PID LOOP LAB
CONTROL FROM FIRST PRINCIPLESLIVE SIMULATION · ORIGINAL VISUAL GUIDES

UNDERSTAND THE
LOOP FROM INSIDE.

Isolate controller terms, create known failure modes, and connect equations to the signals they generate.

THE FEEDBACK LAW

Error becomes action.

e(t)=r(t)−y(t)e(t)=r(t)-y(t)u(t)=Kp(βr−y)+Ki∫(r−y)dt+Kdd(γr−y)dtu(t)=K_p(\beta r-y)+K_i\int(r-y)dt+K_d\frac{d(\gamma r-y)}{dt}
01 / P

Present

uP=Kp(βr−y)u_P=K_p(\beta r-y)

Proportional action scales current weighted error. Its practical effect depends on plant gain, dynamics, delay, and all other loop elements.

02 / I

Past

uI=Ki∑e[k]Tsu_I=K_i\sum e[k]T_s

Integral action accumulates sampled error. It can remove offset, yet needs explicit limits or anti-windup when actuators saturate.

03 / D

Rate

uD=Kd TsTf+Ts x˙Du_D=K_d\,\frac{T_s}{T_f+T_s}\,\dot{x}_D

Filtered derivative estimates rate. Using measurement avoids a derivative kick from a setpoint step when γ = 0.

PID TERM EXPLORER

Change one idea at a time.

The plots below come from the same deterministic engine as the main lab. Select a controlled experiment and inspect its stated mechanism.

PID: 1401 samples from 0.00 to 14.00 seconds.

TIME CONSTANT EXPLORER

One τ is 63.2% of the journey.

For a first-order step response, y(t)=K(1−e−t/τ)y(t)=K(1-e^{-t/\tau}). At 5τ the response is about 99.3% of its final value.

ReIm

SECOND-ORDER EXPLORER

underdamped

Poles move as natural frequency and damping ratio change. For ζ < 1, the complex pole pair produces an oscillatory mode; ζ = 1 is the repeated critical boundary.

s1,2=−ζωn±ωnζ2−1s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}

SEARCHABLE FIELD NOTES

Learn one mechanism at a time.

Each guide connects the equation, the physical effect, and a reproducible experiment you can open in the lab.

01 · FOUNDATIONSPID controller: the complete visual guide

Understand proportional, integral and derivative control through the signals inside a real closed loop.

Read visual guide 02 · THE PRESENTProportional control without the guesswork

See how proportional gain changes speed, offset, damping and actuator demand in a feedback loop.

Read visual guide 03 · THE MEMORYIntegral windup and how to stop it

Learn why an integral term keeps growing during saturation and compare practical anti-windup methods.

Read visual guide 04 · THE DIRECTIONDerivative control, filtering and noise

Understand derivative kick, measurement mode and why an unfiltered derivative term amplifies sensor noise.

Read visual guide 05 · CLASSIC TUNINGZiegler–Nichols PID tuning explained

Use reaction-curve and ultimate-cycle tuning responsibly, with assumptions and limitations made explicit.

Read visual guide 06 · REACTION CURVECohen–Coon tuning for delayed processes

Learn how Cohen–Coon uses process gain, time constant and dead time to tune an FOPDT model.

Read visual guide 07 · ROBUST TUNINGSIMC tuning and the closed-loop time constant

Understand how SIMC exposes the speed-versus-robustness decision through one interpretable parameter.

Read visual guide 08 · PLANT DYNAMICSFirst-order systems and time constants

Connect gain and time constant to a process step response and closed-loop tuning decisions.

Read visual guide