Control Systems: PID Control

PID control sits at the center of modern industrial automation. From temperature loops in chemical plants to servo positioning in manufacturing equipment, the proportional–integral–derivative controller remains the default choice because it is simple, robust, and effective across a wide range of processes. When properly tuned, a PID controller can reject disturbances, track setpoints smoothly, and compensate for steady-state errors without requiring an accurate mathematical model of the plant.

This article explains what PID control does, why each term matters, and how common tuning methods including Ziegler-Nichols are used in practice.

What a PID Controller Does

A control system compares a desired value (the setpoint) to a measured value (the process variable). Their difference is the error:

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

A PID controller computes a control action $u(t)$ from that error by combining three components:

$$u(t)=K_{p}e(t)+K_i{\int }_0^{t}e(\tau )d\tau +K_d\frac{de(t)}{dt}$$

• Proportional (P) reacts to the current error.
• Integral (I) reacts to the accumulated error over time.
• Derivative (D) reacts to the rate of change of the error.

In most industrial implementations, the controller output drives an actuator: a valve position, heater power, motor torque command, or a speed reference. The objective is usually one of two things: good setpoint tracking (follow changes in $r(t)$) and good disturbance rejection (hold steady when load changes).

The Proportional Term: Fast Correction With Tradeoffs

Proportional control applies an output proportional to error: $u_P=K_{p}e(t)$.

• Increasing $K_p$ generally makes the loop respond faster and reduces error.
• Too much $K_p$ can create oscillations or instability, especially when the process has significant delay, inertia, or resonance.
• With proportional-only control, many processes settle with a nonzero steady-state error (often called offset) because a constant output may be needed to balance the process, and P alone stops increasing output once the error is small.

A classic example is temperature control of a heated tank. If heat losses increase with temperature, a fixed heater output might be required to maintain a target. Proportional-only control often leaves the tank a few degrees below setpoint because it needs error to sustain that output.

The Integral Term: Eliminating Steady-State Error

Integral action accumulates error: $u_I=K_i\int e(t)dt$. It continues to adjust the output as long as error persists.

Benefits:

• Drives steady-state error toward zero in many common processes.
• Improves load disturbance rejection, particularly for slow, steady disturbances.

Risks:

• Too much integral action can cause slow oscillations, overshoot, and long settling times.
• Integral can “wind up” when the actuator saturates (for example, a valve hits 0% or 100%). The integral keeps accumulating because the error remains, and when the actuator comes out of saturation, the stored integral drives an excessive correction.

In industrial control, anti-windup strategies are common. A typical approach is to stop integrating when the actuator is saturated or to back-calculate the integral state so it stays consistent with the actual saturated output.

The Derivative Term: Damping and Prediction

Derivative action responds to the slope of the error: $u_D=K_d\frac{de(t)}{dt}$. Intuitively, it adds damping by reacting to how quickly the process is moving toward or away from the setpoint.

Where it helps:

• Reduces overshoot and improves stability when proportional gain is high.
• Improves response in systems that need quick motion control, such as positioning stages or speed control where inertia matters.

Where it hurts:

• Derivative amplifies measurement noise because differentiation emphasizes high-frequency components.
• For many process control loops (flow, pressure, level), D is used sparingly or not at all because noise and sensor dynamics can outweigh the benefit.

Most practical PID controllers apply derivative to the measured value rather than the error, or use a filtered derivative. A common idea is to approximate the derivative with a low-pass filter to reduce noise sensitivity.

Common PID Forms and Practical Details

Industrial PID controllers are often expressed in terms of proportional gain $K_p$, integral time $T_i$, and derivative time $T_d$:

$$u(t)=K_p\left(e(t)+\frac{1}{T_i}\int e(t)dt+T_d\frac{de(t)}{dt}\right)$$

This form is convenient because:

• $T_i$ has units of time and directly reflects how aggressively integral acts.
• $T_d$ has units of time and corresponds to derivative “look-ahead.”

