National Instruments 370757C-01, NI MATRIX Xmath Robust Control Module Stability Margin (smargin)

Chapter 2 RobustnessAnalysis

Stability Margin (smargin)

Assume that the nominal closed-loop system is stable. That belief raises a

question: Does the system remain stable for all possible uncertain transfer

functions that satisfy the magnitude bounds (Equation 2-1)? If so, the

system is said to be robustly stable. If the magnitude bounds are small

enough, the uncertainties will not destabilize the system; your system will

be robustly stable.

Roughly speaking, the stability margin of your system is defined as the

factor by which you can increase all the magnitude bounds li and still

maintain stability for all possible uncertain transfer functions δi. If this

number is larger than one (0dB), then you know that there are no uncertain

transfer functions that satisfy the magnitude bound and destabilize your

system. Moreover, the number tells you how much more uncertainty your

system could tolerate than the given bounds li(ω). If the margin is less than

one, then there are uncertain transfer functions that satisfy the magnitude

bound (Equation 2-1) and result in an unstable system. In this case, the

margin tells you how much you must reduce the magnitude bounds before

you have robust stability.

More precisely, the stability margin at frequency ω is defined as the

smallest α such that the system can have a pole at jω, with the uncertain

transfer functions satisfying |δi(jω)| ≤ αli(ω):

margin(w) = min{ α| systems can have a pole at jω with magnitude bounds αli(jω)}

The stability margin also can be expressed as:

margin(w) = min{ α| det I–HqrjωΔ ≠ 0 such that |Δii|≤αli(α)}

Note The stability margin only depends on Hqr.

The margin often is expressed in dB. If the margin is greater than zero for

all frequencies, then your system is robustly stable. If the margin is less

than zero for some frequencies, then your system is not robustly stable.

Inparticular, there are uncertain transfer functions that satisfy the

magnitude bound (Equation 2-1) and cause the system to have a pole at

those frequencies where the margin is negative. This does not mean that any

δi values that satisfy the magnitude bound will destabilize the system: it

means that there are some bad δi values that satisfy the magnitude bounds

and destabilize the system.