PID tuning — Ziegler-Nichols, loop shaping

1. What is a PID controller (WHAT / WHY)

P (proportional): reacts to the present error. Bigger error ⇒ bigger push. Alone it leaves steady-state error (offset) and can oscillate if too large.
I (integral): accumulates past error. It kills steady-state error because it keeps pushing until the accumulated error is zero. But it adds phase lag ⇒ can destabilise.
D (derivative): predicts the future from the slope. It adds damping / phase lead, reducing overshoot, but amplifies noise.

2. Ziegler-Nichols: the ultimate-gain method (HOW, derived)

Procedure (closed-loop / ultimate-gain method):

Set $K_i=0,\ K_d=0$ (pure P control).
Slowly increase $K_p$ until the output shows sustained, constant-amplitude oscillation.
Record that gain as the ultimate gain $K_u$ and the oscillation period $T_u$ .
Read the gains from the table.

Where do $K_u,T_u$ come from analytically?

At the stability edge the open loop $L(j\omega)=K_u\,G(j\omega)$ satisfies $\angle G(j\omega_u) = -180°,\qquad K_u = \frac{1}{|G(j\omega_u)|},\qquad T_u=\frac{2\pi}{\omega_u}.$

Derivation for a test plant $G(s)=\dfrac{1}{(s+1)^3}$ :

Phase: $\angle G(j\omega) = -3\arctan\omega$ . Set $=-180°=-\pi$ : $\arctan\omega_u=\pi/3 \Rightarrow \omega_u=\tan 60°=\sqrt3$ .
Magnitude: $|G(j\omega_u)|=(\omega_u^2+1)^{-3/2}=(3+1)^{-3/2}=4^{-3/2}=1/8$ .
So $K_u=1/|G|=8$ and $T_u=2\pi/\sqrt3\approx3.63\,$ s.

3. Ziegler-Nichols: the open-loop (reaction-curve) method

Approximate the step response by a delay $L$ and slope. Model: $G(s)\approx \dfrac{K\,e^{-Ls}}{Ts+1}$ . Let $R=K/T$ (max slope / step-height). Then:

Why these? $L$ (dead time) governs how much phase lag you can tolerate; larger $L$ ⇒ smaller gain. The ratios $2L,\,0.5L$ mirror the closed-loop $T_u$ ratios because $T_u\!\sim\!4L$ for such plants.

4. Loop shaping (the "understand it" method)

PID as a loop-shaping filter:

The integral term $K_i/s$ boosts low-frequency gain (slope $-20$ dB/dec, phase $-90°$ ).
The derivative term adds phase lead near $\omega_c$ — you deliberately place the zero of $C(s)$ below $\omega_c$ so its $+90°$ phase counters plant lag and raises the phase margin.

Worked example Design by phase margin (lead action)

Plant $G(s)=\frac{1}{s(s+1)}$ , want $\omega_c=2$ rad/s and $\text{PM}=45°$ with a PD $C=K_p(1+T_d s)$ .

Plant phase at $\omega_c$ : $\angle G = -90° - \arctan 2 = -90°-63.4°=-153.4°$ .
Needed phase from controller: $\text{PM}-180°-\angle G = 45°-180°+153.4°=18.4°$ lead. Why? We need $\angle L=-135°$ to get $\text{PM}=45°$ .
Lead of a PD zero: $\arctan(T_d\omega_c)=18.4°\Rightarrow T_d\omega_c=\tan18.4°=0.333\Rightarrow T_d=0.166$ .
Set magnitude $=1$ : $|G(j2)|=1/(2\sqrt{5})=0.2236$ ; $|1+T_d j2|=\sqrt{1+0.333^2}=1.054$ . $K_p = 1/(0.2236\times1.054)=4.24$ . Why? forces $|L|=1$ at the chosen $\omega_c$ .

5. Common mistakes (Steel-manned)

6. Active recall

Recall Quick self-test (hide and answer)

What two numbers does closed-loop ZN measure, and how? (→ $K_u$ : gain at sustained oscillation; $T_u$ : its period.)
Why does integral action reduce phase margin? (→ adds $-90°$ lag.)
Define phase margin and target range. (→ $180°+\angle L(\omega_c)$ ; $30$ – $60°$ .)
How does D action help loop shaping? (→ adds phase lead near $\omega_c$ , raises PM, adds damping.)

