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

Laurent Burlion
email: laurent.burlion@onera.fr

⇤⇤ Michael

Maliso↵ ⇤⇤⇤
email: maliso↵@lsu.edu

Backstepping Design for Output Feedback Stabilization for Uncertain Systems using Dynamic Extension ?

Keywords: backstepping, nonlinear, output feedback, stabilization

We revisit the backstepping approach. We show how bounded globally asymptotically stabilizing output feedbacks can be constructed for a family of nonlinear systems. The approach relies on the introduction of a dynamic extension and a converging-input-convergingstate assumption. The technique presents several advantages. It provides control laws whose expressions are simple. It makes it possible to stabilize systems in the presence of uncertain terms which prevent the use of the classical backstepping technique. It applies in cases where only part of the state variable is measured.

INTRODUCTION

The backstepping approach is a fundamental technique for constructing globally asymptotically stabilizing control laws for nonlinear systems in feedback form, i.e., having a lower triangular structure; see [START_REF] Dixon | Global exponential setpoint control of wheeled mobile robots: a Lyapunov approach[END_REF]. Since the pioneering contributions in [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF] and [START_REF] Tsinias | Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation[END_REF], it has been developed in many contributions and is successfully used in many applications, as illustrated for instance by [START_REF] Jiang | A recursive technique for tracking control of nonholonomic systems in chained form[END_REF], [START_REF] Pettersen | Underactuated ship tracking control; theory and experiments[END_REF], [START_REF] Smaoui | A study on tracking position control of an electropneumatic system using backstepping design[END_REF], de Queiroz and [START_REF] De Queiroz | Nonlinear control of active magnetic bearings: a backstepping approach[END_REF] and [START_REF] Lee | Adaptive tracking control of an underactuated aerial vehicle[END_REF]. Presentations of the backstepping technique can be found in [START_REF] Maliso↵ | Constructions of Strict Lyapunov Functions[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF], and many other research monographs and papers.

One of the drawbacks of this technique is the complexity of the formulas it sometimes provides, notably when it is applied repeatedly and when size constraints on the control laws have to be respected. This is a limitation of the applicability of the classical backstepping and of the bounded backstepping results of [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF] and [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF]. Another limitation of the approach is due to the fact that in general, it does not apply when only a part of the state is measured. In addition, it relies on the existence of a fictitious feedback which has to be of class C k when the backstepping is applied k times. The presence of uncertainties in the dynamics and in the output may also be an obstacle.

?

Maliso↵ was supported by US National Science Foundation Grant 1408295.

To overcome these drawbacks, a new technique has been proposed recently, based on control laws in which delays are artificial, meaning there are delays in the state values that are used in the controls even when there are no input delays in the original systems. This technique has been initiated in Mazenc and Maliso↵ (2016) and developed and applied in Mazenc et al. (2016), [START_REF] Mazenc | Bounded backstepping control and robustness analysis for timevarying systems under converging-input-convergingstate conditions[END_REF] and [START_REF] Mazenc | Bounded backstepping through a dynamic extension with delay[END_REF], where in particular, the case of systems with delays in the input is considered and where the initial fictitious control law is not required to be of class C 1 . A first adaptation of this technique to uncertain outputs has also been studied in [START_REF] Mazenc | Stabilization of a system that arises in the context of vision based landing of a civil aircraft[END_REF]. Due to the use of visual information, only imprecise measurement of the first backstepping variable was available.

In the present paper, we propose a new backstepping design for globally uniformly asymptotically stabilizing control laws for partially linearizable systems; see [START_REF] Isidori | Nonlinear Control Systems[END_REF] for an introduction to partially linearizable systems. Its fundamental new aspect with respect to the contributions that use artificial delays is that, instead of introducing delays, a finite dimensional dynamic extension is designed, making it possible to obtain feedbacks without delays, which o↵er the following advantages: (i) they are bounded in the cases where bounded feedbacks can be expected, (ii) they are given by simple formulas, (iii) they apply in cases where the subsystem from which the backstepping starts is not known with accuracy, and (iv) they apply in cases where only a part of the state variable is measured and where some parts of the output are not known with accuracy. The family of partially linearizable systems we consider encompasses many systems that are relevant from an applied point of view, as shown for instance by [START_REF] Spong | Partial feedback linearization of underactuated mechanical systems[END_REF] and [START_REF] Olfati-Saber | Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles[END_REF]. The advantages of the control laws we propose imply that our new result is of interest even when boundedness of the feedback is not desired. Also, with respect to the backstepping with artificial delay methods from the works Mazenc and Maliso↵ (2016) and Mazenc et al. (2016), the main drawback of the main result of the present paper is that it relies on a fictitious feedback of class C 1 . Besides, comparisons between the performances of the control laws of Mazenc and Maliso↵ (2016) and Mazenc et al. (2016) and those of the present paper still have to be made and will be the subject of future studies. This paper is organized as follows. The main result is shown in Section 2. Section 3 provides a technical lemma that can facilitate checking our assumptions from Section 2. An example based on a direct drive manipulator is given in Section 4. Concluding remarks are drawn in Section 5.

Our notation will be simplified whenever no confusion can arise from the context. The dimensions of our Euclidean spaces are arbitrary unless other indicated. The Euclidean norm in R a , and the induced norm of matrices, are denoted by | • |, and | • | 1 is the usual sup norm. Given a constant J > 0, sat J denotes the symmetric saturation function defined by sat J (x) = max{ J, min{J, x}} for all x 2 R, where J is called the saturation level. We use the usual class of comparison functions K 1 from [START_REF] Khalil | Nonlinear Systems[END_REF].

MAIN STABILIZATION RESULT

The studied system

We consider the nonlinear time-varying system

8 > > > > > > < > > > > > > : ẋ = f (t, x, g 1 ) ġ1 = a 1,1 g 1 + a 1,2 g 2 ġ2 = a 2,1 g 1 + a 2,2 g 2 + a 2,3 g 3 . . . ġn = a n,1 g 1 + a n,2 g 2 + • • • + a n,n g n + u + ⌦(Y(t, x), G) (1)
where x is valued in R p , G = (g 1 , ..., g n ) is valued in R n , f is a nonlinear function that is locally Lipschitz with respect to (x, g 1 ) and piecewise continuous with respect to t, u is the scalar-valued input, Y : R ⇥ R p ! R l is a continuous function, each a i,j 2 R is a constant, a i,j+1 6 = 0 holds for i = 1, 2...n 1 (with no sign constraint on the a ij 's), and the function ⌦ is locally Lipschitz. This family of systems has been studied in many papers because systems of this type may result from partial linearization and are frequently encountered in practice.

In this section, we assume that the R l+n valued output is

Y (t) = (Y(t, x(t)), g 1 (t), . . . , g n (t)) > .
(2) Our first assumption is: Assumption 1. There are a Lipschitz continuous scalar valued function that is bounded by a known constant 0 and a constant k > 0 such that for any continuous function d : [0, 1) ! R that converges exponentially to the origin, all solutions (⇠, 1 , ...,

n ) : [0, 1) ! R p+n of 8 > > > < > > > : ⇠ = f (t, ⇠, 1 + d(t)) ˙ 1 = k[ 1 + 2 ] . . . ˙ n = k[ n + (t, Y(t, ⇠))]
(3) converge to the origin as t ! 1. Moreover the function ⌦ in ( 1) is bounded by a known constant ⌦ 0.

In terms of the matrices

A = 2 6 6 6 6 6 4 a 1,1 a 1,2 0 . . . 0 a 2,1 a 2,2 a 2,3 . . . 0 . . . . . . . . . . . . . . . . . . a n 1,n a n,1 a n,2 . . . . . . a n,n 3 7 7 7 7 7 5 2 R n⇥n (4)
for n > 1 and A = a 1,1 for n = 1 and B = (0, 0, . . . , 1) > 2 R n , our second assumption is: Assumption 2. There is a locally Lipschitz function $ : R n ! R that is bounded by a known constant $ 0, such that the origin of the system 5) is a globally asymptotically and locally exponentially stable equilibrium.

˙ (t) = A (t) + B$( (t)) (

Statement of and discussion on theorem

We are ready to state and prove the following result: Theorem 1. Let the system (1) satisfy Assumptions 1-2 and let ✏ > 0 be any constant. Then there are a constant matrix L and constants c j for j = 1 to n + 1 such that all solutions of (x, G) : [0, 1) ! R p+n of (1) in closed loop with the dynamic output feedback

u(G, Z, x) = sat Z 0 @ n X j=1 c j z j + c n+1 (t, Y(t, x)) 1 A + $ (G + LZ) ⌦ (Y(t, x), G) ż1 = k[ z 1 + z 2 ] . . . żn 1 = k[ z n 1 + z n ] żn = k[ z n + (t, Y(t, x))] (6)
with Z = (z 1 , ..., z n ) and the saturation level

Z = (1 + ✏) n+1 X j=1 |c j |
asymptotically converge to 0 as t ! 1. ⇤

Before turning to the proof of Theorem 1, we make several remarks on its motivation and value.

1) The feedback ( 6) is bounded by the constant Z +$ +⌦.

2) The formula (6) for the control law does not incorporate the first derivative of (t, Y(t, x(t))), which plays the role of the fictitious control of the classical backstepping approach. Hence, it applies even when (t, Y(t, x)) is not of class C 1 . However, in practice Theorem 1 can only be applied when is of class C 1 because checking that Assumption 1 is satisfied frequently necessitates that be of class C 1 , as we shall show in the next section. By contrast, in Mazenc et al. (2016) and [START_REF] Mazenc | Bounded backstepping control and robustness analysis for timevarying systems under converging-input-convergingstate conditions[END_REF], fictitious feedbacks of class C 0 can be used.

3) One can use changes of variables and an appropriate choice of feedback to transform the system (1) into a system of the form (3) with d = 0 whose stability is ensured by Assumption 1. This follows from our assumption that the a i,i+1 's are nonzero. However, in a sense, this result is not satisfactory because the feedback obtained that way may possess inappropriate properties. For instance, in general they would be unbounded. This motivates our alternative approach, based on dynamic extensions. 5) None of the assumptions of this paper imply that f has to be known with accuracy. Also, we only require measurements of (t, Y(t, x)) and ⌦(Y(t, x), G), instead of Y itself.

Proof of Theorem 1

A key aspect of the proof consists of making g 1 (t) z 1 (t) converge to zero instead of making g 1 (t) (t, Y(t, x(t))) converge to zero, as traditionally done in classical backstepping. To achieve our goal, several changes of coordinates are needed, as follows. We first assume that n > 1, and we explain how these changes of variables can be constructed by induction. Set l 1,1 = 1 and r 1 = g 1 z 1 .

Induction assumption: For all j 2 {1, ..., i} and 1 < i < n, there are constants l j,m for m = 1 to j such that the variables

r j = g j + j X m=1 l j,m z m (7) for j = 1 to i satisfy 8 > > < > > : ṙ1 = a 1,1 r 1 + a 1,2 r 2 ṙ2 = a 2,1 r 1 + a 2,2 r 2 + a 2,3 r 3 . . . ṙi 1 = a i 1,1 r 1 + a i 1,2 r 2 + • • • + a i 1,i r i .
(8)

First step: i = 2. Since r 1 = g 1 z 1 , we have ṙ1 = a 1,1 g 1 + a 1,2 g 2 k[ z 1 + z 2 ] = a 1,1 r 1 + a 1,2  g 2 + a 1,1 + k a 1,2 z 1 k a 1,2 z 2 . (9)
Then the variable

r 2 = g 2 + a 1,1 + k a 1,2 z 1 k a 1,2 z 2 (10) satisfies ṙ1 = a 1,1 r 1 + a 1,2 r 2 (11)
Thus the induction assumption is satisfied at the first step.

