Available online at www.sciencedirect.com Automatica 39 (2003) 2157–2167 www.elsevier.com/locate/automatica BriefPaper Newresultsfornear-optimalcontroloflinearmultiparameter singularlyperturbedsystems  HiroakiMukaidani a ; ∗ ,HuaXu b ,KoichiMizukami c a Graduate School of Education, Hiroshima University, Kagamiyama Higashi-Hiroshima, Hiroshima 739-8524, Japan b Graduate School of Business Sciences, The University of Tsukuba, Otsuka, Bunkyou-ku, Tokyo 112-0012, Japan c Faculty of Engineering, Hiroshima Kokusai Gakuin University, Nakano, Aki-ku, Hiroshima 739-0321, Japan Received 6 November 2001; received in revised form 30 June 2003; accepted 18 July 2003 Abstract In this paper, we consider the linear quadratic optimal control problem for multiparameter singularly perturbed systems in which N lower-level fast subsystems are interconnected through a higher-level slow subsystem. Dierent from the existing methods, a new method is developed to design a near-optimal controller which does not depend on the unknown small parameters. It is shown that the resulting controller in fact achieves an O(‖‖ 2 ) approximation to the optimal cost of the original optimal control problem. ? 2003 Published by Elsevier Ltd. Keywords: Multiparameter singularly perturbed systems (MSPS); Multiparameter algebraic Riccati equation (MARE); Near-optimal control; -independent controller 1. Introduction The deterministic and stochastic multimodeling stability, control,lteringanddynamicgameshavebeeninvestigated extensively by several researchers (see e.g., Khalil, 1979, 1980, 1981; Khalil & Kokotovi c, 1978, 1979a, b;  Ozg uner, 1979; Salman, Lee, & Boustany, 1990; Coumarbatch & Gaji c, 2000a, b; Gaji c, 1988; Gaji c & Khalil, 1986; Wang, Paul, & Wu, 1994). In order to obtain the optimal solution to the multimodeling problems, we must solve the multi- parameter algebraic Riccati equation (MARE), which is parameterized by the small positive same order parame- ters  j ;j =1;:::;N . Various reliable approaches for solv- ing the MARE have been well documented in literatures (see e.g., Coumarbatch & Gaji c, 2000a, b; Mukaidani, Xu, & Mizukami, 2002). However, a limitation of these  This paper was partially presented at IFAC workshop on singular solutions and perterbations, Bucharest, October 2001. This paper was recommended for publication in revised form by Associate Editor Thor I. Fossen under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +81-824-24-7155; fax: +81-824-24-7155. E-mail addresses: mukaida@hiroshima-u.ac.jp (H. Mukaidani), xuhua@gssm.otsuka.tsukuba.ac.jp (H. Xu), mizukami@cs.hkg.ac.jp (K. Mizukami). approaches is that the small parameters are assumed to be known. Thus, it is not applicable to a large class of prob- lemswheretheparametersrepresentsmallunknownpertur- bations whose values are not known exactly. On the other hand, although it is well known that a popular approach to deal with the multiparameter singularly perturbed systems is the two-time-scale design method (see e.g., Kokotovi c, Khalil, & O’Reilly, 1986; Wang et al., 1994), the existing controller only achieves O(‖‖) (where ‖‖ denotes the normofthevector[ 1 ···  N ])approximationoftheoptimal cost. In this paper, we study the linear quadratic optimal con- trol problem for nonstandard multiparameter singularly perturbed systems (MSPS). The considered MSPS is more general compared with Mukaidani and Mizukami (2001) and is based on the specic structure of the lower-level multi-fast subsystems and a higher-level slow subsys- tem (  Ozg uner, 1979). We rst investigate the unique and boundedsolutionoftheMAREandestablishitsasymptotic structure.Usingtheasymptoticstructure,anewnear-optimal controllerwhichdoesnotdependonthevaluesofthesmall parameters is obtained. It is newly shown that the resulting controller achieves O(‖‖ 2 ) approximation of the optimal cost. As another important feature, we prove that the new near-optimal controller is equivalent to the existing one in 0005-1098/$-see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0005-1098(03)00248-6