PID Tuning Basics: What P, I, and D Do and How to Tune Them in the Field

What P, I, and D Do Individually

Proportional (P) action produces an output that is proportional to the current error (the difference between setpoint and process variable). If the error is large, the output is large. If the error is small, the output is small. The proportional gain (Kp) determines how aggressively the controller responds. High gain = aggressive response (risk of oscillation). Low gain = sluggish response (slow to correct disturbances). Proportional action alone cannot eliminate steady-state error — there is always a residual offset because the controller needs some error to produce an output.

Integral (I) action eliminates the steady-state offset that proportional action leaves behind. It accumulates the error over time and adjusts the output until the error reaches zero. The integral time (Ti, in minutes or seconds) determines how fast the accumulation occurs. Short integral time = fast elimination of offset but risk of overshoot and oscillation. Long integral time = slow elimination of offset but more stable response. Integral action is essential for most process loops because operators do not want to see a permanent offset between setpoint and process variable.

Derivative (D) action responds to the rate of change of the error. It predicts where the error is heading and applies a corrective output before the error gets worse. The derivative time (Td) determines the strength of this anticipatory action. Derivative action improves response to fast disturbances and reduces overshoot on setpoint changes. However, it amplifies noise in the process variable signal, which can cause the controller output to chatter. For this reason, derivative action is used sparingly and is often set to zero on noisy processes like flow and liquid pressure loops.

In practice, most loops are tuned as PI controllers (P + I, with D = 0). Derivative action is added only for slow processes (temperature, composition) where the extra anticipation provides a meaningful improvement in response speed. Level control is often P-only (with a wide proportional band) to allow the level to float and absorb flow disturbances.

FOPDT Process Model: Gain, Dead Time, and Time Constant

Before tuning, you need to understand the process dynamics. The most practical way to characterize a process is the FOPDT (First Order Plus Dead Time) model, which describes the process response using three parameters: process gain (Kp), dead time (Td), and time constant (Tau).

Process gain (Kp) is the ratio of the change in process variable to the change in controller output, in engineering units. If a 10% change in controller output causes a 5 PSI change in pressure, the process gain is 5/10 = 0.5 PSI/%. Process gain tells you how sensitive the process is to the controller output.

Dead time (Td or theta) is the delay between when the controller output changes and when the process variable first begins to respond. Dead time comes from transportation delays (fluid moving through a pipe), measurement delays (sensor response time), and actuation delays (valve stroke time). Dead time is the enemy of control — the controller is "flying blind" during the dead time, and its output changes cannot have any effect until the dead time expires.

Time constant (Tau) is how long it takes the process variable to reach 63.2% of its final value after the dead time expires. A process with a 60-second time constant reaches 63.2% at 60 seconds, 86.5% at 120 seconds, 95% at 180 seconds, and 99.3% at 300 seconds (five time constants). The time constant characterizes how fast or slow the process responds — a small time constant means a fast process, a large time constant means a slow process.

The ratio of dead time to time constant (Td/Tau) determines how difficult the process is to control. Processes with Td/Tau less than 0.3 are easy to control. Processes with Td/Tau between 0.3 and 0.8 are moderately difficult. Processes with Td/Tau greater than 0.8 are "dead-time dominant" and very difficult to control with PID. Dead-time dominant processes may require Smith predictor or model predictive control strategies.

Step Response Testing in the Field

The step response test (also called a bump test or open-loop test) is the standard method for identifying FOPDT model parameters. Put the controller in manual mode, wait for the process variable to stabilize, then make a step change in the controller output and record the process variable response over time.

Procedure: (1) Put the controller in manual. (2) Wait for the PV to reach steady state (no trending up or down). (3) Record the PV value and the CO value. (4) Make a step change in the CO — typically 5-10% for a sensitive process, 10-20% for a sluggish process. (5) Record the PV response continuously until it reaches a new steady state. (6) From the response curve, extract the three FOPDT parameters.

Reading the response curve: the dead time is the time from the CO step to when the PV first begins to change (draw a tangent line at the steepest part of the response and extend it back to the original PV value — where it crosses is the dead time point). The time constant is the time from the end of dead time to when the PV reaches 63.2% of its total change. The process gain is the total PV change divided by the CO step size.

Step response tips: make the step large enough to produce a measurable PV change (at least 2-3 times the noise level), but small enough that it does not upset the process (keep within safe operating limits). Perform the test during stable conditions — avoid testing during startup, shutdown, or when upstream processes are changing. If possible, perform step tests in both directions (step up, wait for steady state, step down) and average the results to account for process non-linearity.

