) ẋ2 = sat L 2 (x 3 ) ẋ3 = sat L 3 (u), (1)
where x = (x 1 , x 2 , x 3 ) is valued in R 3 , the input u is valued in R, and L i > 0 is a constant for i = 1, 2, 3, where we use the usual saturation function sat L (x) = max{-L, min{L, x}} for each constant L > 0. 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), (2)
where δ 1 , δ 2 and η are unknown but piecewise continuous functions for which there are known constants η > 1, δ 1 ≥ 0, and δ 2 ≥ 0 such that for all t ≥ 0, we have

η(t) ∈ [1, η] and |δ i (t)| ≤ δ i for i = 1, 2 (3)
(but see our conclusions section below for remarks in more general cases with sampling or output delays). Requiring η(t) ≥ 1 for all t ≥ 0 is not restrictive because in practice, η will have known positive upper and lower bounds, and then we can divide the formula for the output component y 1 in (2) by inf{η( ) : ≥ 0} so the rescaled η and δ 1 are such that the rescaled η is bounded below by 1.

Given any constant u > 0, our goal is to construct an output feedback u that is bounded by u and that is such that the closed loop system (1) is input-to-state stable with respect to δ = (δ 1 , δ 2 ), under the assumption that δ 2 < L 1 . This contrasts with the objectives in [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF], which studied the same system (1) with a sampled version of the same outputs (2), because in [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF], the control was not required to be bounded, and in addition, [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF] required the more stringent condition δ2 < L 1 (1e -1 ) 2 /(40(1 + 2e -1 + e -2 )) as well as a scaling constant λ > 0 in the control, which are not needed here. Hence, the present paper provides potential advantages over [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF], which are made possible by our new dynamical extension in this work which was not present in [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF].

Since [START_REF] Burlion | Vision based anti-windup design with application to the landing of an airliner[END_REF] is not in feedback form, classical backstepping results (e.g., [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]) do not apply. Since η is unknown and not necessarily differentiable, bounded backstepping results in [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF] and [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] do not apply. Other approaches (e.g., [START_REF] Hu | Control Systems with Actuator Saturation[END_REF], [START_REF] Li | Stability and Performance of Control Systems with Actuator Saturation[END_REF], and [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF]) do not apply here either. Moreover, one cannot apply [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF], which does not allow uncertain measurements. Thus, we believe the problem solved in this work was open and worth solving because of its implications for the visual landing of aircraft that we discuss in Section V below.

III. MAIN THEOREM

Our main result is as follows, in which we can always satisfy our conditions [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF] by choosing the positive constants α 1 , β 1 , and L 4 small enough and then choosing the p i 's large enough (because of our assumption δ 2 < L 1 ), and then for any constant ū > 0 we can choose these constants such that the control u is bounded by L 4 + α 3 + β 2 < ū to satisfy any amplitude restriction on the control u:

Theorem 1: Consider the model (1) with the outputs (2), and assume that δ 2 < L 1 . Choose the control

u(t) = 0, t ∈ [0, 2T ) G (t, y t , z t ), t ≥ 2T
where (4)

G (t, y t , z t ) = -sat L 4 x 3 (t) -v 1 (z t ) -β (t) +v 2 (y t , z t ) + v 3 (y t , z t ), (5)
v 1 (z t ) = k (1-e -kT ) 2 z 1 (t) -z 2 (t) -2e -kT z 1 (t -T ) +e -kT z 2 (t -T ) + e -2kT z 1 (t -2T ) , (6)
v 2 (y t , z t ) = k 2 (1-e -kT ) 2 -2z 1 (t) + z 2 (t) +4e -kT z 1 (t -T ) -e -kT z 2 (t -T ) -2e -2kT z 1 (t -2T ) + φ (t) -2e -kT φ (t -T ) +e -2kT φ (t -2T ) , and (7)
v 3 (y t , z t ) = k 1-e -kT -z 3 (t) + e -kT z 3 (t -T ) +ω(t) -e -kT ω(t -T ) (8)
where the z i 's are the states of the dynamic extension

   ż1 (t) = k[-z 1 (t) + φ (t)] ż2 (t) = k[-z 2 (t) + z 1 (t) -e -kT z 1 (t -T )] ż3 (t) = k[-z 3 (t) + ω(t)] (9)
with the initial condition z( ) = 0 for all ∈ [-2T, 0] and

φ (t) = -α 1 p 1 sat p 1 y 1 (t)
and

ω(t) = - β 1 p 2 sat p 2 y 2 (t) -α(t) (10)
and

α(t) = z 2 (t)-e -kT z 2 (t-T ) (1-e -kT ) 2
and β (t) = z 3 (t)-e -kT z 3 (t-T )

1-e -kT [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] and where the positive constants p i , T , k, L 4 , α 1 , and β 1 are chosen such that

