Question 5.1: Find how many positive roots the following characteristic eq...
Find how many positive roots the following characteristic equation has:
s^{6} – 4s^{5} – 7s^{4} + 44s^{3} – 214s^{2} + 140s + 400 = 0 (5.150)
The Routh array is as follows:
| s^{6} | 1 | -7 | -214 | 400 |
| s^{5} | – 4 | 44 | 140 | 0 |
| s^{4} | \frac{(- 7) \times (-4) – 1 \times 44}{(- 4)} = 4 | \frac{(-214) \times (- 4) – 1 \times 140}{(- 4)} = – 179 | \frac{400 \times (- 4) – 1 \times 0} {(- 4)} = 400 | 0 |
| s³ | \frac{44 \times 4 – (- 4) \times (- 179)}{4} = – 135 | \frac{140 \times 4 – (-4) \times 400}{4} = 540 | 0 | 0 |
| s² | \frac{(- 179) \times (- 135) – 4 \times 540}{(- 135)} = – 163 | \frac{400 \times (- 135) – 4 \times 0}{(- 135)} = 400 | 0 | 0 |
| s | \frac{540 \times – (- 135) \times 400}{(- 163)} = 208.7 | 0 | 0 | 0 |
There are four changes of sign in the left-hand column indicating four roots with positive real parts. In fact, this example of the characteristic function was generated by expanding the left-hand side of the following equation:
(s +1)(s – 2)(s + 4)(s – 5)(s – 1 + 3j)(s – 1 – 3j) = 0 (5.151)
so the roots are – 1, 2, – 4, 5, 1 – 3j and 1 + 3j, with the four roots 2, 5, 1 – 3j and 1 + 3j all having positive real parts. A system with this characteristic equation would clearly be extremely unstable!
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