Optimization, Simulation, and Control pp 143-174 | Cite as

# Power Control in Wireless Ad Hoc Networks: Stability and Convergence Under Uncertainties

## Abstract

A successful distributed power control algorithm requires only local measurements for updating the power level of a transmitting node, so that eventually all transmitters meet their QoS requirements, i.e. the solution converges to the global optimum. There are numerous algorithm which claim to work under ideal conditions in which there exist no uncertainties and the model is identical to the real-world implementation. Nevertheless, the problem arises when real-world phenomena are introduced into the problem, such as uncertainties (such as changing environment and time delays) or the QoS requirements cannot be achieved for all the users in the network. In this chapter, we study some distributed power control algorithms for wireless ad hoc networks and discuss their robustness to real-world phenomena. Simulations illustrate the validity of the existing results and suggest directions for future research.

## Keywords

Mobile Node Power Control Positive System Power Control Algorithm Interference Function## References

- 1.R. Sanchez, J. Evans, and G. Minden. Networking on the battlefield: challenges in highly dynamic multi-hop wireless networks. In
*Military Communications Conference Proceedings, 1999. MILCOM 1999. IEEE*, volume 2, pages 751–755, 1999.Google Scholar - 2.Gil Zussman and Adrian Segall. Energy efficient routing in ad hoc disaster recovery networks.
*Ad Hoc Networks*, 1(4):405 – 421, 2003.CrossRefGoogle Scholar - 3.Chee-Yee Chong and S.P. Kumar. Sensor networks: evolution, opportunities, and challenges.
*Proceedings of the IEEE*, 91(8):1247–1256, August 2003.Google Scholar - 4.Alan Mainwaring, David Culler, Joseph Polastre, Robert Szewczyk, and John Anderson. Wireless sensor networks for habitat monitoring. In
*WSNA ’02: Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications*, pages 88–97, New York, NY, USA, 2002. ACM.Google Scholar - 5.V. Kawadia and P.R. Kumar. Principles and Protocols for Power Control in Wireless Ad Hoc Networks.
*Journal on Selected Areas in Communications (JSAC)*, 23(1): 76–88, January 2005.Google Scholar - 6.S. Shakkottai, T. S. Rappaport, and P. C. Karlsson. Cross-Layer Design for Wireless Networks.
*IEEE Communications Magazine*, 41(10):74–80, October 2003.Google Scholar - 7.V. Srivastava and M. Motani. Cross-layer design: a survey and the road ahead.
*Communications Magazine, IEEE*, 43(12):112–119, Dec. 2005.CrossRefGoogle Scholar - 8.G. Foschini and Z. Miljanic. A Simple Distributed Autonomous Power Control Algorithm and its Convergence.
*IEEE Transactions on Vehicular Technology*, 42(4):641–646, November 1993.CrossRefGoogle Scholar - 9.Jens Zander. Distributed co-channel interference control in cellular radio systems.
*IEEE Transaction on Vehicular Technology*, 41(3):305–311, August 1992.CrossRefGoogle Scholar - 10.S. Grandhi, J. Zander, and R. Yates. Constrained power control.
*Wireless Personal Communications*, 2(3): 257–270, August 1995.Google Scholar - 11.R. D. Yates. A framework for uplink power control in cellular radio systems.
*IEEE Journal on Selected Areas in Communications*, 13:1341–1347, September 1995.CrossRefGoogle Scholar - 12.T. ElBatt and A. Ephremides. Joint Scheduling and Power Control for Wireless Ad-hoc Networks. In
*Proceedings of IEEE INFOCOM*, 2002.Google Scholar - 13.Zoran Gajic, Dobrila Skataric, and Sarah Koskie. Optimal SIR-based Power Updates in Wireless CDMACommunication Systems. In
*IEEE Conference on Decision and Control*, volume 5, pages 5146–5151, December 2004.Google Scholar - 14.Jens Zander. Performance of Optimum Transmitter Power Control in Cellular Radio Systems.
