Routh–Hurwitz and Root Locus

Syllabus: Routh-Hurwitz stability criterion; root-locus plots.

6 hand-picked visual resources on this topic, 2 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Routh array, including the two special cases

Necessary condition first: all coefficients present and of the same sign. If any is missing or of opposite sign the system is already unstable and no array is needed. Otherwise the number of sign changes in the first column equals the number of right-half-plane roots.

$\#\text{RHP roots}=\#\text{sign changes in column 1}$
$\text{zero in column 1 only}\Rightarrow\text{replace by }\epsilon\to0^{+}$
then continue and take limits
$\text{row of zeros}\Rightarrow\text{use }\frac{dA(s)}{ds}$
$A(s)$ is the auxiliary polynomial from the row above; its roots are symmetric about the origin.

Where marks are lost

A full row of zeros means roots on the $j\omega$ axis or symmetric pairs — the system is marginally stable at best. Solving the auxiliary polynomial gives the exact $j\omega$ -crossing frequency, which is often what the question actually wants.

See it

Finding the range of $K$

The standard question. Form $1+KG(s)H(s)=0$ , build the array with $K$ symbolic, and force every first-column entry positive. The limiting $K$

Root locus rules that decide answers

The locus is the set of $s$ where the angle condition holds; the gain condition only fixes which point corresponds to which $K$ . Branches start at open-loop poles ( $K=0$ ) and end at open-loop zeros or at infinity ( $K\to\infty$

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Routh–Hurwitz and Root Locus

Syllabus: Routh-Hurwitz stability criterion; root-locus plots.

6 hand-picked visual resources on this topic, 2 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Routh array, including the two special cases

$\#\text{RHP roots}=\#\text{sign changes in column 1}$
$\text{zero in column 1 only}\Rightarrow\text{replace by }\epsilon\to0^{+}$
then continue and take limits
$\text{row of zeros}\Rightarrow\text{use }\frac{dA(s)}{ds}$
$A(s)$ is the auxiliary polynomial from the row above; its roots are symmetric about the origin.

Where marks are lost

See it

Finding the range of $K$

The standard question. Form $1+KG(s)H(s)=0$ , build the array with $K$ symbolic, and force every first-column entry positive. The limiting $K$

Routh–Hurwitz and Root Locus

Routh array, including the two special cases

Finding the range of KKK

Root locus rules that decide answers

Routh–Hurwitz and Root Locus

Routh array, including the two special cases

Finding the range of KKK

Root locus rules that decide answers

Finding the range of $K$

Finding the range of $K$