Parametric Method To Design Proportional-Integral Observers For Linear Time-Varying Systems

In modern control theory,the description and expression of state space is a direct reflection of system characteristics.However,due to some reasons,some states can not be directly measured,resulting in the control engineering related problems can not be solved,so researchers proposed to design observers for the purpose of dynamic system state reconstruction.From the perspective of classical control theory,the steady-state accuracy of the system can be improved by introducing integration.Then,by introducing the integration terms,the observer can use the past and current state informations of the proportional and integral to conduct state estimation,and this kind of observer is called the proportional-integral(PI)observer.By inserting the integral path,more design freedom can be provided for the observer system.This paper proposes a parametric method to design PI observers for linear time-varying system.The sufficient conditions for the existence of the PI observers are given.The completely parametric expressions of the PI observer gain matrices and the left eigenvector matrix are constructed based on three sets of designed free parameters,namely,the eigenvalue _is of the closed-loop error system,the parameter matrix Z(t)and the integral gain matrix K,These parameters represent all the design degrees of freedom for the system.The main research contents are as follows:1.A parametric design method of full-order PI observers for linear time-varying systems is proposed.Firstly,the sufficient conditions for the existence of full-order PI observer are given.Then,the closed-loop error system is transformed into a linear time-invariant system with the desired eigenstructure by establishing the form of the left eigenvector matrix.Finally,based on a class of generalized time-varying Sylvester matrix equations,the completely parametric solutions for the gain matrices of the PI observers are established.The completely parametric expressions of the PI observer gain matrices and the left eigenvector matrix are constructed based on three sets of designed free parameters,namely,the eigenvalue _is of the closed-loop error system,the parameter matrix Z(t)and the integral gain matrix K.These three groups of free parameters provide all the designed degrees of freedom for the system,which can be used to satisfy more performance of the system.2.A parametric design method of reduced-order PI observers for linear time-varying systems is proposed.Firstly,the sufficient conditions for the existence of the reduced-order PI observers are given.Then,by establishing a nonsingular matrix,the original system is equivalent to a new system with special structure based on linear transformation.Using the known output information of the system,the reduced-order PI observers can be constructed to meet the requirements and reduces the complexity of PI observer design.Finally,based on a class of generalized time-varying Sylvester matrix equations,the completely parametric solutions for the gain matrices of the reduced-order PI observers are established.The completely parametric expressions of the PI observer gain matrices and the left eigenvector matrix are constructed based on three sets of designed free parameters,namely,the eigenvalue s_i of the closed-loop error system,the parameter matrix Z(t)and the integral gain matrix K,which provide more degrees of design freedom for the system.3.A parametric design method of PI observers for linear time-varying descriptor systems is proposed.Firstly,the sufficient conditions for the existence of PI observers for linear time-varying descriptor systems are given.Then,the design problem of the PI observers is transformed into the problem of solving a class of generalized time-varying Sylvester matrix equations by establishing the form of the left eigenvector matrix.The completely parametric solutions for the gain matrices of the PI observers are established,and the sufficient conditions for the regularization of the observer system are given.Finally,the completely parametric expressions of the PI observer gain matrices and the left eigenvector matrix are constructed based on three sets of designed free parameters,namely,the eigenvalue _is of the closed-loop error system,the parameter matrix Z(t)and the integral gain matrix K.This parametric method not only ensures the regularity of the observer system,but also realizes the elimination of the impulse response for the observer system.