Step i: Assume that the induction assumption is satisfied at a step i with 1 < i < n. Then since (7) holds for

j = 1, 2, . . . , i we have ṙi = a i,1 g 1 + a i,2 g 2 + ... + a i,i+1 g i+1 + i X m=1 l i,m k[ z m + z m+1 ] = a i,1 r 1 + a i,2 r 2 + ... + a i,i r i + a i,i+1 g i+1 +a i,1 z 1 a i,2 (l 2,1 z 1 + l 2,2 z 2 ) ... a i,i i X m=1 l i,m z m + i X m=1 l i,m k[ z m + z m+1 ]. (12)
Thus, taking

r i+1 = g i+1 + ai,1 ai,i+1 z 1 ai,2 ai,i+1 (l 2,1 z 1 + l 2,2 z 2 ) ... ai,i ai,i+1 i P m=1 l i,m z m + 1 ai,i+1 i P m=1 l i,m k[ z m + z m+1 ] (13) we obtain ṙi = a i,1 r 1 + a i,2 r 2 + ... + a i,i+1 r i+1 , (14)
so the induction assumption is satisfied at the step i + 1.

Taking the time derivative of ( 13) with the choice i = n 1, we can then find a linear change of coordinates R = G+LZ with R = (r 1 , ..., r n ) > that transforms the system (1) into

8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ẋ = f (t, x, z 1 + r 1 ) ṙ1 = a 1,1 r 1 + a 1,2 r 2 ṙ2 = a 2,1 r 1 + a 2,2 r 2 + a 2,3 r 3 . . . ṙn = a n,1 r 1 + a n,2 r 2 + • • • + a n,n r n + u + n X j=1 l n,j z j + l n,n+1 (t, Y(t, x)) + ⌦(Y(t, x), G). ( 15
)
If instead n = 1, then we again obtain (15) with only the dynamics for x and r n present. Let c j = l n,j for j = 1 to n + 1. Then the closed-loop system is

8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : ẋ = f (t, x, z 1 + r 1 ) Ṙ = AR + B [$(R) sat Z 0 @ n X j=1 c j z j + c n+1 (t, Y(t, x)) 1 A + n X j=1 c j z j + c n+1 (t, Y(t, x)) 3 5 ż1 = k[ z 1 + z 2 ] . . . żn = k[ z n + (t, Y(t, x))]. (16)
Consider any solution (x, R, z) : [0, 1) ! R p+2n of (16).

Since Assumption 1 ensures that is bounded by , it follows from ( 16) that there is a finite value t a 0 such that for all t t a , the inequality

n X j=1 c j z j (t) + c n+1 (t, Y(t, x(t)))  Z (17)
is satisfied. Hence, when t t a , the closed-loop system is 8

> > > > > > < > > > > > > : ẋ = f (t, x, z 1 + r 1 ) Ṙ = AR + B$(R) ż1 = k[ z 1 + z 2 ] . . . żn = k[ z n + (t, Y(t, x))]. (18)
Assumption 2 ensures that the R-subsystem of the system (18) is globally asymptotically and locally exponentially stable. Then Assumption 1 allows us to conclude.

TECHNICAL LEMMA

Assumption 2 is a simple classical assumption that can often be checked easily; see our illustration below. However, checking Assumption 1 can be nontrivial. In this section, we give conditions ensuring that Assumption 1 holds.

Assumptions and lemma

Consider the system

8 > > > < > > > : ⇠ = f (t, ⇠, 1 + d(t)) ˙ 1 = k[ 1 + 2 ] . . . ˙ n = k[ n + (t, y ⇠ )] (19)
where y ⇠ = C⇠ and C 2 R q⇥p is a constant matrix, the constant k > 0 will be further restricted below, and the function d : [0, 1) ! R exponentially converges to 0. Then ( 19) is a special case of the system (3) from Assumption 1. We add the following assumptions on (19): Assumption 3. There is a function

 1 2 K 1 such that | (t, y ⇠ )| + |f (t, ⇠, w)|   1 (|(⇠, w)|) (20
) holds for all t 0, ⇠ 2 R p and w 2 R. Also, the function is of class C 1 and bounded by a known constant . Finally, there are two functions  2 and  3 of class K 1 , a function

V of class C 1 , and a positive definite function W such that  2 (|⇠|)  V (t, ⇠)   3 (|⇠|) for all (t, ⇠) 2 [0, 1)⇥R p (21)
and such that for any choice of the piecewise continuous function , the time derivative of V along all solutions ⇠(t) = f (t, ⇠(t), (t, C⇠(t)) + (t)) ( 22)

satisfies the inequality V (t)  W (⇠(t)) + | (t)| 2 for all t 0, and f is locally Lipschitz. Assumption 4. There are positive constants b f and b s such that for the function W from Assumption 3, the inequality @ @t (t, C⇠)

+ @ @y ⇠ (t, C⇠)Cf (t, ⇠, (t, C⇠) + ) 2  b f W (⇠) + b s | | 2 (23)
holds for all t 0, ⇠ 2 R p and 2 R.

We are ready to state and prove the following result: Lemma 1. Let the system (19) satisfy Assumptions 3 and 4. Then for any k k with

k = 3n ⇣ n + n(n+1) 2 b f + bs(n+1) 4 ⌘ ( 24
)
all solutions of (19) converge to 0 as t ! 1.

Proof of Lemma 1

Set s i = i (t, y ⇠ ) for i = 1, 2, . . . , n and S = (s 1 , ..., s n ). Then ( 19) can be transformed into

8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ⇠ = f (t, ⇠, (t, y ⇠ ) + s 1 + d(t)) ṡ1 = k[ s 1 + s 2 ] @ @t (t, y ⇠ ) @ @y ⇠ (t, y ⇠ )Cf (t, ⇠, (t, y ⇠ ) + s 1 + d(t)) . . . ṡn = ks n @ @t (t, y ⇠ ) @ @y ⇠ (t, y ⇠ )Cf (t, ⇠, (t, y ⇠ ) + s 1 + d(t)) . ( 25
)
From Assumption 3, it follows that V (t)  W (⇠(t)) + |s 1 (t) + d(t)| 2 for all t 0. Assumption 4 gives

@ @t (t, y ⇠ )+ @ @y ⇠ (t, y ⇠ )Cf (t, ⇠, (t, y ⇠ ) + s 1 + d(t)) 2  b f W (⇠) + b s |s 1 + d(t)| 2 (26)
for all t 0. Consider the positive definite quadratic function

Q(S) = 1 2 n X i=1 is 2 i . ( 27
)
Then we can use the triangle inequality to obtain

n 1 X i=1 is i s i+1  1 2 n 1 X i=1 i(s 2 i + s 2 i+1 ) = 1 2 n 1 X i=1 is 2 i + n X i=2 (i 1)s 2 i ! (28)
and then the subadditivity of the square root to conclude that along all solutions of (25) for all t 0, we have

Q(t)  k n 1 X i=1 is i ( s i + s i+1 ) kns 2 n + n X i=1 i|s i | q b f W (⇠) + b s |s 1 + d(t)| 2  k 2 n 1 X i=1 is 2 i n X i=2 (i 1)s 2 i ! kns 2 n + n X i=1 i|s i | q b f W (⇠) + b s |s 1 + d(t)| 2  k 2 n X i=1 s 2 i + n X i=1 i|s i | q b f W (⇠) + b s |s 1 + d(t)| 2  k 2 n X i=1 s 2 i + p b f n X i=1 i|s i | p W (⇠) + p b s n X i=1 i|s i ||s 1 | + n X i=1 i|s i | p b s |d(t)|. (29)
Consider the candidate Lyapunov function

U (t, ⇠, S) = n(n + 1) 2 b f V (t, ⇠) + Q(S) (30)
and any trajectory of (25). Along the trajectories of (25),

U (t)  n(n + 1) 2 b f W (⇠) + n(n + 1) 2 b f |s 1 (t) + d(t)| 2 k 2 n X i=1 s 2 i + p b f n X i=1 i|s i | p W (⇠) + p b s n X i=1 i|s i ||s 1 | + n X i=1 i|s i | p b s |d(t)| . (31)
Using the triangle inequality to obtain

p b f i|s i | p W (⇠)  1 2 b f W (⇠) + i 2 s 2 i , (32)
we deduce that

U (t)  nb f 2 W (⇠) k 2 n X i=1 s 2 i + n(n + 1) 2 b f s 2 1 + 1 2 n X i=1 i 2 s 2 i + p b s n X i=1 i|s i ||s 1 | + n(n + 1) 2 b f d 2 (t) +n(n + 1)b f s 1 d(t) + n X i=1 i|s i | p b s |d(t)| . ( 33
) Using p b s i|s i ||s 1 |  i 2 s 2 i + i 2 b s s 2 1 , we obtain U (t)  nb f 2 W (⇠) k 2 n X i=1 s 2 i +n ✓ n(n + 1) 2 b f + b s (n + 1) 4 ◆ s 2 1 + n X i=1 n 2 s 2 i + n(n + 1) 2 b f d 2 (t) +n(n + 1)b f s 1 d(t) + n X i=1 i|s i | p b s |d(t)| . (34)
From ( 24), we deduce that

U (t)  nb f 2 W (⇠) k 6 n X i=1 s 2 i + n(n + 1) 2 b f d 2 (t) +n(n + 1)b f s 1 d(t) + n X i=1 i|s i | p b s |d(t)| . (35)
Since each i (and therefore also each s i ) enters a fixed compact set after a finite time (by the boundedness of and the structure of the system (19)), there are constants ⇤ > 0 and T l such that for all t T l , U (t) 

" nb f 2 W (⇠) + k 6 n X i=1 s 2 i # + d ] (t) (36) with d ] (t) = n(n+1)b f d 2 (t)+⇤|d(t)|.
From this inequality, we can conclude from Barbalat's Lemma (applied to the function in squared brackets in (36), using the fact that d ] (t) converges exponentially to the origin) that all the solutions of ( 19) converge to the origin.

ILLUSTRATION

Consider a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor, which produces the following model from [START_REF] Dawson | Integrator backstepping control of a brush DC motor turning a robotic load[END_REF] (after a change of coordinates which removes a constant):

( ẋ1 = x 2 ẋ2 = g 1 q 1 sin(x 1 ) q 2 x 2 ġ1 = u q 3 arctan(x 2 ) a 1,1 g 1 (37)
with the output Y = (x, g 1 ) and the R 2 -valued state

x = (x 1 , x 2 ), where a 1,1 > 0 and q i > 0 for i = 1 to 3 are constants. Assumption 2 is satisfied with $ = 0, and we choose the bounded function ⌦(x, g 1 ) = q 3 arctan(x 2 ).

