Nonlinear Dyn (2009) 56: 13–22 DOI 10.1007/s11071-008-9375-x ORIGINAL PAPER Robust adaptive nonlinear control for uncertain control-affine systems and its applications Hamid R. Koofigar · Saeed Hosseinnia · Farid Sheikholeslam Received: 16 March 2008 / Accepted: 18 May 2008 / Published online: 11 June 2008 © Springer Science+Business Media B.V. 2008 Abstract This paper addresses the robust tracking control problem for a class of uncertain nonlinear sys- tems with time-varying parameters, perturbed by ex- ternal disturbances. The unknown time-varying para- meters and disturbances are neither required to be pe- riodic nor to have known bounds. Depending on the characteristics of disturbance signals, two adaptive- based control algorithms are developed. First, an adap- tive H ∞ control is designed that achieves: (i) an H ∞ tracking performance when the external disturbances are L 2 signals, and (ii) the convergence of tracking er- ror to zero if the disturbances are bounded and L 2 sig- nals. Then a novel adaptive control algorithm is pro- posed, only with the assumption of boundedness of disturbances, to drive the tracking error to zero. The designed tracking controllers are then used for con- trolling a cart-pendulum system, as an underactuated mechanical system, and chaos synchronization of un- certain Genesio–Tesi chaotic system. Numerical simu- lations are also given to demonstrate the effectiveness of the proposed control schemes. H.R. Koofigar · S. Hosseinnia · F. Sheikholeslam ( ) Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156, Iran e-mail: sheikh@cc.iut.ac.ir H.R. Koofigar e-mail: koofigar@ec.iut.ac.ir S. Hosseinnia e-mail: hoseinia@cc.iut.ac.ir Keywords Chaos synchronization · Robust adaptive control · Time-varying parameters · Tracking control · Uncertain nonlinear systems 1 Introduction Designing robust tracking control for uncertain non- linear systems is considered as a challenging problem in the field of control. In many applications, a nom- inal model can be derived for the system, but model uncertainties and parameter variations cause the de- sired performance not be achieved. In the past years, considerable research efforts have been devoted to tackle this problem. Developing tracking controllers for servo systems [6, 28], magnetic levitation [29], some classes of chaotic systems [5, 15], cart pendulum system [17, 23], optical disk drives [14] and a class of underactuated mechanical systems [11] are sam- ples of tremendous efforts devoted to practical con- trol problems. The assumptions made on the system uncertainties motivate researchers to propose various tracking control methodologies. Among the reported methods, adaptive-based control techniques are pow- erful tools, especially when the variations of unknown parameters are slow enough [1, 12, 18]. In fact, con- ventional adaptive methods including adaptive control laws together with some parameter adjusting mecha- nisms may fail for the case of arbitrarily fast time- varying perturbations. Investigating into this field, sev- eral results have been reported when the time-varying