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

Continuous-Discrete Sequential Observers for Time-Varying Systems under Sampling and Input Delays

Keywords: Delays, observers, sampling. Supported

We provide continuous-discrete observers for a large class of time-varying linear systems where the inputs and outputs have sampling and delays, and where the systems and outputs contain uncertainties.

We allow the delays in the output and input to differ and to be arbitrarily long. We use the observers to design controls that ensure input-to-state stability, under delays and sampling. The observers and controls have no distributed terms. We illustrate our work in two examples including a DC motor model.

I. INTRODUCTION

Global asymptotic stabilization of systems with delays or outputs can be challenging when the output is not available for continuous measurement. This has motivated a large literature on stabilization problems under discrete sampling of an output, which have important engineering implications [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF]. Notable tools in this literature include prediction and reduction model approaches (and other methods) for delay compensation, which can be advantageous in applications because they can compensate for arbitrarily long input delays. Prediction and reduction model approaches construct dynamic extensions that generally lead to distributed terms in controls, so the control formulas often incorporate an integral of past control or state values.

Reduction model methods in [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF] were confined to cases with continuous measurements of the complete state, but [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF] covered linear systems with piecewise constant inputs. For systems with only discrete output measurements, several methods exist. For example, if the input delay is a multiple of the constant length c of all sampling time intervals in the input and output measurements, then one can apply a discrete time reduction model method from [START_REF] Gonzalez | Robustness of a discrete-time predictor-based controller for time-varying measurement delay[END_REF]; see [START_REF] Kazantzis | Control-relevant discretization of nonlinear systems with time-delay using Taylor-Lie series[END_REF].

The seminal works [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] and [START_REF] Deza | High gain estimation for nonlinear systems[END_REF] and more recent works (e.g., [START_REF] Mazenc | Reduction model approach for linear systems with sampled delayed inputs[END_REF], [START_REF] Goffaux | Improving continuous-discrete interval observers with application to microalgae-based bioprocess[END_REF]) used continuous-discrete observers, which apply in cases that are not amenable to static piecewise constant state feedback or to continuous observers. The work [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] used continuous-discrete observers to build globally asymptotically stabilizing output feedback under input delays, and [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] combined reduction model methods with continuous-discrete observers. Other works treated delays, discrete measurements, and sampling separately (but see the nonlinear stabilization work [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] for systems with inputs with zero-order hold under discrete delayed inputs). By contrast, here we use the sequential observers and subsequential predictors approach that was started in [START_REF] Najafi | Closed-loop control of dead time systems via sequential sub-predictors[END_REF] and that was further developed in [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF], and that eliminates the need for distributed terms in the control. For the first time, this paper adapts this approach to cases where continuous-discrete observers are appropriate, and we allow time-varying systems with different delays in the input and output, under sampling.

The work [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] for LTI systems with sampling in the input and the output uses a dynamic extension. However, the distributed terms in works like [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] and [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] may not always lend themselves to implementations, since the controls are not in a closed form. In the present work, our closed form stabilizing controls apply to linear time-varying systems with sampling in the input and in the output, and uncertainties in the output measurements and in the original plant. Then we can prove input-to-state stability under arbitrarily long delays in the input and output. This work adds significant value relative to our conference version [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF] by (i) covering linear time-varying systems (which arise from tracking reference trajectories), (ii) providing sufficient conditions for our Schur matrix condition to hold, and (iii) including an application to a DC motor model. These three features were not present in [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF], which was confined to LTI systems.

II. STATEMENT OF MAIN RESULT

We study systems having the form 8 < :

ẋ(t) = A(t)x(t) + B(t)u(g(t) ⌧ ) + 1 (t) y(t i ) = C(t i )x(t i r ) + 2 (t i ) , i r (1)
where t i = i⌫ for a known constant ⌫ > 0 for all integers i 0, the constant input and output delays ⌧ 0 and r 2 N and the bounded continuous matrix valued functions A, B, and C are known, the state x and the output y are valued in R nx and R ny respectively for any dimensions n

x and n y , g is a right continuous nondecreasing piecewise continuous unbounded function that satisfies g(t)  t for all t 0 and so can represent sampling in the control u (e.g., by choosing g(t) = s i for all t 2 [s i , s i+1 ), where s 0 = 0 and where the sample times s i admit positive constants s and s such that s  s i+1 s i  s for all integers i 0), and the unknown = ( 1 , 2 ) is piecewise continuous and bounded and has finite right limits at each t 0. Including g(t) is motivated by sampling that commonly occurs when controls are implemented, and by the challenge of proving robustness to sampling [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF]. We use standard definitions for the input-to-state stable (or ISS) property; see [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]. We assume that our initial functions for (1) are constant at the initial time t 0 = 0, I denotes an identity matrix, and C is the fundamental matrix for ż(t) = C(t)z(t) for any continuous choice of C. 1 [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] allows outputs and nonperiodic sampling, [START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF] leads to distributed terms in controls, even when A, B, and C are constant matrices. Finally, we assume: Assumption 2. The ratio ⌧ /⌫ is a rational number.

(i) the equalities A(t) = A 0 + A (t) and C(t) = C 0 + C (t) hold for all t 2 R; (ii) the matrix M 0 (⌫) = (I+⌫L 0 C 0 e (⌧ /d)A 0 )e ⌫A 0 is such that M > 0 (⌫)P 0 M 0 (⌫) P 0  I; and (iii) the pair ⇤ = ( A , C ) is such that |P 0 |G( ⇤ )H( ⇤ ) < 1, where G( ⇤ ) = e ⌫|A 0 | (e ⌫| A |1 1
 e ⌫| A |1 ⌫| A | 1 , condition (iii) from Assumption 1 then holds if ⌫| ⇤ | 1 is sufficiently small. For instance, if ⌫ 2 (0, 1) and | ⇤ | 1  1, then G( ⇤ )  Ḡ| ⇤ | 1 ⌫ and H( ⇤ )  H, where Ḡ = e |A 0 |+1 [1 + |L 0 |(|C 0 | + 1)] + |L 0 |e |A 0 | and H = e |A 0 | e(1 + |L 0 |(|C 0 |+1))+1 + |L 0 C 0 |e ⌧ |A 0 | , so part (iii) of Assumption 1 holds if ⌫ and ⇤ also satisfy ⌫| ⇤ | 1 < 1 |P 0 | Ḡ H . While
We prove the following, where + refers to the right limit, and | • | I is the supremum over any interval I; see Remark 1 for the definition of a solution for (2a)-(2b).

Theorem 1. Let Assumptions 1-2 hold for some positive integer d > max{3, ⌧/⌫} and some constant `> 1, and choose L 0 from Assumption 1. Set p = d + `r, and consider the families of continuous-discrete systems (

ż1 (t) = A( 1 (t))z 1 (t) + B( 1 (t))u g(t + ⌧ d r⌫) ⌧ 8t 2 J i z 1 (t + i ) = z 1 (t i ) + ⌫L 0 ⇥ C(t i )z 1 t i ⌧ d y(t i ) ⇤ , i r (2a)
and

( żj (t) = A( j (t))z j (t) + B( j (t))u g(t + j⌧ d r⌫) ⌧ 8t 2 J i z j (t + i ) = z j (t i ) + ⌫L 0 C(t i ) j t i ⌧ d , i r (2b
| (t)|  c 1 e c 2 t | (t + r )| + | | [0,t] holds for all t t
r and for all solutions of ( 1) and ( 2a)

-(2b). (b) The estimate |z p (t ⌧ ) x(t)|  pc 1 e c 2 (t ⌧ ) | (t + r )| + | | [0,t] (3)
holds for all t t r + ⌧ along all solutions of ( 1) and ( 2a)-(2b). (c) If, in addition, there is

constant matrix K such that ẋ(t) = A(t)x(t) + B(t)Kx(g(t)) + a (t)
) for all t > 0), except their states are reset to z j (t + i ) at the times t i . Then z p provides an observer for the state x of (1), by conclusion (b). A guide for finding

K in part (c) is to choose K such that ẋ(t) = [A(t)+B(t)K]x(t)
is uniformly globally exponentially stable to 0 (which is possible, e.g., if (A(0),B(0)) is a controllable pair and K is such that A(0)+B(0)K is Hurwitz, under suitable bounds on the time variation of A and B, by simple calculations), since then the requirements hold if sup t 0 |t g(t)| is small enough, by converse Lyapunov function theory (e.g., [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Theorem 4.14]) and [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF], which give an ISS Lyapunov function for

ẋ(t) = A(t)x(t) + B(t)Kx(g(t)) + a (t).
Our condition d > max{3,⌧ /⌫} shows that smaller ⌫'s produce higher dimensional observers (but our control and observers are free of distributed terms for all r and ⌧ ). It is tempting to guess that robustness in [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF] can be used to directly extend [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF] to linear time-varying systems, but this would not be right because the robustness in [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF] is to added uncertainties on the right side, not uncertainties in coefficient matrices. ⇤

III. PROOF OF THEOREM 1

We divide the proof into three parts. In the first part, we show that part (ii) of Assumption 1 remains true (up to a scaling of P 0 by a positive constant) when M 0 (⌫) in its inequality is replaced by a suitable analog of M 0 (⌫) for time-varying linear systems. The second part provides key relations that enable us to prove the conclusions (a)-(c) of the theorem in the third part.

First Part. Setting L = ⌫L 0 , we prove that there is a constant c 0 > 0 such that the matrices

M ij = A j (t i , t i 1 ) + LC(t i ) A j (t i ⌧ /d, t i 1 ) satisfy M > ij P 0 M ij P 0  c 0 I for all
integers i 0 and j 2 {1, 2, . . . , p}. We will use the facts (which follow from Appendix A.2 with the choices A = A j ) that

A j (t, s) e A 0 (t s)  e (t s)|A 0 | (e (t s)| A |1
1) and | A j (t, s)|  e |A|1(t s) hold for all s 0 and t s and j. By the triangle inequality, for all integers i 0 and j 2 {1, 2, . . . , p}, our choice of G in part (iii) of Assumption 1 is such that

|M ij M 0 (⌫)|  e ⌫A 0 A j (t i , t i 1 ) + L [C 0 C(t i )] e (⌫ ⌧ /d)A 0 + LC(t i ) e (⌫ ⌧ /d)A 0 A j t i ⌧ d , t i 1  e ⌫|A 0 | e ⌫| A |1 1 + |L|| C | 1 e ⌫|A 0 | +⌫|L 0 ||C 0 + C | 1 e ⌫|A 0 | e ⌫| A |1 1 = G( ⇤ ) by using the estimates |L C (t i )e (⌫ ⌧ /d)A 0 |  |L|| C | 1 e |⌫ ⌧ /d||A 0 |  |L|| C | 1 e ⌫|A 0 |
L = ⌫L 0 . Also, |M ij |  e ⌫|A|1 (1 + |LC| 1 ) and so also |M ij + M 0 (⌫)|  H( ⇤ )
for all i and j. Hence, we can use (iii) from Assumption 1 and the matrix difference of squares formula to find a constant c 0 2 (0, 1)

such that V > (M > ij P 0 M ij P 0 )V = V > (M > 0 (⌫)P 0 M 0 (⌫) P 0 )V + V > (M > ij M > 0 (⌫))P 0 (M ij + M 0 (⌫))V  c 0 |V | 2
for all i 0, V 2 R nx , and j = 1, 2, . . . , p.

