Section: New Results
Communication and control co-design in feedback systems
Traditional control theory often disregards issues of connectivity, data transmission, coding and many other items of central importance in wireless sensor networks. In this topic we study new methodologies to design control for systems in which signals are exchanged through a communication network with limited capacity. Some of the general questions addressed here are:
How the signal (source) coding algorithm can be designed jointly and simultaneously with the control law ?
What are the mutual inter-dependencies between the control and communication design?
How can one overcome the limitations of the wireless medium, and
How energy of the sensors and the associated transmission media, can be optimised, by the appropriated energy-aware design of the coding and the control
More specifically, we have studied the following problems:
Passivity design for asynchronous interconnected systems  ,
Differential Coding for networked controlled systems
Delta modulation is a well-known differential coding technique used for reducing the data rate required for voice communication. The standard technique is based on synchronising a state predictor on emitter and receiver and just sending a one–bit 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 [Oops!] we propose an adaptive extension a of fixed-gain differential coding scheme previously introduced by us (see  ). For a constant it was shown in  that only a limited domain of attraction can be 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 user–defined parameter . By making an adaptive quantity, more effective schemes of -modulation can be obtained. 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. In fact, we show that the unstable eigenvalues of the open-loop plant are used to compute the -update law resulting in global asymptotic stable scheme.
In [Oops!] , we propose a gain-scheduling multi-bit Delta-modulator. We first analyse modifications for the - M algorithm proposed by  in order to stabilise discrete-time systems with eigenvalues | a|>2 . This is motivated by the previously mentioned cost issues associated with the simple - M schemes. Secondly, we analyse the packet-loss issue, and determine a maximum allowable number of consecutive bits lost while keeping stability. This analysis is innovative since previous work on the subject have dealt with the limited-rate and the packet losses separately. Recent works considered packet losses but assumed unlimited channel rate, while research dealing with limited-rate channels have not included packet losses. Our results show that the maximum number of packets that can be lost sequentially depends on the region where a certain estimation error lives. Using this fact, we redesign the - M scheme used in  so the system can handle at least a minimum number of packet losses. This mechanism is based on the re-synchronisation between the encoder and decoder.
Further research on this issue will be carried out in this project, addressing quantisation of multiple samples and a comparative analysis with similar techniques. Additionally, the use of transform coding in the context of feedback systems is an interesting open lines of research to explore.
Energy-aware and entropy coding in NCS
In sensor network, each individual sensors will be packaged together with communication protocols, RF electronics, and energy management systems. Therefore, the development of such integrated sensors will be driven by constraints like: low cost, ease of replacement, low energy consumption, and efficient communication links. In particular, low cost technology will induce sensors with low resolution (binary sensors, at the extreme), low consumption (efficient sensor energy management, with; sleep and wake-up modes). As a consequence, communication protocols, modulation strategies, and control laws should be designed to account for substantial energy savings.
Event-driven communication protocols (and control), are usually adjusted for minimum transmission requirements (i.e. transmit and update control laws only when significant information for control is available), is a natural candidate to be used in this context. To this aim, we have propose to use a coding strategy with the ability to quantify and to differentiate stand-still signal events from changes in the source (level crossing detector) [Oops!] . Coding is effectuated by defining a 3-valued alphabet. Then, the stand-still signal event is modulated with a low energy carrier, whereas the changes of levels will be modulated with enough energy. In [Oops!] we study the closed-loop properties of such arrangement for one-dimensional system. In particular, we derive conditions required so that this coding algorithm preserves closed loop stability.
Other possible variant to improve data transmission efficiency by reducing the mean number of bits per unit of time needed for the transmission, and hence the mean energy, is to use entropy coding. Combination of entropy coding with non uniform (event-based) sampling has been demonstrated to result in efficient coding strategies [Oops!] . Entropy coding is a source coding that assigns some probability distribution to the events. 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 (level 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. 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 level crossing detector 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 differential coding algorithms mentioned in section 6.2.1 is that the error is coded on the basis of a 3-valued alphabet rather than a 2-valued one. The role of the entropy coding here is to render more efficient the transmission by improving in the mean number of bits per unit of time needed for the transmission.
Passivity design for asynchronous feedback-interconnected systems
In this topic we have studied the passivity properties of asynchronously non-uniformly sampled systems. The idea of studying these systems comes from the necessity of developing theoretical tools for the analysis of networked and embedded control systems, which usually operated under variable resources like communication rates and computational loads. This results in an asynchronous sub-systems interconnection, as the sampling time may be adapted on the fly as a function of the available resources at the moment. Examples of these systems can be found in many application fields such as remotely-operated systems, interconnected vehicle control loops, and more generally in component-based control design where synchronous exchange of information is not feasible.
