Signal Flow Graph in Control Systems

In control systems engineering, complex systems are often represented using block diagrams. However, as the number of components and feedback loops increases, analyzing these diagrams becomes difficult. A Signal Flow Graph (SFG) provides a simpler graphical method for representing and analyzing the relationships between system variables. By using SFGs and Mason’s Gain Formula, engineers can efficiently determine the transfer function of a system.

A Signal Flow Graph (SFG) is a graphical representation of the mathematical relationships among variables in a system. It consists of nodes connected by directed branches that indicate the direction of signal flow and the gain associated with each connection.

Nodes and Branches are the two basic elements of a Signal Flow Graph.

Node

It is a point that represents a variable or a signal. There are three types of nodes:

• Input Node (Source Node): A node that only provides inputs and has outgoing branches. If the arrow starts from the node and points away from it.
• Output Node (Sink Node): A node that only has incoming branches. If the arrows point toward the node and end there.
• Mixed Node: A node that has both incoming and outgoing branches. If arrows both enter and leave the node.

Branch

A branch represents a line segment connecting two nodes in a signal flow graph. Electronic equipment may produce positive or negative gain, and signal movement follows a specific direction. Branch representation therefore includes both gain and direction to complete a signal flow graph. A simple diagram helps in understanding branch connection and signal direction clearly.

Example

Here, the black dots (y₁, y₂, y₃, y₄) represent the nodes of the Signal Flow Graph, while the lines connecting them are called branches. Each branch has a direction indicated by an arrow. Based on the branch gains, a, b, and c are positive-gain branches, while d is a negative-gain branch, which is why it is represented with a negative sign.

Characteristics of a Signal Flow Graph

1. Nodes
2. Branches or Edges
3. Forward Paths
4. Single Loops
5. Non-touching Loops
6. Mason Gain Formula

$\frac{C(s)}{R(s)} = \frac{\sum_{i=1}^{N} P_i \Delta_i}{\Delta}, \quad \text{where} \quad \begin{aligned} N &= \text{total number of forward paths} \\ P_i &= \text{gain of the } i^{th} \text{ forward path} \\ \Delta &= \text{determinant of the graph} \\ \Delta_i &= \text{path factor for the } i^{th} \text{ forward path} \end{aligned}$

The determinant of the graph (∆) and the path-factor for the i^th path (∆_i) are defined as follows:

In this formula the loops of the Signal Flow Graph is very important. In the next example we will see how can we get a transfer function from this formula.
Transfer function T, R is input, C is output, G are the gains and H are the feedbacks of a transfer function.

Here, two paths are available. The transfer function will be:

$\frac{C(s)}{R(s)} = \frac{\sum_{i=1}^{N} P_i \Delta_i}{\Delta} = \frac{P_1 \Delta_1 + P_2 \Delta_2}{\Delta} = \frac{G_1 G_2 G_4 + G_1 G_3 G_4} {1 – G_1 G_4 H_1 + G_1 G_2 G_4 H_2 + G_1 G_3 G_4 H_2} = \frac{G_1 G_4 (G_2 + G_3)} {1 – G_1 G_4 H_1 + G_1 G_2 G_4 H_2 + G_1 G_3 G_4 H_2}$

Signal Flow Graph from Block Diagram

In the field of engineering, block diagrams are used to simplify intricate circuits. The block diagram shows numerous electronic components as well as input and output. As a result, we must carefully comprehend this before drawing the Signal Flow Graph. Prior to that, we must comprehend the jargon.

• R(s) is the input point, C(s) is the output point.
• The circle with a cross is called summing point(S), and the branches meet at the dot point is called take-off point.
• The “G” inside a box called the gain, there can be many gain blocks in a single block diagram.
• Since electronics components also provide feedback, “H” inside the box is referred to as the feedback.

We shall follow a few rules while trying to obtain a SFG from a block diagram of a system.

1. All the variables, summing points and take off points are represented by nodes.
2. If a summing point is placed before the take off point, then the summing point and the takeoff point are represented by a single node.
3. If a summing point is placed after the take off point, then the summing point and the takeoff point are represented by separate nodes.

These rules are based on the fact that the value of a variable at a node is the sum of incoming nodes and the outgoing nodes do not affect the value of the variable at that node

Now, connect the nodes in the same sense as the block diagram and then indicate the direction on the branches. This completes the drawing of the signal flow graph.

this is how to obtain the signal flow graph from a block diagram