On the preservation of observability under sampling

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Systems & Control Letters 52 (2004) 7 – 15

www.elsevier.com/locate/sysconle

On the preservation of observability under sampling S. Ammar, J.-C. Vivalda∗ Inria-Lorraine (projet Conge), ISGMP Bat. A, Universite de Metz-Ile du Saulcy, 57045 Metz Cedex 01, France Received 2 May 2002; received in revised form 11 July 2003; accepted 22 August 2003

Abstract In this paper, we investigate the problem of the preservation of observability under sampling. c 2003 Elsevier B.V. All rights reserved.  Keywords: Nonlinear systems; Discretization; Observability

1. Introduction When a system is regulated by a digital computer, control decisions are often restricted to be taken at 7xed times 0; ; 2; : : : : For a continuous time system, the resulting situation can be modelled through the constraint that the inputs applied are constant on intervals [0; ), [; 2); : : : : Moreover, if the system is given with a measurement map, the state is (partially) measured only at the 7xed times 0; ; 2; : : : : The goal of this paper is to investigate the problem of the preservation of the observability for such systems; notice that an analogous problem: the preservation of the controllability is studied in [2]. Before stating our main theorem, we will give some notations and de7nitions. We consider a continuous-time system x˙ = f(x; u);

y = h(x);

(1)

de7ned on a di@erentiable manifold M , the control u takes values in a subset U of Rm and for every u ∈ U, the vector 7eld f(·; u) is supposed to be smooth. We consider a sampling time  ¿ 0, if we denote by ’(t; x0 ; u(·)) the solution of (1) with control t → u(t) and initial condition x0 , the -sampled system of (1) is xk+1 = ’(; xk ; uk (·));

yk = h(xk );

(2)

where uk (·) is the control constant on [0; ] and equal to u(k). The solution of (1) issued from x0 with control t → u(t) is de7ned on a semi-open interval that we will denote by [0; e(x0 ; u)[ (0 ¡ e(x0 ; u) 6 + ∞). There exist many notions of observability (see for example [3]), we will work with this basic de7nition: ∗

Corresponding author. Tel.: +33-03-87-54-72-73; fax: +33-03-87-54-72-77. E-mail address: [email protected] (J.-C. Vivalda).

c 2003 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter
doi:10.1016/j.sysconle.2003.08.008

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S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

Denition 1. We say that system (1) is observable for all input if for every triple (u; x0 ; xI0 ), where x0 = xI0 and t → u(t) is an admissible input, x0 and xI0 are initial conditions, there exists t belonging to the interval [0; min(e(x0 ; u); e(xI0 ; u))[ such that h(’(t; x0 ; u(·))) = h(’(t; xI0 ; u(·))). For discrete-time systems, we adopt this de7nition: Denition 2. We say that system (2) is observable for all input if for every triple (u ; x0 ; xI0 ), where u is an admissible input, x0 and xI0 are initial conditions, the set of indices k such that xk and xIk are de7ned and h(xk ) = h(xIk ) is nonempty. Of course, not all the discrete systems are observable, we refer the reader to our fourth example (Eq. (11)): if we consider the system with only the function h1 as an observation function, the points (0; 0; 1) and (0; 0; −1) are undistinguishable. We recall the notion of distinguishability. Denition 3. We will say that the points x0 and xI0 (or that the pair (x0 ; xI0 )) are distinguishable by system (1) if there exists a control u and a time t0 satisfying h(’(t0 ; x0 ; u(·))) = h(’(t0 ; xI0 ; u(·))). The de7nition is analogous for the discrete-time system (2). Assuming the observability of system (1), is it true that system (2) is also observable? (at least for sampling times  small enough). The answer is yes for linear system (see e.g. [4]) but, in general, the assumption of observability is not suKcient to guarantee the preservation of observability as we will see later. In [3], the preservation of observability under sampling is also studied, we will make a comparison between this result and the one of this paper. In [3], the author has introduced a particular notion of local observability for continuous- and discrete-time systems. The 7rst result is concerned with the local observability for continuous-time system whereas the second result states the equivalence between local observability for the continuous-time system and its sampled. This equivalence is realized under the hypothesis that the ideal (of the Lie algebra generated by all the f(·; u)) generated by the vector 7elds f(·; u) − f(·; v) is of full rank at the point of interest; these Lie algebras take place in the theory of controllability. As far as we are concerned with sampled observability, notice that the sampled-system is locally observable for all sampling time 0 ¡  ¡ , the parameter  depending on the neighborhood of the point of interest. In this paper, we do not investigate the observability property in relation with accessibility property (through the consideration of Lie algebras generated by the vector 7elds f(·; u)) and we are interested by the notion of global observability. So we do not need condition on the rank of Lie algebras but we make use of the notion of in7nitesimal observability which is recalled below. Moreover, we will prove that the observability of the sampled system holds for all sampling time small enough. We need the notion of in7nitesimal observability which was 7rst studied by Gauthier and Kupka [1] and we recall that below. Suppose that the set of admissible controls is L∞ (U), the set of all measurable essentially bounded U-valued functions (U ⊂ Rm ), the space of our output functions will be L(Rp ), the set of all measurable Rp -valued functions. The “lift of system (1) on TM ” is de7ned as follows: the mapping f : M × U → TM induces the tangent mapping df : TM ×U → T (TM ), then if ! denotes the canonical involution of T (TM ), ! ◦df de7nes a parametrized vector 7eld on TM . If  : TM → M denotes the canonical projection, let t → (t; 0 ; u(·)) be a trajectory of this vector 7eld associated with the control u and with initial condition 0 , then t → ( (t; 0 ; u(·))) is the trajectory of (1) starting from x0 = (0 ). Moreover, the function h : M → Rp has a di@erential dh : TM → Rp . The lift of (1) is then the system: ˙ = ! ◦ df(; u);

