This paper studies the problem of the relations between existence conditions of common quadratic and those of common infinity-norm Lyapunov functions for sets of discrete-time linear time-invariant (LTI) systems. Based on the equivalence between the robust stability of a class of time-varying systems and the existence of a common infinity-norm Lyapunov function for the corresponding set of LTI systems, the relations are determined. It turns out that although the relation is an equivalent one for single stable systems, the existence condition of common infinity-norm type is strictly implied by that of common quadratic type for the set of systems. Several existence conditions of a common infinity-norm Lyapunov functions are also presented for the purpose of easy checking.

This paper presents an adaptive control design for the ABR traffic congestion control in ATM networks. Firstly, we consider a control-based mathematical model to the ABR traffic congestion control problem. Then the feedback pole placement control design is applied to the ATM ABR traffic congestion control problem for the case of known delays. Finally, by using the online plant parameter estimation algorithm and modifying the controller parameters adaptively in real time, a method to treat the case of unknown time-varying delays is proposed. Several design modifications are introduced to solve practical control issues such as bounded command rate constraint, output buffer saturation and bounded values to the plant parameter estimation algorithm. Simulations are implemented to verify the proposed control design. It is shown that while considering these practical control issues, the control method satisfies the requirements of fairness to users, network efficiency, unknown time-varying delays, queue length control and good convergence performance at an acceptable computation effort.