Second Part. Let P = P 0 /c 0 , where c 0 is from the first part of the proof. Then M > ij P M ij P  I for all i 0 and j = 1, 2, . . . , p. From the definition of {t i }, we can rewrite (2a) as

ż1 (t) = A( 1 (t))z 1 (t) + B( 1 (t))u g(t + ⌧ d r⌫) ⌧ for all t 2 J i , and z 1 (t + i ) = z 1 (t i ) + LC(t i )[z 1 (t i ⌧ d ) x(t i r⌫)] L 2 (t i ) for all i r, because L = ⌫L 0 . Since ẋ( 1 (t)) = A( 1 (t))x( 1 (t)) + B( 1 (t))u (g ( 1 (t)) ⌧ ) + 1 ( 1 (t)
) for all t 0, it follows from the preceding formulas for ż1 (t) and z 1 (t + i ) that 1 as defined in our theorem satisfies 8 > < > :

˙ 1 (t) = A( 1 (t)) 1 (t) 1 t + ⌧ d r⌫ , 8t 2 J i 1 (t + i ) = z 1 (t i ) + ⌫L 0 C(t i ) ⇥ z 1 t i ⌧ d x(t i r⌫) ⇤ x t i + ⌧ d r⌫ ⌫L 0 2 (t i ) = 1 (t i ) + ⌫L 0 C(t i ) 1 t i ⌧ d ⌫L 0 2 (t i ) (4)
for all i r. Since

t i ⌧ d 2 (t i 1 , t i ) for all i 1 (because t i+1 t i = ⌫ > ⌧/d
(t + i ) = M i1 1 (t + i 1 ) + N 0 ( , i) , where N 0 ( , i) = R t i t i 1 M ⇤ ( , t i , w)dw ⌫L 0 2 (t i ) ⌫L 0 C(t i ) R t i ⌧ /d t i 1 M ⇤ ( , t i ⌧ /d, w)dw (5)
and

M ⇤ ( , s, w) = A 1 (s, w) 1 (w + ⌧ /d r⌫). Let z 0 (t) = x(t r⌫). Then we have żj (t) = A( j (t))z j (t) + B( j (t))u (G ⇤ (t, j) ⌧ ) 8t 2 J i z j (t + i ) = z j (t i ) ⌫L 0 j (t i ) + ⌫L 0 ⇥ C(t i )z j t i ⌧ d C(t i )z j 1 (t i ) ⇤ , i r (6)
for all j 2 {1, . . . , p}, where G ⇤ (t, s) = g t + s⌧ d r⌫ , and where k = 2 if k = 1 and k = 0 otherwise. Consequently, since ⌧ /d < ⌫, it follows that for all j 2 {2, . . . , p}, we have

˙ j (t) = A( j (t)) j (t) 8t 2 J i , and j (t + i ) = j (t i ) + ⌫L 0 C(t i ) j (t i ⌧ /d) 8i r. ( 7
)
By integrating the first equation in [START_REF] Goffaux | Improving continuous-discrete interval observers with application to microalgae-based bioprocess[END_REF] over

t i ⌧ d , t i ⇤ , we obtain j (t i ) = A j (t i , t i ⌧ d ) ⇥ z j t i ⌧ d z j 1 t + i ⇤ . By (6), it follows that j (t i ) = A j (t i , t i ⌧ d )[z j (t i ⌧ d ) z j 1 (t i ) ⌫L 0 C(t i )(z j 1 (t i ⌧ d ) z j 2 (t i )) + ⌫L 0 j 1 (t i )
] holds for all j 2 {2, . . . , p} and i r. Hence,

j (t i ) = A j (t i , t i ⌧ d ) j (t i ⌧ d ) + ⌫L 0 j 1 (t i ) A j (t i , t i ⌧ d )⌫L 0 C(t i ) j 1 (t i ⌧ d
) holds for all j 2 {2, . . . , p} and i r, so by [START_REF] Goffaux | Improving continuous-discrete interval observers with application to microalgae-based bioprocess[END_REF],

⇢ ˙ j (t) = A( j (t)) j (t) , 8t 2 J i j (t + i ) = F j (i) j t i ⌧ d A j t i , t i ⌧ d LC(t i ) j 1 t i ⌧ d + ] j 1 (t i ), i r (8)
for all j 2 {2, . . . , p}, where F

j (i) = A j (t i , t i ⌧ /d)+⌫L 0 C(t i ) and ] j 1 (t i ) = A j (t i , t i ⌧ /d)⌫L 0 j 1 (t i ).
Integrating the first equality of ( 8) over

(t i 1 , t i ⌧ /d] now gives ⇢ ˙ j (t) = A( j (t)) j (t) , 8t 2 J i j (t + i ) = M ij j t + i 1 A j t i , t i ⌧ d LC(t i ) j 1 t i ⌧ d + ] j 1 (t i ), i r (9)
for all j 2 {2, . . . , p}, by the definition of the M ij 's from the start of the first part of the proof. Since the first lines of ( 4) and ( 9) (integrated along the J i 's) and the fact that ⌧ /d < ⌫ give

1 t i ⌧ d = A 1 t i ⌧ d , t i 1 1 (t + i 1 ) R t i ⌧ /d t i 1 A 1 t i ⌧ d , w 1 w + ⌧ d r⌫ dw (10) and j t i ⌧ d = A j t i ⌧ d , t i 1 j (t + i 1
) for all integers i r and j 2 {2, . . . , p}, we can combine [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] with the second equality of (9) to obtain the discrete time system

⇢ 1 (t + i ) = M i1 1 (t + i 1 )+N 0 ( , i) j (t + i ) = M ij j (t + i 1 )+N 1 (i, j) j 1 (t + i 1
)+N 1 ( , j, i) for all i r and j 2 {2, . . . , p}

where

N 1 (i, j) = A j t i , t i ⌧ d LC(t i ) A j t i ⌧ d , t i 1 and N 1 ( , j, i) = ] 1 (t i ) + A j t i , t i ⌧ d LC(t i ) R t i ⌧ /d t i 1 A j t i ⌧ d , w 1 w + ⌧ d r⌫ dw when j = 2 and N 1 ( , j, i) = 0 if 3  j  p.
Third Part. We first use the function V (x) = x > P x (where P = P 0 /c 0 is from the start of the second part of the proof) to prove that the dynamics for the discrete time variable

D(i) = (t + i ) = ( 1 (t + i ), . . . , p (t + i ))
are exponentially ISS with respect to = ( 1 , 2 ). By substituting in the formula from the second equation in [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF], and then recalling that M > ij P M ij P  I for all i and j and Young's inequality, and setting

V (j, i) = V ( j (t + i )) V ( j (t + i 1 )), we obtain V (j, i)  > j (t + i 1 ) j (t + i 1 ) + 2(M ij j (t + i 1 )) > P N 1 (i, j) j 1 (t + i 1 ) + N 1 ( , j, i) + N 1 (i, j) j 1 (t + i 1 ) + N 1 ( , j, i) > P N 1 (i, j) j 1 (t + i 1 ) + N 1 ( , j, i)  | j (t + i 1 )| 2 + |N 1 ( , j, i)| 2 |P | + | j 1 (t + i 1 )| 2 |N 1 (i, j)| 2 |P | + ⇥ 2{|N 1 (i, j)||P ||N 1 ( , j, i)|}{| j 1 (t + i 1 )|} +2(M ij j (t + i 1 )) > P N 1 (i, j) j 1 (t + i 1 ) + N 1 ( , j, i) ⇤  1 2 | j (t + i 1 )| 2 + N 2 | j 1 (t + i 1 )| 2 + N 2 ( , j, i) for j = 2, . . . , p and i r, (12)
where

N 2 = sup{|N 1 (i, j)| 2 |P | + 1 4 + 4|M ij | 2 |P | 2 |N 1 (i, j)| 2 : i 0, 1  j  p} and N 2 ( , j, i) = |N 1 ( , j, i)| 2 (|P | + 4|N 1 (i, j)| 2 |P | 2 + 4|M ij | 2 |P | 2
), and where the second inequality followed from using the inequality 2ab  1 4 a 2 + 4b 2 three times with appropriate a and b (once with a = | j 1 (t + i 1 )|, and then twice using a = | j (t + i 1 )|) to upper bound the terms in squared brackets in [START_REF] Mazenc | Continuous-discrete sequential observers under sampling and input delays[END_REF]. The sup in the definition of N 2 is finite because Lemma A.2 implies that sup{|

A j (t i , t i ⌧ /d)| : i 1, 1  j  p} and sup{| A j t i ⌧ d , t i 1 | : i 1, 1  j  p}
are finite and because C is bounded. Also, for each i r, we have

V (1, i)  > 1 (t + i 1 ) 1 (t + i 1 )+2(M i1 1 (t + i 1 )) > P N 0 ( , i)+N 0 ( , i) > P N 0 ( , i)  1 2 | 1 (t + i 1 )| 2 +N 3 ( , i), where N 3 = (2 max i |M i1 | 2 |P | 2 + |P |)|N 0 | 2 ,
and using [START_REF] Mazenc | Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach[END_REF] and

2(M i1 1 (t + i 1 )) > P N 0 ( , i)  1 2 | 1 (t + i 1 )| 2 +2|M i1 | 2 |P | 2 |N 0 ( , i)| 2 .
Since the last upper bound on the right side of ( 12) depends linearly on | j 1 (t + i 1 )| 2 , we can recursively build a strict Lyapunov function for the variable D to complete the proof; see Appendix A.3 below for complete details.

