Routh Array for Stability Analysis

To determine the number of roots in the right-half of the $s$-plane, we use the Routh-Hurwitz criterion. We construct the Routh array from the characteristic equation:

$s^6+s^5-2s^4-3s^3-7s^2-4s-4 = 0$

Routh Array Construction

The array is formed as follows:

Calculating Array Coefficients

1. $s^4$ Row:
   • $b_1 = \frac{(1)(-2) - (1)(-3)}{1} = \frac{-2+3}{1} = 1$
   • $b_2 = \frac{(1)(-7) - (1)(-4)}{1} = \frac{-7+4}{1} = -3$
   • $b_3 = \frac{(1)(-4) - (1)(0)}{1} = -4$
2. $s^3$ Row:
   • $c_1 = \frac{(1)(-3) - (1)(-3)}{1} = \frac{-3+3}{1} = 0$
   • $c_2 = \frac{(1)(-4) - (1)(-4)}{1} = \frac{-4+4}{1} = 0$
   Special Case: Since $c_1 = 0$, we use the auxiliary polynomial from the row above ($s^4$). Auxiliary Polynomial: $A(s) = b_1s^4 + b_2s^2 + b_3 = s^4 - 3s^2 - 4$. Derivative: $\frac{dA(s)}{ds} = 4s^3 - 6s$. The coefficients are 4 and -6. These replace the $s^3$ row.
   • $c_1 = 4$
   • $c_2 = -6$
3. $s^2$ Row:
   • $d_1 = \frac{(4)(-3) - (1)(-6)}{4} = \frac{-12+6}{4} = -\frac{6}{4} = -1.5$
   • $d_2 = \frac{(4)(-4) - (1)(0)}{4} = \frac{-16}{4} = -4$
4. $s^1$ Row:
   • $e_1 = \frac{(-1.5)(-6) - (4)(-4)}{-1.5} = \frac{9+16}{-1.5} = \frac{25}{-1.5} = -\frac{50}{3}$
5. $s^0$ Row:
   • $f_1 = \frac{(-\frac{50}{3})(-4) - (-1.5)(0)}{-\frac{50}{3}} = -4$

Final Routh Array

Determining Roots in Right-Half Plane

According to the Routh-Hurwitz criterion, the number of roots in the right-half $s$-plane is equal to the number of sign changes in the first column of the Routh array.

The first column elements are: $1, 1, 1, 4, -1.5, -50/3, -4$.

The sequence of signs is: +, +, +, +, -, -, -.

Sign changes:

• From +4 to -1.5 (1st change)
• From -1.5 to $-50/3$ (2nd change)
• From $-50/3$ to -4 (3rd change)

There are 3 sign changes in the first column.

Therefore, the system has 3 roots in the right-half of the $s$-plane.