|Explicit Solution and Stability of Linear Time-Varying Differential State Space Systems
International Journal of Control, Automation, and Systems, vol. 15, no. 4, pp.1553-1560, 2017
Abstract : "Linear time-varying (LTV) systems naturally arise when one linearizes nonlinear systems about a trajectory.
In contrast the linear time-invariant (LTI) cases which have been thoroughly understood in the analysis
and synthesis technologies, many features of the LTV systems are still limited and not clear. This paper addresses
the problems of solution and stability of a general unforced LTV differential state space system. Unlike most of
the work based on the Lyapunov theory, numerical simulations, or specific constraint systems, the paper proposes
the spectral decompositions of the LTV systems by employing extended eigenpairs and with simple mathematical
derivation. The spectral decompositions reveal the mechanisms of inherent characterization in general LTV
systems, rather than a particular class. Moreover, a novel set of auxiliary equations is developed for guiding and
obtaining the extended eigenpairs of its system matrix which completely characterize the LTV systems. The solutions
to perform the commutative systems and the second-order systems with companion form are straightforward.
The proposed innovative thinking provides a novel guided way to analyze the LTV systems. These findings are
easily extended to LTI cases. Examples from the literature demonstrate the effectiveness and the superiority of the
proposed approaches when compared with other methods. The proposed results may be of great interest in both for
scientific research and application."
"Asymptotic stability, linear time-varying systems, Lyapunov theory, Riccati equation, state transition matrix."