Classifications of Linear Controlled Systems

Classifications of Linear Controlled Systems

This work is devoted to providing complete solutions to the linear, differential and topological classification problems for linear controlled systems governed by ordinary differential equations. The main novelty and difficulty lie in the topological classification, which is a long standing unsolved problem. By means of an elementary approach to analyze the special form of the topological transformation function, we determine all principal topological invariants for the linear controlled system, i.e., the numbers (counting the algebraic multiplicity) of eigenvalues with negative and positive real parts for the completely uncontrollable subsystem and the Jordan block matrices corresponding to the eigenvalues with vanishing real parts for the same subsystem, and the structure of the completely controllable subsystem. The determination of the last topological invariant is the most difficult part, which is equivalent to show that completely controllable systems are topologically equivalent if and only if they are linearly equivalent. Also, it is shown that there are only finitely many topological equivalence classes for stabilizablesystems with given numbers of state and control variables, which should be useful for some engineering applications. (This talk is based on one of my Ph D students, Jing Li's dissertation).