Let us check that Assumption 1 is satisfied with n = 1, under the added assumptions that q 1 q 2 1 + q 1 < 1 and

q 1 1 + q 1 < q 2 2 . ( 38
) Consider (⇠) = q 1 sin(⇠ 1 ) ⇠ 1 p 1 + ⇠ 2 1 ( 39
)
and any continuous function d : [0, 1) ! R that exponentially converges to 0 and

8 > > > < > > > : ⇠1 = ⇠ 2 ⇠2 = q 1 sin(⇠ 1 ) q 2 ⇠ 2 + 1 + d(t) ˙ 1 = k " 1 + q 1 sin(⇠ 1 ) ⇠ 1 p 1 + ⇠ 2 1 # (40)
with k > 0 being a constant to be specified. The choice

s 1 = 1 q 1 sin(⇠ 1 ) + ⇠ 1 p 1 + ⇠ 2 1 gives ⇠1 = ⇠ 2 ⇠2 = q 2 ⇠ 2 ⇠ 1 p 1 + ⇠ 2 1 + s 1 + d(t) ṡ1 = ks 1 + " q 1 cos(⇠ 1 ) + 1 (1+⇠ 2 1 ) p 1+⇠ 2 1 # ⇠ 2 . (41)
Let us choose

V (⇠) = p 1 + ⇠ 2 1 1 + 1 2 ⇠ 2 2 + q1 4(1+q1) ⇠1 p 1+⇠ 2 1 ⇠ 2 . ( 42
)
Then the first inequality in (38) and the triangle inequality combine to give

q 2 q 1 4(1 + q 1 ) ⇠ 1 ⇠ 2 p 1 + ⇠ 2 1  q 1 4(1 + q 1 ) ✓ 1 2 ⇠ 2 1 1 + ⇠ 2 1 + 1 2 q 2 2 ⇠ 2 2 ◆  q 1 8(1 + q 1 ) ⇠ 2 1 1 + ⇠ 2 1 + q 2 8 ⇠ 2 2 , (43)
and so also

V (t) = ⇠1⇠2 p 1+⇠ 2 1 + q1 4(1+q1) ⇠ 2 2 (1+⇠ 2 1 ) p 1+⇠ 2 1 + ✓ ⇠ 2 + q1 4(1+q1) ⇠1 p 1+⇠ 2 1 ◆ ⇥ ✓ q 2 ⇠ 2 ⇠1 p 1+⇠ 2 1 + s 1 + d(t) ◆ = q 2 ✓ ⇠ 2 2 + q1 4(1+q1) ⇠1⇠2 p 1+⇠ 2 1 ◆ q1 4(1+q1) ⇠ 2 1 1+⇠ 2 1 + q1 4(1+q1) ⇠ 2 2 (1+⇠ 2 1 ) p 1+⇠ 2 1 + ✓ ⇠ 2 + q1 4(1+q1) ⇠1 p 1+⇠ 2 1 ◆ (s 1 + d(t))  3q2 4 ⇠ 2 2 q1 8(1+q1) ⇠ 2 1 1+⇠ 2 1 + ⇠ 2 (s 1 + d(t)) + q1 4(1+q1) ⇠1 p 1+⇠ 2 1 (s 1 + d(t))  q2 4 ⇠ 2 2 q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + ⇣ 1 2q2 + q1 4(1+q1) ⌘ (s 1 + d(t)) 2  q2 4 ⇠ 2 2 q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + ⇣ 1 q2 + q1 2(1+q1) ⌘ s 2 1 + ⇣ 1 q2 + q1 2(1+q1) ⌘ d 2 (t), (44)
along all solutions of (41) for all t 0. where the first inequality in (44) used the second inequality from ( 38) and ( 43), the second equality in (44) used the relations

⇠ 2 (s 1 + d(t))  1 2 q 2 ⇠ 2 2 + 1 2q2 (s 1 + d(t)) 2 and ⇠1 p 1+⇠ 2 1 (s 1 + d(t))  1 4 ⇠ 2 1 1+⇠ 2 1 + (s 1 + d(t)) 2 ,
and the last inequality in (44) use the inequality (a + b) 2  2a 2 + 2b 2 for suitable a and b.

Next let U (⇠, s 1 ) = V (⇠) + 1 2 s 2 1 . Then (44) gives 2 q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + ⇣ 1 q2 + q1 2(1+q1) ⌘ s 2 1 + ⇣ 1 q2 + q1 2(1+q1) ⌘ d 2 (t) ks 2 1 +  q 1 cos(⇠ 1 ) + 1 (1+⇠ 2 1 ) p 1+⇠ 2 1 s 1 ⇠ 2  q2 4 ⇠ 2 2 q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + (q 1 + 1)|s 1 ⇠ 2 | + h 1 q2 + q1 2(1+q1) k i s 2 1 + c ⇤ d 2 , (45)
where c ⇤ = 1 q2 + 1 2 . Since (q 1 +1)|s

1 ⇠ 2 |  k 4 s 2 1 + 1 k (q 1 +1) 2 ⇠ 2 2 , it follows that if k max n 8(q1+1) 2 q2 , 2 ⇣ 1 q2 + q1 2(1+q1) ⌘o , ( 46
) then U (t)  q2 4 ⇠ 2 2 q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + h 1 q2 + q1 2(1+q1) 3 4 k i s 2 1 + 1 k (q 1 + 1) 2 ⇠ 2 2 + c ⇤ d 2 (t)  n q2 8 ⇠ 2 2 + q1 16(1+q1) ⇠ 2 1 1+⇠ 2 1 + k 4 s 2 1 o + c ⇤ d 2 (t) . (47)
It follows from integrating (47) and applying Barbalat's Lemma to the function in curly braces in (47) that Assumption 1 is satisfied. Hence, Theorem 1 applies to ( 40) and provides the dynamic feedback u(Z, x) = sat Z ( z 1 k (x 1 )) + q 3 arctan(x 2 ) ż1 = k  z 1 + q 1 sin(x 1 ) x1 p 1+x 2 1 with Z = (1 + ✏)(1 + k)(q 1 + 1) and the choice (39) of , since the proof of Theorem 1 gives c 1 = 1 and c 2 = k.

CONCLUSIONS

We developed a new backstepping approach whose key ingredient is a finite dimensional dynamic extension. Our work is motivated by the ubiquity of engineering applications that produce the required cascade forms. Our variant of backstepping o↵ers possible advantages in terms of the design of output feedback control and boundedness. Since it does not require artificial delays, it di↵ers from our prior bounded backstepping works that used converginginput-converging-state We hope to develop local versions of our results for systems that are only locally asymptotically stabilizable, and for cases where the constants a i,j in (1) can depend on t, and to allow delays in the input. We will also investigate what family of systems can be stabilized by applying Theorem 1 repeatedly.


Alessio Iovine

Frédéric Mazenc

Bounded Control for DC/DC Converters: Application to Renewable Sources

Keywords: Bounded control, Nonlinear control, Control for Power Converters

Advanced control methods are nowadays increasing in interest for power electronics devices. This paper proposes a nonlinear control law for a DC/DC boost converter dedicated to extract the maximum power from a photovoltaic (PV) array, taking into account the constraints of the control action. System stability analysis is provided by a proper Lyapunov function. Simulations on SimPowerSystems validate how the developed control strategy is able to properly control the converter with good performances.

I. INTRODUCTION

Nowadays, the utilization of complex control techniques for regulating power converters is becoming a common standard. Indeed, the technological progress and its rise in complexity of the required tasks, together with an always increasing demand of better performance, requires more advanced control methods. As a result, the development and application of control techniques different from the classic Proportional Integral Derivative (PID) for power converters is a well established research field, and its related literature is continuously enriched [START_REF] Sira Ramirez | Control design techniques in power electronics devices[END_REF], [START_REF] Bacha | Power electronic converters modeling and control[END_REF], [START_REF] Mariethoz | Comparison of Hybrid Control Techniques for Buck and Boost DC-DC Converters[END_REF], [START_REF] Olalla | Optimal State-Feedback Control of Bilinear DC/DC Converters With Guaranteed Regions of Stability[END_REF], [START_REF] Vazquez | Model predictive control: A review of its applications in power electronics[END_REF], [START_REF] Theunisse | Robust global stabilization of the dc-dc boost converter via hybrid control[END_REF], [START_REF] Konstantopoulos | Nonlinear control of DC/DC power converters with inherent current and power limitation[END_REF], [START_REF] Iovine | Nonlinear Control of a DC MicroGrid for the Integration of Photovoltaic Panels[END_REF], [START_REF] Hernandez-Mendez | A backstepping approach to decentralized active disturbance rejection control of interacting boost converters[END_REF], [START_REF] Alsmadi | Sliding mode control of power converters: Dc/dc converters[END_REF], [START_REF] Moreno-Valenzuela | On Control of a Boost DC-DC Power Converter under Constrained Inpu[END_REF].

Although the variety of advanced control results for DC/DC converters, only few of them take into account the physical limitations of the systems and how they impact on the performances and the operating region [START_REF] Olalla | Optimal State-Feedback Control of Bilinear DC/DC Converters With Guaranteed Regions of Stability[END_REF], [START_REF] Hernandez-Mendez | A backstepping approach to decentralized active disturbance rejection control of interacting boost converters[END_REF], [START_REF] Moreno-Valenzuela | On Control of a Boost DC-DC Power Converter under Constrained Inpu[END_REF], [START_REF] Tucci | Line-independent plugand-play controllers for voltage stabilization in dc microgrids[END_REF], [START_REF] Olalla | Robust LQR Control for PWM Converters: An LMI Approach[END_REF]. To develop an unbounded control law for systems for which large values for the control are prohibited, is quite accepted in literature. This is due to the complexity of the problem of constructing bounded feedbacks. As a consequence, the bounds are taken into account in an empiric way for selecting the gains of the control law and to try to provide an estimation of the operating region of the system related to the acting disturbances and the control law.

When constraints on the control input are taken into account, the used control methods have some drawbacks: for example, calculation time due to the Linear Matrix Inequalities (LMI) methods (see [START_REF] Olalla | Optimal State-Feedback Control of Bilinear DC/DC Converters With Guaranteed Regions of Stability[END_REF], [START_REF] Tucci | Line-independent plugand-play controllers for voltage stabilization in dc microgrids[END_REF], [START_REF] Olalla | Robust LQR Control for PWM Converters: An LMI Approach[END_REF]) or dynamics cancellation due to backstepping (see [START_REF] Hernandez-Mendez | A backstepping approach to decentralized active disturbance rejection control of interacting boost converters[END_REF]) can generate a high error in the control with respect to small measurement errors.

In this paper, a bounded nonlinear control design is used to correctly control a boost DC/DC converter connected to a photovoltaic (PV) panel. Our objective is similar to the one of [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF]: ensuring stability and at the same time the satisfaction of some performance requirements. However, it is worth mentioning that the design we propose does not rely on the main result of [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF] nor on classical techniques of constructions of bounded control laws, such as the ones of [START_REF] Teel | A nonlinear small gain theorem for the analysis of control systems with saturation[END_REF], [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF], [START_REF] Mazenc | Adding an integration and global asymptotic stabilization of feedforward systems[END_REF]. The equilibrium point of the system is provided by a Maximum Power Point Tracking (MPPT) algorithm, which is supposed to be placed in a higher hierarchical control scheme [START_REF] Iovine | Management Controller for a DC MicroGrid integrating Renewables and Storages[END_REF]. The used control technique imposes a desired equilibrium point to the systems, respecting the bounds of the control law while imposing a certain desired convergence rate. Such technique allows for an online calculation of the control input, increasing system performances. Moreover, the problem of nonminimum phase of the boost converter does not affect the introduced control law.

The paper is organized as follows. Section II introduces the considered model, while Section III describes preliminary properties of the system. In Section IV the control law is developed, and Section V outlines simulations to validate it. Conclusions are provided in Section VI.

II. MODELING

The considered model for a DC/DC converter connected to a DC bus and a PV source is taken from [START_REF] Walker | Cascaded DC-DC converter connection of photovoltaic modules[END_REF]. The equivalent circuit representation for a DC/DC boost converter connected to a strong DC grid and the PV source can be expressed using a state space average model, under the hypothesis to then use a Pulse Width Modulation (PWM) technique for control purposes [START_REF] Middlebrook | A general unified approach to modelling switching-converter power stages[END_REF], [START_REF] Bindra | Pulsewidth modulated controller integrated circuit: Four decades of progress [a look back[END_REF]. Three state variables are needed for the system model: the capacitor voltages V C1 : R → R + and V C3 : R → R + (x 1 and x 3 respectively) and the inductor current I L2 : R → R (x 2 ). Let X(t) = (x 1 (t), x 2 (t), x 3 (t)). In the remaining part of the paper, time dependence will be omitted due to lack of space. C 1 , C 3 , R 1 , R 3 , L 2 , R 01 , R 02 are known positive values of the capacitors, resistances and the inductor while V DC ∈ R + is the constant grid voltage.

1 1 R 1 C 1 a 2 1 C 1 a 3 1 L 2 a 4 R 01 L 2 a 5 1 R 3 C 3 a 6 1 C 3

TABLE I THE POSITIVE VALUES OF THE PARAMETERS.

