University of Melbourne · FACULTY OF ENGINEERING

ELEN90055 · Control Systems

Q: Are the phase margin and the gain margin enough to declare a loop robust?

Not on their own. Both margins are measured at a single frequency each: the phase margin at the gain cross-over and the gain margin at the phase cross-over. A Nyquist plot can show a comfortable phase margin and a comfortable gain margin yet still pass close to -1 at some frequency in between, which means a tall sensitivity peak and poor robustness. That is why the peak sensitivity ||S0||_inf, the reciprocal of the closest approach to -1 over all frequencies, is the honest single number, and why the cross-over relation |S0(jwc)| = 1/(2 sin(Mf/2)) links a small phase margin to a large peak.

Q: Why does a time delay limit how fast I can make the loop?

A transport delay of T seconds multiplies the plant by exp(-sT). Treated as a pure output delay, the complementary sensitivity is T0 = exp(-sT), whose magnitude is exactly 1 at every frequency, so a delay can never be pushed small the way an ordinary high-frequency roll-off can. In margin terms the delay adds a phase lag equal to w times T radians that grows with frequency, so at the cross-over it eats the phase margin. The loop tolerates a delay only up to T_max = Mf(radians)/wc, which caps how high the cross-over frequency can go.

Q: Can AI help me with relative stability and robustness in ELEN90055?

Yes, as a study aid. An AI tutor like Sia can explain phase margin, gain margin and the sensitivity peak step by step, walk you through the geometry behind |S0(jwc)| = 1/(2 sin(Mf/2)), and check your arithmetic and units on a small-gain or delay-margin calculation. Use it to understand the method, the sign conventions and the factor of 20 in the gain-margin definition, and to practise; it does not sit your open-book exam or guarantee a mark, and you should confirm every formula against the unit notes on Canvas.

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Chapter 8 of 12 · ELEN90055

Relative stability and robustness

This chapter of ELEN90055 Control Systems at the University of Melbourne asks not just whether a feedback loop is stable, but how far it sits from instability and whether it survives the fact that the real plant never matches the model. Every answer is a distance from the critical point −1 + j0 on the Nyquist plot: the phase margin, the gain margin and the sensitivity peak.

It maps to Part III of the course (frequency-domain analysis of loops, loosely following Goodwin, Graebe & Salgado) and introduces the robustness tools — the small-gain theorem, multiplicative and additive model error, and the hard limit a transport delay imposes on loop speed.

In this chapter

What this chapter covers

01Locate the critical point -1 + j0 and read every stability margin as a distance from it
02Define the phase margin Mf = 180 degrees + angle A0(jwc) at the gain cross-over frequency
03Define the gain margin Mg = 20 log10(1/|A0(jw180)|) dB at the phase cross-over, and know why the factor is 20
04Read the peak sensitivity ||S0||_inf as the reciprocal of the closest approach of the plot to -1
05Derive and use the cross-over relation |S0(jwc)| = 1/(2 sin(Mf/2)) and see why a small phase margin means a tall peak
06Distinguish additive model error G = G0 + Ge from multiplicative error G = G0(1 + G_Delta)
07Apply the small-gain robust-stability test |G_Delta(jw)|*|T0(jw)| < 1 at every frequency
08Explain why a transport delay keeps |T0| = 1 at all frequencies and so caps the achievable cross-over frequency
09Compute a delay margin T_max = Mf(radians)/wc as the largest dead time the loop tolerates

Worked example · free

Read the margins off a Bode plot and check the delay

Q [8 marks]. For a unity-feedback loop you read the open-loop frequency response A0(jw) off a Bode plot. At the gain cross-over frequency wc = 2 rad/s the open-loop phase is angle A0(jwc) = -130 degrees. At the phase cross-over frequency w180 = 5 rad/s the open-loop magnitude is |A0(jw180)| = 0.4. Find the phase margin, the gain margin in dB, the sensitivity magnitude at cross-over |S0(jwc)|, and the largest transport delay the loop can tolerate.

+2Phase margin. Mf = 180 degrees + angle A0(jwc) = 180 - 130 = 50 degrees. This is the extra phase lag that can be added at wc = 2 rad/s before the plot reaches -1; 50 degrees clears the usual 30-degree rule of thumb.
+2Gain margin. Mg = 20 log10(1/|A0(jw180)|) = 20 log10(1/0.4) = 20 log10(2.5) = 20 x 0.398 = 7.96 dB. Keep the factor 20 (an amplitude ratio, not power); the loop tolerates a gain increase of 2.5x before the phase cross-over reaches -1.
+2Sensitivity at cross-over. |S0(jwc)| = 1/(2 sin(Mf/2)) = 1/(2 sin(25 degrees)) = 1/(2 x 0.4226) = 1/0.8452 = 1.18. So the closed-loop sensitivity at wc is about 1.18 (a modest peak floor, roughly 1.5 dB); the true peak ||S0||_inf is at least this.
+2Delay margin. Convert the phase margin to radians: 50 degrees = 50 x pi/180 = 0.873 rad. The delay margin is T_max = Mf(rad)/wc = 0.873/2 = 0.436 s. A transport (dead-time) delay above about 0.44 s adds enough lag at wc to push the phase past -180 degrees and destabilise the loop.

Mf = 50 degrees, Mg = 7.96 dB, |S0(jwc)| = 1.18, and delay margin T_max = 0.436 s. The loop has healthy phase and gain margins, a low sensitivity floor at cross-over, and tolerates a dead time up to about 0.44 s.

