Theodoridis et al. / J Zhejiang Univ-Sci C (Comput & Electron) 2011 12(1):1-16 1 Journal of Zhejiang University-SCIENCE C (Computers & Electronics) ISSN 1869-1951 (Print); ISSN 1869-196X (Online) www.zju.edu.cn/jzus; www.springerlink.com E-mail: jzus@zju.edu.cn Direct adaptive regulation of unknown nonlinear systems with analysis of the model order problem Dimitrios THEODORIDIS †1,2 , Yiannis BOUTALIS †‡1,3 , Manolis CHRISTODOULOU 4,5 ( 1 Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi 67100, Greece) ( 2 Department of Industrial Informatics, Technological Education Institute of Kavala, Kavala 65404, Greece) ( 3 Chair of Automatic Control, University of Erlangen-Nuremberg, Erlangen 91058, Germany) ( 4 Department of Electronic and Computer Engineering, Technical University of Crete, Chania 73100, Greece) ( 5 Dipartimento di Automatica et Informatica, Politecnico di Torino, Torino 10129, Italia) † E-mail: {ybout, dtheodo}@ee.duth.gr Received June 27, 2010; Revision accepted Aug. 5, 2010; Crosschecked Dec. 6, 2010 Abstract: A new method for the direct adaptive regulation of unknown nonlinear dynamical systems is proposed in this paper, paying special attention to the analysis of the model order problem. The method uses a neuro- fuzzy (NF) modeling of the unknown system, which combines fuzzy systems (FSs) with high order neural networks (HONNs). We propose the approximation of the unknown system by a special form of an NF-dynamical system (NFDS), which, however, may assume a smaller number of states than the original unknown model. The omission of states, referred to as a model order problem, is modeled by introducing a disturbance term in the approximating equations. The development is combined with a sensitivity analysis of the closed loop and provides a comprehensive and rigorous analysis of the stability properties. An adaptive modification method, termed ‘parameter hopping’, is incorporated into the weight estimation algorithm so that the existence and boundedness of the control signal are always assured. The applicability and potency of the method are tested by simulations on well known benchmarks such as ‘DC motor’ and ‘Lorenz system’, where it is shown that it performs quite well under a reduced model order assumption. Moreover, the proposed NF approach is shown to outperform simple recurrent high order neural networks (RHONNs). Key words: Neuro-fuzzy systems, Direct adaptive regulation, Model order problems, Parameter hopping doi: 10.1631/jzus.C1000224 Document code: A CLC number: TP183 1 Introduction It is common knowledge that neural networks and fuzzy inference systems are universal approxi- mators (Hornik et al., 1989; Wang, 1994; Passino and Yurkovich, 1998), in their ability to approximate nonlinear functions to any degree of accuracy if suf- ficient numbers of hidden neurons and fuzzy rules are available. The combination of these two different technologies has given rise to neuro-fuzzy (NF) ap- proaches (Lin, 1995; Nounou and Passino, 2004; Li ‡ Corresponding author c Zhejiang University and Springer-Verlag Berlin Heidelberg 2011 et al., 2009; Theodoridis et al., 2009a; 2009b), which are capable of capturing the advantages of both fuzzy logic and neural networks. Many researchers have been active in the adap- tive control area (Tong and Chai, 1999; Ordonez and Passino, 2001; Diao and Passino, 2002; Ge and Jing, 2002; Haddada et al., 2003; Kim and Bien, 2004; Nounou and Passino, 2004; Yang, 2004; Ioannou and Fidan, 2006; Chemachema and Belarbi, 2007; Zhang et al., 2007). In NF adaptive control, two main approaches are followed. First, we have the indirect adaptive control schemes separated into two steps: (1) the dynamics of the system are identi-