Two practical features matter in real deployments:

Setpoint Weighting

Many controllers use setpoint weighting so that a setpoint step does not produce an excessive proportional or derivative kick. This improves operator-facing behavior without sacrificing disturbance rejection.

Output Limits and Actuator Constraints

Every actuator has limits: valves saturate, heaters have maximum power, motors have current limits. Good tuning respects these constraints. Aggressive gains that look good in simulation can cause saturation, windup, and poor recovery in the field.

PID Tuning: What “Good” Looks Like

Tuning means selecting $K_p$, $K_i$ (or $T_i$), and $K_d$ (or $T_d$) to meet a goal. Typical tuning objectives include:

• Minimal overshoot on setpoint steps (important in thermal systems to avoid overheating).
• Fast settling time after a disturbance (important in flow and pressure loops).
• Acceptable oscillation and robustness to process changes (important in industrial plants where operating conditions vary).

There is no universal “best” tuning because the right answer depends on process dynamics, noise, delays, and what performance matters most.

Ziegler-Nichols Tuning: The Classic Starting Point

Ziegler-Nichols methods are widely taught because they provide a systematic way to get workable PID settings quickly. They are usually treated as a starting point, not the final tuning.

Closed-Loop (Ultimate Gain) Method

In the closed-loop approach:

1. Set integral and derivative action to zero (P-only control).
2. Increase $K_p$ until the loop exhibits sustained oscillations.
3. Record the gain at this point (ultimate gain $K_u$) and the oscillation period (ultimate period $P_u$).
4. Compute PID settings using Ziegler-Nichols rules.

The appeal is that it requires no process model. The downside is significant: deliberately driving a real process to sustained oscillation can be unsafe or disruptive. It is often avoided on critical equipment or used only under controlled conditions.

Open-Loop (Reaction Curve) Method

Another Ziegler-Nichols technique uses an open-loop step test to estimate a process reaction curve. From the response, you infer approximate process delay and time constant and then compute controller settings.

This is often more practical in process industries when a temporary manual step can be applied safely. However, it still assumes the process behaves roughly like a simple low-order model with delay. Real plants can deviate due to nonlinearities, constraints, and interacting loops.

Other Tuning Approaches Used in Practice

While Ziegler-Nichols is common, experienced practitioners often adjust tuning based on observed behavior:

Manual Tuning by Loop Behavior

A typical sequence:

• Increase $K_p$ until response is reasonably fast but not oscillatory.
• Add integral to remove offset, increasing gradually until steady-state error is corrected without inducing slow oscillations.
• Add a small amount of derivative only if overshoot or oscillations persist and measurement noise is acceptable.

This approach relies on understanding the physical process and reading trends. It is often the fastest path to stable operation in the field.

Conservative Tuning for Robustness

Industrial processes drift. Fluid properties change, valves wear, loads vary. Conservative tuning trades speed for stability margins. For temperature loops with long time constants, slow and steady is often preferable to aggressive settings that overshoot and cycle.

Model-Based and Software-Assisted Tuning

Modern controllers and DCS/PLC tools may offer autotuning based on relay tests or model fitting. These can be effective, but results still need validation against real operating constraints, especially actuator limits and noise.

Practical Examples of PID Use

Flow Control

Flow loops are often fast and relatively linear. PI control is common. Derivative is usually unnecessary because sensors can be noisy and the process is already quick.

Temperature Control

Temperature loops are slow and often have delays. PI is common; derivative may help reduce overshoot in some cases, but careful filtering is needed. Anti-windup is especially important because heaters and valves saturate.

Position and Speed Control (Motion)

Servo systems frequently use PID or related structures. Derivative-like damping is valuable due to inertia and the need for crisp response. Noise handling and sampling effects become more prominent because loops run fast.

Common Pitfalls and How to Avoid Them

• Chasing noise with derivative: If the measurement is noisy, derivative can make the output jitter. Use filtering, reduce $K_d$, or omit D.
• Too much integral: Slow oscillations and long recovery often point to aggressive integral settings. Increase $T_i$ (weaken integral) or implement anti-windup properly.
• Ignoring dead time: Processes with significant delay can become unstable with high