The PV source is here represented as a voltage source, which is piecewise constant; V P V ∈ R + is the photovoltaic panel voltage. Fig. 1 depicts the considered framework. The measured signals are the states x 1 , x 2 , x 3 and the PV array and DC grid constant voltages

d 1 = V P V , d 2 = V DC . v = 1 -u is the control input,
which is defined as the duty cycle of the circuit. The electrical scheme in Fig. 1 can then be modeled as follows:

     ẋ1 = -a 1 x 1 -a 2 x 2 + a 1 d 1 , ẋ2 = a 3 x 1 -a 3 x 3 v -a 4 x 2 , ẋ3 = -a 5 x 3 + a 6 x 2 v + a 5 d 2 .
(1) Here x, v, d 1 , d 2 are measurable. The parameters' value are depicted in Table I. It must to be noticed that the system has the property to have 2 time-scales, since it satisfies the motion separation principle derived from singular perturbation theory [START_REF] Khalil | Nonlinear systems[END_REF]. Furthermore, to have feasible solutions, the disturbances are such that

d 1 > 0, d 2 > 0, d 2 > d 1 .
The control input is bounded as follows:

v ∈ [0, 1]. (2)

A. Equilibrium point

Let us determine the equilibrium point of the system (1) with v ≡ v * where v * > 0 is a constant. Let E * = (x 1 * , x 2 * , x 3 * ) denote an equilibrium point. Hence

x 1 * = a6v 2 * a5 + a4 a3 a 1 d1-v * d2 a1 a 6 v 2 * a 5 + a 4 a 3 +a2 + v * d 2 , x 2 * = a 1 d1-v * d2 a1 a 6 v 2 * a 5 + a 4 a 3 +a2 , x 3 * = a6v * a5 a 1 d1-v * d2 a1 a 6 v 2 * a 5 + a 4 a 3 +a2 + d 2 .
(3) It follows that the equilibrium point E * is positive if the following inequality is satisfied:

v * < d 1 d 2 . (4)

III. PRELIMINARY ANALYSIS

Let us consider a constant v * ≥ 0 (not fixed yet, such that condition (4) is satisfied). As a consequence, the system in (1) admits an equilibrium point

E * = (x 1 * , x 2 * , x 3 * ) with x 3 * > 0.
The common used technique when dealing with power systems is the one of creating a fast inner loop dealing with the current (x 2 in this case) and an outer control loop dealing with the desired voltage, to be used as a reference for the current loop. This because the system satisfies the separation principle conditions, even if such conditions are not really used when developing the control law. In this work, a similar approach is used: the current x 2 will be the directly controlled variable, and the other two voltages will react as a consequence. To this purpose, some preliminary analysis has to be done about the voltage x 3 in order to test its boundedness, and to allow to focus for the control law development just on the other dynamics.

A. Boundedness of x 3

Now, let us perform a change of feedback: v = v * +µ and impose on µ to be such that |µ| ∞ ≤ µ, where µ > 0 is to be defined later. Noticing that

e 1 = x 1 -x 1 * , e 2 = x 2 -x 2 * , e 3 = x 3 -x 3 * satisfy    ė1 = -a 1 e 1 -a 2 e 2 , ė2 = a 3 e 1 -a 3 v * e 3 -a 4 e 2 -a 3 µ(t)(e 3 + x 3 * ), ė3 = -a 5 e 3 + a 6 v * e 2 + a 6 µ(t)(e 2 + x 2 * ).
(5) Let us consider the positive definite quadratic function:

V (e 1 , e 2 , e 3 ) = a 3 a 6 2 e 2 1 + a 2 a 6 2 e 2 2 + a 2 a 3 2 e 2 3 . (6)
Its time derivative is

V (t) = -a 1 a 3 a 6 e 2 1 -a 2 a 4 a 6 e 2 2 -a 2 a 3 a 5 e 2 3 + (7) -a 2 a 3 a 6 x 3 * e 2 µ(t) + a 2 a 3 a 6 x 2 * e 3 µ(t).
As an immediate consequence, Notice that it follows that if µ = 0, then E * is GES.

V ≤ -a 1 a 3 a 6 e 2 1 -
Considering ς = min{a 1 , a 4 , a 5 } (9)
and the equalities in (3), we obtain

V ≤ -ςV + a 6 √ 2a2a3a1d1 a1 a 4 a 3 +a2 µ √ V + +a 3 √ 2a 2 a 6 a1a6 a5 d1 a1 a 4 a 3 +a2 + d 2 µ √ V . (10)
Let ∈ 0, 1 2 be any constant. As an immediate consequence, there is a value t ≥ 0 such that for all t ≥ t * ,

√ V ≤ (1 + ) √ 2 √ a 2 a 3 c 1 ς µ (11)
where

c 1 = √ a 3 a 6 a 1 a 6 a 5 d 1 a 1 a4 a3 + a 2 + d 2 + a 6 a 1 d 1 a 1 a4 a3 + a 2 . (12)
From the definition of V , it follows that

a 2 a 3 2 |e 3 | ≤ (1 + ) √ 2 √ a 2 a 3 c 1 ς µ. (13)
It follows that

|x 3 (t) -x 3 * | ≤ 2(1 + ) c 1 ς µ. (14)
Thus

x 3 * -2(1 + ) c 1 ς µ ≤ x 3 (t) ≤ x 3 * + 2(1 + ) c 1 ς µ. (15)
We deduce that when

2(1 + ) c 1 ς µ ≤ x 3 * (16) then (1 -)x 3 * ≤ x 3 (t) ≤ (1 + )x 3 * . (17)
Considering ( 3) and ( 12) in [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF], it has to be remarked that it is possible to take the following expression for µ,

µ = 2(1 + ) d 2 ς c 1 (18)
which is independent from v * . Furthermore, due to [START_REF] Iovine | Management Controller for a DC MicroGrid integrating Renewables and Storages[END_REF],

select a v * > 0 such that v * 1 - < µ (19)
implies to satisfy also the following inequality:

v * < d 2 ς 2c 1 . (20)
B. Change of control for the (x 1 , x 2 )-subsystem

Now, let us focus our attention on the (x 1 , x 2 )-subsystem of (1):

ẋ1 = -a 1 x 1 -a 2 x 2 + a 1 d 1 , ẋ2 = a 3 x 1 -a 3 x 3 v -a 4 x 2 . (21)
Let ψ be defined as follows:

ψ(s) =    (1 -)x 3 * s ≤ (1 -)x 3 * , s s ∈ [(1 -)x 3 * , ( 1
+ )x 3 * ], (1 + )x 3 * s ≥ (1 + )x 3 * . ( 22
)
The inequality [START_REF] Walker | Cascaded DC-DC converter connection of photovoltaic modules[END_REF] ensures that there is ν > 0 such that

1 1 - ν + |v * | 1 - ≤ µ. (23)
Notice for later use that this inequality rewrites as

ν ≤ (1 -)µ -v * . (24)
Now, let us perform the following change of feedback

v = x 3 * ν ψ(x 3 ) (25) with ν such that |ν -v * | ≤ ν.
Then, the control input v(t) can be rewritten as

x 3 * ν(t) ψ(x 3 (t)) = v(t) = v * + µ(t) (26)
and consequently µ(t) results

µ(t) = x 3 * ν(t) ψ(x 3 (t)) -v * . (27)
With the following calculations

µ(t) = x 3 * (ν(t) -v * ) ψ(x 3 (t)) + x 3 * v * ψ(x 3 (t)) -v * (28) = x 3 * (ν(t) -v * ) ψ(x 3 (t)) + x 3 * -ψ(x 3 (t)) ψ(x 3 ) v *
and the possibility to write

x 3 * -ψ(x 3 ) ψ(x 3 ) v * ≤ x 3 * (1 -)x 3 * v * = v * 1 - , (29)
the boundedness of µ(t) with respect to its maximum value µ is proven, as in (23):

|µ(t)| ≤ x 3 * ν ψ(x 3 ) + v * 1 - ≤ ν 1 - + v * 1 - ≤ µ. (30)
Then, the (x 1 , x 2 )-subsystem of (1) becomes:

ẋ1 = -a 1 x 1 -a 2 x 2 + a 1 d 1 , ẋ2 = a 3 x 1 -a 3 x 3 x3 * ν ψ(x3) -a 4 x 2 , (31)
and from (30) and the analysis in Section III-A ,we deduce that, when t ≥ t , then

x 3 ∈ [(1 -)x 3 * , (1 + )x 3 * ].
It follows that for all t ≥ t , the (x 1 , x 2 )-subsystem can be written as

ẋ1 = -a 1 x 1 -a 2 x 2 + a 1 d 1 , ẋ2 = a 3 x 1 + bν -a 4 x 2 , (32)
where b = -a 3 x 3 * .

IV. FEEDBACK STABILIZATION A. Control law design

Let us now rewrite the subsystem in (32) with respect to its equilibrium point, defining xi = x i -x i * , i = 1, 2. Also, let us perform the following change of feedback:

ω = bν + a 3 x 1 * -a 4 x 2 * . (33)
It results in

ẋ1 = -a 1 x1 -a 2 x2 , ẋ2 = a 3 x1 -a 4 x2 + ω, (34)
where

|ω| ≤ a 3 x 3 * ν (35) because bν + a 3 x 1 * -a 4 x 2 * = -a 3 x 3 * (ν -v * ) (36)
and |ν -v * | ≤ ν.

To facilitate the control design, we introduce ξ as

ξ = x2 -ρ kx 1 1 + k 3 (x 2 1 + x2 2 ) ( 37
)
where ρ is a small tuning positive parameter and k a positive gain, and calculate its dynamical behavior:

ξ = a 3 x1 -a 4 ξ + η (38) with η = -a4 ρ kx1 1 + k 3 (x 2 1 + x2 2 ) + 1 + 2ρk 3 x1 x2 [1 + k 3 (x 2 1 + x2 2 )] 2 ω + ρk [a1 x1 + a2 x2][1 + k 3 (x 2 2 -x2 1 )] + 2k 2 x1 x2[a3 x1 -a4 x2] [1 + k 3 (x 2 1 + x2 2 )] 2 . ( 39
)
Let us introduce the positive tuning parameter h and define the saturation function sat 1 (m) as

sat 1 (m) = m |m| ≤ 1 m |m| m > 1 (40)
Now we select

η = -hsat 1 k h ξ (41) and obtain ẋ1 = -a 1 x1 -a 2 ρ kx1 1+k 3 (x 2 1 +x 2 2 ) -a 2 ξ, ξ = a 3 x1 -a 4 ξ -hsat 1 k h ξ .
(42)

For this system, we can achieve locally arbitrarily fast speed of convergence by choosing k sufficiently large and we can do this with a feedback ω which respects the needed size constraint.

Then the formula of v is

v = -a3x1 * +a4x2 * b ψ(x 3 ) x 3 * - hsat1( k h ξ) b ψ(x 3 ) x 3 * 1+ 2ρk 3 x1 x2 [1+k 3 (x 2 1 +x 2 2 )] 2 + + a4 ρ k x1 1+k 3 (x 2 1 +x 2 2 ) b ψ(x 3 ) x 3 * 1+ 2ρk 3 x1 x2 [1+k 3 (x 2 1 +x 2 2 )] 2 + - ρk [a 1 x1 +a 2 x2 ][1+k 3 (x 2 2 -x 2 1 )]+2k 2 x1 x2 [a 3 x1 -a 4 x2 ] [1+k 3 (x 2 1 +x 2 2 )] 2 b ψ(x 3 ) x 3 * 1+ 2ρk 3 x1 x2 [1+k 3 (x 2 1 +x 2 2 )] 2 . ( 43
)

B. Stability analysis

We analyze the stability properties of the system (42) via the positive definite function:

V (x 1 , ξ) = 1 2a 2 x2 1 + 1 2a 3 ξ 2 . ( 44
)
Its derivative along the trajectories of the system (42) is

V (t) = 1 a2 x1 -a 1 x1 -a 2 ρ kx1 1+k 3 (x 2 1 +x 2 2 ) -a 2 ξ + + 1 a3 ξ a 3 x1 -a 4 ξ -hsat 1 k h ξ = -W (x 1 , ξ) (45) with W (x 1 , ξ) = 1 a2 a 1 x2 1 + ρ kx 2 1 1+k 3 (x 2 1 +x 2 2 ) + + 1 a3 a 4 ξ 2 + 1 a3 hsat 1 k h ξ ξ. ( 46
)
Since W is definite positive, we deduce that the origin of the system is globally asymptotically stable. Besides, W is approximated in a neighborhood of the origin by

a1 a2 + ρk x2 1 + a4+k a3 ξ 2 .
We deduce that the speed of convergence can be increased by increasing the constant k.

