Fixed points and convergence in monotone systems under positive or negative feedback

In this paper a theorem is discussed that unifies two lines of work in I/O monotone control systems. Under a generalized small gain hypothesis, it is shown that almost all solutions of closed loops of MIMO monotone systems are convergent, regardless of whether the feedback is positive or negative. This result is based on a topological argument showing that any monotonically decreasing n-dimensional map that has convergent iterations must have a unique fixed point. The paper also generalizes the standard small gain theorem by replacing the small gain condition with a weaker hypothesis. An example and simulations are given involving a simple cyclic system under arbitrary feedback.