Carlos Canudas De Wit

Kelit Crisanto

Carlos Canudas-De-Wit
email: carlos.canudas-de-wit@inpg.fr.

Jonathan Jaglin

Non uniform sampling coding in networked controlled linear systems

Keywords: Entropy coding, Networked controlled systems, NCS

come

I. INTRODUCTION

T His paper deals with systems interconnected by a com- munication network where information is transmitted via a particular coding algorithm. Many of such type of control architectures has been studied recently. Some examples are: [START_REF] Ishii | Remote control of lti systems over networks with state quatization[END_REF], [START_REF] Elia | Stabilization of linear systems with limited information[END_REF], [START_REF] Liberzon | On stabilization of linear systems with limited information[END_REF], [START_REF] Lemmon | Control system performance under dynamic quatization: the scalar case[END_REF], [START_REF] Tan | Numerical study of joint quatization and control under block-coding[END_REF], [START_REF] Li | Robust quatization for diginal finite communication bandwidth (dfcb) control[END_REF], [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF], [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF], [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], and [START_REF] Rubio | Adaptive delta-modulation coding in networked controlled systems[END_REF] among others.

In particular, delta modulation (∆-M ) is a well-known differential coding technique used for reducing the data rate required for voice communication, see [START_REF] Proakis | Digital Communications[END_REF]. The standard technique is based on synchronizing a state predictor on emitter and receiver and just sending a onebit error signal corresponding to the innovation of the sampled data with respect to the predictor. The prediction is then updated by adding a positive or negative quantity (determined by the bit that has been transmitted) of absolute value ∆, a known parameter shared between emitter and receiver.

We have recently investigate the closed-loop properties of the ∆-M algorithm when used in the feedback loop. Our results in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], have suggested some modification of the original form of the ∆-M algorithm to improve the closed-loop properties when used in feedback within the context of Networked controlled systems (NCS). The results showed that the stability domain and the resulting precision of the ∆-M is limited by the position of the largest unstable pole of the system. Although this can be improved by increasing the sampling rate, or by the use of extra bits [START_REF] Abdallah | Compensation schemes for a delta-modulation-based ncs[END_REF], this possibility is clearly limited by the maximum permissible data transmission rate. Further studies, have also shown that it is possible to make the modulation gain adaptive so as to improve the global stability results [START_REF] Rubio | Adaptive delta-modulation coding in networked controlled systems[END_REF].

The aim of this work is to explore yet another variant of the ∆-M structure to improve data transmission efficiency in the context of NCS. High compression rates can only be reached by the use of entropy coding. Entropy coding introduce redundancy and assigns some probability distribution to the events. In that way, the mean code length can be reduced.

A pre-requisite for the entropy coding strategy is to design a mechanism with the ability to quantify and to differentiate stand-still signal events, to changes in the source (label crossing detector). For instance, this can be done by defining an alphabet where the source signal information is contained in the time interval between level crossing and in the direction of the level crossing. As mentioned in [START_REF] Mark | A nonuniform sampling approach to data compression[END_REF], by assigning strings of the 2-tuple 00 to represent the time between signal level crossing, and 01 and 10 to denote the direction of level crossing, the output of the nonuniform sampling encoder contains a high probability of the 0 symbol with makes it suitable for an entropy encoder to attain a "good" overall compression ratio. A fundamental difference with the classical ∆-M algorithm is that the error is coded on the basis of a 3-valued alphabet rather than a 2-valued one. These features point toward the possibility of efficient encoding strategies.

The overall coding strategy studied here is composed of two main blocks: 1) a non uniform sampling encoder, NSE, (i.e. asynchronous delta modulation), including a model-based predictor, MBP, similar to the one proposed in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], and 2) a variable length-block encoding scheme, VLE,(i.e. Runlength distortion-less encoding strategy) in which buffering is only requited at the input only.

The overall scheme is shown in Figure 1. The paper aims at studding the closed-loop properties of such arrangement. Due to the fact that the VLE block is a distortion less coding, and for simplicity reasons, the focus of this study is directed toward the study of the stability properties of the first coding block only.

A. Definitions

B. Assumptions

The hypothesis used in the results presented in this paper, are the following:

• The transmitted information is binary • Only encoder-to-decoder information transmission is allowed (feedback between decoder to encoder is forbidden), • Reliable noiseless channel transmission is considered (no data lost, or information distortion is considered), • Transmission delays are neglected, II. PROBLEM SET UP We consider the following SISO discrete-time linear system (possible unstable), of the form,

x k = B(q -1 ) A(q -1 ) u k (1)
together with a RST controller,

u k = R(q -1 ) S(q -1 ) γ T (q -1 ) r k -xk (2)
where r k is the reference, xk is the estimated of the system output x k , and R(q -1 ), S(q -1 ), T (q -1 ) are the control polynomials in the delay operator q -1 . They also satisfy:

T = RB, SA + RB = A cl , γ △ = A cl (1)
with A cl being the closed-loop polynomial, and γ the static gain needed to reach unitary zero-frequency gain. For simplicity, we will omit the use of the argument (q -1 ) when needed.

The coding process consists in: 1) encoding the system output x k , 2) transmitting the coding sequence through the communication channel, and 3) decoding the received information to produce the estimated xk . The complete sequence can be seen as estimation process.

When xk ≡ x k , the above controller give the following closed-loop relation,

x k = γ A cl (q -1 ) r k else (x k = x k ), we have, x k = γ A cl (q -1 ) r k + W (q -1 )x k
where xk = x k -x k is the estimation error, and W = BR/A cl .

As A cl defines a stable polynomial, the output x k is keep bounded as long as xk is bounded as well.

The problem is then to design the coding process that defines the output xk preserving closed-loop properties. This process is described next.

III. CODING PROCESS

The coding (encoding/decoding) process is composed of several steps, described by the following operations:

x k → N U-enc. ν k → E-enc. η k ⇒ channel ηk → E-dec. νk → N U-dec.
xk As shown in Figure 1, the encoder (respectively the inverse decoder) operation is composed of two separate blocks:

• The non uniform sampler encoder (NUE) together with a model-based predictor (MBP), this operation maps x k → ν k (respectively the decoder maps νk → xk ), and • A variable length encoder (VLE) mapping the 3-valued signal, ν k to the string binary signal η k (respectively, the decoder maps ηk → νk ). Each of these components are described next.

A. Description of the NUE and the MBP

Elements composing the Non-Uniform Entropy coding and the model-based predictor are the following:

1) The Level Detector: (ϕ LD : xk → ν k ), take the error signal and codes the output signals into a 3-valued one δ k ∈ {-1, 0, 1} (or equivalent to the binary signal ν k ). That is:

δ k =    1 if one level is crossed uppwards 0 if signal stay at the acual level -1 if one level is crossed dawnwards
Equations behind this are:

l k =    xk ∆ if, xk > 0 0 if, xk = 0 xk ∆ if, xk < 0 δ k = f (x k ) = 0 if, l k = l k-1 sign (l k -l k-1 )
else the level threshold is ∆, and the floor operator ⌊•⌋ rounds to the smaller integer, and the ceil operator ⌈•⌉ rounds to the larger integer.

Finally, the 3-valued signal δ k is transform into a a 2-bits binary number ν k ∈ {00, 01, 10}, by the following operation.

ν k =    00 if δ k = 0 01 if δ k = -1 10 if δ k = 1
The combination '11 ′ is not used in this process. Note that δ k is just a dummy internal variable useful to simplify the formalization of the error equations in stability studies to be presented latter, which can be formalized using real numbers.

Operation principle of the LD. The operation principe of the level detector is shown in Figure 2. The signal detection levels are uniformly spaced by the quantum ∆. The level detector device produces a signal (identified by ′ 01 ′ or ′ 10 ′ ) whenever a level crossing takes place. As level changes are not uniformly spaced in time, this device normally works on the basis of non-uniform sampling. However a complete asynchronous problem formulation, will also require to describe the system to be controlled in this time-asynchronous framework. Instead, and following [START_REF] Mark | A nonuniform sampling approach to data compression[END_REF], we consider quantizing the time between level crossing and encoding the quantizing time interval in uniform steps. By doing this the information transmission in synchronous.

While two symbols are used to characterize the level changes, one more symbol can be used to quantize time intervals. Then 01 indicates upward crossing, 10 downward crossing, and 00 is used to code the time-interval between crossing. To illustrate this consider the example of Figure 2. We assume that uniform samples are taken every time T s , then m samples are taken in the time interval T i = t i -t i-1 before a cross level takes place. As two level (upward)crossing happen within this interval, the information the binary representation of this situation by the level crossing detector produce the following signal, 01, 00, 00, . . . 00 m-pairs , 01

This sequence has then high probability of 0 ′ s, and thus suited for entropy coding.

