Pid Controller⚠ unverified

Understanding the Parameters

• Past, present, future — the elegance of PID is that its three terms cover the error's history in time:

• Proportional ($k_p e$) reacts to the error now — the bigger the error, the harder the correction. Alone it leaves a residual steady-state error (it needs some error to produce output).

• Integral ($k_i\int e\,dt$) accumulates past error — it keeps pushing until the error is driven to exactly zero, eliminating steady-state offset (e.g. holding altitude precisely against a constant disturbance). Its cost is a tendency to overshoot and integrator windup.

• Derivative ($k_d\,\dot e$) anticipates the future from the error's trend — it applies damping, braking the response before it overshoots. It smooths and stabilises but amplifies sensor noise.

• Tuning is the art — the three gains are chosen to trade speed vs stability vs accuracy: raise $k_p$ for a faster response (but risk oscillation), add $k_i$ to kill steady-state error (but risk overshoot), add $k_d$ to damp (but risk noise sensitivity). Classic methods (Ziegler–Nichols) and modern optimisation set them; in aircraft they are often gain-scheduled with airspeed and altitude because the plant changes with flight condition.

• This calculator sums pre-computed terms — it takes $e$, $\int e\,dt$, and $de/dt$ as given inputs and forms the weighted sum. A real controller computes the integral and derivative itself from the time history of the error; here that bookkeeping is assumed done, so the calculator is really "evaluate the PID law at this instant."

• Where it flies — heading hold (Autopilot Heading Hold, a PI special case), altitude hold, airspeed hold, attitude hold, and the inner loops of every autopilot are PID (or PID-derived) laws. The same three terms also run yaw dampers, autothrottles, and engine controls.

Derivation (Approaching a Proof)

PID is a designed control law rather than a derived physical law, but each term has a clear rationale from feedback theory. Let the error be the difference between the desired setpoint $r$ and the measured output $y$:

$$e(t) = r(t) - y(t).$$

The goal is to choose the actuator command $u(t)$ so that $e \to 0$ with good transient behaviour. PID builds $u$ from three functionals of the error:

$$u(t) = \underbrace{k_p\,e(t)}_{\text{present}} + \underbrace{k_i\int_0^t e(\tau)\,d\tau}_{\text{past (removes offset)}} + \underbrace{k_d\,\frac{de(t)}{dt}}_{\text{future (adds damping)}}.$$

Why each term. In the Laplace domain the controller transfer function is $C(s) = k_p + k_i/s + k_d s$. The integrator's $1/s$ pole forces zero steady-state error for step commands (infinite DC gain), the derivative's $s$ zero adds phase lead (damping) to stabilise the loop, and the proportional term sets the overall loop gain and response speed. Placed in a feedback loop with the plant $G(s)$, the closed-loop response is shaped by tuning $k_p, k_i, k_d$ to place the poles for the desired damping and speed. The registry evaluates the time-domain sum at one instant:

$$u = k_p\,e + k_i\,I + k_d\,D, \qquad I = \int e\,dt,\; D = \frac{de}{dt}. \qquad\blacksquare$$

Dimensional check. Each term is a gain times an error functional; for the sum to be a consistent command, $k_p$ carries [command]/[error], $k_i$ carries [command]/([error]·s), and $k_d$ carries [command]·s/[error], so all three products share the command's units ✓. (The registry labels everything dimensionless, folding the units into the gains.)

History and Development

• From ship steering to theory. The PID structure emerged empirically in early automatic ship-steering gear; Nicolas Minorsky analysed it mathematically in 1922 for the automatic steering of the USS New Mexico, articulating the three-term (proportional–integral–derivative) action from observing how a good helmsman steers.

• Ziegler–Nichols (1942). John Ziegler and Nathaniel Nichols published practical tuning rules that made PID deployable across industry without a plant model — cementing PID as the default controller for process control and, later, flight control.

• Ubiquity and endurance. Despite decades of advanced control theory (optimal, robust, adaptive, model- predictive), PID remains the most widely deployed control law in the world — in autopilots, engine controls, disk drives, thermostats, and quadcopters — because it is simple, robust, tunable without a model, and effective. Modern aircraft wrap PID inner loops inside more sophisticated outer architectures, but the three-term core endures.

Related Concepts: Pid Output, Autopilot Heading Hold, Damping Ratio, Peak Overshoot, Settling Time 2%, Phase Margin, Rise Time, Bode Magnitude

Notes: The three-term feedback law $u=k_p e+k_i\!\int e\,dt+k_d\dot e$ — present/past/future of the error (P sets speed but leaves offset; I removes steady-state error but can overshoot/wind up; D damps but is noise- sensitive). Near-duplicate of Pid Output (Physics/Controls). This calc sums pre-computed e, ∫e, de/dt (does not itself integrate/differentiate). Tuning trades speed vs stability vs accuracy; often gain-scheduled. Gains carry the units; defaults ⇒ $u=3$.