# GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance

### Gustavo Sánchez, Miguel Strefezza, Minaya Villasana

#### Abstract

One of the most relevant problems for control engineers is the so-called “mixed H2/H∞”. To solve it, different convexifying strategies became popular in the later 1990s, mainly based on Linear Matrix Inequalities (LMIs). On the other hand, genetic algorithms have also been applied for H2/H∞ synthesis. Indeed, several authors agree that they are able to find good solutions to this important control problem. However, in most of the published papers, only low-order SISO models have been considered. In the present paper a LMI-based algorithm is compared against a genetic algorithm, with respect to three performance indicators: Set Coverage, Maximum Distance and Efficient Set Spacing. Five open-loop MIMO models extracted from COMPleib are studied, for which the degree varies between 5 and 10. Based on numerical results, the genetic algorithm is not able to improve LMI solutions for problems with more than 42 variables, restricted to a budget of 20.000 function evaluations.

#### References

- Apkarian, P., Noll, D., and Rondepierre, A. (2008). Mixed H 2/H ¥ control via nonsmooth optimization. SIAM Journal on Control and Optimization, 47(3):1516- 1546.
- Astrom, K., Panagopoulos, H., and Hagglund, T. (1998). Design of PI controllers based on non-convex optimization. Automatica, 34(5):585-601.
- Leibfritz, F. (2004). COMPlib - Constrained Matrixoptimization Problem Library. Technical report, University of Trier. Department of Mathematics, Germany.
- Sánchez, G., Villasana, M., and Strefezza, M. (2007). Multi-Objective Pole Placement with Evolutionary Algorithms. Springer-Verlag. Berlin. Series: Lecture Notes in Computer Science, 4403:417-427.
- Scherer, C., Gahinet, P., and Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control, 42(7):896-910.
- Takahashi, R., Palhares, R., Dutra, D., and Goncalves, L. (2001). Synthesis and characterization of Paretooptimal solutions for the mixed H2/Hinf problem. Proceedings of the 40th IEEE conference on decision and control. Orlando,FL. USA, pages 3997-4002.
- Takahashi, R., Palhares, R., Dutra, D., and Goncalves, L. (2004). Estimation of Pareto sets in the mixed H 2/H ¥ control problem. International Journal of System Science, 35(1):55-67.

#### Paper Citation

#### in Harvard Style

Sánchez G., Strefezza M. and Villasana M. (2011). **GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance** . In *Proceedings of the International Conference on Evolutionary Computation Theory and Applications - Volume 1: ECTA, (IJCCI 2011)* ISBN 978-989-8425-83-6, pages 282-285. DOI: 10.5220/0003640702820285

#### in Bibtex Style

@conference{ecta11,

author={Gustavo Sánchez and Miguel Strefezza and Minaya Villasana},

title={GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance},

booktitle={Proceedings of the International Conference on Evolutionary Computation Theory and Applications - Volume 1: ECTA, (IJCCI 2011)},

year={2011},

pages={282-285},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003640702820285},

isbn={978-989-8425-83-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Evolutionary Computation Theory and Applications - Volume 1: ECTA, (IJCCI 2011)

TI - GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance

SN - 978-989-8425-83-6

AU - Sánchez G.

AU - Strefezza M.

AU - Villasana M.

PY - 2011

SP - 282

EP - 285

DO - 10.5220/0003640702820285