Delta-Modulation Coding Redesign for Feedback-Controlled...

2684 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 7, JULY 2009

Delta-Modulation Coding Redesign forFeedback-Controlled Systems

Carlos Canudas-de-Wit, Member, IEEE, Fabio Gómez-Estern, Member, IEEE, andFrancisco Rodríguez Rubio, Member, IEEE

Abstract—This paper investigates the closed-loop properties ofthe differential coding scheme known as Delta-modulation (Δ-M )when used in feedback loops within the context of feedback-controlled systems. We propose a new modified scheme of theoriginal form of the Δ-M algorithm which is better suited forapplications where the sensed information is used in feedback. Astate feedback controller is implemented with the state estimatedby a predictor-based differential decoder. Stability of the resultingclosed-loop system (controller–coder–decoder) is studied. Theseproperties (stability and performance) depend on the quantizationparameter Δ, which is assumed constant in the first part of thispaper. In a further step, parameter Δ is made adaptive, by defin-ing an adaptation law exclusively in terms of information availableat both the transmitter and receiver side. With this approach bothstability and performance is improved.

Index Terms—Delta-modulation (Δ-M ), differential coding,networked controlled systems (NCS).

I. INTRODUCTION

D ELTA-MODULATION (Δ-M ) is a differential codingtechnique long used for reducing the data rate required

for voice communication (see [1]). The standard techniqueis based on synchronizing a state predictor on emitter andreceiver and just sending a 1-b error signal corresponding tothe innovation of the sampled data with respect to the predictor.The prediction is then updated by adding a positive or negativequantity (determined by the bit that has been transmitted) ofabsolute value Δ, a known parameter shared between emitterand receiver.

This paper studies different aspects of the use of differentialcoding as a means for transmitting sensor signals in feedbackcontrolled linear systems. In particular, we focus on stabilityissues that appear when the Δ-modulation scheme is embeddedin a control system between the measuring device and the con-troller. The problem is of interest in the area of networked con-

Manuscript received July 30, 2008; revised January 5, 2009. First publishedApril 17, 2009; current version published July 1, 2009. This work was sup-ported in part by the European Commission under project FeedNetBack underContract INFSO-ICT-223866 and in part by the Spanish MCINN under GrantsDPI2007-64697 and DPI2006-07338. The work of F. R. Rubio was supportedby French INRIA-RA, September–October 2008, Grenoble.

C. Canudas-de-Wit is with the Laboratoire d’Automatique de Grenoble,Unite Mixte de Recherche, Centre National de la Recherche Scientifique5528, Department of Automatic Control, Polytechnic of Grenoble, 38031Grenoble, France (e-mail: [email protected]).

F. Gómez-Estern and F. R. Rubio are with the Department of SystemsEngineering and Automation, University of Seville, 41092 Seville, Spain(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2009.2020079

trolled systems (NCS) calling for data-compression algorithms,but also in digital control applications where sensed informa-tion is converted using analog-to-digital converters (ADCs)with few bits (i.e., 1-b ADC).

A. Δ-M as a Coding Mechanism in NCS

NCS is a multidisciplinary field where control and commu-nications technologies and information theory meet. Whereastheir advantages are easily understood, the problems arisenare complex. In one sense, stability may be compromised bythe inclusion of a network, and new analysis must be done,with specific quantization–modulation considerations (see, forinstance, [2] and the references therein). Another approachis to monitor the network quality of service parameters, toconsequentially tune the controller gains for maximum stableperformance, as is presented in [3] and [4]. Finally, codesign(see [5]) addresses simultaneously the modification of the con-trol strategy and the message transmission plan.

Real-time wireless control applications, as shown in Fig. 1,seek for data-compression algorithms for reducing the infor-mation transmitted throughout the communication channel,leading to better resource allocation and/or an improvement ofthe necessary data rates. Moreover, recent advances are stronglyfocused on energy consumption, which is probably the key tocheap distributed sensing. Vendors of technologies like ZigBeeand Bluetooth now claim that devices may be powered onthe same battery for years, opening a way to nonreplaceablebattery devices into the market. Energy consumption is thusa goal together with bandwidth when dealing with modernsensor networks, and new communications technologies. Thesearch for minimal data transmission in control has also foundmotivation in underwater applications, where data rates arenormally bounded below 100 b/s due to the strongly dissipativemedium.

The Δ-M algorithm can be understood as the coarsest two-level (1-b) quantizer. Optimization of the quantization levels ismandatory in large-scale systems, and may be of great interestwhile designing low-cost transmitter/receiver components. Dif-ferential coding in feedback system is thus a simple alternativeto reported control schemes concerning the use of quantizers inthe context of NCS, i.e., [2], [6]–[10]. More precisely, the useof delta-modulators for networked control has been reported in[11], as a practical implementation without theoretical stabilityanalysis.

The modified form of the (Δ-M ) algorithm proposed here,has the ability of enforce the separation principle between the

0278-0046/$25.00 © 2009 IEEE

Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 13, 2009 at 03:36 from IEEE Xplore. Restrictions apply.

CANUDAS-DE-WIT et al.: DELTA-MODULATION CODING REDESIGN FOR FEEDBACK-CONTROLLED SYSTEMS 2685

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

Fig. 2. Block diagram of MEMS accelerometer using Δ-M both in the inner analog capacitance-to-digital conversion and in the outer force-feedback loop.

control law and the estimation process (coding strategy). As aresult, the (linear) feedback law is first designed disregardingthe coding algorithm (x ≡ x). Then, the coding scheme isdesigned in order to preserve stability when embedded in thefeedback loop. It is also worth to mention that this modifiedform of the Δ-M coding structure explicitly contains informa-tion about the model plant and the controller feedback gain.

Others works leading with some variants of the Δ-M codingin feedback systems can be found in [12] for energy-awarecoding, [13] for the combination of Δ-M with entropy variablelength coding, [14] for the use of more than 1 b, and [15] formultivariable coding in the fixed quantization case.

B. Δ-M as 1-b Converter

The other possible scenario where our results will be ofinterest concerns embedded control systems where the sensorsand digital controller are embedded in the same electronics(system on chip). With the aim of saving energy, cost, andspace in the silicon wafers, low-resolution ADCs are preferableto high-resolution ones. Numerous examples of the use ofDelta-modulators embedded, in general, microelectronic de-vices have been reported in the literature. For instance, [16]and [17] use field-programmable gate arrays and applicationspecified integrated circuits for implementing Delta and Sigma-Delta-modulation A/D converters for current and voltage mea-surement. The need for oversampling the analog signals, iscompensated by the technological advantage of reducing thenumber of ADC channels and comparators.

Interesting applications using Δ-M inside control loops arealso found in gyroscopic and acceleration sensors in MEMS,which are configured, as shown in Fig. 2. The principle, asdescribed in [18] and [19], is based on feedback—sensing thecapacitive mass displacement produced by an acceleration on

the device. In [18] and [20], this analogous mass displacementis converted to a digital signal by using 1-b Sigma-Delta mod-ulation encoders. Then, the 1-b signal is decoded and used infeedback to “cancel” the external acceleration that disturbs thesmall mass motion. If the control is successfully designed, thenthe control signal equals the external acceleration providingthe desired measure. Note that this example is similar to theframework shown in Fig. 1 with the difference that the encodersignals are transmitted by wire.

