--- title: "PID controller block diagram, transfer function, characteristics" subject: "Control Systems" chapter: 7 key: "semester_3/control_systems:pid-bd-tf-characteristics" source: https://www.bctengineer.com/q/semester_3/control_systems/pid-bd-tf-characteristics site: BCT Engineering --- # PID controller block diagram, transfer function, characteristics Control Systems · Chapter 7: Performance Specifications and Compensation Design - Asked in: 2080 Baisakh, 6 marks - Video: [PID Controller Explained: Basics, Block Diagram, Significance, Transfer Function, Pros, and Cons](https://www.youtube.com/watch?v=A6_h2YacMr4) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/pid-bd-tf-characteristics - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7 ## Question Explain PID controller with block diagram and its transfer function, along with its characteristics. ## Short answer A PID controller output is u(t)=Kp e(t)+Ki∫e dt+Kd de/dt, with transfer function Gc(s)=Kp+Ki/s+Kd s = (Kd s^2+Kp s+Ki)/s, adding a pole at the origin and two zeros. Proportional action reduces rise time, integral action eliminates steady-state error but may cause overshoot, and derivative action adds damping and improves stability without affecting steady-state error. ## Answer A **PID controller** produces a control signal that is the sum of three terms: proportional to the error, to the integral of the error, and to the derivative of the error. $$ u(t) = K_p e(t) + K_i \int_0^t e(t)\,dt + K_d \frac{de(t)}{dt} $$ #### Block diagram ```text +----------+ +-->| Kp |----+ | +----------+ | r + e | +----------+ + v u +-------+ c --->(O)--+-->| Ki / s |-->(O)---->| Plant |--+--> -^ | +----------+ ^ +-------+ | | | +----------+ | | | +-->| Kd s |----+ | | +----------+ | +-----------------------------------------+ ``` #### Transfer function $$ G_c(s) = \frac{U(s)}{E(s)} = K_p + \frac{K_i}{s} + K_d s = \frac{K_d s^2 + K_p s + K_i}{s} $$ or, in standard form, $G_c(s) = K_p\left(1 + \dfrac{1}{T_i s} + T_d s\right)$ with integral time $T_i = K_p/K_i$ and derivative time $T_d = K_d/K_p$. It adds **one pole at the origin and two zeros** to the open-loop transfer function. #### Characteristics | Action | Rise time | Overshoot | Settling time | Steady-state error | |---|---|---|---|---| | Increase $K_p$ | Decreases | Increases | Small change | Decreases | | Increase $K_i$ | Decreases | Increases | Increases | Eliminated | | Increase $K_d$ | Small change | Decreases | Decreases | No change | - Raises the system type by one, so steady-state error to a step is zero. - Derivative action adds damping and improves stability and transient response. - Gives good overall performance; used in most industrial loops (temperature, pressure, flow, motor speed). - Tuning is done by trial, by Ziegler–Nichols rules or by root locus/frequency methods. - In practice the derivative term is filtered, $K_d s/(1+\tau s)$, to limit noise, and anti-windup is added to the integral term. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the PID transfer function? A. Kp + Ki/s + Kd*s B. Kp*Ki*Kd/s C. Kp + Ki*s + Kd/s D. Kp + Ki/s^2 + Kd*s^2 Answer: A. U/E = Kp + Ki/s + Kd s = (Kd s^2 + Kp s + Ki)/s. (Key point: Transfer function) 2. What does a PID controller add to the open loop? A. One pole at the origin and two zeros B. One pole and one zero C. Two poles at the origin and one zero D. Three zeros Answer: A. The numerator is a quadratic (two zeros) and the denominator is s (one pole at the origin). (Key point: Singularities) 3. Increasing Ki mainly does what? A. Has no effect on steady-state error B. Eliminates steady-state error but increases overshoot and settling time C. Reduces overshoot and settling time D. Only reduces rise time Answer: B. Integral action removes steady-state error at the cost of overshoot and slower settling. (Key point: Effect of Ki) 4. Why is the derivative term filtered, Kd*s/(1+tau*s)? A. To remove steady-state error B. To add a pole at the origin C. To increase the system type D. To limit amplification of high-frequency noise Answer: D. A pure derivative amplifies noise, so a small filter pole is added; anti-windup is added to the integral term for similar practical reasons. 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