We have studied the following issues. First we introduce the notion of (MASP) MAximum Sampling time preserving Passivity for linear systems; given a continuous-time system with some dissipation properties specified, the notion of MASP give a maximum sampling time, T* after which passivity is lost. A second aspect studied here concern the case of a system locally asynchronous but globally synchronous feedback interconnected systems. The notion of globally synchronous comes from the fact that we limit this study to samples Ti of each i -subsystem that are multiple integers among them, nevertheless we allows the sampling time of each individual sub-systems to be time-varying. Finally, we use these results as a design guidelines for the control design, and we propose a numerical algorithm to compute local feedback loops providing a MASP compatible with the maximum sampling-time upper-bound of each sub-system. Details are given in 
Remote Stabilisation via Communication Networks and Tele-operation
The networked control systems constitute a new class of control systems including specific problems such as delays, loss of information and data process. The problem studied here concerns the remote stabilisation of unstable open-loop systems. The sensor, actuator and system are assumed to be remotely commissioned by a controller that interchanges measurements and control signals through a lossless communication network (all lost packets are re-emitted). We assume that this communication network has its own dynamics, and that a model for the average induced time-delay is available. As an example, such a model can be derived for local networks where the transfer protocol (TP) is set by the users and where a router (which can possibly inform the emitters of the instantaneous queue length) manages the packets. Another option is the estimation the average value of the delay with a simple algorithm based on the measurement of the round trip time
In  , and in [Oops!] , we proposed to use a time-varying horizon predictor to design a stabilising control law that sets the poles of the closed-loop system. The computation of the horizon of the predictor is investigated and the proposed control law takes into account the average delay dynamics explicitly. The resulting closed loop system robustness with respect to some uncertainties on the delay model is also considered. Tele-operation subject to time-varying delays has been considered in  .
An Hinfty approach to robust control of bilateral teleoperation systems under communication time-delay is considered and applied on an experimental setup. Using a small gain approach an Hinfty controller is first designed in the nominal case and a robust design is then proposed for any communication delay in the case of environment uncertainties. When delay independent stability cannot be achieved, a way to determine the maximal allowed time-delay is provided. Simulation and experimental results are provided on the teleoperation platform of the NeCS project [Oops!] .
Multi-carrier modulation for underwater acoustic communication
CONNECT (CONtrol of NEtworked Cooperative sysTems) is a project granted by the ANR. In collaboration with IFREMER, GIPSA-lab, PROLEXIA and PGES, CONNECT aims at studying the problem of multi-agent control (AUVs) and coordination with heterogeneous networks, including underwater communication. In this context, we will interest to underwater acoustic communication.
Current underwater acoustic modems are based on very classical single-carrier modulation with a very low bit rate. In the same time, wireless radio-communications have been significantly improved during the latest ten years, in particular thanks to multicarrier modulations. These modulations are indeed now used in several high rate applications (ADSL, DVB-T  , IEEE 802.11a/g, ...). These applications are all based on the same modulation, OFDM (Orthogonal Frequency Division Multiplex). Thanks to the use of a guard time  (or cyclic prefix), it is possible under certain conditions to simplify considerably the equalisation step, so that OFDM is particularly performing. That is why OFDM has also recently been considered for underwater communications : the underwater acoustic channel  ,  is indeed particularly frequency selective, so that OFDM is a potential interesting solution  .
In spite of its advantages, OFDM also suffers from several drawbacks : the guard time induces a spectral efficiency loss, and, what is more, the pulse shape used for each carrier is rectangular, and therefore badly frequency localised. This spectral efficiency loss remains small if we use long duration symbols, but is may cause inter-carrier interferences if the transmission channel is not stationary during a symbol interval. Compensation techniques have then to be studied  . Lastly, the modulated signal has an amplitude which may by very high which is problematic for linear amplification.
Several alternatives have been proposed to solve these problems, among them oversampled OFDM/QAM modulations  and OFDM/OQAM modulations  ,  , and their bi-orthogonal extensions (BFDM)  . For each of these types of modulation, it is possible to use non rectangular pulse shapes, optimised according to a given criterion (time-frequency localisation, frequency localisation...). They appear then to be promising and give more freedom degrees in their conception than classical OFDM. Nevertheless, the equalisation is more complex (in the OQAM case) and their implementation more expensive (in terms of computational complexity).
Our objective is to study multicarrier modulation systems (OFDM, oversampled OFDM/QAM, OFDM/OQAM as well as their bi-orthogonal extensions) and to determine the corresponding receptors, in the framework of underwater acoustic communications. We will then have develop a complete communication chain using and comparing these different modulations to classical underwater communication systems. We will thus interest to various problems : channel estimation, equalisation, synchronisation, non-linearity of digital to analog and analog to digital converters.