β 1 + α 2 < L 2 (12a) 2α 1 + δ 2 < L 1 (12b) L 4 + α 3 + β 2 < L 3 (12c) ηT α 1 + δ 1 < p 1 (12d) 2α 1 η 2 T < p 1 (12e) T β 1 + δ 2 < p 2 (12f) 2β 1 T < p 2 (12g) 2T α 1 η p 1 α 1 + ξ + α 1 p 1 δ 1 + ξ + α 1 p 1 δ 1 < α 1 η (12h)
where

ξ = δ2 β1( β 1 T +p 2) p 2 2 1-2 p 2 β 1 T , α 2 = 2kα 1 1-e -kT ,
α 3 = 4k 2 α 1 (1-e -kT ) 2 , and β 2 = 2kβ 1 1-e -kT . (13)
Then the control u in ( 4) is bounded by

L 4 + α 3 + β 2 , and (1)
in closed loop with the control ( 4) is input-to-state stable with respect to δ = (δ 1 , δ 2 ).

IV. PROOF OF THEOREM

Simple calculations show that the functions v i from ( 6)-( 8)

satisfy v 1 (z t ) = α(t), v 2 (y t , z t ) = α(t), and v 3 (y t , z t ) = β (t)
for all t ≥ 0. This allows us to rewrite the control u from (4) as

u(t) = -sat L 4 x 3 (t) -α(t) -β (t) + α(t) + β (t) (14)
for all t ≥ 2T . Hence, the bound for u follows from the fact that α and β in [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] satisfy

|α(t)| ≤ α 1 , | α(t)| ≤ α 2 , | α(t)| ≤ α 3 , |β (t)| ≤ β 1 , and | β (t)| ≤ β 2 (15)
for all t ≥ 2T . Our proof of the formulas (15) (in Appendix 1 below) use the fact that we can apply variation of parameters (to the system q = k(-q + b) with the choice

(q(t), b(t)) = (z 3 (t), ω(t)), then with the choice (q(t), b(t)) = (z 2 (t), z 1 (t) -e -kT z 1 (t -T ))
, and finally with the choice

(q(t), b(t)) = (z 1 (t), φ (t))) to prove that α(t) = k 2 (1-e -kT ) 2 t t-T m m-T e k( -t) φ ( )d dm and β (t) = k 1-e -kT t t-T e k( -t) ω( )d (16)
for all t ≥ 2T . We also use the formulas [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF] in the rest of the proof as well.

The remainder of the proof is organized as follows. In the first step, we use a change of variables that produces a useful cascaded system with a globally asymptotically stable subsystem. In the second step, we perform a Lyapunov function analysis using states of this new system. In the third step, we use results from the first two steps to find useful bounds on the states of the original system. In the final step, we use linear growth properties of the closed loop system to transform the preceding estimates into the required input-tostate stability estimate.

First step. We use the new variables

ξ 2 (t) = x 2 (t) -α(t) ξ 3 (t) = x 3 (t) -α(t) -β (t). ( 17
) Direct calculations give    ẋ1 (t) = sat L 1 (α(t) + ξ 2 (t)) ξ2 (t) = sat L 2 (β (t) + ξ 3 (t) + α(t)) -α(t) ξ3 (t) = sat L 3 (u) -α(t) -β (t). (18)
Notice that the system (18) in closed-loop with ( 4) is forward complete, by the forward completeness of our dynamic extension (9) (which follows from the boundedness of the nonlinear terms in ( 9)). Also, ( 15) and (12c) imply that

|u(t)| = | -sat L 4 (ξ 3 (t)) + α(t) + β (t)| ≤ L 4 + α 3 + β 2 < L 3
holds for all t ≥ 2T . As an immediate consequence, the system (18) in closed loop with the feedback defined in (4) admits the representation

   ẋ1 (t) = sat L 1 (α(t) + ξ 2 (t)) ξ2 (t) = sat L 2 (β (t) + ξ 3 (t) + α(t)) -α(t) ξ3 (t) = -sat L 4 (ξ 3 (t)) (19)
for all t ≥ 2T . Thus ξ 3 (t) converges asymptotically to 0. We deduce from (12a) and the bounds (15) that there is a class

M function T b : [0, +∞) → [2T, +∞) (depending on β1 and ᾱ2 ) such that for all t ≥ T b (|ξ (0)|), we have |β (t) + ξ 3 (t) + α(t)| ≤ β 1 + α 2 + |ξ 3 (t)| < L 2 . Hence, ẋ1 (t) = sat L 1 (α(t) + ξ 2 (t)) ξ2 (t) = β (t) + ξ 3 (t) (20)
for all t ≥ T b (|ξ (0)|).