IV. ILLUSTRATIONS

Two-Dimensional Example. Consider the special case where

A(t) = " 0 q + A1 (t) q + A2 (t) 0 # , C(t) = [ a + C1 (t) b + C2 (t) ] (13)
and

⌫ = 1 q ⇡ 2 + 2k⇡
, where a, b, and q > 0 are constants and k is any nonnegative integer, for any choice of B. We show how we can satisfy Assumption 1 with

A 0 = " 0 q q 0 # , C 0 = [ a b ], A (t) = " 0 A1 (t) A2 (t) 0 # , and
C (t) = " C1 (t) C2 (t) # > . ( 14
)
If we set ⌫L 0 = [ L 1 L 2 ] > for real constants L 1 and L 2 to be determined, then

M 0 (⌫) = " L 1 a sin(q⌧ /d) L 1 b cos(q⌧ /d) 1+L 1 a cos(q⌧ /d) L 1 b sin(q⌧ /d) L 2 a sin(q⌧ /d) L 2 b cos(q⌧ /d) 1 L 2 a cos(q⌧ /d) L 2 b sin(q⌧ /d) # . (15)
For ⌧ = 0, the eigenvalues

1 2 (L 2 a L 1 b ± p (L 2 a L 1 b) 2 4(1 + L 1 a + L 2 b)) of (15) are valued in ( 1, 1) if L 1 a + L 2 b = 1 and L 1 b L 2 a = c
are satisfied for some c 2 ( 1, 1). For each c 2 ( 1, 1), we can find values L i that satisfy preceding system of equations if a and b are not both zero. Hence, for each ⌧ > 0 and q > 0, we can find a d > 0 so that M 0 (⌫) is Schur stable when a and b are not both zero (by choosing d so that q⌧ /d is small enough, and using continuity of eigenvalues as functions of the entries of a matrix), which allows us to find the P 0 required by Assumption 1. For instance, if

q = 3.8, a = 1, b = 0, ⌧ = 1, d = 30, `= 12, L 1 = 1, L 2 = 1
2 , A1 = 0, A2 = 0.04 sin(t), C = 0, k = 0, and ⌫ = 0.4, then we can use Mathematica [START_REF] Mathematica | The world's definitive system for modern technical computing[END_REF] to solve M > 0 (⌫)P 0 M 0 (⌫) P 0 = I for P 0 and then use the G and H formulas from Remark A.2 (and the fact that A > 0 = A 0 ) to check that Assumptions 1-2 are satisfied. Hence, for any output delay r 2 N, Theorem 1 provides the observer z p , where z p is the state of the last subsystem in (2b) and p = d + `r = 30 + 12r. Moreover, with B = [ 0.52 0] > and K = [0.18 0], we can use [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] (with V (t, x) = x > P ⇤ x where P ⇤ is the solution of (A ] 0

) > P ⇤ + P ⇤ A ] 0 = I and A ] 0 = A 0 + BK, g(t, x) = B, u s (t, x) = Kx, f (t, x) = A(t)
x, W (x) = 0.82|x| 2 , c 1 = 0.01, c 2 = 248.11, and c 3 = 0.01) to check that we can allow any sample rate s 2 [0, 0.93] in the input (in the g(t) as defined in Section II) and establish an ISS result, using part (c) of Theorem 1.

Robotic DC Motor. Consider the case of a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor, which produces this model from [START_REF] Dawson | Integrator backstepping control of a brush DC motor turning a robotic load[END_REF]:

Mq + B q + N sin(q) = I, L İ = V e RI K B q, where M = J K⌧ + mL 2 0 3K⌧ + M 0 L 2 0 K⌧ + 2M 0 R 2 0 5K⌧ , N = mL 0 G 2K⌧ + M 0 L 0 G K⌧ , and B = B 0 K⌧ (16)
and where B 0 is the viscous friction coefficient at the joint, G is the gravitational constant, I(t) is the motor armature current, J is the rotor inertia, K B is the back-emf coefficient, the coefficient K ⌧ characterizes the electromagnetic conversion of armature current to torque, L is the armature inductance, L 0 is the length of the link, m is the mass of the link, M 0 is the mass of the load, q(t) is the position of the load (which is the angular motor position), R 0 is the radius of the load, R is the armature resistance, and V e is the input current voltage. All of the constants in [START_REF] Dawson | Integrator backstepping control of a brush DC motor turning a robotic load[END_REF] are positive. Our work [START_REF] Mazenc | Design of continuous-discrete observers for time-varying nonlinear systems[END_REF] provided continuous-discrete observers for [START_REF] Dawson | Integrator backstepping control of a brush DC motor turning a robotic load[END_REF].

However, [START_REF] Mazenc | Design of continuous-discrete observers for time-varying nonlinear systems[END_REF] left open the problem of observer design under input and output delays. Here we help address these remaining challenges for a linearization of ( 16) around a C 1 reference December 26, 2018 DRAFT trajectory x r (t) = (q r (t), qr (t), I r (t)), which produces the tracking error dynamics

⇢ ẋ1 (t) = x 2 (t), ẋ2 (t) = b 1 x 3 (t) a 1 cos(x 1r (t))x 1 (t) a 2 x 2 (t) ẋ3 (t) = b 0 u(g(t) ⌧ ) a 3 x 2 (t) a 4 x 3 (t), y(t i ) = x 1 (t i r ) + 2 (t i ) , (17)
where

x 1 = q q r , x 2 = q qr , x 3 = I I r , u = V e is the control, a 1 = N /M, a 2 = B/M, a 3 = K B /L, a 4 = R/L, b 0 = 1/L, and b 1 = 1/M [17]
. We can now apply our theorem with

A(t) = 2 4 0 1 0 a 1 cos(x 1r (t)) a 2 b 1 0 a 3 a 4 3 5 , A 0 = 2 4 0 1 0 0 a 2 b 1 0 a 3 a 4 3 5 (18)
and C(t) = C 0 = [1 0 0] and C = 0, where the only nonzero entry of A is a 1 cos(x 1r (t)), so

| ⇤ | 1 = |( A , C )| 1 = a 1 .
For instance, using Mathematica [START_REF] Mathematica | The world's definitive system for modern technical computing[END_REF] to solve M > 0 (⌫)P 0 M 0 (⌫) P 0 = I for P 0 , we can check that if a 2 = b 1 = a 3 = 0.5, a 4 = 0.1, and ⌧ = 1, then Assumptions 1-2 are satisfied with ⌫ = 0.3, L 0 = (1/⌫)[ 0.2 0.18 0.05 ] > , `= 9, d = 30, and any

| ⇤ | 1 2 [0, 0.0725),
and then Theorem 1 gives the observer z p where p = 30 + 9r and r 2 N is the output delay.

V. CONCLUSIONS

We provided a new continuous-discrete observer for linear time-varying systems with input delays, uncertainties, sampling in the output observations, and additive uncertainties on the control and outputs. Our work is new even in the special case of LTI systems, and is motivated by many engineering applications where only sampled output values are available. Our sequential observer method produced observers and stabilizing feedbacks without distributed terms. We allow arbitrarily long constant input and output delays, which can differ. We provided sufficient conditions for the sampling rate ⌫ in the output to satisfy our assumptions, and in future work, we hope to find ways to allow even larger values of ⌫. We also hope to extend our work to aperiodic sampling in the output, and to use ideas from [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] to cover nonlinear systems.

APPENDICES

Appendix A.1: Checking Part (ii) from Assumption 1. We prove the following result:

Lemma A.1. Let (A 0 , C 0 ) 2 R nx⇥nx ⇥ R ny⇥nx
be an observable pair, L 0 be any constant matrix such that H = A 0 + L 0 C 0 is Hurwitz, and the positive values ⌧ , d, and ⌫ be such that ⌧ /d < ⌫.

Let the positive definite symmetric matrix P a be such that P a H + H > P a  I. Set (⌫), where 1) for all s 0 and t s. ⇤

(⌫) = 1 ⌫ h e ⌫|A 0 | 1 ⌫|A 0 | ⌫ + |L 0 C 0 | e ⌫|A 0 | 1 i (A.
(⌫) = A 2 0 R 1 0 R m 0 e `⌫A 0 d`dm + 1 ⌧ ⌫d L 0 C 0 A 0 R 1 0 e (⌫ ⌧
Proof. The function z(t, s) = A (t, s) e A 0 (t s) satisfies @ @t z(t, s) = A(t) A (t, s) A 0 e A 0 (t s) = A 0 z(t, s) + A (t) A (t, s) and z(s, s) = 0 if t s 0. Hence, by variation of parameters,

z(t, s) = R t s e A 0 (t r) A (r) A (
(s) 2 R nx , this gives | A (t, s)|  e |A|1(t s) (by the definition | A (t, s)| = inf{c 2 [0, 1) : | A (t, s)x|  c|x| for all x 2 R nx } of the matrix norm of A (t, s)), which we can combine with (A.3) to get |z(t, s)|  R t s e |A 0 |(t r) e (|A 0 |+| A |1)(r s) dr| A | 1  e (t s)|A 0 | | A | 1 R t s e | A |1(r s)
dr, which one integrates to get the second conclusion of the lemma.

Remark A.2. We can sometimes obtain less conservative bounds on | A (t, s)| and | A (t, s) e A 0 (t s) |, using special properties of A 0 . For instance if

A > 0 = A 0 , then | A (t, s)|  e | A |(t s)
holds for all s 0 and t s (by checking that the time derivative of V + 1 + ⌫|L 0 C 0 |. We illustrate these points in Section IV.

(x) = 1 2 |x| 2 along all solutions of ẋ = A(t)x satisfies V  2| A | 1 V (x(t)),
Appendix A.3: End of Proof of Theorem 1. Applying (12) for j = p and then j = p 1, and then adding the results, provides the bound V (p, i)

+ (2N 2 + 1) V (p 1, i)  1 2 (| p 1 (t + i 1 )| 2 +| p (t + i 1 )| 2 )+2(N 2 +1) N2 ( , i)+(2N 2 +1)N 2 | p 2 (t + i 1 )| 2 , where N2 ( , i) = max j N 2 ( , j, i) (which is finite valued because A is bounded), so the V (p 1, i) cancelled the N 2 | p 1 (t + i 1 )| 2 in the V (p, i) formula.
Repeating this process for V (p s, i) for larger integers s 2 provides constants N 3 0 and ! i > 0 for which this exponentially ISS Lyapunov decay estimate holds along all solutions of (11): by writing the closed loop x system from (c) as ẋ(t) = A(t)x(t)+B(t)Kz p (g(t) ⌧ )+ 1 (t) =

P p j=1 ! i V (j, i)  ! 0 P p j=1 ! i V ( j (t + i 1 )) + N 3 | |
1 This follows from the orthogonality of the matrix e A 0 s for all s 0, which follows because A0 = A > 0 gives A0A > 0 = A > 0 A > 0 = A > 0 A0 and so also I = e sA > 0 +sA 0 = e sA > 0 e sA 0 . We also used the relation x > A0x = 1 2 x > A0x + 1 2 x > A > 0 x = 0. 2 Condition (A.5) was used to convert (A.4) into a continuous time decay estimate. To find c, note that since (4) and ( 9 ) for all t 0 along solutions of this closed loop system, then using (a), the ISS assumption, and (A. ) along all solutions of this closed loop system for all t 0.

Remark 1 .

