ELEN90055 · Control Systems
Design: specifications and P/lag/lead/PID compensation
This chapter of ELEN90055 Control Systems at the University of Melbourne is where analysis turns into design: instead of asking whether a given feedback loop is good, you build the controller that makes it good. You start from time-domain specifications (overshoot and settling time), read off the desired second-order damping ζ and natural frequency ωn, translate those into a loop target (cross-over frequency and phase margin), then pick a compensator — proportional for gain, lag/PI for steady-state accuracy, lead/PD for phase margin and speed, and lead-lag/PID for both.
It forms Part IV of the subject and drives the two large design questions on the final paper, together with the experimental Ziegler-Nichols tuning rules for a PID when no clean model is at hand.
What this chapter covers
- 01Convert an overshoot spec into a minimum damping ratio using Mp = exp(-πζ/√(1-ζ²))
- 02Convert a 2% settling-time spec into a natural frequency using t_s ≈ 4/(ζωn)
- 03Place the desired dominant pole pair at s = -ζωn ± jωn√(1-ζ²)
- 04Turn (ζ, ωn) into a target cross-over and phase margin via Bode's ideal open loop
- 05Choose proportional, lag or PI to raise loop gain and remove steady-state error
- 06Design a lead/PD for maximum phase φmax = arcsin[(τz-τp)/(τz+τp)] at cross-over
- 07Combine both actions in a lead-lag, and recognise PID as its limiting three-term form
- 08Tune a PID from a reaction curve (Ziegler-Nichols Method 1) or the ultimate gain (Method 2)
- 09Re-check the actual step response, since a compensated loop rarely stays exactly second order
Tune a PID by Ziegler-Nichols Method 2 (ultimate gain)
- +1Identify the two measurements. Method 2 needs only the critical (ultimate) gain K_cr = 10 and the critical period T_cr = 3 s of the sustained oscillation at the stability boundary. No transfer function is required.
- +1Proportional gain. The PID rule is Kp = 0.6·K_cr = 0.6 × 10 = 6 (dimensionless, output-per-error in the scaled units).
- +1Integral time. T_I = 0.5·T_cr = 0.5 × 3 = 1.5 s, so the integral gain is Ki = Kp/T_I = 6/1.5 = 4 s⁻¹.
- +1Derivative time. T_D = 0.125·T_cr = 0.125 × 3 = 0.375 s, so the derivative gain is Kd = Kp·T_D = 6 × 0.375 = 2.25 s.
- +1Assemble the controller. C(s) = 6(1 + 1/(1.5 s) + 0.375 s) = 6 + 4/s + 2.25 s. Because the ideal derivative term is improper, a real implementation rolls it off as T_D s/(T_D s + 1) to keep C proper.
- +1Compare and caveat. The P-only rule gives Kp = 0.5·K_cr = 5; the PI rule gives Kp = 0.45·K_cr = 4.5 with T_I = T_cr/1.2 = 3/1.2 = 2.5 s. Ziegler-Nichols targets a quarter-amplitude decay and usually leaves a large overshoot, so treat these as a starting point and detune (lower Kp or lengthen T_I) if the transient spec is tight.
Key terms
- Damping ratio ζ and natural frequency ωn
- The two parameters of the standard second-order system T(s) = ωn²/(s² + 2ζωn s + ωn²). The dimensionless damping ratio ζ (0 < ζ < 1 underdamped) sets the shape (overshoot); the natural frequency ωn [rad/s] sets the speed. The poles sit at s = -ζωn ± jωn√(1-ζ²).
- Peak overshoot Mp
- The fractional overshoot of the unit-step response, Mp = exp(-πζ/√(1-ζ²)) (multiply by 100 for a percentage). It depends on ζ alone: more damping means less overshoot, e.g. ζ = 0.5 gives about 16.3% and ζ = 0.6 about 9.5%.
- 2% settling time t_s
- The time for the step response to enter and stay within a 2% band, t_s ≈ 4/(ζωn) seconds. The decay rate is ζωn, so poles further to the left (larger ζωn) settle faster.
- Phase margin Mf
- The extra phase lag the loop can tolerate before instability, Mf = 180° + ∠A0(jωc) at the cross-over frequency ωc. For Bode's ideal open loop Mf = arctan(2ζωn/ωc), with the rule of thumb Mf(deg) ≈ 100ζ for 0 < ζ < 0.6; phase margin and overshoot are linked through ζ.
- Lag / PI compensator
- Lag C = K(τz s + 1)/(τp s + 1) with 0 < τz < τp raises low-frequency gain (better steady-state accuracy) and adds negative phase everywhere. PI = K(τz s + 1)/s pushes the lag pole to the origin, giving zero steady-state error to a step; neither helps the phase margin.
