Signal Flow Graphs

Signal-flow graphs offer a streamlined and intuitive approach to representing control systems, providing an alternative to traditional block diagrams. These graphs use branches to symbolize systems and nodes to represent signals, effectively illustrating the relationships and interactions within the system.

In a signal-flow graph, branches denote the system's transfer functions, while nodes represent the signals. The direction of signal flow is indicated by arrows, with the corresponding transfer function written adjacent to each arrow. Unlike block diagrams, where summing junctions include negative signs to denote subtraction, signal-flow graphs incorporate these negative signs directly into the transfer functions.

The first step in converting a block diagram into a signal-flow graph is to identify the system signals. These signals are then represented as nodes in the signal-flow graph. Identifying and drawing the first nodes for each signal in the system is crucial in this conversion process. Then, connect these nodes with branches that represent the system's transfer functions, ensuring the direction of signal flow is accurately depicted.

For example, in a block diagram with multiple feedback loops:

1. Start by establishing nodes for each signal within the feedback loops.
2. Link the nodes with branches that capture the transfer functions and their directions. Negative signs at summing junctions in the block diagram are incorporated into the transfer functions within the signal-flow graph.

The final step in converting block diagrams to signal-flow graphs involves simplifying the graph. This simplification is achieved by eliminating intermediate signals that have only one incoming and one outgoing branch, reducing the complexity of the graph and making it easier to analyze.

Once the signal-flow graph is established, Mason's rule can be applied to calculate the system's transfer function. This involves:

1. Determining all possible forward paths from the input to the output node and calculate their gains.
2. Identifying all loops and their gains, and determining which loops are non-touching.
3. Computing Δ using an alternating series of sums of loop gains and non-touching loop gains. Δ_k is calculated by excluding loop gains intersecting with the k^th forward path.
4. Substituting these values into Mason's rule to obtain the transfer function.

By converting block diagrams to signal-flow graphs, control systems can be analyzed more efficiently, leveraging the systematic approach provided by Mason's rule to determine transfer functions.