Relative Stability: Gain and Phase Margins

The Nyquist criterion of Chapter 16 returns a verdict: the closed loop is stable, or it has \(Z\) poles in the right half-plane. That verdict is exact and it is also, on its own, of limited engineering value. Consider two loops, both stable, whose Nyquist plots cross the negative real axis at \(-0.98\) and at \(-0.2\). Both satisfy \(Z=0\). But the first is a hair's breadth from disaster: a two per cent rise in gain — a warmer amplifier, a stiffer actuator, a component tolerance — tips it over. The second would survive a five-fold gain error.

Real plants are never known exactly. Their parameters drift with temperature and age, their models omit high-frequency dynamics, and their gains change with operating point. A design that is stable only for the nominal model is not a design at all. What we need is a measure of relative stability: how much the loop can be perturbed before the plot reaches the critical point.

The phrase "reaches the critical point" is the key. Stability is lost when the Nyquist plot passes through \(-1\). So the natural measure of safety is the distance, in some sense, between the plot and that point. The two classical margins are the two simplest such distances — one measured radially, one measured along the unit circle — and each corresponds to a physically meaningful kind of model error.

The point \(-1\) is the complex number of magnitude 1 and angle \(-180^\circ\). A point on the Nyquist plot coincides with it only if both conditions hold at the same frequency. That observation splits the approach to instability into two independent directions.

A loop can fail because its gain is too large at the frequency where the phase is already \(-180^\circ\). The frequency at which \(\angle G(j\omega)H(j\omega) = -180^\circ\) is the phase crossover frequency \(\omega_{pc}\). At that frequency the plot lies on the negative real axis; whether the loop is safe depends on how far along that axis it lies.

Alternatively, a loop can fail because its phase lag is too great at the frequency where the gain is already unity. The frequency at which \(|G(j\omega)H(j\omega)| = 1\) — that is, \(0\) dB — is the gain crossover frequency \(\omega_{gc}\). At that frequency the plot lies on the unit circle; whether the loop is safe depends on how far round that circle it has swung.

Each margin answers a question of the form "how much of one thing can change before the plot lands on \(-1\), the other thing being held fixed?" That is exactly the structure of a tolerance specification, which is why the margins survived from Bode's laboratory notebooks into every control textbook since.

One warning to plant early: the two are computed at different frequencies. A perfectly respectable gain margin at \(\omega_{pc}\) tells you nothing about the phase at \(\omega_{gc}\). This is not a technicality; it is the source of most of the errors students make with margins, and Example 1 makes it concrete.

Suppose the loop is open-loop stable and its Nyquist plot crosses the negative real axis at \(-a\), so that \(|G(j\omega_{pc})H(j\omega_{pc})| = a\). Section 16-7 showed that multiplying the loop gain by a factor \(\lambda\) scales that crossing to \(-\lambda a\). The crossing lands exactly on \(-1\) when \(\lambda a = 1\). The critical multiplier is therefore \(1/a\), and it has a name.

Three readings of the same number are worth keeping straight. As a ratio, \(\mathrm{GM}=2\) means the gain may double. In decibels the same margin is \(20\log_{10}2 = 6.02\) dB. As a position on the plot, it says the negative-real-axis crossing sits at \(-0.5\), halfway to the critical point.

The sign convention deserves a moment. If the crossing is at \(-0.5\), then \(a=0.5\), \(\mathrm{GM}=2\), and \(\mathrm{GM}_{\text{dB}} = -20\log_{10}0.5 = +6\) dB: positive, and the loop is stable with room to spare. If the crossing is at \(-2\), then \(\mathrm{GM}=0.5\) and \(\mathrm{GM}_{\text{dB}} = -6\) dB: negative, and for an open-loop stable minimum-phase plant that means the loop is already unstable, the gain needing to be halved to recover. A negative gain margin in decibels is not a small margin; it is a failed design.

Two degenerate cases are common enough to name. If the phase never reaches \(-180^\circ\) — true of every first- and second-order loop with no delay and no right-half-plane zero — there is no \(\omega_{pc}\), no real-axis crossing, and the gain margin is infinite. Such a loop cannot be destabilised by gain alone, and its phase margin is then the only margin that means anything. Conversely, if the plot crosses the negative real axis at several frequencies, the gain margin is quoted at the crossing closest to \(-1\) from the right, since that is the one the gain reaches first.

The gain margin collapses to unity — 0 dB — exactly at \(K=6\), the stability boundary found in Chapter 16. The margin and the boundary are the same fact stated twice.

