An energy amplification condition for decentralized adaptive stabilization

We are interested here in the problem of global decentralized adaptive regulation of square multivariable linear time-invariant systems without any restrictions on relative degrees nor matching assumptions. The first solution to this problem was recently reported by the author using Morse´s new dynamic certainty equivalent adaptive controller to prove that global stabilization is possible if the unmodelled interconnections do not induce “amplification of the energy of the signals in all channels”. In this paper we show that, to preserve global convergence, it is actually enough to have only one “non-amplifying channel”. A corollary of this result is that all systems where one of the outputs is affected only by its corresponding input are decentralized adaptively stabilizable. Instrumental for the establishment of our result is the fundamental S-procedure losslessness theorem of Megretsky and Treil, together with some basic loop-transformations and D-scalings