Step-by-step Routh-hurwitz Analysis for a Fourth-order Polynomial System

The Routh-Hurwitz criterion is a fundamental tool in control system engineering used to determine the stability of a system without explicitly calculating its roots. When dealing with a fourth-order polynomial characteristic equation, the Routh-Hurwitz table provides a systematic approach to assess stability efficiently.

Understanding the Characteristic Equation

The characteristic equation of a fourth-order system typically takes the form:

a₄ s⁴ + a₃ s³ + a₂ s² + a₁ s + a₀ = 0

where a_i are real coefficients. To analyze stability, all roots of this polynomial must have negative real parts, which is what the Routh-Hurwitz criterion helps determine.

Constructing the Routh-Hurwitz Table

The process begins by arranging the coefficients into the first two rows of the Routh array:

• First row: a₄, a₂, and any other coefficients with even powers
• Second row: a₃, a₁, and any other coefficients with odd powers

For example, if the polynomial is 2s⁴ + 3s³ + 4s² + 5s + 6, the table starts as:

First row: 2, 4, 6

Second row: 3, 5, 0

Completing the Routh-Hurwitz Table

Next, calculate the elements of the subsequent rows using determinants based on the previous two rows. The general formula for the element b_i,j is:

b_i,j = (determinant of a 2×2 matrix) / (element of the previous row)

This process continues until all rows are filled. The key is to check the first column of the table.

Determining System Stability

Once the table is complete, analyze the first column:

• If all the elements in the first column are positive, the system is stable.
• If any element is zero or negative, the system is unstable or marginally stable.

This method allows engineers to quickly assess stability without solving for roots explicitly, saving time and effort in control system design.