Using the formulas y

1 (t) = η(t)x 1 (t) + δ 1 (t) and y 2 (t) - α(t) = x 2 (t) + δ 2 (t) -α(t) = ξ 2 (t) + δ 2 (t)
for our output components from (2) (which follow from ( 17)), and also using our formulas for φ and ω from ( 10) and ( 16), we can then rewrite [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF] as

ẋ1 (t) = sat L 1 -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) sat p 1 η( )x 1 ( ) +δ 1 ( ) d dm + ξ 2 (t) (21) and ξ2 (t) = - β 1 k p 2 (1-e -kT ) t t-T e k( -t) sat p 2 ξ 2 ( ) + δ 2 ( ) d + ξ 3 (t) (22) for all t ≥ T b (|ξ (0)|).
We also rewrite the ξ 2 -subsystem as ξ2 (t) = -

β 1 k p 2 (1-e -kT ) t t-T e k( -t) sat p 2 (ξ 2 ( )) d + J 2 (t), (23)
where

J 2 (t) = β 1 k p 2 (1-e -kT ) t t-T e k( -t) [sat p 2 (ξ 2 ( )) -sat p 2 ξ 2 ( ) + δ 2 ( ) d + ξ 3 (t) . ( 24
)
We rewrite the x 1 subsystem as

ẋ1 (t) = J 1 (t)+ sat L 1 -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) P 1 ( )d dm , (25)
where P 1 ( ) = sat p 1 η( )x 1 ( ) and

J 1 (t) = sat L 1 -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) sat p 1 η( )x 1 ( ) +δ 1 ( ) d dm + ξ 2 (t) -sat L 1 -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) P 1 ( )d dm . (26)
By (12b), we have α 1 < L 1 , and 2 holds for all t ≥ 0 and sat p 1 is bounded by p 1 , so (25) gives

t t-T m m-T e k( -t) d dm = 1 k 2 (1 -e -kT )
ẋ1 (t) = J 1 (t) -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) sat p 1 η( )x 1 ( ) d dm (27)
for all t ≥ T b (|ξ (0)|). Finally, since saturations have the global Lipschitz constant 1, we can check that

|J 2 (t)| ≤ |ξ 3 (t)| + β 1 p 2 δ 2 and |J 1 (t)| ≤ |ξ 2 (t)| + α 1 p 1 δ 1 for all t ≥ T b (|ξ (0)|), (28)
by the formula t t-T e k( -t) d = 1 k (1-e -kT ). We next analyze the stability properties of the system [START_REF] Zhu | Linear backstepping output feedback control for uncertain linear systems[END_REF].

Using [START_REF] Zhu | Linear backstepping output feedback control for uncertain linear systems[END_REF], it follows that, for all

t ≥ T b (|ξ (0)|) + T , ξ2 (t) = - β 1 p 2 sat p 2 (ξ 2 (t)) + R 1 (t) + J 2 (t), where (29a) R 1 (t) = β 1 k p 2 (1-e -kT ) t t-T e k( -t) [sat p 2 (ξ 2 (t)) -sat p 2 ξ 2 (t) -t ξ2 (s)ds d . (29b) From (22), we deduce that | ξ2 (t)| ≤ |ξ 3 (t)| + β 1 . As a consequence, for all t ≥ T b (|ξ (0)|) + T , we have |R 1 (t)| ≤ β 1 k p 2 (1-e -kT ) t t-T e k( -t) (t -)d |ξ 3 | [t-T,t] + β 1 ≤ β 1 T p 2 |ξ 3 | [t-T,t] + β 1 . ( 30
)
This inequality and the first inequality in (28) yield

|R 1 (t)| + |J 2 (t)| ≤ β 1 p 2 T |ξ 3 | [t-T,t] + β 1 + δ 2 + |ξ 3 (t)| (31)
for all t ≥ T b (|ξ (0)|) + T . Since the ξ 3 subsystem is globally asymptotically stable to 0, we deduce from (12f) that there is a class M function T c : [0, +∞) → [0, +∞) and a constant δ 0 > 0 such that

sup ≥T c (|ξ (0)|) |R 1 ( )| + |J ( )| ≤ β 1 p 2 T β 1 + δ 2 + δ 0 < β 1 . (32)
Hence, by (29a), there is a class M function T d : [0, +∞) → [0, +∞) such that T d (s) ≥ T b (s)+T for all s ≥ 0 and such that |ξ 2 (t)| < p 2 for all t ≥ T d (|ξ (0)|), because ξ2 > 0 (resp., < 0) when ξ 2 (t) ≤ -p 2 (resp., ≥ p 2 ). Then for all t ≥ T d (|ξ (0)|), we have sat p 2 (ξ 3 (t)) = ξ 3 (t) and therefore also

