On necessary and sufficient conditions for stability for n-input n-output convolution feedback systems with a finite number of unstable poles

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| < ??. Thus the subsystem represented by G is an open-loop unstable distributed parameter system. One theorem is obtained. It gives necessary and sufficient conditions for stability. A basic device is the following: the sum of the principal parts of the Laurent expansions of ?? at the unstable poles is factored as a ratio of two right coprime polynomial matrices. There are two necessary and sufficient conditions, the first is the usual infimum one, i.e. infRe s ?? 0 |det[I+??(s)]| > 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.