On the Reachability of Linear Time Varying Systems

In this paper system properties of generalized linear time varying (LTV) systems are discussed where, in addition to the control, its certain derivatives also appear both in the dynamics and the observation equation. Developing an adequate version of the Cauchy formula, a necessary and suffiecient condition for complete reachability of generalized LTV systems is obtained in terms of a generalized Gram matrix. Starting from the expansion of coefficient functions in the corresponding Lie algebra basis, we derive an appropriate condition of persistent excitation. The latter leeds to a general condition of complete reachability in terms of quasi-polynomials of the solution of the Wei-Norman equation and differential polynomials of the coefficient functions of the generalized LTV system. Also applying the well-known duality theory of LTV systems, other basic system properties such as controllability, reconstructability and observability can be also treated.