SME
Control Systems · Problem type
Right-half-plane poles by Routh
Use this for “How many roots of s³ + s² + 2s + 8 = 0 lie in the right half-plane?”
Diagram
First column 1, 1, −6, 8: each sign change is a pole in the right half-plane.
Tap a worked-out cell to see its cross product.
This is an example. Type your own values to start a new problem.
s³
+
a₂
s²
+
a₁
s
+
a₀
=
0
b₁
First-column entry of the s¹ row
worked out (not typed)
N
Poles in the right half-plane
worked out (not typed)
Formulas
b₁
=
a₂
×
a₁
−
a₀
a₂
−6
=
1 × 2 − 8
1
N
=
1 −
a₂
÷ |
a₂
|
2
+
1 −
a₂
×
b₁
÷ |
a₂
×
b₁
|
2
+
1 −
b₁
×
a₀
÷ |
b₁
×
a₀
|
2
2
=
1 − 1 ÷ |1|
2
+
1 − 1 × (−6) ÷ |1 × (−6)|
2
+
1 − (−6) × 8 ÷ |−6 × 8|
2
Assumptions
•
The s¹ entry is (a₂a₁ − a₀) ÷ a₂ for s³ + a₂s² + a₁s + a₀, so the first column is 1, a₂, b₁, a₀.
•
Each sign change down the first column is one pole in the right half-plane.
•
No sign changes and no zeros in the column means every pole is in the left half-plane.
•
A 0 in the column (a₂, b₁ or a₀) needs ε, the auxiliary polynomial or a pole at 0, so no count is given here.
Step-by-step
WE KNOW
a₂ = 1, a₁ = 2, a₀ = 8
FIND
first-column entry of the s¹ row (b₁), poles in the right half-plane (N)
Step 1 · Find first-column entry of the s¹ row (b₁)
Use b₁ = (a₂ × a₁ − a₀) ÷ a₂
The
s
1
entry
is
the
2
×
2
cross
product
of
the
two
rows
above,
over
the
s
2
row’s
lead.
b₁
=
1 × 2 − 8
1
b₁
=
−6
Step 2 · Find poles in the right half-plane (N)
Use N = (1 − a₂ ÷ |a₂|) ÷ 2 + (1 − a₂ × b₁ ÷ |a₂ × b₁|) ÷ 2 + (1 − b₁ × a₀ ÷ |b₁ × a₀|) ÷ 2
Each bracket is 1 where the sign flips and 0 where it stays: count the sign changes down the column.
N
=
1 − 1 ÷ |1|
2
+
1 − 1 × (−6) ÷ |1 × (−6)|
2
+
1 − (−6) × 8 ÷ |−6 × 8|
2
Signs down the column: +, +, −, +
N
=
0
+
1
+
1
N
=
2
Check
Put the numbers back in. Both sides should match.
−6
=
1 × 2 − 8
1
✓
2
=
1 − 1 ÷ |1|
2
+
1 − 1 × (−6) ÷ |1 × (−6)|
2
+
1 − (−6) × 8 ÷ |−6 × 8|
2
✓
Topic check
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