 = dh( ◦ (t; 0 ; u))(t);

where  denotes the canonical projection from TM into M .

(3)

S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

9

We have the following de7nition for the in7nitesimal observability: Denition 4. Given (u; x) ∈ L∞ (U) × M , we consider the application Pu; x between Tx M and L(Rp ) which associates to 0 ∈ Tx M the function t → dh(’(t; x; u(·)))(t) ((t) being the solution of (3) with initial condition 0 ). System (1) is called in7nitesimally observable at (u; x) if the linear mapping Pu; x is injective and uniformly in7nitesimally observable if it is in7nitesimally observable at all (u; x) ∈ L∞ (U) × M . 2. Main result The systems under consideration are of the form (1) where the state x belongs to a compact smooth di@erentiable manifold M of dimension d and the inputs u take value in a compact subset U of Rm . We suppose that the u-parametrized vector 7elds f(·; u) are smooth; the space of control functions will be the space L∞ (U) of all essentially bounded applications from U to Rp . Under these conditions, remark that the solutions of system (1) are de7ned for all t ¿ 0 and so the solutions of (4) exist for all k. Given a positive number , we denote by u , a piecewise constant control which takes constant value on intervals [k; (k + 1)[ and we let u (t) = uk if t ∈ [k; (k + 1)[. The -sampled system of (1) (with time sample ) is the discrete-time system: xk+1 = ’(; xk ; uk (·));

yk+1 = h(xk+1 );

uk (·)

(4) uk (t)

uk

’(; xk ; uk (·))

is the application de7ned on [0; ] by = and is the solution of (1) with where initial condition xk , control uk (·) at time . We will say that u (·) is M D-bounded if there exists a di@erentiable application u such that u (t) = u(k) for t ∈ [k; (k + 1)[ and the norm of the derivative of u is bounded by M (so u(t + h) − u(t) 6 Mh for all h). Our main result is the following: Theorem 1. Assume that system (1) is observable for every input u(·) and uniformly in8nitesimally observable, then for all M ¿ 0, there exists a 0 ¿ 0 such that the -sampled system of (1) is observable (in the sense of De8nition 1) for all  6 0 and all M D-bounded input u . Before proving this result, we want to discuss the assumptions of this theorem in order to show that they are all essential. First, we consider the following system (without control) in R2 : x˙1 (t) = −x(t)2 x2 (t);

x˙2 (t) = x(t)2 x1 (t);

y(t) = x1 (t);