 1 is exponentially ISS with respect to the arbitrary locally bounded piecewise continuous uncertainty a , then the (x, )dynamics, consisting of (1) and the = ( 1 , . . . , p ) system in closed loop with u(t) = Kz p (t), are ISS with respect to = ( 1 , 2 ). ⇤ Solutions of (2a)-(2b) are understood as follows. The z j subsystems (called sequential observers) are continuous time ones (whose dynamics are defined by the first equations in (2a)-

  which follow because d ⌧ /⌫, and recalling the ⌫|L 0 || C | 1 e ⌫|A 0 | in the G( ⇤ ) formula and the fact that

1 )

 1 and R(⌫) = 2 (⌫)|P a | + |H > P a H| + 2⌫ (⌫)|P a H| + ⌫ 2 |P a | (⌫) 2 . Then if ⌫R(⌫)  1/2, then the matrix P 0 = (2/⌫)P a satisfies the requirements of part (ii) of Assumption 1. ⇤ Proof. For any M 2 R nx⇥nx , we have e sM = I + M R s 0 e `M d`for all s 0 and e M = I + M + M 2 R 1 0 R m 0 e `M d`dm; the first equality is by the Fundamental Theorem of Calculus, and then the second equality followed by setting s = 1 and then s = `in the first equality. Specializing the second equality to the case where M = ⌫A 0 , and specializing the first equality to the case where M = (⌫ ⌧ d )A 0 and s = 1, we then have M 0 (⌫) = I + ⌫H + ⌫ 2

  2 [0,t] . Now the existence of positive constants c a and c b such that the ISS estimate| (t + i )|  c a e c b (t i tr) | (t + r )| + | | [0,t] (A.4)holds for all integers i r follows from standard ISS results[START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. Combining (A.4) with the fact that the first equalities in (4) and (9) provide a positive constant c such that| (t)|  c(| (t + i )| + | | [0,t]) for all t 2 J i and i 0 (A.5) now allows us to prove part (a) of the theorem. 2 To prove the remaining conclusions (b)-(c) of the theorem, first note that z p (t) = x t + p⌧ d ⌫r + p P s=1 s (t + (p s)⌧ /d) (A.6) for all t 2 R (by an induction on p). Hence, since `2 N satisfies d⌫ = `⌧ , and since p `r = d, it follows that z p (t) = x (t + ⌧ ) + p P s=1 s (t + (p s)⌧ /d) , (A.7) which we evaluate at t ⌧ and combine with (a) to get (b); the ⌧ in e c 2 (t ⌧ ) in (3) was used to take the (p s)⌧ d ⌧ ⌧ into account. We prove (c) by combining (a) and (A.7),

  [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] and (A.7) to find a constant c ⇤ > 0 and a⇤ 2 K 1 such that |x(t/2)|  ⇤ (|x(0)|+| | [ ⌧,0] +| | [0,t] ) and | a | [t/2,t]  c ⇤ (e c 2 t | | [ ⌧,0] +| | [0,t]

  We assume the following, where M is nonnegative definite, | • | is the usual Euclidean norm and the induced matrix norm, and | • | 1 is the essential supremum: Assumption 1. There are constant matrices A 0 2 R nx⇥nx , C 0 2 R ny⇥nx , and L 0 2 R

a  M b for square matrices M a and M b of the same size means M b M a nx⇥ny ; an integer d and a constant `> 1 such that d = `⌧ /⌫; bounded piecewise continuous functions A : R ! R nx⇥nx and C : R ! R ny⇥nx ; and a symmetric positive definite P 0 2 R nx⇥nx such that:

  for all i), we can integrate the first equality in (4) along (t i 1 , t

	i	⌧ /d) into the
	last expression for 1 (t + i ) in (4), to obtain
	1

i ] and then on (t i 1 , t i ⌧ /d] using variation of parameters, and then substitute the resulting formulas for 1 (t i ) and 1 (t

  (⌫), and we can bound and then evaluate the integrals from (A.2) to get| (⌫)|  ¯ (⌫) and |R(⌫)|  R(⌫). Since ⌫ R(⌫)  1/2, we conclude that M 0 (⌫) > P a M 0 (⌫) P a  (⌫/2)I, so the lemma follows from our choice of P 0 .

										d )`A0	d`.	(A.2)
	This gives M 0 (⌫) > P	a M 0 (⌫) P	a = ⌫ H > P	a + P	a H + ⌫ 2	(⌫) > P	a + P	a (⌫) + H > P	a H +
	⌫ 3	(⌫) > P	a H + H > P	a (⌫) + ⌫ 4	(⌫) > P	a (⌫), by collecting terms. Hence, our choice of P	a
	gives M 0 (⌫) > P ⌫ (⌫) > P a H + H > P a M 0 (⌫) P a (⌫) +⌫ 2 a  ⌫ [ I + ⌫R(⌫)], where R(⌫) = (⌫) > P (⌫) > P	a + P	a (⌫) + H > P	a H +

a Appendix A.2: Comparison Lemma. We used this lemma in our proof of Theorem 1:

Lemma A.2. Let A 0 2 R nx⇥nx

and A : R ! R nx⇥nx be piecewise continuous and bounded. Then the fundamental solution A associated with

A(t) = A 0 + A (t) satisfies | A (t, s)|  e |A|1(t s)

and A (t, s) e A 0 (t s)  e (t s)|A 0 | (e (t s)| A |1

  and then integrating the result), which(t r) || A | 1 e | A |1(r s) dr = e | A |1(t s)1.1 It follows from the first part of our proof of our theorem that the theorem remains true in the A > 0 = A 0 case if we replace e ⌫|A 0 | in the G( ⇤ ) formula by 1 and replace the formula for H by H( ⇤ ) = (1 + ⌫|L 0 C| 1 )e |

	we substitute into (A.3) to get |z(t, s)| 	R	t s |e A 0
	December 26, 2018		DRAFT

A |1⌫

  (g(t) ⌧ + (p s)⌧ /d)+ 1 (t), and then finding ¯ 2 K 1 and constants ci > 0 such that |x(t)|  c1 e c2 t/2 |x(t/2)|+ ¯ (| a | [t/2,t]

	A(t)x(t)+B(t)Kx(g(t))+	a (t), where	a (t) = B(t)K	P	p s=1 s
					p 2⌫|A|1	max{⌫	p	2p, 1}.

) give | ˙ j (t)|  |A|1| j (t)| + | 1|[0,t] for all t 2 Ji, i, and j, we get | ˙ (t)|  p 2(|A|1| (t)| + p p| 1|[0,t]), so Gronwall's inequality gives | (t)|  (⌫ p 2p| 1|[0,t]+| (t + i )|)e p 2⌫|A|1 for all t 2 Ji and i, so we can pick c = e

December 26, 2018 DRAFT

by US National Science Foundation Grant 1711299. Mazenc is with EPI DISCO


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

Laurent Burlion
email: laurent.burlion@onera.fr.

Michael Malisoff
email: malisoff@lsu.edu.

Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Systems *

Keywords: backstepping, nonlinear, output feedback, stabilization. A summary [17]

We construct bounded globally asymptotically stabilizing output feedbacks for a family of nonlinear systems, using a dynamic extension and a converging-input-converging-state assumption. We provide sufficient conditions for this assumption to hold, in terms of Lyapunov functions. The novelty is that our construction provides formulas for the control bounds while allowing uncertainties that prevent the use of classical backstepping, and cases where only part of the state variable is available for measurement, without requiring the time lagged states in the feedback control that were required in the artificial delays approach. We illustrate the relevance of our work to engineering in an application to a single-link direct-drive manipulator.

Introduction

Backstepping is a fundamental approach that is often used to construct globally asymptotically stabilizing feedback controls for nonlinear systems having a suitable lower triangular structure called feedback form [START_REF] Dixon | Global exponential setpoint control of wheeled mobile robots: a Lyapunov approach[END_REF]. Important early contributions to backstepping include [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], and its numerous applications can be found in [START_REF] Bialy | An adaptive backstepping controller for a hypersonic airbreathing missile[END_REF], [START_REF] Chen | Exponential tracking control of a hydraulic proportional directional valve and cylinder via integrator backstepping[END_REF], [START_REF] Farrell | Command filtered backstepping[END_REF], [START_REF] Jiang | A recursive technique for tracking control of nonholonomic systems in chained form[END_REF], [START_REF] Lee | Adaptive tracking control of an underactuated aerial vehicle[END_REF], [START_REF] Pettersen | Underactuated ship tracking control; theory and experiments[END_REF], [START_REF] Sezkin | Boundary backstepping control of flow-induced vibrations of a membrane at high Mach numbers[END_REF], [START_REF] Smaoui | A study on tracking position control of an electropneumatic system using backstepping design[END_REF], and many other works. See also the discussions in the texts and monographs [START_REF] Isidori | Nonlinear Control Systems[END_REF] and [START_REF] Khalil | Nonlinear Systems[END_REF], and many other research articles such as [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF].

One challenge in applying traditional backstepping is that it is not always amenable under magnitude constraints on the control laws, which can limit the applicability of classical backstepping or of the bounded backstepping results of [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] and [START_REF] Mazenc | Backstepping with bounded feedbacks[END_REF]; see [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF], [START_REF] Hu | Control Systems with Actuator Saturation[END_REF], [START_REF] Iovine | Nonlinear control of a DC microgrid for the integration of photovoltaic panels[END_REF], [START_REF] Li | Stability and Performance of Control Systems with Actuator Saturation[END_REF], [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF], and [START_REF] Yin | Event-triggered constrained control of positive systems with input saturation[END_REF] for the importance of control bounds. Another limitation of the approach is due to the fact that in general, it does not apply under outputs, when only a part of the state is measured. Moreover, backstepping generally requires a fictitious feedback that is required to be of class C k when the backstepping is applied k times. Uncertainties in the dynamics or in the output may also be an obstacle (but see [START_REF] Zhu | Linear backstepping output feedback control for uncertain linear systems[END_REF] for backstepping for certain linear systems that contain uncertainty).

One recent backstepping technique produces controls that contain delayed state observations, even when the current state is available for measurement. This produces delayed state values in the control even when there are no delays in the original systems, so the delays are called artificial. The artificial delays method was introduced in [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF], and developed in [START_REF] Mazenc | Bounded backstepping through a dynamic extension with 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]. The preceding works do not require C 1 conditions on the fictitious feedback control. See [START_REF] Mazenc | Stabilization of a system that arises in the context of vision based landing of a civil aircraft[END_REF], which used artificial delays to solve a problem involving visual information, where only imprecise measurements of the first backstepping variable are available for use in the control.

The present paper helps to overcome significant remaining challenges in backstepping-based feedback control, by providing globally asymptotically stabilizing feedback controls for a large class of partially linearizable systems; see [START_REF] Isidori | Nonlinear Control Systems[END_REF] for discussions on partially linearizable systems. A novel and valuable feature of the present work is its use of a dynamic extension instead of artificial delays, which produces the following advantages as compared to existing backstepping works. First, our new backstepping controls are bounded in situations where bounded feedbacks can be expected (which contrasts with [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] where the controls obtained for the original systems are not bounded controls), and our method produces closed form expressions for the control bounds. Second, our new method of this work ensures global asymptotic stabilization in cases where the system from which the backstepping begins contains uncertainties, rather than the weaker condition of input-to-state stability with respect to uncertainties in the system that was provided in works such as [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF]. Finally, this work applies in cases where only a part of the state variable is measured and where some parts of the dynamics are not known with accuracy, while still allowing cases where the fictitious feedback is not necessarily C 1 .

This work is organized as follows. Our main theorem is shown in Section 2. Section 3 provides methods to verify that our assumptions from Section 2 are satisfied. An illustration based on a direct drive manipulator and an additional example are given in Section 4, and Section 5 summarizes the value of our paper and our suggestions for future research. This article adds value as compared with our preliminary conference version [START_REF] Mazenc | Backstepping design for output feedback stabilization for a class of uncertain systems using dynamic extension[END_REF] by (i) providing explicit formulas for our control bounds in our theorem, (ii) showing how the converging-inputconverging-state assumption of our theorem can be satisfied with fictitious feedbacks that are not necessarily C 1 , and (iii) analyzing the effects of multiple uncertain constants in the manipulator illustration. These three features were not contained in [START_REF] Mazenc | Backstepping design for output feedback stabilization for a class of uncertain systems using dynamic extension[END_REF], which required the fictitious feedbacks to be C 1 in its sufficient conditions, did not provide formulas for the control bounds, and only allowed one unknown constant in its one illustration.

