Proportional, integral and derivative control actions

What proportional, integral and derivative actions each do, the PID equation and proportional band, offset under P control, and how P and PI control change the closed-loop response of a first-order process.

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Why it matters

Over 90 % of industrial loops run on some form of PID controller. Knowing exactly what each of the three actions does (speed, offset removal, damping), how the controller output responds to an error, and what each action does to the closed-loop poles is the core skill of a control engineer and a favourite area for GATE questions.

Key ideas

Error and action. The error is e(t) = set point − measured value. A reverse-acting controller decreases its output when the measurement rises (for example, a heating-steam valve on a temperature loop); a direct-acting controller increases it. The choice must make the overall loop negative feedback, taking the valve's fail action into account.

Proportional (P) action. p(t) = p̄ + K_c·e(t), where p̄ is the bias (output at zero error).

  • Immediate action proportional to the error.
  • Often expressed as proportional band PB = 100/K_c (%), the percentage of the measurement span that drives the output through its full range (K_c dimensionless).
  • Leaves a steady-state offset for set-point or load changes on self-regulating processes, because a non-zero error is needed to hold the output away from its bias. Larger K_c reduces offset but reduces damping.

Integral (I) action. Adds (K_c/τ_I)·∫e dt. The output keeps changing as long as any error remains, so it eliminates offset. τ_I is the integral (reset) time; 1/τ_I is the reset rate (repeats per unit time). For a step error, a PI controller repeats the proportional action once every τ_I. Integral action adds a pole at s = 0, raises the order of the loop and reduces stability (smaller τ_I means more oscillation). With a saturated valve the integral keeps growing: reset windup, prevented by anti-windup logic.

Derivative (D) action. Adds K_c·τ_D·de/dt. It anticipates the error from its rate of change, adds phase lead and damping, and allows a higher K_c. It does nothing at steady state, cannot remove offset, amplifies measurement noise, and gives a spike ("derivative kick") on a set-point step unless it acts on the measurement instead of the error. Real controllers filter it: τ_D·s/(α·τ_D·s + 1) with α about 0.05–0.2. It is used on slow, smooth loops such as temperature and composition, seldom on noisy flow loops.

Which to use. P only for loose level control where offset is acceptable; PI for most flow, pressure and level loops; PID for slow multi-capacity loops (temperature, some composition).

Effect on a first-order process K/(τs + 1).

  • P: closed loop stays first order, faster (τ/(1 + K_c·K)), with offset 1/(1 + K_c·K) for a unit set-point step.
  • PI: closed loop becomes second order, no offset, can be underdamped but is never unstable for a pure first-order process.
  • Adding dead time or more lags makes instability possible at high gain.

Formulas

p(t) = p̄ + K_c·[e(t) + (1/τ_I)·∫₀ᵗ e dt + τ_D·de/dt]

  • Ideal (parallel, ISA) PID; p output (% or mA), K_c gain (output/input units), τ_I and τ_D in s or min.

G_c(s) = K_c·(1 + 1/(τ_I·s) + τ_D·s)

  • Ideal PID transfer function; P: K_c; PI: K_c·(1 + 1/(τ_I·s)); PD: K_c·(1 + τ_D·s).

PB = 100/K_c (%)

  • K_c dimensionless (% output per % of measurement span).

p(t) − p̄ = K_c·E·(1 + t/τ_I)

  • PI response to a constant error step of size E.

Offset = 1/(1 + K_c·K_p)

  • P control, unit set-point step, first-order (self-regulating) process, unity valve and measurement gains.

τ_I·τ·s² + τ_I·(1 + K_c·K_p)·s + K_c·K_p = 0

  • Closed-loop characteristic equation, PI control of K_p/(τs + 1).

Worked examples

Example 1 (standard): PI controller output. A PI controller with K_c = 2 and τ_I = 4 min sees a sudden, sustained error of 5 % at t = 0. Find the change in output just after t = 0 and at t = 6 min.

  1. PI response to a constant error E: Δp = K_c·E·(1 + t/τ_I).
  2. At t = 0⁺: Δp = 2·5·(1 + 0) = 10 % (proportional kick).
  3. At t = 6 min: Δp = 2·5·(1 + 6/4) = 2·5·2.5 = 25 %.
  4. Check: every 4 min (one τ_I) the integral adds another 10 %, repeating the proportional action.

Δp = 10 % at t = 0⁺ and 25 % at t = 6 min

Example 2 (GATE level): P versus PI on a first-order process. Process K_p = 2, τ = 5 min; valve and transmitter gains are 1. (a) P control with K_c = 3: find the closed-loop time constant and offset for a unit set-point step. (b) PI control with K_c = 3, τ_I = 1 min: find ζ and the offset.

  1. (a) C/R = K_cK_p/(τs + 1 + K_cK_p) = 6/(5s + 7) = (6/7)/((5/7)s + 1).
  2. τ_CL = 5/7 = 0.714 min; offset = 1 − 6/7 = 1/7 = 0.143.
  3. (b) Characteristic equation: τ_I·τ·s² + τ_I·(1 + K_cK_p)·s + K_cK_p = 1·5·s² + 1·7·s + 6 = 0.
  4. Normalise: (5/6)s² + (7/6)s + 1 = 0, so τ = √(5/6) = 0.913 min and 2ζτ = 7/6 = 1.167, giving ζ = 1.167/(2·0.913) = 0.639.
  5. Offset: C/R = K_cK_p(τ_I s + 1)/(5s² + 7s + 6), and C/R(0) = 6/6 = 1, so offset = 0.

