Routh-Hurwitz Criterion for System Stability Analysis

The Routh-Hurwitz criterion is a mathematical test used to determine the stability of a linear time-invariant (LTI) control system by examining the roots of its characteristic equation. A system is considered stable if all the roots of its characteristic equation have negative real parts. The Routh-Hurwitz criterion allows us to determine this without explicitly calculating the roots.

Characteristic Equation and Routh Array Setup

The given characteristic equation for the closed-loop control system is:

$$\text{s}^3 + 2.4\text{s}^2 + 1.8\text{s} + 0.5 = 0$$

To apply the Routh-Hurwitz criterion, we first construct the Routh array. The coefficients of the characteristic equation are arranged in a specific pattern to form the initial two rows of the array.

The given incomplete Routh array is:

Calculating Routh Array Elements (s¹ Row)

We need to calculate the values for 'a' and 'c' for the s¹ row. These are calculated using the elements from the two rows above it (s³ and s² rows).

• Calculation for 'a':

  The formula for 'a' is:

  $$a = \frac{(2.4 \times 1.8) - (1 \times 0.5)}{2.4}$$

  Let's perform the calculation:

  $$a = \frac{4.32 - 0.5}{2.4}$$

  $$a = \frac{3.82}{2.4}$$

  $$a \approx 1.59166$$

  Rounding to two decimal places, we get:

  $$a \approx 1.59$$

• Calculation for 'c':

  The formula for 'c' is derived similarly, using the next column of coefficients (which are effectively zeros beyond the given terms):

  $$c = \frac{(2.4 \times 0) - (1 \times 0)}{2.4}$$

  $$c = \frac{0 - 0}{2.4}$$

  $$c = 0$$

Calculating Routh Array Elements (s⁰ Row)

Next, we calculate the values for 'b' and 'd' for the s⁰ row. These are calculated using the elements from the s² and s¹ rows.

• Calculation for 'b':

  The formula for 'b' is:

  $$b = \frac{(a \times 0.5) - (2.4 \times c)}{a}$$

  Substitute the calculated values of 'a' and 'c':

  $$b = \frac{(1.59166 \times 0.5) - (2.4 \times 0)}{1.59166}$$

  $$b = \frac{0.79583 - 0}{1.59166}$$

  $$b = \frac{0.79583}{1.59166}$$

  $$b = 0.5$$

• Calculation for 'd':

  The formula for 'd' is:

  $$d = \frac{(a \times 0) - (2.4 \times 0)}{a}$$

  $$d = \frac{0 - 0}{a}$$

  $$d = 0$$

Final Routh Array

After calculating all the unknown elements, the complete Routh array is:

Conclusion on Routh Array Values

Based on our calculations using the Routh-Hurwitz criterion, the values for a, b, c, and d are:

• $$a \approx 1.59$$
• $$b = 0.5$$
• $$c = 0$$
• $$d = 0$$

For a system to be stable, all the elements in the first column of the Routh array must have the same sign (and generally, all positive for common engineering systems). In this completed array, the first column elements are 1, 2.4, 1.59, and 0.5, all of which are positive, indicating a stable system.