2) The model-based predictor: Has the role to recover the encoded signal x k for the 3-valued binary signal ν k . It is composed of:

• The inverse of the level detector: ϕ -1 LD : ν k → δ k → xk , which equations are:

δ k =    0 if ν k = 00 -1 if ν k = 01 1 if ν k = 10
and, a basis for it design. This structure is inspired by our previous works in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], [START_REF] Rubio | Adaptive delta-modulation coding in networked controlled systems[END_REF], and also in [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF]. The predictor is a dynamic linear discrete-time operator that maps the output of the inverse level detector, to the signal prediction xk . Its structure depends upon the particular control used (state feedback or output feedback). For instance, for the RST-control discussed here, it has the following form:

xk = xk-1 + ∆ • δ k • The
xk = W γ T r k + xk , W △ = BR A cl (3)
Which results in the following error equation:

xk = W xk -xk (4)

B. Description of the entropy coding VLE

As mentioned in the introduction, high compression rates can only be reached by the use of entropy coding. Entropy coding introduce redundancy and assigns some probability distribution to the events. In that way, the mean code length can be optimized. Run-length codes 1 , are a class of variablelength codes that are sub-optimal (when compared to the Huffman code), but have the advantage of avoiding buffering at the decoder side, and therefore reducing data transmission latency. An example used in [START_REF] Mark | A nonuniform sampling approach to data compression[END_REF] is described below.

Let ϕ RL : ν k → η k , describe the VLE map. Then, when a run-length code of length n is applied, the result is the output η k which a variable-length binary signal. An example of a coding scheme of block length n = 2, and n = 3 is shown below.

Assuming that the coding sequence is independent and identically distributed, and that the upward crossing frequency equals the downwards crossing frequency, i.e. p = P (00), P (01

) = P (10) = 1 2 (1 -p)
then, according to [START_REF] Mark | A nonuniform sampling approach to data compression[END_REF], the mean coding length, C L , of this scheme is:

C L = 2 1 -p (2 (n-1) -1)
1 -p bits and the compaction ratio,C R , is:

C R = C L n = 2 n 1 -p (2 (n-1) -1)
1 -p which has the limiting value:

lim p→1 C R = 2 n -2 n .
IV. ERROR SYSTEM Following the assumptions maid in this paper (lossless channel transmission), we then have that ν k = νk , and that δ k = δk . In this case binary variables are not needed, and hence error equation can be described by real variables only.

Introducing the following error definitions:

• e k = x k -γ A cl
e k = W (q -1 )x k (5) xk = W (q -1 )ε k (6)
with W = BR/A cl being the stable operator defined previously. Note that the ε k = ε k (x k ), and thereby the above error equation can be seen as two systems in cascade, i.e. the output of the autonomous system ( 6) is the input of the stable system [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF]. For stability purposes it is thus sufficient to demonstrate the stability properties of the sub-system (6).

A. Properties of the LD and it inverse

Note that ε k writes as:

ε k = xk -xk = xk -ϕ RL • ϕ -1 RL {x k } = xk -φRL {x k } where φRL △ = ϕ RL • ϕ -1 RL : xk → xk .
Note that this map is dynamic, defined by the following relation:

xk = xk-1 + ∆ • δ k (7)
with δ k = f (x k ) as defined before. System ( 6)-( 7) can be then seen as a feedback system, as shown by Figure 3, i.e.

xk = W (q -1 )ε k = W (q -1 ) xk -xk = W (q -1 ) xk - ∆ 1 -q -1 f (x k )
Ideally we would like that the map φRL be a linear map with

- - W + xk xk ε k ∆ 1-z -1 δ k f
Fig. 3. Closed-loop interconnection of System ( 6)- [START_REF] Ishii | Remote control of lti systems over networks with state quatization[END_REF]. Jonathan: merci de faire cete figure.

unitary gain. This ideal goal is hampered by several factors:

• unknown initial conditions of x0 ,

• badly chosen T s , and ∆, and • chattering in the neighborhood of the quantum ∆. In particular, large sampling times T s , and too small quantum ∆ may results in signal variation of more than one level, which may leads to unrecovered bias in the estimated, leading to potential instabilities for unstable open-loop systems.

The qualitative valuation of this algorithm is shown next trough a simulation examples, and some qualitative discussions.

V. QUALITATIVE EVALUATION Consider the following simple system

A(q -1 ) B(q -1 ) = bq -1 1 -aq -1 ( 8
)
The controller is:

R = r 0 +r 1 q -1 , S = 1-q -1 , T = A cl (1) B(1) A cl = (1-a cl ) 2
obtained from the closed-loop specification given by

A cl = (1 -a cl ) 2 ,

A. Initial conditions

Initial conditions of the predictor x0 need to be synchronized with initial condition of the system, i.e. x0 = x 0 , in particular at the decoder side. This requires a specific initialization procedure that send this initial information before the coding algorithm is triggered. Figure 4 shows the algorithm behavior when this synchronization is performed correctly. When this synchronization is not done correctly, Figure 5 shows a bias in the estimates, which produces an error in the tracking signal. Note however, that this ill-synchronization does not causes instabilities, but just a tracking performance degradation, proportional to the initial mismatch between initial condition in the decoder estimates. The value of ∆ has an important impact on the quality of the estimates and the system stability. Larger values of ∆ are suited for enhance stability, but it degrades the estimated quality. Inversely, small values for ∆ will results in better estimated quality, but it jeopardize the system stability. Evidently, also the selection of the sampling period T s play also an important role, and more specifically, the ratio ∆/T s , as it will be discussed below.

Note that the coding scheme is based on the hypothesis that information is coded by a 3-valued signal δ k , or equivalently that level changes are restricted to a maximum of one level of magnitude ∆. This can bee seen from the inverse of the level detector equation, that aims at reconstructing the estimation error trough the relation xk = xk-1 + ∆ • δ k . When the this equation is correctly initialized ( x0 = x0 ) in particular at the decoder side, and the change of level is limited to one, then the maximum distance between xk , and x0 is given by ∆. Looking closely the error equation ( 5)-( 6), we can see that as a consequence, the norm of the tracking error becomes proportional to ∆, i.e.

|e| |W | 2 ∆ the system precision is then proportional to ∆ as long as |l kl k-1 | 1. In order to this to happen, it is necessary that to select ∆ such as the net change between two consecutive samples of x is bounded by this value, i.e.

|x k -xk-1 | < ∆ (9)
In this relation the sampling time does not appears explicitly because is embedded in the particular discrete-time notation, which have the following signification:

xk = x(t k ), xk-1 = x(t k -T s )
Thus equation ( 9) can also be read as:

| d dt x(t)| ≈ 1 T s |x(t k ) -x(t k -T s )| < ∆
T s the rate of change of the estimated error signal should thus be bounded by the ration ∆ Ts . The Figure 6, and Figure 7 shown this event. In Figure 6 ∆ Ts is sufficient small to keep

|l k -l k-1 | 1, whereas in Figure 7 |l k -l k-1 | 2.
As a consequence in this second case the estimation performance is degraded, and in certain case for even larger ∆ Ts (not shown here) instabilities may occur.

C. Chattering effect

Note that practically all parameters of the control scheme affects the spectrum of the evolution of δ k , and in particular the ration ∆/T s , but also the level of open-loop instabilities of the considered system. Figure 8 show the resulting histogram of δ k for different simulations with different parameters. The results shows that by increasing T s the frequency spectrum of δ k is reduced, and hence the event δ k = 0 has higher probability to occur. We recall distributions with high roll-off will be benefic for data compression, as illustrated by the VLE algorithm. from the same figure and similar reasons, we observe that open-loop instable systems with a high degree of instability are less adapted for entropy coding. The resulting compression ratio are reported in Table II. Higher compression rates are thus obtained for the cases where p is higher, i.e. the bottom simulation in Fig. 8.

VI. CONCLUSIONS

In this paper we have investigated the possibility to use asynchronous coding in the context of networked controlled systems. The main motivation has been to explore the benefits in terms of coding compression ratio when the NUE is combined with entropy coding strategies. In particular we have analyzed the case of run-length coding, which in spite of its sub-optimality do not require buffering at the decoder side, and hence reduce latency.

We have shown also that the predictor initial conditions, seems to be critical for the well operation of the coding algorithm. When these are not exactly known the predictor produce a bias, although stability is not happened. We have also presented some preliminary studies showing that the choice of ∆ and T s are critical for the well functioning of scheme. In particular "high" ratios of ∆/T s are suited for stability, whereas in opposition "small" ratios of ∆/T s are better for data compression. This underline a trade-off between stability and performance of the proposed scheme.

Finally, further analytic studies are in progress to find analytic conditions for stability, optimal selection of the ratio ∆/T s , and eventually making this ratio time (state) varying.

Fig. 1 .

 1 Fig. 1. Block diagram of the non uniform entropy coding in the feedback loop.

Fig. 2 .

 2 Fig.2. Illustration of the level detector working operation principe.

  r k : the tracking error, • xk = x k -xk : the prediction error, and • ε k = xk -xk : the LD error. we have the closed-loop error system:

  and r 0 = a+1-2a cl b , r 1 = -a-a 2 cl b . Note that this controller introduced a integral action as well. Parameter used in simulations are: a = 1.1, b = 1, a c = 0.96, T s = 0.05 (sec), ∆ = 0.02, and x 0 = 0.1.

Fig. 4 .

 4 Fig. 4. Simulations with synchronized initial conditions; x 0 = x0 = 0.1. reference and output (top), output and it estimated (middle), and error variables x(red), and x(blue)

Fig. 5 .