C. Selected equilibrium point

To highlight the generality of the introduced controller and the possibility to apply it to the boost converter, a further remark is needed. Indeed, the equilibrium point in (3), x * , is function of the value of the control at the equilibrium, v * , i. e.

x * = f (v * ). Since in reality the available degree of freedom is used to impose an equilibrium, while the equilibrium for the other dynamics is calculated as a consequence of it, it must be remarked that the previous introduced analysis still holds when fixing the equilibrium according to a desired reference, with just the hypothesis to consider equilibria that are in the admissible set Ω * . The possibilities are:

Parameter Value Parameter Value R 1 100 mΩ R 3 100 mΩ C 1 100 mF C 3 10 mF L 2 33 mH R 01 10 mΩ

TABLE II THE VALUES OF THE PARAMETERS.

• a value x e 1 is imposed;

x 1 * = x e 1 x 2 * = a1 a2 (d 1 -x e 1 ) x 3 * = 1 2 d 2 + d 2 2 + 4β 1 φ 1 (x e 1 -γ 1 φ 1 ) , (47)
where

β 1 = a6a1 a5a2 , γ 1 = a4a1 a3a2 , φ 1 = (d 1 -x e 1
). • a value x e 2 is imposed;

x 1 * = d 1 -a2 a1 x e 2 x 2 * = x e 2 x 3 * = 1 2 d 2 + d 2 2 + 4 a6 a5 x e 2 [d 1 -γ 2 x e 2 ] , (48)
where γ 2 = a2 a1 + a4 a3 . Without loss of generality, it must to be remarked that similar calculations apply also in the case where x e 3 is imposed.

V. SIMULATIONS

The simulations are developed in a realistic electric framework, which is SimPowerSystem, a MatlabR2016b toolbox. Both the situations described in Section IV-C are considered, in order to show the effectiveness of the proposed control law either when fixing x e 1 or x e 2 . The value of the parameters are introduced in Table II. The considered simulation time is about 5 seconds, the discrete simulation step is of 0.01 ms and the frequency for the PWM is 20kH.

The fact to use a realistic toolbox for electrical systems results in a more complex framework, since the considered hypothesis of fixed disturbances cannot be met in a real scenario. Nevertheless, the introduced control law is shown to properly perform even in case of slowly (with respect to the state variables) time-varying disturbances d 1 and d 2 . The reference value for x e 1 or x e 2 is considered piecewise constant, and varying each one second; it is supposed to be provided by a MPPT algorithm, which is not modeled in this work.

A. Case 1: choice of a value x e 1 Fig. 2 depicts both the reference x e 1 and the dynamical behavior of the state variable x 1 when the system is controlled by the control input introduced in equation (43) with target to impose a desired x e 1 . As shown, the controller perfectly meets its target, since the error dynamics converges to zero. Furthermore, since the choice of the desired reference x e 1 has been properly done, the controller fits its boundaries, as it is depicted in Fig. 5. The dynamical evolutions of the other two equilibria for x 2 and x 3 , together with their dynamics, are introduced in Fig. 3 and4: they are time varying since d 1 and d 2 are time varying. As already mentioned, the reference is piecewise constant and changes its value every seconds. Since the source is a PV panel, the reference is selected to provide only positive currents. The oscillations taking place in Fig. 4 are due to the switching nature of the implemented system, and the different way of representing them with the average model here used: nevertheless, it is easy to verify that x 3 follows its reference.

B. Case 2: choice of a value x e 2

Here the reference x e 2 is imposed, and the resulting controller implemented. As shown in Fig. 9, it results in a feedback that is bounded in its operating region. Fig. 6, and 8 depict the dynamical behaviours of the state variables with respect to their equilibria: convergence is assured for all of them.

C. Discussion on the implemented control

From Fig. 5 and 9 it must be noted that the developed control action allows for a "smooth" curve in the obtained duty cycle. It is important since it results in less stress for the components and the devices.

This outcome is due to the proper choice of the parameters , ρ and k with respect to the considered set Ω * of the equilibria and a desired convergence rate. The resulting controller is shown to be bounded even when dealing with the transient due to the new reference, and not only in steady state.

The calculation of such set Ω * is out of the scope of this paper. Future works will focus on an analytical description of this set and how to calculate it, as well as optimal choices of the aforementioned parameters.

VI. CONCLUSIONS

In this work a new nonlinear control law for DC/DC boost converters connected to renewable power sources as PV is introduced. The control input is calculated online, and takes into account physical limitations while providing desired performances. Stability analysis is investigated by a proper Lyapunov function, and the system is shown to be asymptotically stable around the chosen equilibrium point. Realistic simulation results are given in SimPowerSystems to show the effectiveness of the proposed control action. Future works will deal with the optimization of such control law with respect to the broadest set of possible equilibria.

Fig. 1 .

 1 Fig. 1. The adopted electrical scheme. The boost DC/DC converter is introduced in the blue rectangular region, and is connected to a strong DC grid and to the PV array.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Case 1: The dynamical behavior of x 1 with respect to its chosen reference x e 1 .

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig.[START_REF] Olalla | Optimal State-Feedback Control of Bilinear DC/DC Converters With Guaranteed Regions of Stability[END_REF]. Case 1: The dynamical behavior of x 3 with respect to its reference x * 3 , calculated as a consequence of the choice of x e 1 .

Fig. 7 .Fig. 8 .

 78 Fig. 7. Case 2: The dynamical behavior of x 2 with respect to its chosen reference x e 2 .

  a 2 a 4 a 6 e 2 2 -a 2 a 3 a 5 e 2 3 + (8) +a 2 a 3 a 6 |x 3 * ||e 2 |µ + a 2 a 3 a 6 |x 2 * ||e 3 |µ.

Efficacity, R&D Institute for Urban Energy Transition, Paris, France. (e-mail: a.iovine@efficacity.com).

Inria EPI DISCO, L2S-CNRS-CentraleSupélec,

rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: frederic.mazenc@l2s.centralesupelec.fr).


Frédéric Mazenc

Saeed Ahmed
email: ahmed@ee.bilkent.edu.tr

Michael Malisoff
email: malisoff@lsu.edu.

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

Reduced Order Finite Time Observers for Time-Varying Nonlinear Systems

Keywords: Observer, stability, time-varying

We construct finite-time reduced order observers for a broad class of nonlinear time-varying continuous-time systems. We illustrate our results using a tracking problem for nonholonomic systems in chained form.

I. INTRODUCTION

The problem of estimating the values of solutions of systems when some variables cannot be measured is of great relevance from the theoretical and applied points of view. Asymptotic observers, such as for instance the celebrated Luenberger observer [START_REF] Luenberger | Observers for multivariable systems[END_REF], [START_REF] Luenberger | Introduction to Dynamic Systems[END_REF], are very popular and many observers for families of nonlinear systems have been constructed. However, they provide a useful estimate only after a transient period during which they cannot be used, which can be a disadvantage in some applications like fault detection where a finite-time state estimation is desirable [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous time system in predetermined fnite time[END_REF].

To obtain an exact estimate of the solutions of a system in an arbitrary short amount of time, finite time observers have been proposed. Some use nonsmooth functions; see for instance [START_REF] Lopez-Ramirez | Finitetime and fixed-time observer design: Implicit Lyapunov function approach[END_REF], [START_REF] Perruquetti | Finite time observers: application to secure communication[END_REF]. Their designs are based on homogeneous properties which preclude the possibility of deriving smooth observers from this technique. Another type of finite time observers has been developed. They are smooth and use past values of the output or dynamic extensions. They have been proposed a few decades ago for linear systems; see in particular [START_REF] Engel | A continuous time observer which converges in finite time[END_REF], [START_REF] Raff | An observer that converges in finite time due to measurement-based state updates[END_REF], and our other references below. More recently, finite time observers were designed for nonlinear systems, e.g., in [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], [START_REF] Menold | Finite time convergent observers for nonlinear systems[END_REF], and [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF]. They apply to systems whose vector field is time-invariant when the output is set to zero and provide estimates of all the state variables.

Since systems are frequently time-varying and tracking problems can be recast into stabilization problems of equilibria of time-varying systems, and since the measured components of the state do not need to be estimated, this paper adapts the main results of [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF] and [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF] to construct finite-time reduced order observers for a family of nonlinear time-varying systems. The observer we will build only gives estimates of the unmeasured variables, as does the asymptotic observer proposed for instance in [START_REF] Friedland | Reduced-order state obervers[END_REF] and [START_REF] Bonnans | Commande et Optimisation de Systèmes Dynamiques[END_REF]Chapt. 4,Sec. 4.4.3]. This feature presents the following technical advantages. This provides a technical advantage over other observers that would require fundamental solutions of timevarying linear systems whose dimensions equal that of the original systems, because of the difficulty of finding formulas for fundamental solutions of higher dimensional timevarying linear systems. To the best of our knowledge, finitetime reduced order observers for nonlinear time-varying systems are proposed for the first time in the present paper.

After providing an introductory result in Section II, we state and prove our main reduced order finite-time observer design for time-varying systems in Section III. Our illustration in Section IV demonstrates our approach, in a tracking problem for a nonholonomic system in chained form, and we conclude with ideas for future research in Section V.

We use standard notation, which will be simplified whenever no confusion can arise from the context and the dimensions of our Euclidean spaces are arbitrary, unless otherwise noted. Any k ⇥ l matrix whose entries are all 0 is denoted 0. The Euclidean norm, and the induced norm of matrices, are denoted by | • |, and I is the identity matrix. For any constant ⌧ > 0, any continuous function ' : [ ⌧, +1) ! R n and all t 0, we define ' t by ' t (m) = '(t + m) for all m 2 [ ⌧, 0]. We denote by C in the set C([ ⌧, 0]) of all continuous functions ' : [ ⌧, 0] ! R n . For any continuous function ⌦ : [ ⌧, +1) ! R n⇥n , we let ⌦ denote the function such that

@ ⌦ @t (t, t 0 ) = ⌦ (t, t 0 )⌦(t) (1)
and ⌦ (t 0 , t 0 ) = I for all t 2 R and t 0 2 R. Then M(t, s) =

II. INTRODUCTORY RESULT

Before proving our general result for time-varying systems in Section III, we provide an introductory result that shows how to construct reduced order finite-time observers for a simple family of systems. We consider the system

ẋ(t) = Ax(t) + (t) (2)
with x valued in R n and where : [0, +1) ! R n is a piecewise-continuous locally bounded function. We assume that the output

y(t) = Cx(t) (3)
is valued in R p for any dimension p  n and C is of full rank. Also, we assume that the pair (A, C) is observable. Since C is of full rank, there exist matrices A 1 and A 2 , a linear change of coordinates

x T = C T x =  y x r (4)
and functions i for i = 1, 2 that are piecewise-continuous with respect to their first argument and linear with respect to y such that the x T system can be written as follows:

⇢ ẏ(t) = A 1 x r (t) + 1 (t, y(t)) ẋr (t) = A 2 x r (t) + 2 (t, y(t)) . (5)
Since the pair (A, C) is observable, it follows that the pair Lemma 1] to prove that there are a matrix L 2 R (n p)⇥p and a constant ⌧ > 0 such that

