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Controls and Systems · Chapter 9 of 10 · Advanced

Stability Margins and Nyquist

A loop can be stable yet on the brink. The gain and phase margins measure how much extra gain or lag the loop can take before it tips over, the practical language of robustness.

Readiness check

This chapter measures how close to instability a loop is. Tick only what you can do closed-notes.

Find the gain and phase crossover frequencies.
Evaluate the magnitude and angle of G(jω).
Convert a gain ratio to decibels.
Recall what −180 degrees of phase means for feedback.
Read a polar (Nyquist) curve.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit crossovers in Chapter 8.

3 or more weak itemsReview complex magnitude and angle before continuing.

The core idea

Instability happens when the loop gain reaches one with 180 degrees of lag. The gain and phase margins measure how far the loop is from that point at the two crossover frequencies.

GM = 1/|L(jω_pc)|, ∠L(jω_pc) = −180°PM = 180° + ∠L(jω_gc), |L(jω_gc)| = 1Nyquist: encircle −1 ⇒ unstable

Negative feedback becomes positive when the loop adds 180 degrees of phase; if the gain is also one there, the loop sustains an oscillation. The gain margin is how much the gain can be raised before the magnitude reaches one at the phase crossover (where the lag is already −180). The phase margin is how much more lag the loop can take before reaching −180 at the gain crossover (where the magnitude is already one). The Nyquist criterion says the same thing geometrically: stability depends on how the polar plot of L(jω) circles the critical point −1. Comfortable margins, a gain margin above about 6 dB and a phase margin near 45 degrees, keep the loop robust to the modeling errors that real plants always carry.

The skill works when: you read the gain at the phase crossover and the phase at the gain crossover.

The skill breaks down when: the two margins are read at the same frequency, or the −1 point is ignored.

The concept. The Nyquist plot of the open-loop L(jω). Stability hinges on the critical point −1: how closely the curve approaches it sets the gain and phase margins.

The skills, taught in order

Five skills define the two crossovers, the gain and phase margins, the Nyquist criterion, and the link to robustness.

9.1 The two crossover frequencies

The gain crossover ω_gc is where |L| = 1 (0 dB); the phase crossover ω_pc is where ∠L = −180 degrees. The margins are read at these two frequencies, one for gain and one for phase, so finding them is the first step.

9.2 The gain margin

At the phase crossover the lag is already −180 degrees, so the only thing keeping the loop stable is that the gain is below one. The gain margin GM = 1/|L(jω_pc)| is the factor by which the gain could rise before instability, usually quoted in decibels.

9.3 The phase margin

At the gain crossover the magnitude is already one, so stability depends on the lag being less than −180 degrees. The phase margin PM = 180° + ∠L(jω_gc) is the extra lag the loop can absorb, and it correlates closely with damping: a phase margin near 45 to 60 degrees gives a well-damped response.

Margin	Read at	Healthy value
Gain margin	phase crossover ω_pc	greater than about 6 dB
Phase margin	gain crossover ω_gc	about 30 to 60 degrees

9.4 The Nyquist criterion

The Nyquist plot maps L(jω) into the complex plane. The criterion counts how many times the curve encircles the −1 point and relates it to open-loop poles; for a stable open loop, any encirclement of −1 means the closed loop is unstable. It handles cases Bode margins cannot, such as conditionally stable systems.

9.5 Margins and robustness

Margins are not just a stability check; they are a measure of how much modeling error, delay, or gain drift the loop can survive. A design with thin margins may be stable on paper yet ring or go unstable in hardware, which is why robust margins are a specification in their own right.

Engineering connection: the phase margin sets the closed-loop damping, tying this chapter back to the transient specs of Chapter 4, and lead-lag compensators are designed to hit a target margin.

Worked example 1: the gain margin

An open-loop transfer function is L(s) = 8/[s(s + 2)(s + 4)]. Find the phase crossover frequency and the gain margin.

Figure 1. At the phase crossover (−180 degrees), the magnitude is 15.6 dB below 0 dB. That gap is the gain margin: the gain could rise sixfold before the loop reaches the brink.

ProblemFind ω_pc and the gain margin for the loop in Figure 1.
Given / findL(s) = 8/[s(s + 2)(s + 4)]. Find ω_pc (∠L = −180°) and GM.
AssumptionsSinusoidal steady state; gain margin from the magnitude at the phase crossover.
ModelSet the phase to −180 degrees to get ω_pc, then GM = 1/|L(jω_pc)|.
Equations∠L = −90° − arctan(ω/2) − arctan(ω/4) = −180°GM = 1/|L(jω_pc)|
SolveThe phase reaches −180° when arctan(ω/2) + arctan(ω/4) = 90°, i.e. (ω/2)(ω/4) = 1, so ω_pc = √8 = 2.83 rad/s. There |L| = 8/[2.83·√12·√24] = 8/48 = 0.167, so GM = 1/0.167 = 6.0, which is 20 log 6 = 15.6 dB.
CheckThe phase crossover frequency 2.83 rad/s and the gain factor of 6 match the Routh result from Chapter 7: instability arrives at K = 48 = 6 × 8, exactly the gain margin times the current gain.
ConclusionA gain margin of 15.6 dB means the loop tolerates a sixfold gain increase before oscillating, a healthy margin.

