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Robustness of network flow control against disturbances and time-delay. (English) Zbl 1157.93503
Summary: This paper studies robustness of F. P. Kelly et al’s source and link control laws in [J. Oper. Res. Soc. 49, No. 3, 237–252 (1998; Zbl 1111.90313)] with respect to disturbances and time-delays. This problem is of practical importance because of unmodelled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show \(L_{p}\)-stability, for \(p\in [1,\infty ]\), with respect to additive disturbances. We pursue \(L_{\infty }\)-stability within the input-to-state stability (ISS) framework of E. D. Sontag [IEEE Trans. Autom. Control 34, No. 4, 435–443 (1989; Zbl 0682.93045)], which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small time-delays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in F. Paganini et al. [Proceedings of 2001 Conference on Decision and Control, Orlando, FL, pp. 185–190 (2001)].

MSC:
93D21 Adaptive or robust stabilization
93C23 Control/observation systems governed by functional-differential equations
93D25 Input-output approaches in control theory
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