On Robust Adaptive Control Of Nonlinear Systems With Dynamic Uncertainties

In the past few years, with the development of control theories, control research of nonlinear system has attracted wide attention of scholars both at home and abroad. Especially adaptive dynamic surface control technique, combined adaptive control and dynamic surface technology, has provided a new approach to the research of nonlinear systems. Time-delays and unmodeled dynamics exist in many practical nonlinear systems, and they are frequently a source of instability and performance degradation. Therefore, research on nonlinear systems with time-delays and unmodeled dynamics has very important theoretical significance and practical application value. Using RBF neural network to approximate unknown continuous function, some adaptive control methods are presented for several classes of nonlinear systems with dynamic uncertainties by combining Lyapunov stability theory and adaptive control as well as dynamic surface control. The main contributions are as follows:Firstly, by utilizing backstepping method, the dynamic surface control technique and the approximation capability of neural networks and the Lyapunov function of integral type, an adaptive neural network control is proposed for a class of nonlinear systems in pure feedback form. The proposed approach contains only one adaptive parameter that needs to be updated online, and cancels the assumption of the supper bound of the optimal approximation error. Based on the above system, considering the case with unmodeled dynamics, an adaptive control scheme is proposed by giving relevant assumptions. The proposed scheme loosens the restrictions for control system by removing the assumption with respect to the derivative of the virtual control coefficients, and relaxes the assumptions with respect to unmodeled dynamics by combining Young’s inequality and RBF neural network technique. The Approximation for the unknown continuous function mainly use two methods, system function direct approximation and black-box function approximation after the reasoning by applying Young’s inequality, compared with the two algorithm, the effect and advantages and disadvantages can be seen clearly. By theoretical analysis, the closed-loop systems are proven to be stable with the tracking error converging to a small neighborhood of the origin. Simulation results are given to illustrate the effectiveness of the proposed control scheme.Secondly, adaptive neural network dynamic surface control is presented for a class of nonlinear systems in strict-feedback form with unmodeled dynamics. The design scheme makes the approach of dynamic surface control be extended to the strict-feedback nonlinear systems with unmodeled dynamics, and broadens the extent of application of the approach of dynamic surface control. The effects of unmodeled dynamics is overcome by utilizing the defined compact set in dynamic surface control design.Using Young’s inequality, two adaptive parameter tuning schemes are developed. Compared with the existing results, the proposed approach reduces the number of adjustable parameters effectively, and relaxes the condition of dynamic uncertainties, and does not require the derivative of the virtual control coefficients. By theoretical analysis, the closed-loop control system is shown to be seme-globally uniformly ultimately bounded, with the tracking error converging to a small neighborhood of the origin. Simulation results are given to illustrate the effectiveness of the proposed control scheme.Thirdly, based on backstepping, an adaptive neural control scheme is proposed for a class of perturbed strict-feedback nonlinear systems with unknown state time-varying delays and unmodeled dynamics. By constructing appropriate Lyapunov-Krasovskii functionals, the upper bound functions of the time-varying delay uncertainties are relaxed. Using RBF neural networks to approximate the unknown nonlinear functions, the controller singularity problem was resolved by constructing continuous approximation functions.The virtual control gain was effectively dealt with by utilizing Young’s inequality. It is demonstrated that the proposed controller guarantees that all the signals in the closed-loop system are semi-global uniformly ultimately bounded and the tracking error eventually converges to a neighborhood of zero. Simulation results are given to illustrate the effectiveness of the proposed control scheme.Lastly, using the technique of RBF neural network and the sliding mode control method, a novel adaptive neural network control scheme is proposed for a class of MIMO nonlinear systems. Compared with the existing literature, the proposed approach deals with several uncertainties, including unmodeled dynamics, time-varying delay uncertainties, completely unknown control directions, and unknown dynamic disturbances.Using appropriate Lyapunov-Krasovskii functionals not only effectively compensates for the delayed states, but also relaxes the assumption of the upper bound functions of time-varying delayed uncertainties. Moreover, the controller does not require the delay integral term, which significantly alleviates the computational burden. By theoretical analysis, the closed-loop control system is proved to be semi-globally ultimately bounded. Simulation results are given to illustrate the effectiveness of the proposed control scheme.