Result. ω_pc = 2.83 rad/s; GM = 6 (15.6 dB).

Worked example 2: the phase margin

For L(s) = 100/[s(s + 10)], with gain crossover frequency ω_gc = 7.86 rad/s (from Chapter 8), find the phase margin and state whether the closed loop is stable.

Figure 2. At the gain crossover the phase is −128 degrees, so it sits 52 degrees above the −180 line. That gap is the phase margin, here a healthy 51.8 degrees.

ProblemFind the phase margin for the loop in Figure 2 and judge stability.
Given / findL(s) = 100/[s(s + 10)], ω_gc = 7.86 rad/s. Find PM and the stability conclusion.
AssumptionsSinusoidal steady state; phase margin from the phase at the gain crossover.
ModelEvaluate ∠L at ω_gc, then PM = 180° + ∠L(jω_gc).
Equations∠L = −90° − arctan(ω/10)PM = 180° + ∠L(jω_gc)
Solve∠L(j7.86) = −90° − arctan(7.86/10) = −90° − 38.2° = −128.2°. PM = 180° − 128.2° = 51.8°.
CheckThe phase margin is positive and there is no finite phase crossover (the phase only approaches −180 as ω → ∞), so the gain margin is infinite. Both margins positive means the closed loop is stable, and a 52 degree margin implies a well-damped response.
ConclusionA phase margin near 50 degrees is a robust, well-damped design. The margin doubles as a predictor of overshoot, linking back to the damping ratio.

Result. PM = 51.8°, so the closed loop is stable and well damped.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Margins at one frequency	Gain and phase margins confused	"Am I at the gain or phase crossover?"	Gain margin at ω_pc, phase margin at ω_gc.
Sign error in phase margin	Negative PM for a stable loop	"Did I add 180 to a negative angle?"	PM = 180° + ∠L, where ∠L is negative.
Ignoring the −1 point	Nyquist conclusion wrong	"How does the curve circle −1?"	Count encirclements of −1 relative to open-loop poles.
Treating margins as just stability	Thin margins accepted	"Is there room for modeling error?"	Keep robust margins; thin ones ring or fail in hardware.

Practice ladder

Level 1 · Direct skill

At the phase crossover, |L| = 0.25. Find the gain margin in decibels.

Show answer

GM = 1/0.25 = 4, which is 20 log 4 = 12 dB. The gain could quadruple before instability.

Level 2 · Mixed concept

At the gain crossover the open-loop phase is −150 degrees. Find the phase margin and comment on damping.

Show answer

PM = 180° − 150° = 30°. A 30 degree margin is on the low side, implying noticeable overshoot and limited robustness.

Level 3 · Independent problem

For the Worked Example 1 system, raise the gain to K = 48. What is the new gain margin?

Show answer

At K = 48 the magnitude at ω_pc is 48/48 = 1, so GM = 1/1 = 0 dB: the loop is exactly marginally stable, oscillating at 2.83 rad/s, as Routh predicted.

Level 4 · Transfer to real engineering

Take a real feedback loop and argue what gain or phase margin it should be designed for, given the uncertainty in its plant model.

What good work looks like

A target gain margin (around 6 to 12 dB) and phase margin (around 45 degrees) justified by expected delay, gain drift, and modeling error.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I read the gain margin at the phase crossover and vice versa."

"Give me three loops; I will say which has the most robust margins."

"What is the phase margin?" Evaluating ∠L at the crossover yourself is the skill.

"Is this robust?" Interpreting the margins is the point.

Portfolio task

For a real loop, find both crossover frequencies, compute the gain and phase margins, and state whether the design is robust enough for its plant uncertainty.

Must include: both crossover frequencies, a gain margin and a phase margin, and a robustness judgement.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Where is the gain margin read?

At the phase crossover, GM = 1/|L(jω_pc)|.

2. Where is the phase margin read?

At the gain crossover, PM = 180° + ∠L(jω_gc).

3. What is the critical point in Nyquist?

The −1 point; how the curve encircles it decides closed-loop stability.

4. What are healthy margins?

Gain margin above about 6 dB, phase margin around 45 degrees.

5. What does the phase margin predict?

The closed-loop damping and overshoot, linking to the damping ratio.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive a gain margin and a phase margin from a blank page.

+3 daysFind the margins of two new loops.

+7 daysCarry margins into PID tuning, Chapter 10.

+30 daysRelate phase margin to the damping ratio from Chapter 4.

Textbook mapping

Item	Mapping
Primary source	Ogata, Modern Control Engineering, Chapter 7 (Nyquist Stability and Stability Margins)
Cross-reference	Nise, Ch. 10 · Dorf and Bishop, Ch. 9
Core topics	9.1 Crossover frequencies · 9.2 Gain margin · 9.3 Phase margin · 9.4 Nyquist criterion · 9.5 Margins and robustness
Engineering connection	Phase margin sets closed-loop damping; compensators target a chosen margin.
Read next	Chapter 10: PID Control and Tuning.