On Adaptive Dynamic Surface Control For Pure Feedback Systems With Unmodeled Dynamics

An actual nonlinear system often has a variety of uncertainties such as unknown system function,unknown control gain and unmodeled dynamics.Among them,unmodeled dynamics may destroy the stability condition of the system and cause the system to be unstable.The adaptive control of uncertain nonlinear systems has been widely studied.Because its system state and system function are inseparable in the nonlinear system with pure feedback form,it is more complicated to study the pure feedback system than the strict feedback form.In recent years,with regard to the design of nonlinear control system,the error constraint or dynamic performance is put forward under the premise of satisfying the stability requirement.It requires the tracking error to satisfy the constant constraint or converge to the preset residual set with a certain convergence rate,which is called output constraint and prescribed performance control.By applying these constraints to measurable states of a strict feedback or pure feedback system,a state constraint problem is formed.In this regard,the current commonly used solution is to introduce model predictive control,error conversion,barrier Lyapunov function and so on.In this paper,the Lyapunov stability theory,the adaptive control theory,the dynamic surface control and the decentralized control technique are combined with the approximation ability of the radial basis function neural network for the adaptive control hotspot problem.And the state or output constraints of the nonlinear system,respectively,proposed several adaptive control program,the main results are as follows:Firstly,for a class of pure-feedback nonlinear systems with state and input unmodeled dynamics and unknown control gain sign,two kinds of adaptive dynamic surface control schemes are proposed by using nonlinear transformation,improved dynamic surface control method and Nussbaum function property.The normalization signal is used to constrain the input unmodeled dynamics,thereby effectively suppressing the resulting perturbations.By introducing dynamic signals,the dynamic uncertainties caused by the state unmodeled dynamics are effectively handled.By adding the nonnegative normalization signal to the whole Lyapunov function,the boundedness of the control signal is effectively handled by the characteristics of the dynamic surface control.The closed-loop control system is proved to be semi-globally uniformly ultimately bounded by theoretical analysis.The effectiveness of the proposed scheme is verified by numerical simulation of inverted pendulum model and third-order system.Secondly,for a class of pure-feedback nonlinear systems with unmodeled dynamics and output constraint as well as state constraints,the pure feedback nonlinear system is transformed into a strict feedback nonlinear system in form by nonlinear transformation.A new adaptive control scheme is proposed by using the modified dynamic surface control method.Using the dynamic signal to deal with unmodeled dynamics,the barrier Lyapunov function(BLF)is introduced at first step or each recursive step,and the corresponding virtual control and adaptive control laws are designed and the closed-loop system is shown to be bounded.The output and state constraints are never violated.The numerical simulations verify the effectiveness of the proposed scheme.Thirdly,a decentralized adaptive control method for a class of pure-feedback coupled system with state unmodeled dynamic and output constraints is proposed by using improved dynamic surface control method.In the first step of the recursive design,the integral Barrier Lyapunov Function(iBLF)is introduced.The virtual controller is designed by using Young's inequality and separation theorem.Based on the compact set from the stability analysis and the stability analysis in the dynamic surface control,the coupling effect term is effectively dealed with.It is proved that the signal in the closed-loop system is semi-global uniformly ultimate bounded and the output signal satisfies the output constraint requirement.The numerical simulations verify the effectiveness of the proposed scheme.