## Section: New Results

### Stability and control design of asynchronous interconnected systems

Participants : C. Canudas-de-Wit [ contact person ] , N. Marchand, S. Durand.

Networked and embedded control systems are 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.

#### 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 systems that are asynchronously interconnected. Imposing certain passivity properties to each sub-system it is possible to design a local controller each sub-system disregarding the particular characteristic of the other system (modular design).

In particular we have studied the following items. 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
T_{i} 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
[17] .

Further studies consider systems that are either Input/Output Strictly passive (IOSP), or systems which have bounded
L_{2} -gains less than one. The analysis is performed by using the concept of MAximum Sampling time preserving Dissipation (MASD), for each interconnected system. We investigate the impact of using the scattering transformation in the computation of the MASD, and we provide a numerical
algorithm (based on a set of LMI's) that allows to choose the most suitable configuration for the interconnection (see
[48] )

#### Event-based control design

Asynchronicity is becoming more and more meaningful in modern control architectures and some new control strategies are being developed by some research teams in the world. The principle of these control laws is to compute the control law only when some event occurs, this event characterizing a change in the system and therefore a need for a new control. These approaches are supposed to reduce the number of times the control is computed and to remove the real-time hard constraint on the computational system. In this domain, our contribution is twice. First, based on [50] , we proposed a fully asynchronous control scheme (without any time information) for chain of integrators that insures the global stability of the system with only measures when the states cross a priori defined level. This work presented at the IFAC world congress in Korea [5] . Secondly, we removed the safety limit condition introduced by K-E. Årzén in his event-based PID controller [25] . This safety limit was added to prevent the system to be sampled less than what Shannon theorem requires. This work therefore reveals that the Shannon sampling condition is no more needed in the context of event based systems. This was submitted to the next European Control Conference [37] .