Necessary and sufficient conditions stability for<tex>n</tex>-input,<tex>n</tex>-output convolution feedback systems with a finite number of unstable poles

Abstract

We consider n-input n-output convolution feedback systems characterized by y = G * e and e = u - y, where the open-loop transfer function ? contains a finite number of unstable poles and the expansion ?a(s) + ?i=0 ? Gi exp(-sti). The latter is such that, i) ?a is the Laplace transform of L1 functions, ii) t0 = 0, iii) the delay-times ti, for i = 1, 2, ..., are positive and iv) the coefficient matrices Gi form an absolutely convergent series, i.e. ?i=0 ? |Gi| 0, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition has a nice interpretation in terms of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate. The same is true if, instead of unity feedback, constant nonunity feedback represented by a nonsingular matrix is used. This allows us, through the use of the small gain theorem or the contraction mapping theorem, to consider non-linear systems, derived from the original one by introducing a non-linear time-varying gain in the forward loop.