ELEN90055 · Control Systems
Classical loop-shaping and Bode gain-phase relation
This chapter of ELEN90055 Control Systems at the University of Melbourne designs a feedback controller by shaping the open-loop gain directly: large at low frequency for tracking and disturbance rejection, small at high frequency for noise and robustness, and crossing 0 dB at a chosen cross-over frequency with a slope of about -20 dB/dec.
It sits in Part IV.c of the course (loosely following Goodwin, Graebe & Salgado) and rests on Bode's gain-phase relation — for a minimum-phase loop the phase is fixed by the magnitude slope — together with the direct-synthesis recipe C = A/G and the way right-half-plane poles and zeros fence in the achievable cross-over.
What this chapter covers
- 01Read loop-shaping as choosing |A0(jw)| = |G(jw)C(jw)|: big at low w, small at high w, cross 0 dB at wc
- 02State Bode's gain-phase relation for a minimum-phase loop: phase is a weighted average of the magnitude slope
- 03Use the working rule: a -20*l dB/dec magnitude slope gives about -l*(65 to 90) degrees of phase
- 04Explain why the loop-gain slope through cross-over should be about -20 dB/dec for a healthy phase margin
- 05Apply the RHP-zero limit wc < z and the RHP-pole limit wc > p, and the feasible window p < wc < z
- 06Compute the extra phase lag -2*arctan(wc/z) a right-half-plane zero adds at cross-over
- 07Synthesise a controller by direct inversion C = A/G for a minimum-phase plant, checking properness nr(A) >= nr(G)
- 08Explain why C = A/G is restricted to minimum-phase plants (unstable controller or forbidden cancellation otherwise)
- 09Read cross-over frequency and phase margin off a Bode plot in the Q4 exam archetype
Design a controller by C = A/G for a minimum-phase plant
- +1Confirm minimum-phase and relative degree. All poles (-1, -5) and all zeros (none) are in the open left-half plane, and there is no transport delay, so G is minimum-phase and the inversion recipe C = A/G is allowed. Its relative degree is nr(G) = 2 - 0 = 2.
- +2Choose the desired open loop A(s). For zero step error put a pole at s = 0 (an integrator); for a -20 dB/dec cross-over at wc = 4 rad/s make the integrator cross 0 dB at 4; add one far pole a decade above (at s = -40) so the controller is realisable: A(s) = 4 / (s (0.025 s + 1)).
- +1Check the properness condition. A has relative degree 2, equal to the plant's relative degree, so C = A/G has relative degree 2 - 2 = 0. That is a proper (physically realisable) controller, satisfying nr(A) >= nr(G).
- +2Invert to get the controller. C = A/G = [4 / (s (0.025 s + 1))] * [(s+1)(s+5) / 10] = 4 (s+1)(s+5) / (10 s (0.025 s + 1)) = 0.4 (s+1)(s+5) / (s (0.025 s + 1)). The left-half-plane plant poles -1 and -5 are cancelled by controller zeros, which is a stable (permitted) cancellation.
- +1Cross-over and phase margin. Solve |A(jwc)| = 1: 4 / (wc*sqrt(1 + (0.025 wc)^2)) = 1 gives wc^2 (1 + 0.000625 wc^2) = 16, so wc^2 = 15.84 and wc = 3.98 rad/s (essentially 4). The phase is angle A(jwc) = -90 - arctan(0.025*3.98) = -90 - 5.68 = -95.7 degrees, so Mf = 180 - 95.7 = 84.3 degrees.
- +1Confirm the steady state and slope. A has a pole at s = 0, so A(0) is infinite and S0(0) = 0: zero steady-state step error. The far pole at -40 sits a decade above wc, so the magnitude passes 0 dB at about -20 dB/dec, consistent with the ~84 degree phase margin.
Key terms
- Open-loop gain A0(jw)
- The product A0 = G0*C of plant and controller evaluated on the imaginary axis. Loop-shaping designs the magnitude |A0(jw)| directly: large at low frequency, small at high frequency. Units: dimensionless.