ξ2 (t) = - β 1 p 2 ξ 2 (t) + R 1 (t) + J 2 (t). ( 33
)
Second step. This step will use the candidate Lyapunov function ϒ(ξ 2 ) = 1 2 ξ 2 2 . By (29b), we get

|R 1 (t)| ≤ β 1 k p 2 (1-e -kT ) t t-T e k( -t) t | ξ2 (s)|dsd ≤ β 1 p 2 t t-T | ξ2 (s)|ds (34)
for all t ≥ T d (|ξ (0)|). By using [START_REF] Zhu | Linear backstepping output feedback control for uncertain linear systems[END_REF] to upper bound | ξ2 (s)| in (34), we deduce from the second inequality in (34) that for all t ≥ T d (|ξ (0)|), we have

|R 1 (t)| ≤ β 2 1 k p 2 2 (1-e -kT ) t t-T s s-T e k( -s) d ds × sup m∈[t-2T,t] |ξ 2 (m)| + β 1 p 2 t t-T |J 2 (s)| ds ≤ β 2 1 T p 2 2 sup m∈[t-2T,t] |ξ 2 (m)| + β 1 T p 2 β 1 p 2 δ 2 + |ξ 3 | [t-T,t] , (35)
where the last inequality is a consequence of the first inequality in (28). By using the first inequality in (28) and (35) to bound R 1 and J 2 from (33), we easily deduce that for any constant ω 0 > 0, we have

Υ(t) ≤ β 1 p 2 -ξ 2 2 (t) + β 1 T p 2 sup m∈[t-2T,t] ξ 2 2 (m) + |ξ 2 (t)| β 1 T p 2 + 1 β 1 p 2 δ 2 + µ(|ξ (0)|,t) ≤ β 1 p 2 -ϒ(ξ 2 (t)) + 2β 1 T p 2 sup m∈[t-2T,t] ϒ(ξ 2 (m)) + B * (t) 2 , (36)
along all solutions of (33) for all t ≥ T d (|ξ (0|) + T , where

B * (t) = (1 + ω 0 ) β 1 T p 2 + 1 2 β 2 1 p 2 2 δ 2 2 + 1 + 1 ω 0 µ 2 (|ξ (0)|,t) , (37)
and where the last inequality in (36) used the triangle inequality ab ≤ 1 2 a 2 + 1 2 b 2 with a = |ξ 2 (t)| and b being the quantity in squared brackets in (36) (followed by a use of the relation (r + s) 2 ≤ (1 + ω 0 )r 2 + (1 + (1/ω 0 ))s 2 for suitable r and s), and where µ is a class K L function. Since µ ∈ K L , we can then find a class M function T d : [0, +∞) → [0, +∞) such that T d (s) ≥ T d (s) for all s ≥ 0 and such that µ(|ξ (0)|,t) ≤ ω 2 0 for all t ≥ T d (|ξ (0)|), and therefore such that the quantity in squared brackets in (37) is bounded above by ω 0 + ω 2 0 for all t ≥ T d (|ξ (0)|). Therefore, for any ε > 0, we can choose ω 0 > 0 small enough such that

B * (t)β 1 /(p 2 (c 1 -c 2 )) < ξ (1 + ε) (38)
for all t ≥ T d (|ξ (0)|), where ξ was defined in ( 13), c 1 = β1 /p 2 , and c 2 = 2 β 2 1 T /p 2 2 . Since (12g) ensures that 1 > 2 β1 T /p 2 , it follows from Lemma A.1 in Appendix 2 (applied to v(t) = ϒ(ξ 2 (t + T d (|ξ (0)|) + T )), and with the preceding choices of c 1 and c 2 and the choice

∆ 1 = 1 2 ( β1 /p 2 ) sup{B * (t) : t ≥ T d (|ξ (0)|}) that |ξ 2 (t)| ≤ ξ (1 + ε) + µ 0 (|x(0)| + |δ | ∞ ,t) (39)
holds for all t ≥ T d (|ξ (0)|); this is done by using the formulas ( 16)-( 17) to find a function γ

∈ K ∞ such that |ξ (0)| ≤ γ(|x(0)| + |δ | ∞ )
for all initial states, in order to find µ 0 .

The next part of the proof is devoted to the x 1 -subsystem (27). First, we deduce from (27

) that | ẋ1 (t)| ≤ α 1 +|J 1 (t)| ≤ α 1 + |ξ 2 (t)| + α 1
p 1 δ 1 , using the second inequality in (28), so (39) gives

| ẋ1 (t)| ≤ α 1 + ξ (1 + ε) + α 1 p 1 δ 1 + µ 0 (|x(0)| + |δ | ∞ ,t) (40)
for all t ≥ T d (|ξ (0)|). From (27), it follows that for all t ≥ T d (|ξ (0)|), we have

ẋ1 (t) = R 2 (t) + J 1 (t) -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) sat p 1 (η( )x 1 (t)) d dm, (41a)
where R 2 (t) =