Recall Feynman: explain to a 12-year-old

Imagine steering a toy car to a line on the floor. P: turn the wheel more the farther you are from the line. I: if a wind keeps blowing you off, keep turning a little extra until you finally sit on the line. D: if you're rushing at the line fast, ease off early so you don't zoom past. Tuning is picking how strong each of these reactions is. Ziegler-Nichols is like: "turn the P knob up until the car wobbles side to side forever, note how fast it wobbles, then use that to set all three knobs." Loop shaping is: "draw a picture of how the car reacts to fast vs slow wiggles, and adjust the knobs so slow mistakes get fixed strongly but fast jitter is ignored."

Connections

PID controller basics — origin of the three terms
Bode plot & frequency response — the canvas for loop shaping
Stability margins (gain & phase margin)
Nyquist stability criterion — why $-180°$ / gain 1 is the edge
Integral windup and anti-windup
State-space control & LQR — alternative to hand tuning
Root locus method — another gain-selection tool

Flashcards

What is the PID control law in the time domain?

u(t)=K_p e + K_i\int e\,dt + K_d\,\dot e

, with

e=r-y

In closed-loop ZN, what are

K_u

and

T_u

The proportional gain that produces sustained constant-amplitude oscillation, and the period of that oscillation.

ZN PID gains from

K_u,T_u

K_p=0.6K_u,\ T_i=T_u/2,\ T_d=T_u/8

(then

K_i=K_p/T_i,\ K_d=K_p T_d

Why does adding integral action reduce stability margin?

Each

1/s

term adds

-90°

phase lag, cutting the phase margin near crossover.

Define phase margin.

\text{PM}=180°+\angle L(j\omega_c)

where

\omega_c

is the gain-crossover frequency (

|L|=1

What condition marks the edge of instability for

L(s)

|L(j\omega)|=1

and

\angle L(j\omega)=-180°

simultaneously.

For

G=1/(s+1)^3

, find

\omega_u,K_u,T_u

\omega_u=\sqrt3

K_u=8

T_u=2\pi/\sqrt3\approx3.63

What is "derivative kick" and its fix?

A step in reference makes

de/dt

spike; fix by taking derivative of the measurement

-K_d\dot y

and filtering it.

What does high open-loop low-frequency gain buy you in loop shaping?

Small steady-state error and good disturbance rejection.

Target gain margin in loop shaping?

\ge 6

dB (factor of 2).

Classic ZN targets what response shape?

Quarter-amplitude decay (~25% overshoot) — aggressive, often needs detuning.

ZN open-loop PID from FOPDT (

L

=delay,

R

=slope/step)?

K_p=1.2/(RL),\ T_i=2L,\ T_d=0.5L

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

PID tuning ka matlab hai teen knobs — $K_p, K_i, K_d$ — ko aise set karna ki system fast bhi ho, stable bhi rahe, aur overshoot zyada na kare. P present error dekhta hai, I past errors ko jodta jaata hai (isliye steady-state offset khatam karta hai), aur D future ko predict karta hai (slope se), damping deta hai. Dhyaan rakho: I zyada karoge to phase lag ( $-90°$ ) aata hai aur system oscillate karne lagta hai.

Ziegler-Nichols ek recipe hai. $K_i$ aur $K_d$ ko zero karke sirf $K_p$ badhaao jab tak output constant-amplitude oscillation dikhaye. Us gain ko $K_u$ (ultimate gain) aur oscillation ke period ko $T_u$ bolo. Phir table se: $K_p=0.6K_u$ , $T_i=T_u/2$ , $T_d=T_u/8$ . Yeh seconds mein starting point de deta hai, par yeh aggressive hota hai (~25% overshoot), to baad mein thoda detune karna padta hai.

Loop shaping samajhdaari wala tareeka hai. Yahan hum open-loop $L(s)=C(s)G(s)$ ka Bode plot dekhte hain. Chahiye: low frequency pe high gain (accuracy ke liye), high frequency pe low gain (noise reject karne ke liye), aur crossover $\omega_c$ pe achha phase margin ( $30$ – $60°$ ) taaki stable rahe. Integral term low-frequency gain badhaata hai, aur derivative term $\omega_c$ ke aas-paas phase lead deta hai jisse margin badhta hai. Matlab tum jaante ho kyun har knob kya kar raha hai — yeh ZN se zyada control deta hai. Practical tip: $K_i/K_p$ ko $\omega_c$ se kaafi neeche rakho, aur derivative ko measurement pe lagaao (error pe nahi) taaki "derivative kick" na ho.

PID tuning — Ziegler-Nichols, loop shaping

1. What is a PID controller (WHAT / WHY)

2. Ziegler-Nichols: the ultimate-gain method (HOW, derived)

Where do $K_u,T_u$ come from analytically?

3. Ziegler-Nichols: the open-loop (reaction-curve) method

4. Loop shaping (the "understand it" method)

5. Common mistakes (Steel-manned)

6. Active recall

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections