| This dissertation presents some results of solution problem and stability property for linear time-varying systems. The contents contain the following three parts:; 1. Solution Problem. First, we present a theorem that is:; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); is a solution of linear time-varying systems x(t) = A(t)x(t) if, and only if (lamda)(,i)(t) and u(,i)(t) satisfy (UNFORMATTED TABLE FOLLOWS); A(t)u(,i)(t) = (lamda)(,i)(t)u(,i)(t)+u(,i)(t) (FOR ALL)t(TABLE ENDS); where (lamda)(,i)(t) is said to be an extended eigenvalue of A(t) associated with the extended eigenvector u(,i)(t).; Based on the properties of this new definition, we present: a spectral representation for state transition matrix of linear time-varying systems which reduces to that of linear time-invariant system when the extended eigenvector u(,i) is constant and a previously unknown solvable class which is called modified A(,h) class.; Next, we present an algorithm for finding a transformation to transform any given system such that the resultant system matrix being a phase-variable canonical form, which facilitates the analysis or synthesis in many system applications.; Finally, we present a transformation that will reduce high-order system matrix in phase-variable canonical form to a lower order one which is also in phase-variable canonical form provided that a particular integral of the equivalent ordinary differential equation can be found. Therefore we concentrated our focus on the solution of second-order linear time-varying systems, and several new solvable classes were identified.; 2. Stability Problem. A conjecture for the sufficient condition for stability of a second-order linear time-varying system is given. This criterion gives better stability results than those of Wazewski's and Mori, et al's. We also give interesting examples to show that even for a stable system, its transposed system may be unstable, however, their composed system can still be stable.; 3. Design Problem. A direct design technique for designing state feedback controller and observer is introduced. First, a desired system matrix is chosen based upon the solution problems and stability problems discussed earlier. Then, the simplest matrix-generalized inverse technique is applied to achieve this purpose. Because of a consistency condition should be satisfied when we apply that method, choice of a desired system matrix should have some restrictions for the given system matrix A(t) and input matrix B(t). It is noted that this technique can also be applied to linear time-invariant system. |
