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Communication Networks

This course explores the stability of communication networks, covering topics such as linear systems, Nyquist analysis, functional differential equations, and network controls. It examines the stability issues in routing and transport protocols, providing key concepts and results.

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Communication Networks

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  1. Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley

  2. Stability • Motivation • Overview of results • Linear Systems • Nyquist • Functional Differential Equations

  3. Motivation • Network is a controlled system • Controls: MAC, Routing,Transport, … • The system is nonlinear and has delays; the stability of the control system is non-trivial • Many examples of instability of routing and transport • We review key concepts and results on the stability of systems and we apply them to the transport protocols

  4. Overview of Results • Linear System • Poles: x(n+1) = ax(n) + u(n) … |a| < 1  bibo • Nyquist: feedback system, L(s) = K(s)G(s).Stable if L(jw) does not encircle – 1.(If L(jw0) = - 1 – e < - 1, then input at w0 blows up.)

  5. Overview of Results • Nonlinear system • Linearize around equilibrium x0. • If linearized system is stable, then x0 is locally stable for original system • Nonlinear system: Lyapunov • Assume V(x(t)) decreases and level curves shrink • Then the system is stable

  6. Overview of Results • Markov Chain: Lyapunov • Let x(t) be an irreducible Markov chain • Assume V(x(t)) decreases by at least – e < 0, on average, when x(t) is outside of a finite set A • Then x(t) is positive recurrent

  7. Overview of Results • Functional Differential Equation: • Assume V(x(t)) decreases whenever it reachesa maximum value over the last r seconds, thenthe system is stable…. [Razumikhin]

  8. Linear Systems Laplace Transform

  9. Linear Systems

  10. Linear Systems Example

  11. Linear Systems Example

  12. Linear Systems Observation

  13. Nyquist Slide from a tak by Glenn Vinnicombe

  14. Nyquist

  15. Nyquist Slide from a tak by Glenn Vinnicombe

  16. Nyquist MIMO Case:

  17. Nyquist Example 1  Closed-Loop is stable

  18. Nyquist Example 2

  19. Nyquist Example 2 …  Stable if T < 1.35s

  20. Nyquist and Transport: 1 G. Vinnicombe, “On the stability of end-to-end control for the Internet.”

  21. Nyquist and Transport: 2 Linearized System: Theorem: F. Paganini, J. Doyle, S. Low, “Scalable Laws for Stable Network Congestion Control,” Proceedings of the 2001 CDC, Orlando,FL, 2001.

  22. Functional Differential Equations Consider the following nonlinear system with delay: FDE We want a sufficient condition for stability of x(t) = x*.

  23. FDE: Example

  24. FDE

  25. FDE Lyapunov Approach:

  26. Razumikhin

  27. Razumikhin

  28. FDE

  29. FDE and Transport Recall linearized: Nonlinear: Theorem: Proof: Razumikhin …. Z. Wang and F. Paganini, “Global Stability with Time-Delay in Network Congestion Control.”

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