--- title: "Relative stability using R-H criterion" subject: "Control Systems" chapter: 4 key: "semester_3/control_systems:rh-relative-stability" source: https://www.bctengineer.com/q/semester_3/control_systems/rh-relative-stability site: BCT Engineering --- # Relative stability using R-H criterion Control Systems · Chapter 4: Stability - Asked in: 2074 Chaitra, 3 marks; 2072 Chaitra, 4 marks (asked 2 times) - Video: [Relative Stability Analysis](https://www.youtube.com/watch?v=jUX-3XIyH2E) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-relative-stability - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4 ## Question Explain how RH (Routh Hurwitz) method is used for determining relative stability. ## Short answer Relative stability is checked by shifting the imaginary axis: substitute s = z − σ into the characteristic equation. Apply Routh–Hurwitz to the new polynomial in z. If the first column has no sign changes, all closed-loop poles lie to the left of s = −σ, guaranteeing a minimum decay rate. ## Answer **Relative stability** tells how far the closed-loop poles lie to the left of the imaginary axis, i.e. how stable a stable system is. The basic Routh–Hurwitz test only checks for poles on or to the right of the $j\omega$ axis. To find out whether all roots lie to the left of a line $s=-\sigma$, the axis is shifted. #### Method (axis shifting) 1. Take the characteristic equation $q(s)=0$. 2. Substitute $s = z - \sigma$, where $\sigma>0$ is the required margin. This moves the vertical line $s=-\sigma$ to the new imaginary axis ($z=0$). 3. Expand to get a polynomial $q(z)$. 4. Form the Routh array of $q(z)$. - Number of sign changes = number of roots to the **right** of $s=-\sigma$. - No sign change: all roots lie to the left of $s=-\sigma$, so every root has a real part more negative than $-\sigma$. Each mode then decays at least as fast as $e^{-\sigma t}$. - A zero row (or a zero constant term): some roots lie exactly on the line $s=-\sigma$. 5. Repeat for other $\sigma$ values, or keep $\sigma$ (or a gain $K$) as a variable to find the limit. ```text jw' jw | | jw' = new axis (z = 0) x | | x | | Shift: s = z - sigma -------+-----------+--------> sigma x -sigma 0 | | ``` #### Example $q(s)=s^3+6s^2+11s+6$ (roots $-1, -2, -3$). Check whether all roots lie to the left of $s=-0.5$. Put $s=z-0.5$: $$ q(z) = z^3 + 4.5z^2 + 5.75z + 1.875 $$ | Row | Col 1 | Col 2 | |---|---|---| | $z^3$ | 1 | 5.75 | | $z^2$ | 4.5 | 1.875 | | $z^1$ | 5.333 | | | $z^0$ | 1.875 | | No sign change, so all roots lie to the left of $s=-0.5$. With $s=z-1$, $q(z) = z^3+3z^2+2z$; the constant term becomes zero, showing a root exactly at $s=-1$. #### Uses - Ensures a minimum decay rate, i.e. a settling time of about $t_s \approx 4/\sigma$ or less. - Lets the designer find the range of gain $K$ that keeps all poles to the left of a chosen line. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Relative stability measures A. How far the closed-loop poles lie to the left of the imaginary axis B. Whether the system has zeros C. The number of poles D. The gain at zero frequency Answer: A. It tells how stable a stable system is, i.e. the distance of poles into the LHP. (Key point: Meaning of relative stability) 2. To test whether all roots lie left of s = -sigma, substitute A. s = z + sigma B. s = z - sigma C. s = sigma*z D. s = 1/z Answer: B. With s = z - sigma the line s = -sigma becomes the new imaginary axis z = 0. (Key point: Axis shift) 3. In the Routh array of q(z), the sign changes equal A. Roots on the imaginary axis B. Roots left of s = -sigma C. Roots to the right of s = -sigma D. Number of zeros Answer: C. Sign changes count the roots in the right half of the z-plane, i.e. right of s = -sigma. (Key point: Interpreting sign changes) 4. In the shifted array, an all-zero row means A. All roots are real B. The system is unstable C. sigma is too small D. Roots lie exactly on the line s = -sigma Answer: D. Imaginary-axis roots in z correspond to roots on s = -sigma. 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