--- title: "Effect of adding poles and zeros" subject: "Control Systems" chapter: 3 key: "semester_3/control_systems:effect-add-pole-zero" source: https://www.bctengineer.com/q/semester_3/control_systems/effect-add-pole-zero site: BCT Engineering --- # Effect of adding poles and zeros Control Systems · Chapter 3: System Transfer Function and Responses - Asked in: 2080 Bhadra, 4 marks - Video: [Effects of adding poles and zeros](https://www.youtube.com/watch?v=RzqVU4XsUvE) — Leena Timothy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/effect-add-pole-zero - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3 ## Question Discuss the effect of addition of pole and zero in a system. ## Short answer Adding a pole shifts root locus right, reduces stability, slows response and increases rise time. Adding a zero shifts root locus left, improves stability, speeds up response but a closed-loop zero increases overshoot. Far-left poles/zeros have little effect. ## Answer Adding a pole or a zero changes the shape of the root locus and the time response. Poles slow the system down; zeros speed it up. #### Addition of a pole - **Open-loop pole (to $G(s)H(s)$):** the root locus bends to the **right**, towards the imaginary axis. The system becomes less stable and the range of $K$ for stability shrinks. - Closed-loop response becomes slower: rise time and settling time increase. - Peak overshoot usually increases (lower effective damping). - A pole at the origin raises the system type, which reduces steady state error. - **Closed-loop pole:** adds a slow mode; overshoot falls but rise time rises (more sluggish response). #### Addition of a zero - **Open-loop zero:** pulls the root locus to the **left**, away from the imaginary axis. Relative stability improves (this is the idea behind PD and lead compensation). - Closed-loop response becomes faster: rise time and peak time decrease. - **Closed-loop zero:** adds a derivative term $\frac{1}{a}\frac{dc}{dt}$ to the response, so overshoot **increases**, especially when the zero is close to the origin. - Bandwidth increases. ```text jw pole added zero added | locus bends right locus bends left | \ / --+-------x----- -----o------- ``` | Effect on | Adding pole | Adding zero | |---|---|---| | Root locus | shifts right | shifts left | | Stability | decreases | increases | | Rise time | increases | decreases | | Bandwidth | decreases | increases | A pole or zero far from the dominant poles (more than about 5 times further left) has little effect. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does adding an open-loop pole do to the root locus? A. Bends it to the right, reducing relative stability B. Bends it to the left C. Has no effect D. Moves it onto the real axis Answer: A. Added poles push the locus toward the imaginary axis. (Key point: Open-loop pole) 2. Which statement about adding an open-loop zero is correct? A. It pulls the locus left and makes the response faster B. It makes the response slower C. It reduces bandwidth D. It lowers system type Answer: A. Zeros improve stability and speed (PD or lead idea) and increase bandwidth. (Key point: Open-loop zero) 3. Effect of a closed-loop zero near the origin on overshoot? A. Increases overshoot B. Decreases overshoot C. No change D. Removes overshoot Answer: A. It adds a derivative term to the response, increasing overshoot. (Key point: Closed-loop zero) 4. What is the effect of adding a pole at the origin? A. Raises system type and reduces steady-state error B. Lowers system type C. Increases steady-state error D. Makes response faster Answer: A. Each origin pole raises the type, which improves steady-state accuracy. (Key point: Pole at origin) ## Close variants set in other exams - [Effect of adding a zero](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aeffect-add-zero) — Control Systems, chapter 3, asked 2× ## Related in Control Systems - [Poles and zeros from state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-zeros-poles-5-6-1) — Control Systems, chapter 8 - [Transfer function, poles and zeros from 3×3 state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-poles-zeros-3x3) — Control Systems, chapter 8, asked 3× - [Root locus: poles −2±j3, zero at origin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-2j3-zero-0-critical) — Control Systems, chapter 5 - [Nyquist criterion from given pole-zero and Nyquist plot](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-criteria-from-plot) — Control Systems, chapter 6 - [Root locus with poles 0, −1 and zero −3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-0-1-zero-3) — Control Systems, chapter 5 - [Root locus with poles 0, −6, −2 ± j3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-0-6-2j3) — Control Systems, chapter 5 ## Related in other subjects - [Transmission zeros and zero shifting by partial pole removal](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atransmission-zeros-zero-shifting) — Filter Design, chapter 4, asked 3× - [Lattice structure of second order pole-zero system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alattice-iir-pole-zero-2nd) — Digital Signal Analysis and Processing, chapter 4 - [Difference equation example; poles/zeros response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Apole-zero-define-difference-eq) — Digital Signal Analysis and Processing, chapter 3 - [Inverse Chebyshev characteristics, order, poles and zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-poles-zeros) — Filter Design, chapter 2, asked 2×