- Cross-over frequency wc
- The frequency where the open-loop gain is unity, |A0(jwc)| = 1 (0 dB). The phase margin is read here, and the local magnitude slope at wc sets that margin. Units: rad/s.
- Minimum-phase system
- A transfer function with no right-half-plane poles or zeros and no transport delay. For a minimum-phase loop the phase is uniquely fixed by the magnitude curve (Bode's gain-phase relation), and direct inversion C = A/G is allowed.
- Bode gain-phase relation
- For a minimum-phase transfer function the phase at any frequency is a weighted average of the log-magnitude slope, weighted most heavily at that frequency. Working rule: a slope of -20*l dB/dec gives about -l*(65 to 90) degrees of phase.
- Loop-shaping
- A design method that chooses the shape of the open-loop gain |A0(jw)| to meet specifications: high low-frequency gain for accuracy, a -20 dB/dec cross-over for phase margin, and steep high-frequency roll-off for noise rejection and robustness.
- Direct synthesis (C = A/G)
- For a minimum-phase plant, pick the desired stable open loop A(s), then set C = A/G, since A = G*C. Requires the properness condition nr(A) >= nr(G). Not valid for non-minimum-phase plants, which would need an unstable controller or a forbidden cancellation.
- RHP-zero constraint
- A right-half-plane zero at z adds extra phase lag -2*arctan(w/z), so it caps the cross-over: wc < z (rule of thumb wc <= z/2). It limits the achievable loop speed and cannot be removed by any controller.
- RHP-pole constraint
- A right-half-plane (unstable) pole at p must be actively stabilised, which needs enough loop gain out past it: wc > p. Combined with a RHP zero this gives the feasible window p < wc < z, which is empty if z is at or below p.
Classical loop-shaping and Bode gain-phase relation FAQ
Why should the loop-gain slope be about -20 dB/dec at cross-over?
By Bode's gain-phase relation, the phase near cross-over is set by the magnitude slope there. A -20 dB/dec slope corresponds to roughly -90 degrees of phase, giving a phase margin near 90 degrees (in practice about 65 to 90 degrees) - healthy. A -40 dB/dec slope corresponds to about -180 degrees, leaving essentially no phase margin, so the loop rings or goes unstable. You can roll off faster well above cross-over for noise, just not through it.
What is the difference between a minimum-phase and a non-minimum-phase plant for loop-shaping?
A minimum-phase plant has no right-half-plane poles or zeros and no delay, so its phase is fixed by its magnitude and you may synthesise the controller directly as C = A/G. A non-minimum-phase plant (a RHP zero, RHP pole, or delay) adds extra phase lag on top of what the magnitude dictates, so the same magnitude slope buys less phase margin, and C = A/G is not usable - the RHP zero at z instead caps the cross-over at wc < z, and a RHP pole at p forces wc > p.
Can AI help me with classical loop-shaping in ELEN90055?
Yes, as a study aid. Sia can walk through the method step by step - how the gain-phase relation links slope to phase, how to read cross-over and phase margin off a Bode plot, why -20 dB/dec is the target, and how the RHP bounds wc < z and wc > p arise - using worked practice problems so you can follow the reasoning yourself. It is a tutor for understanding the derivations and checking your own working, not a source of ready-made exam answers or a guaranteed grade; the final exam is a 3-hour open-book written exam, run as a secure task where generative-AI help is not permitted.
Studying with AI? Sia — free AI electrical engineering tutor works through ELEN90055 step by step.
Exam move
Loop-shaping is best learned as one connected story rather than a list of formulae. Start from the loop-gain map (big gain low, small gain high, cross-over in between), then anchor everything on Bode's gain-phase relation so you can convert a magnitude slope into a phase and hence a phase margin - this is what makes -20 dB/dec at cross-over the target. Practise the direct-synthesis recipe C = A/G on a few minimum-phase plants, always stating 'minimum-phase' first and checking the properness condition, then practise the non-minimum-phase case where a RHP zero caps wc < z and a RHP pole forces wc > p. The exam is open book, so drill the method and the sign and factor conventions (the 20 in dB, the -2*arctan(wc/z) lag, the direction of each inequality) rather than memorising results, and always end a design by re-checking the actual step response.
