Adaptive Dynamic Programming-based Control For A Class Of Nonlinear System

For a class of continuous-time nonlinear systems,this paper adopts adaptive dynamic programming(ADP)method,and considers the logic from a single system to a multi-agent system,from a single input to multiple inputs,and from general to special,respectively.The solutions to approximately optimal control problems are applied to single-input single-output systems,multi-input systems,strict-feedback systems,and multi-agent systems.The main work includes the following four aspects:1)For a class of single-input single-output nonlinear systems,an event-triggered output feedback ADP control strategy is proposed.In the case of incompletely measurable states in a system with disturbances,a neural network-based state observer is designed.For unknown disturbances,a nonlinear disturbance observer is constructed,which only depends on output information of the system.In order to avoid the communication load caused by real-time communication and continuous data transmission,an event-triggered mechanism is introduced in the communication link from a plant to a controller.Combined with state and disturbance observers and event-triggered mechanisms,an event-triggered output feedback ADP control strategy,only including a critic network,is designed.This strategy not only reduces unnecessary actions of the controller,but also reduces the measurement cost for the whole states,and makes performance index reach approximately optimal.Finally,by virtue of Lyapunov function,the stability of the system is analyzed,and all signals of the closed-loop system are bounded.Also,the effectiveness of the proposed control strategy is verified by two simulation examples.2)For a class of multi-input nonlinear systems with input constraints,an event-triggered ADP control strategy based on zero-sum game is proposed.By treating disturbances as disturbance inputs to the system,the optimal control problem of this system is transformed into a multi-player zero-sum game problem,where the performance index consists of multiple control inputs and disturbances.In the case of unknown dynamics,a neural network is used to estimate the unknown dynamics to realize system identification.In order to further save the communication resources from a plant to a controller,an event-triggered function with time-varying gain is introduced.In this function,when the triggered threshold is small enough,the gain will be correspondingly larger,so that the event will be triggered when the norm of the triggered error reaches a larger threshold.Under the ADP framework with only critic network,a switching gain-based event-triggered ADP control strategy is designed,which not only does not depend on the detailed dynamic information,but also enables the performance index of the multi-player system approximately optimal under the worst disturbance.Finally,with the help of the Lyapunov function,the stability of the system is analyzed,and it is proved that all signals of the closed-loop system are bounded,and the Zeno phenomenon does not exist.Also,the effectiveness of the strategy is verified by the simulation results.3)For a class of strict feedback nonlinear systems with the case of known/unknown control gain,an event-triggered tracking control strategy is proposed via backstepping and ADP.For the situation where the control gain is known,a barrier Lyapunov function is introduced to address the state constraints,and the fuzzy logic system is used to estimate the unknown dynamics.Under the framework of backsteeping method,a fuzzy logic system-based feedforward control strategy is designed.Subsequently,the tracking control problem of a strict feedback system with state constraints is transformed into a zero-sum differential game problem of a tracking error system.For the transformed tracking error system,the disturbance is regarded as the disturbance input of the transformed system,and a zero-sum differential game control strategy is designed by ADP technology,where a critic network is established to approximate the solution of the related Hamilton–Jacobi–Isaacs equation.In order to avoid unnecessary updates and waste of resources caused by continuous and real-time communication,an event-triggered mechanism is introduced on the link from a controller to a plant.The designed control strategy makes the full states satisfy the constraints,and eliminates the Zeno phenomenon.Further,for the situation where the control gain is unknown,a new uncertainty term is formed by combining the unknown control gain term with the unknown dynamic term,and the fuzzy logic system is used for estimation.Under the action of the feedforward control,the tracking control problem of a strict feedback system with unknown gain is transformed into a zero-sum differential game problem of a tracking error system.For the transformed tracking error system,it is similar to the above-mentioned situation where the control gain is known.Finally,by virtue of the Lyapunov function,the stability of the system is analyzed.Then,it is obtained that all signals of the closed-loop system are bounded,the output of the system can track the desired signal,and the Zeno phenomenon can be excluded.The effectiveness of this strategy is verified by two simulation results.4)For a class of nonlinear multi-agent systems with input and output constraints,a state consensus distributed ADP control strategy is proposed.The nonlinear mapping function is used to realize system transformation,and the state consensus control problem of multi-agent systems with output constraints is transformed into an unconstrained one.By considering the external disturbance as the disturbance input of the system,the pre-transformation state consensus tracking control problem is transformed into a multi-player zero-sum game problem.For the transformed game problem,a distributed state consensus control strategy is designed.This strategy makes the performance index of the transformed system approximately optimal under the worst case of disturbance.Finally,by analyzing the stability of the system with Lyapunov function,it is proved that all signals of the multi-agent system are bounded,the state of the system can achieve consensus,and the input and output signals of the followers meet the constraints.Simulation results verify the effectiveness of this strategy.