(A 2 , A 1 ) is observable; see [6, pp. 304-306]. Since (A 2 , A 1 ) is observable, one can use [9,
M ⌧ = e A2⌧ e H⌧ (6)
with the choice H = A 2 + LA 1 is invertible. In terms of the new variable

x s = x r + Ly (7)
we can then use simple calculations to obtain

ẋs (t) = (A 2 + LA 1 )x r (t) + 2 (t, y(t)) + L 1 (t, y(t)). (8) From the definition of H, it follows that ẋs (t) = Hx s (t) + Ky(t) + 3 (t, y(t)) (9)
where

K = (A 2 + LA 1 )L and 3 = 2 + L 1 .
By integrating the second equation in ( 5) and ( 9), we obtain

x r (t ⌧ ) = e A2⌧ x r (t) R t t ⌧ e A2(t m ⌧ ) 2 (m, y(m))dm (10)
and

x s (t ⌧ ) = e H⌧ x s (t) R t t ⌧ e H(t m ⌧ ) [Ky(m) + 3 (m, y(m))]dm (11)
for all t ⌧ . From the definition (7), we deduce that

x r (t ⌧ ) = e H⌧ x r (t) + e H⌧ Ly(t) Ly(t ⌧ ) R t t ⌧ e H(t m ⌧ ) [Ky(m) + 3 (m, y(m))]dm (12)
for all t ⌧ . Subtracting ( 12) from [START_REF] Menold | Finite time convergent observers for nonlinear systems[END_REF] and then using the definition of M ⌧ in (6), we deduce that

M ⌧ x r (t) = e H⌧ Ly(t) Ly(t ⌧ ) R t t ⌧ e H(t m ⌧ ) [Ky(m) + 3 (m, y(m))]dm + R t t ⌧ e A2(t m ⌧ ) 2 (m, y(m))dm (13)
for all t ⌧ . Since M ⌧ is invertible, it follows that

x r (t) = xr (t) (14)
holds for all t ⌧ , where

xr (t) = M 1 ⌧ R t t ⌧ e H(t m ⌧ ) [Ky(m) + 2 (m, y(m)) +L 1 (m, y(m))]dm +M 1 ⌧ R t t ⌧ e A2(t m ⌧ ) 2 (m, y(m))dm +M 1 ⌧ ⇥ e H⌧ Ly(t) Ly(t ⌧ ) ⇤ . (15)
Thus, when 1 and 2 are known, the formula (15) provides the exact value of x r (t) for all t ⌧ .

III. MAIN RESULT FOR TIME-VARYING SYSTEMS A. Statement of Main Result and Remarks

In this section, we show how the finite-time observer design of the previous section adapts to time-varying nonlinear systems of the type

⇢ ż(t) = A 1 (t)x r (t) + 1 (t, z(t)) ẋr (t) = A 2 (t)x r (t) + 2 (t, z(t)) ( 16
)
where z is valued in R p , x r is valued in R n p , the output is

y(t) = z(t) + ✏(t) (17)
where ✏(t) is a piecewise continuous function that is bounded by a constant ✏ 0, the functions A i for i = 1 and 2 are piecewise continuous and bounded, and 1 and 2 are functions that are piecewise continuous with respect to t and locally Lipschitz with respect to z and such that there is a nonnegative valued continuous function such that

| 1 (t, z)| + | 2 (t, z)|  (|z|) (18)
for all t 0 and z 2 R p . Remark 1: The special structure of the system (16) does not limit the family of linear systems to which our approach applies because, as explained in the previous section, any system of the type

Ẋ = A(t)X + F(t, Y ) (19)
with an output Y = CX with C of full rank can be transformed through a linear time-invariant change of coordinates into a system of the form (16). The term ✏(t) in (17) represents a disturbance in the measurements, which is of interest because in practice, measurements are frequently affected by disturbances. ⇤ We introduce the following assumption: Assumption 1: There are a constant ⌧ > 0 and a bounded function L of class C 1 with a bounded first derivative such that for all t 2 R, the matrix

⇤(t) = A2 (t, t ⌧ ) H (t, t ⌧ ) (20)
with

H(t) = A 2 (t) + L(t)A 1 (t) is invertible.
⇤ See the remarks below on ways to check the preceding assumption. Let us define

3 (t, z) = L(t) 1 (t, z) + 2 (t, z) and 4 (t, z) = [D(t)z + 3 (t, z)], (21)
where

D(t) = L(t) H(t)L(t). (22)
Let

xr (t) = ⇤(t) 1 R t t ⌧ [ A2 (m, t ⌧ ) 2 (m, y(m) ✏(m)) + H (m, t ⌧ ) 4 (m, y(m) ✏(m))] dm +⇤(t) 1 [ H (t, t ⌧ )L(t)(y(t) ✏(t)) L(t ⌧ )(y(t ⌧ ) ✏(t ⌧ ))] (23)
for all t 0. We will prove:

Theorem 1: Let (16) satisfy Assumption 1. Then

x r (t) = xr (t) (24)
holds for all t ⌧ . Remark 2: In general, one can check easily that Assumption 1 is satisfied when n p = 1 because then A2 and H take the simple one dimensional forms

A2 (t, t 0 ) = e R t t 0 A2(m)dm and H (t, t 0 ) = e R t t 0 H(m)dm .
When n p > 1, checking this assumption may be more difficult because then formulas for A2 and H would be more difficult to compute. However, it is easier to determine explicit expressions for A2 and H than for A where A is the function in (19) because the dimension of A is larger than those of A 2 and H. This can be an advantage of the reduced-order approach over the full order one. ⇤ Remark 3: If there are a function L, two constants ⌧ > 0 and $ 2 (0, 1) such that | A2 (t, t ⌧ ) 1 H (t, t ⌧ )|  $ for all t 0, then Assumption 1 is satisfied and ⇤ 1 is bounded. Indeed, in this case the matrix

I A2 (t, t ⌧ ) 1 H (t, t ⌧ ) is invertible for all t 0 and ⇥ I A2 (t, t ⌧ ) 1 H (t, t ⌧ ) ⇤ 1 = 1 P k=0 ⇥ A2 (t, t ⌧ ) 1 H (t, t ⌧ ) ⇤ k (25)
for all t 0, which implies that

⇥ I A2 (t, t ⌧ ) 1 H (t, t ⌧ ) ⇤ 1  1 1 $ (26)
for all t 0. We deduce that ⇤(t) 1 is well-defined for all t 0 and

⇤(t) 1 = I A2 (t, t ⌧ ) 1 H (t, t ⌧ ) 1 A2 (t, t ⌧ ) 1  | A 2 (t,t ⌧ ) 1 | 1 $
, which is bounded by a constant because A2 (t, t ⌧ ) 1 is bounded. Also, the function ⇤ 1 is bounded if the system is periodic because then ⇤ 1 is continuous and periodic. ⇤ Remark 4: When ✏ is unknown, the exact estimate (24) cannot be used because ✏ is present in (23). Fortunately, we can use equation ( 24) to obtain the approximate observer

x ⇤ r (t) = ⇤(t) 1 R t t ⌧ [ A2 (m, t ⌧ ) 2 (m, y(m)) + H (m, t ⌧ ) 4 (m, y(m))] dm +⇤(t) 1 [ H (t, t ⌧ )L(t)y(t) L(t ⌧ )y(t ⌧ )] . (27)
Since the functions A 2 and H are bounded, we can easily use Gronwall's inequality to check that sup m2[t ⌧,t] | H (m, t ⌧ )| and sup m2[t ⌧,t] | A2 (m, t ⌧ )| are bounded functions of t. We deduce that if 1 and 2 are globally Lipschitz with respect to their second argument and ⇤(t) 1 is bounded, then there is a constant l a > 0 such that

|x ⇤ r (t) x r (t)|  l a |✏| 1 (28)
for all t ⌧ . ⇤

B. Proof of Theorem 1

The variable

x v (t) = A2 (t, 0)x r (t) (29)
is such that

ẋv (t) = A2 (t, 0)A 2 x r (t) + A2 (t, 0)[A 2 (t)x r (t) + 2 (t, z(t))] = A2 (t, 0) 2 (t, z(t)) . (30)
By integrating (30) between t ⌧ and t ⌧ , we obtain

x v (t) = x v (t ⌧ )+ Z t t ⌧ A2 (m, 0) 2 (m, z(m))dm . ( 31
)
From the definition of x v , it straightforwardly follows that

A2 (t ⌧, 0) 1 A2 (t, 0)x r (t) = x r (t ⌧ ) + R t t ⌧ A2 (t ⌧, 0) 1 A2 (m, 0) 2 (m, z(m))dm. (32)
For any continuous function ⌦ : R ! R n⇥n , the function

⌦ defined as ⌦ (t, t 0 ) = ⌦ (t, t 0 ) > satisfies @ ⌦ @t (t, t 0 ) = ⌦(t) > ⌦ (t, t 0 ) . ( 33
)
It follows from the semigroup property of flow maps that

⌦ (t, 0) = ⌦ (t, t ⌧ ) ⌦ (t ⌧, 0). Consequently ⌦ (t, 0) > = ⌦ (t ⌧, 0) > ⌦ (t, t ⌧ ) > . Also, we have ⌦ (m, 0) = ⌦ (m, t ⌧ ) ⌦ (t ⌧, 0), which implies that ⌦ (m, 0) > = ⌦ (t ⌧, 0) > ⌦ (m, t ⌧ ) > for all m t ⌧ . It follows that ⌦ (t, 0) = ⌦ (t ⌧, 0) ⌦ (t, t ⌧ ) and ⌦ (m, 0) = ⌦ (t ⌧, 0) ⌦ (m, t ⌧ ).
From the preceding identities and (32), we deduce that

A2 (t, t ⌧ )x r (t) = x r (t ⌧ ) + R t t ⌧ A2 (m, t ⌧ ) 2 (m, z(m))dm . (34)
Also, the choice x s (t) = x r (t) + L(t)z(t) gives

ẋs (t) = A 2 (t)x r (t) + 2 (t, z(t)) + L(t)z(t) +L(t)[A 1 (t)x r (t) + 1 (t, z(t))] = H(t)x r (t) + L(t)z(t) + 3 (t, z(t)) = H(t)x s (t) + [ L(t) H(t)L(t)]z(t) + 3 (t, z(t)) . (35)
Arguing as we did to prove (34), we obtain

H (t, t ⌧ )x s (t) = x s (t ⌧ )+ R t t ⌧ H (m, t ⌧ )[D(m)z(m) + 3 (m, z(m))]dm (36)
where D is the function defined in ( 22), for all t ⌧ . From the definition of x s , we deduce that

H (t, t ⌧ )[x r (t) + L(t)z(t)] = x r (t ⌧ ) + L(t ⌧ )z(t ⌧ ) + R t t ⌧ H (m, t ⌧ )[D(m)z(m)+ 3 (m, z(m))]dm. (37)
By reorganizing the terms, we obtain

H (t, t ⌧ )x r (t) = x r (t ⌧ ) H (t, t ⌧ )L(t)z(t) + L(t ⌧ )z(t ⌧ ) + R t t ⌧ H (m, t ⌧ )[D(m)z(m)+ 3 (m, z(m))]dm. (38)
By subtracting (38) from (34), we obtain

⇤(t)x r (t) = R t t ⌧ A2 (m, t ⌧ ) 2 (m, z(m))dm R t t ⌧ H (m, t ⌧ )[D(m)z(m)+ 3 (m, z(m))]dm + H (t, t ⌧ )L(t)z(t) L(t ⌧ )z(t ⌧ ) (39)
for all t ⌧ . Assumption 2 ensures that for all t 2 R, ⇤(t) is invertible, which implies that (24) is satisfied.

IV. ILLUSTRATION A. The studied problem

To illustrate Theorem 1, let us consider the following system from [7, p. 137]: 40) is a nonholonomic system in chained form.

