On the use of multi-loop circle criterion for saturating control synthesis

This paper deals with regional stabilization of linear time-invariant systems by dynamic output feedback controllers subject to known bounds on the magnitudes of the control inputs. Specifically, we consider the achievable region of attraction, i.e., the set of vectors with the following property: there exists a (nonlinear) controller such that any closed-loop state trajectory converges to the origin as long as the initial state belongs to the set. Two subsets of such set are characterized: one is derived from the linear analysis that considers the behavior of the states in the linear (nonsaturated) region only; while the other is based on the nonlinear analysis using the multi-loop circle criterion. The main result obtained shows that the two sets are exactly the same. Thus we conclude that the circle criterion does not help, within our framework, to increase the size of the region of attraction in saturating control synthesis when compared with that resulting from the linear analysis