On the poles and zeros of linear, time-varying systems

Definition of poles and zeros are presented for continuous-time, linear, time-varying systems. For a linear, time-varying state equation, a set of time-varying poles defines a stability-preserving variable change relating the original state equation to an upper triangular state equation, A zero is a function of time corresponding to an exponential input whose transmission to the output is blocked. Both definitions are shown to be generalizations of existing definitions of poles and zeros for linear, time-varying systems and are consistent with the definitions for linear, time-invariant systems. A computation procedure is presented using a QR decomposition of the transition matrix for the state equation. A numerical example is given to illustrate this procedure.