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 optimized, 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
In a networked controlled system, the output signals have to be digitalized before transmission. Our objective is then to use the minimum quantization bits necessary to maintain stability on the closed loop system, that is to say the minimum bandwidth.
Several quantization methods have already been proposed during the last decade. Delta modulation is one alternative to minimize the numbers of bits to be coded. The reason is that innovation increments (with a granularity depending on a quantization factor ) are coded rather than the absolute value of the signal. Recent works in  have re-adapted the standard form of the delta modulation structure to their use in a feedback setup. One advantage of this type of strategy is that the coding algorithm can be built in a methodological and simple manner. A limitation is that re-synchronization may be needed, if the signal track is lost. Inspired by this approach several variants of  have been studied as for example gain scheduling 2-bit coding  . Except for the trivial case of diagonalizable multi-variable system that can be reformulated as a set of n -scalar ones, all these works deal exclusively with scalar system.
We have generalized the delta-modulation coding presented in  , to MIMO systems in  . In particular we have introduced a vector coding structure for multi-variable centralized linear systems. The notion of centralization refers here to the fact that both the encoder-decoder and the control law use the full available information from all sensors. The idea is shown in Fig. 5 , where we can see that all the sensed system outputs are collected in a central point, then transformed into a different coordinate-basis (using a transform matrix) before they are coded using a vector-coding algorithm. At the receiver side, it is similarly assumed that the transmitted information arrives to a central receiver, then decoded, and finally the control is computed using this centralized information. It is worth to notice that decentralized case is clearly much more constrained, even in absence of a coding process. We have also shown that this fixed-gain simple and methodical coding strategy results in a ultimately uniformly (local) stability. We have also provided an estimation of the attraction domain, and a new method to tune the coding gains, resulting in closed-loop precision improvements. Simulation results have also been presented validating the proposed approach.
Previously, in  authors have presented a quantization method based on a one-bit-adaptive modulation. The interest of this technique is that it permits global stability in the scalar case provided that the open loop eigenvalue is such that: | |<1.3 . Nevertheless it can be shown that in this case (one bit quantization) it is theoretically possible to just have | |<2 . That is why we have introduced a new adaptive Delta modulation variant called D-ZIZO (Dwell Time Zoom In Zoom Out)(not yet published). This algorithm is inspired by the contributions of Liberzon  and  (ZIZO algorithm). D-ZIZO needs the previous and the actual information but constraints the maximal open loop eigenvalue. We have determined how many past information we need (dwell time) to assure that for any open loop eigenvalue it is possible to ensure global stabilization.
Even if the essence of the algorithms is the same, it seems that the dwell time phase permits to have some improvements. This comparison could be summarized with these following items.
This change of coordinates and the quantization steps tuning method enable D-ZIZO to obtain the minimum bandwidth on the communication channel for all linear systems whereas ZIZO may sometimes reach the theoretical limit for certain classes of linear systems.
ZIZO generates high frequencies on the quantization noise though this signal is smoother for D-ZIZO.
For same regulation detection, D-ZIZO is faster than ZIZO. Moreover, with the same convergence rate, the detection is faster for D-ZIZO.
One drawback for D-ZIZO is the generalization to non-linear systems that seems difficult. Without the change of coordinates which is the key point of D-ZIZO, it is impossible to obtain an adaptive algorithm in D-ZIZO case. With ZIZO, the issue is solved.
Energy-aware and entropy coding in NCS
Wireless low-cost sensor networks are an expanded technology in many new and varied areas such as: traffic monitoring and control (urban, highways), undersea monitoring/exploration, environment sensing (forest, farms, etc.), building services, large instruments with distributed sensing and actuators (Tokamak, telescopes), etc.
In this context, future generation of this type of sensors are expected to 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 energy-efficient communication links. In turn, these constraints bring new problems to be considered in the exploitation of this information. For instance, low cost will induce sensors with low resolution (binary sensors, at the extreme) advocating for minimum bit coding strategies, low consumption will impose issues on efficient sensor energy management (sleep and wake-up modes, differentiation of stand-still event), ease of replacement will imply the system ability to keep safe operation in a failure of one or several sensors, and finally communication links and protocols should be designed to account for energy savings, information loss, and varying fading characteristics.
