Signal Flow Graphs (with Examples) - Control Systems - Electrical Engineering

Introduction

Block diagram reduction is a standard method for determining the transfer function of a control system. For complex systems this method becomes time-consuming and prone to error. An alternate and compact graphical method called the Signal Flow Graph (SFG) was developed by S. J. Mason. The SFG relates input and output variables of a set of linear algebraic equations directly and visually. In the SFG approach the transfer function is often called the transmittance of the system.

Characteristics of SFG

• Graphical representation: SFG is a graphical representation of the relationship between variables of a set of linear algebraic equations. Each variable is represented by a node and directed branches represent algebraic relations (gains) between nodes.
• Direct mapping from equations: SFG can be constructed directly from the equations and therefore does not require repeated block-diagram reductions.
• Flow direction: Each branch in the graph has an arrow showing the direction of signal flow and a weight showing the branch gain.
• Linear systems only: SFG is applicable only to linear time-invariant algebraic representations (or linearised systems represented by linear algebraic equations).
• Nodes and summation: Each node variable is the algebraic sum of the contributions arriving along incoming branches; outgoing branches do not alter the node value.

Terms used in Signal Flow Graphs

Node

A node represents a system variable (for example x₁, x₂, ...). The value at a node equals the sum of all signals entering that node. Signals leaving the node do not change the node value; they only carry copies of the node variable downstream.

Branch

A branch is a directed connection from one node to another. The branch arrow indicates the direction of signal flow, and the branch is assigned a gain (a numerical or symbolic multiplier) which multiplies the signal as it passes along the branch.

Summing (junction) point and transmitting point

A node can act as a summing point where contributions from several branches add algebraically. For example, if a node x₁ receives signals x₂, x₃, x₄ then

x₁ = x₂ + x₃ + x₄

A node can act as a transmitting (outgoing) point which supplies the same node value to several outgoing branches. For example, if x₁ splits into two outgoing signals x₅ and x₆, then

x₁ = x₅ + x₆

Input node (source) and Output node (sink)

• Input node or source: A node with only outgoing branches (no incoming branches) representing an independent excitation.
• Output node or sink: A node with only incoming branches (no outgoing branches) representing an observable output variable.

Forward path

A forward path is any path that starts at an input node and ends at an output node following the direction of the branch arrows and not visiting any node more than once. The forward path gain is the product of gains of all branches along the forward path.

Loop

A loop is a closed path that starts and ends at the same node and travels in the direction of the branch arrows without passing through any node more than once. The loop gain is the product of all branch gains around that loop.

Non-touching loops

Two loops are said to be non-touching if they share no common node. Non-touching loops can be multiplied together when forming the determinant used in Mason's Gain Formula.

Forward path gain and loop gain (summary)

• Forward path gain: Product of branch gains along a forward path from input to output.
• Loop gain: Product of branch gains around a closed loop.

Construction of an SFG and Mason's Gain Formula

To construct an SFG from a set of linear algebraic equations, represent each variable by a node and draw directed branches between nodes according to the coefficients in the equations. A coefficient a_ij multiplying x_j in the expression for x_i yields a branch from node j to node i with gain a_ij.

Mason's Gain Formula (transmittance)

Mason's Gain Formula gives the overall transfer function (transmittance) from a specified input node to a specified output node directly from the SFG.

The formula is

G = Σ (P_k · Δ_k) / Δ

where

• P_k is the gain of the k^th forward path from input to output.
• Δ is the determinant of the graph defined as

Δ = 1 - (sum of individual loop gains) + (sum of gains of all possible pairs of non-touching loops) - (sum of gains of all possible triplets of non-touching loops) + ...

• Δ_k is the value of Δ for the portion of the graph that remains when all loops touching the k^th forward path are removed. In other words, Δ_k is computed using only the loops that do not touch the k^th forward path.

To apply Mason's formula in practice:

1. Identify all forward paths from the chosen input node to the chosen output node and compute each forward path gain P_k.
2. Identify all loops in the SFG and compute each loop gain.
3. Determine combinations of non-touching loops and compute sums of products as required to obtain Δ.
4. For each forward path, compute Δ_k by excluding any loop that touches that forward path.
5. Substitute the values into Mason's formula to obtain the overall transfer function.