C. Main Paper Contribution

This paper’s contributions are presented in the following or-der. First, we study the case of the Δ-M algorithm under fixed-gain Δ. We initially address the continuous-time formulationcase, which is relevant because it captures the limiting caseof the discrete-time formulation (finite data rate transmission),and provides a better understanding of the maximal stabilityproperties that can be attained. It also shows the substantialsimplification that can be reached in the stability analysis, aswell as the separation properties between states and estimationobtained by using the proposed Δ-M modified form. In thispart, we study the closed-loop stability properties of the originalform of the Δ-M algorithm. Then, we propose a new codinglaw that improves over the original form of the Δ-M algorithm,and it allows for the separation principle mentioned previously.

Based in this modified form, we then study the discrete-timecase, which permits a more clear assessment of the data rateconstraints. It is shown that the system states converge to a ballenclosing the origin, and that the result is semiglobal. The sizeof the attraction region and the system precision, depends onthe coding gain Δ linearly; it is also a function of the locationof the unstable discrete-time open-loop poles, which, in turn,are related to the sampling time.



The last part of this paper illustrates an adaptation mecha-nism, consisting of making time variant the quantization inter-val Δ. In our adaptive approach, closed-loop global asymptoticstability is proved for the noiseless case. The effect of randomnoise in the performance of the adaptive quantization scheme isalso analyzed via numerical simulations.

D. Comparison With Other Approaches

The problem of quantization with time-varying resolutionin feedback loops has been addressed in [21] and [22]. Thefirst of these works presents, in the case of fixed resolution, ascheme similar to [23], in the sense that the state estimationis computed trough a filter built upon the closed-loop systemmatrix. However, the extension to variable-scale quantizationis only defined in the zooming-in direction, and hence theinitial states are upper bounded by the initial choice of thezoom factor. As a consequence, only semiglobal stabilizationis achieved. Here, we propose a Δ-update law that works wellfor both in-and-out zooming directions, thus providing a meansto capture unbounded initial states in the zooming-out stage andguaranteeing global asymptotic stability. Moreover, by definingan explicit Δ-update law in both directions, if the state isdriven temporarily out of the domain of attraction (DOA) at anytime due to unattended disturbances, the system will recoverstability, unlike the case of [21].

On the other hand, the work of [22] also guarantees globalasymptotic stability with time-varying full-state quantization intwo subsequent stages. Although our approach can be roughlyunderstood as a particular case of the theorems provided there,an important new feature introduced here is a state predictorthat reduces the amount of data transmitted per sample by onlysending the quantized prediction error. This reduces the datarate to a minimum of 1 b per sample, while the data rate in[22, Th. 3] is log2(2M), a value that cannot be taken arbitrarilylow. M is actually defined as a function of the matrices A, B,K of the control system, and it must be large enough to makethe (thereby defined) scale factor small. In [22, Th. 4], a 1-bper sample data-rate transmission scheme is also discussed, buthere the separation principle is not present, as the feedback u =H(q(x)) is no longer a linear feedback of the state estimatedon the receiver side. However, probably the most significantdifference between that approach and the one presented hereis the zooming factor. In [22], it is calculated in terms ofthe convergence of the state to an attractive ellipsoid, whosegeometry depends in a complex way on the system matrices.In our approach, the zoom factor is updated at all times with asimple law that only depends on the last two state estimations,irrespective of the system matrices (and hence not subject to theside effects of bad identification).

E. Definitions and Notation

Let x ∈ Rn, A ∈ R

n×n. The following definitions and nota-tion are in order.

1) ‖x‖2 =∑n

i=1 x2i .

2) |x| =∑n

i=1 |xi|, where |x| ≥ ‖x‖.

Fig. 3. Block diagram of the continuous-time version of the problem.

3) sgn(x) = [ sgn(x1), sgn(x2), . . . , sgn(xn)]T, withsgn(0) = ±1 arbitrarily.1

4) ‖A‖ is the induced Euclidean norm of A.

F. Control Setup

The control setup under study is shown in Fig. 3, whichdescribes a closed-loop system under one-way communicationchannel. The control computations are assumed to be doneat the plant side, whereas the sensor is remotely located. In-formation from sensor to controller is transmitted through anetwork, and coded by the differential encoder/decoder mapsΦE : x �→ δ, and ΦD : δ �→ x, where δ(t) is the encoded signalvector of dimension n, with only two-valued elements δi(t) ∈{−1, 1} ∀n = 1, 2, . . . , n, and x(t) ∈ R

n is the estimated valueof x as obtained from the decoding map ΦE . This architectureis commonly found in the NCS literature where a commonactuator is used for controlling a system with distributed sensors(probably a large number of low-cost wireless devices). Somemotivating practical applications are as follows.

1) Energy-efficiency control in buildings, where a set oflow-power temperature and humidity sensors are placedall over the premises, in order to gather distributed infor-mation and use it together with heat flow or energy con-sumption models to design optimal control strategies. Inthose cases, the actuators are normally few and large insize (air conditioning units, window openers,. . .), and theyare wired to the controller (when not directly built-in).

2) Another application where a great research effort is beingdone is in underwater vehicle control. In a scenario wherea set of underwater explorer robots are remotely con-trolled and coordinated by a surface vessel, the limitationson data rate appear exclusively on the upward link, as thesurface ship enjoys a significantly larger source of energyfor transmitting.

G. Assumptions

The hypotheses used all along this paper, are the following.1) The coding strategy is assumed to be scalar, i.e.,

each component xik of the vector signal xk is coded

1When our encoding algorithm requires that value, we will allow it to issuea + 1 or −1 arbitrarily.



Fig. 4. Block diagram of standard continuous-time version of the Δ-Mcoding scheme. The encoder ΦE is depicted at the left, where the decoder ΦD

is shown at the right. For n > 1, δ is defined as δ = [δ1, δ2, . . . , δn]T, withδi ∈ −1, 1.

independently of each other. Therefore, the encoding(respectively decoding ΦD) process is defined as a n-dimensional map, ΦE = [Φ1

E ,Φ2E , . . . Φn

E ]T, with ele-ments Φi

E : xi �→ δi.2) The encoded system outputs are binary, i.e., δi

k ∈{−1, 1}.

3) The maximum bit rate per unit of time is R (in bits persecond).

4) Transmission and ADC delays are neglected. Aspects re-lated to transmission delays have been studied elsewhere(see, for instance, [24] and [25]).

II. CONTINUOUS-TIME FORMULATION

To simplify matters, we first consider a fully continuous-time formulation. The discrete-time case, including data-rateconstraints, will be studied in subsequent sections. Consider alinear system, with linear feedback of the following form:

x(t) =Ax(t) + Bu(t) (1a)

u(t) = −Kx(t) (1b)

where x(t) ∈ Rn, A ∈ R

n×n, B ∈ Rn×m are controllable ma-

trices, u(t) is the input vector.

A. Differential Coding

There exists a large class of source coding algorithms aimingat compressing information for a more efficient data trans-mission. Differential coding schemes belong to the temporalwaveform coding class of algorithms. In differential coding,differences between successive samples are encoded rather thanthe samples themselves. Since the difference between samplesis expected to be smaller than the actual sample amplitude,fewer bits are required to represent the differences. Δ-M ,shown in Fig. 4, is the simplest form of differential coding(see [1]), in which a two-level (1-b) quantizer is used inconjunction with a first-order predictor. The continuous-timeversion of the coding algorithm is as follows.

1) Encoder map ΦE : x �→ δ

˙x(t) = Δ(t) · δ(t) x(0) = 0 (2a)

δ(t) = sgn (x(t) − x(t)) (2b)

where sgn(ζ) = [sgn(ζ1), sgn(ζ2), . . . , sgn(ζn)]T.2) Transmitted information: δi(t) ∈ {−1, 1}.

