Design of mixed H 2 =H 1 control systems using algorithms inspired by the immune system Frederico G. Guimara ˜es a , Reinaldo M. Palhares a, * , Felipe Campelo b , Hajime Igarashi b a Department of Electronics Engineering, Federal University of Minas Gerais, Av. Anto ˆnio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil b Research Group of Informatics for System Synthesis, Graduate School of Information Science and Technology, Hokkaido University, Kita 14 Nishi 9, Kita-ku, Sapporo 060-0814, Japan Received 24 November 2006; received in revised form 16 February 2007; accepted 20 February 2007 Abstract We utilize optimization algorithms inspired by the immune system for treating the mixed H 2 =H 1 control problem. Both precisely known systems and uncertain systems with polytopic uncertainties are investigated. For the latter, a novel methodology is proposed to compute the worst case norms within the polytope of matrices. This methodology consists in defining the worst case norm computation as an implicit optimization problem with a special structure. We exploit this structure of the problem for its solution. The paper presents both mono and multiobjective optimization algorithms devel- oped from the clonal selection principle. The former is the real-coded clonal selection algorithm (RCSA) and the latter is the multiobjective clonal selection algorithm (MOCSA). The complete design process involves the combination of synthe- sis and analysis. The RCSA is used for analysis, through the worst case norm computation for a given provided controller. The MOCSA is used for synthesis, working on a population of candidate controllers, until providing an estimate of the Pareto set for the mixed H 2 =H 1 control problem. The numerical examples illustrate the power and the validity of the proposed approach for robust control design. Moreover, our approach for worst case norm evaluation is compared with other approaches available in literature. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Robust control; Mixed H 2 =H 1 control, Polytope-bounded uncertainty; Artificial immune systems; Multiobjective optimiz- ation 1. Introduction The mixed H 2 =H 1 control problem is an important problem in control system design [31–34,25]. Also, this problem is still a very difficult multiobjective optimization problem, which may be characterized by (i) 0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.02.046 * Corresponding author. Tel.: +55 31 3409 3457; fax: +55 31 34994850. E-mail address: palhares@cpdee.ufmg.br (R.M. Palhares). Information Sciences xxx (2007) xxx–xxx www.elsevier.com/locate/ins ARTICLE IN PRESS Please cite this article in press as: F.G. Guimara ˜es et al., Design of mixed H 2 =H 1 control systems using ..., Informat. Sci. (2007), doi:10.1016/j.ins.2007.02.046