Conclusions

In this paper, we demonstrated how genetic algorithms can be applied to aid robust stability analysis. Our approach only provides a sufficient condition for instability for uncertain nonlinear parametric polynomials. We modified the GA using novel genetic operators and applied the GA to problems that are difficult to solve using conventional methods due to nonlinearities in the coefficients of their characteristic polynomials. Our computational results indicate that GAs can test the sufficiency condition for system instability with a given accuracy. Furthermore, GAs can be applied to a wider class of systems with large numbers of uncertain parameters. This perspective makes GAs rather unique and promising compared with other approaches.

We also note that if the GA search is unsuccessful we cannot make any conclusions about stability or instability. Our last example illustrates this fact. However, the paucity of tools for dealing with nonlinear parametric polynomials makes our genetic algorithm based approach a viable alternative.

A necessary and sufficient condition for robust stability of discrete-time systems is that all roots of its characteristic polynomial lie inside the unit circle. However, to use genetic algorithms to test this stability condition further investigation is needed. It is our belief that further research in this direction is justified.