Chapter 3

Scalability in the Frequency Domain

Tao generates one, one generates two, two generates three, three generates all things.

—Lao Dan (580–500 BC), Tao Te Ching

This chapter studies the problem of the scalability of the stability results for distributed control systems. The problem is closely related to some differential geometric properties of frequency response plots of local dynamics of agents (nodes) in networks, such as clockwise property, modulus monotonicity, slope monotonicity, phase velocity, critical point of clockwise property, etc. The chapter starts from the clockwise property which plays a key role in scalability analysis. Then, detailed geometric analysis of scalability is conducted for first-order and second-order time-delayed systems which are often encountered in coordinated control systems and end-to-end congestion control systems. Finally, based on the notion of convex directions in the space of stable quasi-polynomials, a frequency sweeping method of scalability test is provided for high-order time-delayed systems.

3.1 How the Scalability Condition is Related with Frequency Responses

Denote by γ the gain margin the transfer function

equation

where W(s) is a rational function of s, and T > 0 is the delay constant. Actually, γ is defined by

(3.1) equation

where ωc > 0 is ...

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