*IEEE Transaction on Vehicular Technology*, 41(1):57–62, February 1992.CrossRefGoogle Scholar - 15.D. Mitra. An asynchronous distributed algorithm for power control in cellular radio systems. In
*4th WINLAB Workshop*, Rutgers University, New Brunswick, NJ, 1993.Google Scholar - 16.Nicholas Bambos, Shou C. Chen, and Gregory J. Pottie. Channel Access Algorithms with Active Link Protection for Wireless Communication Networks with Power Control.
*IEEE/ACM Transactions on Networking*, 8(5):583–597, 2000.CrossRefGoogle Scholar - 17.Fredrik Gunnarsson. Power control in cellular radio systems: Analysis, design and estimation, PhD Thesis, Linköping universitet 2000.Google Scholar
- 18.K.K. Leung, C.W. Sung, W.S. Wong, and T.M. Lok. Convergence theorem for a general class of power control algorithms.
*IEEE Transactions of Communications*, 52(9):1566–1574, September 2004.CrossRefGoogle Scholar - 19.C.W. Sung and K.K. Leung. A generalized framework for distributed power control in wireless networks.
*IEEE Transactions on Information Theory*, 51(7):2625–2635, 2005.MathSciNetCrossRefGoogle Scholar - 20.H. Boche and M. Schubert. A unifying approach to interference modeling for wireless networks.
*IEEE Transactions on Signal Processing*, 58(6):3282–3297, June 2010.MathSciNetCrossRefGoogle Scholar - 21.H. R. Feyzmahdavian, M. Johansson, and Themistoklis Charalambous. Contractive Interference Functions and Rates of Convergence of Distributed Power Control Laws. In
*International Conference on Communications (ICC)*, 2012.Google Scholar - 22.M. Xiao, N.B.Shroff, and E.K.P. Chong. A utility-based power-control scheme in wireless cellular systems.
*IEEE/ACM Transaction on Networking*, 11(2):210–221, April 2003.CrossRefGoogle Scholar - 23.Roger A. Horn and Charles R. Johnson.
*Matrix*Analysis. Cambridge University Press, Cambridge CB2 2RU, UK, 1985.Google Scholar - 24.Roger A. Horn and Charles R. Johnson.
*Topics in*Matrix Analysis. Cambridge University Press, Cambridge CB2 2RU, UK, 1994.Google Scholar - 25.S.U. Pillai, T. Suel, and Seunghun Cha. The Perron-Frobenius theorem: some of its applications.
*Signal Processing Magazine, IEEE*, 22:62–75, March 2005.Google Scholar - 26.Fredrik Gunnarsson and Fredrik Gustafsson. Control theory aspects of power control in UMTS.
*Control Engineering Practice*, 11(10):1113 – 1125, 2003.CrossRefGoogle Scholar - 27.D. ONeill, D. Julian, and S. Boyd. Seeking Foschini’s genie: Optimal rates and powers in wireless networks.
*IEEE Transaction on Vehicular Technology*, 2003 (accepted but not presented).Google Scholar - 28.Xingwen Liu, Wensheng Yu, and Long Wang. Stability analysis of positive systems with bounded time-varying delays.
*Trans. Cir. Sys.*, 56(7):600–604, 2009.Google Scholar - 29.Lorenzo Farina and Sergio Rinaldi.
*Positive Linear Systems: Theory and Applications*. New York: Wiley, July 2000.MATHCrossRefGoogle Scholar - 30.M.A. Rami and F. Tadeo. Controller synthesis for positive linear systems with bounded controls.
*IEEE Transactions on Circuits and Systems II: Express Briefs*, 54(2):151–155, 2007.CrossRefGoogle Scholar - 31.O Mason and R Shorten. On linear copositive lyapunov functions and the stability of switched positive linear systems.
*IEEE Transactions on Automatic Control*, 52(7), 2007.Google Scholar - 32.Florian Knorn, Oliver Mason, and Robert Shorten. On linear co-positive lyapunov functions for sets of linear positive systems.
*Automatica*, 45(8):1943 – 1947, 2009.MathSciNetMATHCrossRefGoogle Scholar - 33.Themistoklis Charalambous, Ioannis Lestas, and Glenn Vinnicombe. On the stability of the foschini-miljanic algorithm with time-delays. In
*CDC*, pages 2991–2996, 2008.Google Scholar - 34.C. A. Desoer and Y. Yang. On the generalized Nyquist stability criterion.
