Topological techniques for the solution of multi-loop sampled systems

This paper presents a flow-graph approach for analyzing multiloop sampled-data systems. Two techniques for finding the sampled output are examined. These are, first, construction of a "sampled" signal flow graph from the original system; second, a general gain formula which can be directly applied to the original system. The theorems necessary to establish the formula are rigorously proved. The first technique allows two possible modes of solution. The sampled output can be found directly from the sampled flow graph by the use of Mason's formula, or in case of a more complicated multi-loop system, the problem of enumerating nontouching feedback loops can be simplified by the use of topological matrices. Techniques developed in the paper are also applied to the solution of multirate systems. This paper has been published by the American Society of Mechanical Engineers as ASME Paper No. 59-NAC-1.