Signal Flow Graphs - Basics

An in depth discussion of signal flow graphs and their manipulation can be found in:

• Howard R., Dynamic probabilistic systems, John Wiley, New York, 1971.
• Truxal, JG., Automatic feedback control system synthesis. McGraw-Hill, New York, 1955.

A. Rules and Definitions of Signal Flow Graphs

Signal flow graphs follow four rules:

• Signals travel along branches only in the direction of the arrows.
• A signal travelling along any branch is multiplied by the transmission of that branch.
• The value of any node variable is the sum of all signals entering the node.
• The value of any node variable is transmitted on all branches leaving that node.

A path is a continuous succession of branches, traversed in the indicated branch directions. The path transmission is defined as the product of branch transmissions along the path. A loop is a simple closed path, along which no node is encountered more than once per cycle. The loop transmission is defined as the product of the branch transmissions in the loop.

The transmission T of a flow graph is defined as the signal appearing at some designated dependent node per unit of signal originating at a specified source node. Specifically, T_ik is defined as the signal appearing at node k per unit of external signal injected at node j. There are a number of ways of computing transmissions.

B. Basic Operations on Signal Flow Graphs

Solution of signal flow graphs requires knowledge of certain of their topological properties. The basic operations of addition, multiplication, distribution and factoring can be used to reduce the number of branches and nodes in the system. At first glance, it might appear that by successive application of such transformations a graph could be reduced to a single branch connecting any two given nodes. However, if the graph contains a closed loop of dependencies, as when modeling iterations, one or more self loops will eventually appear.

C. The Effect of a Self Loop

The effect of a self loop at some node on the transmission through that node is analyzed in the figure below.

Effect of a self loop

The node signal at the first node is x and the signal returning around the self loop is xt. Since the node signal is the algebraic sum of the signals entering that node, the external signal arriving from the left must equal y(1-t). Hence, the effect of a self loop t is to divide an external signal by the factor (1-t) as the signal passes through the node. This holds for all t.

D. Solution by Node Absorption

Node absorption corresponds to the elimination of a variable by substitution in the associated algebraic equations. With the aid of the basic transformations and the self loop replacement, any node in a graph can be absorbed and the equivalent expressions for the transmission between two other nodes calculated. Although the branch is no longer shown, its effects is included in the new branch transmission values, as shown below.

Absorption of a node

To compute the overall graph transmission, all the intermediate nodes are absorbed in turn, yielding the transmission between the start and finish nodes. reduction of graphs is computationally intensive and manual solution of graphs of even moderate size can be difficult.

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