*IEEE Transaction on Automatic Control*, 25(1):187–196, 1980.MATHCrossRefGoogle Scholar - 35.Xingzhe Fan, M. Arcak, and J.T. Wen. Robustness of CDMA power control against disturbances and time-delays. In
*American Control Conference*, volume 3, pages 3622–3627, 2004.Google Scholar - 36.Wassim M. Haddad and VijaySekhar Chellaboina. Stability theory for nonnegative and compartmental dynamical systems with time delay.
*Systems and Control Letters*, 51(5):355 – 361, 2004.MathSciNetMATHCrossRefGoogle Scholar - 37.Themistoklis Charalambous and Yassine Ariba. On the stability of a power control algorithm for wireless networks in the presence of time-varying delays. In
*The 10th European Control Conference (ECC)*, August 2009.Google Scholar - 38.S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan.
*Linear Matrix Inequalities in System and Control Theory*. SIAM, Philadelphia, USA, 1994. in Studies in Applied Mathematics, vol.15.Google Scholar - 39.Themistoklis Charalambous. A lyapunov krasovskii method for the stability of the foschini-miljanic algorithm under time-varying delays: An independent of delays approach. In
*CUED/F-INFENG/TR.646*, January 2010.Google Scholar - 40.Annalisa Zappavigna, Themistoklis Charalambous, and Florian Knorn. Unconditional stability of the Foschini-Miljanic algorithm.
*Automatica*, 48(1):219 – 224, 2012.MathSciNetMATHCrossRefGoogle Scholar - 41.F. Berggren.
*Power control and adaptive resource allocation in DS-CDMA*. PhD thesis, KTH, Stockholm, Sweden, 2003.Google Scholar - 42.Themistoklis Charalambous.
*Power Control for Wireless Ad-Hoc Networks*. PhD thesis, University of Cambridge, July 2010.Google Scholar - 43.Mohammed M. Olama, Seddik M. Djouadi, and Charalambos D. Charalambous. A general framework for continuous time power control in time varying long term fading wireless networks. In
*Proceedings of the Ninth IASTED International Conference on Control and Applications*, CA ’07, pages 69–74, Anaheim, CA, USA, 2007. ACTA Press.Google Scholar - 44.G. C. Rota and G. Strang. A note on the joint spectral radius.
*Proceedings of the Netherlands Academy*, 22:379–381, 1960.MathSciNetGoogle Scholar - 45.T. Holliday, A. Goldsmith, N. Bambos, and P. Glynn. Distributed power and admission control for time-varying wireless networks. IEEEINTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, 2004.Google Scholar
- 46.Adam Czornik. On the generalized spectral subradius.
*Linear Algebra and its Applications*, 407:242 – 248, 2005.MathSciNetMATHCrossRefGoogle Scholar - 47.A. Moller and U.T. Jonsson. Stability of high order distributed power control. In
*Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on*, pages 4963 –4970, December 2009.Google Scholar - 48.A. Moller and U.T. Jonsson. Input Output Analysis of Power Control in Wireless Networks. In
*Decision and Control,. Proceedings of the 49th IEEE Conference on*, pages 6451 – 6456, December 2010.Google Scholar - 49.M. Abbas-Turki, F.de S. Chaves, H. Abou-Kandil, and J.M.T. Romano. Mixed \({H}_{2}/{H}_{\infty }\) power control with adaptive qos for wireless communication networks. In
*European Control Conference (ECC), Budapest, Hungary*, August 2009.Google Scholar - 50.Chi Wan Sung and Kin Kwong Leung. Opportunistic power control for throughput maximization in mobile cellular systems. In
*Communications, 2004 IEEE International Conference on*, volume 5, pages 2954–2958, June 2004.Google Scholar - 51.
- 52.Charles R. Johnson. Sufficient conditions for d-stability.
*Journal of Economic Theory*, 9(1):53 – 62, 1974.MathSciNetCrossRefGoogle Scholar - 53.D. P. Bertsekas and J. N. Tsitsiklis.
*Parallel and Distributed Computation*. New Jersey 07458, USA, Prentice-Hall, 1989.Google Scholar - 54.M. Edelstein. On fixed and periodic points under contractive mappings.
*J. London Math. Soc.*, 37:74–79, 1962.MathSciNetMATHCrossRefGoogle Scholar