Stability of Power Control in Multiple Coexisting Wireless Networks: An $\mathscr {L}_{2}$ Small-Gain Perspective

Owing to the emerging techniques such as heterogeneous network and cognitive radio that allow a large number of wireless networks to intensively and flexibly use resources of radio spectrum, the network densification and dynamic spectrum access have become two important trends of wireless networking systems, so that the spectral environments will be increasingly crowded in the near future. Under such a background, this study addresses the stability related problem of the power control in multiple coexisting wireless networks. We develop an analytical framework that is capable of utilizing the small-gain theorem to show the considered power control is bounded-input bounded-output (BIBO) stable in the sense of the ℒ
₂
gain if proper sufficient conditions are satisfied. Through modeling the power control as a feedback system and making use of the uncertainty representation, this framework can derive generalized results for the power-control algorithms that emerge in previous works. The framework developed in this study also admits the incorporation of multipath propagation that is a very common phenomenon in wireless communication systems. Within our derivation, making use of the theory for positive systems dramatically simplifies the analysis and leads to many refined results.