--- title: "Range of PI gains Kp, KI for stability" subject: "Control Systems" chapter: 4 key: "semester_3/control_systems:pi-kp-ki-range-stable" source: https://www.bctengineer.com/q/semester_3/control_systems/pi-kp-ki-range-stable site: BCT Engineering --- # Range of PI gains Kp, KI for stability Control Systems · Chapter 4: Stability - Asked in: 2073 Shrawan, 4 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/pi-kp-ki-range-stable - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4 ## Question For a closed loop system presented by the block diagram below, determine the range of controller gain (Kp, KI) so that the PI controller provides the stable output. [Figure: R → summing point (+, −) → PI controller (Kp + KI/s) → plant 1/[(s+1)(s+2)] → y; unity negative feedback from y] ## Short answer The closed-loop characteristic equation is s³ + 3s² + (2+Kp)s + KI = 0. Routh array yields KI > 0 and 3(2+Kp) − KI > 0, giving KI < 3(Kp+2). The stability range is 0 < KI < 3(Kp+2) with Kp > −2. ## Answer The PI controller $K_p + \frac{K_I}{s}$ adds a pole at the origin. Find the closed-loop characteristic equation and apply Routh's criterion. #### Characteristic equation $$ \begin{aligned} G(s) &= \left(K_p + \frac{K_I}{s}\right)\frac{1}{(s+1)(s+2)} = \frac{K_ps + K_I}{s(s+1)(s+2)} \\ 1+G(s) = 0 &\Rightarrow s(s^2+3s+2) + K_ps + K_I = 0 \\ &\Rightarrow s^3 + 3s^2 + (2+K_p)s + K_I = 0 \end{aligned} $$ #### Routh array | Row | Col 1 | Col 2 | |---|---|---| | $s^3$ | 1 | $2+K_p$ | | $s^2$ | 3 | $K_I$ | | $s^1$ | $\frac{3(2+K_p)-K_I}{3}$ | 0 | | $s^0$ | $K_I$ | | #### Conditions for stability - $s^0$ row: $K_I > 0$ - $s^1$ row: $3(2+K_p) - K_I > 0 \Rightarrow K_I < 3(K_p+2)$ - This also requires $K_p > -2$. $$ 0 < K_I < 3(K_p + 2) $$ **Example:** with $K_p = 1$, the integral gain must satisfy $0 < K_I < 9$; with $K_p = 4$, $0 < K_I < 18$. A larger $K_p$ allows a larger $K_I$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The PI controller Kp + KI/s adds A. A zero at the origin only B. A pole at s = -1 C. No poles D. A pole at the origin Answer: D. The 1/s term introduces a pole at the origin, making the loop type 1. (Key point: Effect of PI) 2. The characteristic equation is A. s^3+3s^2+(2+Kp)s+KI=0 B. s^3+3s^2+2s+KpKI=0 C. s^2+3s+2+Kp+KI=0 D. s^3+3s^2+(2+KI)s+Kp=0 Answer: A. s(s+1)(s+2)+Kp s+KI. (Key point: Characteristic equation) 3. The s^1 row condition gives A. KI > 3(Kp+2) B. KI < 3(Kp+2) C. KI < Kp+2 D. KI < 3Kp Answer: B. 3(2+Kp) - KI > 0. (Key point: Upper limit) 4. The stable range is A. KI > 3(Kp+2) B. 0 < KI < Kp C. 0 < KI < 3(Kp+2) D. KI < 0 Answer: C. Combining KI>0 (s^0) and KI<3(Kp+2) (s^1). (Key point: Final range) ## Related in Control Systems - [Block diagram reduction: G1–G4, H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1-g4-h1h2-2075a) — Control Systems, chapter 3, asked 4× - [Closed loop control system block diagram explained](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-closed-loop) — Control Systems, chapter 1, asked 4× - [Control system block diagram and closed loop advantages](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-cl-advantages) — Control Systems, chapter 1 - [Block diagram reduction: cascaded G1–G3, H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2g3-h1h2-bei2081) — Control Systems, chapter 3, asked 2× - [Open and closed loop systems with block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aol-cl-bd-role) — Control Systems, chapter 1, asked 2× - [Define control system and closed loop block roles](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adefine-cs-bd-role) — Control Systems, chapter 1, asked 5× ## Related in other subjects - [Closed loop diagram; P, PI, PD, PID](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aclosed-loop-block-diagram-pid-types) — Embedded System, chapter 7 - [Closed-loop TF with digital PID, plant 1/(s(s+1))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apid-closed-loop-plant-s1) — Digital Control System, chapter 3, asked 2× - [Response sequence with digital PI controller](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aresponse-kronecker-delta-pi) — Digital Control System, chapter 3 - [Digital PI design, plant e^(−5s)/(s+0.4), step response](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Api-design-dead-time-5s-step) — Digital Control System, chapter 4