(5)

where x denotes the point of components (x1 ; x2 ) and  the euclidean norm. The trajectories of this system are circles centered at the origin; to be more precise, the solutions are given by x1 (t) = x1 (0) cos(x(0)2 t) − x2 (0) sin(x(0)2 t);

x2 (t) = x1 (0) sin(x(0)2 t) + x2 (0) cos(x(0)2 t):

This system is obviously observable. Given  ¿ 0, consider now its -sampled: x1 ((k + 1)) = x1 (k) cos(x(k)2 ) − x2 (k) sin(x(k)2 ); x2 ((k + 1)) = x1 (k) sin(x(k)2 ) + x2 (k) cos(x(k)2 ); y(k) = x1 (k)

(6)
0 and the initial conditions x =(0; R) and xI =(0; −R). If we choose R= 2=, we have obviously h(x )=h(xI0 ) and x(k) = x0 , x(k) I = xI0 for all k ¿ 0; this proves that it is not possible to 7nd  such that system (6) is observable. 0

0

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S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

The same, slightly modi7ed, idea provides the same result: consider the following system given on S 1 , the circle of radius 1 and center (0; 0) in R2 : x˙1 = −

1 x2 ; 1−u

x˙2 =

1 x1 ; 1−u

h(x) = x1 ;

(7)

where the space U is equal to [0; 1[. We can give explicitly the solutions of this system: x1 (t) = x1 (0) cos U (t) − x2 (0) sin U (t); x2 (t) = x2 (0) cos U (t) + x1 (0) sin U (t); t with U (t) = 0 ds=(1 − u(s)); so the -sampled of system (7) is x1 ((k + 1)) = x1 (k) cos

  − x2 (k) sin ; 1 − uk 1 − uk

x2 ((k + 1)) = x2 (k) cos

  + x1 (k) sin ;  1 − uk 1 − uk

y(k) = x1 (k)

(8)

take now uk = ( − )= for all k; all pairs of points (x10 ; x20 ) and (x10 ; −x20 ) are indistinguishable (notice also that u is M D-bounded for all M ). These two examples show that it seems reasonable to work in the frame of compact manifold with a compact set U. We now want to show that it is not possible to relax the hypothesis of in7nitesimal observability. Consider the following system given on the sphere S 2 : x˙1 = −(x3 + 1)x2 ;

x˙2 = (x3 + 1)x1 ;

x˙3 = 0; y1 = h1 (x);

y2 = h2 (x);

(9)

where the observation function h1 is equal to x3 and h2 is de7ned by      x2   exp − 1 if (x1 ; x2 ) = (0; 0); 1 2 (x1 ) x12 + x22 (x12 + x22 )2 h2 (x) =   0 if (x1 ; x2 ) = (0; 0) with 1 a smooth function with compact support [ − 1; 1], positive on ] − 1; 1[ and on ] − ∞; 0], positive on ]0; +∞[. If we use spherical coordinates: x1 = cos # cos ’;

x2 = cos # sin ’;

2

a smooth function null

x3 = sin #;

we can easily see that, on the circle of altitude x3 = sin #0 (#0 ∈ ] − =2; =2[), the function h2 is zero excepted when −arcsin(cos3 #0 ) 6 ’ 6 arcsin(cos3 #0 ). It is not diKcult to prove that system (9) is observable: the solution of this system is easily expressed as x1 (t) = x1 (0) cos(x3 (0) + 1)t − x2 (0) sin(x3 (0) + 1)t; x2 (t) = x2 (0) cos(x3 (0) + 1)t + x1 (0) sin(x3 (0) + 1)t;

x3 (t) = x3 (0):

So if we take two trajectories x(t) and x(t), I issued from initial conditions x(0) and x(0), I such that h(x(t)) = h(x(t)), I we see that x3 (0) = xI3 (0), then if we had (x1 (0); x2 (0)) = (xI1 (0); xI2 (0)), it would be possible to 7nd a time t0 ¿ 0 such that h2 (x(t0 )) = 0 while h2 (x(t I 0 )) = 0. We will see now that the -sampled of (9) is not observable. A sampling time  ¿ 0 being given, the -sampled of system (9) is given by x1 ((k + 1)) = x1 (k) cos(x3 (k) + 1) − x2 (k) sin(x3 (k) + 1); x2 ((k + 1)) = x2 (k) cos(x3 (k) + 1) + x1 (k) sin(x3 (k) + 1);

