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

Stability Motivation Overview of results Linear Systems Nyquist Functional Differential Equations

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

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(j  ) does not encircle – 1. (If L(j  0 ) = - 1 –  < - 1, then input at  0 blows up.)

Overview of Results Nonlinear system Linearize around equilibrium x 0. If linearized system is stable, then x 0 is locally stable for original system Nonlinear system: Lyapunov Assume V(x(t)) decreases and level curves shrink Then the system is stable

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

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

Linear Systems Laplace Transform

Linear Systems

Example

Linear Systems Example

Linear Systems Observation

Nyquist Slide from a tak by Glenn Vinnicombe

Nyquist

Slide from a tak by Glenn Vinnicombe

Nyquist MIMO Case:

Nyquist Example 1  Closed-Loop is stable

Nyquist Example 2

Nyquist Example 2 …  Stable if T < 1.35s

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

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

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

FDE: Example

FDE

Lyapunov Approach:

Razumikhin

FDE

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