Ziegler-Nichols and Lambda Tuning Methods

Ziegler-Nichols (Z-N) open-loop method: This classic method uses the FOPDT model parameters to calculate initial tuning values. For a PI controller: Kc = 0.9 × Tau / (Kp × Td) and Ti = 3.33 × Td. For a PID controller: Kc = 1.2 × Tau / (Kp × Td), Ti = 2.0 × Td, Td_derivative = 0.5 × Td. Z-N tuning produces an aggressive response with approximately 25% overshoot on setpoint changes. It provides fast disturbance rejection but the overshoot may be unacceptable for some processes.

Lambda tuning: This method allows you to specify the desired closed-loop response speed by choosing a Lambda value (λ, the desired closed-loop time constant). The Lambda tuning formulas for PI are: Kc = Tau / (Kp × (λ + Td)) and Ti = Tau. The Lambda value is your design choice: larger Lambda = slower, more conservative response; smaller Lambda = faster, more aggressive response. A common starting point is Lambda = 3 × Td (three times the dead time), which gives a stable response with minimal overshoot.

Lambda tuning is preferred for most process applications because it gives the engineer direct control over the response speed. It also produces less overshoot than Z-N tuning. The tradeoff is slower disturbance rejection. For processes where overshoot is dangerous (exothermic reactors, pH control near discharge limits), Lambda tuning with a conservative Lambda value is the standard approach.

The Z-N closed-loop method (also called the ultimate gain method) involves increasing the proportional gain with the controller in automatic until the loop oscillates continuously at a constant amplitude. The gain at this point is the ultimate gain (Ku) and the oscillation period is the ultimate period (Pu). Tuning values are calculated from Ku and Pu. This method requires the process to oscillate, which is unacceptable for many processes. It is rarely used in modern practice but remains in textbooks and certification exams.

Step Response Metrics: Overshoot, Settling Time, and Rise Time

After tuning, evaluate the controller performance using standard step response metrics measured from a setpoint change or disturbance recovery.

Overshoot is the amount the PV exceeds the setpoint before settling, expressed as a percentage of the setpoint change. Zero overshoot means the PV approaches the setpoint from one side and never crosses it. 25% overshoot means the PV goes 25% past the setpoint before coming back. For most process loops, 5-15% overshoot is acceptable. For temperature control of exothermic reactors, zero overshoot is required. For level control, overshoot is usually not a concern.

Settling time is the time from the disturbance or setpoint change until the PV remains within a specified band (typically ±2% or ±5%) of the final value. Shorter settling time means faster recovery from upsets. Settling time is often the most important metric for disturbance rejection — how fast does the loop recover when something changes?

Rise time is the time from 10% to 90% of the final value on a setpoint change. Fast rise time means the loop responds quickly to operator commands. Rise time and overshoot are in tension: faster rise time typically produces more overshoot. The tuning tradeoff is choosing how much overshoot is acceptable to achieve the desired speed of response.

Integrated error metrics (IAE, ISE, ITAE) integrate the error over time to produce a single performance number. IAE (Integral of Absolute Error) weights all errors equally. ISE (Integral of Squared Error) penalizes large errors more than small errors. ITAE (Integral of Time-weighted Absolute Error) penalizes errors that persist for a long time. These metrics are used by auto-tuning software to optimize the tuning parameters numerically.

Practical Tips for Field Tuning

Start conservative. If you are unsure of the tuning, start with low gain and long integral time. A sluggish loop is safe; an oscillating loop is not. You can always increase the gain and decrease the integral time to speed up the response. Going the other direction (reducing gain to stop oscillation) is stressful because the loop is oscillating while you are trying to fix it.

Fix the valve before tuning. No amount of tuning will fix a loop with a sticky valve, an oversized valve, or a valve with excessive dead band. If the valve does not respond smoothly to small signal changes (test by making 1-2% output steps in manual), fix the valve first. Tuning on top of a mechanical problem just masks the real issue and produces tuning values that will not work when the valve is eventually repaired.

Tune integral after proportional. Start with integral action turned off (Ti set to maximum). Adjust the proportional gain until the loop responds to setpoint changes with an acceptable speed and no more than one or two oscillation cycles that damp out quickly. Then gradually decrease the integral time until the steady-state offset is eliminated without introducing sustained oscillation.

Watch out for integral windup. If the controller output saturates (hits 0% or 100%) for an extended period, the integral term accumulates a large value. When the PV finally enters the controllable range, the accumulated integral causes a massive overshoot. Most modern controllers have anti-windup features, but verify they are enabled. If not, set output limits and enable anti-windup clamping.

Document everything. Record the FOPDT parameters, the tuning values, the date, and the performance (overshoot, settling time). When the loop needs re-tuning later (process changes, equipment modifications), you have a baseline to compare against. Keep a tuning log for critical loops.