Question

From the following Routh array table, which tells us that there are

\( S^n \)	Col 1	Col 2	Col 3
\( S^5 \)	2	1
\( S^4 \)	3	2	1
\( S^3 \)	\(-\frac{4}{3}\)	\(-\frac{2}{3}\)
\( S^2 \)	\(\frac{1}{2}\)	1
\( S^1 \)	2
\( S^0 \)	1

• A. One root in the left half s-plane
• B. Two roots in the left half s-plane
• C. Two roots in the right half s-plane
• D. One root in the left half S-plane and one root in the right half S-plane

Hint:

Routh-Hurwitz Criterion: The number of roots of the characteristic equation with positive real parts (in the RHP) is equal to the number of sign changes in the first column of the Routh array.
For a system to be stable, all elements in the first column of the Routh array must be positive (no sign changes).
If a zero appears in the first column (but not the entire row is zero), replace it with a small positive \(\epsilon\) and proceed.
If an entire row is zero, it indicates roots on the \(j\omega\)-axis or symmetrically located roots. Form an auxiliary polynomial from the row above and differentiate.

Answer 1

The Correct Option is C

The Routh-Hurwitz stability criterion uses the Routh array to determine the number of roots of the characteristic polynomial that lie in the right half of the s-plane (RHP). The number of sign changes in the first column of the Routh array corresponds to the number of roots in the RHP.

The first column of the given Routh array is: 

• S^5 : 2
• S^4 : 3
• S^3 : -4/3
• S^2 : 1/2
• S^1 : 2
• S^0 : 1

Let's count the sign changes in this first column:

• From \(S^5\) (2, positive) to \(S^4\) (3, positive): No sign change.
• From \(S^4\) (3, positive) to \(S^3\) (-4/3, negative): One sign change (positive to negative).
• From \(S^3\) (-4/3, negative) to \(S^2\) (1/2, positive): One sign change (negative to positive).
• From \(S^2\) (1/2, positive) to \(S^1\) (2, positive): No sign change.
• From \(S^1\) (2, positive) to \(S^0\) (1, positive): No sign change.

There are a total of two sign changes in the first column.

According to the Routh-Hurwitz criterion, the number of sign changes in the first column of the Routh array is equal to the number of roots of the characteristic equation that are in the right half of the s-plane.

Therefore, there are two roots in the right half s-plane. This means the system is unstable.

The characteristic polynomial is of order 5 (from \(S^5\)), so there are 5 roots in total.

• Number of RHP roots = 2
• Number of LHP roots = Total roots - RHP roots - \(j\omega\)-axis roots. Assuming no \(j\omega\)-axis roots (as no row of zeros occurred and was handled), then LHP roots = 5 - 2 = 3.

The question asks what the table tells us. It tells us there are two roots in the right half s-plane.

This matches option (c).

Final Answer:

Two roots in the right half s-plane