 5 Fig. 5. Simulations with ill-synchronized initial conditions; x 0 = x0 = 0.1. reference and output (top), output and it estimated (middle), and error variables x(red), and x(blue)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Simulation results with ∆ = 0.02 ; T e = 0.05 yielding |l k -l k-1 | 1. Output and reference (upper), x vs. x (middle), and l kl k-1 (lower)

Fig. 8 .

 8 Fig. 8. Simulations with ∆ = 0.02;Te = 0.05;a = 1.6 (upper), ∆ = 0.02; Te = 0.05; a = 1.1 (middle), and with ∆ = 0.02; T e = 0.15; a = 1.1(bottom). Time evolution of δ k (left), histogram of events (right)

TABLE II COMPRESSION

 II RATIOS AND CODE LENGTH OBTAINED WITH n = 3 FROM DATA SHOWN IN FIG. 8.

	case (n=3)	p	C L	C R
	upper	.4	3.4208	1.1403
	middle	.75 4.6250	1.5417
	lower	.8	4.8800	1.6267

Class of coding strategy that can decode information instantaneously.


Carlos Canudas-De-Wit

Francisco R Rubio

José Fornés

Fabio Gómez-Estern

Differential Coding in Networked Controlled Linear Systems

Keywords: Differential coding, delta modulation, stabilization of linear systems in Networked controlled systems

This paper investigates the closed-loop properties of the differential coding scheme known as Delta-Modulation (Δ-M ) when used in feedback loops within the context of linear systems controlled through some communication network. We propose a new modified scheme of the original form of the Δ-M algorithm which improves the closed-loop properties. The new coding structure explicitly uses information from the system model. Stability properties for both: the continuoustime and the discrete-time version of this new algorithm are assessed. The results shows that the stability domain and the resulting precision of the Δ-M is limited by the position of the unstable poles of the system.

I. INTRODUCTION

This paper studies different aspects of the use of differential coding in as a mean for transmitting sensing signals in feedback controlled linear systems interconnected through some transmission network. The problem is of interest in the area of Networked Controlled Systems (NCS), where we find several applications calling for data-compression algorithms aiming at reducing the amount of information that may be transmitted throughout the communication channel, and therefore permitting a better resource allocation and/or an improvement of the permissible closed-loop system bandwidth (data-rate).

In particular, we study the closed-loop properties resulting from the Delta modulation (Δ-M ) algorithm, which is the simplest form of the differential coding, in which a twolevel (1-bit) quantizer is used in conjunction with a firstorder predictor. As such, the Δ-M algorithm can be seen as special case of the family of signal quantizers. Although quantization may not be the largest factor contributing to data compression, and hence to high data rates, optimization of the quantization levels is mandatory in large-scale systems, and may be of great interest while designing low cost transmitter/receiver components [START_REF] Ishii | Remote control of lti systems over networks with state quatization[END_REF].

Quantized feedback design, is a current topic of interest in the NCS field, see for instance [START_REF] Mitter | Stabilization of linear systems with limited information[END_REF], [START_REF] Liberzon | On stabilization of linear systems with limited information[END_REF], [START_REF] Lemmon | Control system performance under dynamic quatization: the scalar case[END_REF], [START_REF] Tan | Numerical study of joint quatization and control under block-coding[END_REF], [START_REF] Ishii | Remote control of lti systems over networks with state quatization[END_REF], [START_REF] Li | Robust quatization for diginal finite communication bandwidth (dfcb) control[END_REF].

The Δ-M algorithm is a two-level dynamic quantizer, which can be seen as the coarsest dynamic quantizer, although this does not necessarily mean that the Δ-M encoder yields the minimum data-rate stabilizing control strategy. The C. Canudas-de-Wit is with The Laboratoire d'Automatique de Grenoble, UMR CNRS 5528, ENSIEG-INPG, B.P. 46, 38 402, ST. Martin d´Hères, FRANCE. Email: carlos.canudas-de-wit@inpg.fr.

Francisco R. Rubio José Fornés, and Fabio Gómez-Estern are with the Department of Automatic Control and Systems Engineering at the University of Sevilla, Spain. Email: rubio@esi.us.es, fornes@trajano.us.es, fabio@esi.us.es dynamic property of the Delta modulation, and its variants1 have been designed mainly from the signal transmission perspective, but no much as part of a feedback system, and as a consequence not much is known about the ability of this algorithm in the NCS context. This paper is devoted to this problem framed as shown in Figure 1.

First we present several results concerning the continuoustime formulation, which is relevant because it captures the limiting case of the discrete-time formulation, and provides a better understanding of the maximal stability properties that can be attained. Then, we study the discrete-time case, which permits a more clear assessment of the data rate constraints. It is shown that the system states converge to a ball enclosing the origin, and that the result is semi-global. The size of the attraction region, i.e. the system precision, depends on the coding gain linearly, and are also functions of the location of the unstable discrete-time open-loop poles. These results also show that the stability properties improve as the sampling time is reduced, or equivalently as the data rate is increased.

A. Assumptions

The hypothesis used in the results all along the paper, are the following:

• The transmitted information is binary δ k ∈ {-1, 1} • Data is sent at a bounded rate (we select the sampling frequency in order to transmit only one δ k at a time).

B. Definitions and notation

Let x ∈ R n , A ∈ R n×n , then : ||x|| 2 = n i=1 x 2 i , |x| = n i=1 |x i |, where -|x| -||x||, sgn(x) = [sgn(x 1 ), sgn(x 2 ), • • • , sgn(x n )] T . ||A|| is the induced Eu- clidian norm of A.

II. CONTINUOUS-TIME FORMULATION

We consider linear systems, with a linear feedback of the following form:

ẋ(t) = Ax(t) + Bu(t) (1a) u(t) = -K x(t) (1b)
where

x(t) ∈ R n , A ∈ R n×n , B ∈ R n×1 are controllable matrices, u(t) is the input of dimension one, δ(t) ∈ R n is
the encoded signal, with only two-valued elements δ i (t) ∈ {-1, 1}. x(t) ∈ R n is the estimated value of x as obtained from the process of differential coding described next. We consider here that the linear feedback gain K is designed such that the matrix A c = A -BK, is strictly stable. To simplify matters, we first consider a fully continuous-time formulation. The more realistic discrete-time case, including data-rate constraints, will be studied in latter sections. This control setup is shown Figure 1, which describes a closed-loop system under one-way communication channel. The controller is assumed to be located at the plant side, whereas the sensor is remotely located. Information is transmitted through some data communication network. In order to focus the study on the impact of the differential coding, we only consider lossless transmission channels with neglected delay. Aspects related to the transmission delay have been studied elsewhere, see for instance [START_REF] Wiltrant | Remote stabilization via time-varying communication network delays[END_REF], and [START_REF] Wiltrant | output stabilization via two-channel transmission communication[END_REF].

A. Differential coding

Delta modulation (Δ-M ) shown in Figure 2 is the simplest form of differential coding (see [START_REF] Proakis | Digital Communications[END_REF]), in which a two-level (1-bit) quantizer is used in conjunction with a first-order predictor. The continuous-time version of the Δ-M coding algorithm is, with

δ(t) = sgn(x(t) -x(t)), ẋ(t) = Δ(t) • sgn (x(t) -x(t)) = Δ(t) • δ(t) (2a) x(t) = t 0 Δ(τ ) • δ(τ )dτ (2b)
The first equation (2a) describes the encoder dynamic equation, whereas the second one (2b) describes the decoder. Δ(t) is the step size, which in general is chosen to be constant, but time-varying gains can also be considered. A mandatory choice is that the gain Δ(t) must be the strictly the same in both sides of the coding process, in order to the decoder to work properly.

To simplify notation, in this section we drop the timearguments in all the variables when its explicit mention is not needed.

B. Closed-loop stability properties of the standard Δ-M coding algorithm

Consider the problem of finding a suitable value for Δ so as to stabilize system (1), under the coding law [START_REF] Mitter | Stabilization of linear systems with limited information[END_REF]. It is easy to see that the error equations read as:

ẋ = A c x + BK x (3a) ẋ = A c x + BK x ẋ -Δsgn(x) (3b)
with x = xx. Which means that Δ should be chosen large enough to locally dominate the rate of change of the system state, ẋ. Clearly, this can be done only under relative "high-values" for Δ, as is demonstrated in the following result.

Proposition 1: STANDARD Δ-M CODING. Consider system (1) where x ∈ R n is the state estimate computed according to the Δ-M scheme shown in [START_REF] Mitter | Stabilization of linear systems with limited information[END_REF]. Let x = xx, and assume that Δ ∈ R n×n is constant, and has the following form:

Δ = Δ 0 • I n×n , Let ζ = (x T , xT ) T
, then for any possible value of initial conditions ζ(0), there exists a corresponding scalar constant value for Δ 0 given by

Δ 0 c • ||ζ(0)|| such that ζ(t) → 0, as t → ∞.
The constant c > max(c 5 , c 5 c 6 ) > 0, and the c i given by the following relations:

c 1 = ||2P BK||, c 2 = ||A c ||, c 3 = ||BK||, c 4 = max{c 1 , c 2 }, c 5 = c 3 + c 2 4 4q , c 6 = λmaxP +1
λminP +1 . λ max P , λ min P , and q = λ min Q, are respectively, the maximum and minimum eigenvalues of P = P T > 0, and the minimum eigenvalue of Q > 0, solutions of P A c + A T c P = -Q.

