Design Of Command Filtered Backstepping Control For Uncertain High-Order Strict-Feedback Systems

The real world is predominantly governed by fundamental physical laws such as Newton’s Laws,Theorem of Linear and Angular Momentum,Lagrange’s Equations,and Kirchhoff’s Laws.As a result,the original models of many systems are naturally secondorder,and complex models that are composed of these models are naturally high-order.While most physical models are comprised of second-or high-order differential equations,these models are often transformed into first-order systems and designed within the framework of first-order state-space approaches,despite the fact that its contribution to controller design is limited.In practical applications,nonlinear systems often exhibit uncertainties such as disturbances,unmodeled dynamics,or complex systems that make it difficult to fully understand their dynamics.This study focuses on a class of secondand high-order strict feedback systems(SFSs)with uncertainties,using high-order fully actuated(HOFA)system approaches to directly design controllers without converting the system into a first-order state space.This greatly reduces the number of steps required in the backstepping approach.Moreover,to address the problem of the differential explosion that occurs in the traditional designed first-order backstepping method and it also appears in high-order backstepping,a high-order command filtering backstepping(CFB)controller is designed.Controllers are designed for systems with parameter uncertainties,nonlinear function uncertainties,and both types of uncertainties.The main research content of this thesis includes:A high-order robust command filtered backstepping(HORCFB)controller is designed for second-order and high-order SFSs with dynamic uncertainties.Although CFB has achieved fruitful research results,its processing method so far has been to transform the physical model into a first-order state-space form,ignoring its original fully-actuated characteristics under high-order differential equations.The HORCFB controller proposed for second-order and high-order SFSs with dynamic uncertainties has the following characteristics: by fusing HOFA system approaches and robust CFB technique,the HORCFB controller is constructed directly using the inherent full-actuated characteristics of the high-order system,and the high-order subsystem does not need to be transformed into a first-order subsystem,which greatly reduces the steps of backstepping and avoids the differential explosion phenomenon caused by repeated differentiation of virtual control laws.Due to the fact that the HORCFB scheme is an improvement of the first-order CFB approach,it has the problem of imprecise estimation of the derivatives of virtual control law.To address this issue,a differentiator based on high-gain observer(HGO)is proposed for the estimation problem of virtual control law,where the form of the high-gain observer as a differentiator has structural similarities to the form of the integral chain of the high-order full-actuated subsystem,and it is a novel method that truly fits the highorder lower-triangular system to solve its differential explosion.Moreover,by adjusting the corresponding parameters of the differentiator based on HGO,the estimation error can be sufficiently small.When the relevant parameters of the HGO tend to infinity,the estimation error will tend to zero.Finally,the proposed method for improving the estimation of derivatives of virtual control law at each order is applied to the trajectory tracking control problem of a single-link mechanical arm with an elastic rotary joint driven by a DC motor,and simulation results demonstrate the effectiveness of the proposed improved scheme.Based on the theory of HOFA system approaches,a class of high-order adaptive CFB(HOACFB)controllers is designed for second-order and high-order SFSs with parameter uncertainties.The proposed HOACFB scheme differs from the HORCFB approach in its organizational structure,introducing high-order filters to generate compensation signals required for computing compensation tracking errors.Although the introduced dynamics render the original stability analysis framework inapplicable,since it is impossible to select a Lyapunov function containing all dynamic states.This research can also avoid the problem of complexity explosion caused by repeated differentiation of virtual control law while achieving direct design based on the high-order SFSs to reduce the number of backstepping steps.Theoretically,it is rigorously proved that under these two situations,the states of the closed-loop system under the proposed high-order controller are ultimately uniformly bounded,and the tracking error can be made arbitrarily small by adjusting the pre-designed parameters.Traditional adaptive control methods require regression vectors to satisfy persistent excitation(PE)conditions in order to converge to their true values and PE is often difficult to be verified online.Thus,a novel concurrent learning-based high-order adaptive CFB(CL-HOACFB)control is proposed by skillfully combining the techniques of concurrent learning(CL),adaptive CFB,and HOFA system approaches.For many online applications,it is usually impossible to monitor whether the signal is persistently excited since the external reference input is event-based and its prior information is unknown.Therefore,for many adaptive control applications,parameter convergence cannot be guaranteed in practice.The key feature of the proposed parameters update law is that it only requires the linear independence of the related matrix of recorded data and the rank of its data matrix to satisfy certain conditions,which greatly relaxes the requirement of PE in previous methods.Additionally,the convergence of the system is theoretically proven,and the tracking error can be made arbitrarily small using pre-designed parameters.Finally,a benchmark example in electromechanical systems is presented to demonstrate the effectiveness and practical potential of the proposed approach.This study developed a high-order robust adaptive CFB(HORACFB)controller to address systems with both parametric and dynamic uncertainties.Additionally,the study considered the problem of unknown parameter convergence to the actual values.Compared to systems with only parametric uncertainties,systems with dynamic uncertainties in addition to parametric uncertainties introduce challenges to parameter estimation since the parameterized model is perturbed by disturbance or noise.Methods that avoid the use of high-order derivative information,which are constructed solely to address parametric uncertainties,are no longer applicable in this case.To address this issue,the study introduced new filtering variables to construct a new perturbed parameterized model.Furthermore,a robust adaptive control law was designed to achieve trajectory tracking control,and the idea of CL was tailored to construct the parameter update law,which enables the system parameters to converge to their true values.