Arrays of Systems, Micro-Systems and Nano-Systems
Our method extends the well known homogenization method for continuum physics to complex multiphysic systems e.g. mechatronic systems. It is based on the so-called two-scale transform and allows to formulate systematically simplified multi-scale models for systems having a multi-scale nature.
Multiscale models show a dramatic reduction of computation cost. They split into a system of PDEs at the macroscopic level and classical multiphysic equations at the level of a unit cell. All fields are well described as for instance stresses, strains, temperature, currents, voltages in each part of a system. Model quality increases with the size of the array. The mathematical foundations of the method are rigorous and well justified.
The problem of regulation of a spatially distributed mechatronic system is formulated like a standard control problem. The designer has to choose devices for observation and actuation, to develop a simplified model for the system and the perturbations and to define the regulation objectives. He does his best for finding a tradeoff between efficiency and robustness and checks that the realization of the command may be implemented in real time.
A part of the phenomena are regulated locally in each subsystem by using existing command strategies. But often another part of them are global and the usual command tools are not well fitted for them. This comes mainly because the number of degree of freedom is too large and a real time control law cannot be implemented on a straightforward manner.
For this class or problems, I am developing approximations of standard regulators (LQR, LQG, H-infinity) that are made for being implemented on arrays of computational units. Those units are allowed to communicate only with their neighbors and may be embedded in the system. They constitute a so-called semi-decentralized computational architecture. The methods are formulated on the Operatorial Riccati Equations (ie not discretized) that are associated to the partial differential equations governing the system. With these approaches, the Operational Riccati equations play a central role when building the regulator.