Fractional Sliding Mode Control And Simulation For The Spherical Robot Like Underactuated System

Underactuated systems are widely used in practical engineering.Such as bridge crane,inverted pendulum,and surface ship are typical underactuated systems.Compared with actuated systems,underactuated systems have the advantages such as low complexity,lightweight,and low energy consumption.Due to some special characteristics of underactuated systems,the conventional nonlinear control methods for actuated systems are not suitable for the underactuated systems.Among the existing control methods for underactuated systems,the sliding mode control method is insensitive to parameter changes and easy to implement.The features of the sliding mode control method make it play an important role in all the control methods for underactuated systems.With the development of research on fractional order calculus,it has attracted a lot of researchers'attentions.Now the fractional order sliding mode control has been an important point in the area of fractional order control.The combination of sliding mode control and fractional order control can not only further improve the control performance but also further improve the robustness.In this paper,a kind of underactuated system represented by a spherical robot is selected as the research object,and the application of fractional order sliding mode control in this type of underactuated system is studied.The main research work of this paper is as follows(1)Focus on the problem that large overshoot and long adjustment time when the traditional sliding surface is applied to the underactuated system,a new fractional order PI~?D~?sliding surface with a higher degree of freedom is proposed by introducing fractional order differential and fractional order integral operators scientific.Combining the hierarchical sliding mode control method and the proposed fractional order PI~?D~?surface,a fractional order sliding mode controller is designed for the underactuated system.The stability of the whole system is proved by the Lyapunov method and relevant fractional order theory.The conditions of parameters selection are given.The simulation results of linear motion speed control of the spherical robot show the effectiveness and superiority of the proposed controller.Compared to a conventional one,the dynamic performance and robustness of the system are improved significantly.(2)Considering the external disturbances and model error in the practical system.The proposed fractional order PI~?D~?sliding surface is easy to bring input saturation.A new anti-saturation fractional order sliding mode controller is proposed rely on the fast convergence rate of the fractional order PI~?D~?sliding surface.Firstly,a nonlinear disturbance observer is designed.Secondly,an auxiliary system is designed to ensure the stability of the system.Finally,to further improve the control performance,a novel adaptive filter is introduced to pre-process the input signal.The proposed adaptive filter can adjust the parameter with the change of desired control target and output of the controller.The stability of the whole underactuated system is proved with relevant theory.The effectiveness of the controller is verified by simulation.(3)Aiming at the problem that the convergence rate of the nonsingular terminal sliding surface in the existing terminal sliding mode control methods for underactuated systems is slow when state variables near the equilibrium point of the system.A novel fractional order nonsingular fast terminal sliding surface with faster convergence rate is proposed via introducing a fractional integral operator.According to the hierarchical sliding mode control method,a global fast convergent fractional terminal sliding mode controller is designed.The stability of the whole control system is proved.To further accelerate the convergence rate of the proposed fractional order nonsingular fast terminal sliding surface,an improved segmented fractional order nonsingular sliding surface is proposed.Then a controller is designed by combining the improved segmented fractional order nonsingular sliding surface with a nonlinear disturbance observer.The convergence domain and corresponding time of each sliding surface and tracking error are given.The effectiveness and superiority of the two controllers are verified in the spherical robot.(4)To improve the control performance of the underactuated system under the action of conventional integer order fixed time sliding surface,a novel fractional order fixed time sliding surface is proposed by introducing fractional differential operator.The fixed time convergence of the proposed sliding surface is proved,and the upper bound of the settling time is given.Then a new fractional fixed time controller is designed for the underactuated system.The stability of the whole system is obtained.The simulation results in the spherical robot system show that the fractional order fixed time sliding mode controller has better control performance than a traditional one.but there are singularity problems in this fractional order fixed time sliding mode controller.To solve the singularity problem of the fractional order fixed time sliding surface,a fractional order PI~?D~?terminal sliding surface is proposed by introducing a fractional order integral operator in the nonlinear term with singularity and add fractional order differential operator to the high order nonlinear term.The control singularity is solved,and better control performance is achieved at the same time.