Now hold the gain fixed and let phase lag creep in. At the gain crossover frequency the plot has magnitude 1, so it sits on the unit circle at some angle \(\phi = \angle G(j\omega_{gc})H(j\omega_{gc})\). For a stable design this angle is somewhere between \(-90^\circ\) and \(-180^\circ\). Adding a pure phase lag rotates that point clockwise around the unit circle without changing its magnitude; it reaches \(-1\) after a rotation of \(180^\circ + \phi\) — remembering that \(\phi\) is negative, so this is a positive quantity.

Where the gain margin guards against errors in gain — amplifier tolerance, actuator authority, a changing load — the phase margin guards against errors in timing. Unmodelled high-frequency poles, computational delay in a digital controller, sampling, transport lag: all of these subtract phase while leaving the low-frequency magnitude essentially untouched. The phase margin is the budget from which they are paid, and it is the margin that is usually the binding constraint in practice.

Because a lag of \(\mathrm{PM}\) degrees at \(\omega_{gc}\) is what kills the loop, and because a dead time \(T\) produces a lag of \(\omega T\) radians, the phase margin converts directly into a tolerable dead time. That conversion is the delay margin of Section 17-8.

The Bode plot is where margins are actually read, because the two crossover frequencies appear as two horizontal reference lines rather than as geometric constructions. The magnitude panel carries the \(0\) dB line; the phase panel carries the \(-180^\circ\) line. The recipe is four steps and no arithmetic.

Find \(\omega_{gc}\) where the magnitude curve crosses \(0\) dB. Drop down to the phase panel at that frequency and read the phase; the amount by which it exceeds \(-180^\circ\) is the phase margin. Then find \(\omega_{pc}\) where the phase curve crosses \(-180^\circ\), go up to the magnitude panel at that frequency and read the magnitude in dB; the amount by which it lies below \(0\) dB is the gain margin.

Notice the crossing pattern in the figure: the gain margin is measured on the upper panel at the frequency identified on the lower one, and the phase margin the other way round. Reading both on the same panel, or both at the same frequency, is the single most common mistake in the subject.

The asymptotic Bode construction makes this quick even by hand. Because the magnitude asymptotes are straight lines of slope \(-20\) dB/decade per pole, \(\omega_{gc}\) can usually be located by inspection, and the phase there estimated from the pole and zero corner frequencies. A design rule follows directly and is worth memorising: a loop whose magnitude curve crosses \(0\) dB on a \(-20\) dB/decade slope has a healthy phase margin; one that crosses on \(-40\) dB/decade is marginal, and one that crosses on \(-60\) dB/decade is almost certainly unstable. The reason is Bode's gain–phase relation: for a minimum-phase system, a sustained slope of \(-20n\) dB/decade is accompanied by a phase of roughly \(-90n\) degrees, so a \(-40\) dB/decade crossover implies a phase near \(-180^\circ\) and a phase margin near zero.

The \(-40\) dB/decade entry is deliberately vague, and honestly so: nearby corner frequencies pull the true phase well away from the asymptotic estimate, which is exactly why that case must be computed rather than guessed. The rule is a screening tool, not a substitute for evaluating \(\angle GH(j\omega_{gc})\).

On the Nyquist plot both margins live in one picture, which is why the plot remains the better vehicle for understanding even though the Bode plot is the better vehicle for measurement.

Draw the unit circle centred at the origin. The plot crosses it at \(\omega_{gc}\); the angle between the negative real axis and the radius to that crossing is the phase margin, measured as a rotation the plot would still have to make to reach \(-1\). The plot crosses the negative real axis at \(\omega_{pc}\), at the point \(-a\); the gain margin is \(1/a\), which is the factor by which the whole curve would have to be inflated for that crossing to reach \(-1\).

Seeing them together exposes what neither reveals alone. The gain margin is measured strictly along the real axis; the phase margin strictly along the unit circle. Between those two directions lies a whole arc of the plot that neither margin inspects. If the curve makes an excursion toward \(-1\) at some intermediate frequency, both classical margins can be large while the true distance to the critical point is small.

That true distance has a name: the vector margin, the shortest Euclidean distance from \(-1\) to the Nyquist plot. Since the distance from \(-1\) to a point on the plot is \(|1+G(j\omega)H(j\omega)|\), the vector margin is

The quantity inside the maximum is the sensitivity function, and its peak \(M_s\) is a single number that no excursion of the plot can hide from. A common modern specification is \(M_s \le 2\), equivalently a vector margin of at least \(0.5\). It is a useful fact that \(M_s\le 2\) automatically guarantees a gain margin of at least \(2\) and a phase margin of at least about \(29^\circ\), so the vector margin is the stronger requirement, and the classical pair are the more easily measured consequence of it.

