Question

For the following Routh's array, the system is

• A. Unstable
• B. Stable
• C. Marginally stable
• D. Conditionally stable

Hint:

For Routh-Hurwitz problems, always inspect only the first column. \[ \text{Number of sign changes} = \text{Number of right-half-plane poles} \] No sign change \(\Rightarrow\) Stable One or more sign changes \(\Rightarrow\) Unstable Entire row of zeros \(\Rightarrow\) Check for marginal stability.

Answer 1

The Correct Option is A

Concept:& nbsp;
According to the Routh–Hurwitz stability criterion, the stability of a system can be determined by examining the signs of the elements in the first column of the Routh array.

• If all the elements in the first column have the same sign, the system is stable.
• Each sign change in the first column corresponds to one pole in the right-half of the \(s\)-plane.
• The presence of one or more right-half-plane poles implies that the system is unstable.

Step 1: Write the first column of the Routh array.
From the given Routh table, \[ \begin{array}{|c|c|} \hline \text{Row} & \text{First Column Element} \\ \hline s^3 & 1 \\ s^2 & 2 \\ s^1 & -4.5 \\ s^0 & 9 \\ \hline \end{array} \] Thus, the sequence of signs is \[ +,\quad +,\quad -,\quad +. \]& nbsp;

Step 2: Count the sign changes.
Moving down the first column: \[ +\rightarrow+ \qquad \text{(No sign change)} \] \[ +\rightarrow- \qquad \text{(First sign change)} \] \[ -\rightarrow+ \qquad \text{(Second sign change)} \] Therefore, the total number of sign changes is \[ \boxed{2}. \]& nbsp;

Step 3: Determine the number of right-half-plane poles.
According to the Routh–Hurwitz criterion, \[ \text{Number of sign changes} = \text{Number of poles in the RHP}. \] Hence, \[ \boxed{\text{Number of RHP poles} = 2}. \]& nbsp;

Step 4: Conclude the stability.
Since the system has two poles in the right-half \(s\)-plane, \[ \boxed{\text{The system is unstable}.} \] A stable system must have all poles located in the left-half plane.& nbsp;

Final Answer:
\[ \boxed{\text{System is Unstable}} \] Hence, the correct option is \[ \boxed{\text{(A)}} \]