Proof: Consider the quadratic Lyapunov function V = x T P x + xT x, Evaluating V along solutions of system (3), and using -|x| -||x||, gives: -

V -q||x|| 2 + c 1 ||x|| ||x|| + c 2 ||x|| ||x|| + +c 3 ||x|| 2 -Δ 0 |x| -q||x|| 2 + c 1 ||x|| ||x|| + c 2 ||x|| ||x|| + +c 3 ||x|| 2 -Δ 0 ||x|| -q||x|| 2 + c 4 ||x|| ||x|| + c 3 ||x|| 2 -Δ 0 ||x|| + - + + + + Δ Δ δ = ±1 δ x xx x × × Ac Ac Encoder Decoder
(||x||, ||x||) q -c4 2 -c4 2 Δ0 ||x|| -c 3 ||x|| ||x||
From here we can see that V is negative as long as the matrix in the equality above is positive definite, i.e. for values of Δ 0 , such that q

( Δ0 ||x|| -c 3 ) > c 2 4
4 , or equivalently if Δ 0 > c 5 ||ζ|| c 5 ||x||. Nevertheless, this condition by itself does not define the domain of attraction. For this we proceed further as follows.

Define λ M = λ max P + 1, and λ m = λ min P + 1. With this definition we have the following bounds on V (ζ),

λ M ||ζ|| 2 V (ζ) λ m ||ζ|| 2 -λ M ||ζ|| 2 -V (ζ) -λ m ||ζ|| 2
using these bounds, and assuming that Δ 0 > c 5 ||ζ||, then there exists a scalar function (||ζ||) > 0, such that

V -(||ζ||) • ||ζ|| 2 -(||ζ||) V λ M (4)
Integration on both sides of this equation along the timeinterval [0, t], gives,

V (t) V (0) exp(-ϕ( (||ζ||)),
where

ϕ(t) = t 0 (||ζ(τ )||) λM
dτ. Note that ϕ(t) > 0, as long as Δ 0 > c 5 ||ζ||. Using again the bound on V in the expression above, we obtain

||ζ(t)|| 2 λ m λ M ||ζ(0)|| 2 exp(-ϕ(t)) ||ζ(t)|| λ m λ M ||ζ(0)|| exp(-ϕ(t)) = c 6 ||ζ(0)|| exp(-ϕ(t)/2),
and using the relation given in Proposition 1, i.e. Δ 0 c • ||ζ(0)|| > c 5 c 6 ||ζ(0)|| in the above inequality, we get ||ζ(t)|| < Δ0 c5 exp(-ϕ(t)/2). Therefore, under the relation Δ 0 > c 5 ||ζ(0)||, as proposed in Proposition 1, and using exp(-ϕ(t)/2) < 1 ∀t, we conclude that |ζ(t) < Δ 0 /c 5 as required for Eq. ( 4) to hold. As a final consequence, the time derivative of the Lyapunov function is negative definite and the convergence ||ζ(t)|| → 0 is guaranteed.

C. A new structure for the Δ-M coding

The result in Proposition 1 shows that relative large gains for Δ 0 will be required to stabilize the system. Note that in this continuous-framework high gains will result in a important "chattering" effect, thus increasing the estimation error variance when the algorithm is discretized, or adding noise to the measured signal. Although the analysis is conservative, and probably lower values Δ 0 may still be possible, it is interesting to study new structures of the Δ-M coding such as to reduce the necessary gain for stabilization. The algorithm presented next is a modification of the original form of the Δ-M algorithm.

Consider now the following modified Δ-M algorithm together with system (1):

ẋ(t) = A c x + Δ(t) • δ(t) (5)
with, δ(t) = sgn (x(t)x(t)), where a new term A c x has been included. This equation describes both: the encoder and the decoder, as shown the block diagram of Figure 3.

It can be shown that the new error equation resulting from this new structure is described as two systems in cascade interconnection, i.e.

ẋ = A c x + BK x (6a) ẋ = Ax -Δsgn(x) (6b)
note that equation (6b) describes an autonomous system which solution is the input of the stable linear system (6a).

The stability properties of this algorithm are simplest than the one presented previously. They are also more tractable, and certainly less conservatives. They are given next, and also results in a system (6) which is semi-globally asymptotically stable.

Proposition 2: MODIFIED Δ-M CODING. Consider system (1) together with the modified Δ-M coding scheme [START_REF] Li | Robust quatization for diginal finite communication bandwidth (dfcb) control[END_REF]. As before, assume that Δ ∈ R n×n is constant and has the following form:

Δ = Δ 0 • I n×n ,
and that Δ 0 fulfills the following inequality,

Δ 0 > a • ||x(0)||, a = 1 2 λ sup A + A T , then ζ(t) = x T (t), xT (t)
T is bounded and tends to zero as t → ∞.

Proof: The proof is straightforward. Let V = xT x/2, and from equation (6b), we have that

V = 1 2 xT (Ax -Δsgn(x)) + 1 2 (Ax -Δsgn(x)) T x -Δ 0 |x| + 1 2 xT A + A T x -Δ 0 ||x|| + 1 2 xT A + A T x -Δ 0 ||x|| + a||x|| 2 = -||x|| (Δ 0 -a||x||)
where we have used the relation -|x| -||x||. From here we can see that the condition Δ 0 > a • ||x(0)||, with a given as in the last Proposition, makes V decrease, and hence ensures that ||x(t)|| < ||x(0)||, and that x(t) remains bounded and tends to zero in finite time. Finite-time convergence is typical in switching systems of the form (6b), and will not be demonstrated here. From this analysis we can also conclude that x(t) ∈ L 2 ∩ L ∞ .

To complete the proof note that equation (6b), describes an strictly stable linear system with input x(t) ∈ L 2 ∩ L ∞ , because x(t) is bounded and tends to zero, we can conclude that x(t) is also bounded and tends to zero, as the Proposition states.

Remark 1: Note that the proposed new coding form, in addition to simplify the stability conditions by making them only depend on the estimation initial error, the new structure introduces a side effect a low pass filtering action than will improve the filtering properties of the decoding dynamic equations.

Remark 2: Clearly, it is also interesting to study the case where only output signals y ∈ R m , are measurable. This case has also been analyzed and a further modified coding algorithm can be designed, see [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF]

III. DISCRETE-TIME ALGORITHM

In this section we present the extension of the continuoustime differential coding to the more realistic case of discretetime framework. We first introduce a simple case of a onedimensional unstable system for which optimal gain Δ, and attraction domain are easily found. We then extend this result to higher dimension systems.

A. One-dimension system example

Consider the following one-dimensional discrete timesystem, together with the control law, and the differential encoding modified law:

x k+1 = ax k + bu k (7) u k = -K xk (8) xk+1 = a c xk + Δsgn(x k ) (9) with, K ∈ R, a 1, a c = (a -bK); |a c | < 1, and xk = x k -xk .
The differential encoder ( 9) is designed on the same basis than the continuous-time version aiming at decoupling the coding error equation from the resulting closed-loop system. This gives the following error equations, with cascade structure:

x k+1 = a c x k + bK xk (10) xk+1 = ax k -Δsgn(x k ) (11)
stability of the whole system can thus be tacked by only studying the stability of the coding error equation (11). Let V k = x2 k , and

∇V k = V k+1 -V k , then ∇V k = x2 k+1 -x2 k = (a 2 -1)x 2 k -2aΔ|x k | + Δ 2
The right hand side of the equality defines a second order polynomial of the form αr 2 + βr + Δ 2 = 0, with r = |x k | with roots r 1 , r 2 given by:

r 1 = a -1 a 2 -1 Δ, r 2 = a + 1 a 2 -1 Δ
Note that the roots are always real and positive since a > 1, and that these values defines three zones, where ∇V k changes signs, i.e.

∇V k = ⎧ ⎨ ⎩ 0 if |x k | r 1 < 0 if r 1 < |x k | < r 2 0 if r 2 |x k |
For some constant Δ, this means that if the initial condition |x 0 |, is taken within the region where ∇V k is negative, then the function V k will decrease until |x k | enters in the region |x k | < r 1 , where ∇V k changes sign. In other words, the error coding variable is locally attracted to a threshold delimited by the value of r 1 , which depends linearly on the coding gain Δ. This gains also delimits the stability border r 2 , as shown the previous expressions. Therefore, larger values for Δ will make the system more stable but less precise, and inversely reduction of the gain Δ will lead to small estimation error, but will at the same time reduce the domain where the system is keep stable.

