Conversion of Block Diagrams into Signal Flow Graphs

In this article, we will discuss the method of converting the block diagram into a signal flow graph in a control system. We will first discuss about signal flow graph and its terminologies. We also discuss the construction of signal flow graphs from linear equations. We will then discuss about block diagram and its components. We will then discuss the steps for conversion and then see an example. We will discuss the Mason gain formula and its example. Later in the article, we will discuss the advantages, disadvantages, and applications of this method.

What is a Signal Flow Graph?

A signal flow graph is a graphical representation of the control systems. It consists of a network in which nodes represent system variables. These nodes are connected by direct branches.

Terminologies Related to the Signal Flow Graph

We will now discuss the common terminologies that are there when the signal flow graph is discussed. These terminologies are:-

• Node: It is a system variable equal to the sum of all signals arriving at a node. Output signals do not affect the value of the node.
• Branch: A branch is a line that connects two nodes in the direction of the signal flow.
• Transmittance: Transmittance is gain between nodes, also known as branch gain.
• Input or source node: It is a node that has only outgoing branches.
• Output or sink node: It is a node that has only incoming branches.
• Chain or mixed node: It is a node having both incoming and outgoing nodes.
• Path: It is the traversal from one node to another in the direction of the signal such that no node is traversed more than once.
• Forward Path: It is the path from the input node to the output node.
• Closed loop: It is the loop that starts from a particular node and ends at the same node.
• Self loop: It starts from one end and ends at the same node. It has only one branch.
• Path gain: It is the product of all branch gain in a path.
• Loop gain: It is the product of branch gain in a closed loop.
  

  

Construction of Signal Flow Graph from Linear Equation

Let us consider a system which is described by a set of linear equations

x_2=a_{12}x_1+a_{12}x_3+a_{12}x_4 \newline x_3=a_{23}x_2 \newline x_4=a_{24}x_2+a_{32}x_3+a_{44}x_4 \newline x_5=a_{25}x_2+a_{45}x_4

Where the input node is x₁ and output node is x₅

Now constructing the SFG

Step 1: First placing the nodes

Step 2: Graph from 1st Equation

Step 3: Graph from 1st and 2nd equation

Step 4: Graph from 1st, 2nd and 3rd equation

Step 5: Combing all the four equations we get the final signal flow graph

What is Block Diagram?

A system consists of number of components. The function of each component is represented by a block. All the blocks are interconnected by the lines with arrows indicating the flow of signal from output of onw block to another. These block diagram gives the idea of the system and the interrelation of various components of the system.

Different parts of Block Diagram

There some parts that are used to show different functions in a block diagram. These are:-

• Functional Block: This symbol represents the transfer function G(s) of a system.
• Summing Point: This is the point where different output signal from previous block or different signals of the system are added to form a single signal
• Take off point: It is tapping point in the system where the desired signal is tapped off to be utilized elsewhere in the diagram.

Steps to Draw Signal Flow Graph from Block Diagram

We need to follow the steps given below to convert block diagram into signal flow graph.

• Replace the input and output signal by nodes.
• Replace all the summing points by nodes.
• Replace all taking off points by nodes.
• If the branch connecting a summing point and take off point can be combined then it is represented by a single node.
• If there are more takeoff points from the same signal then all the take off points can be combined and represented by a single node.
• If the gain of the link connecting two summing points is one then the two summing points can be combined and replaced by a single node.

Mason's Gain Formula

Mason's gain formula is used to find the overall transmittance or gain of the system from signal flow graph.

T=\frac{\Sigma_{k=1}^kP_k\Delta_k }{\Delta}

Where,

• P_k is the forward path gain of k^th path from a specified input node to an output node
• \Delta_k is the path factor associated with the concerned path and involves all closed loops in the graph which are isolated from the forward path under consideration.
• \Delta = 1 - [sum of all individual loop transmittance] + [sum of loop transmittance products of all possible pairs of non-touching loops] - [sum of loop transmittance products of all possible triplets of non-touching loops] + .......

Solved Example

Convert the block diagram into signal flow graph and find the overall transfer function

The signal flow diagram of the given block diagram is

The forward paths are

P_1=G_1G_3G_4 \:\:\:\:\:P_2=-G_1G_2

The loop gains are

L_1=G_3G_4(-1) \newline L_2=G_1G_3H_1(-1) \newline L_3=G_1G_3H_1(-1) \newline L_4=G_1(-G_2)(-1)G_3H_1(-1) \newline L_5=G_1(-G_2)(-1)G_3H_1(-1)

As we can see that all loops are touching path P_1\:and \:P_2 . therefore the path factors will be unity.

\Delta=1-(L_1+L_2+L_3+L_4+L_5) \newline= 1+G_3G_4+2G_1G_3H_1+2G_1G_2G_3H_1

using mason's gain formula we get,

\frac{C}{R}=\frac{P_1\Delta_1+P_2\Delta_2}{\Delta}=\frac{G_1G_3G_4-G_1G_2}{1+G_3G_4+2G_1G_3H_1+2G_1G_2G_3H_1}

Applications of Conversion of Block Diagrams into Signal Flow Graphs

• Signal Flow Graphs are used to provide a clear representation of signal propagation in a system.
• It is also used for defining paths in a system which allows for analyzing the system and overall behavior of the system.
• It is used for design of control system and helps to analyze and implement feedback loop.
• It is used for creating mathematical models of control system

Advantages of Conversion of Block Diagrams into Signal Flow Graphs

• It provides us with a clear and easy to understand representation of a system in the form of a graph. These helps in easy identification of different components of the system.
• It reduces the complexity of a system. This makes it easy to analyze different components of the system step by step.
• Converting the block diagram to SFG gives us simple representation of the system. It makes the process of path identification easier.

Disadvantages of Conversion of Block Diagrams into Signal Flow Graphs

• They can't be used for non-linear system making their scope limited to only linear time-invariant system.
• They are subjected to manual errors because of many components and loops.
• It only provides the flow information and sometimes don't provide other parameters that are important for some control systems.
• They are sometimes difficult to use in case of complex and large control system.

Conclusion

In conclusion, we have seen the importance of signal flow diagram and the conversion of block diagram to signal flow diagram. We have learnt about the steps required for converting the block diagram into signal flow diagram with the help of examples. This method is a very useful as it provides simple and clear representation of the system. It also reduces the complexity of the system. Although it is quiet advantageous but it still has some limitation like applicable only to linear systems and not effective in case of large and complex systems.