Example

Consider the system described by the following equations where x₁ is the input and x₅ is the output.

x₂ = a₂₂x₁ + a₃₂x₃ + a₄₂x₄ + a₅₂x₅

x₃ = a₂₃x₂

x₄ = a₃₄x₃ + a₄₄x₄

x₅ = a₃₅x₃ + a₄₅x₄

The SFG is constructed by following the algebraic coefficients and drawing nodes and branches accordingly.

1. Draw all nodes representing variables x₁, x₂, x₃, x₄, x₅.

2. Draw branches for equation x₂ = a₂₂x₁ + a₃₂x₃ + a₄₂x₄ + a₅₂x₅. Each term gives a branch to node x₂ with the corresponding gain.

3. Draw branches for x₃ = a₂₃x₂. This yields a branch from x₂ to x₃ with gain a₂₃.

4. Draw branches for x₄ = a₃₄x₃ + a₄₄x₄. This gives a branch x₃ → x₄ (gain a₃₄) and a self-loop at x₄ (gain a₄₄).

5. Draw branches for x₅ = a₃₅x₃ + a₄₅x₄. This produces branches x₃ → x₅ (gain a₃₅) and x₄ → x₅ (gain a₄₅).

6. Combine all partial graphs to form the complete SFG of the system.

Identifying forward paths and loops for this example

From the constructed SFG we can identify forward paths from x₁ to x₅ and loops. Typical forward paths are:

• P₁: x₁ → x₂ → x₃ → x₅ with gain P₁ = a₂₂ · a₂₃ · a₃₅.
• P₂: x₁ → x₂ → x₃ → x₄ → x₅ with gain P₂ = a₂₂ · a₂₃ · a₃₄ · a₄₅.

Representative loops include (list expressed symbolically):

• L₁: self-loop at x₄ with gain a₄₄.
• L₂: two-node loop between x₂ and x₃ with gain a₂₃ · a₃₂.
• L₃: loop x₂ → x₃ → x₄ → x₂ with gain a₂₃ · a₃₄ · a₄₂.
• L₄: loop x₂ → x₃ → x₅ → x₂ with gain a₂₃ · a₃₅ · a₅₂.
• Other larger loops are possible by following branches and returning to the start node; list them consistently when analysing the graph.

Forming Δ and Δ_k

Compute Δ using the series

Δ = 1 - (L₁ + L₂ + L₃ + L₄ + ...) + (sum of products of all distinct pairs of non-touching loops) - (sum of products of all distinct triplets of non-touching loops) + ...

For each forward path P_k, compute Δ_k by omitting any loop that shares a node with P_k and then recomputing Δ with the remaining (non-touching) loops.

Using Mason's formula (symbolic result)

Applying Mason's Gain Formula to obtain G(s) = x₅/x₁ gives

G = (P₁ · Δ₁ + P₂ · Δ₂ + ...) / Δ

Substitute the symbolic expressions for P₁, P₂, Δ, Δ₁, etc., to obtain the transfer function in terms of the a_ij coefficients. For many practical problems the algebra simplifies because some a_ij are zero or small; the symbolic method allows exact manipulation without ad-hoc block reduction.

Worked method summary and tips

• Always draw clear, labelled nodes and arrows. Write the branch gain alongside each arrow.
• Systematically list all forward paths; ensure a path does not repeat a node.
• Systematically list all loops; for each loop list the nodes it includes and the product of branch gains.
• Identify non-touching loop combinations (pairs, triplets, ...). Mark which loops touch which forward paths for Δ_k computation.
• Construct Δ using the alternating sum rule. Compute Δ_k by removing loops touching P_k.
• Apply Mason's Gain Formula to obtain the transfer function without resorting to repeated algebraic elimination.

Applications

• Derivation of transfer functions for multi-variable linear systems.
• Analysis of feedback loops and interaction between subsystems.
• Useful in control engineering, signal processing and network theory to obtain compact symbolic relations.

Final remarks

The signal flow graph method and Mason's Gain Formula provide a rigorous and often shorter path to the overall transfer function for linear systems represented by simultaneous algebraic equations. Learning to identify forward paths, loops and non-touching combinations reliably is the key skill; practice with a variety of examples (including the one above) to build fluency.