We have proved the following result. Proposition 3: DISCRETE ONE-DIMENSIONAL SYSTEM. Consider system ( 7)-( 8), with constant Δ, then if the initial conditions of the coding error are such that |x 0 | < r 2 then the following hold:

• |x k | < r 2 , ∀k 0, • ∃k 0 : |x k | r 1 , ∀k k 0 , and • lim k→∞ d(x k , B γ ) = 0.
where r 1 and r 2 have been previously defined, and d(x k , B γ ) is the minimum distance from x k to any point within the interval

B γ := {x ∈ R : |x| < γ}, γ = Kb 1 -a c r 1
Proof: The first two items result from the previous development. The last one derives from the following arguments. First note that x k = Kb z-ac xk . Then, defining the discrete time transfer function

H(z) = Kbz z-ac , with impulse response h(n) = Kba n c , n 0, yields the input-output relation x k = H(z)z -1 xk which in time domain corresponds to the convolution x k = ∞ m=0 xm-1 h(k -m) = ∞ m=0 xm h(k -m -1)
because xm = 0 ∀m < 0 is assumed. Now using the fact that x k is absolutely bounded (|x k | < r 2 , ∀k 0) and also ultimately bounded by r 1 (second item of the Proposition), we can split the convolution as follows

x k = k0 m=0 xm h(k -m -1) + ∞ m=k0+1 xm h(k -m -1) hence |x k | r 2 k0 m=0 |h(k -m -1)| + r 1 n-1 m=k0+1 |h(k -m -1)|
where the causality of h(n) has been used. Substituting the impulse response into this expression yields

|x k | < Kbr 2 a k c (1 -a -k0-1 c ) a c -1 + Kbr 1 a k-k0-1 c -1 a c -1 = Kbr 1 1 -a c + v(k)
where v(k) stands for

v(k) = Kb r 2 (1 -a -k0-1 c ) + r 1 a -k0-1 c a c -1 a k c ,
and as a c < 1, we conclude that lim k→∞ v(k) = 0 and the state x asymptotically approaches the interval

|x k | γ with γ = Kb 1 -a c r 1 . ( 12
)
Remark 3: This result displays an inherent trade-off between stability and precision for discrete-time differential coding when the gain Δ is fixed. This suggests the search for other coding strategies with variant gains. Note also that, as the sampling time2 is chosen small, a approaches 1 and the precision is increased. Indeed, lim a→1 r 1 (a) = 0, thus by making T s infinitely small, the limit case of the continuoustime precision is approached.

Remark 4: The above analysis will be extended to the case where the open loop model is affected by bounded noise, see [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF]

B. n-dimensional systems

The previous study can be generalized to system with higher dimension of the form

x k+1 = Ax k + Bu k (13) u k = -Kx k (14)
where

x k ∈ R n , A ∈ R n×n , B ∈ R n×1 . (A, B) are stabilizable pair.
For the sake of space, the noise considerations will be spared or simplicity, we consider systems whose matrix A has distinct and real eigenvalues, that is there exist a transform matrix T such that Λ = T AT -1 = diag{λ i }, with λ i = 0, ∀ i.

The differential encoding modified law is: The error equations in the x k , and zk coordinates are:

xk+1 = A c xk + Δ sgn(T xk ) (15
x k+1 = A c x k + BKT -1 zk (16) zk+1 = Λz k -T Δ sgn(z k ) (17)
Proposition 4: DISCRETE N-DIMENSIONAL SYSTEM. Let λ m , and λ M , the smaller and the larger eigenvalues of Λ, respectively. Consider the differential encoding law (15), in feedback with system (13)-( 14), with the gain Δ given as

Δ = Δ 0 • T -1 • Δ Δ = diag {sgn(λ 1 ), sgn(λ 2 ), • • • , sgn(λ n )}
where Δ 0 is a positive scalar constant. Then, if λ 2 m > (λ 2 M -1), and the initial condition of the coding state are such that ||z 0 || < r 2 the following holds:

• ||z k || r 2 , ∀k 0, • ∃k 0 : |z k | r 1 , ∀k k 0 , and • lim k→∞ d(x k , B β ) = 0.
where 0 < r 1 < r 2 are given as:

r 1 = Δ 0 λ 2 M -1 λ m -λ 2 m -λ 2 M + 1 r 2 = Δ 0 λ 2 M -1 λ m + λ 2 m -λ 2 M + 1
and d(x k , B β ) is the minimum Euclidean distance from x k to any point within the ball

B β := {x ∈ R n : x < β},
and β is a constant that can be computed as in the proof of Proposition 3.

Proof: The proof follows along similar steps than the previous proof. Introduce V k = zT k zk , and

∇V k = V k+1 -V k , then ∇V k = zT k+1 z -zT k zk = zT k Λ 2 -I zk -2sgn(z k ) ΔΛz k + +Δ 2 0 sgn(z k ) T Δ2 sgn(z k ) = zT k Λ 2 -I zk -2sgn(z k ) ΔΛz k + Δ 2 0 Note that -sgn(z k ) ΔΛz k = - i Δ 0 |z i,k | • |λ i | -Δ 0 λ m i |z i,k | = -Δ 0 λ m |z k | -Δ 0 λ m ||z k || Then, ∇V k (λ 2 M -1)||z k || 2 -Δ 0 λ m ||z k || + Δ 2 0 ≡ ϕ(||z k ||) ϕ(||z k ||) =
0 defines a second order polynomial with roots, r 1 , and r 2 , as defined previously. A necessary condition for this polynomial to have real roots, or equivalently, for the minimum of ϕ(||z k ||) to be negative is that λ 2 m > (λ 2 M -1). However, this condition is only sufficient for making ∇V k negative in the domain r 1 ||z k || r 2 . Other less conservative condition may be found.

We have, as before, three regions: and hence, the first two properties invoked in Proposition 4 follow exactly the same arguments than then ones used in previous sections. They are in consequence omitted here. The last statement follows from the relation x k = G(z)z k , and assuming that the x-subsystem (16) may also be diagonalized, the diagonalization leads to a set of n perturbed systems of the form (10) in new coordinates, from which we conclude boundedness of the state ||x k || based on the same arguments of Proposition 3 (omitted here for the sake of space).

∇V k = ⎧ ⎨ ⎩ 0 if ||z k || r 1 < 0 if r 1 < ||z k || < r 2 0 if r 2 ||z k || 4181
IV. DATA-RATES It is assumed here that samples of x(t), are digitalized with large enough resolution such that x k digitalization errors are neglected. As the transmission is done by only one bit, the data transmission rate (number of bits transmitted per unit of time) associated to this scheme, is R δ = one bit Ts = f s , in [Bits/sec] where f s = 1/T s is the sampling frequency, and the system precision is determined by the size of Δ quantizing the minimum step in the decoding process.

Let R, in [Bits/sec], be the network data-rate, then if the maximum network capabilities are used, i.e. R δ = R = f s , the system precision and the domain of stability can be rewritten as a function of R. For instance, in the onedimensional case, the expression for r 1 and r 2 rewrites as:

r 1 = e α/R -1 e 2α/R -1 Δ, r 2 = e α/R + 1 e 2α/R -1 Δ
where α is the unstable pole of the continuous-time original system. As expected, increasing the transmission rate R, improves precision (lim R→∞ r 1 (R) = 0), and stability (lim R→∞ r 2 (R) = ∞)

V. SIMULATIONS Simulations of the first order unstable system presented in Section III-A have been carried out with the following data: a = e Tsα , α = 1, T s = 0.05, b = 1a, a c = 0.95, and Δ = 0.2. The scheme has been adapted in such way that different set points can be tracked.

Figure 4 shows the obtained results. The upper figure shows the decoded signal xk , and the real signal x(t). The initial condition for the estimated signal are x0 = 0, whereas the system is initialized at x 0 = -0.5. As show this figure, the decoded is able to track reasonably well the source signal. Closed-loop responses are shown in the middle figure. We can observe that the so call granular noise (small oscillations around the source signal of amplitude proportional to the step size Δ) do not have a major impact in the quality of the closed-loop responses. An explanation for that is the high frequency contains of the granular noise an its small amplitude. Finally the lower Figure shown the evolution in time of the one-bit modulated signal δ.

VI. CONCLUSIONS

In this paper we have investigated the stability properties of the Delta-modulation coding rule, when used as a transmission mean in networked controlled linear systems. It was first shown that the standard form of the Δ-M algorithm can be modified, including information about the system and the controller, to enhance the attraction domain to which the closed-loop system is shown to be stable. These results were extended to the case of output regulation, and finally we shown that in the discrete-time case, a trade-off between system precision and size of the stability domain can be assessed.

Fig. 1 .

 1 Fig. 1. Block diagram of the problem set up studied in this paper.

Fig. 2 .

 2 Fig. 2. Block diagram of standard continuous-time version of the Δ-M coder.

Fig. 3 .

 3 Fig. 3. Block diagram of standard continuous-time version of the Δ-M coder with the proposed modification.

  ) with, A c = (A -BK); |λ i {A c }| < 1, and xk = x kxk , ∈ R n , Δ ∈ R n×n . Note that the sign function depends on the transform coordinates zk = T xk , and sgn(z k ) = sgn(T xk ) = [sgn(z 1,k ), sgn(z 2,k ), • • • sgn(z n,k )]T

Fig. 4 .

 4 Fig. 4. Simulation results

Adaptive Delta modulation (AΔ-M ), and Sigma-Delta modulation (ΣΔ-M ).

The pole of the discrete-time system is related to the pole, ω ol , of the open-loop continuous, one as a = e ω ol Ts .

ACKNOWLEDGEMENTS.

Thanks are due to Prof. Sanjoy Mitter for their comments and discussion on the issue of this paper. The authors would like to acknowledge MCYT for funding this work under grants DPI2004-06419, HF2003-0237 and SAB2003-0085. Thanks are also due to the programm PICASSO No. 07261YJ(EGIDE), of the French Minister of foreign affaires.


