Fractional Order Sliding Mode Control Of Lagrangian Systems

Due to the development and application of fractional calculus theory in industrial field,people have recognized its superiority and began to synthesize fractional calculus and other advanced applications to seek for better performance,such as high-effective fractional order source management,high-precision and high-speed fractional order control,high-accurate fractional order economics analysis and flexible deployment of space system.Sliding mode theory has been a hottest start-up in the research on fractional calculus,dut to its advantages,such as being insensitive to parameters,robust to disturbance and easy to combine with other algorithms,and many novel applications and improved methods about fractional order sliding mode sprung up.As Lagrangian system is the most popular abstract model in industrial application,the study on its fractional order sliding mode control is burgeoning,and has achieved some remarkable results.To address the fractional order sliding mode control for Lagrangian system,a good idea is to develop the existing integer-order system with fractional calculus to directly obtain the enhancement in performance.Some researchers also tried to design the fractional order system based on the original system's characteristics and the requirement of control performance.The former is a convenient way to lower the cost and shorten development times to yield new fractional order applications,however the scientific achievements about fractional order sliding mode control for Lagrangian system are rare,because of Lagrangian systems' complexity,nonlinearity and interconnection making fractional order sliding surface and control law hard to design.To break through the limits mentioned above and to find a systematic design approach to stabilizing Lagrangian system with fractional order sliding mode control has been a hot topic in fractional order control field.This thesis focuses on fractional order sliding mode controller design to cope with the stabilization of Lagrangian system,and the main research contents are as follows:As everyone knows,the dynamics of finite-time stable system is better than the asymptotically stable system's,and to address the finite-time stability of Lagrangian system,a new terminal sliding mode controller,named as dual terminal sliding mode has been proposed,which can be also constructed as an adaptive law.To eliminate the singularity in the the control law,a fractional order saturation has been designed,and corresponding theoretical analysis about the solution of Lagrangian system has been presented.The fractional order sign function has been utilized to increase the reaching speed.The numerical simulation of rigid manipulator has verified these proposed methods.Input limitation is very common in industrial application,because a real actuator can merely provide bounded force or torque.Considering Lagrangian system with limited inputs,there exists an issue on stability analysis when using Lyapunov's second method,that is the controller command is impossible to be directly expressed,which has been the main obstacle in controller design for systems with limited inputs.To address this issue,a novel adaptive fractional order sliding mode controller has been designed,and the fractional order sliding surface ensures the asymptotical stability of the reduced system,and meanwhile the adaptive law can help to express the entire controller command during the stability analysis.Furthermore,the dynamics of the adaptive law also guarantees that the constraints on the expression of limited inputs are satisfied.To address the stabilization of underactuated Lagrangian system,the uniformly ultimate boundedness of fractional order system has been investigated,and then an nonlinear integer-order sliding mode controller has been presented to prove the solution of the underactuated Lagrangian system is uniformly ultimate bounded.A novel fractional order controller is developed based on the integer-order one,and to yield a continuous input,the fractional order disturbance observer and adaptive law are also synthesized into the controller.The theoretical analysis indicates that the limited input benefits from the adaptive law to make the solution of underactuated Lagrangian system uniformly ultimate bounded.All the proposed methods are tested via numerical simulation on the deployment of space tethered spacecraft system.To address the high-precision motion control of the flat Lagrangian system,a continuous-time adaptive fractional order integral terminal sliding mode controller is proposed based on the dual terminal sliding mode.The adaptive law has been designed to remove the partial loss of precision caused by the state-related uncertainties.The sliding surface can reduce the steady state error effectively.Moreover,to degenerate the error caused by fractional order approximation and digital compute application,a discrete-time fractional order integral terminal sliding mode controller is investigated,and the rigorous analyses about the reaching phase and stable precision of reduced system have both been given out.The proposed continuous-time and discrete-time methods are well tested on the real experimental facility,the linear-motor-based motion control system,and the experimental results indicate that the proposed methods have better performance on response and tracking precision than existing methods.