Our notation will be simplified whenever no confusion would arise from the context. The dimensions of our Euclidean spaces are arbitrary unless we indicate otherwise. The Euclidean norm in R a , and the induced norm of matrices, are both denoted by | • |, and | • | ∞ is the usual sup norm. For each constant J > 0, we let sat J denote the symmetric saturation function that is defined by sat J (x) = max{-J, min{J, x}} for all x ∈ R, where J is called the saturation level. Finally, we use the usual classes of comparison functions KL and K ∞ from [START_REF] Khalil | Nonlinear Systems[END_REF].

Main Theorem

The studied system

We consider the nonlinear time-varying system

             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 + Ω(Y(t, x), G) (1)
where x is valued in R p , the state components G = (g 1 , ..., g n ) are valued in R n , f is locally Lipschitz with respect to (x, g 1 ) and piecewise continuous and locally bounded with respect to t, u is the scalar-valued input, Y : R × R p → R l is a continuous function, each a i,j ∈ R is a constant, a i,i+1 = 0 holds for i = 1, 2...n -1 (with no sign constraint on the a ij 's), and Ω is a locally Lipschitz function that is bounded by a known constant Ω ≥ 0. Systems of this type 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)
which is realistic in practice. We next state our main assumption; see Section 3 for Lyapunov function based sufficient conditions for this assumption to hold.

Assumption 1. There exist a locally Lipschitz scalar valued function ψ that is bounded by a known constant ψ ≥ 0, and a constant k > 0, such that for any continuous function d : [0, +∞) → R that converges exponentially to the origin, all solutions (ξ,

λ 1 , ..., λ n ) : [0, +∞) → R p+n of          ξ = f (t, ξ, λ 1 + d(t)) λ1 = k[-λ 1 + λ 2 ] . . . λn = k[-λ n + ψ(t, Y(t, ξ))] (3)
converge to the origin as t → +∞.

In terms of the matrices

A =         a 1,
        ∈ R n×n (4)
for n > 1 and A = a 1,1 for n = 1 and B = (0, 0, . . . , 1) ∈ 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

Γ(t) = AΓ(t) + B (Γ(t)) (5)
is a globally asymptotically and locally exponentially stable equilibrium.

See [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF] for general conditions under which Assumption 2 is satisfied, in terms of the eigenvalues of A.

Statement of and discussion on theorem

To state our theorem, we first recursively define the following real constants, in terms of the constant k from Assumption 1 and the a i,j 's from the system dynamics (1) and ( 4). We set

c 1 = - n i=1 a n,i l i,1 -kl n,1 , and c j = - n i=j a n,i l i,j -kl n,j + kl n,j-1 if 2 ≤ j ≤ n, where (6a)
l 1,1 = -1; l 2,1 = a1,1+k a1,2 ; l j,j = k aj-1,j l j-1,j-1 if 2 ≤ j ≤ n; (6b)
l j,1 = - j-1 s=1 aj-1,s aj-1,j l s,1 -k aj-1,j l j-1,1 if 3 ≤ j ≤ n; l j,w = 0 if j < w ≤ n; (6c)
and l j,w = -j-1 s=w aj-1,s aj-1,j l s,w -k aj-1,j l j-1,w + k aj-1,j l j-1,w-1 if 3 ≤ j ≤ n and 2 ≤ w ≤ j -1. (6d) Formulas (6b)-(6d) define a lower triangular matrix L = [l i,j ] ∈ R n×n . We are ready to state and prove the following result, where sat was defined in the previous section:

Theorem 1. Let the system (1) satisfy Assumptions 1-2, let be any positive constant, choose the constants l i,j and c j from (6a)-(6d), and choose the n × n matrix L = [l i,j ]. Then all solutions (x, G) : [0, +∞) → R p+n of (1), in closed loop with the dynamic output feedback

u(G, Z, x) = -sat Z n j=1 c j z j + kl n,n ψ(t, Y(t, x)) + (G + LZ) -Ω (Y(t, x), G) ż1 = k[-z 1 + z 2 ] . . . żn-1 = k[-z n-1 + z n ] żn = k[-z n + ψ(t, Y(t, x))] (7)
with Z = (z 1 , ..., z n ) and the saturation level

Z = ψ(1 + )   n j=1 |c j | + k| n,n |   ,
are such that lim t→+∞ (x(t), G(t)) = 0. Also, the feedback u in ( 7) is bounded by the constant Z + + Ω.

Before turning to the proof of Theorem 1, we make several remarks on its motivation and value. 1) Our assumptions do not ensure forward completeness of the closed loop (x, G) dynamics from our theorem, but the theorem is true without such forward completeness because it only applies to solutions that are defined on [0, +∞). In particular, it allows finite time escape phenomena. On the other hand, we can ensure forward completeness of this closed loop system under additional linear growth conditions on f , ψ, and Y in the state.

2) The formula [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF] for the control law does not incorporate the first derivative of ψ(t, Y(t, x(t))), which plays the role of the fictitious control of the classical backstepping approach. Hence, it applies even when ψ is not of class C 1 . While the works [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF][START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] also did not require the fictitious feedback ψ to be C 1 , these previous works use a very different artificial delay approach that did not allow outputs and they required full knowledge of the nonlinear subsystem to ensure uniform global asymptotic stability of the zero equilibrium.

3) One can use changes of variables and a feedback to transform (1) into a system of the form (3) with d = 0 whose exponential stability property is ensured by Assumption 1. However, such a result would not be satisfactory, because the feedback obtained that way may possess inappropriate properties. For instance, in general it would be unbounded. This motivates our alternative approach, based on dynamic extensions. 4) None of our assumptions imply that f has to be known with accuracy. Also, we only require measurements of ψ(t, Y(t, x)) and Ω(Y(t, x), G) in the control, instead of Y itself. This is a weaker requirement than requiring measurements of Y(t, x) or G, because the ψ allows uncertainties in the measurements, and because both ψ and Ω allow cases where only some (instead of all) components of Y are available for measurements, which can occur in autonomous vehicle/robotic guidance, navigation and control applications with unknown states.

Proof of Theorem 1

An important feature of the proof consists of making g 1 (t) -z 1 (t) converge to zero, instead of the classical backstepping approach of making g 1 (t) -ψ(t, Y(t, x(t))) converge to zero. To achieve our goal, we use the constants l i,j from (6b)-(6d) to make the change of coordinates R = G + LZ. We first assume that n > 1. Later we explain what occurs when n = 1.

Induction assumption: We prove the following by induction on i: For all i ∈ {2, ..., n}, we have

         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)
for all t ≥ 0.

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 + a1,1+k a1,2 z 1 -k a1,2 z 2 . (9)
Since our formulas (6b) give

r 2 = g 2 + l 2,1 z 1 + l 2,2 z 2 = g 2 + a1,1+k a1,2 z 1 -k a1,2 z 2 , (10)
we obtain ṙ1 = a 1,1 r 1 + a 1,2 r 2 . Thus the induction assumption is satisfied at the first step.

Step i: Assume that the induction assumption is satisfied at a step i ∈ {2, ..., n -1}. Then since

r j = g j + j m=1 l j,m z m (11)
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 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 m=1 l i,m z m + i m=1 l i,m k[-z m + z m+1 ]. (12)
Thus, since our recursive formulas from (6b)-(6d) (with the choice j = i + 1) also give

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 m=1 l i,m z m + 1 ai,i+1 i m=1 l i,m k[-z m + z m+1 ] (13)
we can multiply both sides of (13) by a i,i+1 and then combine ( 12)-( 13) to obtain ṙi = a i,1 r 1 + a i,2 r 2 + ... + a i,i+1 r i+1 , (

so the induction assumption is satisfied at the step i + 1.

Computing the time derivative of r n = g n + l n,1 z 1 + . . . + l n,n z n , it follows from our formulas for the c j 's in (6a) that the system (1) can be written as

                   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 j=1 c j z j + kl n,n ψ(t, Y(t, x)) + Ω(Y(t, x), G). (15)
If instead n = 1, then we again obtain [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] with only the dynamics for x and r n present. Then the closed-loop system from the statement of our theorem is

                     ẋ = f (t, x, z 1 + r 1 ) Ṙ = AR + B (R) -sat Z n j=1 c j z j + kl n,n ψ(t, Y(t, x)) + n j=1 c j z j + kl n,n ψ(t, Y(t, x)) ż1 = k[-z 1 + z 2 ] . . . żn = k[-z n + ψ(t, Y(t, x))]. (16)
Consider any solution (x, R, z) : [0, +∞) → R p+2n of ( 16).

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

n j=1 c j z j (t) + kl n,n ψ(t, Y(t, x(t))) ≤ Z (17)
is satisfied. Hence, when t ≥ t a , the closed-loop system is

             ẋ = f (t, x, z 1 + r 1 ) Ṙ = AR + B (R) ż1 = k[-z 1 + z 2 ] . . . żn = k[-z n + ψ(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.

Lemma for Checking Assumptions

Assumption 2 is a simple classical assumption that can often be checked easily [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF]; see our illustrations below. However, checking Assumption 1 can be nontrivial. Our work [START_REF] Mazenc | Backstepping design for output feedback stabilization for a class of uncertain systems using dynamic extension[END_REF] provided sufficient conditions for Assumption 1 to hold, under the assumption that the fictitious feedback ψ is C 1 . In this section, we provide an alternative method that allows the fictitious feedback ψ to be Lipschitz but not necessarily C 1 . We consider the system

   ẋ = f (t, x, λ 1 + d(t)) λi = k[-λ i + λ i+1 ], if 1 ≤ i ≤ n -1 λn = k[-λ n + ψ(t, x)] (19)
where the constant k > 0 will be further restricted below, and the continuous function d : [0, +∞) → R exponentially converges to 0. Then ( 19) is a special case of (3). Our lemma will assume the following three assumptions, where the second assumption is a variant of the standard input-to-state stability Lyapunov function decay condition, and where the third one becomes a linear growth condition on f in the significant special case where W is a quadratic (but see Remark 3.1 for a way to relax the uniform global Lipschitzness requirement from Assumption 3 so that we instead only assume a uniform local Lipschitzness condition on ψ): Assumption 3. There is a constant q 1 > 0 such that |ψ(t, a) -ψ(t, b)| ≤ q 1 |a -b| holds for all a and b in R p and all t ≥ 0, and the function f is such that ( 19) is forward complete. Also, ψ(t, 0) = 0 for all t ≥ 0. Assumption 4. There exist a C 1 function V , a uniformly continuous positive definite function W , a constant L > 0, and class K ∞ functions γ 1 and γ 2 such that γ 1 (|x|) ≤ V (t, x) ≤ γ 2 (|x|) hold for all (t, x) ∈ [0, +∞) × R p and such that the time derivative of V along all solutions of ẋ(t) = f (t, x(t), ψ(t, x(t)) + δ(t)) for all choices of the continuous function δ satisfies V (t) ≤ -W (x(t)) + L|δ(t)| 2 for all t ≥ 0.

