Non-stationary linear systems

Non-stationary systems are described by a model whose coefficients are explicitly time-varying. Consideration of these system models is of considerable practical importance. Indeed, when, for example, the drift of a component is known during operation, or when the operating principle implies periodically varying coefficients [9], or when one seeks to linearize a non-linear process, not around an operating point, but along a trajectory, a linear model with time-varying coefficients may be sufficient to correctly describe the observed behavior. It should also be noted that the principle of Kalman filtering was successfully established using a state equation with time-dependent parameters [5][6].

All these models, whose coefficients depend on time, are sometimes referred to as time-varying parameter systems or unsteady systems, and can therefore be used to represent or describe a wide range of behaviors. In this case, it is of interest to have methods for applying the techniques usually used on linear systems with constant coefficients [8]. This is what we're going to see, confining ourselves to the case of single-input, single-output systems, as multi-input, multi-output cases don't present any particular difficulties.