3) Decoder map ΦD : δ �→ x

x(t) =

t∫0

Δ(τ) · δ(τ)dτ, x(0) = 0. (3)

Δ(t) ∈ Rn×n is the step size matrix, which, in general, ischosen to be constant (time-varying, or adaptive gains can alsobe considered, as shown later in this paper). For designing sim-plicity, Δ(t) is frequently chosen to be diagonal. A mandatorychoice is that the gain Δ(t) must be strictly the same on bothsides of the coding process, in order for the decoder to workproperly.

To simplify notation, time arguments will be dropped inthis section in all variables when its explicit mention is notnecessary.

B. Coding Separation Principle

The complete coding process can be seen as an estima-tion process defined by the coding map ΦC : x �→ x, for theproblem of state feedback stabilization. In the case of reliablenoiseless channel transmission, ΦC is simply defined as ΦC =ΦE ◦ ΦD.

Behind the separation principle that will be advocated herelies the idea of first designing the feedback gain K disregardingthe coding algorithm; assuming that x ≡ x, and then designingthe coding map ΦC in such a way to preserve stability whenΦC is embedded in the feedback loop. In fact, the feedback lawdesign is separated from the design of the coding algorithm,while the converse is not true, as it will be demonstrated later.

With this in mind, we first consider here that the linearfeedback gain K is designed so that the matrix

Ac = A − BK

is strictly stable under the controllability assumption of thepair (A,B). Then, we will see that some modifications onthe standard Δ-M algorithm are in order, for this separationprinciple to be possible. Before introducing the modified formof the Δ-M algorithm, we first discuss the stability propertiesachievable with the standard form.

C. Standard Δ-M Coding Algorithm in Feedback

Consider the problem of finding a suitable value for Δ tostabilize system (1), under the coding law (2) and (3). It is easyto see that the error equations read as

x = Acx + BKx (4a)˙x = Acx + BKx︸︷︷︸

x

−Δsgn(x) (4b)

with x = x − x, and sgn(x) as defined in Section I-e. A firstobservation is that this system is fully coupled, and that thecontrol and coding law cannot be designed independently ofeach other. In particular, we note that the gain Δ should,somehow, be chosen large enough to locally dominate the rateof change of the system state, x. Clearly, this can be done onlyunder relatively “high-values” for Δ.



Fig. 5. Block diagram of standard continuous-time version of the Δ-M coderwith the proposed modification.

Putting aside technicalities in the sense of existence of solu-tions (i.e., Filippov’s type, etc.) due to the discontinuous right-hand side (RHS) of (4), which are not central for the discussionhere, the computation details given in Appendix I, show that ifΔ is constant, and of the form

Δ = Δ0 · In×n, Δ0 > 0

with Δ0 being a scalar, one can select a sufficiently largeΔ0, such as to ensure that the state and error trajectories ζ =(xT, xT)T tend to zero. That is, Δ0 must satisfy a condition ofthe type

Δ0 ≥ c · ‖ζ(0)‖

where c > 0 is a constant depending on the system and con-troller parameters.

D. New Structure for the Δ-M Coding

The previous discussion (summarized in Proposition 5)shows that relatively large gains for Δ0 will be required tostabilize the system, as the estimation of the bound on Δ0

would tend to be conservative, and the size of the region ofattraction is suited to be large. Moreover, in this continuous-time framework, high gains will result in an important “chatter-ing” effect, thus increasing the estimation error variance whenthe algorithm is discretized, or adding noise to the measuredsignal. Although the analysis is conservative, and probablylower values Δ0 may still be possible, it is interesting tostudy new structures of the Δ-M coding such as to reduce thenecessary gain for stabilization. The algorithm presented nextis a modification of the original form of the Δ-M algorithm,and most importantly, it yields an estimation error equationdecoupled from the system state x.

Consider now the following modified Δ-M algorithm:

˙x(t) = Acx + Δ(t) · δ(t) (5)

together with system (1), and δ(t) as defined before. Note that anew term Acx has been included in this case. As a consequence,the coding algorithm depends explicitly on the system modeland the feedback gain used. This equation describes both: theencoder and the decoder, as shown in the block diagram ofFig. 5. It can be shown that the new error equation resultingfrom this new structure is described as two systems in cascadeinterconnection, i.e.,

x = Acx + BKx (6a)˙x = Ax − Δ · sgn(x) (6b)

note that (6b) describes an autonomous system whose solutionis the input of the stable linear system (6a).

The stability properties of this algorithm are simpler toanalyze than the one presented previously. They are also moretractable, and certainly less conservative. They are given next,and also result in a system (6) which is semiglobally asymptot-ically stable.

Proposition 1—Modified Δ-M Coding: Consider system (1)together with the modified Δ-M coding scheme (5). As before,assume that Δ ∈ R

n×n is constant and has the following form:

Δ = Δ0 · In×n

and that Δ0 fulfills the following inequality:

Δ0 > a · ‖x(0)‖ , a =12λsup{A + AT}

then, ζ(t) = (xT(t), xT(t))T is bounded and tends to zero ast → ∞.

Proof: The proof is straightforward. Let V = xTx/2, andfrom (6b), we have that

V =12xT (Ax − Δsgn(x)) +

12

(Ax − Δsgn(x))T x

≤ − Δ0|x| +12xT(A + AT)x

≤ − Δ0‖x‖ +12xT(A + AT)x

≤ − Δ0‖x‖ + a‖x‖2 = −‖x‖ (Δ0 − a‖x‖)

where we have used the relation −|x| ≤ −‖x‖. From here,it can be seen that the condition Δ0 > a · ‖x(0)‖, with agiven as in the last Proposition, makes V decrease, and henceensures that ‖x(t)‖ < ‖x(0)‖, and that x(t) remains boundedand tends to zero in finite time. Finite-time convergence istypical in switching systems of the form (6b), and will not bedemonstrated here. From this analysis, we can also concludethat x(t) ∈ L2 ∩ L∞.

To complete the proof note that (6a) describes an strictlystable linear system with input2 x(t) ∈ L2 ∩ L∞, hence we canconclude that x(t) is also bounded and tends to zero, as theProposition states. �

Remark 1: Note that the proposed new coding form, inaddition to simplify the stability conditions by making themonly depend on the estimation initial error, the new structureintroduces a side effect of a low-pass filtering action thatwill improve the filtering properties of the decoding dynamicequations.

III. DISCRETE-TIME ALGORITHM

In this section, we present the extension of the continuous-time differential coding to the more realistic case of a discrete-time framework. We first introduce a simple case of a 1-Dunstable system for which the optimal gain Δ and the attraction

2Because x(t) is bounded and tends to zero with Lyapunov function V <β‖x‖2 for some β, locally.



domain are easily found. Then, we extend this result to systemsof higher dimension.

A. One-Dimensional System Example

Consider the following 1-D discrete-time system, togetherwith the control law, and the differential coding modified law.

1) Open-loop system, and encoder

xk+1 = axk + buk (7)

xk+1 = [a − bK]xk + Δ · δk (8)

δk = sgn(xk). (9)

2) Transmitted information δk ∈ {−1, 1}.3) Decoder and control law

xk+1 = [a − bK]xk + Δ · δk (10)

uk = − Kxk (11)

with, K ∈ R, a ≥ 1, ac = (a − bK); |ac| < 1, and xk =xk − xk.

