Relative stability using R-H criterion

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.

         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 $s=z-0.5$:

$$q(z) = z^3 + 4.5z^2 + 5.75z + 1.875$$

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.