Fabio Gomez Estern

Carlos Canudas De Wit

Francisco R Rubio

Jose Fornes

Fabio Gómez-Estern

Carlos Canudas-De-Wit

José Fornés

Adaptative delta-modulation coding for networked controlled systems

Keywords: Differential coding, delta modulation, stabilization of linear systems in Networked controlled systems

come

I. INTRODUCTION

This paper proposes an adaptive extension a of fixed-gain differential coding scheme previously introduced by the same authors (see [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF]) in the context of linear systems interconnected through some transmission network. The problem is of interest in the area of Networked Controlled Systems (NCS), where we find several applications calling for datacompression algorithms aiming at reducing the amount of information that may be transmitted throughout the communication channel.

Delta modulation (∆-M ) is a well-known differential coding technique used for reducing the data rate required for voice communication, see [START_REF] Proakis | Digital Communications[END_REF]. The standard technique is based on synchronizing a state predictor on emitter and receiver and just sending a onebit error signal corresponding to the innovation of the sampled data with respect to the predictor. The prediction is then updated by adding a positive or negative quantity (determined by the bit that has been transmitted) of absolute value ∆, a known parameter shared between emitter and receiver.

In [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] we have analyzed stability issues that appear when the (fixed-gain) ∆-modulation scheme was embedded in a control system between the device measuring the state and the actuator. Delta modulation (∆-M ) algorithm can also be understood as the coarsest two-level (1-bit) quantizer. Thus, this technique is a simple alternative to works concerning the C. Canudas-de-Wit is with The Laboratoire d'Automatique de Grenoble, UMR CNRS 5528, ENSIEG-INPG, B.P. 46, 38 402, ST. Martin d´Hères, FRANCE. Email: carlos.canudas-de-wit@inpg.fr.

Fabio Gómez-Estern, Francisco R. Rubio and José Fornés, are with the Department of Automatic Control and Systems Engineering at the University of Sevilla. Email: rubio@esi.us.es, fornes@trajano.us.es, fabio@esi.us.es use of quantizers in the context of NCS, i.e. [START_REF] Ishii | Remote control of lti systems over networks with state quatization[END_REF], [START_REF] Mitter | Stabilization of linear systems with limited information[END_REF], [START_REF] Liberzon | On stabilization of linear systems with limited information[END_REF], [START_REF] Lemmon | Control system performance under dynamic quatization: the scalar case[END_REF], [START_REF] Tan | Numerical study of joint quatization and control under block-coding[END_REF], [START_REF] Li | Robust quatization for diginal finite communication bandwidth (dfcb) control[END_REF], among others. Compared to these approaches, the modified form of the (∆-M ) algorithm proposed in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], has the ability of providing a kind of separation principle, where the feedback gain if first designed assuming instantaneous unconstrained communication, and then the coding scheme is designed in order to preserve stability when embedded in the feedback loop. It is worth to remark that this modified form of the ∆-M coding structure explicitly contains information about the model plant and the controller feedback gain.

For a constant ∆ it was shown in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] that only a limited domain of attraction was obtained. In addition, the state was only guaranteed to converge asymptotically to a finite ball, begin its size related to the parameters of the open-loop plant, and on the the userdefined parameter ∆.

By making ∆ an adaptive quantity, more effective schemes of ∆-modulation have been already proposed in the communication community [START_REF] Proakis | Digital Communications[END_REF]. The idea is to design an update law for ∆, defined exclusively in terms of the information available both at the receiver and transmitter, aiming at improving the resolution of the differential coding by reducing the gain ∆ for slowly varying signals, while enlarging ∆ in case of rapid change of the input, and hence allowing for faster signal tracking and higher bandwidth of the transmitted signals. So far in the communication area, adaptation laws for ∆ have been proposed under somewhat heuristic criteria, as little information is supposed to be available on the dynamics of the source signal. However, when dealing with feedback systems, the dynamical properties of the plant become very useful in designing the adaptive law. This problem, to which this paper is devoted, is framed as shown in Figure 1.

The problem of quantization with time-varying resolution in feedback loops has been addressed in [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF] and [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF]. The first of these works presents, in the case of fixed resolution, a scheme similar to [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], in the sense that the state estimation is computed trough a filter built upon the closed-loop system matrix. However, the extension to variable-scale quantization is only defined in the zooming-in direction, and hence the initial states are upper bounded by the initial choice of the zoom factor. As a consequence, only semi-global stabilization is achieved. Here we propose a ∆-update law that works well for both in-and-out zooming directions, thus providing a means to capture unbounded initial states in the zooming-out stage and guaranteeing global asymptotic stability. Moreover, by defining an explicit ∆-update law in both directions, if the state is driven temporarily out of the domain of attraction at any time due to unattended disturbances, the system will recover the stability, unlike the case of [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF]. On the other hand, the work of [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF] also guarantees global asymptotic stability with time-varying full-state quantization in two subsequent stages. Although our approach can be roughly understood as a particular case of the Theorems provided there, an important new feature introduced here is a state predictor that reduces the amount of data transmitted per sample by only sending the quantized prediction error. This reduces the data-rate to a minimum of 1 bit per sample, while the data-rate in Theorem 3 of [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF] is log 2 (2M ), a value that cannot be taken arbitrarily low. M is actually defined as a function of the matrices A,B,K of the control system, and it must be large enough to make the (thereby defined) scale factor small. In Theorem 4 of [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF] a 1-bit per sample data-rate transmission scheme is also discussed, but here the separation principle is not present, as the feedback u = H(q(x)) is no longer a linear feedback of the state estimated on the receiver side. But probably the most significant difference between that approach and the one presented here is the zooming factor. In [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF], it is calculated in terms of the convergence of the state to an attractive ellipsoid, whose geometry depends in a complex way on the system matrices. In our approach the zoom factor is updated at all times with a simple law that only depends on the last two state estimations, irrespective of the system matrices (and hence not subject to the side effects of bad identification).

x k+1 = Ax k + Bu k u k K xk δ k = ±1 δ k = ±1 xk x k
First, we recall some of the results in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] showing the main modification imparted to the standard ∆-M algorithms, and its resulting stability properties under fixed-gain ∆. The main paper contribution is the introduction of an adaptation mechanisms consists of making time variant the quantization interval ∆. This type of adaptation law, although well known in the communication field, is used and analyzed for the first time in the feedback configuration shown in Figure 1. It is also shown that this adaptive coding structure, modified as proposed in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], is proven to yield closed-loop global asymptotic stability for a class of open-loop unstable linear systems.

II. FIXED-GAIN ∆-M CODING IN FEEDBACK

In this section we recall some of the results in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] which are pertinent for introducing the need for an adaptation law for ∆. To make the presentation as simplest as possible, we just present the one-dimension noiseless case. In reference [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] readers can find the extension of this results to the ndimensional case, the consideration of noise, and the case of output feedback. These results concerns the case of fixed modulation gain, under the feedback configuration shown in Figure 1.

A. Assumptions

The hypothesis used in the results presented in this section, but also all along the paper, are the following:

• The transmitted information is binary δ k ∈ {-1, 1}

• Only encoder-to-decoder information transmission is allowed (feedback between decoder to encoder is forbidden), • Reliable noiseless channel transmission is considered (no data lost, or information distortion is considered),

• Transmission delays are neglected, • Channel rate capacity is R[b.p.s.],
• Data is sent at a bounded rate (we select the sampling frequency in order to transmit only one δ k at a time).

B. One-dimension system example

Consider the following one-dimensional discrete timesystem, together with the control law, and the differential coding modified law:

• Open-loop system, and encoder:

x k+1 = ax k + bu k (1) xk+1 = [a -bK]x k + ∆ • δ k (2) δ k = sgn(x k ) (3)
• Transmitted information:

δ k ∈ {-1, 1},
• Control law and decoder:

xk+1 = [a -bK]x k + ∆ • δ k (4) u k = -K xk (5) with, K ∈ R, a 1, a c = (a -bK); |a c | < 1, and xk = x k -xk .
The modified differential coder (2)-( 4) differs from its standard form in that the term within the square brackets depends on the system model parameters a, and b, and on the control gain K. In the standard form this term is equal to one, i.e. the encoder is described by a delayed integral term. The advantages of this modification is that the coding error equations become decoupled from the system state equations, allowing for a simple analysis, and more important, for a design of the feedback gain K independent to the coding structure (sort of separation principle). Therefore, the closedloop matrix a c can be designed by the sole and well known stabilization conditions for linear systems.

This algorithm gives the following error equations, with cascade structure:

x k+1 = a c x k + bK xk (6) xk+1 = ax k -∆sgn(x k ) (7)
as |a c | < 1, stability of the whole system can thus be tacked by only studying the stability of the coding error equation [START_REF] Li | Robust quatization for diginal finite communication bandwidth (dfcb) control[END_REF], as is presented in Proposition 3 of [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF]).