The modified differential coder (8)–(10) differs from its stan-dard form in that the term within the square brackets dependson the system model parameters a, and b, and on the controlgain K. In the standard form, this term is equal to one, i.e.,the encoder is described by a delayed integral term. As pre-viously mentioned, the advantages of this modification is thatthe coding error equations become decoupled from the systemstate equations, making the design of the feedback gain Kindependent of the coding structure.

This algorithm gives the following error equations, in cas-cade form:

xk+1 = acxk + bKxk (12)

xk+1 = axk − Δsgn(xk) (13)

stability of the whole system can thus be tackled by onlystudying the stability of the coding error equation (13).

Let Vk = x2k, and ∇Vk = Vk+1 − Vk, then

∇Vk = x2k+1 − x2

k = (a2 − 1)x2k − 2aΔ|xk| + Δ2. (14)

The RHS of the equality defines a second-order polynomial ofthe form αr2 + βr + Δ2 = 0, with r = |xk| and roots r1, r2

given by

r1 =a − 1a2 − 1

Δ r2 =a + 1a2 − 1

Δ.

Note that, for unstable systems,3 these roots are always real andpositive since a > 1, and that these values define three zones,where ∇Vk changes sign, i.e.,

∇Vk =

⎧⎨⎩

≥ 0, if |xk| ≤ r1

< 0, if r1 < |xk| < r2

≥ 0, if r2 ≤ |xk|.

3The case of stable systems is simpler. We will only discuss here the moreinvolved case of unstable ones.

For some constant Δ, this means that if the initial condition|x0| is taken within the region where ∇Vk is negative, thenthe function Vk will decrease until |xk| enters in the region|xk| < r1, where ∇Vk changes sign. At the next sampling time,the system is pushed away from the region r1 ≤ |xk|. Due tothe discrete nature of the problem, special care must be takenin order to check that this repulsive jump (at k + 1) does notlead the system state out of |xk+1| < r2, and hence guaranteethat the region |x| < r2 is indeed an invariant set. The conditionensuring such a property depends on a and Δ as follows.From (14), we have that the maximum possible jump of xk+1

obtained from any value in |xk| < r1 happens at xk∗ = 0, andhas a magnitude equal to Δ. Therefore, it is straightforward tosee that the state remains within the region |xk| < r2 for allk > k∗ + 1, as long as the following relation holds:

Δ < r2 =a + 1a2 − 1

Δ.

This inequality is satisfied if and only if a < 2, which is thewell-known necessary and sufficient condition for stabilization(see [26]).

Summarizing, we have that if the initial error coding variableverifies |x0| < r2, then the error coding state |xk| is locallyattracted to a threshold delimited by the value of r1. After that,the state |xk| is ultimately bounded by Δ, which is below r2 aslong as a < 2.

It is important to remark that the value of Δ plays animportant role in both; the system stability and the systemprecision:

1) |xk| < (a + 1)/(a2 − 1)Δ defined the DOA of the tra-jectories delimiting the system stability domain. It isenlarged by increasing Δ.

2) |xk| < Δ is an invariant set (included in the previousDOA) defining the system precision. Better precision isreach with small values for.

Therefore, larger values for Δ will make the system morestable but less precise; conversely, the reduction of the gainΔ will lead to small estimation errors, but at the same time,it will reduce the domain where the system remains stable. Thisanalysis is summarized below.

Proposition 2—Discrete One-Dimensional System: Con-sider system (7)–(11), with constant Δ and a < 2. Then, if theinitial conditions of the coding error are such that |x0| < r2

then, the following holds:1) |xk| < r2 ∀k ≥ 0;2) ∃k0 : |xk| ≤ Δ ∀k ≥ k0;3) limk→∞ d(xk,Bγ) = 0

where r2 = Δ(a + 1)/(a2 − 1) and d(xk,Bγ) is the minimumdistance from xk to any point within the interval

Bγ := {x ∈ R : |x| < γ} , γ =Kb

1 − acΔ.

Proof: See Appendix II. �Remark 2: This result displays an inherent tradeoff between

stability and precision for discrete-time differential codingwhen the gain Δ is fixed. This suggests the search of othercoding strategies with varying gains. Note also that, as the



sampling time4 is chosen small, a approaches one and the preci-sion is increased. Indeed, lima→1 r1(a) = 0, thus by making Ts

infinitely small, the limit case of the continuous-time precisionis approached.

B. Noisy One-Dimensional System

The aforementioned analysis will be extended to the casewhere the open-loop model is affected by bounded noise. Inthis case, the state equations turn into

xk+1 = axk + buk + wk

uk = −Kxk

xk+1 = acxk + Δsgn(xk)

where wk is a bounded noise (|wk| < W ). With the proposedfeedback, the dynamics of the error variable xk turns into

xk+1 = axk − Δsgn(xk) + wk.

Using again the Lyapunov function Vk = x2k, and ∇Vk =

Vk+1 − Vk, then

∇Vk = (a2 − 1)x2k − 2(axk + wk)Δsgn(xk)

+ 2axkwk + Δ2 + w2k

≤ (a2 − 1)x2k − 2a|xk|(Δ − W ) + (Δ + W )2. (15)

There is an interval of |xk| such that ∇Vk is negative as longas the last second-order polynomial on |xk| has real roots(otherwise it would be an everywhere positive parabola). Thiscondition is implied by

4a2(Δ − W )2 − 4(a2 − 1)(Δ + W )2 > 0

for which it is necessary that

(Δ − W )2

(Δ + W )2>

a2 − 1a2

= 1 − 1a2

. (16)

It is clear that the RHS of the inequality is less than one for alla ∈ IR. Now, we will consider that Δ is a tuning parameter andlimΔ→∞(Δ − W )2/(Δ + W )2 = 1.

Hence, the left-hand side of (16) can be made arbitrarily closeto one, and greater than any value of the RHS. Rearrangingterms, we have that the set of values of Δ such that the poly-nomial (15) has real roots is

Δ > W1 +

√1 + 1

a2

1 +√

1 − 1a2

. (17)

This inequality makes sense for unstable open-loop plants(a > 1). Otherwise, (16) is trivially satisfied for all Δ.

The roots of polynomial (15) determine the region of attrac-tion in x of the proposed scheme, and its steady-state error.

4The pole of the discrete-time system a is related to the open-loop continuousone as a = eωolTs .

These roots are

r1,2 =a(Δ−W ) ±

√a2((Δ−W )2−(Δ+W )2)+(Δ+W )2

2(a2−1).

From the analysis of this expression, the following observa-tions are in order.

1) Considering that the variable of polynomial (15) is theabsolute value of |x|, the existence of positive real rootsguarantees the existence of an interval on x such thatΔV < 0.

2) A necessary condition for the polynomial to have positivereal roots is (17).

3) If the latter holds, there are real values r1, r2 suchthat r1 < |x| < r2 implies ΔV < 0, i.e., the DOA isdefined by |x| < r2, and the estimation error is ultimatelybounded by |x| < r1.

Regarding the variable xk, an analysis analogous to the previ-ous section can be made by observing the following expression,xk = 1/(z − ac)(Kbxk + wk) = 1/(z − ac)wk. Now, consid-ering that the new input wk is ultimately bounded as |wk| <r1 + W (from the upper bounds on |wk| and |xk|), the samearguments of the previous section lead to the conclusion thatthe state x asymptotically approaches the interval |xk| ≤ γ withγ = (Kbr1 + W/1 − ac).