- Lead / PD compensator
- Lead C = K(τz s + 1)/(τp s + 1) with 0 < τp < τz adds positive phase peaking at ω = 1/√(τzτp), with maximum lead φmax = arcsin[(τz - τp)/(τz + τp)]. It raises phase margin (and damping) without lowering cross-over; PD is the improper limit with the pole at infinity.
- PID controller (parallel form)
- C(s) = Kp + Ki/s + Kd s = Kp(1 + 1/(T_I s) + T_D s), the limiting lead-lag with the lag pole at s = 0 and the lead pole at infinity. Kp [output/error] sets gain, Ki [per second] removes the step offset, Kd [seconds] adds phase lead; T_I is the integral time [s] and T_D the derivative time [s], with Ki = Kp/T_I and Kd = Kp·T_D.
- Ziegler-Nichols tuning
- Two experimental recipes for a PID. Method 1 fits the open-loop step response to K e^(-sT)/(τp s + 1) and reads Kp = 1.2·τp/(KT), T_I = 2T, T_D = 0.5T. Method 2 uses the critical gain K_cr and period T_cr at the stability boundary: Kp = 0.6·K_cr, T_I = 0.5·T_cr, T_D = 0.125·T_cr.
Design: specifications and P/lag/lead/PID compensation FAQ
How do I decide between a lag/PI and a lead/PD compensator?
Match the compensator to the specification it fixes. A lag or PI raises the loop gain at low frequency, which improves steady-state accuracy (a lag leaves a small residual offset; a PI removes the step offset entirely because its pole sits at the origin), but both add negative phase and so do not help the phase margin. A lead or PD adds positive phase around cross-over, which raises the phase margin and hence the damping, and it lets you push the cross-over higher for a faster response — but it does nothing for low-frequency accuracy and it amplifies high-frequency noise. So if the spec is about steady-state error, reach for lag/PI; if it is about overshoot, phase margin or speed, reach for lead/PD; if you need both at once, cascade them as a lead-lag, which in its limiting three-term form is a PID.
Why re-check the step response after designing the compensator?
The whole design is built on the fiction that the closed loop behaves like a single second-order pole pair, so a target damping and natural frequency can stand in for overshoot and settling time. A compensated loop rarely stays exactly second order: a PID closed loop has five or more poles, and extra slow poles (often near a PI zero) can add overshoot or a sluggish tail that the two-pole picture never predicted. Dominance of the design pole pair is never guaranteed, so you confirm the actual step response against the overshoot and settling-time spec before declaring the design finished. Confirm the exact modelling assumptions expected against the current unit notes on Canvas.
Can AI help me with design and PID compensation in ELEN90055?
Yes, as a study aid. An AI tutor such as Sia can explain the spec-to-(ζ, ωn) step, walk through a lead or lag design and the Ziegler-Nichols tables, and check your algebra on a phase-margin calculation or a set of PID gains. Use it to learn the method, the sign conventions and the factors, and to practise with your own numbers; it does not sit your open-book exam for you or guarantee a mark, and you should confirm every rule and formula against the current unit notes on Canvas.
Studying with AI? Sia — free AI electrical engineering tutor works through ELEN90055 step by step.
Exam move
Design questions all share one skeleton, so make it automatic: spec → (ζ, ωn) → target cross-over and phase margin → pick a compensator → place it → re-check the step response. Drill the two conversions until they are reflexes — invert Mp = exp(-πζ/√(1-ζ²)) for the minimum damping, then use t_s ≈ 4/(ζωn) for the natural frequency — and keep straight that ζ controls shape (overshoot) while ωn controls speed. Learn the compensator library as a table of what each buys you and what phase it adds: proportional (gain, no phase), lag/PI (low-frequency accuracy, negative phase), lead/PD (phase margin and speed, positive phase), lead-lag/PID (both). The single most common slip is swapping lead and lag: a lead has its zero below its pole and adds positive phase, a lag has its zero above its pole and adds negative phase, so check the sign of the phase before you commit. Get the Ziegler-Nichols factors exactly right (Method 1: 1.2, 2, 0.5; Method 2: 0.6, 0.5, 0.125) and remember Ki = Kp/T_I. The final exam is 3 hours, open book, worth 70% of the subject with a hurdle you must pass, and you answer all four questions summing to 50 marks — budget about 3.6 minutes per mark, so a 15-mark design question earns about 54 minutes. Because it is open book, marks reward method shown with correct signs and units, not memorised formulae: state the spec-to-(ζ, ωn) step, evaluate the plant magnitude and phase at cross-over explicitly, place zeros and poles with numbers, carry units on every time constant [s] and frequency [rad/s], and finish with a one-line time-domain re-check. Confirm the exact date of the next (Semester 1, ~June 2027) sitting on Canvas.
