On Adaptive Event-triggered Control For Nonlinear Systems With Constraints

In order to make the system to run stably,some constraints should be imposed on the state or output.Otherwise it will have some negative effects on the dynamic performance of the system.In several cases,the components will be damaged.In addition,compared with the usual time-driven control method,event-trigger control(ETC)has advantaes in reducing energy consumption and occupation rate of transmission bandwidth.However,the traditional ETC is based on the assumption that the closed loop system is input-to-state stable(ISS).How to relax the ISS assumption is a problem that deserves further study.In this paper,some adaptive DSC-ETC methods are proposed for several classes of single-input single-output(SISO)and multiple-input multiple-output(MIMO)nonlinear systems with unmodeled dynamic and time-varying states or output constrains.The main content of this paper is explained as follows:Firstly,an adptive dynamic surface control method is proposed for a class of pure-feedback systems with symmetric full-state time-varying constraints and unmodeled dynamics.By introducing a one-to-one nonlinear mapping,the symmetric full-state time-varying constraints can be handled.Auxiliary signals are designed to solve the problem unmodeled dynamics,respectively.In the stability analysis,it is proved that the close-loop system can be semi-globally uniformly ultimately bounded(SGUUB).A numerical example is provided to demonstrate the effectiveness of the proposed approach.Secondly,an event-triggered adaptive neural dynamic surface control is proposed for uncertain nonlinear strict-feedback systems with asymmetrical full state time-varying constraints and dynamic uncertainties.Using hyperbolic tangent function as invertible mapping,the constrained uncertain strict-feedback nonlinear system is transformed into a novel uncertain system without constraints.A dynamic signal is employed to estimate the dynamical uncertainties.At recursive each step,the unknown smooth nonlinear functions produced in the derivation process are approximated by RBFNNs.Based on the transformed novel system,event-triggered controller is developed by utilizing modified DSC.With the help of defined compact set in the stability analysis,all signals in the adaptive closed-loop system is bounded.Moreover,all states can obey the time-varying constraint conditions.Simulations of two numerical examples are employed to verify and clarify the theoretical findings.Thirdly,an event-triggered adaptive neural control is proposed for uncertain MIMO nonlinear systems with dynamic output constraints and static state constraints as well as dynamical uncertainties.By introducing a nonlinear invertible mapping with hyperbolic tangent function,the original system is converted into a novel unconstrained MIMO system.In recursive each step,the unknown smooth nonlinear function vectors are estimated by RBFFNs.A dynamic signal is employed to estimate the dynamical uncertainties in the novel system.Based on the transformed system,event-triggered adaptive controller is formed by a modified dynamic surface control technique.Exploiting the defined compact set,the SGUUB of all signals in the adaptive closed-loop system is proved.Furthermore,all the output and states can strictly comply with the defined constraint conditions.Simulations of two numerical examples are adopted to verify and clarify the theoretical results.Fourthly,an adaptive neural event-triggered control method is proposed for uncertain block-structure MIMO constrained non-affine nonlinear systems with unmodeled dynamics.A dynamic signal produced by the auxiliary system based on the property of unmodeled dynamics is employed to solve the dynamical disturbances.The unknown continuous function obtained at each step of recursion is estimated by using RBFNNs.Utilizing logarithmic function as an invertible mapping,the uncertain constrained MIMO non-affine system is changed into a novel unconstrained block-structure MIMO non-affine system.Using improved DSC strategy,adaptive ETC scheme is developed for the transformed non-affine system based on relative threshold mechanism.According to the Lyapunov method,all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded.A constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are used to illustrate the theoretical findings.