SME
Process Dynamics & Control · Problem type
Stable or not: the Routh test for a cubic
Use this for “Is the loop with characteristic equation s³ + 6s² + 11s + 6 = 0 stable?”
Diagram
First column 1, 6, 10, 6: each sign change is a pole in the right half-plane.
Tap a worked-out cell to see its cross product.
Routh test a₂a₁ − a₃a₀ = 60
Stability Stability = stable
This is an example. Type your own values to start a new problem.
a₃
Coefficient of s³
example
a₂
Coefficient of s²
example
a₁
Coefficient of s
example
a₀
Constant term
example
a₂a₁ − a₃a₀
Routh test
calculated
Stability
Stability
worked out (not typed)
Formulas
a₂a₁
−
a₃a₀
=
a₂
×
a₁
−
a₃
×
a₀
60 = 6 × 11 − 1 × 6
Stability:
stable
if
a₂
,
a₁
,
a₀
>
0
and
a₂
a₁
−
a₃
a₀
>
0;
on
the
edge
if
a₂
a₁
−
a₃
a₀
=
0;
else
unstable
6 > 0 and 11 > 0 and 6 > 0 and 60 > 0, so stable
Assumptions
•
The characteristic equation a₃s³ + a₂s² + a₁s + a₀ = 0, with a₃ positive.
•
Stable needs every coefficient positive and the Routh s¹ row positive: a₂a₁ > a₃a₀.
•
The s¹ row at 0 means two roots on the imaginary axis: steady cycling.
•
To count the roots in the right half-plane, or find the K at the edge, use the Routh pages of Control Systems.
Step-by-step
WE KNOW
a₃ = 1, a₂ = 6, a₁ = 11, a₀ = 6
FIND
Routh test (a₂a₁ − a₃a₀), stability (Stability)
Step 1 · Find Routh test (a₂a₁ − a₃a₀)
Use a₂a₁ − a₃a₀ = a₂ × a₁ − a₃ × a₀
The
Routh
s
1
row
is
this
test
a₂
,
so
with
a₂
positive
they
share
their
sign.
a₂a₁
−
a₃a₀
=
6
×
11
−
1
×
6
a₂a₁
−
a₃a₀
=
66
−
6
a₂a₁
−
a₃a₀
=
60
Step 2 · Find stability (Stability)
Use Stability: stable if a₂, a₁, a₀ > 0 and a₂a₁ − a₃a₀ > 0; on the edge if a₂a₁ − a₃a₀ = 0; else unstable
The first column of the Routh array keeps one sign: no root crosses into the right half-plane.
6 > 0 and 11 > 0 and 6 > 0 and 60 > 0: stable
Stability
=
stable
if
a₂
,
a₁
,
a₀
>
0
and
a₂a₁
−
a₃a₀
>
0
Stability = stable if 6, 11, 6 > 0 and 60 > 0
Stability = stable
Check
Put the numbers back in. Both sides should match.
60 = 6 × 11 − 1 × 6 ✓
6 > 0 and 11 > 0 and 6 > 0 and 60 > 0, so stable ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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