-k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) [sat p 1 (η( )x 1 ( )) -sat p 1 (η( )x 1 (t))] d dm. (41b)
As an immediate consequence, for all t ≥ T d (|ξ (0)|), we have

|R 2 (t)| ≤ k 2 α 1 (1-e -kT ) 2 p 1 η t t-T m m-T e k( -t) |x 1 (t) -x 1 ( )| d dm ≤ α 1 η p 1 t t-2T | ẋ1 (s)
|R 2 (t)| ≤ 2T α 1 η p 1 α 1 + ξ (1+ε) + µ 0 (|x(0)|+|δ | ∞ ,t) + α 1 p 1 δ 1 , (43)
where µ 0 (s,t) = µ 0 (s,t -2T ). Combining (43) and the second inequality in (28) with (39), we obtain a class M function

T d : [0, +∞) → [0, +∞) such that sup t≥T d (|ξ (0)|) (|R 2 (t)| + |J 1 (t)|) < α 1 η , (44)
where the last inequality is a consequence of (12h) and by choosing ε ∈ (0, 1) to be small enough. Next observe that when x 1 (t) ≥ p 1 /η, then, bearing in mind that η is lower bounded by 1, we deduce from (41a) that

ẋ1 (t) ≤ -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) p 1 η d dm + R 2 (t) + J 1 (t) = -α 1 η + R 2 (t) + J 1 (t) < 0 (45) for all t ≥ T d (|ξ (0)|),
ẋ1 (t) = -k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) η( )x 1 (t)d dm + R 2 (t) + J 1 (t) (46)
for all t ≥ T e (|ξ (0)|). We can assume that T e (s) ≥ T d (s) + 2T for all s ≥ 0.

Let us consider the candidate Lyapunov function ν(x 1 ) =

1 2 x 2 1 .
Since η is lower bounded by 1, we deduce from the second inequality in (28) and (42) that

ν(t) ≤ -α 1 p 1 x 2 1 (t) + x 1 (t) R 2 (t) + J 1 (t) ≤ -α 1 p 1 x 2 1 (t) + |x 1 (t)| α 1 η p 1 t t-2T | ẋ1 (s)|ds + |x 1 (t)| |ξ 2 (t)| + α 1 p 1 δ 1 (47)
holds along all solutions of (46) for all t ≥ T e (|ξ (0)|). From (27), we deduce that

| ẋ1 (t)| ≤ k 2 α 1 (1-e -kT ) 2 p 1 t t-T m m-T e k( -t) η|x 1 ( )|d dm + |J 1 (t)| ≤ α 1 η p 1 sup m∈[t-2T,t] |x 1 (m)| + |ξ 2 (t)| + α 1 p 1 δ 1 (48)
for all t ≥ T e (|ξ (0)|), by our bound η on η, and where the second inequality is by (28), so (47) and the choice

H (s) = α 1 η p 1 sup m∈[s-2T,s] |x 1 (m)| + |ξ 2 (s)| (49) give ν(t) ≤ -α 1 p 1 x 2 1 (t) + |x 1 (t)| α 1 η p 1 t t-2T H (s)ds +|x 1 (t)| 2T α 1 η p 1 α 1 p 1 δ 1 + |ξ 2 (t)| + α 1 p 1 δ 1 ≤ -α 1 p 1 x 2 1 (t) + 2T α 2 1 η 2 p 2 1 sup m∈[t-4T,t] |x 1 (m)| 2 +|x 1 (t)| α 1 η p 1 t t-2T |ξ 2 (s)|ds + 2T α 2 1 η p 2 1 δ 1 + |ξ 2 (t)| + α 1 p 1 δ 1 ( 50
)
for all t ≥ T e (|ξ (0)|) + 2T . Setting ξ = ξ (1 + ε), it follows from (39) that, for all t ≥ T e (|ξ (0)|) + 2T ,

ν(t) ≤ -α 1 p 1 x 2 1 (t) + 2T α 2 1 η 2 p 2 1 sup m∈[t-4T,t] |x 1 (m)| 2 + {|x 1 (t)|} 2T α 1 η p 1 (ξ + µ 0 (|x(0)|+|δ | ∞ ,t)) + 2T α 2 1 ηδ 1 p 2 1 +ξ + µ 0 (|x(0)|+|δ | ∞ ,t)+ α 1 δ 1 p 1 ≤ -2(1-ω 0 )α 1 p 1 ν(x 1 (t)) + 4T α 2 1 η 2 p 2 1 sup m∈[t-4T,t] ν(x 1 (m)) + p 1 4ω 0 α 1 2T α 1 η p 1 (ξ + µ 0 (|x(0)| + |δ | ∞ ,t)) + 2T α 2 1 ηδ 1 p 2 1 + ξ + µ 0 (|x(0)|+|δ | ∞ ,t) + α 1 p 1 δ 1 2 (51)
where the last inequality applied Young's inequality ab ≤