8 > > < > > : ⇠4 = ⇠ 3 v 1 ⇠3 = ⇠ 2 v 1 ⇠2 = v 2 ⇠1 = v 1 (40) with (⇠ 1 , ⇠ 2 , ⇠ 3 , ⇠ 4 ) valued in R 4 and the input (v 1 , v 2 ) valued in R 2 . Then (
We assume that the variables ⇠

1 ⌦

 1 (t, s) is the fundamental solution associated to ⌦ for the system ẋ = ⌦(t)x; see[START_REF] Sontag | Mathematical Control Theory[END_REF] Lemma C.4.1].

  We require v 2 to be bounded, i.e., that there is a constantv 2 > 0 such that |v 2 (t)|  v 2 for all t 0.As an immediate consequence, it follows that the finite escape time phenomenon does not occur and ⇠ 2 (t) is bounded by an affine function of t. By integrating the last equation of (44), we deduce that

									4 , ⇠	3 and ⇠	1 are measured,
	but that ⇠	2 is not measured. Also, we assume that there is
	no disturbance in the measurement of ⇠	2 . Let us consider the
	problem of making the system (40) track
	(⇠	1r (t), ⇠	2r (t), ⇠	3r (t), ⇠	4r (t)) = (2 sin(t), 0, 0, 0) . (41)
	We can recast this as a problem of stabilizing the origin of a
	time-varying system using the classical time-varying change
	of variables
							x	1 = ⇠	1	⇠	1r (t)	(42)
	and by selecting the feedback
	v where sign is the function defined by sign(m) = m 1 (t, x 1 ) = 2sign(x 1 ) p |x 1 | + 2 cos(t) |m| when (43) m 6 = 0 and sign(0) = 0. They result in
	8 > > > > < > > >	h h 2 (t) 3 2 ⇠2 = v ⇠4 = ⇠ ⇠3 = ⇠	2sign(x 2sign(x	1 ) 1 )	p |x p |x	1 | + 2 cos(t) 1 | + 2 cos(t)	i i	(44)
	> :	ẋ1 =	2sign(x	1 )	p |x	1 | .
	p |x 1 (t) = 0 for all t 1 (t)| = p p 1 (0)|] and x deduce that for all t t 2 [0, p |x 1 (0)|, we have ⇠ 1 (0)| t when |x p 1 (0)|. We |x 2 (t)x 1 (t) = 0. |x B. Observer Design
	Since ⇠ since only ⇠ 2 (t)x	1 (t) = 0 holds for all t	p |x	1 (0)|, and

2 is not measured, our next objective is now to construct an observer for the two dimensional system ⇢ ⇠3 = 2 cos(t)⇠

Malisoff was supported by NSF Grant 1711299.

for all t p |x 1 (0)|. With the notation of the previous section, this system can be rewritten as

with x r (t) = ⇠ 2 (t), the output z(t) = ⇠ 3 (t), A 1 (t) = 2 cos(t) , A 2 (t) = 0 , ✏(t) = 0, 1 (t, z) = 0 , and 2 (t, z) = v 2 (t).

(47)

We next check that Theorem 1 applies, with ⌧ = 2⇡ and y = ⇠ 3 . Choosing L(t) = 2 cos(t), we obtain

We conclude that Assumption 1 is satisfied. Thus Theorem 1 applies, and provides the estimate

We conclude that

for all t max n 2⇡,

C. Output feedback tracking

In this section, we illustrate how our estimate (50) can be used to solve a tracking problem that we described in Section IV-A. We design a state feedback for

Let us define

Simple calculations (based on the double angle formula for the sine function) give

Then the derivative of the positive definite quadratic function

where (s) = s p 1+s 2 , we obtain

We now use the LaSalle Invariance Principle to check that (53) in closed loop with (54) is uniformly globally asymptotically stable to 0, as follows. Consider any solution

) of ( 53)-( 54) such that cos(2t)⇣ 1 (t) 2 sin(t)⇣ 2 (t) + ⇣ 3 (t) = 0 for all t 0, Then v 2 (t) = 0 and so also ⇣i (t) = 0 for all t 0 and i = 1 to 3. Consequently,

for all t 0. We deduce easily that necessarily, ⇣ i (0) = 0 for i = 1 to 3 (by differentiating through (56) with respect to t and then setting t = 0 in the result). Consequently (⇣ 1 (t), ⇣ 2 (t), ⇣ 3 (t)) = (0, 0, 0) for all t 0. Then, we deduce from the LaSalle Invariance Principle that (54) renders the origin of (53) uniformly globally asymptotically stable. It follows that the bounded feedback

renders the origin of (51) uniformly globally exponentially stable. By grouping terms, we obtain

Then (57) and (50) lead us to the stabilizing output feedback

where M(m, t) = 2(m t + 2⇡) + sin(2m) sin(2t) and

Notice that v 2 is a solution of an implicit equation.

D. Simulation

We performed simulations, which show the efficiency of the approach. Fig. 1 shows the simulation of closed loop nonlinear time varying system (44) with v 2 defined in (58). Since our simulation shows good stabilization, it helps to illustrate our general theory in the special case of the system (40). We choose x 1 (0) = 0.5 which implies that

2] and x 1 (t) = 0 for all t 1/ p 2. This is evident from the simulation as well.

V. CONCLUSIONS

We proposed a new type of reduced order finite time observers. The observer applies to time-varying systems. We conjecture that it can be used to solve a problem of constructing interval observers that is similar to those in [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF]. We plan to apply our observer to solve a dynamic output feedback stabilization problem for a MIMO nonlinear system. We will study other extensions. In particular, we will combine the main result of the present paper with the result of [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF] and to the case where there are a delay and a disturbance in the input and where the outputs are only available on some finite time intervals.


Michael Malisoff
email: malisoff@lsu.edu

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

Sequential Predictors for Linear Time-Varying Systems with Delays in the Vector Field and in the Input

Keywords: Delays, stability, time-varying systems I

We provide new sequential predictors for a large class of linear time-varying systems that contain constant delays in the vector fields and also constant delays in the inputs. We allow the input delays to be arbitrarily large. We prove global exponential stability of the origin for an augmented system that includes the original system in closed loop with our sequential predictors based feedback control. We illustrate our new theorem in an example from identification theory.

Feedback stabilization can be challenging from the theory side, and also has many engineering motivations. See, e.g., the work [START_REF] Merad | Isometric torque control for neuromuscular electrical stimulation with time-varying input delay[END_REF], and [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], and [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] which provide useful surveys of delay compensating control.

The usual prediction approach eliminates feedback delays, by replacing the delayed states in the feedback by predicted values. While standard prediction generally yields controls with distributed terms, see, e.g., [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF], [START_REF] Cacace | Memoryless approach to the LQ and LQG problems with variable input delay[END_REF], [START_REF] Cacace | Optimal control of linear systems with large and variable input delays[END_REF], and [START_REF] Zhou | Truncated predictor feedback for linear systems with long time-varying input delays[END_REF] for controllers for special cases of time invariant systems that do not produce distributed terms and that are based on prediction, and the works [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF] and [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF] on chain observers that do not allow different delays in the vector fields and in the input, and so do not address the problems that we help solve here. To address cases with different delays in the input and the vector fields, this work provides a new class of sequential predictors, consisting of two interconnected subsets of subpredictors. The first subset of p subpredictors compensates for the delay in the vector field. Then a second set of p(k -1) predictors compensates for the input delays, where we assume that the input delay is an integer multiple kτ of the delay τ in the vector field for some integer k ≥ 2.

II. MAIN RESULT

We study systems of the form

ẋ(t) = A(t)x(t) + B(t)u(t -kτ ) + C(t)x(t -τ ) (1)
where x is valued in R n for any dimension n, τ ≥ 0 is a constant input delay, k ≥ 2 is a positive integer, and the matrix valued functions A, B, and C are assumed to be continuous and bounded. The feedback control u will be specified later, and we write it as a function of t to keep the notation simple. We assume that the initial functions are constant at the initial time, and that the initial time is always zero. Time-varying linear systems of the form (1) arise when linearizing a nonlinear system around a reference trajectory, and can represent a closed loop system with a relatively short delay τ > 0 and a longer delay kτ with the integer k as large as desired. The following assumption agrees with the assumptions from [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] and [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] in the special case of timevarying linear systems with constant delays when C = 0:

Assumption 1: There is a bounded continuous matrix valued function K such that the origin of

ẋ(t) = A(t)x(t) + B(t)u s (t, x(t)) + C(t)x(t -τ ) (2)
with u s (t, x) = K(t)x is uniformly globally exponentially stable.

In many cases, Assumption 1 can be satisfied in practice. For instance, in some cases, one can treat the term δ(t) = C(t)x(t -τ ) as a disturbance and use high gains to dominate this term. Then no condition on the size of τ is needed. In other situations, one can often determine K(t) so that ẋ(t) = (A(t) + C(t) + B(t)K(t))x(t) is uniformly globally exponentially stable to 0 and prove that ẋ(t) = (A(t) + B(t)K(t))x(t) + C(t)x(t -τ ) is uniformly globally exponentially stable to 0, when τ is smaller than a constant τ , where τ can be found by an approach that is similar to [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] and [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. See also Section IV for an example.

On the other hand, since we allow k in (1) to be as large as desired, emulation would not cover our feedback problem for (1), so we use sequential predictor controls. Our main result is as follows, where I n is the identity matrix and | • | ∞ is the essential supremum:

Theorem 1: Let Assumption 1 hold, let > 0 be a constant, and choose any integer

p > max 2, 2 √ 2τ |A + I n | ∞ max {|A| ∞ , |A + I n | ∞ } . (3)
Set L(t) = -[A(t) + I n ] and φ i (t) = t + iτ p for all i ∈ {1, 2, . . . , pk}. Then (1) in closed loop with the control

u new (t) = K(t + kτ )z pk (t), (4)
where z pk is the last n components of the npk-dimensional system

                                    żi (t) = A(φ i (t))z i (t) +B(φ i (t))u new (φ i (t) -kτ ) +L(φ i (t)) z i t -τ p -z i-1 (t) + C(φ i (t))x t -τ + iτ p , 1 ≤ i ≤ p żp+j (t) = A(φ p+j (t))z p+j (t) +B(φ p+j (t))u new (φ p+j (t) -kτ ) +L(φ p+j (t)) z p+j t-τ p -z p+j-1 (t) + C(φ p+j (t))z p t -τ + jτ p , 1 ≤ j ≤ p(k -1) (5)
and z 0 = x, is such that the dynamics for (x(t), z 1 (t)-x(t+ τ /p), z 2 (t) -z 1 (t + τ /p), . . . , z pk (t) -z pk-1 (t + τ /p)) is uniformly globally exponentially stable to 0.

Remark 1: The z i dynamics in ( 5) is called the ith sequential predictor for each i. Formula (3) implies that when A is the zero function, we can choose any p ≥ 2, by picking > 0 small enough. However, Theorem 1 also applies under nonzero A's, by choosing a large enough p such that (3) holds (since the right side of (3) does not depend on p).

