Laboratory for Analysis and Architecture of Systems
The group has a long-term tradition in the non-linear domain as well as that of infinite dimension. One may distinguish:
This technique was developed through industrial collaboration for defense applications as soon as 1989. It has given rise during the past ten years to purely deterministic versions, which supersede the old random technique or Monte-Carlo type. This allows revisiting classical defense applications such as RADAR, SONAR; GPS or Telecommunications, with significant performance improvement. Besides, this deterministic particle technique applies without change to optimal control synthesis through Pontryagin approach.
Polynomial filters with infinite time-horizon better known as Volterra filters, allow economical non-linear solutions, which are well-suited to embarked systems. When constrained to finite dimensional realizations (separable kernel), the optimal parameters may be computed without any approximations for bilinear stochastic systems. This generic class can achieve effective approximations of most SNL encountered in practice.
The group has also developed techniques for the optimal identification of stochastic dynamical systems, based on hereditary algorithms (linearly increasing memory, with respect to data length). Being essential to exact resolution, such an hereditary memory happens to be crucial for critical applications (e.g. modal proximity), and provides easily for adequate truncating, as a function of criticity. First developed for ARMAX signals under the form of lattices, it has been recently extended to homogeneous bilinear systems and applied to the identification of human locomotion.
Diffusive representation is a new operator theory devoted to general dynamic problems. It was introduced and developed at LAAS during the last ten years. Its main interest lies in analysis, approximation and synthesis of dynamic operators of pseudodifferential nature. Diffusive formulations are well adapted to various situations encountered in many current problems of physics, control or signal, namely when the underlying complexity is excessive for standard formulations, due in particular to hereditary dynamic behaviours intimately associated with pseudodifferential components, often of long memory type. In the framework of diffusive representation, such major shortcomings are by-passed by use of suitable state realisations especially devoted to this purpose, and whose numerical approximations are straightforward.