α 1 ω 0 p 1 a 2 + p 1 4α
1 ω 0 b 2 to the terms in curly braces. Third step. By (12e), we can assume that ω 0 ∈ (0, 1) is small enough so that

(1-ω 0 )α 1 p 1 > 2T α 2 1 η 2 p 2 1 .
From Lemma A.1 below and the second inequality in (51), there is a µ 1 ∈ K L and a class M function T e : [0, +∞) → [0, +∞) such that for all t ≥ T e (|ξ (0)|), we have

ν(x 1 (t)) ≤ p 3 1 (1+ω 0 ) 8α 2 1 ω 0 (p 1 (1-ω 0 )-2T α 1 η 2 ) 2T α 1 ηξ p 1 + 2T α 2 1 ηδ 1 p 2 1 +ξ + α 1 δ 1 p 1 2 + µ 1 (|x(0)|+|δ | ∞ ,t), (52)
by the argument that led to (39). Consequently, with the choice µ 2 = √ 2µ 1 , we get

|x 1 (t)| ≤ γ a + µ 2 (|x(0)| + |δ | ∞ ,t), (53)
where γ a = √ 2B a and B a is the quantity in curly braces in (52), by the subadditivity of the square root. We next find analogous bounds for x 2 and x 3 .

From (39) and the fact that ξ 2 = x 2α, and also using ( 16), we deduce that

x 2 (t) + k 2 (1-e -kT ) 2 t t-T m m-T e k( -t) α 1 p 1 sat p 1 (η( )x 1 ( ) +δ 1 ( ))d dm ≤ ξ + µ 0 (|x(0)| + |δ | ∞ ,t) (54)
and so also

|x 2 (t)| ≤ ξ + µ 0 (|x(0)| + |δ | ∞ ,t) + k 2 (1-e -kT ) 2 α 1 p 1 t t-T m m-T e k( -t) [η|x 1 ( ) + δ 1 ]d dm (55)
for all t ≥ T d (|ξ (0)|). From this inequality and (53), we deduce that for all t ≥ T e (|ξ (0)|) + 2T , we have

|x 2 (t)| ≤ ξ + µ 0 (|x(0)| + |δ | ∞ ,t) + α 1 p 1 ηγ a + δ 1 ( 56
)
where

µ 0 = µ 0 + ( ᾱ1 /p 1 ) η µ 2 ∈ K L . Moreover, since x 3 (t) = α(t) + β (t) + ξ 3 (t), we have |x 3 (t)| ≤ k 3 (1-e -kT ) 2 t t-T m m-T e k( -t) |φ ( )|d dm + k 2 (1-e -kT ) 2 t t-T e k( -t) |φ ( )|d + k 2 (1-e -kT ) 2 t-T t-2T e k( -t) |φ ( )|d + k 1-e -kT t t-T e k( -t) |ω( )|d + |ξ 3 (t)|, (57)
by [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF]. Next observe that our choices of y 1 and

y 2 from (2) give |φ (t)| ≤ α 1 p 1 |η(t)x 1 (t) + δ 1 (t)| and |ω(t)| ≤ β 1 p 2 |x 2 (t) + δ 2 (t) + k 2 (1-e -kT ) 2 t t-T m m-T e k( -t) φ ( )d dm (58)
for all t ≥ 2T . It follows from ( 53) and (56) that we can find a function

µ 3 ∈ K L such that |φ (t)| ≤ φ + µ 3 (|x(0)| + |δ | ∞ ,t), where φ = α 1 p 1 ηγ a + δ 1 ( 59
)
and |ω(t)| ≤

β 1 p 2 ξ + α 1 p 1 ηγ a + δ 1 + β 1 p 2 s t t-T m m-T e k( -t) d dmφ + δ 2 +µ 3 (|x(0)| + |δ | ∞ ,t)) ≤ ω + β 1 p 2 µ 3 (|x(0)| + |δ | ∞ ,t), (60)
by enlarging T e as needed without relabeling, where s = k 2 /(1e -kT ) 2 and where

ω = β 1 p 2 ξ + α 1 p 1 (ηγ a + δ 1 ) + φ + δ 2 . ( 61
)
Since ( 16) also gives

| α(t)| ≤ 2k|φ | [t-2T,t]
1-e -kT for all t ≥ 2T , we deduce from the formula x 3 (t) = α(t) + β (t) + ξ 3 (t) and our bounds on φ and ω from (59) and (60) that we can find a class M function T f : [0, +∞) → [0, +∞) and a function

µ 4 ∈ K L such that |x 3 (t)| ≤ 2k 1-e -kT φ + ω + µ 4 (|x(0)| + |δ | ∞ ,t) (62)
for all t ≥ T f (|ξ (0)|), using the formula for β from ( 16).

