Section: Scientific Foundations
Multi-disciplinary nature of the project
The project propose to investigate problems in the area of NCS with the originality of integrated aspects on computation, communication and control. The combination of these three disciplines requires the interplay of the multi-disciplinary fields of: communication, real-time computation, and system theory (control). Figure 2 , shows the natural interaction between disciplines that concern the NeCS project. The arrows describe the direction in which these areas interact, i.e.
Complexity and energy-management are additional features to be considered as well. Complexity here refers to the problems coming from: wireless networks with varying interconnection topologies, multi-agent systems coordination, scale of the number of sensors, etc. Energy management concerns aspect related to the efficient handling of energy in wireless sensors. That is the efficient way to send information, and perform computations.
(a) Control in Communication
This area concerns more control applications where control methods are used to solve problems found in the communication field. Examples are: the Power control in cell telephones, and the optimal routing of messages in communication networks (Internet, sensor networks).
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(b) Communication in Control
This area concerns problems where communication and information theory interacts with system theory (control). A typical scheme of a networked controlled system (NECS) is shown in Fig. 3 . As an example, of a classical paradigm we can mention the stabilization problem under channel (communications) constraints. A Key result here [75] was to show that it was generically impossible to stabilize a linear system in any reasonable sense, if the feedback channel's Shannon classical capacity C was smaller than the sum of the logarithms, base 2, of the unstable eigenvalues. In other words, in order to be able to cope with the stabilization problem under communication constraints, we need that
where the 
i's are unstable eigenvalue of the open loop
system. Intuitively, this means that the rate of information
production (for discrete-time linear systems, the intrinsic rate
bits/time equals 
) should be smaller than
the rate of information that can be transmitted throughout the
channel. In that way, a potentially growing signal can be cached
out, if the information of the signal is send via a channel with
fast enough transmission rate. In relation to this, a problem of
interest is the coding and control co-design. This issue is
motivated by applications calling for data-compression algorithms
aiming at reducing the amount of information that may be transmitted
throughout the communication channel, and therefore allowing for a
better resource allocation and/or for an improvement of the
permissible closed loop system bandwidth (data-rate).
Networked control systems also constitute a new class of control systems including specific problems concerned by delays. In NCS, the communication between two agents leads unavoidably to transmission delays, discrete-time sending whereas the controlled processes evolves in continuous time, loss of information and data process. Our objectives 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 due to time-varying communications delays, asynchronous samplings, packets losses or lake of synchronization. All those phenomena can be modelled as the introduction of time-delays in the closed loop system. Even if this time-delay approaches can be easily proposed, they require careful attention and more complex analysis. In general, the introduction of delays in a controlled loop leads to a reduction of the performances with respect to the non delay situation and could even makes the systems unstable. Our objective is to provide specific modelling of these phenomena and to develop dedicated tools to and methodologies to cope with stability and stabilization of such systems.
(c) Computation in Control
This area concerns the problem of redesigning the control law such as to account for variations due to the resource allocation constraints. Computation tasks having different levels of priority may be handled by asynchronous time executions. Hence controllers need to be re-designed as to account for non-uniform sampling times resulting in this framework. Questions on how to redesign the control laws while preserving its stability properties are in order. This category of problems can arise in embedded systems with low computation capacity or low level resolution.
(d) Control in Computation
The use of control methods to solve or to optimize the use of computational resources is the key problem in this area. This problem is also known as a scheduling control. The resource allocations are decided by the controller that try to regulate the total computation load to a prefixed value. Here, the “system” to be regulate is the process that generate and use the resources, and not any physical system. Hence, internal states are computational tasks, the control signal is the resource allocation, and the output is the period allowed to each task.
(c + d) Integrated control/scheduling co-design
Control and Computation co-design describes the possibility to study the interaction or coupling between the flows (c) and (d) . It is possible, as shown in Figure 4 , to re-frame both problems as a single one, or to interpret such an interconnection as the cascade connection between a computational system, and a physical system.
In our framework the feedback scheduling is designed w.r.t a QoC (Quality of Control) measure. The QoC criterion captures the control performance requirements, and the problem can be stated as QoC optimization under constraint of available computing resources. However, preliminary studies suggest that a direct synthesis of the scheduling regulator as an optimal control problem leads, when it is tractable, to a solution too costly to be implemented in real-time [56] . Practical solutions will be found in the currently available control theory and tools or in enhancements and adaptation of current control theory. We propose in Figure 4 a hierarchical control structure : besides the usual process control loops we add an outer control loop which goal is to manage the execution of the real-time application through the control of the scheduling parameters of the inner loops. Together with the outer loop (working on a periodic sampled time scale) we also need a scheduling manager working on a discrete events time scale to process exception handling and admission control.
The task periods directly affect the computing load, they have been
chosen as actuators. They can be implemented through software
variable clocks. As timing uncertainties cannot be avoided and are
difficult to model or measure, we currently design robust control
algorithms using the 
control theory, which have been
successfully simulated and experimentally
validated [73] .
This methodology is supported by Orccad (see Section 5.1 ) where a run-time library for multi-rate multitasking has been developed and integrated. It will be further improved using a QoS-based management of the timing constraints to fully benefit from the intrinsic robustness of closed loop controllers w.r.t. timing uncertainties.
