Industrial Applications of Genetic Algorithms:Genetic Algorithms for H<SUB>2</SUB> Controller Synthesis

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RESULTS

Using the example of the four disk system, the compensator gain matrix was found, and the minimum cost functional value was 0.0699905. The objective of the GA was to find solutions that contain the matrix elements deviations of the original gain matrix that minimize the cost functional to a value of 0.0699905. For all three cases, the GA was run for 35 generations, using two-point crossover and a mutation operator that changes one of the parameters of a string based on a uniform probability distribution. The results that follow are the fitness values of the best string, averaged from ten different runs, plotted against the number of function evaluations.

For the first case, the exact gain matrix elements were within the range of ± 5% different from the initial gain matrix that was used. An initial population of 50 strings was randomly selected, where each parameter represented the amount of deviation between ± 5% different than the exact gain matrix elements. The results of this case are displayed in Figure 3.4, showing that the GA obtained a fitness value that was less than one percent larger than the exact minimal cost functional value. The second and third cases contained an initial population of 100 strings with each parameter representing the amount of deviation between ± 10% for the second case and ± 25% for the third case. The results of the second case are displayed in Figure 3.5. The results show that the GA obtained a fitness value that was less than one percent larger than the exact minimal cost functional value. The results of the third case, shown in Figure 3.6, reveal that the GA did not perform as well as the other cases, obtaining a fitness value that was about six percent larger than the exact minimal cost functional value.

Figure 3.4 GA performance for 5% deviation.

Figure 3.5 GA performance for 10% deviation.

Figure 3.6 GA performance for 25% deviation.

Figure 3.7 Disk structure impulse response.

The results of using a GA for H2 compensator synthesis produced solutions that fulfilled the requirement of closed-loop stability. Figure 3.7 shows the impulse response of the four disk system for the extreme case when the initial gain matrix is deviated within the range of ± 25%. Output responses are displayed for the initial gain matrix used with the dynamic compensator, and the gain matrix obtained by the GA used with the dynamic compensator. As shown, the response of the system using the dynamic compensator with the initial gain matrix is unstable and diverges quickly. However, the response of the system using the dynamic compensator with the gain matrix obtained from the GA displays a stabilizing effect.

ing the dynamic compensator with the gain matrix obtained from the GA displays a stabilizing effect.

CONCLUSIONS

The results of this study show that the GA led to near-exact solutions, which lowered the value of the cost function, well beyond that which would have been obtained by a random search procedure. The GA obtained fitness values that were less than one-percent of the exact minimal cost functional value for gain matrices that were within the range of ± 10%, and fitness values that were about six-percent of the exact minimal cost functional value for the extreme case where the gain matrices were on the range of ± 25%. Also, the requirement of closed loop stability was maintained up to the extreme case of a ± 25% deviation of the exact gain matrix. Furthermore, the GA was able to begin with an unstable initial solution, and find a near-optimal stable solution. This is a very important aspect for compensator design since traditional methods have problems with unstable initial conditions. From these studies, the GA has proven to be a valid algorithm to determine the gain matrix of a dynamic compensator for H2 compensator synthesis.

ACKNOWLEDGMENTS

I would like to express my appreciation to Dr. Michael Freeman and Dr. Charles Karr for their expertise, support, and encouragement. I would also like to thank Dr. Mark Whorton of NASA/Marshall Space Flight Center for his help and expertise during this research project. This research has been supported by the AIAA/Francois-Xavier Bagnoud Foundation and the Department of Aerospace Engineering and Mechanics at the University of Alabama.

REFERENCES

  1 Whorton, M., Calise, A., and Buschek, H. (1996). Homotopy algorithm for fixed order mixed H2/H∞ design. Journal of Guidance, Control, and Dynamics, 19(6, November-December), 1262-1269.
  2 Results of the STABLE Microgravity Vibration Isolation Flight Experiment. (1998). Internet, http://zaphod.msfc.nasa.gov/∼httpser/papers/design/whorton5.html.
  3 Sharkey, J., Nurre, G., Beals, G., and Nelson. (1992). A chronology of the on-orbit pointing control system changes on the Hubble space telescope and associated pointing improvements. Proceedings of the 1992 AIAA Guidance, Navigation and Control Conference, Aug. 10-12, Hilton Head Island, SC.
  4 Seinfeld, D., Haddad, W., Berstein, D., and Nett, C. (1991). H2/H∞ controller synthesis: Illustrative numerical results via quasi-Newton methods. Proceedings of the 1991 American Control Conference, Boston, MA, June 26-28, 1155-1156.
  5 Goldberg, David E. (1989). Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison-Wesley.

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