Parameterized Design Of Robust Observers For High-Order Singular Linear System

In recent decades,generalized systems have been applied in many fields,such as transportation,financial system,power system,biological system and so on.With the continuous development of higher-order linear system theory,the research of higher-order descriptor linear system observers has also gradually attracted people’s attention.In view of the fact that the observer can reconstruct the state of the system and solve the problem that the state cannot be directly measured in engineering,this paper studies the design of robust observers for higher-order descriptor linear systems on the basis of the theory of descriptor systems.The main research contents are as follows:1.Three parametric solutions of higher-order generalized Sylvester matrix equation are given based on left eigenvector matrix,right coprime decomposition and singular value decomposition.2.The parametric design of PI and PD observers is proposed for two different kinds of higher-order descriptor time-invariant linear systems.Firstly,the requirement that the system-based estimation error tends to zero(the PI observer also ensures that its integral term tends to zero)gives constraints that make the system regular and pulse-free.Then,the Jordan decomposition of the augmented error system matrix pair is carried out,and the observer design is further transformed to solve the higher-order generalized Sylvester matrix equation.Finally,the parametric expressions of the observer gain matrices and the left eigenvector matrix are constructed by using the free parameters,namely the closed-loop eigenvalues s_i and a group of free parameter vectors f_i(the PI observer also needs to consider the degree of freedom contained in the integral gain matrix F),which provide all the degrees of freedom for the design of the system and can also be used to satisfy other performance of the system.3.The parametric design of PD observer is studied for two different kinds of higher-order descriptor time-varying linear systems.Firstly,the PD observer is designed for the system under the condition that the system is fully observable.Then,according to the established left eigenvector matrix,the augmented error system matrix pair is decomposed by Jordan,and the PD observer design is further transformed to solve the higher-order Sylvester matrix equation.Finally,parameterized expressions of the observer gain matrices and the left eigenvector matrix are constructed using the degrees of freedom contained in the solution of the equation,namely the closed-loop eigenvalues s_i and a group of free parameter vectors f_i(the degrees of freedom contained in the matrix L_d(θ)should also be considered in Section 4.3).4.The parametric design problem of robust observer is proposed for higher-order descriptor time-invariant linear system.Firstly,the assumptions for the existence of robust observer are given,and a suitable robust observer is designed for the system.Secondly,the Jordan decomposition is carried out for the augmented error system matrix pairs,and the robust observer design is further transformed into solving higher-order Sylvester matrix equations.Then,parameterized expressions of the gain matrices and the left eigenvector matrix of the robust observer are constructed according to three sets of free parameters,namely the closed-loop eigenvalues s_i,a set of free parameter vectors f_i and a parameter matrix_dL.Finally,the design parameters with degrees of freedom are optimized by minimizing the sensitivity of closed-loop poles,so that the sensitivity of closed-loop eigenvalues to the change of system coefficient matrices are minimized,and a set of optimal parameters are obtained.