III. PROOF OF MAIN RESULT

The proof has three steps. First, we prove that the system

ξ(t) = A(φ i (t))ξ(t) + L(φ i (t))ξ t -τ p (6)
is uniformly globally exponentially stable to 0 for each fixed choice of i, when (3) is satisfied. In the second step, we show that the dynamics for (z 1 , . . . , z p ) with the choices

z i (t) = z i (t) -z i-1 t + τ p (7)
for i = 1, 2, . . . , p is uniformly globally exponentially stable to 0. Combining the first two steps allows us to prove the conclusion of the theorem in our third step. We use the fact that for each constant c * > 0 and each continuous function

q : [-c * , ∞) → [0, ∞), the conditions d dt t t-c * t q(r)drd = c * q(t) - t t-c * q(r)dr (8a) and t t-c * t q(r)drd ≤ c * t t-c * q(r)dr (8b)
hold for all t ≥ 0. First Step. We first show that (3) implies that ( 6) is uniformly globally exponentially stable to 0, using the following variant of a Lyapunov-Krasovskii functional argument from [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. Our strategy is to first rewrite [START_REF] Cacace | Memoryless approach to the LQ and LQG problems with variable input delay[END_REF] in the form

ξ(t) = -ξ(t) -[A(φ i (t))+ I n ] ξ t -τ p -ξ(t) (9)
and to note that the time derivatives of V (ξ) = 1 2 |ξ| 2 and

V (ξ t ) = V (ξ(t)) + 2τ p |L| 2 ∞ (max{|A| ∞ , |L| ∞ }) 2 t t-2τ /p t s |ξ( )| 2 d ds (10)
along all solutions of (9) for all t ≥ 0 satisfy

d dt V (ξ(t)) = -|ξ(t)| 2 -ξ(t)[A(φ i (t)) + I n ] × ξ t -τ p -ξ(t) ≤ -|ξ(t)| 2 +|ξ(t)||A + I n | ∞ t t-τ /p | ξ( )|d ≤ -|ξ(t)| 2 + |ξ(t)||A + I n | ∞ × max{|A| ∞ , |L| ∞ } t t-2τ /p |ξ( )|d ≤ -2 |ξ(t)| 2 + 1 2 |A + I n | 2 ∞ × (max{|A| ∞ , |L| ∞ }) 2 2τ p × t t-2τ /p |ξ( )| 2 d , (11)
by Hölder's inequality ab ≤ 2 a 2 + 1 2 b 2 with a = |ξ(t)|, followed by Jensen's inequality, and therefore also

d dt V (ξ t ) ≤ -2 -4τ 2 p 2 |A+ I n | 2 ∞ max{|A| 2 ∞ , |L| 2 ∞ } |ξ(t)| 2 -τ p |A+ I n | 2 ∞ max |A| 2 ∞ ,|L| 2 ∞ t t-2τ /p |ξ( )| 2 d (12)
where ξ t is defined by ξ t (s) = ξ(t + s) for all values t ≥ 0 and s ≤ 0, by applying (8a) with the choices c * = 2τ /p and q(r) = |ξ(r)| 2 . Since (3) ensures that the quantity in squared brackets in ( 12) is positive, we can then use (8b) with the same choices of c * and q to find a positive constant c a such that d ds V (ξ s ) ≤ -c a V (ξ s ) along all solutions of (9) for all s ≥ 0, which we can then integrate on any interval [0, t] to get the desired uniform global exponential stability condition on (6), using the quadratic structure of V and our assumption that the initial functions are constant.

Second Step. We prove that the dynamics for z a = (z 1 , . . . , z p ) defined in terms of the error components ( 7) is uniformly globally exponentially stable to zero, which will follow from the first step of the proof and our choices of the arguments φ i (t) in the time-varying coefficients A, B, C, and L in our sequential predictor dynamics [START_REF] Besancon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF]. First note that the closed loop x subsystem in Theorem 1 satisfies

d dt x(φ 1 (t)) = A(φ 1 (t))x(φ 1 (t)) +B(φ 1 (t))u new (φ 1 (t) -kτ ) + C(φ 1 (t))x(φ 1 (t) -τ )
which produces the subsystem ż1 (t) = A(φ 1 (t))z 1 (t) + L(φ 1 (t))z 1 (t -(τ /p)), which is uniformly globally exponentially stable to 0, by the first step of the proof. Reasoning inductively then shows that the dynamics

       ż1 (t) = A(φ 1 (t))z 1 (t) + L(φ 1 (t)z 1 t -τ p żi (t) = A(φ i (t))z i (t) + L(φ i (t))z i t -τ p -L(φ i (t))z i-1 (t), i = 2, . . . , p (13)
for z a are uniformly globally exponentially stable to 0, by noting that each of the systems

żi (t) = A(φ i (t))z i (t) + L(φ i (t))z i t -τ p + δ(t) (14)
for i = 2, . . . , p is exponentially input-to-state stable with respect to the disturbance δ, since V from the first step of the proof is also an input-to-state stability Lyapunov-Krasovskii functional for ( 14) and L is a bounded function.

Third

Step. We prove the theorem using the second step of the proof, by setting ζ(t) = z p (t -τ ) -x(t) and first rewriting the (z p+1 , . . . z pk ) subsystem of (5) as

żp+ (t) = A(φ p+ (t))z p+ (t) + B(φ p+ (t))u new (φ p+ (t) -kτ ) + L(φ p+ (t)) z p+ t-τ p -z p+ -1 (t) + C(φ p+ (t)) x t + τ p + ζ t + τ p , 1 ≤ ≤ p(k -1). ( 15
)
We also use the fact that for all t ≥ 0, we have

z p t -τ + τ p + z p-1 t -τ + 2τ p + . . . + z 1 (t) = z p t -τ + τ p -x t + τ p = ζ t + τ p ( 16
)
which follows from the pairwise cancellation of terms in the sum of the p terms in a telescoping sum. Using [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], we can rewrite the dynamics for z b = (z p+1 , . . . , z pk ) as

żp+1 (t) = A(φ p+1 (t))z p+1 (t) +L(φ p+1 (t)) z p+1 t-τ p -z p (t) +C(φ p+1 (t))ζ t + τ p żp+ (t) = A(φ p+ (t))z p+ (t) +L(φ p+ (t))z p+ t -τ p -L(φ p+ (t))z p+ -1 (t) , 2 ≤ ≤ p(k -1), (17)
since our formulas for the last p(k -1) sequential predictors in [START_REF] Besancon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF] ensure that there are no C terms in the dynamics for zp+2 , . . . , zpk . Using the uniform global exponential stability of the dynamics ( 13) for z a , the telescoping sum ( 16), and the exponential input-to-state stability of ( 14) with respect to δ, it follows that the dynamics for (z a , z b ) = (z 1 , . . . , z pk ) is uniformly globally exponentially stable to 0.

Moreover, we can write z pk (t) = x(t + kτ ) + σ(t) + ζ(t + kτ ) in terms of the sum

σ(t) = p(k-1)-1 s=0 z pk-s t + sτ p (18)
for all t ≥ 0 (using a telescoping sum argument as we did to obtain ( 16)). Then the dynamics for the combined variable (x(t), z 1 (t), . . . , z pk (t)) = (x(t), z 1 (t) -x(t + τ /p), z 2 (t)z 1 (t + τ /p), . . . , z pk (t) -z pk-1 (t + τ /p)) are

ẋ(t) = A(t)x(t) + C(t)x(t -τ ) +B(t)u s (t, x(t)+σ(t -kτ )+ζ(t)) (19)
interconnected with ( 13) and [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF]. Since ( 16) and ( 18) express σ and ζ as linear combinations of the components of (z a , z b ), and the (z a , z b ) subsystem of ( 19) is uniformly globally exponentially stable to 0, it follows from Assumption 1 and the linearity of u s in x that ( 19) is uniformly globally exponentially stable to 0, which proves the theorem.

IV. ILLUSTRATION

Consider the dynamics from identification theory (from [START_REF] Anderson | Exponential stability of linear equations arising in adaptive identification[END_REF], [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF] and several references contained therein)

ẋ = -m(t)m (t)u (20)
with a control u where x is valued in R n for any dimension n, under the following assumption from [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]: Assumption 2: The function m : R → R n is continuous, there are constants α ∈ (0, 1), β > 0, and c > 0 such that α I n ≤ t+c t m(τ )m (τ )dτ ≤ β I n for all t ∈ R, [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and |m(t)| = 1 for all t ∈ R.

We can build a strict Lyapunov function V for (20) in closed loop with u s (t, x) = x, using this lemma from [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]: Lemma 1: Let m, α , β , and c be such that Assumption 2 is satisfied. Then, with the choices κ = 1 + c 2 + 1 4α c4 and

P (t) = κI n + t t-c t s m(l)m (l) dl ds, (22)
the function

V (t, x) = x P (t)x satisfies V ≤ -α |x| 2 /2 along all trajectories of ẋ(t) = -m(t)m (t)x(t) for all t ≥ 0. Moreover, |P | ∞ ≤ κ + c2 .
Under the preceding assumptions, [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] provided a bound τ such that ẋ(t) = -m(t)m (t)u s (t, x 1 (t -τ 1 ), . . . , x n (tτ n )) is uniformly globally asymptotically stable to 0 for all constant delays τ i ∈ [0, τ ] for i = 1, 2, . . . , n. Also, since [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF] has no drift term, [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]Remark 4] showed how to allow arbitrarily large τ by scaling the control, but this control scaling approach may not be viable in practice. Therefore, we illustrate Theorem 1 above for the input delayed version

ẋ(t) = -M a (t)u(t -kτ ) -M b (t)x b (t -τ ) (23)
where x = (x a , x b ) for any dimensions n a and n b for x a and x b respectively such that n = n a + n b , and M a (t) (resp., M b (t)) consists of the first n a (resp., last n b ) columns of m(t)m (t) for all t ∈ R. We use the following lemma: Lemma 2: Let the assumptions of Lemma 1 hold, and M a , M b , α , κ, and c be defined as above and τ > 0 be a constant such that τ < α 4 √ 5(κ + c2 ) .

Then Assumption 1 is satisfied with

A = 0 n×n , B = -[M a 0 n×n b ], C = -[0 n×na M b ],
and u s (t, x) = x. Proof: By writing

ẋ(t) = -M a (t)x a (t) -M b (t)x b (t -τ ) (25)
in the form ẋ

(t) = -m(t)m (t)x(t) + M b (t)[x b (t) -x b (t - τ )
] and using the bound |M | ∞ ≤ 1, it follows that the time derivative of the function V from Lemma 1 along all solutions of (25) for all t ≥ 0 satisfies

V ≤ -α |x(t)| 2 2 + {|x(t)|} 2(κ+c 2 ) t t-2τ |x( )|d ≤ -α |x(t)| 2 4 + 4(κ+c 2 ) 2 1 α (2τ ) t t-2τ |x( )| 2 d , (26)
by applying the inequality

c 1 c 2 ≤ α 4 c 2 1 + 1 α c 2 2
where c 1 and c 2 are the terms in curly braces in [START_REF] Zhou | Stabilization of linear systems with both input and state delays by observer-predictors[END_REF] followed by Jensen's inequality. Therefore, the time derivative of

V (x t ) = V (x(t)) + 5(κ + c2 ) 2 2τ α t t-2τ t |x(s)| 2 dsd
along all solutions of (25) for all t ≥ 0 satisfies

d dt V (x t ) ≤ -α 4 -5(κ + c2 ) 2 4τ 2 α |x(t)| 2 -(κ + c2 ) 2 2τ α t t-2τ |x( )| 2 d , (27)
by (8a) with c * = 2τ and q( ) = |x( )| 2 . The constant in curly braces in (27) is positive, by [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF]. Hence, we can reason as in the last part of the first step in the proof of Theorem 1 to get an exponential decay estimate on V and so also on |x(t)| all solutions of [START_REF] Weston | Sequential predictors under time-varying feedback and measurement delays and sampling[END_REF], which proves the lemma. Since Assumption 1 is satisfied, the uniformly globally exponentially stabilizing feedback control for [START_REF] Weston | Sequential predictors under time-varying feedback and measurement delays and sampling[END_REF] is given by Theorem 1, for any k ≥ 2 and any τ satisfying [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF]. This allows arbitrarily large delays kτ . Moreover, since Assumption 1 is satisfied with A = 0, we can choose p = 2, which produces 2k sequential predictors; see Remark 1.

V. CONCLUSION

We provided uniformly globally exponentially stabilizing sequential predictor feedbacks for time-varying linear systems that have delays in the vector field and a longer delay in the feedback. We illustrated how such systems arise when different components of the feedback control can have different delays, which were beyond the scope of the existing sequential predictor based feedback control designs. We plan to extend this work to time-varying delays using analogs of our methods for time-varying delays from [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF]. We also plan generalizations with nonlinear systems and measurement delays and sampling. This would extend our sequential predictors works [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] and [START_REF] Weston | Sequential predictors under time-varying feedback and measurement delays and sampling[END_REF] which did not allow different delays in the vector fields.

Supported by NSF Grant 1711299.