Distributed control of spatially distributed systems
The current mainstream paradigm in control systems is that all the relevant measured variables are brought to a central computer (controller), processed all at once and finally the resulting commands are distributed to the individual actuators. The last decades have witnessed a growing number of applications in which the number of such measured and command variables is huge, easily above a few hundred or thousand signals. With the advent of MEMS devices, more then a hundred thousand variables can easily appear in a given application. The traditional centralized approach fails in these scenarios. Instead, a spatially distributed control paradigm has been (re)emerging. Each local controller measures the local variables and computes the local commands for the actuators. Since the dynamics of many systems is such that local changes necessarily induce a dynamic response in some neighbourhood, the local controller should communicate with the neighbors. Restricting the communication to the nearest neighbors might be preferrable from an energetic viewpoint in wireless networks.
In our research we propose new approaches for control design and implementation for such systems. Our goal is to develop new control design algorithms based on multidimensional systems and signals theory. Assuming an unbounded spatial domain as an approximation of a very large spatial domain (and dense network of sensors and actuators), spatial shift operator can be defined. For spatially homogeneous systems with linear(ized) dynamics this suggests use of a convenient framework of multidimensional transfer functions, that is, fractions of two n-dimensional polynomials; one variable corresponding to time and one, two or three variables corresponding to the spatial variable(s). Formulating the problems in the language of n-d polynomials, we can take advantage of research conducted by some of our senior colleagues in 1980s. That polynomial research has ultimately lead to a commercial Polynomial toolbox for Matlab, developed and distributed by PolyX.
We also struggle hard to base our methods on a realistic ground by building experimental setups for verification of our algorithms. One such experimental benchmark system is a distributed control of temperature profile in a long aluminium rod. The other experimental setup is an autonomous vehicular platoon composed of racing slot cars (also LEGO Mindstorms NXT based robotic vehicles).