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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
##### Keywords:
Network flow control; Time delay; ISS
Full Text:
##### References:
 [1] S. Deb, R. Srikant, Global stability of congestion controllers for the Internet, University of Illinois, Urbana, IL, Internal Report, February 2002. · Zbl 1364.93560 [2] Hale, J; Verduyn Lunel, S, Introduction to functional differential equations, (1993), Springer Berlin · Zbl 0787.34002 [3] C. Hollot, Y. Chait, Nonlinear stability analysis for a class of TCP/AQM networks, in: Proceedings of the IEEE Conference on Decision and Control, Orlando, FL, December 2001, pp. 2309-2314. [4] Johari, R; Tan, D, End-to-end congestion control for the internetdelays and stability, IEEE/ACM trans. networking, 9, 818-832, (2001) [5] Kelly, F; Maulloo, A; Tan, D, Rate control in communication networksshadow prices, proportional fairness and stability, J. oper. res. soc., 49, 237-252, (1998) · Zbl 1111.90313 [6] Khalil, H, Nonlinear systems, (1996), Prentice-Hall Englewood Cliffs, NJ [7] S. Kunniyur, R. Srikant, End-to-end congestion control: utility functions, random losses and ecn marks, in: Proceedings of INFOCOM 2000, Tel Aviv, Israel, March 2000. [8] Low, S; Lapsley, D, Optimization flow control—ibasic algorithm and convergence, IEEE/ACM trans. networking, 7, 6, 861-874, (1999) [9] S. Low, F. Paganini, J. Wang, S. Adlakha, J. Doyle, Dynamics of TCP/RED and a scalable control, in: Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communication Societies, INFOCOM, 2002, pp. 239-248. [10] Massoulié, L, Stability of distributed congestion control with heterogeneous feedback delays, IEEE trans. automat. control, 47, 6, 895-902, (2002) · Zbl 1364.93868 [11] Ortega, J.M; Rheinboldt, W.C, Iterative solution of nonlinear equations in several variables, (2000), Siam Philadelphia · Zbl 0949.65053 [12] Paganini, F, A global stability result in network flow control, Systems control lett., 46, 165-172, (2002) · Zbl 1031.93137 [13] F. Paganini, J. Doyle, S. Low, Scalable laws for stable network congestion control, in: Proceedings of 2001 Conference on Decision and Control, Orlando, FL, December 2001, pp. 185-190. [14] Sontag, E, Smooth stabilization implies coprime factorization, IEEE trans. automat. control, 34, 435-443, (1989) · Zbl 0682.93045 [15] Teel, A, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE trans. automat. control, 41, 9, 1256-1271, (1996) · Zbl 0863.93073 [16] G. Vinnicombe, On the stability of end-to-end congestion control for the internet. University of Cambridge, Technical Report CUED/F-INFENG/TR.398, December 2000. [17] Z. Wang, F. Paganini, Global stability with time-delay in network congestion control, in: Proceedings of Conference on Decision and Control, Las Vegas, NV, December 2002. [18] Wen, J; Arcak, M, A unifying passivity framework for network flow control, IEEE trans. automat. control, 49, 162-174, (2004) · Zbl 1365.90042 [19] J. Wen, M. Arcak, A unifying passivity framework for network flow control, in: INFOCOM 2003, San Francisco, CA, April 2003. · Zbl 1365.90042
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