Open-loop unstable convolution feedback systems with dynamical feedbacks

Abstract: This paper considers distributed multivariable convolution feedback systems characterized byy"1 = G"1 * e"1,y"2 = G"2 * e"2,e"1 = u"1 - y"2 ande"2 = u"2 + y"1 where the subsystem transfer functionsG@?"1 andG@?"2 both admit a pseudo-coprime factorization in the subalgebra of absolutely summable distributions of order zero. The most general result, Theorem 1, gives necessary and sufficient conditions for stability of the system. This condition is specialized to the lumped case in Theorem 1L. Finally for distributed systems which have a finite number of open-loop unstable poles Theorem 1D gives an algorithmic test for stability. The graphical interpretation of both Theorem 1 and 1D is given in detail, and illustrated by examples. The transfer functions considered here are used in modeling transportation lags and communication lags, as well as in distributed systems such as steam in long pipe.