Assumption 5. There are constants q 2 > 0 and q 3 > 0 such that f from [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] satisfies

|f (t, x, ψ(t, x) + ∆)| 2 ≤ q 2 W (x) + q 3 ∆ 2 (20)
for all t ≥ 0, x ∈ R p , and ∆ ∈ R, where W is from Assumption 4.

We emphasize that the requirement (20) will be checkable in practice, because W will be known from the decay estimate from Assumption 4; see the examples below. We now prove: Lemma 1. Let Assumptions 3-5 hold, and choose any constants β ∈ (0, 1) and k > 0 such that 2e 2 (n -1)β < e -1 and k > max ka , kb , where

ka = q 1 e 2nq 3 β(β -n -1) [e -1 -2e 2 (n -1)β](1 -β) and kb = nq 2 1 q 2 e 3/2 (β -n -1)L (e -1)( √ e -1)(β -1 -1) . (21)
Then all solutions (x, λ) : [0, +∞) → R p+n of ( 19) converge to 0 as t → +∞.

Proof. To make the proof easy to follow, we first summarize the main ideas of the proof, before providing all needed technical details. In the first part of the proof, we prove an input-to-state stability type decay condition on a suitable nonnegative valued function κ of the differences λ i (t) -λ i+1 (t) of the states of the λ subsystem of [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF], with an overshoot depending on the function d 2 and W (x(t)). Then, we use Assumption 4 to ensure that x(t) and so also λ(t) converge to zero as t → +∞, using the decay condition from the first part of the proof. We now provide the needed technical details. Fix any solution (x, λ) : [0, +∞) → R p+n of [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF]. For the rest of the proof, we write ψ(t) to mean ψ(t, x(t)). Set ρ i (t) = λ i (t) -λ i+1 (t) for i = 1, 2, . . . , n -1, ρ n (t) = λ n (t) -ψ(t), and h = 1 k . Then we can apply variation of parameters to the λ n subsystem of [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF] to get

ρ n (t) = e -1 λ n (t -h) + k t t-h e -k(t-m) ψ(m)dm -ψ(t) = e -1 ρ n (t -h) + e -1 [ψ(t -h) -ψ(t)] + k t t-h e k(m-t) [ψ(m) -ψ(t)]dm, (22)
by writing ψ(m) as [ψ(m) -ψ(t)] + ψ(t) in the first integral in [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF]. Here and in the sequel, all equalities and inequalities should be understood to hold for all t ≥ h along our arbitrary chosen solution of [START_REF] Mazenc | New control design for bounded backstepping under input delay[END_REF].

Moreover, for i = 1 to n -1, we can use the preceding argument with ρ n and ψ replaced by ρ i and λ i+1 respectively and the fact that λi+1 (t) = -kρ i+1 (t) to obtain

ρ i (t) = e -1 ρ i (t -h) + e -1 [λ i+1 (t -h) -λ i+1 (t)] + k t t-h e k(m-t) [λ i+1 (m) -λ i+1 (t)]dm = e -1 ρ i (t -h) + ke -1 t t-h ρ i+1 (r)dr + k 2 t t-h e k(m-t) t m ρ i+1 (r)drdm, (23)
where the last equality used the Fundamental Theorem of Calculus to express λ i+1 (t-h)-λ i+1 (t) and λ i+1 (m)λ i+1 (t) as integrals of -kρ i+1 over [t -h, t] and [m, t] respectively. Let i (t) = |ρ i (t)|. Then, for i = 1 to n -1, we can use [START_REF] Pettersen | Underactuated ship tracking control; theory and experiments[END_REF] to obtain

i (t) ≤ e -1 i (t -h) + ke -1 + k 2 0 -h e km dm t t-h i+1 (r)dr = e -1 i (t -h) + k t t-h i+1 (r)dr, ( 24
)
by computing the absolute values of both sides of ( 23), then applying the triangle inequality to the resulting right side, and finally using the bound

| t m ρ i+1 (r)dr| ≤ t t-h i+1 (r)dr for all m ∈ [t -h, t].
It follows from Young's and Jensen's inequalities that, for any constant p > 0, the inequality

2 i (t) ≤ (1 + p) e -2 2 i (t -h) + 1 + 1 p k 2 h t t-h 2 i+1 (r)dr (25) is satisfied if 1 ≤ i ≤ n -1, by using the relation (a + b) 2 ≤ (1 + p)a 2 + (1 + 1/p)b 2
where a and b are the second to last and last terms in [START_REF] Sezkin | Boundary backstepping control of flow-induced vibrations of a membrane at high Mach numbers[END_REF], respectively. Similarly, it follows from applying the upper bound

sup t-h≤m≤t |ψ(m) -ψ(t)| ≤ q 1 t t-h | ẋ(r)|dr (26)
twice to the last terms in [START_REF] Mazenc | New bounded backstepping control designs for time-varying systems under converging input converging state conditions[END_REF] (where (26) used the Lipschitz constant q 1 from Assumption 3), and then using Jensen's inequality in the result, that for any constant p > 0,

2 n (t) ≤ (1 + p) e -2 2 n (t -h) + 1 + 1 p q 2 1 h t t-h | ẋ(r)| 2 dr. ( 27
)
Let us use the simplifying common notation µ i (t) = 2 i (t) for i = 1 to n and set p = e -1. Then

µ i (t) ≤ e -1 µ i (t -h) + 1 + 1 p k 2 h t t-h µ i+1 (r)dr, 1 ≤ i ≤ n -1 and µ n (t) ≤ e -1 µ n (t -h) + 1 + 1 p q 2 1 h t t-h | ẋ(r)| 2 dr. ( 28
) Set ς(t) = n j=1
β n-j µ j (t) and κ(ς t ) = t t-h e -b1(t-r) ς(r)dr , where b 1 = 1 2h .

Then it follows from using [START_REF] Tsinias | Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation[END_REF] to upper bound the terms being summed in [START_REF] Wei | Stability criteria of linear systems with multiple input delays under truncated predictor feedback[END_REF], and then using the inequalities µ j+1 (t) ≤ β j+1-n ς(t) for j = 0 to n -1, that

ς(t) ≤ n-1 j=1 β n-j e -1 µ j (t -h) + 1 + 1 p k 2 h t t-h µ j+1 (r)dr + e -1 µ n (t -h) + 1 + 1 p q 2 1 h t t-h | ẋ(r)| 2 dr ≤ e -1 ς(t -h) + n-1 j=1 β n-j 1 + 1 p k 2 h t t-h β j-n+1 ς(r)dr + 1 + 1 p q 2 1 h t t-h | ẋ(r)| 2 dr ≤ e -1 ς(t -h) + (n -1) ek 2 e-1 hβ t t-h ς(r)dr + e e-1 q 2 1 h t t-h f r, x(r), ψ(r) + n l=1 ρ l (r) + d(r) 2 dr,
where the last inequality followed because n l=1 ρ l (t) = λ 1 (t) -ψ(t).

By applying Assumption 5 to the function ∆

(r) = n =1 ρ (r) + d(r), it follows that ς(t) ≤ e -1 ς(t -h) + (n-1)hβek 2 e-1 t t-h ς(r)dr + q4he e-1 t t-h W (x(r))dr + q 2 1 q3he e-1 t t-h n l=1 ρ l (r) + d(r) 2 dr, (30)
where q 4 = q 2 1 q 2 . Then for any constant 0 ∈ (0, 1), we can use the inequality (a+b) 2 ≤ (1+ 0 )a 2 +(1+(1/ 0 ))b 2 for suitable nonnegative values a and b to find a constant d > 0 (depending on 0 , k, and n) such that

ς(t) ≤ e -1 ς(t -h) + ek 2 (n-1) e-1 hβ t t-h ς(r)dr + eq4h e-1 t t-h W (x(r))dr + (1+ 0)e e-1 q 2 1 q 3 h t t-h n l=1 ρ l (r) 2 dr + d * (t) ≤ e -1 ς(t -h) + ek 2 (n-1) e-1 hβ t t-h ς(r)dr + e e-1 q 4 h t t-h W (x(r))dr + (1+ 0)e e-1 q 5 h t t-h n l=1 µ l (r)dr + d * (t) ≤ e -1 ς(t -h) + (n -1)k 2 β + q 5 (1+ 0 )β 1-β (β -n -1) eh e-1 t t-h ς(r)dr + eq4h e-1 t t-h W (x(r))dr + d * (t),
where q 5 = nq 2 1 q 3 and d * (t) = d t t-h d 2 (r)dr, the second inequality used the consequence (ρ 1 (r) + . . .

+ ρ n (r)) 2 ≤ n(ρ 2 1 (r) + . . . + ρ 2 n (r)) = n(µ 1 (r) + . . . µ n (r)
) of Chebyshev's inequality, and the last inequality used the bound µ l (r) ≤ β l-n ς(r) for all l and h and all r ≥ 0 and then the geometric sum formula. Let q 6 = e e-1 q 4 . Then

ς(t) ≤ e -1 ς(t-h) + e e-1 (n-1)β + (1+ 0)q5β 1-β (β -n -1)h 2 1 h t t-h ς(r)dr + q 6 h t t-h W (x(r))dr + d * (t) ≤ e -1 ς(t-h) + e -1 (1-0) 2h t t-h ς(r)dr + q 6 h t t-h W (x(r))dr + d * (t) , (31)
since k = 1/h, by choosing 0 > 0 small enough and using our assumption that k > ka to check that our choices q 5 = nq 2 1 q 3 and h = 1/k give e e-1 (n -

1)β + q5β 1-β (β -n -1)h 2 < 1 2e . (32)
Since n l=1 ρ l (t) = λ 1 (t)-ψ(t), and since ρ 2 l = µ l for all l, it follows from Assumption 4 (and from reapplying Chebyshev's inequality, as before) that along all solutions of (19), we have

V (t) ≤ -W (x(t)) + L n l=1 ρ l (t) + d(t) 2 ≤ -W (x(t)) + Lq 7 n l=1 µ l (t) + 1 + 1 0 Ld 2 (t), ( 33
)
where q 7 = (1 + 0 )n. Using the lower bounds ς(t) ≥ β n-j µ j (t), it follows from the geometric sum formula that

V (t) ≤ -W (x(t)) + Lq 7 n l=1 β l-n ς(t) + L d 2 (t) = -W (x(t)) + Lq 7 β -n -1 β -1 -1 ς(t) + L d 2 (t) , (34)
where [START_REF] Yoon | Truncated predictor feedback control for active magnetic bearing systems with input delay[END_REF] and our choice b 1 = 1 2h , we deduce that

L = (1 + (1/ 0 )) L. Let b 2 = √ e -1. Then our choice of κ(ς t ) in (29) gives κ(t) = -b 1 κ(ς t ) -b 2 ς(t) + (1 + b 2 )ς(t) -e -b1h ς(t -h). Using
κ(t) ≤ -b 1 κ(ς t ) -b 2 ς(t) + (1 + b 2 ) e -1 ς(t -h) + e -1 (1-0 ) 2h t t-h ς(r)dr + q 6 h t t-h W (x(r))dr -e -b1h ς(t -h) + √ ed * (t) = -b 1 t t-h e -b1(t-r) ς(r)dr -b 2 ς(t) + (1+b2)e -1 (1-0 ) 2h t t-h ς(r)dr +(1 + b 2 )q 6 h t t-h W (x(r))dr + √ ed * (t) ≤ -1 2h t t-h e -1 2h (t-r) ς(r)dr -b 2 ς(t) + 1-0 2h t t-h e -1 2h (t-r) ς(r)dr + q 6 √ eh t t-h W (x(r))dr + √ ed * (t) = -0 2h t t-h e -1 2h (t-r) ς(r)dr -b 2 ς(t) + q 6 √ eh t t-h W (x(r))dr + √ ed * (t), (35)
where we also used the fact that (1 + b 2 )e -1 -e -b1h = 0 and our definitions of κ and b 1 in [START_REF] Wei | Stability criteria of linear systems with multiple input delays under truncated predictor feedback[END_REF].

