Routh Array Example

The Routh array, a fundamental concept in control systems engineering, is used to determine the stability of a system by analyzing its characteristic equation. Developed by Edward John Routh in the late 19th century, this method provides a straightforward way to assess the stability of a system without solving the characteristic equation directly. Let’s dive into an example to understand how the Routh array works.

Consider a system with the following characteristic equation: [s^4 + 2s^3 + 3s^2 + 4s + 5 = 0]

To construct the Routh array, we start by arranging the coefficients of the polynomial in a specific pattern, where the first two rows are filled with the coefficients of the polynomial, and then we calculate subsequent rows based on a set of rules.

1. The first row consists of every other coefficient starting from the highest degree of (s), which for our polynomial would be 1, 3, 5.
2. The second row consists of the remaining coefficients, which are 2, 4.
3. For each subsequent row, the first element is calculated as the determinant of a 2x2 matrix formed by the first two elements of the two rows immediately above it, divided by the first element of the row above it. The rest of the elements are calculated similarly but using the appropriate elements from the rows above.

Here’s how the array looks initially:

s^4 | 1  3  5
s^3 | 2  4

To calculate the next row (for (s^2)), we apply the Routh rule: - The first element is ((2*3 - 1*4)/2 = (6 - 4)/2 = ²⁄₂ = 1). - The second element is ((2*5 - 1*0)/2 = ¹⁰⁄₂ = 5), but since there’s no (s^1) term in the second row to interact with the (s^3) row for our calculation, we simply use the formula directly for each element needed.

Thus, the updated array is:

s^4 | 1  3  5
s^3 | 2  4
s^2 | 1  5

Continuing this process for (s^1) and (s^0), we calculate: - For (s^1): Using the formula, the first element is ((1*4 - 2*5)/1 = (4 - 10)/1 = -⁶⁄₁ = -6). Since we’re calculating the row below (s^2) and there’s only one element in the (s^2) row that we use for calculation (because the pattern repeats every two rows), the second element would not be calculable in a traditional sense without additional terms, but in practice, you’d look at the remaining terms and the pattern of calculation, which might involve zeros or other coefficients not directly provided in the initial setup for a complete analysis.

Given the initial setup and focusing on the methodological example, our array so far looks like this, with the understanding that actual calculations for subsequent rows depend on precise application of the formula and availability of terms:

s^4 | 1  3  5
s^3 | 2  4
s^2 | 1  5
s^1 | -6

Analysis

In a typical Routh array analysis, the presence of any zeros in the first column or sign changes indicates the existence of roots of the characteristic equation on the imaginary axis or in the right half of the s-plane, respectively, which can imply instability or marginal stability of the system.

For simplicity and adherence to the example’s scope, we’ve illustrated the initial steps of constructing a Routh array. In practice, each row’s calculation depends on the previous rows, and specific rules dictate how to handle special cases such as zeros or negative numbers appearing in the first column of any row.

Conclusion

The Routh array offers a systematic approach to assessing the stability of control systems by evaluating the roots of the characteristic equation without explicitly solving it. Through a step-by-step calculation process, engineers can determine the presence of unstable roots, which is crucial for designing and analyzing control systems. While this example simplifies the process for illustrative purposes, the Routh array method is a powerful tool in the field of control systems engineering.

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