Example 3

We now consider a simple example that has been specifically constructed to mislead the genetic algorithm approach. Consider a fourth order $z^{\prime}$polynomial

$$\begin{eqnarray*}\lefteqn{ P(z^{\prime}, {\bf q}) = a_0 + a_1z^{{\prime}} + a_2z... ...q_3\\ & & a_3 = q_1^{2}q_2^{2}q_3^{3}\\ & & a_4 = -q_1q_2q_3 \end{eqnarray*}$$

The system's uncertain parameter ranges are
$Q = \{ {\bf q}: \: q_1\in[1.4; 3.0], \: q_2\in[1.0; 2.8], \: q_3\in[1.8; 3.2], \: q_4\in[1.2; 2.8]\}$. We constructed this polynomial to stress that our approach only provides sufficiency conditions. A genetic algorithm using the same GA parameters as in the previous two examples was unable to find any roots on or within the unit circle even after 1000 generations. We thus cannot conclude anything about the system's stability from our approach and will need to use some other tool for stability analysis - in this case, an analytic solution is possible.