A margin would be a bureaucratic quantity if it only certified safety. Its real value is that it predicts the closed-loop transient, and it does so through the prototype second-order system of Chapter 8. Take the unity-feedback loop whose open-loop transfer function is

This is the standard second-order closed loop, so its overshoot and settling time are exactly those of Chapter 8. Now compute its phase margin. First locate the gain crossover by setting the magnitude to unity:

The quadratic in \(\omega^{2}\) has one positive root, taken above. Now evaluate the phase there. The open-loop phase is \(-90^\circ\) from the integrator plus the lag of the real pole:

The result depends on \(\zeta\) alone — \(\omega_n\) has cancelled. That is the whole point: the phase margin is a measure of damping, not of speed. Evaluating the expression gives a relation between a frequency-domain margin and a time-domain overshoot that no amount of algebra could have made obvious.

If the phase margin fixes the damping, what fixes the speed? The gain crossover frequency. From the expression for \(\omega_{gc}\) above, at \(\zeta=0.5\) we get \(\omega_{gc}=0.786\,\omega_n\), and Chapter 8's two per cent settling time \(t_s \approx 4/(\zeta\omega_n)\) then gives

The pairing is the central design insight of frequency-domain design: \(\omega_{gc}\) sets how fast the loop is; PM sets how well behaved it is. Compensator design in Part 5 is then a matter of raising \(\omega_{gc}\) for speed and restoring the phase margin the increase destroyed — which is exactly what a lead network does — or of raising the low-frequency gain for accuracy while leaving \(\omega_{gc}\) and PM alone, which is what a lag network does.

The phase margin becomes concrete the moment it is converted into time. A pure dead time \(T\) contributes a phase lag of \(\omega T\) radians, and Section 17-4 established that the loop can absorb a lag of PM before the plot reaches \(-1\). Since the critical frequency is \(\omega_{gc}\), equating the two gives the largest dead time the loop can survive.

The formula carries an uncomfortable message. A fast loop — large \(\omega_{gc}\) — spends its phase margin quickly, because the same delay costs more phase at a higher frequency. Doubling the bandwidth halves the tolerable delay at fixed PM. Speed and delay tolerance are traded directly against one another, and no choice of controller structure repeals the exchange rate. Bandwidth is limited by the loop's slowest un-modelled lag, and it is limited hard.

A practical corollary: in a digital control loop the sample-and-hold alone contributes an average dead time of half a sample period, \(T_s/2\). Requiring that this consume no more than about a fifth of the delay margin is the usual justification for the rule that the sampling frequency should be at least twenty to thirty times the gain crossover frequency — a far more demanding requirement than the Nyquist–Shannon sampling rate, and one derived from stability rather than from information.

The two classical margins are summaries, and every summary discards something. Four situations are worth recognising on sight, because in each the margins are computed correctly and the conclusion drawn from them is still wrong.

Multiple crossovers. If the magnitude curve crosses \(0\) dB more than once, there are several gain crossover frequencies and therefore several phase margins; if the phase crosses \(-180^\circ\) more than once, several gain margins. Quoting one of them as "the" margin hides the others. The honest procedure is to list all of them and take the worst, and to check the encirclement count of Chapter 16 rather than trusting any margin at all.

Conditional stability. Section 16-7 met loops that are destabilised by decreasing the gain. Their gain margin, quoted as a factor by which gain may be increased, says nothing whatever about the lower boundary. For such a system the margin must be quoted as an interval, \(K_1 \lt K \lt K_2\), or it is worse than useless.

Open-loop unstable plants. With \(P\ge1\) a positive phase margin does not imply stability; only \(N=-P\) does. Margins remain meaningful as measures of distance from the critical point, but the sign conventions invert and the verdict must always come from the encirclement count first.

Resonant excursions. A lightly damped mode can send the Nyquist plot on a loop that passes close to \(-1\) at a frequency far from either crossover. Both classical margins can then be textbook-perfect while the loop rings violently and a small parameter change destabilises it. This is precisely the failure the vector margin of Section 17-6 detects, and it is the reason \(M_s\) has largely replaced GM and PM in modern robust design.

In every problem identify which crossover frequency the question needs before computing anything. Assume unity feedback and an open-loop stable minimum-phase plant unless stated otherwise. Difficulty rises down the list.