Gain margin, phase margin — stability margins

WHY do we even need margins?

Consider a closed loop with open-loop transfer function $L(s) = G(s)H(s)$ (controller × plant × sensor). The closed-loop transfer function is:

$T(s) = \frac{G(s)}{1 + L(s)}$

So stability is about how close the curve $L(j\omega)$ passes to the critical point $-1$ . Gain and phase margins are two rulers measuring that distance along two specific directions.

The two critical frequencies

Because the point $-1 = 1\angle(-180^\circ)$ needs both magnitude 1 and phase $-180^\circ$ , we ask two separate questions:

At the frequency where phase is already $-180^\circ$ , how far below 1 is the gain? → Gain Margin (GM)
At the frequency where gain is already 1, how far above $-180^\circ$ is the phase? → Phase Margin (PM)

Deriving Gain Margin from first principles

The extra multiplicative gain $k$ that would push it exactly to $-1$ satisfies: $k \cdot |L(j\omega_{pc})| = 1 \quad\Rightarrow\quad k = \frac{1}{|L(j\omega_{pc})|}$

Deriving Phase Margin from first principles

At $\omega_{gc}$ the phase is $\angle L(j\omega_{gc})$ (some value like $-135^\circ$ ). The gap to $-180^\circ$ :

$\text{PM} = \angle L(j\omega_{gc}) - (-180^\circ) = 180^\circ + \angle L(j\omega_{gc})$

Figure — Gain margin, phase margin — stability margins

WHY these are good stability rulers (Dual Coding)

Nyquist view: Plot $L(j\omega)$ in the complex plane. GM = how much you can scale the plot outward before it swallows $-1$ (measured along the negative-real axis). PM = angle you can rotate the plot before the unit-circle intersection reaches $-180^\circ$ .
Bode view: Two stacked plots (magnitude dB, phase deg). Draw a vertical line where phase $=-180^\circ$ ; GM is the distance from the magnitude curve up to 0 dB. Draw a vertical line where magnitude $=0$ dB; PM is the distance from the phase curve up to $-180^\circ$ .

Worked example Worked Example 1 —

$L(s)=\dfrac{K}{s(s+1)(s+2)}$ with $K=1$ Find phase crossover. $\angle L = -90^\circ - \arctan\omega - \arctan(\omega/2) = -180^\circ$ . Why this step? We need the frequency where phase hits the danger angle to compute GM. Set $\arctan\omega + \arctan(\omega/2) = 90^\circ$ ⟹ $\tan$ of sum $\to\infty$ ⟹ $1 - \frac{\omega^2}{2}=0$ ⟹ $\omega_{pc}=\sqrt2$ . Magnitude there: $|L| = \dfrac{1}{\omega\sqrt{\omega^2+1}\sqrt{\omega^2+4}}$ at $\omega=\sqrt2$ : $= \dfrac{1}{\sqrt2\cdot\sqrt3\cdot\sqrt6} = \dfrac{1}{\sqrt{36}} = \dfrac{1}{6}$ . Why? $|L|$ at $\omega_{pc}$ is exactly what GM inverts. GM $= 1/|L| = 6 = 20\log_{10}6 \approx 15.6$ dB. Comfortably stable in gain.

Worked example Worked Example 2 — Phase margin of

$L(s)=\dfrac{1}{s(s+1)}$ Gain crossover: $|L| = \dfrac{1}{\omega\sqrt{\omega^2+1}} = 1$ ⟹ $\omega^2(\omega^2+1)=1$ ⟹ $\omega^4+\omega^2-1=0$ . Let $u=\omega^2$ : $u=\frac{-1+\sqrt5}{2}\approx0.618$ , so $\omega_{gc}\approx0.786$ . Why this step? PM is measured at the frequency where gain is unity. Phase there: $\angle L = -90^\circ - \arctan(0.786) = -90^\circ - 38.2^\circ = -128.2^\circ$ . PM $= 180^\circ - 128.2^\circ = 51.8^\circ$ . Healthy (rule of thumb: aim for $30^\circ$ – $60^\circ$ ).

Rules of thumb (80/20)

Common mistake Steel-manning the classic errors

1. "Bigger gain is always safer / faster, so raise $K$ ." Why it feels right: higher gain gives faster response and smaller steady-state error. The fix: raising $K$ shifts the whole magnitude curve up, shrinking GM. Past GM the loop goes unstable. Speed and stability trade off.

2. "Positive GM alone means stable." Why it feels right: GM is literally called a stability margin. The fix: You need both GM $>0$ dB and PM $>0$ . Also, for non-minimum-phase or conditionally-stable systems the simple margins can mislead — trust the full Nyquist criterion.

3. Confusing which frequency goes with which margin. Why it feels right: both are "crossings." The fix: GM lives at the phase crossover ( $\angle=-180^\circ$ ); PM lives at the gain crossover ( $|L|=1$ ). Memorize: margin is measured where the OTHER quantity is at its critical value.

