Special Case 1 of Routh-Hurwitz (RH) Criteria
📌 Special Case 1 occurs when the first element of any row in the Routh array is zero, while the rest of that row contains at least one non-zero element.
💥 The immediate effect of this zero in the first column is that the term in the next row becomes infinite, causing the standard Routh test to fail.
🔢 An example characteristic equation given was $s^5 + 2s^4 + 3s^3 + 6s^2 + 2s + 1 = 0$, which resulted in $0$ in the $s^3$ row, leading to division by zero for the $s^2$ row elements.

Method 1: Substituting Epsilon ($\epsilon$)
1️⃣ This method requires replacing the zero element in the first column with a small positive number, $\epsilon$.
🔄 Complete the entire Routh array using $\epsilon$, and then examine the sign changes in the first column by taking the limit as $\epsilon \rightarrow 0$.
📉 For the example, after applying the limit, the first column of the modified array showed two sign changes (from positive to $-\infty$ and then to $+1.5$), indicating the system is unstable with two poles in the Right Half Plane (RHP).

Method 2: Inverse Polynomial Method
1️⃣ This approach involves replacing $s$ with $1/z$ in the original characteristic equation, $f(s)=0$.
➿ The resulting equation, $f(z)=0$, must be rearranged into decreasing powers of $z$ after taking the Least Common Multiple (LCM).
📊 The stability is then analyzed by forming the Routh array for the new characteristic polynomial in $z$; for the example, this also resulted in two sign changes, confirming instability.

Key Points & Insights
➡️ When the RH test fails due to a zero in the first column, utilize either the $\epsilon$ substitution or the inverse polynomial method to proceed with stability analysis.
➡️ The number of sign changes observed in the first column of the final Routh array (either $\epsilon$-based or $z$-based) directly corresponds to the number of system poles located in the Right Half of the $s$-plane.
➡️ In the provided unstable example, both methods concluded that the system has two poles in the RHP.

📸 Video summarized with SummaryTube.com on Nov 12, 2025, 13:44 UTC