Parametric Design Method For Observer Of Quasi-Linear Systems

In practical working environments,due to considerations of cost and convenience,acquiring complete state information of a dynamic system is often unfeasible.To tackle this challenge,the emergence of state observers has garnered significant attention and research from the academic community.While current research on observers primarily focuses on linear systems,establishing a general research approach for non-linear system observers remains elusive.Quasi-linear systems,which can be expressed in the form of linear systems but still retain their inherent nonlinearity,serve as a bridge between linear and nonlinear systems,thereby possessing considerable research significance.Consequently,this paper concentrates on investigating quasi-linear systems as the subject of study,addressing the issue of challenging state variable measurement.The paper begins by analyzing the stability of the observer error dynamical system,establishing the dynamical equation for a full-order state observer of a typical quasi-linear system.To solve the observer design problem,the stability condition of the observer error system is utilized,transforming the problem into Sylvester equation.A parametric design method is then introduced based on the parametric solution of Sylvester equation.This design method is further extended to observer design for descriptor quasi-linear systems.Additionally,a parametric design method for reducedorder state observers is proposed,specifically tailored for normal quasi-linear systems with partially measurable states.Finally,the paper demonstrates the implementation of state feedback control using the state observer.The primary contents of this paper encompass the following aspects:(1)For the standard quasi-linear system,when all the states of the system are not measurable,a parametric design method of the state observer is proposed to estimate the real states of the system,which effectively solves the problem that the states of the quasi-linear system are difficult to measure directly,and the observed states are applied to the design of the feedback control law.Firstly,a parametric method for the design of a full-order observer is proposed using the parametric solution of Sylvester equation,and the general parametric form of the observer is given.Secondly,the separation principle of the first-order linear system is extended to the quasi-linear system,in other words,the independent design of the observer and the controller is realized.Further,the general parametric form of the feedback controller is established using the observed states.In addition,the degrees of freedom provided by the free parameters are used to optimize the performance by selecting appropriate optimization index.Finally,the effectiveness of the proposed method is verified by numerical arithmetic examples and simulations.(2)Considering the standard quasi-linear system under the premise of partial state measurability of the system,a reduced-order state observer is designed to achieve the estimation of the unknown state,which not only effectively solves the problem that the state of the quasilinear system is difficult to measure,but also reduces the implementation cost of the observer and is easy for engineering applications compared with the full-order state observer.Firstly,the target system is transformed into the desired chunked form by using equivalent linear variation.Second,the stability conditions of the observer are derived by analyzing the stability of the error dynamic system.Further,a parametric design method for the reduced-order observer is given with the help of the solution of Sylvester equation.Then,the state estimation of the observer is applied to the controller design of the system.Finally,the effectiveness of the proposed method is verified by numerical arithmetic examples and simulations.(3)For the descriptor quasi-linear system,when all the states of the system are not measurable,the normal parametric design method of the full-order observer for the quasi-linear system is extended,and a parametric design method of the full-order state observer for the descriptor quasi-linear system is proposed to ensure the regularity of the closed-loop system,and the observed states are applied to the design of the feedback control law.Firstly,based on the parametric solution of the generalized Sylvester equation,a parametric method for the design of the full-order state observer is proposed,and the general parametric form of the observer is given.Secondly,the separation principle of the first-order linear system is extended to the descriptor quasi-linear system,in other words,the independent design of the observer and the controller is realized.Further,the general parametric form of the feedback control law is established using the observed states.Finally,the validity of the proposed method is verified by a numerical arithmetic example and a circuit example,which are of certain guidance for practical engineering applications.