In fact, ( 9)-( 11) provide a constant c > 0 such that |ż(t)| ≤ c(|z| [t-T,t] + |x(t)| + |δ | ∞ ) and therefore also

|z| [t-T,t] ≤ c t 0 |z| [ -T, ] + |x( )| d + ct|δ | ∞ , (65)
for all t ≥ 0 (since we assumed that the initial functions for the z dynamics are 0). Therefore, we can apply Gronwall's inequality (e.g., from [7, Lemma A.1, p.651]) to the function

F (t) = |z| [t-T,t] to find a constant c > 0 such that |z(t)| ≤ c e cT g (|x(0)|) (|x| [0,t] + T g (|x(0)|)|δ | ∞ ) (66)
for all t ∈ [0, T g (|x(0)|)], which we can use to find positive constants c > 0 and c * such that the right side of ( 1) is bounded by c(e c * T g (|x(0

)|) |x| [0,t] + |δ | 2 ∞
) for all t ∈ [0, T g (|x(0)|)] (by applying the triangle inequality to upper bound the term in (66) containing the |δ | ∞ and using the class M structure of T g ), and then we can apply Gronwall's inequality to the x system (as we did for the z system) to get the required constant L > 0. The final input-to-state stability estimate now follows by adding the bounds for (63)-(64) for |x(t)| to find a bound that holds for all t ≥ 0.

V. APPLICATION TO VISUAL LANDING OF AIRCRAFT

To illustrate our results, we consider the lateral dynamics of an Airbus airliner in a glide phase which must align with a runway using a body fixed monocular camera [START_REF] Burlion | Vision based anti-windup design with application to the landing of an airliner[END_REF]. This problem is a challenge of strong relevance in cases where 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. See Figure 1, and see [START_REF] Bras | Approach maneuvers for autonomous landing using visual servo control[END_REF] for valuable research on vision based aircraft control, which does not provide the input-to-state stability to uncertainty that we provide in this work. As noted in [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF], this research was motivated by this 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 ), (67)
where u lat is the input, V = 72m.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). Then (67) can be transformed into the system (1) by applying the change of coordinates x 1 = ∆ Y , x 2 = V ∆ ψ , x 3 = gϕ while changing the saturation limits accordingly. In [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF], the time varying parameter η(t) is chosen to satisfy η(t) ∈ [0.3, 1], and in our simulations to follow, we scale it so that the new η is bounded below by 1 (as explained in Section II above). Note that η is minimum when the aircraft is close to the runway and starts its flare.

The saturation limits are

L 1 = 25m.s -1 , L 2 = 7m.s -2 , and L 3 = 6m.s -3 (68)
For comparison purposes, the controller studied in this paper is called 'controller 1' in our simulations, and we compare its performance with the control from [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] which we refer to as 'controller 2'. We use the folloing parameters the satisfy the assumptions of Theorem 1 when δ1 ≤ 0.9 and δ2 ≤ 0.3:

k = 0.1, T = 2, L 4 = 1, ᾱ = 2, p 1 = 10, β = 0.2, and p 2 = 2.5. ( 69
)
It is assumed that the runway suddenly appears in the image at instant t = 5s and will remain in the camera field of view throughout the descent. As such, the x i 's are reinitialized at time t = 5s. Doing so, our controller can be fairly compared to any another one (possibly reacting from t = 0). Indeed, according to (4), controller 1 is 0 for t ∈ (0, 4) seconds. Figure 2 shows that controller 1 succeeds to lower a lateral deviation of 30m for several initial heading deviations ∆ ψ ∈ {-5, 0, 5}deg. Figure 3 shows the responses when the outputs are noisy.

On the other hand, our figures also show the performance of the 'controller 2' from [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF]. We made these choices of the parameters in 'controller 2' in our simulations: k 1 = 0.3, k 2 = 0.59, k 3 = 0.8, τ = 2.5, L 4 = 1, and v = 1.4, and φ (x) = 0.15 tanh(0.4x).

(70)

Using these parameters, the assumptions of [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] are satisfied when δ1 = δ2 = 0. Moreover, looking at Figures 2 and3 below, we observe the settling times are almost similar when one uses controllers 1 and 2. However, the assumptions of [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] seem to be far more conservative when the outputs are noisy. Indeed, the assumptions of [START_REF] Mazenc | Stabilization with imprecise measurements: application to a vision based landing problem[END_REF] would not be satisfied with δ1 = 0.9 and δ2 = 0.3, and in order to satisfy them it would be necessary to multiply these bounds by 4.10 -3 .