S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

x3 ((k + 1)) = x3 (k);

y1 ((k + 1)) = h1 (x((k + 1)));

y2 ((k + 1)) = h2 (x((k + 1))):

11

(10)

We will prove that there exist two distinct initial conditions (x10 ; x20 ; x30 ) and (xI01 ; xI02 ; x30 ) with same altitude x30 such that h2 (x(k)) = h2 (x(k)) I = 0 for all k ¿ 0. From number theory, we know that there exists a rational number r = a=b such that 0 ¡ 1=r − = ¡ 1=a2 ; moreover, the numerator of r can be chosen arbitrarily large. Take such a number r and let x30 such that (x30 + 1) = 2r, with this choice of x30 , the trajectory issued from a point x0 with altitude x30 is constituted by points on the circle of altitude x3 = x30 on S 2 . The maximum of the angular distance between two consecutive points on this trajectory is at least equal to 2=b, so if we can 7nd a and b such that 2=b ¿ 2 arcsin(cos3 #0 ) (where #0 ∈ ] − =2; =2[) we will be able to 7nd a point (x10 ; x20 ; x30 ) such that h2 (x(k)) = 0 for all k ¿ 0 (indeed, it will be possible to choose this point such that none of the points x0 (k) comes in the angular sector −arcsin(cos3 #0 ) 6 ’ 6 arcsin(cos3 #0 )) and if we take xI01 = x1 (), xI02 = x2 () and xI03 = x30 , we will also have h2 (x(k)) I = 0 for all k. Notice that we have the following equivalences: 

3

3  arcsin(cos3 #) ∼ cos3 # ∼ −# = − arcsin x3 ∼ (1 − x32 )3=2 ∼ 23=2 (1 − x3 )3=2 : x3 =1 x3 =1 #==2 #==2 2 2 So, if a is chosen large enough, we will have arcsin(cos3 #0 ) 6 2 · 23=2 (1 − x30 )3=2 , now   1  2r r 2 0 = 2r − 6 2 = 1 − x3 = 2 − 2  r  a ab therefore b arcsin(cos3 #0 ) 6

24 a3=2 b1=2

;

which proves that 2 arcsin(cos3 #0 ) can be made less than 2=b if a is large enough. Now we will prove that we cannot relax the hypothesis on the D-boundedness of the control. Consider the following system given in S 2 : x˙1 = −u(2 + x3 )x2 ;

x˙2 = u(2 + x3 )x1 ; d x˙3 = 0;

h1 (x) = x1 ;

(11)

the space U being equal to {−1; 1}. As far as we are concerned by the observability property, we can see that all pairs of points are distinguishable, excepted the pair constituted by the points (0; 0; 1) and (0; 0; −1). So, if we add the (smooth) observation function h2 such that  −1 if x3 ¡ − 1=4; h2 (x) = 0 if x3 ¿ 0; system (11) becomes observable. Besides, let us denote by Lf h1 the Lie derivative of function h1 with respect to the vector 7eld f de7ned above and t → ’(t; x; u(·)) the solution of (11) starting from x with control t → u(t). An easy computation shows us that the di@erential of the mapping x → (h1 (’(t; x; u(·))); (Lf h1 )(’(t; x; u(·)))) between S 2 and R2 is one-to-one (excepted, perhaps, for a 7nite number of t), so system (11) is uniformly in7nitesimally observable. We will now show that if the sampling time  is small enough, the sampled system is not observable (in fact it would be possible to show that the sampled system is unobservable whatever the sampling time).

