A Mathematical Technique for the Analysis of Linear Systems

The design of a linear system often takes the form of its response in the time domain to a test function. Computation of such response, particularly in the case of feedback systems, is usually complex and time consuming. Numerical methods have been applied successfully but to date, generally, they have not been related closely to the physical system being analyzed. It is shown in this paper that the z-transformation developed originally for the analysis and synthesis of sampled-data systems is applicable to numerical solutions of continuous linear systems. A model of the continuous linear system is devised in which sampling is introduced at some convenient point. The sampled time function is then reconstructed into a polygonal approximation by means of a holding operator and the output of the system is readily computed as a train of pulses giving the values of the output at sampling instants. Both analytic and arbitrary inputs can be handled by this model. The errors in the output which result by making computations on the sampled model rather than the actual system are obtainable by the application of a procedure quite similar to that of the main computation. An illustrative example using feedback is given where it is shown that the theoretical predictions concerning error produced by the model are within very close tolerances to the actual error.