Large-scale systems

When dynamical systems grow too big, it is impossible to use standard control design techniques, as the number of inputs and outputs becomes unmanageable. Moreover, a centralized controller would need a central processing unit which can input all the measures of the system and decide all the command actions.

Typically, such problems emerge in control of formations or platoons. Think of an highway full of thousands of cars: the approach of having a central unit knowing all about all of them is not feasible. What is feasible, is having controllers implemented on each car, which can decide their actions on the base of a limited knowledge of what the others are doing.

If we schematize the formation / platoon with a graph, where each node represents a vehicle and each arrow a communication link, then we would like to design controllers which have the same interconnection structure:

In the picture, on the left we see a system made of the interconnections of smaller identical subsystems (vehicles). The goal is design local controllers (the red circles on the right picture) which keep the same limited interconnections as the original system. In this way, there is no need of a central unit.

Another approach can be of having a completely decentralized local controller, which acts only locally with no knowledge of the "neighbourhood". Decentralized controllers may be easier to design, however they will have poor performance; decentralized controllers are of course sub optimal with respect to centralized ones, but still their performance can be quite close to them.

References

P. Massioni, G. Scorletti, Consensus analysis of large-scale nonlinear homogeneous multi-agent formations with polynomial dynamics. International Journal of Robust and Nonlinear Control, Volume 28, Issue 17, November 2018, pages 5605–5617.

P. Massioni Distributed control for alpha-heterogeneous dynamically coupled systems Systems & Control Letters 72, October 2014, pages 30-35. 

P. Massioni, T. Keviczky, E. Gill, M. Verhaegen A Decomposition Based Approach to Linear Time-Periodic Distributed Control of Satellite Formations IEEE Transactions on Control Systems Technology, Volume 19, Issue 3, May 2011, pages: 481 - 492.

P. Massioni, M. Verhaegen Distributed Control for Identical Dynamically Coupled Systems: a Decomposition Approach IEEE Transactions on Automatic Control, Volume 54, Issue 1, January 2009, pages: 124 - 135.