Left Conjugate Product Of Polynomial Matrices With Its Applications

For years, system state estimation and observer design have attractedwidespread concern, and the design of high-order observer for linear systems hasbeen developed in recent years. The key issue of various high-order observer have abearing on the so-called high-order Sylvester polynomial matrices, and the previouscommon matrix equation such as Lyapunov equation、Stein equation、normalSylvester equation、Yakubovich equation、Kalman-Yakubovich equation all can beregarded as special cases of high-order Sylvester matrix equation. So some intensivestudy of high-order Sylvester equation is of great importance.This thesis proposes a new form of dual high-order Sylvester polynomialconjugate matrix equation based on the predecessors and two new mathematicaloperators called left conjugate product and dual Sylvester conjugate sum. The twonew mathematical operators are made to contribute to the deduction of the completeparametric solutions to the dual high-order Sylvester polynomial conjugate matrixequation. The main contents of this article contain theoretical section and Matlabsimulation section.In the theoretical section this thesis proposes the concepts of polynomial leftconjugate product and dual Sylvester conjugate sum, then explores several basicoperation properties including distributive law and associative law of these newoperators. Based on this foreshadowing conclusion this paper explores variousproperties of polynomial matrices mainly including unimodularity、invertibility andnormal form under the frame of left conjugate product. Base on the special operationproperties of the two new operators in the last parts of the theoretical section somecomplete parametric solutions are proposed for the dual high-order Sylvesterpolynomial conjugate matrix equation.In the Matlab simulation section a operation toolbox is developed under theframe of left conjugate product. This toolbox contains several specific usefulfunction such as the operation of polynomial equation and dual Sylvester conjugatesum under the frame of left conjugate product、the solution to the coprime matrcesand the solution to the dual high-order Sylvester conjugate matrix equations.