Brief paper A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems ✩ Charalampos P. Bechlioulis a , George A. Rovithakis b,1 a Control Systems Laboratory, School of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Zografou, Athens 15780, Greece b Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece article info Keywords: Nonaffine systems Uncertain systems Global approximation-free control Low-complexity Prescribed performance abstract A universal, approximation-free state feedback control scheme is designed for unknown pure feedback systems, capable of guaranteeing, for any initial system condition, output tracking with prescribed performance and bounded closed loop signals. By prescribed performance, it is meant that the output error converges to a predefined arbitrarily small residual set, with convergence rate no less than a certain prespecified value, having maximum overshoot less than a preassigned level. The proposed state feedback controller isolates the aforementioned output performance characteristics from control gains selection and exhibits strong robustness against model uncertainties, while completely avoiding the explosion of complexity issue raised by backstepping-like approaches that are typically employed to the control of pure feedback systems. In this respect, a low complexity design is achieved. Moreover, the controllability assumptions reported in the relevant literature are further relaxed, thus enlarging the class of pure feedback systems that can be considered. Finally, simulation studies clarify and verify the approach. 1. Introduction During the past several years, controlling systems with com- plex and uncertain nonlinear dynamics has attracted considerable research effort. Significant progress has been achieved through adaptive feedback linearization (Sastry & Isidori, 1989), adaptive backstepping (Krstic, Kanellakopoulos, & Kokotovic, 1995) and adaptive neural network/fuzzy logic control (Farrell & Polycar- pou, 2006; Ge, Hang, Lee, & Zhang, 2002; Lewis, Jagannathan, & Yesildirek, 1999; Rovithakis & Christodoulou, 2000; Spooner, Mag- giore, Ordonez, & Passino, 2002). The aforementioned results were obtained for systems in affine form, that is, for plants linear in the control input variables. However, there exist applications such as chemical processes and flight control systems, which cannot be ex- pressed in an affine form. The difficulty associated with the con- trol design of such systems arises from the fact that an explicit E-mail addresses: chmpechl@mail.ntua.gr (C.P. Bechlioulis), robi@eng.auth.gr (G.A. Rovithakis). 1 Tel.: +30 2310995820; fax: +30 2310996124. inverting control design is, in general, impossible, even though the inverse exists. Initially, nonaffine systems in low triangular canon- ical form (i.e., system nonlinearities satisfy a matching condition) were considered (Ge & Zhang, 2003; Hovakimyan, Lavretsky, & Cao, 2008; Hovakimyan, Nardi, & Calise, 2002; Labiod & Guerra, 2007; Leu, Wang, & Lee, 2005; Park, Huh, Kim, Seo, & Park, 2005; Park, Park, Kim, & Moon, 2005; Wang, Chien, & Lee, 2011; Yang & Calise, 2007; Zhao & Farrell, 2007). Subsequently, as the problem became more apparent, the significantly more complex as well as more general class of pure feedback nonaffine systems (i.e., all system states and control inputs appear implicitly in the system nonlinear- ities) was tackled (Chien, Wang, Leu, & Lee, 2011; Ge & Wang, 2002; Ren, Ge, Su, & Lee, 2009; Wang, Chien, Leu, & Lee, 2010; Wang, Ge, & Hong, 2010; Wang, Hill, Ge, & Chen, 2006; Wang & Huang, 2002; Wang, Liu, & Shi, 2011; Zhang & Ge, 2008; Zhang, Wen, & Zhu, 2010; Zhang, Zhu, & Yang, 2012; Zou, Hou, & Tan, 2008). More specifi- cally, in case of single-input single-output nonaffine systems with unknown nonlinearities, fuzzy systems and neural networks have been utilized to approximate an ‘ideal controller’, whose existence is guaranteed by the Implicit Function Theorem. Works incorpo- rating the Mean Value Theorem (Chien et al., 2011; Ge & Wang, 2002; Ge & Zhang, 2003; Labiod & Guerra, 2007; Ren et al., 2009; Wang, Chien et al., 2011, 2010; Wang, Ge et al., 2010; Wang et al.,