Solution Problem and Stabilization Problem of Linear Time-Varying Systems.

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.