The Zeros of Transient Responses

This paper considers the zeros of the transient responses corresponding to rational Laplace transforms whose poles are all real and whose zeros are allowed to be either real or complex. The possible values for the number of zeros of the transient responses are determined under various assumptions on the Laplace transforms. For instance, when the poles of the transform are allowed to have any multiplicity, the possible values for the number of zeros of the transient response are found in terms of the number of zeros of the transform. On the other hand, when the poles are all simple and the pole and zero locations of the transform are known, a stronger result is obtained. These results are developed for the function obtained by extending to all time the expression for the transient response that holds for positive time. The possible values for the number of its zeros are determined for positive time and for negative time separately.