4. "Phase margin is $\angle L$ itself." The fix: PM $=180^\circ + \angle L(j\omega_{gc})$ — it's the gap to $-180°$ , not the angle.

Recall Feynman: explain to a 12-year-old

Imagine pushing a kid on a swing. If you push at just the right rhythm, tiny pushes build up into a huge swing — that's a loop feeding itself. A control system can accidentally do that: its own correction comes back at the wrong time and the wrong size and makes things worse and worse. Phase margin asks "how badly can my timing be off before pushes start adding up?" Gain margin asks "how much harder can I push before pushes start adding up?" Big margins = you're nowhere near making the swing go crazy. Small margins = one nudge and it's out of control.

Flashcards

What point on the complex plane defines instability for

1+L(s)

-1

, i.e. magnitude 1 at angle

-180^\circ

Define the phase crossover frequency

\omega_{pc}

The frequency where

\angle L(j\omega) = -180^\circ

Define the gain crossover frequency

\omega_{gc}

The frequency where

|L(j\omega)| = 1

(0 dB).

Formula for gain margin?

\text{GM} = 1/|L(j\omega_{pc})|

, or

-20\log_{10}|L(j\omega_{pc})|

dB.

Formula for phase margin?

\text{PM} = 180^\circ + \angle L(j\omega_{gc})

At which frequency is gain margin evaluated, and why?

At phase crossover (

\angle=-180°

), because there only extra gain is needed to reach

-1

At which frequency is phase margin evaluated?

At gain crossover (

|L|=1

), because there only extra phase lag is needed to reach

-180°

Typical healthy design values for GM and PM?

\gtrsim 6

dB (factor ~2), PM

\approx 30^\circ

–

60^\circ

Approximate relation between PM and damping ratio?

\zeta \approx \text{PM}^\circ/100

Max time delay tolerable before instability?

\tau_{max} = \text{PM (radians)}/\omega_{gc}

Why does increasing loop gain

K

reduce gain margin?

It shifts the magnitude curve up, moving

|L|

closer to 1 at the phase crossover.

Is positive gain margin alone sufficient for stability?

No — you need both GM

>0

dB and PM

>0

(and Nyquist for tricky systems).

Connections

Nyquist stability criterion — the parent theorem; encirclements of $-1$ .
Bode plots — where GM and PM are read graphically.
Root locus — how poles move as gain increases toward the stability limit.
Time delay and Padé approximation — how delay eats phase margin.
Damping ratio and overshoot — PM ↔ transient quality link.
PID controller tuning — margins as design targets in GNC loops.
Loop shaping — deliberately sculpting $L(j\omega)$ for target margins.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, feedback loop tab unstable hoti hai jab signal wapas aake khud ko hi feed karne lage — bilkul swing pe bachche ko sahi timing pe push karne jaisa. Danger point complex plane pe $-1$ hai, matlab magnitude $1$ aur angle $-180^\circ$ . Agar loop ka response $L(j\omega)$ is point ko chhoo le, system marginal unstable ho jaata hai. Stability margins basically batate hain ki hum is danger point se kitne door hain.

Do rulers hain. Gain margin us frequency pe naapte hain jahan phase already $-180^\circ$ ho chuka hai (phase crossover). Wahan poochte hain — gain ko aur kitna guna badha sakte hain jab tak $|L|$ ek ho jaaye? Formula: $\text{GM}=1/|L(j\omega_{pc})|$ . Phase margin us frequency pe naapte hain jahan gain already ek hai (gain crossover), aur poochte hain — kitni extra phase lag daal sakte hain jab tak $-180^\circ$ pahunch jaaye? Formula: $\text{PM}=180^\circ + \angle L(j\omega_{gc})$ .

Yaad rakhne ka trick "GaP": Gain margin Phase-crossover pe, Phase margin Gain-crossover pe — matlab har margin wahan naapo jahan doosri quantity apni critical value pe ho. Common galti: log sochte hain gain badhane se system fast aur safe ho jaayega, lekin gain badhaoge to magnitude curve upar shift hoti hai, GM chhoti ho jaati hai, aur ek point ke baad system unstable. Isliye speed aur stability ke beech trade-off hota hai.

GNC (rockets, drones, autopilot) mein ye margins bahut important hain kyunki sensor delay, actuator lag, model uncertainty sab phase kha jaate hain. Design rule: PM around $30^\circ$ – $60^\circ$ aur GM kam se kam $6$ dB rakho, taaki real-world disturbances mein bhi loop stable rahe aur zyada overshoot na ho.

Gain margin, phase margin — stability margins

WHY do we even need margins?

The two critical frequencies

Deriving Gain Margin from first principles

Deriving Phase Margin from first principles

WHY these are good stability rulers (Dual Coding)

Rules of thumb (80/20)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections