A Unified Framework for Determining Post-Jump Initial Conditions for Biproper LTI Systems Subjected to Analytic Causal and Dirac Delta Inputs

The determination of post-jump initial conditions is essential for the accurate analysis and numerical simulation of causal linear time-invariant (LTI) systems subjected to discontinuous inputs. Although conventional time-domain methods require these initial conditions immediately after a switching event, no general analytical procedure exists for deriving them directly from the governing equations. This paper presents a unified analytical framework for determining the post-jump initial conditions of causal single-input single-output LTI systems governed by constant-coefficient nonhomogeneous ordinary differential equations with known pre-jump initial conditions and subjected to either analytic causal or Dirac delta inputs. Three complementary methodologies are developed: an ordinary differential equation (ODE)-based approach that derives explicit recursive closed-form expressions, a state-space model (SSM)-based approach that establishes an equivalent vector formulation, and a Laplace transform Initial Value Theorem (IVT)-based approach that determines not only the output value but also the higher-order derivative initial conditions at t=0+. The three methodologies are shown to be mathematically equivalent, and several examples validate the proposed framework and demonstrate its applicability to the analysis and numerical simulation of LTI systems.