--- title: "Maximum K for poles left of −0.5" subject: "Control Systems" chapter: 4 key: "semester_3/control_systems:rh-k-s1-s-s2-s3" source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s1-s-s2-s3 site: BCT Engineering --- # Maximum K for poles left of −0.5 Control Systems · Chapter 4: Stability - Asked in: 2070 Chaitra, 8 marks - Video: [Applications of R-H Criteria (Part 1)](https://www.youtube.com/watch?v=S0_Hd5M7aJY) — Neso Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s1-s-s2-s3 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4 ## Question The open loop transfer function of a closed loop system is G(s) = K(s+1)/[s(s+2)(s+3)]; find maximum possible K for which the poles lie on left of point −0.5. ## Short answer Shift the imaginary axis to −0.5 by setting s = z−0.5. The shifted characteristic equation is z³+3.5z²+(1.75+K)z+(0.5K−1.875)=0. Routh on z requires 0.5K−1.875>0 ⇒ K>3.75. All poles lie left of −0.5 for K>3.75; there is no finite maximum, only the boundary K=3.75. ## Answer **Result:** all poles lie left of $s=-0.5$ when $K > 3.75$. There is no finite maximum; $K = 3.75$ is the limiting value. To check that all poles lie to the left of $s=-0.5$, shift the imaginary axis to $s=-0.5$ by putting $s = z - 0.5$. Then apply the Routh criterion in $z$. Unity feedback is assumed. #### Characteristic equation $$ \begin{aligned} s(s+2)(s+3) + K(s+1) &= 0 \\ s^3 + 5s^2 + (6+K)s + K &= 0 \end{aligned} $$ #### Shift: $s = z - 0.5$ $$ \begin{aligned} (z-0.5)(z+1.5)(z+2.5) &= z^3 + 3.5z^2 + 1.75z - 1.875 \\ K(z-0.5+1) &= Kz + 0.5K \end{aligned} $$ $$ z^3 + 3.5z^2 + (1.75+K)z + (0.5K - 1.875) = 0 $$ #### Routh array in z | Row | Col 1 | Col 2 | |---|---|---| | $z^3$ | 1 | $1.75+K$ | | $z^2$ | 3.5 | $0.5K-1.875$ | | $z^1$ | $\frac{3.5(1.75+K)-(0.5K-1.875)}{3.5}=\frac{3K+8}{3.5}$ | 0 | | $z^0$ | $0.5K-1.875$ | | #### Conditions - $z^1$ row: $3K + 8 > 0 \Rightarrow K > -2.67$ (true for $K>0$) - $z^0$ row: $0.5K - 1.875 > 0 \Rightarrow K > 3.75$ So all closed-loop poles lie to the left of $s=-0.5$ when $$ K > 3.75 $$ #### Comment on "maximum K" For this system there is **no finite maximum**. As $K\to\infty$, one pole moves to the zero at $-1$, and the other two follow asymptotes at $\pm90^\circ$ from the centroid $\frac{(0-2-3)-(-1)}{2}=-2$, which is left of $-0.5$. The limiting (boundary) value is $K=3.75$. At $K=3.75$ the poles are $s=-0.5$ and $s=-2.25\pm j1.56$, so one pole is exactly at $-0.5$. **Answer:** all poles lie left of $s=-0.5$ for $K > 3.75$; the boundary value is $K = 3.75$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the requirement 'all poles to the left of s = -0.5' tested with the Routh criterion? A. Put s = z + 0.5 and require roots on the jω axis B. Put s = z - 0.5 and require all z-roots in the left half plane C. Apply Routh directly to s and require K < 0.5 D. Multiply the equation by (s+0.5) and apply Routh Answer: B. Substituting s = z - 0.5 moves the imaginary axis to s = -0.5, so Routh in z tests the shifted condition. (Key point: Shifting the axis) 2. What is the characteristic equation of K(s+1)/[s(s+2)(s+3)] with unity feedback? A. s^3+5s^2+(6+K)s+K=0 B. s^3+5s^2+6s+K=0 C. s^3+6s^2+(5+K)s+K=0 D. s^3+5s^2+6s+K(s+1)^2=0 Answer: A. s(s+2)(s+3)=s^3+5s^2+6s, and adding K(s+1) gives s^3+5s^2+(6+K)s+K. (Key point: Characteristic equation) 3. What condition on K comes from the z^0 row, 0.5K - 1.875 > 0? A. K > 3.75 B. K < 3.75 C. K > -2.67 D. K > 1.875 Answer: A. 0.5K > 1.875 gives K > 3.75, which is the limiting value for poles left of -0.5. (Key point: Limiting K) 4. Why is there no finite maximum K for poles left of -0.5? A. Asymptotes at ±90° from centroid -2 stay left of -0.5, so any K above 3.75 works B. K = 3.75 is a maximum by definition C. The poles move into the RHP for large K D. The z^1 row gives K < 3.75 Answer: A. For large K, one pole goes to the zero at -1 and the others follow asymptotes from -2, all left of -0.5, so 3.75 is a lower limit only. 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