Let U (x t , ς t ) = V (x(t)) + (1+ 0)Lq7(β -n -1) b2(β -1 -1) κ(ς t ) + (1+ 0 )q8(β -n -1) b2(β -1 -1) h t t-h t m W (x(r))drdm , (36)
where q 8 = Lq 6 q 7 √ e. Then for a suitable constant c > 0, we can combine (34) with (35) to obtain

U (t) ≤ -W (x(t)) -Lq7 0(β -n -1) β -1 -1 ς(t) + q8(1+ 0)(β -n -1) b2(β -1 -1) h 2 W (x(t)) + cd * * (t) , (37)
where d * * = Ld 2 + d * . Consequently, since our choice of kb from our assumption on k = 1 h implies that

h < b2(β -1 -1) q8(β -n -1)(1+ 0 ) (38) when 0 > 0 is small enough, we can find a constant c * > 0 such that U (t) ≤ -c * W (x(t)) -0Lq7(β -n -1) β -1 -1 ς(t) + cd * * (t) . (39)
Also, since d(t) exponentially decays to zero, we have

∞ 0 s s-h d 2 (r)drds < +∞ and ∞ 0 d 2 (r)dr < +∞.
Since W is uniformly continuous, the lemma follows from Barbalat's Lemma, by integrating (39) over [0, +∞) (which implies that x(t) is bounded, so (x(t), λ(t)) is bounded, by the boundedness of ψ(t, x(t))), then concluding that lim t→+∞ x(t) = lim t→+∞ ς(t) = 0, then using the definition of ς to also conclude that lim t→+∞ ρ i (t) = 0 and therefore also lim t→+∞ λ i (t) = 0 for each i (because Assumption 3 now implies that lim t→+∞ ψ(t, x(t)) = 0). Remark 3.1. The first condition in Assumption 3 is the requirement of a uniform global Lipschitz constant q 1 for ψ. We can prove a variant of the preceding lemma if we replace this uniform global Lipschitz requirement by the assumptions that (i) ψ is uniformly locally Lipschitz in the state uniformly in time (meaning, for each constant R > 0, there is a constant qR > 0 such that |ψ(t, a) -ψ(t, b)| ≤ qR |a -b| holds for all a and b that satisfy max{|a|, |b|} ≤ R and all t ≥ 0), (ii) ψ is bounded, and (iii) W in Assumption 4 is proper. Condition (ii) is not at all restrictive, because ψ is assumed to be bounded in our Assumption 1 in the statement of our Theorem 1. With the preceding replacements in Assumptions 3-5, we can now prove a semiglobal (instead of a global) version of Lemma 1 on a sequence of sets whose nested union is our entire state space R n+p , using the following algorithm: Step 1: Given any constant r > 0, use the input-to-state stability Lyapunov function decay condition on V from Assumption 4 (and standard input-to-state stability arguments from [START_REF] Khalil | Nonlinear Systems[END_REF]Chapter 4]) to find a constant R > 0 such that sup t≥0 |x(t)| ≤ R holds for each solution x(t) of ẋ(t) = f (t, x(t), ψ(t, x(t))+δ(t)) whose its initial state satisfies |x(t 0 )| ≤ r and each continuous function δ that is bounded by 2 ψ+|d| ∞ +1, namely, R = β 0 (r, 0)+γ 0 (2 ψ +|d| ∞ +1) where the class KL function β 0 and the class K ∞ function γ 0 are the comparison functions in the input-to-state stability estimate. Step 2: Choose constants k > 0 and β ∈ (0, 1) such that ( 21) is satisfied with q 1 replaced by qR (so k and β will now depend on R). Then the same argument we used to prove Lemma 1 shows that with the choice of k from Step 2, all solutions of (x, λ) : [0, +∞) → R p+n of ( 19) having initial states that satisfy |x(0)| ≤ r converge to 0 as t → +∞, except that instead of considering all times t ≥ h in the proof of Lemma 1, we constrain the time argument t in the proof to satisfy t ≥ max{h, T (|λ(0)|)} where the function T is such that |λ 1 (t)| ≤ ψ + 1 for all t ≥ T (|λ(0)|), and then we apply the input-to-state stability condition from Step 1 to the function δ(t) = -ψ(t, x(t)) + λ 1 (t) + d(t) (which satisfies |δ(t)| ≤ 2 ψ + |d| ∞ + 1 for all t ≥ T (|λ(0)|)). Then we can use the semiglobal version of the lemma to find sufficient conditions for a semiglobal version of our theorem, by allowing the k in Assumption 1, in (6a)-(6d), and in the control [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF] to depend on an upper bound on the norm of the initial state. We leave the remaining details to the reader.

Illustrations

Theorem 1 applies to a broad class of nonlinear systems, without requiring time lagged values of the state in the feedback. In this section, we illustrate these points in two examples. In our first example, we apply Lemma 1 to check our assumptions from Theorem 1. In our second example, we use an alternative method to check the assumptions, to cope with the presence of multiple unknown parameters in the system.

First Illustration

We apply Lemma 1 to the weakly nonminimum phase dynamics

     ẋ(t) = ωΛ(x(t)) + g 1 (t) ġ1 (t) = g 2 (t) ġ2 (t) = u(t) (42)
with the state space R 3 where ω > 0 is a constant and Λ is a globally Lipschitz function having global Lipschitz constant 1, which is not amenable to classical backstepping, because the right side of ẋ(t) will not in general be differentiable. In [START_REF] Mazenc | Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions[END_REF], we used an artificial delay approach to construct a globally asymptotically stabilizing bounded backstepping controller for (42) in the special case where ω = 1 and Λ(x) = |x|/(1 + |x|), which required the current and one time lagged value of the state in the feedback. On the other hand, we can satisfy the requirements from Lemma 1 for (42) with n = 2, a 1,1 = a 2,1 = a 2,2 = 0, a 1,2 = 1, and Ω = 0, using

f (t, x, g 1 ) = ωΛ(x) + g 1 , ψ(x) = -ωΛ(x) -ωsat 1 (x), V (x) = 1 ω x 0 sat 1 ( )d , and W (x) = 1 2 sat 2 1 (x), (43)
since along all solutions of ẋ(t) = f (x(t), ψ(x) + δ(t)) for all t ≥ 0, these choices and the triangle inequality give

V (t) = -sat 2 1 (x) + 1 ω sat 1 (x)δ(t) ≤ -1 2 sat 2 1 (x) + 1 2ω 2 δ 2 (t), (44)
and we can choose L = 1 2ω 2 , q 1 = 2ω, q 2 = 6ω 2 , and q 3 = 3/2 (by the inequality (-ωsat 1 (x) + ∆) 2 ≤ 3ω 2 sat 2 1 (x) + (3/2)∆ 2 ). We can also choose β = e-1 4.1e 2 , and then ka and kb from Lemma 1 with the preceding choices are

ka = 2ωe 6β(β -2 -1) (e -1 -2e 2 β)(1 -β) and kb = 2ω 6e 3/2 (β -2 -1) (e -1)( √ e -1)(β -1 -1) (45)
so Assumption 1 holds for any k > max{ ka , kb } = 61.2729ω. Also, Assumption 2 holds by well known nested saturation controls for chains of integrators. Hence, the desired feedback control is provided by Theorem 1.

Second Illustration

We consider the dynamics

   ẋ1 = x 2 ẋ2 = p 0 (g 1 -p 1 sin(x 1 ) -p 2 x 2 ) ġ1 = u -p 3 arctan(x 2 ) + a 1,1 g 1 (46)
with the output Y = (x, g 1 ) and the R 2 -valued state components x = (x 1 , x 2 ), where a 1,1 < 0 and p i > 0 for i = 0 to 3 are constants. This includes the dynamics of a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor from [START_REF] Dawson | Integrator backstepping control of a brush DC motor turning a robotic load[END_REF], which are (46) in the special case where p 0 = 1 (after a change of coordinates which removes a constant). Assumption 2 is satisfied with = 0 and n = 1 (because in this case, (5) with = 0 is a linear system that is exponentially stable to 0), and we choose Ω(x, g 1 ) = -p 3 arctan(x 2 ). To illustrate how Theorem 1 covers cases where the nonlinear part of the dynamics may not be known, let us assume that a 1,1 , p 1 , and p 3 are known, but that p 0 and p 2 are not known (so we may use a 1,1 , p 1 , and p 3 in the control, but we cannot use p 0 or p 2 in the control). The uncertainty of p 0 (resp., p 2 ) can represent an uncertain mass (resp., viscous friction coefficient) in the manipulator dynamics. However, we assume that

Let us choose V (ξ) = 1 + ξ 2 1 -1 + 1 2 ξ 2 2 + p1 4(1+p1) ξ 1 ξ 2 and s 1 = λ 1 -p 1 sin(ξ 1 ) + ξ 1 .

(51)

In the new variables (ξ, s 1 ), the system (50) becomes ξ1 = ξ 2 , ξ2 = -p 2 ξ 2 -ξ 1 + s 1 + d(t) + (p 0 -1)M(t), ṡ1 = -ks 1 + -p 1 cos(ξ 1 ) + In Fig. 2, we plot MATLAB simulations for the system (46) in closed loop with the control provided by our theorem, with = 0.05, p 0 = p 1 = p 3 = 1, p 2 = 1.5, and a 1,1 = -1. The figure shows the good performance of our control, and therefore helps illustrate the value of our theorem in a specific case of the dynamics (46). Moreover, with the choice p 1 = 1, our conditions (47) imply that our control is also valid for any constants p 2 ∈ (1, 2) and p 0 ∈ (0.95, 1.05), which means that the viscous friction coefficient (resp., mass) measurement can tolerate a 50% (resp., 5%) error. This illustrates the robustness of our control design to uncertainties.

