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Range of PI gains Kp, KI for stability

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.

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]

The PI controller Kp+KIsK_p + \frac{K_I}{s} adds a pole at the origin. Find the closed-loop characteristic equation and apply Routh's criterion.

Characteristic equation

G(s)=(Kp+KIs)1(s+1)(s+2)=Kps+KIs(s+1)(s+2)1+G(s)=0⇒s(s2+3s+2)+Kps+KI=0⇒s3+3s2+(2+Kp)s+KI=0\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

RowCol 1Col 2
s3s^312+Kp2+K_p
s2s^23KIK_I
s1s^13(2+Kp)−KI3\frac{3(2+K_p)-K_I}{3}0
s0s^0KIK_I

Conditions for stability

  • s0s^0 row: KI>0K_I > 0
  • s1s^1 row: 3(2+Kp)−KI>0⇒KI<3(Kp+2)3(2+K_p) - K_I > 0 \Rightarrow K_I < 3(K_p+2)
  • This also requires Kp>−2K_p > -2.
0<KI<3(Kp+2)0 < K_I < 3(K_p + 2)

Example: with Kp=1K_p = 1, the integral gain must satisfy 0<KI<90 < K_I < 9; with Kp=4K_p = 4, 0<KI<180 < K_I < 18. A larger KpK_p allows a larger KIK_I.

✓ Answer checked by Claude Opus 5.5 (high effort)

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