Closed Form Expressions of Linear Continuous Time System Responses

Abstract

General closed form expressions of linear continuous time system responses of an arbitrary order are derived, by first relating them to basic responses, i.e., responses corresponding to unity numerator transfer functions. Those are then related to the fundamental solutions of the underlying differential equations. These expressions apply to all systems without any restrictions on the poles or the zeros, further the systems may be noncausal. We derive responses for all regular types of inputs, impulse, step, ramp, parabola, etc., in addition for all generalized derivatives of the impulse. All the presented results have a direct counterpart in results presented in Sigurðsson et al. (2017) on discrete time systems based on the fundamental solution of the associated difference equation. Efficient evaluations of the fundamental solutions along with their derivatives and integrals can thus be extended to the responses and are readily implemented, e.g., as Matlab functions. Such results may be presented symbolically as functions of time or evaluated numerically at any sequence of times without time stepping.