An Approximate Method for Analytically Evaluating the Response of Time-Variable Linear Systems

For all but the most elementary time-variable linear systems, solutions of the response problem must be particular solutions obtained by machine computation. Thus, the insight into system response characteristics, which is gained from analytic procedures, is lost. This paper proposes an approximate representation of the time-variable weighting functionh(t,\tau)(system response at timetto a unit impulse at timet -\tau) which permits a straightforward analytic evaluation of response for both deterministic and stochastic inputs. Although the evaluation of the approximate representation may require machine computation, once the approximate representation is obtained, no further machine calculations are necessary. The approximation has the formh*(t,\tau) = \Sigma_{i = 1}^{N} a_{i}(t)\phi_i (\tau)where the\phi_i (\tau)are arbitrary approximating functions. Methods are given for evaluating thea_{i}(t). Several families of orthonormal approximating functions are discussed which simplify both the evaluation and utilization of the approximation. For example, when the\phi_i (\tau)have rational Laplace transforms, thea_{i}(t)are computed directly from the solutions of certain time-variable linear differential equations. For stationary stochastic inputs, simple series expressions for the output correlation function are derived, and it is shown that propitious choice of the\phi_{i} (\tau)greatly reduces the number of integrations which must be made in evaluating the series. Two example problems illustrate the feasibility of the proposed procedures.