| The main topic of this thesis is the analysis of linear time-varying systems from state space models. Unlike linear time-invariant systems, the analysis of linear time-varying systems is very difficult since the state transition matrix, in general, can not be explicitly obtained except for in some special solvable classes. In this thesis, we present a rather general method for the solution of linear time-varying systems based upon the successive decomposition method combined with the Lie algebraic approach. This method, if applicable, allows the system coefficient matrix to be decomposed successively by a series of scalar functions and constant matrices. In order to find these appropriate scalar functions and constant matrices, the dimension of the Lie algebra generated by the resulting matrix from decomposition is employed as an index to provide a direction for this method. It is demonstrated that, with this method, we can solve most of the previously known solvable classes. Moreover, some new solvable classes are identified by applying this method and systematic procedures for decomposing the system coefficient matrix are given.; Also in this thesis, a structure theory of the commutative class is presented by using the spectral representation. From this, we find that the idempotent matrices in the spectral representation of the integral of the system coefficient matrix are constant and the nilpotent matrices commute with their derivatives. If the integral of the system coefficient matrix is of simple structure, then it has n linearly independent constant eigenvectors and the stability of the system can be completely determined by its eigenvalues.; Finally, it is shown that the stability of certain classes of linear time-varying systems can be directly determined by the eigenvalues of the constant matrices and the associated scalar functions based on the decomposition of their system coefficient matrices by using the presented method. |