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Assume that 0 ¡  ¡ =4 and consider the function f : x3 →

1 − x32 1 + cot 2 (2 + x3 )

an easy computation shows that f (0) ¿ 0 and f (1) ¡ 0 which proves the existence of two numbers, x30 and xI03 , such that 0 ¡ x30 ¡ xI03 ¡ 1 and f(x30 ) = f(xI03 ). Consider now the two points A and AI of coordinates (x10 ; x20 ; x30 ) and (x10 ; xI02 ; xI03 ), respectively, with x10 = f(x30 ) (= f(xI03 )); x20 = x10 cot(2 + x30 ); xI02 = x10 cot(2 + xI03 ): I Take the control u ≡ 1, we have x1 ()= Clearly these two points belong to S 2 and we have h(A) = h(A). I xI1 () = 0 and so h(A) = h(A) = (0; 0); now on the interval ]; 2], we take the control u ≡ −1 and we return I by switching from control u ≡ 1 to u ≡ −1, we see that we will have to our starting points A and A, I I is indistinguishable. h(A(k)) = h(A(k)) for all k ¿ 0, so the pair (A; A)

3. Proof of the main result Our strategy to prove Theorem 1 is quite simple: 7rst we will prove that if x0 and xI0 are two di@erent points in X , there exist two neighborhoods Vx0 and VxI0 of x0 and xI0 respectively and a positive number 0 such that every point of Vx0 is distinguishable from every point of VxI0 for all -sampled system (with  ¡ 0 ) and all M D-bounded control. Then we will prove that, for all x0 , there exists a neighborhood Wx0 of x0 and 0 ¿ 0 such that for all  ¡ 0 and all M D-bounded control, every pair of (di@erent) points of Wx0 is distinguishable

for the -sampled system. We will conclude by using an argument of compactness. The space t∈R+ U endowed with the product topology is compact and so every sequence (fn )n¿0 on this space admits (at least) one limit point but, generally, we cannot extract a subsequence which converges to this limit point; nevertheless this is true if the fn ’s are M D-bounded. Lemma 1. Let (un )n¿0 be a sequence of M D-bounded piecewise constant applications de8ned on R+ with range in a compact subset U of Rm . Assume that limn→∞ n = 0, then there exists a subsequence (unk )k¿0 of (un )n¿0 which converges to a continuous function u.

Proof. Let u be a limit point of the sequence (un (·))n¿0 in the compact space t∈R+ U. For every n, recall that there exists a derivable application fn with derivative bounded by M such that un (t)=fn ([t=n ]n ); by investigating the possible values for the di@erence [(t+h)=n ]−[t=n ] ([x] denotes the integer part of x) and using the uniform boundedness of the derivatives of fn , we see easily that un (t+h)−un (t) 6 2M max(|h|; n ). Set t and h and let , ¿ 0, there exists n such that n ¡ |h|, un (t + h) − u(t + h) ¡ ,=2 and un (t) − u(t) ¡ ,=2, so we have: u(t + h) − u(t) 6 u(t + h) − un (t + h) + un (t + h) − un (t) + un (t) − u(t) 6 ,=2 + 2M max(n ; |h|) + ,=2 = , + 2M |h|: This inequality being true for all , ¿ 0, we have u(t + h) − u(t) 6 2M |h| for all t, h which proves that u is continuous. Take now a sequence of positive number (,n )n¿1 converging to 0; let -1 ¿ 0 be such that 2M-1 ¡ ,1 =3 and p = [1=-1 ]. Consider the numbers 0; -1 ; 2-1 ; : : : ; p-1 , the inequalities |un (k-1 ) − u(k-1 )| ¡ ,1 =3;

k = 0; : : : ; p

(12)

S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

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are true for an in7nity of integers n; choose n1 such that inequalities (12) are satis7ed and such that 2Mn1 ¡ ,1 =3. Now if x ∈ [0; 1], we can write x = k-1 + h with 0 6 h ¡ -1 and 0 6 k ¡ p and we have un1 (x) − u(x) 6 un1 (k-1 + h) − un1 (k-1 ) + un1 (k-1 ) − u(k-1 ) + u(k-1 ) − u(k-1 + h) 6 2M max(n1 ; h) + ,1 =3 + 2Mh 6 ,1 : Suppose inductively that we have p integers n1 ¡ · · · ¡ np such that for all x ∈ [0; k] (1 6 k 6 p) unk (x) − u(x) ¡ ,k , then, by reasoning as above, we can 7nd np+1 ¿ np such that unp+1 (x) − u(x) ¡ ,p+1 for all x ∈ [0; p + 1]. Obviously, the subsequence (unk )k¿1 converges to u. Lemma 2. We assume that system (1) is observable for all input u. Let M ¿ 0 and (x0 ; xI0 ) a pair of distinct points in X , then there exist 0 ¿ 0 and two neighborhoods Vx0 and VxI0 of x0 and xI0 , respectively, such that for all  ¡ 0 , every pair of points in Vx0 × VxI0 is distinguishable by the -sampled system of (1) for all M D-bounded input. Proof. The proof is by contradiction, assume that there exist a sequence of positive numbers (n )n¿1 converging to 0, two sequences (x n )n¿1 and (xIn )n¿1 of points converging respectively to x0 and xI0 and a sequence of M D-bounded controls (un )n¿1 such that h ◦ ’(n ; x n (kn ); ukn (·)) = h ◦ ’(n ; xIn (kn ); ukn (·))