Conclusions

We provided a new backstepping approach using a finite dimensional dynamic extension, which is motivated by the ubiquity of engineering applications that lead to the required cascade forms. Advantages of our approach include the facts that we do not require the fictitious feedback to be C 1 , and the fact that we can satisfy input bounds without using artificial delays. We hope to develop local versions for systems that are only locally asymptotically stabilizable, and to allow measurement delays [START_REF] Wei | Stability criteria of linear systems with multiple input delays under truncated predictor feedback[END_REF][START_REF] Yoon | Truncated predictor feedback control for active magnetic bearing systems with input delay[END_REF]. We will also investigate families of systems that can be stabilized by applying Theorem 1 repeatedly, and possible extensions to PDEs.

Figure 1 :

 1 Figure 1: Feedback Control in First Illustration

2p1 1+p1 < p 2 2 1 2 2

 222 < 1+p1 p1 and |p 0 -1| < min{p 1 , p2 }, where p1 = 3(1+p1)p1 2p +8p1(1+p1)+16(1+p1) 2 and p2 = p2 4p +8p2+8 (47) and we choose any constant k > 0 such that 1+p1 + 2|p 0 -1| and k > 16 3p2 (p 1 + 1) 2 . (48) We will show that (47)-(48) ensure that Assumption 1 is satisfied with n = 1 for any k that satisfies (48). Consider ψ(ξ) = p 1 sin(ξ 1 ) -ξ 1 , where ξ 1 = ξ1 √ function d : [0, +∞) → R that exponentially converges to 0 and ξ1 = ξ 2 , ξ2 = p 0 -p 1 sin(ξ 1 ) -p 2 ξ 2 + λ 1 + d(t) , λ1 = k [-λ 1 + p 1 sin(ξ 1 ) -ξ 1 ] .

  u(Z, x) = -sat Z ((a 1,1 + k)z 1 -kψ(x 1 )) + p 3 arctan(x 2 ), ż1 = k [-z 1 + p 1 sin(x 1 ) -x 1 ] (59)with Z = (1 + )(|a 1,1 + k| + k)(p 1 + 1) and the choice (49) of ψ, since our formulas (6a)-(6b) give c 1 = a 1,1 + k.

Figure 2 :

 2 Figure 2: Feedback Control in Second Illustration

  = -p 2 ξ 2 -ξ 1 + s 1 + d(t) and where we omit the time argument of the state variables to simplify the notation. Also, the second inequality in (47) and the triangle inequality combine to give (57) over [0, +∞) and applying Barbalat's Lemma to the function in curly braces in (57) that Assumption 1 is satisfied (because d 2 (t) exponentially converges to 0 as t → +∞). Hence, Theorem 1 applies to (46) and provides the dynamic feedback

	follows from integrating
												1 1 ) (1+ξ 2 √	1 1+ξ 2	ξ 2 , (52)
	where M(t) -p 2	p1 4(1+p1)	ξ1ξ2 √ 1+ξ 2 1	≤	p1 4(1+p1)	1 2	ξ 2 1 1+ξ 2 1	+ 1 2 p 2 2 ξ 2 2	≤	p1 8(1+p1)	ξ 2 1 1+ξ 2 1	+ p2 8 ξ 2 2 .		(53)

Therefore, for any constant 0 ∈ (0, 1), and with the choice M (t) = (p 0 -1) ∂V (ξ) ∂ξ2 (t)M(t), we obtain

4(1+p1)

along all solutions of (52) for all t ≥ 0, where the first inequality in (54) used the first inequality in (47) and (53), and the second inequality in (54) used the relations

followed by the inequality (s

it follows from (53) with p 2 replaced by 1 that the choice of V in (51) is proper and positive definite.

along all solutions of (50) for all t ≥ 0. Since (p 1 + 1)|s 1 ξ 2 | ≤ 3k 8 s 2 1 + 2 3k (p 1 + 1) 2 ξ 2 2 , it follows from our conditions on |p 0 -1| and k from (47)-(48) that if we choose 0 ∈ (0, 1) close enough to 0, then we can find a constant c 0 > 0 such that

along all solutions of (50) for all t ≥ 0, because the second inequality in (48) implies that 2 3k (p 1 + 1) 2 ≤ p2 8 , and because our bounds p1 and p2 on |p 0 -1| from (47) imply that there are constants c a ∈ (0, p 2 /8) and c b ∈ (0, 3p 1 /(32(1 + p 1 )) such that

for all t ≥ 0, by repeated applications of the triangle inequality; the first set of terms in curly braces in (58) were obtained from using the triangle inequality to upper bound |ξ 2 ξ 1 |, the second set of terms in curly braces in (58) were used to upper bound |ξ 2 (s 1 + d(t))|, and similarly for the other terms in curly braces in (58). It


David Dureisseix

Optimal iso-area 4 x 4 checkerboard

come

Optimal iso-area 4 x 4 checkerboard

Introduction

Color-change has for main goal to make a pattern appear on a folded model, using a sheet with one color per face. Amazing and famous designs are John Montroll's zebra [START_REF] Montroll | African Animals in Origami[END_REF], Satoshi Kamiya's tiger [START_REF] Sony | Tiger design by Satoshi Kamiya[END_REF], Xiaoxian Huang's A-diamond [START_REF] Huang | Poker cards: A Diamond[END_REF], Tanaka Masashi's dice [START_REF] Masashi | Crease pattern challenge: dice[END_REF]… and many more.

Geometric patterns, usually flat folded, leads to many challenges, puzzles… Checkered patterns with many color shifts, are among the more demanding ones, see e.g. [START_REF] Grabarchuk | The Origami Checkerboard Puzzle[END_REF]. Starting from a square sheet of paper is a strong constraint (folding a strip is much easier); therefore, chessboard patterns have been studied for several years [START_REF] Budai | Chequered patterns[END_REF], [START_REF] Larousserie | Echiquiers et damiers[END_REF], [9].

If the challenge is not hard enough, additional constraints can be added, such as optimality: minimize the size of the unfolded sheet, for a given folded n x n checkered model. For instance, the n = 4 case designed by Max Hulme [START_REF] Hulme | Chess sets[END_REF], Figure 1, starts with a 8 x 8 square. You may try to improve it… The classical chessboard (n = 8) has numerous solutions, and up to now, optimal versions use a 32 x 32 sheet [11], [START_REF] Hollebeke | Echiquier[END_REF], [START_REF] Chen | OUSA convention[END_REF].

What happens when increasing n? To estimate the required size of the paper square sheet, one may assume that a color change along a line segment consumes the same amount of paper edge length of the initial sheet perimeter [11], [START_REF] Demaine | Geometric Folding Algorithms: Linkages, Origami, Polyhedra[END_REF]. Nevertheless, 2009 produced a "coup de théâtre" [START_REF] Demaine | Folding a better checkerboard[END_REF], exemplifying that this assumption was too strong and can be relaxed. A smaller sheet can be used for a better solution… at least for n = 16 and beyond… good luck to fold it! (Robert Lang did fold the 8 x 8 version and said: « Wow, this was not one of the easiest things I've done! »)

Can we increase complexity? Of course! Additional constraints can be considered, as: seamless design (each small square of the board should be made with a single continuous surface) [5], [START_REF] Demaine | Folding a better checkerboard[END_REF], pixel-matrix (can we change the color of each board square independently with a single fold?) [START_REF] Demaine | Folding a better checkerboard[END_REF], [START_REF] Dureisseix | Color change and pixel-matrix challenge[END_REF]… Herein, we focus on another challenge: design an iso-area (double faced) model: the pattern is the same on each face of the flat-folded model. Indeed, other one already thought about this! For instance, Jeremy Shafer designed a 4 x 4 pattern from a 18 x 18 square sheet (doubling the paper edge size with respect to Max Hulme) [START_REF] Shafer | Checkerboard -Iso Area 4X4[END_REF], [START_REF] Shafer | Iso area 4x4 checkerboard. Origami Ooh La La![END_REF].

Now, can we do it with a smaller square sheet?

Edge diagram

The edge diagram is the trace on the flat-folded model of the edge of the initial sheet of paper. For small checkerboards, it happens to be an interesting tool for design, so let's use it here (though it relies on the assumption of using the edge for the color-change which is too strong as said before…). Then, imagine a path on a regular grid that constitutes a continuous line, and a closed loop, that should separate the squares on the board. This line should do it on one face (plain line) and on the reverse face (dotted line), Figure 2.

We therefore may well improve the design (with respect to optimality) for attempting using a 12 x 12 initial sheet: some segments on the perimeter of the edge diagram are no more mandatory for having a continuous line. Now, how to design the double-faced model?

Iso-area technique

Geometric origami often relies on symmetries. Here also, symmetries are of interest, especially the one of the iso-area technique [START_REF] Kasahara | Origami for the Connoisseur[END_REF], [START_REF] Maekawa | The definition of iso-area folding[END_REF]. Looking at the crease pattern, if its mountain and valley folds are permuted (i.e. if the paper is reversed recto-verso, and a mirror symmetry applied) the crease pattern is unchanged, eventually up to an isometry (rotation, translation or mirror symmetry). This technique has been used for the n = 2 case [START_REF] Konjevod | Integer programming models for flat origami[END_REF], and also, for instance, for several tessellations [START_REF] Lang | Twists, Tilings and Tessellations[END_REF]. As an illustration, Figure 3 compares the windmill base and its iso-area version. Now we need to fold the model, prescribing the edge of the paper to follow the previous edge diagram.

Proposed solution

Let start with a 12 x 12 square of a dual-colored paper, not too thick, Figure 5.

Step 1: mark the creases. Tessellation addicts would probably pre-crease all the sheet for increasing precision. To get all the 12 x 12 square boards and their diagonals, this would require nevertheless 112 folds with a cumulative length close to 4.3 m for a 12 cm side sheet… A typical difficulty of the iso-area technique concerns the strongly coupled movements during folding: it is therefore difficult to split the movement in several successive steps. Usually, it leads to a collapse of the crease pattern in a single step. Note that, though the final model is flat-folded, this step is often not feasible while respecting the rigid folding issue (keeping paper rigid between the folds) [START_REF] Lang | Twists, Tilings and Tessellations[END_REF].

Steps 2 and 3: performing the X pattern with iso-area technique is not so easy (though feasible with the crease pattern in Figure 4), so we cheat a bit by breaking the symmetry for sake of simplicity.

Step 4: This one is really an iso-area fold. The central square rotates by 90 degrees during the collapse [START_REF] Thomas | Rigid foldability of the augmented square twist[END_REF].

Step 5: the most difficult task has already been done. The remaining steps are finishing touches.

Want more?

Beyond the n = 4 case, the obvious question is: what happens when increasing n? The simplest cases may well be when n is even, but anyway, the model complexity increases rapidly with n [START_REF] Dureisseix | Color change and pixel-matrix challenge[END_REF]. Who wants to set a record? Figure 5: folding the optimal iso-area 4 x 4 checkerboard.

Figure 1 :

 1 Figure 1: Max Hulme's design [10]. Top: edge graph, edge diagram and folded model. Bottom: crease pattern.

Figure 2 :

 2 Figure 2: proposed model. Edge graph, edge diagram (recto: plain lines, verso: dotted lines) and folded model.

Figure 3 :

 3 Figure 3: top: classical windmill base. Bottom: its iso-area version.

Figure 4 :

 4 Figure 4: crease pattern of an iso-area X pattern.