Approximation-Free Prescribed Performance Control for Unknown SISO Pure Feedback Systems Charalampos P. Bechlioulis and George A. Rovithakis Abstract— A universal control scheme is designed for un- known pure feedback systems, capable of guaranteeing, for any initial system condition, output tracking with prescribed performance and bounded signals in the closed loop. In this paper, 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. The proposed approximation-free and low-complexity control scheme isolates the output performance from the control gains selection and exhibits strong robustness against model uncertainties. In fact, any system in pure feedback form obeying certain controllability assumptions can be controlled by the proposed scheme without altering either the controller structure or the control gain values. Finally, a simulation study clarifies and verifies the approach. I. I NTRODUCTION During the past several years, adaptive control of systems possessing complex and unknown nonlinear dynamics has attracted considerable research effort. Significant progress has been achieved through adaptive feedback linearization [1], adaptive backstepping [2], control Lyapunov functions [3] and adaptive neural network/fuzzy logic control [4]. The aforementioned results were obtained for systems in affine form, that is, for plants linear in the control input variables. However, there exist practical systems such as chemical processes and flight control systems, which cannot be expressed in an affine form. The difficulty associated with the control design of such systems arises from the fact that an explicit inverting control design is, in general, impossible, even though the inverse exists. Initially, nonaffine systems in low triangular canonical form (i.e., system nonlinearities satisfy a matching condition) were considered. Subsequently, as the problem became more apparent, the significantly more complex as well as general class of pure feedback nonaffine systems (i.e., all system states and control inputs appear implicitly in the system nonlinearities) was tackled. In case of single-input single-output nonaffine systems with unknown nonlinearities, fuzzy systems and neural networks have been used to approximate an ‘ideal controller’, whose Charalampos P. Bechlioulis is with the School of Mechanical Engineer- ing, National Technical University of Athens, Athens, 15780, Greece. Email: chmpechl@mail.ntua.gr George A. Rovithakis is with the School of Electrical & Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece. Email: robi@eng.auth.gr The work of the second author was co-financed by the EU-ESF and Greek national funds through the operational program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF)-Research Funding Program ARISTEIA I, project PIROS. existence is guaranteed by the Implicit Function Theorem. Works incorporating the Mean Value Theorem [5]–[13], the Taylor series expansion [14] and the contraction mapping method [15], [16] have been proposed to decompose the original nonaffine system into an affine in the control part and a nonaffine part representing generalized modeling errors. Subsequently, standard robust adaptive control tools were employed. However, approximating this “ideal controller” is a difficult task, leading also to complex neural network and fuzzy system structures. In [17], [18], instead of seeking a direct solution to the inverse problem, an analytically invertible model was introduced and a neural network was designed to compensate for the inversion error. Finally, in [19], singular perturbation theory was applied to derive an adaptive dynamical inversion method for uncertain nonaffine systems. Despite the recent progress in the control of unknown nonaffine systems, certain issues still remain open. First, it should be noticed that all aforementioned works have resorted to approximation-based techniques to deal with the model uncertainties of the system. Unfortunately, this approach inherently introduces certain issues affecting closed loop stability and robustness. Specifically, even though the existence of a closed loop initialization set as well as of certain control gain values that guarantee closed loop stability is proven, the problem of proposing an explicit constructive methodology capable of a priori imposing the required stability properties is not discussed. As a consequence, the produced control schemes yield inevitably reduced levels of robustness against modeling imperfections. Moreover, the results are restricted to be local as they are valid only within the compact set where the capabilities of the uni- versal approximators (i.e., neural networks, fuzzy systems, etc.) hold. Furthermore, the introduction of approximating structures increases the complexity of the proposed control schemes in the sense that extra adaptive parameters have to be updated (i.e., nonlinear differential equations have to be solved numerically) and extra calculations have to be conducted to output the control signal, thus making implementation difficult. Finally, all aforementioned works guarantee convergence of the tracking error to a residual set, whose size depends on explicit design parameters and some unknown bounded terms. However, no systematic procedure exists to accurately compute the required upper bounds, thus making the a prio- ri selection of the design parameters to satisfy certain steady