In this paper combined algorithms for the control of nontriangular nonlinear systems with unmatched uncertainties will be presented. The controllers consist of a combination of Dynamical Adaptive Backstepping (DAB) and Sliding Mode Control (SMC) of first and second order. In order to solve a tracking problem, the DAB algorithm (a generalization of the backstepping technique) makes use of virtual functions as well as tuning functions to construct a transformed system for which a regulation problem has to be solved. The new state is extended by an n−ρth order subsystem in canonical form where n is the order of the original system and ρ is the relative degree. The role of the sliding mode control is to replace the last step of the design of the control law to obtain more robustness toward disturbances and unmodeled dynamics. The main advantages of the second-order sliding mode algorithm are the prevention of chattering, higher accuracy, and a significant simplification of the control law. A comparative study of these first and second order sliding controllers will be presented. [S002-0434(00)02604]

We consider a number of dynamical adaptive backstepping control algorithms for the class of observable non-minimum phase nonlinear continuous uncertain systems (triangular and non-triangular), as well as systems with disturbances which can be converted to the parametric semi-strict feedback form. Nonlinear, sliding and second-order sliding control laws are developed. Adaptive backstepping algorithms with tuning functions, and modular parameter identification approaches are presented. BACK, a Maple symbolic algebra package, has been developed as a tool for the design of dynamical adaptive nonlinear controllers for regulation and tracking tasks. IntroductionThe control of nonlinear systems is very important because there are many practical applications in which it is inappropriate t'or the nonlinear equations to be assumed approximately linear. In particular, if there are some unknown parameters, uncertain dynamics, and external and internal disturbances, classical methods do not provide suitable control.Adaptive control is usually employed for the regulation of uncertain systems, when no information is available about the bounds of the unknown parameters. In the 1990s a new family of adaptive control algorithms, using the backstepping approach, was developed [11,16]. This control scheme allows the systematic design of adaptive controllers for triangular nonlinear systems containing unmatched parametric uncertainty. The various backstepping control design algorithms [10,11,16] compiled in [17], provide a systematic framework for the design of tracking and regulation strategies suitable for large classes of nonlinear systems. These static adaptive backstepping (SAB) algorithms enlarged the class of nonlinear systems controlled via a Lyapunov-based control law to uncertain systems transformable into the parametric strict feedback (PSF) form and the parametric pure feedback (PPF) form. In general, local stability is achieved for systems in the PPF form, whilst global stability is guaranteed for systems in the

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