Control For Pure-feedback Nonlinear Systems

In recent years, control of pure-feedback nonlinear systems has been received much at?tention. From the viewpoint of mathematical model, pure-feedback nonlinear systems repre?sent a class of lower-triangular nonlinear systems, which have a more representative form thanstrict-feedback nonlinear systems. On the other hand, many practical systems, such as mechan?ical systems, aircraft systems, and chemical systems, can be modeled as or transformed intopure-feedback nonlinear systems, which have the cascade and nonaiffne properties. Therefore,the research on control for pure-feedback nonlinear systems is an important task both in con?trol theory and in practice. However, because of the nonaffine structures, control approachesfor strict-feedback nonlinear systems can not be directly extended for pure-feedback nonlinearsystems. To address such challenges, we systematically investigate the control design for pure-feedback nonlinear systems by using a new viewpoint. The main contents of the dissertation arecomposed of the following ifve parts:(I)Control for a class of pure-feedback nonlinear systems with affine inputThis part studies the output tracking problem for a class of afifne-in-control pure-feedbacknonlinear systems. To remove the difficulties arising from the nonaffine structures, a novelbackstepping procedure is proposed. Differing with the standard backstepping approach, in theproposed scheme, the nonaffine functions in each subsystem are treated as the virtual controlvariable of the subsystem. The proposed control algorithm not only ensures the global asymp?totic stability of the tracking error, but also guarantees the boundedness of all the closed loopsignals. By following the new backstepping approach, we further investigate adaptive controlfor uncertain afifne-in-control pure-feedback nonlinear systems. The effectiveness and correct?ness of the proposed schemes are validated through two numerical examples.(II)Control for a class of nonafifne pure-feedback nonlinear systemsThis part investigates the control problem for a class of pure-feedback nonlinear systemswith nonaiffne input. To eliminate the obstacle arising from the nonaiffne input, the originalsystem can be transformed into the augmented aiffne-in-control systems via adding an auxiliaryintegrator By employing the backstepping technique, a state feedback controller with arbitrarilyinitial value of control input is developed. It is shown by theoretical analysis that the proposed control algorithm is effect to ensure global asymptotic tracking, as well as boundedness of allsignals of the closed-loop system. In addition, adaptive output tracking problem is studied for aclass of uncertain nonafifne pure-feedback nonlinear systems. A simulation example is providedto demonstrate the effectiveness of the proposed control schemes.(III)Dynamic surface control for nonaffine pure-feedback nonlinear systemsThis part focuses the dynamic surface control for nonaiffne pure-feedback nonlinear sys?tems. Instead of using the mean value theorem, the original nonafifne-in-control systems canbe transformed into the augmented afifne-in-control systems via adding an auxiliary integrator,by incorporating the dynamic surface control technique and a novel backstepping approach, the“proposed algorithm can eliminate the explosion of”complexity problem inherent in the back-stepping design. The proposed controller ensures the semi-global uniformly ultimately bound?edness of all the closed loop signals, with the tracking error converging to a small neighborhoodof the origin by appropriately choosing design parameters. Furthermore, we investigate adap?tive dynamic surface control for nonaffine pure-feedback nonlinear systems. Finally, simulationresults are shown the effectiveness of the proposed schemes.(IV)Adaptive control with prescribed performance for a class of uncertain pure-feedback nonlinear systemsTwo adaptive control algorithms with prescribed performance are proposed for a class ofuncertain pure-feedback nonlinear systems in this part. In the ifrst scheme, by employing anerror transformation function, the output tracking problem with error constraints can be trans?formed into the unconstrained stabilization problem, then an adaptive controller is proposed viaa novel backstepping technique. In the second algorithm, a simple barrier Lyapunov function isused to guarantee the prescribed performance, and an adaptive backstepping controller is direct?ly obtained. It is proven that the proposed control schemes are suiffcient to ensure the prescribedperformance, as well as the boundedness of all the closed loop signals. Finally, simulation re?sults are shown the effectiveness of the proposed schemes.(V)Robust adaptive control for a class of pure-feedback nonlinear systems with un?known backlash-like hysteresisIn this part, adaptive output tracking problem is investigated for a class of uncertain pure-feedback nonlinear systems with unknown backlash-like hysteresis nonlinearity. By combining the solution properties of the hysteresis model with backstepping approach, a robust adaptivecontrol algorithm is developed without constructing a hysteresis inverse. The proposed controlscheme can also be modified to tackle the bounded disturbances by adaptively estimating theirbounds. It is proven that the designed adaptive controllers can guarantee global stability of theclosed-loop system. Two numerical examples are provided to show the effectiveness of theproposed control schemes.