for all n ¿ 1 and all k ¿ 1:

n

The sequence (u )n¿1 being M D-bounded, we can suppose that it converges to a continuous control u even if we have to consider a subsequence of this sequence. For this control u, there exists a t0 such that h ◦ ’(t0 ; x0 ; u(·)) = h ◦ ’(t0 ; xI0 ; u(·)); this inequality remains true if we replace t0 by t ∈ ]t0 −-; t0 +-[ with - ¿ 0 small enough. Now the approximation lemma (see [4]) tells us that ’(t; x n ; un (·)) tends to ’(t; x0 ; u(·)) uniformly on an interval [0; T ] provided that the sequence of controls (un )n¿1 is equibounded and convergent (which is the case of (un )n¿0 ). So, take T such that [0; T ] ⊃ ]t0 − -; t0 + -[, if n is large enough, we have h ◦ ’(t; x n ; un (·)) = h ◦ ’(t; xIn ; un (·))

for all t ∈ ]t0 − -; t0 + -[

and there exists an integer kn such that kn n ∈ ]t0 − -; t0 + -[; this leads us to the contradiction. We want to point out now a relation between in7nitesimal observability and local observability. Lemma 3. Suppose that system (1) is in8nitesimally observable at (u0 ; x0 ), then system (1) is locally observable for all input closed enough to u0 . To be more precise, there exist a neighborhood V of x0 , open intervals J1 ; : : : ; Jd such that ∀x; xI ∈ V; ∃k ∈ {1; : : : ; d} | ∀t ∈ Jk ; h(’(t; x; u(·))) = h(’(t; x; I u(·))) for all control u closed enough to u0 (when restricted to an interval [0; T ] containing the Jk ’s). This lemma is not very original, it is mentioned in [1,4] a slightly weaker result is proved; to prove our lemma, we will follow the reasoning of [4]. Proof. We denote by dhi , i = 1; : : : ; p the components of dh, by 0i (resp. 1i ) the point ’(ti ; x0 ; u0 (·)) ∈ M (resp. (ti ; 0 ; u0 (·)) ∈ TM with (0 ) = x0 ) and we prove by induction the existence of indices i1 ; : : : ; id and times ti1 ; : : : ; tid such that the mapping - which to 0 ∈ Tx0 M associates (dhi1 (0i1 ) · 1i1 ; : : : ; dhid (0id ) · 1id ) ∈ Rd

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S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