C. n-Dimensional Systems

The previous study can be generalized to systems with higherdimension of the form

xk+1 =Axk + Buk (18)

uk = −Kxk (19)

where xk ∈ Rn, A ∈ R

n×n, B ∈ Rn×1. (A,B) are stabili-

zable pair.For the sake of space, the noise considerations will be spared

for simplicity. We consider systems whose matrix A has distinctand real eigenvalues, i.e., there exists a transform matrix T suchthat Λ = TAT−1 = diag{λi}, with λi �= 0 ∀ i.

The differential encoding modified law is

xk+1 = Acxk + Δsgn(T xk) (20)

with Ac = (A − BK); |λi{Ac}| < 1, and xk = xk − xk, ∈R

n, Δ ∈ Rn×n. Note that the sign function depends on the

transform coordinates zk = T xk, and

sgn(zk) = sgn(T xk)

= [sgn(z1,k), sgn(z2,k), . . . , sgn(zn,k)]T .

The error equations in the xk, and zk coordinates are

xk+1 = Acxk + BKT−1zk (21)

zk+1 = Λzk − TΔsgn(zk). (22)

Proposition 3—Discrete n-Dimensional System: Let λm andλM be the smaller and the larger eigenvalues of Λ, respectively.



Consider the differential encoding law (20), in feedback withsystem (18) and (19), with the gain Δ given as

Δ = Δ0 · T−1 · ΔΔ = diag {sgn(λ1), sgn(λ2), . . . , sgn(λn)}

where Δ0 is a positive scalar constant. Then, if λ2m > (λ2

M −1), and the initial condition of the coded state is such that‖z0‖ < r2 the following holds:

1) ‖zk‖ ≤ r2 ∀k ≥ 0;2) ∃k0 : |zk| ≤ r1 ∀k ≥ k0;3) limk→∞ d(xk,Bβ) = 0;

where 0 < r1 < r2 are given as

r1,2 =Δ0

λ2M − 1

[λm ±

√λ2

m − λ2M + 1

]

and d(xk,Bβ) is the minimum Euclidean distance from xk toany point within the ball

Bβ := {x ∈ Rn : ‖x‖ < β}

and β is a constant that can be computed as in the proof ofProposition 2.

Proof: The proof follows along similar steps to the previ-ous proof. Introduce Vk = zT

k zk, and ∇Vk = Vk+1 − Vk, then

∇Vk = zTk+1zk+1 − zT

k zk

= zTk (Λ2 − I)zk − 2sgn(zk)ΔΛzk

+ Δ20sgn(zk)TΔ2sgn(zk)

= zTk (Λ2 − I)zk − 2sgn(zk)ΔΛzk + Δ2

0.

Note that

−sgn(zk)ΔΛzk = −∑

i

Δ0|zi,k| · |λi|

≤ − Δ0λm

∑i

|zi,k|

= − Δ0λm|zk| ≤ −Δ0λm‖zk‖.

Then,

∇Vk ≤(λ2

M − 1)‖zk‖2 − Δ0λm‖zk‖ + Δ2

0 ≡ ϕ (‖zk‖) .

ϕ(‖zk‖) = 0 defines a second-order polynomial with roots, r1

and r2, as defined previously. A necessary condition for thispolynomial to have real roots, or equivalently, for the minimumof ϕ(‖zk‖) to be negative is that λ2

m > (λ2M − 1). However,

this condition is only sufficient for making ∇Vk negative inthe domain r1 ≤ ‖zk‖ ≤ r2. Other less conservative conditionsmay be found.

We have, as before, three regions

∇Vk =

⎧⎨⎩

≥ 0, if ‖zk‖ ≤ r1

< 0, if r1 < ‖zk‖ < r2

≥ 0, if r2 ≤ ‖zk‖

and hence, the first two properties invoked in Proposition 3follow exactly the same arguments than the ones used inprevious sections. They are in consequence omitted here. Thelast statement follows from the relation xk = G(z)zk, andassuming that the x-subsystem (21) may also be diagonalized,

the diagonalization leads to a set of n perturbed systems ofthe form (12) in new coordinates, from which we concludeboundedness of the state ‖xk‖ based on the same argumentsas in Proposition 2. �

D. Data Rates

It is assumed here that samples of x(t), are digitalized withlarge enough resolution such that xk digitalization errors areneglected. As the transmission is done by only 1 b, the datatransmission rate (number of bits transmitted per unit of time)associated to this scheme, is Rδ = (1 b/Ts) = fs, (in bits persecond) where fs = 1/Ts is the sampling frequency, and thesystem precision is determined by the size of Δ, the minimumquantization step in the decoding process.

Let R (in bits per second) be the network data-rate. Then,if the maximum network capabilities are used, i.e., Rδ = R =fs, the system precision and the domain of stability can berewritten as a function of R. For instance, in the 1-D case, theexpression for r1 and r2 is rewritten as

r1 =eα/R − 1e2α/R − 1

Δ r2 =eα/R + 1e2α/R − 1

Δ

where α is the unstable pole of the continuous-time orig-inal system. As expected, increasing the transmission rateR improves precision (limR→∞ r1(R) = 0), and stability(limR→∞ r2(R) = ∞).

IV. ADAPTIVE CODING

The proposed scheme can be improved by making the quan-tization factor Δ variable. We will show that proper choice ofa Δ-adaptation mechanism relating the quantization step to theamplitude of the state results in global asymptotic convergenceof the estimation error and the system states to zero. Thisis a significant achievement with respect to the fixed-gainscheme presented before, which was limited to finite domainsof attraction and convergence to finite balls5. The main ideas ofthe adaptive scheme will be introduced here, while the detailsand the stability proof has been reported in [27].

One possibility is to make the adaptation law for Δ beexplicitly state depended. However, the state x is not availableat the receiver side. Therefore, the adaptation law must definedexclusively in terms of the information available both at thereceiver and transmitter; which is not the state itself, but thecoding signal δk, and also the estimation of the state xk. Thisis an additional difficulty resulting for the hypotheses made inthis paper.

A reasonable approach is to enlarge Δk for large values of theestimated error prediction, and decrease it for smaller values, inthe same way as used in the standard adaptive Δ-M for open-loop scenarios [1], i.e.,

Δk+1 = K(δkδk−1)Δk, K > 1.

5The attraction domain can be enlarged arbitrarily (by increasing Δ), at theprice of increasing also the amplitude of the remaining oscillation (granularity).



Fig. 6. Adaptive coding scheme (only the encoder structure is depicted). Thefigure shows the case of 1-D systems. The selection gain block toggles the valueof φk according to (24).

With this adaptation law, the value of Δk increases whentwo consecutive signals δk, δk−1 have the same sign. K isthen selected to minimize the total distortion. There exist alsoother variations to this law, like the continuously variable slopeΔ-M . However, it is important to stress out that in the contextof feedback systems, instabilities may occur if rate of variationof Δk does not accommodate to the maximum (or minimum)rate of variation of the dynamics of the system to be controlled.This is the main difficulty in designing adaptive Δ-M laws forcontrol.

Under these considerations, a Δ-adaptive mechanism, withminimal storage and computation power requirements, can bedesigned under the following assumptions.

1) If δk = δk−1, the error prediction is assumed to be grow-ing, thus Δk must be increased.

2) If δk �= δk−1, the error prediction is assumed to converge(oscillating close to zero) and Δk must be decreased.

This leads to the following update law:

Δk+1 =φk+1Δk, Δ0 > 0 (23)

φk+1 ={

λ+, if δk = δk−1

λ−, if δk �= δk−1(24)

where 0 < λ− < 1 is the exponential decay rate of Δk, andλ+ > 1 is the exponential growth rate. The block scheme isshown in Fig. 6.

