International Journal of Control, Vol.87, No.2, 301-311, 2014
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 input/output monotone control systems. Under a generalised small-gain hypothesis, it is shown that almost all solutions of closed loops of multiple-input, multiple-output 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 generalises 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.