Dynamic Surface Control Of Incommensurate Nonlinear Fractional Order Systems

With more and more practical physical systems being found to have fractional order features,fractional order system theory has gradually become a focal point of research in the field of control.Research has shown that incommensurate fractional order models describe actual systems more accurately than commensurate fractional order models.Therefore,it is of great significance to research the control issue of incommensurate fractional order systems.Considering the widespread existence of nonlinear phenomena in practical applications,this dissertation conducts in-depth research on the control problem of incommensurate nonlinear fractional order systems(INFOSs).It is well known that the backstepping control(BC)method has a systematic design structure,and the control scheme combined with the fuzzy adaptive method can effectively solve the control problem of uncertain nonlinear systems.However,the traditional BC has two defects in theory:firstly,an explosion of computational complexity appears when BC technique is applied to high-dimensional systems;secondly,BC is only applicable to strict feedback systems,which degrades its practical value seriously.In the face of complex network systems and higher task requirements,the control effect of BC is also unsatisfactory.For integer order systems,the abovementioned issues have been solved to some extent,while it is an enormous challenge to extend these techniques to INFOSs.Therefore,from the perspectives of theory and application respectively,on the one hand,this dissertation makes up for the theoretical deficiency of BC for INFOSs;the other hand,considering the problems of system state constraints,transient performance,and state unmeasurable,it enriches the application scenarios of BC of INFOSs.Firstly,a fuzzy adaptive dynamic surface control(FADSC)method is proposed for strict-feedback INFOSs.The fuzzy logic system is utilized to approximate the uncertain dynamics,and the fractional order filter is designed to avoid the computational explosion caused by repeatedly calculating the derivative of the virtual control signal in the traditional backstepping process.The indirect Lyapunov approach is adopted to analyze the stability of the closed-loop system,the discussion of complex relationship between the real state and the pseudo-state is bypassed,and we can conclude that the pseudo-state is uniformly bounded.Considering that the error between the filtered signal and the virtual control signal will affect the control effect of the system,the error compensation mechanism is introduced.The command filtering controller is designed,and the error compensation effect is illustrated by simulation results.Secondly,two FADSC schemes are presented for nontriangular structure uncertain INFOSs.When the direction of the partial derivative of the system’s nonlinear dynamics to the virtual control variable is known and bounded,the differential mean value theorem is applied to separate the virtual control variable.The FADSC strategy is designed to ensure that all signals of the closed-loop system are bounded.A dynamic controller is proposed for the system which does not match the above assumptions.The control input is regarded as a new state variable,and the augmented system is transformed into a non-strict feedback form through model transformation.A novel adaptive control law is designed with the aid of the properties of fuzzy membership function.The problem that traditional dynamic surface control cannot be applied to nontriangular structure systems will be solved.Thirdly,static and dynamic event-triggered FADSC schemes are designed for nontriangular structure uncertain INFOSs.It is also proved that the designed dynamic event trigger mechanism requires less data transmission times than the corresponding static trigger conditions,which can further save network resources.A nonlinear fuzzy state observer is designed to estimate the unmeasurable state of the system.Based on the output feedback and event trigger mechanism,a FADSC strategy is designed to ensure that all signals in the closed-loop system are bounded and that the system has a minimum transmission time interval.Finally,for the nontriangular structure uncertain INFOSs,the prescribed tracking performance and full state constraints are considered respectively.Coordinate transformation is carried out for the tracking error,so that when the new variable is bounded,the tracking error of the closed-loop system can reach a satisfactory convergence rate and predefined control accuracy.By using an important fractional order differential inequality,the FADSC process is designed to make the tracking error achieve preset performance.In order to deal with the full state constraint problem of the system,the state transformation is designed based on the nonlinear state relevant function.The system state constraint issue is transformed into the boundedness problem of the new state variable.A FADSC algorithm is designed with the aid of the important fractional order differential inequality,so that the feasibility condition of the virtual control variable is removed.