Nonlinear Tracking Control Based On The Fuzzy And Neural Network Approximation

In practice,most systems have nonlinear characteristics and uncertainties.Recently,the control problem of uncertain nonlinear systems has been widely concerned by different researchers,and many mature control methods have been proposed,which are represented by fuzzy system,neural network,backstepping,dynamic surface,adaptive control and so on.However,there are still many unsolved problems,such as the traditional high-order control systems still rely on recursive design ideas,and the strong constraint condition that the control coefficient does not cross zero is required to deal with the problem of unknown control direction,etc.Aiming at a class of uncertain nonlinear systems,this paper mainly studies the problems of single-system tracking control and multi-agent-system cooperative tracking control(containment control).The main contents are as follows:(1)For the uniformly ultimately bounded tracking control problem,the design method of indirect adaptive fuzzy tracking control for a class of non-affine nonlinear pure-feedback systems is studied.Through the system transformation,the non-affine pure-feedback system is transformed into an integral chain system.An adaptive fuzzy controller is designed by using T-S fuzzy system and state observer.Compared with the traditional backstepping design method,the proposed design method does not need a complex recursive design process,and the obtained control input signal is smooth.The stability of the closed-loop system is proved by using the Lyapunov theory,and uniformly ultimately bounded tracking performance is obtained and the sufficient condition for the asymptotic convergence of tracking error is given.Finally,the effectiveness of the control method is verified by two examples.(2)For the finite-time tracking control problem,the design method of finite-time tracking control for a class of uncertain non-affine nonlinear pure-feedback systems is studied.By using a low-pass filter,a non-affine system is transformed into a strict-feedback system with an affine input.In order to ensure the dynamic and steady-state performance of tracking error,a new performance function is defined,and a new finite-time output tracking controller is designed combined with adaptive fuzzy control.According to the traditional finite-time stability theory,the tracking error converges to a predetermined range within a finite time.Finally,the effectiveness of the method is verified by two examples.(3)For the problem of unknown control coefficients,the design method of adaptive fuzzy tracking control for a class of non-affine pure-feedback nonlinear systems is studied.The control coefficient and the dynamics of the reference signal are unknown.Firstly,the system is transformed into an affine strict-feedback form,and the unknown functions(including the unknown control coefficient)of the transformed system is approximated by fuzzy systems.Then a positive adaptive parameter is designed based on the fuzzy system,smooth projection and obstacle function,which yields that the fuzzy system in the control law can cross zero.Compared with the traditional methods based on the Nussbaum function,this method does not require the condition that the unknown control coefficient never crosses zero.The stability of the closed-loop system is analyzed theoretically.The effectiveness of the control method is verified by two examples.(4)For the problem of unknown time-varying parameters,the neural network boundary approximation and tracking control design method for a class of uncertain nonlinear systems with unknown time-varying parameters(called unknown nonlinear spatiotemporal systems)are studied.A time-varying parameter extraction method is proposed to separate the unknown time-varying parameters from the uncertain non-linear spatiotemporal functions.Taking the supremum of the Euler norm of the extracted time-varying parameter vector,the unknown nonlinear spatiotemporal function is mapped to an unknown state-based boundary function,and then the neural network is used to approximate the boundary function.Based on the proposed time-varying parameter extraction method,an adaptive neural network tracking controller is designed for a class of uncertain strict-feedback nonlinear spatiotemporal systems,which ensures the asymptotic convergence of the tracking error.Finally,an example is used to verify the effectiveness of the proposed method.(5)For the containment control problem under the directed topology,the design method of distributed fuzzy containment control for a class of non-affine nonlinear pure feedback multi-agent systems is studied.The model of each agent is modeled as an unknown non-affine nonlinear pure-feedback system.In order to avoid the complex design process caused by backstepping,each unknown nonlinear system is transformed into a simple integral model.A distributed adaptive fuzzy containment control method is proposed by constructing fuzzy systems and fuzzy state observers to approximate unknown nonlinear functions and unmeasurable states of the system.The stability of the closed-loop system is analyzed in the sense of Lya-punov,which ensures that the containment error converges to zero in a neighborhood of zero.Simulation results show that the proposed scheme is effective.(6)For the containment control problem under switching topologies,the design method of distributed fuzzy containment control for a class of uncertain non-affine nonlinear multi-agent systems under the condition of jointly connected topology is studied.First,each agent system is transformed into the strict-feedback form,then the unknown nonlinear functions of the transformed system are approximated by fuzzy sys-tems,and then a positive bounded adaptive parameter is designed via the smooth projection.Aiming at considering the formation between leaders and not considering the formation between leaders,the con-tainment control laws are designed respectively.Different from the existing results,this paper gives the condition of average dwell time for the switching topologies.The stability of the closed-loop system is analyzed by using Lyapunov theory.Finally,the corresponding examples are given for the cases of leaders with and without formation to verify the effectiveness of the proposed method.