Input-Output Jumps of Scalar Linear Systems

In the classic control systems analysis a known difficulty is to determine a transient output response when non-zero (discontinuous) initial conditions are present. This difficulty is overcome by means of the presented input-output jump relations. For a system with order n and relative degree r = n - m, these relations relate the jump discontinuities of the input and its derivatives up to the order m - 1 to the jump discontinuities of the output and its derivatives up to the order n - 1. A simplified behavior theory is used in deducing these straightforward relations. A simple, complete solution to the initial conditions problem that uses neither generalized derivatives nor ad hoc assumptions is then presented. A detailed example illustrating this solution is included. Copyright (C) 2019. The Authors. Published by Elsevier Ltd. All rights reserved.