PID Controller

Overview & Motivation

The Proportional-Integral-Derivative (PID) controller is the workhorse of industrial feedback control. It continuously computes an error $e(t)$ — the difference between a desired setpoint and a measured process variable — and applies a correction based on three terms: one proportional to the current error, one to its accumulated history, and one to its rate of change.

Despite its simplicity, PID handles a remarkably wide range of plants — from temperature regulation to motor speed control — because each term addresses a different aspect of the transient response:

• P reacts immediately to the current error.
• I eliminates steady-state offset by integrating past errors.
• D anticipates future error by acting on the rate of change.

This library implements a discrete recursive form suitable for fixed-sample-rate embedded loops, where the output is computed incrementally rather than from scratch at every step.

Mathematical Theory

Continuous-Time PID

$u(t) = K_p \, e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}$

Discrete Recursive Form

Discretizing with sampling period $T_s$ and applying the difference approximation:

$u[n] = u[n-1] + a_0 \, e[n] + a_1 \, e[n-1] + a_2 \, e[n-2]$

where:

$a_0 = K_p + K_i + K_d$ $a_1 = -(K_p + 2K_d)$ $a_2 = K_d$

This incremental (velocity) form has two key advantages:

1. No integral accumulator — the integral action is implicit in the recursion, reducing overflow risk.
2. Bumpless mode switching — toggling between auto/manual mode does not cause output jumps because the output is updated incrementally.

Output Saturation

The output is clamped to $[u_{\min}, u_{\max}]$ after each computation, which also provides implicit anti-windup: the integral term cannot drive the output beyond the saturation limits.

Complexity Analysis

Case	Time	Space	Notes
Per sample	$O(1)$	$O(1)$	3 multiplications, 3 additions, 1 clamp

The algorithm uses a fixed-size recursive buffer storing $e[n-1]$, $e[n-2]$, and $u[n-1]$. No loops, no dynamic allocation, fully deterministic execution time.

Step-by-Step Walkthrough

Parameters: $K_p = 2$, $K_i = 0.5$, $K_d = 0.1$, limits $[-10, 10]$, setpoint $= 5$.

Coefficients:

• $a_0 = 2 + 0.5 + 0.1 = 2.6$
• $a_1 = -(2 + 0.2) = -2.2$
• $a_2 = 0.1$

Step	Measured	$e[n]$	$e[n-1]$	$e[n-2]$	$\Delta u$	$u[n]$ (clamped)
0	0.0	5.0	0	0	$2.6 \times 5 = 13$	10.0 (clamped)
1	2.0	3.0	5.0	0	$2.6(3) - 2.2(5) + 0.1(0) = -3.2$	6.8
2	4.0	1.0	3.0	5.0	$2.6(1) - 2.2(3) + 0.1(5) = -3.5$	3.3
3	4.8	0.2	1.0	3.0	$2.6(0.2) - 2.2(1) + 0.1(3) = -1.38$	1.92

The output converges toward the steady-state value needed to maintain the setpoint.

Pitfalls & Edge Cases

• Derivative kick. A sudden change in setpoint causes a spike in $e[n] - e[n-1]$. Mitigate by differentiating the process variable instead of the error, or by filtering the derivative term.
• Integral windup. Although the recursive form provides implicit anti-windup through output clamping, very aggressive $K_i$ values can still cause sluggish recovery from saturation. Monitor the output rail time.
• Sample rate dependency. The gains $K_i$ and $K_d$ are implicitly scaled by $T_s$. Changing the control loop rate without retuning will alter the effective integral and derivative contributions.
• Fixed-point saturation. For Q15/Q31 types, the intermediate products $a_0 \cdot e[n]$ etc. can overflow. Ensure gains and error ranges are scaled to stay within the representable range.
• Setpoint jumps. Call Reset() after large setpoint changes to clear stale error history.

Variants & Generalizations

Variant	Key Difference
Standard (positional) PID	Computes $u[n]$ from scratch each step using an explicit integral accumulator
PI controller	$K_d = 0$; simpler, no derivative noise issues
PD controller	$K_i = 0$; no steady-state error correction, used when offset is acceptable
PID with derivative filter	Low-passes the derivative term to reject high-frequency noise
Gain-scheduled PID	Tuning parameters vary as a function of operating point
Cascade PID	Inner and outer loops, each with its own PID — common in motor control

Applications

• Temperature control — Maintaining oven, room, or process temperatures via heater/cooler actuation.
• Motor speed/position control — Servo drives, robotics joints, CNC machines.
• Flow and pressure regulation — Industrial process control (chemical plants, water treatment).
• Voltage/current regulation — Power supply output regulation, battery charging.
• Attitude control — Drone stabilization, satellite pointing.

Connections to Other Algorithms

graph LR
    PID["PID Controller"]
    LQR["LQR Controller"]
    KF["Kalman Filter"]
    PID -.->|"alternative"| LQR
    KF -->|"state estimate"| LQR
    KF -->|"filtered measurement"| PID

Algorithm	Relationship
LQR Controller	Optimal alternative when a state-space model is available; PID is model-free.
Kalman Filter	Can provide filtered state estimates as input to either PID or LQR

References & Further Reading

• Åström, K.J. and Murray, R.M., Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, 2008 — Chapter 10.
• Åström, K.J. and Hägglund, T., Advanced PID Control, ISA, 2006.
• Ziegler, J.G. and Nichols, N.B., "Optimum settings for automatic controllers", Transactions of the ASME, 64, 1942.