22.7 : SFG Algebra

In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.

Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the signals from all incoming branches. The outgoing branches then transmit this value, multiplied by their respective gains, to the subsequent nodes.

Parallel branches that direct signals in the same direction can be consolidated into a single branch. The gain of this new branch is the sum of the gains of the original parallel branches. For example, if two branches with gains G₁ and G₂ are parallel, they can be replaced by a single branch with a gain G=G₁+G₂.

Cascaded branches, or branches connected in series, can similarly be simplified. The gain of the resulting branch is the product of the gains of the original branches. For example, if two branches with gains G₁ and G₂ are cascaded, they can be replaced by a single branch with a gain G=G₁×G₂.

​In feedback systems, specific algebraic equations are used to derive the closed-loop transfer function. This involves recognizing the feedback loop and applying the appropriate formula to determine the system's behavior. To convert a block diagram of a control system into an SFG, follow these steps:

1. Identify Variables: Each variable in the system is associated with a node.
2. Interconnect Nodes: Draw branches between nodes to represent the relationships between variables, ensuring the direction of signal flow is accurately depicted.
3. Label Branches: Assign the appropriate gain to each branch, reflecting the transfer function of the corresponding part of the system.

SFGs can also be derived from a set of algebraic equations. This process involves these steps:

1. Identify and create nodes for each variable in the equations.
2. Connect the nodes based on the relationships defined by the equations, using branches to represent these relationships.
3. Assign gains to the branches according to the coefficients in the equations.

By leveraging these principles, SFGs provide a versatile and intuitive method for analyzing complex control systems, facilitating the derivation of transfer functions and enhancing our understanding of system dynamics.