Heuristic and simple as it may seem, this adaptation lawguarantees global asymptotic stability of the close-loop system,under the conditions stated in the following proposition.

Proposition 4: For any initial condition, the state xk of sys-tem (7)–(11) with the adaptive Δ-modulation coding scheme(24), asymptotically converges to zero as k → ∞ if thereexist parameters λ+ > 1, λ− ∈ (0, 1) satisfying the followinginequalities:

λ+ >a (25)

λ− < (λ+)−β2 (26)

where

β(a, λ−, λ+)�= 1 + logρ

(1 +

a(a − λ−)(ρ − 1)(λ−)2

)

ρ�=

λ+

a.

Moreover, Δk also converges to zero regardless its initialvalue Δ0.

Remark 3: A practical implementation of the algorithmshould avoid too low values of Δk. If the zoom-in stage isallowed to run for a long time, Δk would become very small. Inthat situation, a disturbance driving the state far from the originwould originate a zoom-out stage equally long, as Δk wouldhave to undo all the previous reduction steps, for catching upthe state. This is properly addressed in the Section V.

The details of the proof of this proposition can be found in[27], and are omitted here for the sake of clarity. However, akey consideration is that a choice of parameters λ+ and λ−

fulfilling the stability conditions (25) and (26) is not alwaysfeasible. Indeed, (26) is an implicit equation, and its solvabilitydepends on the particular value of a.

In Fig. 7, we have shown the expression λ−(λ+)−(β/2) from(26) for a = 1.2. The surface is below zero over a small area,and hence there are solutions fulfilling Proposition 4. However,for some values of a, such as a = 1.4, the whole surfaceremains above zero, and hence, the system cannot be properlytuned. Moreover, as the RHS of (26) decreases with a, therewill be no more solutions for any a > 1.4.

A. n-Dimensional Adaptive Coding

The results of Section III-C can be easily extrapolated tothe adaptive coding results. Indeed, under the condition ofdiagonalizing the system matrix A, it is easy to check that(22) become a set of independent scalar equations that can berewritten as

{zi}k+1 = λi{zi}k − {Δi}ksgn ({zi}k) , i = 1, . . . , n.

Now, by substituting each Δi by an adaptive parameter in theform (23)

{Δi}k+1 = {φi}k+1{Δi}k, Δ0 > 0 (27)

{φi}k+1 ={

λ+, if sgn ({zi}k) = sgn ({zi}k−1)λ−, if sgn ({zi}k) �= sgn ({zi}k−1)

(28)

we then conclude that each {zi} will converge to zero by thesame arguments presented in the previous scalar analysis; andfrom such convergence, the stability of the system is directlyimplied.

B. Relation With the N & S Condition for Stabilization UnderChannel Limitations

A further issue that must be addressed with respect to theadaptive Δ-M scheme is the question whether the limitation onthe open-loop poles (parameter a) to be below a certain value isa structural property or a limitation due to the sufficient natureof the result.



Fig. 7. Expression λ− − (λ+)−(β/2) for a = 1.2.

In any case, our limitation should be consistent with thenecessary and sufficient condition for stabilization presentedin [26]. This condition indicates that the minimal data raterequired for stabilizing a discrete-time system via a communi-cation channel of maximum rate capacity R [b.p.u.]6 is relatedto the unstable open-loop poles (λun

i ) as: R >∑

log2(λuni ),

which in our case simplifies to

R > log2 a. (29)

The implicit assumption made within the framework of ourdiscrete-time formulation is that the channel can reliably trans-mit one bit-per-unit of time, i.e., that R = 1. This meansthat with regard to condition (29), the maximum admissiblevalue for a is a < 2, which is consistent with our sufficientcondition a < 1.313, probably due to the technicalities used forthe stability analysis, or either due to the particular structureof the proposed adaptation law. This also indicates that thereis some conservatism in the computation of the admissibleset of the parameter λ+ and λ−. Some alternative adaptationstrategies could be devised in order to improve this bound.For example, a more sophisticated adaptive algorithm based onolder samples of δk could possibly be designed, at the cost ofhigher complexity.

V. SIMULATIONS

System (7)–(11) has been simulated for the set of valuesR = 1[b.p.s.], a = 1.1, b = 1, K = 0.2. Initial conditions forthe states are the following: x0 = 100, x0 = 0, and Δ0 = 5for the adaptive algorithm. Its adaptation gains are λ− = 0.4,λ+ = 1.21, satisfying conditions (25) and (26). For the sakeof comparison with the nonadaptive scheme, a first simulationhas been made with constant Δ. Fig. 8 shows the behavior ofthe closed-loop system when Δ is given fixed values (it onlyswitches at specific times, between values 20, 10, 5, and then20). We have chosen large values of Δ in order to highlight

6R is given in dimensionless units, i.e., bits per unit of time [b.p.u.].

Fig. 8. Simulation results for nonadaptive Δ-M scheme, with Δ fixed alongspecific intervals.

its connection with the chattering amplitude. The plot indicatesclearly that both magnitudes have the same order. An importantissue here is that Δ cannot be fixed too low (thus reducing thegranularity) without compromising the DOA.

Fortunately, this limitation has been successfully tackledwith the new adaptive approach, as is shown in Fig. 9. In thisplot, the state xk, x, and the adaptive quantization parameterΔk for the noiseless system are depicted. On the upper plot,the state xk, x are plotted together along the whole simulation.As expected, convergence of both the estimation and the stateto zero is obtained regardless the initial values. In the lowerplot of that figure, Δk can be compared to the state xk at theinitial stages of the simulation, where the zoom-out and zoom-in periods can be distinguished.

Although we do not have a conclusive theoretical analysison the performance of the adaptive scheme in the presence ofnoise, we have observed via simulations that the inclusion ofwhite noise in the system dynamics, no matter the amplitude,



Fig. 9. Simulation results for adaptive Δ-M scheme, noiseless system.

Fig. 10. Simulation results for adaptive Δ-M scheme, noisy system.

does not cause instability of the system. Yet, naturally, thesteady state presents variations around the origin whose am-plitude is directly related to the amplitude of the added noise(Fig. 10).

In all simulations, some granularity has been allowed byconstraining Δk to remain always above Δmin = 2. This is apractical adjustment done for improving the transient behavior(see Remark 3). Without this saturation, Δk would keep tendingto zero in steady state, and whenever a disturbance drives thestate away from the origin, a large number of samples would berequired for (23) to make Δk large enough to capture againsuch state (i.e., the zoom-out transient time would undesir-ably depend on the length of the previous zoom-in stage). Inthe noisy scenario, it is very important to properly tune the

minimum allowable value of Δk to minimize the steady-statechattering.

VI. CONCLUSION

We have investigated the stability properties of the Δ-Mcoding rule, when used as an ADC or a transmission means innetworked controlled linear systems. It was first shown that thestandard form of the Δ-M algorithm can be modified, includinginformation about the system and the controller, to enlargethe attraction domain of the closed-loop system equilibrium.Then, we have shown that in the discrete-time case, a tradeoffbetween system precision and size of the stability domain canbe assessed.



These results were extended to the case of adaptive Δ-M .An explicit adaptation rule has been discussed, with still sig-nificantly easy implementation. This scheme guarantees globalasymptotic stability under the constrain imposed by a limiton the maximum unstable eigenvalues of the system that arecompatible with the ones given in [26]. The effect of randomnoise has been analyzed for all schemes.

The practical application of the proposed technique is thegrowing field of low-cost wireless sensor networks, whereminimum data transmission per sample results in a signif-icant improvement of battery life and optimal bandwidthmanagement.

