Optimal Control For Constrained-input Nonlinear Systems With State Delays

In recent years,optimal control theory has been widely used in industrial production,economic management,and defense and military fields.In addition,the impact of state delay and control constraints on system stability is also a key consideration.Therefore,this article discusses the adaptive optimal control problem of a constrained input system with state delay.Firstly,for two types of systems,two algorithms are used:an online strategy iteration algorithm(PI)is proposed for continuous time constraint input systems with state delay,and a greedy iterative algorithm based on approximate dynamic programming(ADP),that is,heuristic dynamic programming(HDP)for discrete cases.Secondly,two algorithm structures are constructed by using neural networks(NNs)respectively,and the action-critic neural network is used to approximate bellman’s equation and its corresponding optimal control inputs.Third,by calculating the time difference error(TDE),the parameter adjustment rate of the neural network is derived.Fourth,using the Lyapunov theoretical analysis method,it proves that the system state vector and the weights of the critic neural network is uniformly ultimately bounded(UUB),that is,the new stability conditions of the system.In contrast to the existing literature,the existing literature only considers state delays,and there is little discussion of the amplitude limits that control inputs.These literature results do not guarantee that the control input will remain within the allowable range,which can lead to performance degradation and may even make the system unstable.The results obtained in this article apply to systems that have both state delay and constraint inputs.Finally,numerical examples are used to verify the effectiveness of the proposed method.