Adaptive Backstepping Control Of Uncertain Systems

System uncertainty means that mathematic models which describe systems and their environment are not completely certain, some unknown or random elements may be con-tained. Objectively speaking, every plant in practice has some uncertainty to some extent. Indeterminacy may appear both inside and outside of systems. Internal system uncertain-ties generally refer to unknown structure and parameters of mathematic models describing objects which include unmodeled dynamics, unknown plant parameters and control coef-ficients. While external system uncertainties usually come from unpredictable time-delay of actuators and stochastic disturbances, for example, unknown deterministic constant de-lay in control input and measurement noise possessing unknown statistical character. If we omitted these indeterminacies when we design control schemes, controlled objects may not be up to required performance, even worse, systems would become unstable. In the face of these objectively existing, various kinds of uncertainties, how to design appropriate control feedback to make closed-loop system stable and achieve expected performance is the target of adaptive control research.In this paper, we focus on a variety of uncertain plants involving uncertain nonlinear time-varying systems, uncertain linear minimum-phase systems, uncertain linear delay sys-tems. The method considered here is widely used adaptive backstepping control. Because of diversity of the objects at which we aim, traditional adaptive backstepping will lose its effect. Instead, we make many improvements on backstepping and combine it suitably with other control approaches. Some meaningful research results are listed as follows.(1) This paper presents a modular-based adaptive control scheme for parametric strict feedback nonlinear time-varying systems. The parameters considered include both contin-uous and piece-wise time-varying parameters and they are not necessarily restricted to be slowly time-varying or infrequent jumping. The control design module and the parameter estimation module are totally independent. It is proved that the uniform boundedness of all closed-loop system signals can be guaranteed with the proposed control scheme. The performance of the tracking error in the mean square sense with respect to the parameter variation rate is also established.(2) This paper presents a variation on adaptive backstepping output feedback con-trol design for uncertain minimum phase linear systems. Unlike the traditional nonlinear design, the proposed control method is linear and Lyapunov-based without utilizing over-parametrization, tuning functions or nonlinear damping terms to address parameter estima-tion error. Local stability of the closed-loop system and trajectory tracking are guaranteed. If the system dimension equals to the relative degree, the global stabilization and asymp-totic convergence are achieved.(3) In this paper, concentrating on a class of linear plants whose relative degree equals to system dimension, we develop a Lyapunov-based control scheme to achieve trajec-tory tracking despite some classic difficulties including unmeasurable system state, un-known plant parameters and unknown input time-delay. A comprehensive approach com-bining adaptive backstepping output feedback with prediction-based boundary control is employed in the design. The stability analysis exhibits the global boundedness of all closed-loop system signals and the tracking performance is also guaranteed.