(a) τ_CL ≈ 0.714 min, offset ≈ 0.143; (b) ζ ≈ 0.64 (underdamped), offset = 0

Example 3 (short): proportional band. A controller set to PB = 40 % has K_c = 100/40 = 2.5; halving the PB to 20 % doubles the gain to 5.

Common mistakes

  • Thinking derivative action removes offset; only integral action does.
  • Confusing τ_I with integral gain: a larger τ_I means weaker integral action. Similarly, a larger PB means a smaller gain.
  • Forgetting the bias p̄ and treating the controller output as zero at zero error.
  • Ignoring direct/reverse action: the wrong sign turns negative feedback into positive feedback.
  • Using derivative on noisy flow loops, or derivative on error with frequent set-point steps.
  • Assuming PI on a first-order process can go unstable; it can only become more oscillatory.

For GATE CH

Expect: controller output at a given time for a step or ramp error (P, PI, PD, PID), offset under P control, closed-loop time constant or damping factor under P and PI control, proportional band conversion, and qualitative questions on which action removes offset, adds damping or amplifies noise. Practise writing characteristic equations for PI loops and normalising them.

Quick check

  1. What gain corresponds to a proportional band of 200 %?
  2. Which action produces "repeats per minute"?
  3. Under P control with K_cK_p = 9, what is the offset for a unit set-point step?
  4. A PD controller (K_c = 2, τ_D = 1 min) sees an error ramp e = 3t (%/min). What is the output change at t = 2 min?

Answers: 1. 0.5. 2. Integral action (reset rate 1/τ_I). 3. 0.1. 4. 2·(3·2 + 1·3) = 18 %.

Interview questions

All Process Instrumentation and Control interview questions

Try answering each one aloud before you open it.

  1. 1.What is a Proportional (P) control action in process control?Concept

    A Proportional (P) control action is a type of feedback control where the control signal is proportional to the error signal. The error signal is the difference between the setpoint and the process variable. The proportional gain (Kp) determines the magnitude of the control action. A higher Kp results in a larger control action for a given error, which can lead to faster response but may cause overshoot.

  2. 2.Explain the Integral (I) control action and its purpose in a PID controller.Concept

    The Integral (I) control action is used to eliminate the steady-state error in a control system. It integrates the error over time, meaning it sums up past errors to adjust the control output. This action helps in driving the error to zero, ensuring that the process variable reaches the setpoint. However, excessive integral action can lead to instability and oscillations.

  3. 3.Describe the Derivative (D) control action and its effect on system stability.Concept

    The Derivative (D) control action predicts the future behavior of the error by calculating its rate of change. It provides a control action that is proportional to the rate of change of the error. This action helps in damping the system response, reducing overshoot, and improving stability. However, it can amplify noise in the system, so it must be used carefully.

  4. 4.Why is a PID controller commonly used in industrial process control?Application

    A PID controller is widely used because it combines the benefits of proportional, integral, and derivative control actions. The proportional action provides immediate response to error, the integral action eliminates steady-state error, and the derivative action improves stability and response time. This combination makes PID controllers versatile and effective for a wide range of processes.

  5. 5.What happens if the proportional gain (Kp) is set too high in a PID controller?Application

    If the proportional gain (Kp) is set too high, the system can become overly sensitive to errors, leading to excessive oscillations and instability. The control action may cause the process variable to overshoot the setpoint significantly, resulting in a system that is difficult to stabilize. It is important to tune Kp carefully to balance responsiveness and stability.

  6. 6.How does the integral time constant (Ti) affect the performance of a PID controller?Application

    The integral time constant (Ti) determines how quickly the integral action responds to accumulated error. A smaller Ti results in a faster response to eliminate steady-state error, but it can also lead to instability and oscillations if set too low. Conversely, a larger Ti slows down the response, which may result in a longer time to reach the setpoint.

  7. 7.In what scenarios would you prefer using a PI controller over a PID controller?Application

    PI is the default for fast, noisy loops such as flow, liquid pressure and most level loops: derivative action would amplify measurement noise and give little benefit because the process has little lag to anticipate. Derivative action pays off in slow, smooth, multi-capacity loops such as temperature or some composition loops, where it adds damping and allows a higher gain. PI is also easier to tune and more robust when the model is uncertain.

  8. 8.A discrete PID controller in parallel form has K_p = 2, K_i = 1 s⁻¹ and K_d = 0.5 s. The accumulated integral of error is 3 unit·s, the present error is 3 units, the previous error was 2 units and the sample time is 1 s. What is the controller output (deviation from bias)?Numerical

    Output = K_p·e + K_i·∫e dt + K_d·Δe/Δt. Proportional term = 2 × 3 = 6; integral term = 1 × 3 = 3; derivative term = 0.5 × (3 − 2)/1 = 0.5. The output is 6 + 3 + 0.5 = 9.5 units above the bias. In practice the derivative is usually taken on the measurement and filtered to avoid noise and set-point kick.

  9. 9.A process has a setpoint of 100 units and a current process variable of 90 units. If the proportional gain (Kp) is 4, what is the proportional control action?Numerical

    The proportional control action is calculated as the product of the proportional gain (Kp) and the error. The error is the difference between the setpoint and the process variable. Here, the error is 100 - 90 = 10 units. Therefore, the proportional control action is 4 * 10 = 40 units.

  10. 10.What are the potential drawbacks of using a PID controller in a noisy environment?Application

    In a noisy environment, the derivative action of a PID controller can amplify the noise, leading to erratic control actions. This can cause the system to become unstable and oscillate. Additionally, noise can affect the accuracy of the integral action, leading to incorrect adjustments. To mitigate these issues, filtering techniques or tuning adjustments may be necessary.

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