Communication Networks

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(n1) ax(n) u(n) a lt 1 ? bibo
• Nyquist feedback system, L(s) K(s)G(s).Stable
  if L(jw) does not encircle 1.(If L(jw0) - 1
  e lt - 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 lt 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 lt 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.