A Complete Approach to Time Varying Linear Systems
This work addresses a fundamental theoretical gap in control theory by generalizing eigenvalue-based methods to time-varying systems, potentially impacting system analysis and design.
This paper introduces a unifying theory for linear second-order systems that treats time-varying and time-invariant systems equivalently, providing a transformation to diagonalize time-varying state matrices and a canonical form for the fundamental matrix based on dynamic eigenvalues. Examples demonstrate a unified approach to solving time-invariant, time-varying, and periodic systems.
This paper presents a unifying theory of Linear second order systems that allows time-varying and time invariant systems to be treated in the same way for the first time. In the process, a transformation is given that diagonalizes an arbitrary time varying state matrix in a spectrum invariant way. A canonical form for the fundamental matrix is given that depends on dynamic eigenvalues and related eigenvectors dependent upon the Riccati Characteristic Equation for the system, which intuitively generalizes the standard characteristic equation for time invariant systems. The technique is shown by examples to give a unified approach to the solutions of time invariant, time-varying, and periodic systems.