Example 2

Consider the following characteristic polynomial

$$\begin{eqnarray*}\lefteqn{ P(z,q,k) = a_0 + a_1z + a_2z^2 + a_3z^3 }\\ & & a_0... ...& & a_3 = -\frac{1}{2}[q{e}^{-qT}+ k(1-q{e}^{-qT}-{e}^{-qT})]\\ \end{eqnarray*}$$

where q is a plant parameter and k is the controller gain. The coefficient functions are exponential for the uncertain plant parameter q and affine for the controller gain k. Robustness analysis for such an exponential dependence is difficult using conventional techniques. Let the sampling interval T be unity and the feasible region for the uncertain parameters be ${\bf Q}$ $\{q,k: \; q \in [0.1,2],\; k \in[0.1,2]\}$.

We set all parameters of the GA to be the same as in the first example. We map z to $z^{\prime}$ in the given polynomial by reversing the order of the coefficients then test for the existence of a root on or inside the unit circle. Once again, the tolerance $\epsilon$ is set to $9 \times 10^{-16}$ as a true numeric zero. First, we check for roots on the unit circle. After 58 generations, the tolerance $\epsilon$ converges to 0.00021171124248198 and further improvement is unlikely, as shown in Figure 9. The value of $P(z^{\prime},q,k)$ is $2.111\!E\!-\!4 \pm 3.6812\!E\!-\!11j$. The accuracy is not satisfactory for our requirement, and therefore further search inside the unit circle is necessary. After running 153 generations searching inside the unit circle, the GA finally found a pair of roots at |0.985708500750795|e^{1.57079632679489700j} with q = 0.10000000000000001, k = 1.427358664247987, and $P(z^{\prime}, q,k) = 1.6653\!E\!-\!16 \pm 5.4515\!E\!-\!16j$. Once again, using the GA, we can conclude that the given polynomial is unstable.

  

Figure: The fitness of the polynomial $P(z^{\prime },{\bf q})$ as function of generation (Example 2)