Below we provide a modified version of that Proposition where an additional restriction on parameter a is considered. [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF]). Consider system (1)-( 5), with constant ∆ and a < 2. Then if the initial conditions of the coding error are such that

Theorem 1: ∆-M ALGORITHM WITH FIXED-GAIN (MODIFICATION OF
|x 0 | < r 2
then the following hold:

• |x k | < r 2 , ∀k 0, • ∃k 0 : |x k | r 1 ∀k k 0 , and • lim k→∞ d(x k , B γ ) = 0.
where r 1 and r 2 are:

r 1 = a -1 a 2 -1 ∆, r 2 = a + 1 a 2 -1 ∆ and d(x k , B γ )
is the minimum distance from x k to any point within the interval

B γ := {x ∈ R : |x| < γ}, γ = Kb 1 -a c r 1
Proof: The main argument can be read in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], and the following analysis justifies the upper bound on a. Let

V k = x2
k , and

∇V k = V k+1 -V k . Then, ∇V k = x2 k+1 -x2 k = (a 2 -1)x 2 k -2a∆|x k | + ∆ 2
This expression is negative on the region

r 1 < |x k | < r 2 .
However, when |x k | < r 1 , ∇V k takes positive values, up to a maximum of ∆ 2 at xk0 = 0. In that case, ∇V k = ∆ 2 and at the following sampling time |x k0+1 | = ∆. In order to ensure that the state remains within the region |x k | < r 2 it is required that

∆ < r 2 = a + 1 a 2 -1 ∆
and this is satisfied if and only if a < 2, so this is the condition required for asymptotic stability.

For some constant ∆, this results means that if the initial condition |x 0 | are taken smaller than r 2 , the error coding variable is locally attracted to a threshold delimited by the value of r 1 , which depends linearly on the coding gain ∆. This gains also delimits the stability border r 2 , as shown the above expressions. Therefore, larger values for ∆ will make the system more stable but less precise, and inversely reduction of the gain ∆ will lead to small estimation error, but will at the same time reduce the domain where the system is keep stable.

This result displays an inherent trade-off between stability and precision when the gain ∆ is fixed. This suggests the search for other coding strategies with variant gains. Note also that, as the sampling time1 is chosen small, a approaches 1 and the precision is increased. Indeed, lim a→1 r 1 (a) = 0, thus by making T s infinitely small, the limit case of the continuous-time infinite precision is approached.

III. ADAPTIVE ∆-M CODING SCHEME

In this section we propose a ∆-adaptation law resulting in a global asymptotic convergence of the estimation error and system states to zero. This is a significant achievement with respect to the fixed-gain scheme presented before which was limited to finite domains of attraction (though arbitrarily large) and convergence to finite balls.

A. Adaptation law design

Adaptive ∆-modulation aims at improving the resolution of the differential coding scheme according to the size of the signals to be transmitted (in our case the states), hence a reasonable approach is to enlarge ∆ k for large values of the estimated states, and decrease it for smaller values. This is the main guideline for designing an adaptation law with the premise that the variations assigned to ∆ k should be defined exclusively in terms of the information available both at the receiver and transmitter; that is on δ k .

In order to design an ∆-adaptive mechanism to achieve global asymptotic stability two opposite behaviors must be observed. These stem from the Theorem 1:

• For large values of the state x when the condition for state decrease is not fulfill, i.e. |x k | r 2 , ∆ k must grow monotonically at a higher rate than the plant escape velocity. This suggest an exponential growth law for ∆ k when the state is detected to be escaping from the origin.

• When the state is trapped into a domain of attraction |x k | < r 2 , ∆ k must decrease (for improving resolution) at a slower rate than the state convergence in order to preventing it from getting to small relative to the state and hence breaking again condition |x k | < r 2 needed for convergence. These behaviors indicate a trade-off design of the exponential reduction rate of ∆ k as compared to the potential growing (decreasing) rate of the open-loop plant. A key question underlying these issues is the way receiver supposed to distinguish between the cases where the error escaping or converging towards the origin. As x is not available at the decoder side, the growth law will be designed on the basis of the transmitted data, i.e. δ k = sgn(x k ) and, possible, on its past values.

Based on the previous considerations, we propose an adaptive scheme with minimal storage and computation power requirements for updating ∆ k based on the following criteria:

1) If δ k = δ k-1 then the state is assumed to be escaping, thus ∆ k must be increased. 2) If δ k = δ k-1 then the state is assumed to converge (oscillating close to zero) and ∆ k must be decreased. The following update law is proposed:

∆ k+1 = φ k+1 ∆ k (8)
φ k+1 = λ -+ 1 2 (λ + -λ -) |δ k+1 + δ k | (9)
where 0 < λ -< 1 is exponential decay rate of ∆ k , and λ + > 1 is exponential growth rate. This adaptation law can be seen as a generalization of the (1970) adaptation Jayant's rule which can be consulted in [START_REF] Proakis | Digital Communications[END_REF]. The Jayant's rule is described by

∆ k+1 = λ δ k+1 δ k 0 ∆ k , with λ 0 1.
Hence this is a special case of our proposed rule with λ + = λ 0 , and λ -= 1/λ 0 . As it will be seen later, allowing λ + and λ -to be freely choosing, the stability conditions turnout to be less conservative.

B. Error equations

The complete feedback systems with the adaptive delta ∆-modulation coding scheme is then:

• Open-loop system [START_REF] Tan | Numerical study of joint quatization and control under block-coding[END_REF], and encoder ( 11)-( 13):

x k+1 = ax k + bu k (10) xk+1 = [a -bK]x k + ∆ k • δ k (11) ∆ k+1 = φ k+1 ∆ k ( 12
)
φ k+1 = λ -+ 1 2 (λ + -λ -) |δ k+1 + δ k | (13)
• Decoder ( 14)-( 16) and control law (17):

xk+1 = [a -bK]x k + ∆ k • δ k (14) ∆ k+1 = φ k+1 ∆ k ( 15
)
φ k+1 = λ -+ 1 2 (λ + -λ -) |δ k+1 + δ k | (16) u k = -K xk (17)
With the above definitions, the closed-loop error dynamics become

xk+1 = ax k -∆ k • δ k ∆ k+1 = φ k+1 ∆ k , ∆ 0 > 0 ( 18
)
φ k+1 = λ -+ 1 2 (λ + -λ -) |δ k+1 + δ k |
The causality of the system is guaranteed by the fact that all times the computation of xk+1 is only based on the xk and older values. The following Proposition states the stability of the closed-loop system

C. Main result (stability analysis)

Theorem 2: The error trajectories xk of system (18), resulting from the adaptive ∆-modulation coding scheme ( 10)-( 17), globally asymptotically converge to zero as k → ∞ if there exist parameters λ + > 1, λ -∈ (0, 1) satisfying the following inequalities:

λ + > a (19) λ -< (λ + ) -β 2 , (20)
where

β(a, λ -, λ + ) △ = 1 + log ρ 1 + a (a -λ -) (ρ -1) (λ -) 2 ρ △ = λ + a , Moreover, ∆ k also converges to zero regardless it initial value ∆ 0 .
Proof: The claim will be proved in two steps. First, a new variable is defined in order to capture the proportion between xk and ∆ k , namely

y k △ = xk ∆ k
and boundedness of that variable will be proved. Secondly, it will be shown that ∆ k asymptotically converges to zero. Consequently, convergence of xk to the origin is directly implied.

Along the trajectories (18), variable y k evolves along the dynamics

y k+1 = 1 φ k (ay k -sgn(y k )) . ( 21
)
Fact 1. Trajectories of y k cross the zero axis in finite time.

This means that starting from any initial condition, x0 and ∆ 0 , and thus y 0 , there must be a future time k 0 < ∞ such that y k0-1 • y k0 < 0. This is easily shown by imagining a trajectory with no zero crossings on y k (hence on xk as ∆ k is always positive). Assuming initially positive x, i.e. starting from y 0 > 0, we have,

y k+1 = 1 λ + (ay k -1) , (22)
hence

y k+1 -y k = 1 λ + (ay k -1)-y k = a λ + -1 y k - 1 λ + < - 1 λ +
for the given choice of λ. Then, starting from y 0 > 0, we have

y k < y 0 - k λ + ,
and hence there is some constant k 0 y 0 λ + such that y k0-1 > 0 and y k0 < 0. Due to the symmetry of the system equations, a similar argument applies if the trajectory starts from y 0 < 0.

Fact 2. Trajectories of y k are bounded after finite time.

Immediately after a zero crossing of the system (assuming + to -without loss of generality) , y k0 is bounded as follows

y k0 = 1 λ + (ay k0-1 -1) > - 1 λ + (23)
hence |y k0 | < 1 λ + , as y k0-1 > 0 and y k0 < 0. Now if we search for bounds on subsequent samples we must update the growth factor of ∆ to λ -and compute

y k0+1 = 1 λ -(ay k0 + 1) , ( 24
)
and it is easy to see that the the right hand side of the last expression is positive, as from the bound 0 > y k0 > -1/λ + we have

y k0+1 > 1 λ -- a λ + + 1 > 0.
Moreover, |y k0+1 | < 1/λ -. These observations are summarized in Fig. 2, where the necessity of double zero crossing after a set of positive values is illustrated, as well as the upper bounds inferred by the switching dynamics. In order to conclude the analysis on the bounds, one step further must be taken on the switching dynamics. Using 0 < y k0+1 < 1/λ -, we have

y k0+2 = 1 λ -(ay k0+1 -1) ,
and hence 2, two situations (a) and (b) are then identified, (a) -1 λ -< y k0+2 < 0. This situation (including the norm bound) is exactly the one found at instant k 0 + 1, then it has been already considered (b) y k0+2 > 0. In this case we have two subsequent positive samples, i.e. the situation of y k0-1 is recovered. In that case, the dynamic equation turns into (23) and along it the map y k → y k+1 is contracting, hence the norm will be decrease until a future sign change. The previous analysis yields that after the first sign change, the state y k remains bounded as

- 1 λ -< y k0+2 < 1 λ - a λ --1 , As illustrated in Fig.
|y k | < 1 λ - a λ --1 (25)
With the above facts, the proof of the proposition reduces to show asymptotic convergence to zero of ∆ k , and hence concluding convergence o the state x. With this objective we will use the following definition, Definition 3.1: Given a sequence of positive (negative) samples of y k , the fly-time is the number of sampling instants elapsed between the two zero-crossings that enclose the signal. For its computation, the first and the last positive (negative) samples are considered. This magnitude is viewed in Fig. 3. We will compute a non-conservative upper bound on it. Indeed, considering y 0 as the first positive sample (i.e. resetting the time count), we have along the dynamics (23),

y k = ρ k y 0 - 1 a ρ k -1 ρ -1 (26)
then, the zero crossing occurs at the next sampling instant after the time the right hand side of this equation vanishes. Therefore, the fly-time k * is bounded as

k * 1 + log ρ (1 + y 0 a(ρ -1)) 1 + log ρ 1 + a (a -λ -) (ρ -1) (λ -) 2 = β(a, λ -, λ + )
for which we have used the finite-time bound on y k (and hence y 0 ) computed in (25).

On the other hand, the fly-time can be regarded as the exact number of times the ∆ factor is increased via the exponential law ∆ k+1 = λ + ∆ k , then the net value of ∆ after the flying period, starting from ∆ 0 is

∆ k = (λ + ) k ∆ 0 ,
Hence a condition for asymptotic convergence of ∆ k to zero is that, on zero crossings, the net decrease in ∆ compensates the net amount increase over the flying period. This is guaranteed by choosing λ -such that, considering k 0 as the first negative sample (again as in Fig. 2)

∆ k0+2 = (λ -) 2 ∆ k0 < (λ -) 2 (λ + ) β ∆ 0
where the power in (λ -) 2 has been introduced using the fact that after a flying period, two consecutive zero crossings must occur (see argument of Fact 2). This gives a less restrictive condition on λ -. Hence, the condition of net decrease of ∆ after a flying period is

∆ k0+2 < ∆ 0 ⇐ (λ -) 2 (λ + ) β < 1, or, equivalently, λ -< (λ + ) -β 2 ,
which is the condition (20) stated at the Proposition. ¿From the net convergence of ∆ k to zero and the boundedness of y k , we conclude that

lim k→∞ xk = 0
This completes the proof.

IV. PARAMETER VALIDITY REGION

The choice of parameters λ + and λ -fulfilling the stability conditions ( 19)-( 20) is nontrivial. Indeed, (20) is an implicit equation where λ -appears both at the left and the right hand sides, and the solvability depends on the particular value of a.

Defining ψ(λ + , λ -, a) 20), it is clear that there is a set of valid parameters if and only if ψ(•) becomes negative somewhere in the IR 2 region

△ = λ --(λ + ) -β 2 from (
[λ -∈ (0, 1)] × [λ + ∈ (a, ∞)],
however, some calculations that will be presented elsewhere show that any solution of (20) must lie within the set

Ω a : [λ -∈ (0, 1)] × [λ + ∈ (a, λ + max (a))],
where

λ + max (a) △ = a 1 + 1 + 4 a(a-1)

,

For a given value of the open-loop pole a, if there is no pair (λ + , λ -) ∈ Ω a making ψ(•) negative, then there is no valid solution for ( 19)-(20). Therefore, Ω a is the region that should be numerically scanned for solutions. Unfortunately, for some values a there is no such solution, while, as can be easily calculated, the condition a < 1.313 guarantees the existence of solutions.

Fig. 4 (left) clearly shows that for a = 1.2 there is a set of values of λ + and λ -for which the plotted expression is negative, i.e. the conditions are fulfilled. On the other hand, for values of a 1.313 (a = 1.4 in Fig. 4) (right), nothing can be said about the existence of admissible parameters, but it can be observed that the whole surface ψ is above zero and no solution has been found numerically. Moreover, as the right hand side of (20) decreases with a, there will be no more solutions for greater a.

V. RELATION WITH THE N & S CONDITION FOR STABILIZATION UNDER CHANNEL LIMITATIONS.

A further issue that must be addressed with respect to the adaptive ∆-M scheme is the question wether the limitation on the open-loop poles (parameter a) to be below 1.6180 is a structural property or a limitation due to the sufficient nature of the result.

In any case, our limitation should be consistent with the necessary and sufficient condition for stabilization presented in [START_REF] Tatikonda | Control under communication constraints[END_REF]. This condition indicates that the minimal data rate required for stabilizing a discrete-time system via a communication channel of maximum rate capacity R [b.p.u.] 2 is related to the unstable open-loop poles (λ un i ) as: R > log 2 (λ un i ), which in our case simplifies to: R > log 2 a (27)

The implicit assumption made within the framework of our discrete-time formulation is that the channel can reliably transmit one bit-per-unit of time, that is that R = 1. This means that with regard to the condition (27), the maximum admissible value for a is a < 2, which is consistent with our sufficient condition a < 1.313, probably due to the technicalities used for the stability analysis, or either due to the particular structure of the proposed adaptation law. This also indicates that there is some conservatism in the computation of the admissible set of the parameter λ + and λ -. Some alternative adaptation strategies could be devised in order to improve this bounds. For example, a mores sophisticated adaptive algorithm based on more past samples of δ k could be possible designed. For the sake of comparison with the non-adaptive scheme presented in [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF], a first simulation has been made with a constante value of ∆. Fig. 5 shows the behavior of the closed-loop system when the ∆ is given fixed values, (it is only changed at the times the set-point changes). As can be seen, the granularity of the closed-loop error signal is directly related to the choice of the constant ∆. During the simulation, after the first step change of the reference, ∆ has been made larger and the granularity significantly increases. The main issue of [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] was that ∆ cannot be fixed at a very low value (thus reducing the granularity) without compromising the domain of attraction.

Fortunately, this has been successfully tackled with the new adaptive approach, as is illustrated in Fig. 6. In the upper plots of this figure, the state x k , x and the set-point r k are depicted, while the adaptive quantization parameter ∆ k is plotted on the lower graph. As expected, convergence to zero is obtained in all cases. The granularity in the steady state is due to the fact that ∆ k has not been allowed to take values below ∆ min = 0.05. Without this saturation, ∆ k would keep tending to zero in steady state, and whenever a disturbance drives the state away from the origin, a large number of samples would be required for [START_REF] Liberzon | On stabilization of linear systems with limited information[END_REF] to make ∆ k large enough to capture again such state (i.e. the transient time would undesirably depend on the time the system spends on the steady state).

VII. CONCLUSIONS

In this paper we have investigated the stability properties of the Delta-modulation coding rule, when used as a transmission mean in networked controlled linear systems. It was first shown that the standard form of the ∆-M algorithm can be modified, including information about the system and the controller. These results were extended to the case of adaptive ∆ k . An explicit adaptation rule was proposed and range of parameters were derived to ensure asymptotic stability. These results displayed a limit on the maximum unstable eigenvalues of the system that are compatible with the ones given in [START_REF] Tatikonda | Control under communication constraints[END_REF].

VIII. ACKNOWLEDGEMENTS.

Special thanks are due to Prof. Chaouki Abdallah and Ivan Lopez for indicating the additional condition on parameter a in Theorem 1. Thanks are also due to Prof. Sanjoy Mitter for his comments and discussion on the issue of this paper.

Fig. 1 .

 1 Fig. 1. Block diagram of the problem set up studied in this paper.

Fig. 2 .

 2 Fig. 2. Behavior of y k at zero crossings.

Fig. 3 .

 3 Fig. 3. Definition of fly-time.

Fig. 5 .

 5 Fig. 5. Simulation results for non-adaptive ∆-M scheme, with ∆ fixed along specific intervals.

  VI. SIMULATIONS System (10)-(17) has been simulated for the set of values a = 1.1, b = 1, K = 0.2, x k (0) = -0.5, xk (0) = 0, with ∆ 0 = 5, λ -= 0.4, λ + = 1.21, according to conditions (19) and (20).

Fig. 4 .

 4 Fig. 4. (Left) Expression λ - -(λ + ) -β 2 for a = 1.2; (Right) The same expression for a = 1.4.

Fig. 6 .

 6 Fig. 6. Simulation results for adaptive ∆-M scheme.

The pole of the discrete-time system is related to the pole, ω ol , of the open-loop continuous, one as a = e ω ol Ts .

The authors would like to acknowledge MCYT-FEDER for funding this work under grants DPI2004-06419, HF2003-0237 and SAB2003-0085. Thanks for funding are also due to the programm PICASSO No. 07261YJ(EGIDE), of the French Minister of foreign affaires.