APPENDIX ISTABILITY OF THE CONTINUOUS-TIME

STANDARD Δ-M ALGORITHM

Proposition 5—Standard Δ-M Coding: Consider system(1) where x ∈ R

n is the state estimate computed according tothe Δ-M standard scheme (2) and (3). Let x = x − x, andassume that Δ ∈ R

n×n is constant, and has the form Δ =Δ0 · In×n.

Let ζ = (xT, xT)T, then there exists a constant c > 0 suchthat for any possible value of initial conditions ζ(0), there existsa corresponding scalar constant value Δ0 ≥ c · ‖ζ(0)‖ such thatζ(t) → 0, as t → ∞.

The constant c > max(c5, c5c6) > 0, and the ci givenby the following relations: c1 = ‖2PBK‖, c2 = ‖Ac‖,c3 = ‖BK‖, c4 = max{c1, c2}, c5 = c3 + (c2

4/4q), c6 =√(λM/λm). Where λM = λsup{P}, λm = λmin{P}, with

P = diag{P, I}, and q = λminQ. P = PT > 0, and Q > 0,are solutions of PAc + AT

c P = −Q.Proof: Consider the quadratic Lyapunov function V =

xTPx + xTx. Evaluating V along solutions of system (4), andusing −|x| ≤ −‖x‖, gives

V ≤ − q‖x‖2 + c1‖x‖ ‖x‖ + c2‖x‖ ‖x‖ + c3‖x‖2 − Δ0|x|≤ − q‖x‖2 + c1‖x‖ ‖x‖ + c2‖x‖ ‖x‖ + c3‖x‖2 − Δ0‖x‖≤ − q‖x‖2 + c4‖x‖ ‖x‖ + c3‖x‖2 − Δ0‖x‖

≤ − (‖x‖, ‖x‖)(

q − c42

− c42

Δ0‖x‖ − c3

)(‖x‖‖x‖

).

From here, we can see that V is negative as long as the matrixin the equality above is positive definite, i.e., for values of Δ0,such that q((Δ0/‖x‖) − c3) > (c2

4/4), or equivalently if Δ0 >c5‖ζ‖ ≥ c5‖x‖. Nevertheless, this condition by itself does notdefine the DOA. For this, we proceed further as follows. DefineλM = λsup{P}, and λm = λmin{P}, with P = diag{P, I}.With this definition, we have the following bounds on V (ζ):

λM‖ζ‖2 ≥ V (ζ) ≥ λm‖ζ‖2

−λM‖ζ‖2 ≤− V (ζ) ≤ −λm‖ζ‖2

using these bounds, and assuming that Δ0 > c5‖ζ‖, then thereexists a scalar function ε(‖ζ‖) > 0, such that

V ≤ −ε (‖ζ‖) · ‖ζ‖2 ≤ −ε (‖ζ‖) V

λM. (30)

Integration on both sides of this equation along the time-interval[0, t], gives

V (t) ≤ V (0) exp (−ϕ (ε(‖ζ‖))

where ϕ(t) =∫ T

0 (ε(‖ζ(τ)‖)/λM ) dτ. Note that ϕ(t) > 0, aslong as Δ0 > c5‖ζ‖. Using again the bound on V in theexpression above, we obtain

‖ζ(t)‖2 ≤ λm

λM‖ζ(0)‖2 exp (−ϕ(t))

‖ζ(t)‖ ≤√

λm

λM‖ζ(0)‖ exp (−ϕ(t))

= c6 ‖ζ(0)‖ exp (−ϕ(t)/2)

and using the relation given in Proposition 5, i.e., Δ0 ≥c · ‖ζ(0)‖ > c5c6‖ζ(0)‖ in the above inequality, we get‖ζ(t)‖ < (Δ0/c5) exp(−ϕ(t)/2). Therefore, under the rela-tion Δ0 > c5‖ζ(0)‖, as proposed in Proposition 5, and usingexp(−ϕ(t)/2) < 1 ∀t, we conclude that |ζ(t)‖ < Δ0/c5 asrequired for (30) to hold. As a final consequence, the timederivative of the Lyapunov function is negative definite and theconvergence ‖ζ(t)‖ → 0 is guaranteed. �

APPENDIX IIPROOF OF PROPOSITION 3

Proof: The first two items result from the previous de-velopment. The last item of the claim derives from the fol-lowing arguments. First, note that xk = (Kb/z − ac)xk. Then,defining the discrete-time transfer function H(z) = (Kbz/z −ac), with impulse response h(n) = Kban

c , n ≥ 0, yields theinput–output relation xk = H(z)z−1xk which in time domaincorresponds to the convolution

xk =∞∑

m=0

xm−1h(k − m) =∞∑

m=0

xmh(k − m − 1)

because xm = 0 ∀m < 0 is assumed. Now, using the fact thatxk is absolutely bounded (|xk| < r2∀k ≥ 0) and also ulti-mately bounded by Δ (second item of the Proposition), we cansplit the convolution as follows:

xk =k0∑

m=0

xmh(k − m − 1) +∞∑

m=k0+1

xmh(k − m − 1)

hence,

|xk| ≤ r2

k0∑m=0

|h(k − m − 1)| + Δn−1∑

m=k0+1

|h(k − m − 1)|

where the causality of h(n) has been used. Substituting theimpulse response into this expression yields

|xk| < Kbr2

(ak

c (1 − a−k0−1c )

ac − 1

)+ KbΔ

(ak−k0−1

c − 1ac − 1

)

=KbΔ1 − ac

+ v(k)



where v(k) stands for

v(k) =Kb

(r2(1 − a−k0−1

c ) + Δa−k0−1c

)ac − 1

akc

and as ac <1, we conclude that limk→∞ v(k)=0 and the state xasymptotically approaches the interval |xk|≤(Kb/1−ac)Δ.�

REFERENCES

[1] J.-G. Proakis, Digital Communications, 4th ed. Series in Electrical andComputer Engineering. New York: McGraw-Hill, 2001.

[2] N. Elia and S.-K. Mitter, “Stabilization of linear systems with limitedinformation,” IEEE Trans. Autom. Control, vol. 46, no. 9, pp. 1384–1400,Sep. 2001.

[3] Y. Tipsuwan and M.-Y. Chow, “Gain scheduler middleware: Amethodology to enable existing controllers for networked control andteleoperation—Part I: Networked control,” IEEE Trans. Ind. Electron.,vol. 51, no. 6, pp. 1218–1227, Dec. 2004.

[4] Y. Tipsuwan and M.-Y. Chow, “Gain scheduler middleware: Amethodology to enable existing controllers for networked control andteleoperation—Part II: Teleoperation,” IEEE Trans. Ind. Electron.,vol. 51, no. 6, pp. 1228–1237, Dec. 2004.

[5] P. Martí, J. Yépez, M. Velasco, and J. M. Fuertes, “Managing quality-of-control in network-based control systems by controller and mes-sage scheduling co-design,” IEEE Trans. Ind. Electron., vol. 51, no. 6,pp. 1159–1167, Dec. 2004.

[6] H. Ishii and T. Basar, “Remote control of LTI systems over networks withstate quatization,” Syst. Control Lett., vol. 54, no. 1, pp. 15–31, Jan. 2005.

[7] D. Liberzon, “On stabilization of linear systems with limited informa-tion,” IEEE Trans. Autom. Control, vol. 48, no. 2, pp. 304–307, Feb. 2003.

