Stability Is Not A Yes/No Proposition

How Stable Is This System?
Phase Margin
Gain Margin

Gain And Phase Margins On Bode' Plots

Problems

        It's not enought to know that a system is stable or unstable.  If a system is just barely stable, then a small gain in a system parameter could push the system over the edge, and you will often want to design systems with some margin of error.

        If you're going to do that, you'll need some measure of how stable a system is.  To get such measures - and there are at least two that are widely used - we will have to re-visit the Nyquist stability criterion.

        This lesson has straightforward goals.  You need to be able to determine how to get measures of relative stability, and this lesson will introduce two measures - phase margin and gain margin.  That leads to these goals.

Given a linear feedback system,

  Be able to determine the phase margin for the system using either a Nyquist plot or a Bode' plot.
  Be able to determine the gain margin for the system using either a Nyquist plot or a Bode' plot.

        Here's a Nyquist plot for a system.  Let's say it is a plot of KG(jw)H(jw).

        How stable is this system?

• We've marked two points.
• The zero db/unit circle crossing.
• That's marked with a small orange circle.
• The - 180^o crossing.
• That's marked with a small blue circle.
But, the most important point is the point at -1

 Here's a clip.

• As the clip plays, the gain of the system increases.
• The blue dot shows the -180^o crossing.
• The red dot shows the 0 db - unit circle - crossing.
• As the clip plays, you can see that the system becomes unstable, and the system does not satisfy the Nyquist stability criterion.

        Now, let's answer this question.  For a some given value of gain, "How stable is this system?".

• To answer the question we need a numerical measure of stability.
• It seems very natural to use the  -180^o crossing and the 0 db crossing to form a numerical measure.
• In each case, we can devise a numerical measure of how far the system is from  going unstable.  We will examine two stability margins that measure how far the system is from instability.

        Phase margin is the most widely used measure of relative stability when working in the frequency domain.  On a Nyquist plot we examine the unit circle (which is just all those points that have a magnitude of 1) and we can see that the system we intuitively think of as less stable is closer to the -1 point when we measure distance along the unit circle (which goes through -1).

• We define phase margin as the angle that the frequency response would have to change to move to the -1 point.
• The two different gains shown for the Nyquist plot below would lead to two different phase margins.  The system with the frequency response with the dashed line is less stable.

        To measure phase margin, we measure the angular difference between the point on the frequency response at the unit circle crossing and -180⁰.  Here is an example system.

• Here, we cam compute the phase margin.  We see that:
• tan(f_m) = 1.7
• f_m = 59.5^o.

        Gain margin is another widely used measure of relative stability when working in the frequency domain.  On a Nyquist plot we examine the -180^o crossing and we can see that the system we intuitively think of as less stable is closer to the -1 point when we measure distance along the negative real axis.

• We define gain margin as the amount that the frequency response would have to increase to move to the -1 point.
• The two different gains shown for the Nyquist plot at the right would lead to two different gain margins.  The system with the frequency response with the dashed line is less stable.

• Here, the gain margin is hard to determine when we see the entire plot - a common fault when using Nyquist plots.
• Expand the plot, realizing we will lose part of the plot in the process, but we will focus on the negative axis crossing.

• Now we can see that the Nyquist plot crosses at about -0.18.
• That means that you can increase the gain by a factor of 1/.18 before the system becomes unstable.
• That's a factor of 5.55
• The gain margin is 5.55
• This is an intuitively pleasing definition of a measure of relative stability.

• Gain margin is usually specified in db.  For a gain margin of 5.55 - the factor you can increase gain by - the gain margin, in db, is:
• 20 log10(5.55) = 14.8 db

        Finally, we need to note that most design of this type is done using Bode' plots, and we need to be able to interpret phase margin and gain margin on Bode' plots of frequency response.  Next we will consider how to measure phase and gain margin on a Bode' plot.  Remember these points.

• Phase margin is just the difference between  -180^o and the actual phase angle of the frequency response function, KG(jw)H(jw), measured at the frequency where the magnitude of the frequency response function, |KG(jw)H(jw)|, is equal to one.  On a Bode' plot that frequency is the zero db crossing frequency.
• Gain margin is just the amount of gain that you can add to move the zero db crossing to occur at the same frequency as the -180^o crossing.
• Phase margin is just the difference between  -180^o and the actual phase angle of the frequency response function, KG(jw)H(jw), measured at the frequency where the magnitude of the frequency response function, |KG(jw)H(jw)|, is equal to one.
• On a Bode' plot that frequency is the zero db crossing frequency.

• On this Bode' plot, the low frequency gain - in db - is about 14 db.
• The zero db crossing occurs near f = 0.42 Hz.
• At the zero db crossing, the phase is almost right at -150^o,
• So the phase margin is +30^o.

• Gain margin is just the amount of gain that you can add to move the zero db crossing to occur at the same frequency as the -180^o crossing.
• On this Bode' plot, the -180^o crossing occurs at f = 0.6 Hz.
• At that point, the magnitude plot is at approximately -6.5 db, so the gain can be increased by 6.5 db, and 6.5 db is the gain margin.

        Gain margin and phase margin are measures of relative stability.  As such, they are things that can be specified.  A system may be specified to have a phase margin of at least 40^o, for example, or the gain margin might be specified to be 10 db.  In designing control systems it is not at all uncommon to have to design to specified gain and phase margins.

        The idea here is that someone purchasing a system may have a pre-conceived idea of the amount of stability that they want in a system.  It may be that in their experience certain amounts of phase margin correspond to certain other kinds of behavior in their particular kind of system.  For example, in a certain kind of aircraft attitude control system, and phase margin may be strongly correlated with overshoot in certain maneuvers, and by specifying phase margin the person specifying may really be trying to control other aspects of behavior that cannot easily be computed during the design.

• Problem Nyquist4P01
• Problem Nyquist4P02