3.5.39 · D5Guidance, Navigation & Control (GNC)
Question bank — PID tuning — Ziegler-Nichols, loop shaping
For the underlying machinery, keep PID controller basics, Bode plot & frequency response, and Stability margins (gain & phase margin) open beside this.
0. Symbols and pictures you need before the traps
Everything below reuses a small vocabulary. Rather than send you off to other notes mid-question, here is the minimal self-contained dictionary, each symbol tied to a picture.
Everything from here uses only those symbols and pictures.
True or false — justify
At the edge of sustained oscillation the open loop has gain 1 and phase −180°.
True. A signal fed back inverted (−180°) and unattenuated (gain 1) exactly reinforces itself, giving a constant-amplitude oscillation forever — this is the definition of the stability boundary (Figure s01, middle trace).
Increasing always makes the closed loop faster.
False. It raises the crossover frequency (speed) up to a point, but past the ultimate gain the loop is unstable — "faster" becomes "diverging". Speed and stability trade off.
Integral action improves both steady-state accuracy and phase margin.
False. It kills steady-state error but the pole contributes a flat of phase near crossover (Figure s04), reducing phase margin — accuracy and robustness pull in opposite directions here.
Derivative action adds damping, so more is always safer.
False. Its zero adds phase lead (good for margin, Figure s04 green) but multiplies high-frequency noise by ; too much makes the actuator chase sensor noise and can destabilise via amplified measurement jitter.
Ziegler-Nichols gains are conservative and safe to ship.
False. Classic ZN targets quarter-amplitude decay, which means roughly overshoot — aggressive, not conservative. It's a starting point, not a final answer.
The two ZN methods (ultimate-gain and reaction-curve) should give identical gains.
False. They model the plant differently ( critical point vs FOPDT fit) and only agree approximately, because is an approximation. Expect similar ballpark, not identical numbers.
Gain margin and phase margin measure the same robustness.
False. GM is how much you can scale gain before instability (at ); PM is how much extra lag you can add (at ). A loop can have good GM and poor PM or vice versa (Figure s03).
A loop with high phase margin can still respond sluggishly.
True. Phase margin bounds overshoot/damping, but speed is set by the crossover frequency . You can have a very damped yet very slow loop if is low.
The standard form and , so a ZN table in is interchangeable with one in without care.
False. They are convertible, but the tables are stated in one specific form; plugging where is expected (or forgetting to multiply/divide by ) is the single most common tuning bug.
Loop shaping requires knowing the plant model; ZN closed-loop does not.
True. The ultimate-gain method extracts from an experiment on the real plant — no model needed. Loop shaping reshapes a known , so it needs .
Spot the error
"To hit steady-state accuracy fast, set near ."
The error is placing the integral corner near crossover — that dumps the integral's lag right where you need phase margin. Keep so integral lag has decayed before crossover.
"For , ."
Wrong: the three identical poles each contribute , so the phase is , not . Phases of cascaded factors add (Figure s04); arguments don't merge inside one arctan.
"Sustained oscillation happens when ."
Wrong angle — the stability boundary needs the open loop phase at , because only an inverted feedback signal reinforces itself. At the loop is still comfortably stable.
"Use derivative on the error, , exactly as the formula says."
The formula is right but the implementation isn't: a step in reference makes a huge spike ("derivative kick", Figure s05). Use derivative on the measurement instead — see Integral windup and anti-windup for the analogous clamping fix on the integral side.
"Since GM dB is the target, a loop with GM = 20 dB is even better tuned."
Not necessarily — huge gain margin usually means the loop is over-detuned (low ), so it's sluggish and disturbance rejection is poor. Margins are a window to sit inside, not a "bigger is better" score.
"Anti-windup changes the tuning gains ."
No — anti-windup only prevents the integrator from accumulating while the actuator is saturated. The gains are unchanged; it fixes the nonlinear saturation behaviour, not the linear tuning.
Why questions
Why does pure P control leave a steady-state offset?
With zero error the P term outputs zero, but a nonzero output is needed to hold against a load/disturbance — so the loop settles at whatever small error produces that holding output. Only integral action can supply output at zero error.
Why does the integral term specifically eliminate steady-state error?
The integrator keeps accumulating as long as any error exists, so its output only stops changing when the error is exactly zero — it forces the steady state to have .
Why place the derivative zero below the crossover frequency?
Its phase lead climbs gradually as rises past the zero (Figure s04, green); to have most of that lead already active at (where you need margin), the zero must sit at a lower frequency so its phase has risen by the time you reach crossover.
Why does dead time force smaller controller gain in the reaction-curve method?
Dead time adds pure phase lag that grows with frequency without touching the gain, so it eats phase margin fast. The only defence is to lower (hence lower gain), which is why .
Why can't a proportional-only loop achieve both fast response and small overshoot on an oscillatory plant?
Raising pushes closed-loop poles toward the imaginary axis on the root locus, increasing speed but reducing damping. Without D-action's added lead, you can't buy speed and damping at once.
Why is high open-loop gain wanted at low frequency but not high frequency?
Low-frequency gain suppresses slow disturbances and tracking error (the loop "sees" and corrects them); high-frequency gain would amplify sensor noise and violate robustness where the plant model is least trustworthy.
Why does phase margin usually grow as ?
Lowering slides the crossover leftward on the Bode plot into the region where the plant's poles haven't yet donated their full each (phase is still near , Figure s04). Since PM , a less-negative phase at a lower means a bigger margin — but a slower, disturbance-blind loop.
Edge cases
What does closed-loop ZN do if the plant never sustains oscillation as rises?
For plants with poles and no dead time the phase never reaches (each pole caps at , Figure s04), so there is no finite (equivalently no ) — the method simply doesn't apply, and you must use the reaction-curve or a model-based method.
For a pure integrator plant , what is its phase, and what does that mean for ZN?
Its phase is a constant , never reaching , so no ultimate gain exists. Such a plant already has zero steady-state error to steps, so heavy integral action is often unnecessary anyway.
What happens to phase margin as (very detuned loop)?
PM typically grows toward a large stable value (see the "why" above), but the loop becomes uselessly slow and cannot reject disturbances — a "stable but deaf" controller. Robustness without performance.
If the sensor is very noisy, why might you set entirely?
Derivative multiplies noise by (gain rising with frequency), so on a noisy channel it can dominate the actuator command; dropping D (or filtering it as ) trades a little damping for a much quieter, safer output.
For an FOPDT plant with (no dead time), what does the reaction-curve formula predict?
— the recipe demands infinite gain, which is nonsense. It signals that a delay-free plant isn't limited the way the FOPDT model assumes; use a different design.
What is the danger of tuning at one operating point for a nonlinear plant?
Gains that give good margins near one setpoint can give tiny or negative margins elsewhere (changed local slope/gain), so the loop may be stable at test and unstable in service — gain-scheduling or state-space/LQR methods address this.