Adaptive Prescribed-time Control For Uncertain Nonlinear Systems

Stability,accuracy,and speed are the three key factors for evaluating system performance.Achieving rapid,high-precision convergence of a system is the primary objective pursued by scholars in the control field.Prescribed-time control is capable of stabilizing the controlled object at the origin within a predetermined time frame,independent of the system’s initial conditions and control parameters.This approach finds broad applications in fields such as aerospace,telemedicine,and intelligent driving,making it a subject of considerable academic interest.In practical applications,controlled objects often exhibit nonlinear dynamic characteristics and are subject to parameter uncertainty caused by modeling errors and mechanical wear,posing inherent challenges to system control.Moreover,in specific scenarios,limited network bandwidth and channel capacity necessitate minimizing information exchange to conserve communication resources while ensuring system performance.Given these considerations,this paper investigates the problem of adaptive prescribed-time control for uncertain nonlinear systems under different control inputs,taking into account factors like time delays and time-varying parameters,which impact the design of the system controller and stability analysis.The specific research tasks are as follows:(1)The problem of adaptive prescribed-time control for uncertain nonlinear systems is studied using dynamic gain techniques.By introducing a prescribed-time adjustment function to scale the system’s states,the original system’s prescribed-time stability problem is transformed into a bounded stability problem for the scaled system.Subsequently,a continuous adaptive prescribed-time controller is proposed using dynamic gain techniques.According to Lyapunov stability theory,it is demonstrated that this controller can ensure bounded stability for the scaled system,thereby achieving prescribed-time control for the original system.In comparison to traditional finite/fixed-time control methods,the convergence time in this algorithm is independent of the system’s initial conditions and control parameters and can be pre-determined.(2)The problem of adaptive prescribed-time control for uncertain nonlinear systems is investigated using a non-scaling design approach.Based on a prescribed-time adjustment function,a novel adaptive prescribed-time stability theorem is introduced.Subsequently,for nonlinear systems with unknown parameters,a continuous time-varying state feedback control scheme is proposed using backstepping control and adaptive techniques.Furthermore,utilizing the adaptive prescribed-time stability theorem,it is demonstrated that this scheme can drive the system’s state to converge to the zero equilibrium point within the prescribed time.In contrast to scaling design methods,the non-scaling design approach directly employs the prescribed-time adjustment function in the controller design,without resorting to state scaling or time transformation.Consequently,this approach effectively reduces the order of the prescribed-time adj ustment function,thus alleviating the computational burden on the system and conserving computational resources.(3)The problem of adaptive prescribed-time control for time-delay uncertain nonlinear systems is investigated using Lyapunov-Krasovskii(L-K)functionals.First,the definition of prescribed-time stability for time-delay uncertain nonlinear systems is provided,along with an explanation of the necessary conditions to achieve prescribed-time(including finite-time and fixed-time)stability in these systems.Next,based on the prescribed-time adjustment function and L-K functionals,a prescribed-time stability theorem suitable for time-delay systems is proposed,and the effectiveness of this theorem is discussed under various conditions.Finally,for time-delay uncertain nonlinear systems,a continuous timevarying adaptive controller is designed to achieve prescribed-time stability in the closed-loop system.The impact of different prescribed-time adjustment functions on controller design is also analyzed.(4)The adaptive prescribed-time control problem is studied for uncertain nonlinear systems under the static event-triggering mechanism.First,based on the limit convergence theorem of abnormal integrals and contradiction method,a new prescribed-time stability criterion is proposed,which provides explicit theoretical guidelines for addressing the studied problem.Further,controllers and event-triggering mechanisms are collaboratively designed through the application of adaptive technology and the incorporation of backstepping control idea.Subsequently,building upon the proposed prescribed-time stability criterion,it is demonstrated that the designed controller can effectively regulate the system to the origin within the predetermined time interval,eliminating the occurrence of the Zeno phenomenon.This advancement enhances the earlier prescribed-time stabilization results,which were primarily reliant on continuous control signals,ultimately leading to resource-efficient communication.(5)The adaptive prescribed-time control problem is investigated for uncertain nonlinear systems with time-varying parameters under the dynamic event-triggering mechanism.Firstly,a novel approach to directly demonstrate the prescribed-time stability criterion is presented by applying L’Hopital’s rule.Furthermore,by introducing a dynamic signal into the triggering condition,a novel event-triggering mechanism is proposed,which can effectively reduce the number of trigger moments compared to the traditional event-triggering mechanism.Then,a new event-based adaptive control strategy is presented to achieve prescribed-time stability of nonlinear uncertain systems with time-varying parameters by skillfully utilizing a key lemma and the new event-triggering mechanism,while avoiding the Zeno phenomenon.Moreover,the relationships between the design parameters of the event-triggering mechanism,the trigger time interval and system performance are also characterized.