The objective of our work published in  is specifically to treat aspect related to the energy management, in relation with the particular code to be used. To this aim, we have proposed to use a coding strategy based in the following 3 main ingredients:
Differential codingencodes the differences (error prediction) between successive samples rather than the samples themselves. Since differences between samples are expected to be smaller than the actual samples amplitudes, fewer bits are required to represent the differences. However, in its standard form no specific distinction is made for the stand-still events.
Event-based codinghas the ability to quantify and to differentiate stand-still signal events from changes in the source (level crossing detector). Coding is effectuated by defining at least 3-valued alphabet for the minimum bit case, and ( 2 L+ 1 )-valued alphabet for a general case with a precision inversely proportional to . Hence, the stand-still signal event is modulated with a low-power signal, whereas the changes of levels will be modulated with a high-power one.
Entropy codingcan be added to improve the energy use by assigning a probability distribution to the events. In that way, the mean transmission energy can be substantially improved for systems where the stand-still events may have high probability to occur (i.e. stable systems).
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, from changes in the source (level crossing detector, denoted here as ). 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. We assign 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 which makes it suitable for an entropy encoder to attain a “good” overall compression ratio, and hence (as it will be shown here), a substantial improvements of energy saves. A fundamental difference with the classical Differential algorithm (i.e. delta modulation, see  ) is that the error is coded on the basis of a 3-valued alphabet rather than a 2-valued one. If the levels are uniformly spaced and constant then the signal prediction precision, and hence the resulting closed-loop behavior, will be limited by the size of the level spacing. Instead, we propose here to increase the number of levels to 2 L+ 1 , where , so as to match the required precision. This leads to a 2 L+ 1 -word alphabet, which can still be combined with a energy-efficient variable length (VLE) entropy coding to improve the use of energy for cases and systems where the stand-still events have a substantial probability to arise.
Stabilization under Communication Networks: A Time-Delay Approach
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 stabilization of systems where the sensor, actuator and system are assumed to be remotely commissioned by a controller that interchanges measurements and control signals through a communication network. Additional dynamics are introduced in the system due to the wireless communications such as time-varying communications delays, asynchronous samplings, packets losses or lake of synchronization.
In  , and in  , we proposed to use a time-varying horizon predictor to design a stabilizing 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  . In  and  , we also proposed an observer-based controller to ensures the stabilization of networked controlled systems. The main interest of such a controller concerns the potential to take into account the additional dynamics induced by the networks cited above. Further developments will take into account the quantification and the coding of the transmitted data packets.
Another effort has been devoted to the problem of controlling a set of agents, cooperating under communication constrains. It is well-known that introducing a delay generally leads to a reduce of the performance or to instability. Thus, investigating the impact of time-delays in the consensus problem is an important issue. In our research, we assume that each agent receives instantaneously its own output information but receives the information from its neighbors after a constant delay . The setup we considered leads to study the following equation , where >0 and A is the classical adjacency matrix. These corresponds to a more realistic setup than the one usually considered in the literature  . More especially, in  and  , we investigate the influence of the communication on the location of the agreement point and on the convergence rate, which is not straightforward when delays appear in the network. First, we proved that whatever the delay and whatever the graph, the set of agents will reach a consensus. The consensus equilibrium depends on the delay and on the initial conditions taken in an interval is given by
where U2 is a vector depending on the communication graph. Then, based on Lyapunov-Krasovskii techniques and LMI representation, an estimate of the convergence rate is provided. Figure 6 shows the examples of four communications graphs and Figure 7 shows the corresponding convergence rate.
It can be seen that the convergence rate strongly depends on the connection. Note that an interesting phenomena concerning the full connected network is pointed out. It is now well known that for some systems, delays could improve the performance and even lead to stability  . It thus appears that a set of full connected agents is one of those systems.
NecsCar is an electrical vehicle (scale 1/3) to be used as an experimental platform to study improvement of new control architectures. The vehicle is designed to be remotely tele-operated from our active steering wheel platform, ant it will be equipped of a 3D vision system to provide the operator with stereo vision capabilities. Bilateral teleoperation can be performed using wheel contact torque measurements, feed back for force deflection. Wireless connection will allows us to test coding algorithms, resource sharing, and robustness against transmission delays. First experiments were conducted this summer and visible at http://necs.inrialpes.fr/