Introduction to Routh-Hurwitz (RH) Criteria
📌 The RH criteria is a famous method to determine the stability of a control system by analyzing the roots (poles) of the characteristic equation without solving high-order polynomials.
📐 The generalized closed-loop transfer function is $\frac{N(s)}{D(s)}$, where the characteristic equation is $D(s) = a_0s^n + a_1s^{n-1} + \dots + a_n = 0$.
😥 For high-order equations (e.g., 8th order, $n=8$), finding $n$ roots (closed-loop poles) is practically impossible, making the RH test essential.

Necessary Conditions for Stability
🧐 The characteristic polynomial must satisfy two initial necessary conditions to potentially be stable:
1. Sign Consistency: All coefficients ($a_0, a_1, \dots, a_n$) must have the same sign (all positive or all negative).
2. No Vanishing Coefficients: All powers of $s$ from $s^n$ down to $s^0$ must be present; no coefficient can be zero (vanish).
✅ If a polynomial satisfies both conditions, it is called a Hurwitz polynomial; otherwise, the system is declared unstable immediately.

Routh-Hurwitz Test Fundamentals
📜 The RH criterion is a combination of work by Edward Routh (1876) and Adolf Hurwitz (1895).
📊 The test requires tabulating the coefficients into an array, known as the Routh array.
🛑 Stability is achieved if and only if all terms in the first column of the Routh array have the same sign (zero sign changes). The number of sign changes equals the number of poles in the Right Half Plane (RHP).

Method of Forming the Routh Array
⚗️ The first two rows of the array are filled directly using the coefficients of the characteristic equation ($a_0, a_2, a_4, \dots$ for the first row; $a_1, a_3, a_5, \dots$ for the second row).
🔢 Subsequent elements ($b_1, c_1, \dots$) are calculated using a determinant-like formula based on the two preceding rows, dividing by the first element of the row above the one being calculated.
- For the row $s^{n-2}$ (elements $b_1, b_2, \dots$): $$b_1 = \frac{a_1 a_2 - a_0 a_3}{a_1}$$
- For the row $s^{n-3}$ (elements $c_1, c_2, \dots$): $$c_1 = \frac{b_1 a_3 - a_1 b_2}{b_1}$$

Stability Analysis Examples
🌟 Example 1: $s^3 + 6s^2 + 11s + 6 = 0$
✅ Passed necessary conditions (all coefficients positive and non-zero).
📉 The resulting Routh array's first column was $[1, 6, 10, 6]$, showing zero sign changes.
🏆 Conclusion: The system is stable (all poles in the LHP).

💥 Example 2: $s^3 + 4s^2 + s + 16 = 0$
✅ Passed necessary conditions.
📈 The Routh array's first column yielded coefficients that resulted in two sign changes (e.g., from $4$ to $-3$, and from $-3$ to $16$).
📉 Conclusion: The system is unstable, possessing two poles in the RHP.

Key Points & Insights
➡️ Before constructing the Routh array, always verify the necessary conditions (same sign coefficients and no missing powers of $s$); failure means immediate instability.
➡️ The RH criteria allows determining system stability without calculating the actual roots of the characteristic equation, a major computational advantage for higher-order systems.
➡️ The number of sign changes in the first column of the Routh array directly reveals the number of unstable poles (poles located in the Right Half Plane).
➡️ Coefficients $a_0, a_1, a_2, \dots$ must be arranged alternately in the first two rows of the Routh table ($\mathbf{a_0, a_2, a_4, \dots}$ in row 1; $\mathbf{a_1, a_3, a_5, \dots}$ in row 2).

📸 Video summarized with SummaryTube.com on Nov 12, 2025, 13:39 UTC