On the local input to state practical stability of feedback interconnected systems

Sufficient conditions for "local input-to-state practical stability" of feedback interconnected systems are derived. This concept of stability denotes that the state of the interconnection remains bounded for sufficiently small input and initial conditions, but it does not tend to zero for zero input. The derivation of the sufficient conditions is based on the establishment of a small-gain theorem for interconnections of locally input-to-state stable systems. The criterion, which must be satisfied by the gain functions of the subsystems that constitute the interconnection, can give results for a wide class of systems, since it can also be applied to gain functions that cannot be overbounded by any linear function in a neighborhood of zero. Finally, two illustrative examples are presented that show the application of the introduced small-gain theorem, for the study of the stability of systems' interconnections.