Research On Parametric Control Approaches Of Second-Order Quasi-Linear Systems

The primary precondition of the analysis and design of the system is to establish the model of the dynamical system by using some physical laws,such as Newton's laws of motion,conservation of angular momentum,Lagrangian equation,Euler's equation,and so on,which lead to that the original models of systems are second-or higher-order.The traditional approaches of analysis and design of control systems mostly turn the original model into a first-order system and treat it by the state-space approach.However,this transformation has deficiencies,for example,first-order state-space models no longer possess inherent properties,such as positive definiteness,full-actuation,etc.;the complexity and workload of system design are increased.Therefore,this dissertation focuses on second-order quasi-linear systems which represent a big portion of physical control systems,and proposes parametric design approaches for second-order quasi-linear systems under the second-order framework.The main contents include the following two aspects.One is the controllability of quasi-linear systems.PBH-like criteria for the controllability of quasi-linear systems are proposed for standard,descriptor,and high-order cases.The other is a series of parametric design approaches of control laws for second-order quasi-linear systems,including displacement plus acceleration state feedback,output feedback,and dynamic output feedback.The main results consist of five parts.Firstly,the controllability of quasi-linear systems is difficult to determine,PBH-like criteria are proposed for the controllability of quasi-linear systems in standard and descriptor cases and extended to the high-order case.The definition of controllability of quasi-linear systems is proposed based on the controllability definition paradigm of Kalman.Using Schauder's fixed-point theorem,the original problem is converted into the existing problem of a fixed-point in Banach space.Further,the existence of a fixed point requires that the determinant of the controllability Gramian has a positive lower bound.However,it is relatively difficult to solve the controllable Gramian matrix which depends on the coefficient matrices,a PBH-like criterion is proposed and extended to the descriptor and high-order quasi-linear systems.Finally,the effectiveness of the proposed criteria is verified by simulation.Secondly,it is well-known that the open-loop control is sensitive to disturbances,then,a parametric design approach of displacement plus acceleration state feedback control for second-order quasi-linear systems is proposed in this dissertation.The traditional method of closed-loop control consists of displacement plus velocity state variables,in which the velocity state variable is obtained by integrating the acceleration state variable.This may cause errors when there is a disturbance in the system.Therefore,the displacement plus acceleration state feedback control is adopted.The general parametric forms of displacement plus acceleration state feedback controller and right closed-loop eigenvector are obtained.Further,the conditions that the closed-loop system can be transformed into a given linear time-invariant system are discussed,that is,the closed-loop system with the desired eigenstructure.Using the degrees of freedom in an arbitrary parameter matrix in the proposed approach,the controller can be optimized to achieve regional pole placement and some additional performance.Finally,the proposed approach is applied to the trajectory tracking control of a two-link manipulator to verify its effectiveness.Thirdly,the state variables are not accessible for direct measurement in some cases,a parametric design approach of output feedback control for second-order quasi-linear systems is proposed in this dissertation.In application,the output feedback is more suitable and practical.Thus,based on the solution to a type of second-order generalized Sylvester matrix equations,the general parametric form of the output feedback controller is established concerning the state variables,the time-varying parameter vector,the constant closed-loop system,and another two groups of arbitrary parameters,and also for the left and right closed-loop eigenvectors matrices.With the proposed parametric output feedback control,the closed-loop system is transformed into a constant linear system with the desired eigenstructure.Finally,the proposed approach is compared with the feedback linearization method and applied to the trajectory tracking control of a two-link manipulator to verify its effectiveness.Fourthly,to achieve the effect of state feedback,a parametric design approach of dynamic output feedback control for second-order quasi-linear systems is proposed in this dissertation.The dynamic output feedback scheme is introduced in this dissertation,and the dynamic output feedback control law for descriptor second-order quasi-linear systems is proposed,the proposed parametric approach provides the general parametric forms of dynamic output feedback controllers for non-normalization and normalization cases.On the one hand,the parametric approach allows the closed-loop system to have the desired eigenstructure,whether the closed-loop form is singular or not.On the other hand,normalization design simplifies complexity and reduces the computation load,it is also easy and simple to maintain the regularity of the closed-loop system.Finally,the proposed approach is applied to the trajectory tracking control of a two-link manipulator to verify its simpleness and effectiveness.Finally,the NXT Selective Compliance Assembly Robot Arm(SCARA)system is selected to verify the proposed parametric control method of second-order quasi-linear systems.This paper introduces the hardware and software composition and working principle of the NXT SCARA system.The forward and inverse kinematics,and dynamics equations are established,and how to obtain the desired trajectories of joint angles by using the desired tracking trajectory by inverse dynamics,then two methods of constructing the desired reference trajectories are deduced.Finally,the parametric control method proposed in this paper is compared with the feedback linearization control method on the experimental NXT SCARA platform,and its advantages in tracking performance and error,chattering reduction,and robustness are verified.