is one-to-one. Assume inductively that we have constructed subspaces K0 ⊃ · · · ⊃ Kr of Tx0 M , indices i1 ; : : : ; ir and times ti1 ; : : : ; tir such that K0 = Tx0 M , and for 1 6 j 6 r, Kj = {0 ∈ Tx0 M | dhi1 (0i1 ) · 1i1 = · · · = dhij (0ij ) · 1ij = 0} and so that dim Kj = d − j. Take a nonzero 0 in Kr , due to the injectivity of the mapping Pu0 ;x0 , there exist ir+1 and tir+1 such that dhir+1 (0ir+1 ) · 1ir+1 = 0, this provides the induction step and the existence of the one-to-one application between Tx0 M and Rd follows from the existence of Kn . Consider now the mapping 4 between a neighborhood of x0 and Rd de7ned by 4(x) = (hi1 (’(ti1 ; x; u0 (·))); : : : ; hid (’(tid ; x; u0 (·)))) its di@erential at x = x0 is the linear application - which is one-to-one. Thus by the inverse function theorem, the mapping 4 is one-to-one in a neighborhood of x0 . Moreover, we can notice that, in the inverse function theorem, the size of this neighborhood depends on the norm of the di@erential. So, if the times tik (k =1;  : : : ; d) vary on small intervals J1 ; : : : ; Jd and if u is a control closed enough to u0 on an interval [0; T ] ⊃ Jk , the norm of - remains far from 0 and so we can conclude to the existence of a neighborhood V as in the statement of the lemma. Lemma 4. We assume that system (1) is uniformly in8nitesimally observable for all inputs u. Let M ¿ 0 and x0 be a point in X , then there exist 0 ¿ 0 and a neighborhood Wx0 of x0 such that for all  ¡ 0 , every pair of distinct points in Wx0 × Wx0 is distinguishable by the -sampled system of (1) for all M D-bounded input. Proof. The proof is by contradiction, assume that there exist a sequence of positive numbers (n )n¿1 converging to 0, two sequences (x n )n¿1 and (xIn )n¿1 of distinct points converging to x0 and a sequence of M D-bounded controls (un )n¿1 such that h ◦ ’(n ; x n (kn ); ukn (·)) = h ◦ ’(n ; xIn (kn ); ukn (·))

for all n ¿ 1 and all k ¿ 1:

The sequence (un )n¿1 being M D-bounded, we can suppose that it converges to a continuous control u0 even if we have to consider a subsequence of it. For this control u0 , take a neighborhood V of x0 as in the preceding lemma. If n is large enough, the points x n and xIn belong to V , and for every k ∈ {1; : : : ; d}, there exists an integer lk such that lk n ∈ Jk , moreover, the piecewise constant control un is closed to u0 . Choose such a n, by application of the preceding lemma, there exists an index k such that h ◦ ’(n ; x n (kn ); ukn (·)) = h ◦ ’(n ; xIn (kn ); ukn (·)); which is a contradiction. Proof of the main result. From Lemmas 2 and 4, we know that for every pair (x0 ; xI0 ) of M × M , there exist a neighborhood Vx0 ;xI0 of (x0 ; xI0 ) and a x0 ;xI0 ¿ 0 such that every pair of distinct points in this neighborhood is distinguishable by every -sampled system of (1) and for all inputs if  ¡ x0 ;xI0 . The neighborhoods Vx0 ;xI0 constitute a covering of M × M , since this set is compact, we can 7nd a 7nite subcovering Vx01 ;xI10 ; : : : ; Vx0r ;xIr0 of M × M and it is clear that, if we let 0 = min{x0i ;xIi0 | i = 1; : : : ; r}, every -sampled system of (1) is observable for all inputs if  ¡ 0 . Notice also that, given two distinct initial conditions x0 and xI0 , there exists an index k such that h(x(k)) = h(x(k)), I now either h(x((k + 1))) = h(x((k I + 1))) or h(x((k + 1))) = h(x((k I + 1))), in this last case we can consider x((k + 1)) and x((k I + 1)) as initial conditions and deduce the existence of an index k  ¿ k + 1 I  )). So we can conclude that there exists an in7nite number of indices such that such that h(x(k  )) = h(x(k h(x(k)) = h(x(k)). I

S. Ammar, J.-C. Vivalda / Systems & Control Letters 52 (2004) 7 – 15

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Remark. Notice that our demonstration proves the existence of an in8nite sequence of indices k such that h(xk ) = h(xIk ) which is slightly stronger that what is required in De7nition 2. References [1] J.-P. Gauthier, I. Kupka, Observability and observers for nonlinear systems, SIAM J. Control Optim. 32 (4) (1994) 975–994. [2] E.D. Sontag, On the preservation of certain controllability properties under sampling, in: I.D. Landau (Ed.), Outils et Modules MathSematiques pour l’Automatique, l’Analyse de SystTemes et le Traitement du Signal, Vol. 3, Editions du CNRS, Paris, 1983, pp. 623– 637. [3] E.D. Sontag, A concept of local observability, Systems Control Lett. 5 (1984) 41– 47. [4] E.D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics, Vol. 6, Springer, Berlin, 1990.