VI. CONCLUSIONS

We used a recent backstepping approach to construct a useful new class of bounded controls for a chain of saturated integrators that arises in the visual landing of aircraft. This overcame the challenge of having imprecise output measurements by proving an input-to-state stability estimate. In future work, we plan to generalize this work to allow sampling (as described in [START_REF] Palmeira | Sampled-data control under magnitude and rate saturating actuators[END_REF], [START_REF] Seuret | Taking into account period variations and actuator saturation in sampled-data systems[END_REF]) or delays in the outputs, which would be useful for modeling the effects of image processing. APPENDIX 1: PROOF OF BOUNDS [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF] We use the formulas [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF] to prove the bounds on the derivatives of α and β in [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF]; the bounds on α and β in [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF] follow from the definition of the saturation and ( 16). We use the common notation s = k 2 /(1e -kT ) 2 and the equalities and inequalities to follow should be understood to hold for all t ≥ 2T . We have α(t) = -kα(t) + s t t-T e k( -t) φ ( )d -t-T t-2T e k( -t) φ ( )d , by our choice of α 2 in [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF]. Also, since α(t) = -k α(t) + s(-k) t t-T e k( -t) φ ( )d t-T t-2T e k( -t) φ ( )d + s[φ (t) -2e -kT φ (t -T ) + e -2kT φ (t -2T )], (A.3)

4 ∞

 4 Fourth step. By combining the upper bounds (53), (56), and (62) for the |x i (t)|'s and recalling the formulas for the components of ξ , we can construct functions β 0 ∈ K L and γ 0 ∈ K ∞ and a class M function T g : [0, +∞) → [0, +∞) such that |x(t)| ≤ β 0 (|x(0)|,t) + γ(|δ | ∞ ) (63) for all t ≥ T g (|x(0)|), where the construction of T g used the fact that our formulas (17) imply that there is a class M function Θ : [0, +∞) → [0, +∞) such that |ξ (0)| ≤ Θ(|x(0)|) holds for all initial states ξ (0) and x(0), so we can choose T g (s) = T f (Θ(s)). Also, the linear growth of the x dynamics provides a constant L > 0 such that |x(t)| ≤ Le LT g (|x(0)|)-t |x(0)| + L|δ | for all t ∈ [0, T g (|x(0)|)].

Fig. 1 .

 1 Fig. 1. Notation used in the alignment part of the glide phase

Fig. 2 .Fig. 3 .

 23 Fig. 2. Evolution of x 1 (t) when δ1 = 0, δ2 = 0.

| α| ∞ ≤ kα 1 +sα 1 1 k 1 1 k 1

 1111 -e -kT + 1 k e -kT -e -2kT = kα 1 +sα 1 e -kT 1 + e -kT = α 2 , (A.2)

  where the second inequality is a consequence of (44). Similarly, one can prove that when x 1 (t) ≤ -p 1 /η then ẋ1 (t) > 0. Hence, there is class M function T

e : [0, +∞) → [0, +∞) such that |η( )x 1 (t)| < p 1 for all t ≥ T e (|ξ (0)|) and ≥ 0. It follows from (41a) that

Burlion was partly supported by ANR VISIOLAND project ANR-13-CORD-0012 dedicated to VISIOn based aircraft LANDing techniques. Malisoff was partly supported by NSF Grant 1711299.

our formula for α 3 from [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] gives

1-e -kT + s 1e -kT + s e -kT -e -2kT ᾱ1 +s 1+2e -kT +e -2kT ᾱ1 ≤ 4α 1 s = α 3 .

(A.4) Finally, since β (t) = -kβ (t) + k

1-e -kT ω(t)ω(t -T )e -kT , (A.5) we have

by our choice of β 2 in [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF]. which completes our proof of the bounds (15

is satisfied for all t ≥ 0. Let c s > 0 be the unique positive value such that c s = c 1 -c 2 e c s h , and let

holds for all t ≥ 0. Proof: First, observe that

for all t ≥ 0. Since c 1 > c 2 , the required constant c s exists. Also, the function p(t) = e -c s t satisfies

for all t ≥ 0. It now suffices to prove that l v p(t) ≥ ṽ(t) for all t ≥ -h. To this end, suppose that there is a constant t c > 0 such that l v p(t) > ṽ(t) for all t ∈ [-h,t c ) and l v p(t c ) = ṽ(t c ), for the sake of obtaining a contradiction. Let w(t) = ṽ(t)l v p(t). Then, using l v p(t c ) = ṽ(t c ), we deduce from (A.9) and (A.10) that

where the last inequality in (A.11) follows because if we choose a t * ∈ [t ch,t c ] such that sup m∈[t c -h,t c ] ṽ(m) = ṽ(t * ), then the quantity in squared brackets in (A.11) is ṽ(t * )l v p(t ch), which is negative if t * = t ch (by our choice of t c ) and is also negative if t * ∈ (t ch,t c ] because in that case it is bounded above by l v (p(t * )p(t ch)) < 0. From ẇ(t c ) < 0 and w(t) < 0 when t ∈ [-h,t c ), we deduce that w(t c ) < 0, which is a contradiction. Hence, w(t) < 0 for all t ≥ -h. Thus ṽ(t) ≤ l v p(t) for all t ≥ -h, which gives the conclusion.