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The Routh-Hurwitz criterion is a mathematical method used to determine the stability of a linear time-invariant system. It involves analyzing the characteristic equation of the system to assess whether all roots have negative real parts. This article provides an overview of the method and practical examples to illustrate its application.
Understanding the Routh-Hurwitz Criterion
The Routh-Hurwitz criterion uses the coefficients of the characteristic polynomial to construct a Routh array. The stability of the system depends on the signs of the first column of this array. If all elements in the first column are positive, the system is stable. Any sign change indicates the presence of roots with positive real parts, leading to instability.
Constructing the Routh Array
To build the Routh array, follow these steps:
- Write the coefficients of the characteristic polynomial in the first two rows.
- Calculate the remaining rows using determinants based on the above rows.
- Analyze the first column for sign changes.
Practical Example
Consider the characteristic equation: s^3 + 2s^2 + 3s + 4 = 0. Construct the Routh array:
First row: 1, 3
Second row: 2, 4
Remaining rows are calculated as follows:
Row 3: (2*3 – 1*4)/2 = (6 – 4)/2 = 1
Row 4: 4 (copy of the last coefficient)
The first column is: 1, 2, 1, 4. Since all are positive, the system is stable.