Pole Locations for the Characteristic Equation via Routh Analysis

We want to count the number of roots (poles) of the characteristic polynomial $p(s)=2s^5+3s^4+2s^3+3s^2+2s+1$ in three regions: the left half-plane (LHP), right half-plane (RHP), and on the $j\omega$-axis (imaginary axis). This is a standard “root-location counting” task solved using the Routh-Hurwitz procedure, extended with the imaginary-axis special case.

Key terms:

• Routh table
• Sign changes
• Imaginary-axis roots

We will:

1. Build the Routh table.
2. Use sign changes in the first column to count RHP poles.
3. Detect and count any $j\omega$-axis poles using the auxiliary polynomial procedure when a row becomes all zeros.

Step 1: Routh table setup

Write $p(s)$ in descending powers:

• $a_5=2,\ a_4=3,\ a_3=2,\ a_2=3,\ a_1=2,\ a_0=1$.

For a 5th-order polynomial, the Routh table has rows for $s^5, s^4, s^3, s^2, s^1, s^0$.

Construct:

• $s^5$ row: $2 \quad\ \ 2 \quad\ \ 1$
• $s^4$ row: $3 \quad\ \ 3 \quad\ \ 0$

Then compute the $s^3$, $s^2$, $s^1$ rows using the standard Routh table recurrence: [ b_i=\frac{c_1 a_2-c_2 a_1}{c_1} ] (in row-to-row form: each entry is built from two rows above).

Carrying out the arithmetic:

Routh table (completed)

[ \begin{array}{c|ccc} s^5 & 2 & 2 & 1\ s^4 & 3 & 3 & 0\ s^3 & \frac{7}{3} & \frac{7}{3} & 0\ s^2 & 3 & 1 & 0\ s^1 & \frac{2}{3} & 0 & 0\ s^0 & 1 & 0 & 0 \end{array} ]

So the first column is: [ \left[2,\ 3,\ \frac{7}{3},\ 3,\ \frac{2}{3},\ 1\right] ]

Step 2: Count poles on the RHP using sign changes

In our first column: [ 2 \to 3 \to \frac{7}{3} \to 3 \to \frac{2}{3} \to 1 ] Every entry is positive, so there are 0 sign changes.

Therefore:

• RHP poles count $=0$.

Step 3: Check for $j\omega$-axis poles

A $j\omega$-axis pole typically appears in Routh analysis when:

• there is a row of all zeros (leading to an auxiliary polynomial), or
• an element in the first column becomes exactly zero (requiring an $\epsilon$ perturbation and an auxiliary polynomial).

In our completed table, the first-column entries are strictly positive ($2,3,7/3,3,2/3,1$), and there is no zero row.

So:

• $j\omega$-axis poles count $=0$.

Step 4: Count LHP poles by complement

The polynomial order is $5$, so total number of poles (counting multiplicity) is 5. With:

• RHP poles = 0
• $j\omega$-axis poles = 0

Thus LHP poles: [ 5 - 0 - 0 = 5 ]

Final counts (answer)

From the Routh table for $p(s)=2s^5+3s^4+2s^3+3s^2+2s+1,$ we conclude:

• Number of poles in LHP: $5$
• Number of poles in RHP: $0$
• Number of poles on the $j\omega$-axis: $0$

This means the characteristic polynomial has all 5 roots strictly in the left half-plane, so the associated closed-loop system (if this is a characteristic polynomial) would be asymptotically stable.