Sia tip — Match units on every line: Mf is an angle (convert to radians only inside the delay formula, or you will be off by a factor of about 57), Mg is 20 log10 in dB, and |S0| is dimensionless. And remember |S0(jwc)| = 1/(2 sin(Mf/2)) is the value at cross-over, a lower bound on the true peak ||S0||_inf, not the peak itself.

Glossary

Key terms

Critical point -1 + j0: The point on the Nyquist plot that the open-loop response A0(jw) must not encircle for a stable open loop to give a stable closed loop. Every stability margin measures how far the plot stays from this point.
Gain cross-over frequency wc: The frequency at which the open-loop gain is unity, |A0(jwc)| = 1, so the plot meets the unit circle. The phase margin is read at this frequency. Units: rad/s.
Phase margin Mf: Mf = 180 degrees + angle A0(jwc): the extra phase lag that can be added at the gain cross-over before the loop reaches -1. Quoted in degrees; a healthy loop has Mf greater than about 30 degrees. A negative value means the loop is already unstable.
Gain margin Mg: Mg = 20 log10(1/|A0(jw180)|) dB, evaluated at the phase cross-over frequency w180 where angle A0 = -180 degrees: the extra loop gain that can be applied before instability. The factor is 20 (an amplitude ratio). A rule of thumb is Mg above about 15 dB; if the phase never reaches -180 degrees the gain margin is infinite.
Peak sensitivity ||S0||_inf: The maximum over frequency of |S0(jw)| = 1/|1 + A0(jw)|, equal to the reciprocal of the closest approach of the plot to -1. Dimensionless; a common target is below 4 (about 12 dB). It captures robustness in one number even when Mf and Mg look fine.
Cross-over relation: |S0(jwc)| = 1/(2 sin(Mf/2)): the sensitivity magnitude at the gain cross-over, derived from the chord length between the two unit vectors at -1 and at A0(jwc). A small phase margin forces this value, and hence the peak, to be large.
Multiplicative model error G_Delta: The relative mismatch between the true plant and the model, G = G0(1 + G_Delta), so G_Delta = Ge/G0. Dimensionless and usually small at low frequency but growing past 1 at high frequency. The additive form writes the absolute error as G = G0 + Ge.
Small-gain theorem: A robust-stability test: if C stabilises the model loop and the true loop has the same number of right-half-plane poles, then C stabilises every true plant provided |G_Delta(jw)| times |T0(jw)| is less than 1 at all frequencies. It forces |T0| to be small where the model error is large.
Delay margin T_max: The largest transport (dead-time) delay a loop tolerates, T_max = Mf(in radians)/wc seconds. A pure delay keeps |T0| = 1 at every frequency, so it cannot be made small and it directly caps the cross-over frequency.

FAQ

Relative stability and robustness FAQ

Are the phase margin and the gain margin enough to declare a loop robust?

Not on their own. Both margins are measured at a single frequency each: the phase margin at the gain cross-over and the gain margin at the phase cross-over. A Nyquist plot can show a comfortable phase margin and a comfortable gain margin yet still pass close to -1 at some frequency in between, which means a tall sensitivity peak and poor robustness. That is why the peak sensitivity ||S0||_inf, the reciprocal of the closest approach to -1 over all frequencies, is the honest single number, and why the cross-over relation |S0(jwc)| = 1/(2 sin(Mf/2)) links a small phase margin to a large peak.

Why does a time delay limit how fast I can make the loop?

A transport delay of T seconds multiplies the plant by exp(-sT). Treated as a pure output delay, the complementary sensitivity is T0 = exp(-sT), whose magnitude is exactly 1 at every frequency, so a delay can never be pushed small the way an ordinary high-frequency roll-off can. In margin terms the delay adds a phase lag equal to w times T radians that grows with frequency, so at the cross-over it eats the phase margin. The loop tolerates a delay only up to T_max = Mf(radians)/wc, which caps how high the cross-over frequency can go.

Can AI help me with relative stability and robustness in ELEN90055?

Yes, as a study aid. An AI tutor like Sia can explain phase margin, gain margin and the sensitivity peak step by step, walk you through the geometry behind |S0(jwc)| = 1/(2 sin(Mf/2)), and check your arithmetic and units on a small-gain or delay-margin calculation. Use it to understand the method, the sign conventions and the factor of 20 in the gain-margin definition, and to practise; it does not sit your open-book exam or guarantee a mark, and you should confirm every formula against the unit notes on Canvas.

Studying with AI? Sia — free AI electrical engineering tutor works through ELEN90055 step by step.

Study strategy

Exam move

Anchor everything to one picture: the Nyquist plot and the critical point -1 + j0. Be able to draw it and mark, in order, the gain cross-over where |A0| = 1 (phase margin lives here), the phase cross-over where the angle is -180 degrees (gain margin lives here), and the closest approach of the plot to -1 (the peak sensitivity). Memorise the three definitions with their units: Mf = 180 degrees + angle A0(jwc) in degrees, Mg = 20 log10(1/|A0(jw180)|) in dB with the factor 20, and ||S0||_inf dimensionless. Then drill the two links that carry most of the marks: the cross-over relation |S0(jwc)| = 1/(2 sin(Mf/2)), which shows why a small phase margin is dangerous, and the small-gain test |G_Delta| times |T0| < 1, which shows why uncertainty and delay cap the cross-over frequency. Practise converting a phase margin to a delay margin T_max = Mf(radians)/wc, and always convert to radians first. The exam is 3 hours, open book, worth 70% with a hurdle, and you answer four questions summing to 50 marks, so budget about 3.6 minutes per mark. Because it is open book, marks reward method shown with correct sign, factor and unit conventions rather than memorised formulae, so define your symbols and state which margin measures what. Confirm the exact date of the next (Semester 1, ~June 2027) sitting on Canvas.

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