[8] M. Lemmon and Q. Ling, “Control system performance under dynamicquatization: The scalar case,” in Proc. 43rd IEEE Conf. Decision Control,Nassau, Bahamas, 2004, pp. 1884–1888.

[9] S. Tan, X. Wei, and J.-S. Baras, “Numerical study on joint quantizationand control under block-coding,” in Proc. 43rd IEEE Conf. DecisionControl, Nassau, Bahamas, 2004, pp. 4515–4520.

[10] K. Li and J. Baillieul, “Robust quantization for digital finite communica-tion bandwidth (DFCB) control,” IEEE Trans. Autom. Control, vol. 49,no. 9, pp. 1573–1584, Sep. 2004.

[11] T. Li and Y. Fujimoto, “Control system with high-speed and real-time communication links,” IEEE Trans. Ind. Electron., vol. 55, no. 4,pp. 1548–1557, Apr. 2008.

[12] C. Canudas-de-Wit, J. Jaglin, and C. Siclet, “Energy-aware 3-level codingand control co-design for sensor network systems,” in Proc. Conf. ControlAppl., Singapore, 2007, pp. 1012–1017.

[13] C. Canudas-de-Wit, J. Jaglin, and K. Crisanto Vega, “Entropy coding innetworked controlled systems,” in Proc. Eur. Control Conf., Kos, Greece,2007, pp. 1211–1218.

[14] I. Lopez, C. Abdallah, and C. Canudas-de-Wit, “Gain-scheduling multi-bit delta-modulator for networked controlled systems,” in Proc. Eur. Con-trol Conf., Kos, Greece, 2007, pp. 15–22.

[15] C. Canudas-de-Wit, J. Jaglin, and C. Siclet, “Delta modulation for mul-tivariable centralized linear networked controlled systems,” in Proc. 47thConf. Decision Control, Cancun, Mexico, 2008, pp. 4910–4915.

[16] J. Acero, D. Navarro, L. A. Barragán, I. Garde, J. I. Artigas, andJ. M. Burdio, “FPGA-based power measuring for induction heating ap-pliances using sigma-delta A/D conversion,” IEEE Trans. Ind. Electron.,vol. 54, no. 4, pp. 1843–1852, Aug. 2007.

[17] A. Mertens and D. Eckardt, “Voltage and current sensing in power elec-tronic converters using sigma-delta A/D conversion,” IEEE Trans. Ind.Appl., vol. 34, no. 5, pp. 1139–1146, Sep./Oct. 1998.

[18] M. Kraft, “Closed-loop digital accelerometer employing oversamplingconversion,” Ph.D. dissertation, School Eng., Coventry Univ., Coventry,U.K., 1997.

[19] S. Park and R. Horowitz, “Adaptive control for the conventional modeof operation of MEMS gyroscopes,” J. Microelectromech. Syst., vol. 12,no. 1, pp. 101–108, Feb. 2003.

[20] M. Handtmann, R. Aigner, A. Meckes, and G. K. Wachutka, “Sensitivityenhancement of MEMS inertial sensors using negative springs and activecontrol,” Sens. Actuators A, Phys., vol. 97/98, pp. 153–160, Apr. 2002.

[21] J.-P. Hespanha, A. Ortega, and L. Vasudevan, “Towards the control oflinear systems with minimum bit-rate,” in Proc. 15th Int. Symp. MTNS,Notre Dame, IL, 2002, CD-ROM.

[22] R.-W. Brockett and D. Liberzon, “Quantized feedback stabilization oflinear systems,” IEEE Trans. Autom. Control, vol. 45, no. 7, pp. 1279–1289, Jul. 2000.

[23] C. Canudas-de-Wit, F. Rubio, J. Fornes, and F. Gomez-Estern, “Differen-tial coding in networked controlled linear systems,” in Proc. ACC. SilverAnniversary, Minneapolis, MN, Jun. 2006, pp. 4177–4182.

[24] E. Wiltrant, C. Canudas-de-Wit, D. Georges, and M. Alamir, “Remotestabilization via time-varying communication network delays,” in Proc.IEEE Control Appl. Conf., Sep. 2004, pp. 474–479.

[25] E. Wiltrant, C. Canudas-de-Wit, and D. Georges, “Output stabilizationvia two-channel transmission communication,” in Proc. IFAC WorkshopTime-Delay Syst., Sep. 2003.

[26] S. Tatikonda and S. Mitter, “Control under communication constraints,”IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1056–1068, Jul. 2004.

[27] F. Gómez-Estern, C. Canudas-de-Wit, F. Rubio, and J. Fornes, “Adap-tive delta-modulation coding for networked controlled systems,” in Proc.ACC, New York, 2007, pp. 4911–4916.

Carlos Canudas-de-Wit (M’09) received the B.Sc.degree in electronics and communications from theTechnological Institute of Monterrey, Monterrey,Mexico, in 1980, and the M.Sc. and Ph.D. degrees inautomatic control from the Department of AutomaticControl, Polytechnic of Grenoble, Grenoble, France,in 1984 and 1987, respectively.

In 1985, he was a Visiting Researcher with LundInstitute of Technology, Lund, Sweden. Since 1987,he has been the Director of Research with the Na-tional Center for Scientific Research (CNRS), De-

partment of Automatic Control, Polytechnic of Grenoble, where he teachesand conducts research in the area of nonlinear control of mechanical systemsand networked controlled systems. He is the Leader of the joint CNRS/INRIAteam project on Networked Controlled Systems. His research interests includenetworked controlled systems, vehicle control, adaptive control, identification,control of robots, and systems with friction. His research publications includesmore than 120 conference papers and more than 47 papers published ininternational journals.

Dr. Canudas-de-Wit served as an Associate Editor for the IEEETRANSACTIONS ON AUTOMATIC CONTROL (1992–1997) and Automatica(1999–2002).

Fabio Gómez-Estern (M’06) received the M.Sc.degree in communications engineering from theEscuela Superior de Ingenieros de Sevilla, Seville,Spain, in 1998, and the Ph.D. degree in automaticcontrol, in 2003, which was jointly received from thesame institution and the Laboratoire des Signaux etSystèmes, CNRS, Gif-sur-Yvette, France.

He held a postdoctoral position with the Univer-sity of Twente, Enschede, The Netherlands. He iscurrently an Associate Professor with the Depart-ment of Systems Engineering and Automation, Uni-

versity of Seville, Seville, where he teaches in the areas of nonlinear control andautomation and leads public-financed and industrial research projects. He hasbeen a Visiting Researcher at several laboratories across Europe. His researchinterests include nonlinear control systems and networked control systems.

Francisco Rodríguez Rubio (M’89) received theIndustrial Electrical Engineering and Ph.D. degreesfrom the Escuela Técnica Superior de IngenierosIndustriales de Sevilla, Seville, Spain, in 1981 and1985, respectively.

He is currently a Professor with the Department ofSystems Engineering and Automation, University ofSeville, Seville. He has worked on various researchand development projects in cooperation with indus-try. His current interests are in the areas of networkedcontrolled systems, adaptive control, robust process

control, nonlinear control systems, and robotics. He has written two books: Ad-vanced Control of Solar Plants (Springer-Verlag, 1997) and Control Adaptativoy Robusto (Seville University, 1996). He has authored or coauthored more than150 technical papers published in international journals and conferenceproceedings.

Dr. Rubio received the CITEMA award for the best work on automation bya young engineer in 1980.


Delta-Modulation Coding Redesign for Feedback-Controlled...

Documents

Transcript of Delta-Modulation Coding Redesign for Feedback-Controlled...