Block Diagrams of Control Systems 1.4

In the last tutorial, we learnt about transfer functions. In this tutorial we shall learn about block diagrams in control systems. A block diagram is an intuitive way of representing a system. It is a graphical representation that shows us how the systems are interconnected and how the signal flows between them. In other words, block diagrams are mathematical drawings of the system. While a transfer function does not give the internal details of the system, block diagrams can be made to show the required internal details.

Let’s now get started with learning about block diagrams by understanding the basic terminologies involved.

1) Blocks

A block represents a system or a sub system. The block usually has the transfer function of that particular system of subsystem which it represents.

2) Arrows

Arrows represent the direction of the flow of signal or information. This will tell us how the individual systems/subsystems are connected.

3) Summing Point

The summing points add or subtract signals. It is a small circle with a small "+" or "-" near the entry of each signal telling us if the signals are being added or subtracted.

We can also include a small X inside the circle and write the "+-" signs inside. It would look like,

You can follow whichever representation you are comfortable with.

4) Take Off Point

When we need to use the same signal to feed into multiple systems, we make use of take off points.

With this, we are all set to draw our first block diagram.

Remember the example of a microwave cooking a potato from our first tutorial? Its block diagram would be:

This was quite simple and should help you get familiarized with the terminologies.

Now, let’s move on to something more complex. We shall now draw the block diagram of a series RLC circuit which has been our go-to example!

In general, drawing a block diagram for a system involves breaking down the equations and presenting it in the form of blocks. While doing so, we always consider the input variable first and then we end with the output variable. This example of a series RLC circuit will make this clear.

Applying Kirchhoff’s voltage law to the loop shown above,

Laplace transformation of the above equations with initial conditions assumed zero will be:

Here V_i(s) is the input variable, V_o(s) is the output variable and I(s) is an intermediate variable. Usually a system has more intermediate variables and block diagrams helps us in visualizing these intermediate variables.

A general procedure that is followed is that we start with making use of the input variable and other required variables to form the intermediate variable(s) and use them to form the output.

In this case, we shall form I(s) using V_i(s) and V_o(s) and then using this I(s), we shall form V_o(s) all in accordance with the system equations which we formed above.

Now,

First we shall use a summing point.

The output of the summing point is passed through a block of transfer function:

Next, we shall use the other equation,

We combine the above two blocks and then with the help of a take off point, we connect the output to the summing point where we need the output variable as one of the inputs.

The above is the block diagram representation of the series RLC circuit.

Though block diagrams are simple, it becomes really confusing when a large number of blocks are present. This gives us a need to reduce the block diagrams for our convenience. We shall now discuss a few rules that will help us reduce complex block diagrams. In fact, using these rules, we can reduce any number of blocks into a single block which eventually represents the transfer function of the entire system. These rules are often referred to as block diagram algebra and we shall now look at them one by one.

1) Blocks in cascade.

When there are two or more blocks in a cascade (one next to the other), the resultant block would just be the product of the transfer functions of individual blocks.

2) Blocks in parallel.

When there are two or more blocks in parallel, the resultant block would just be the sum of the transfer functions of individual blocks.

3) Eliminating a feedback loop.

Consider a simple feedback loop with a system block G(s) and feedback block H(s).

If we just look at the block G(s) with E(s) as input and C(s) as output,

Where E(s) is the difference or sum of the input and the feedback depending upon the type of feedback. For a feedback that is negative, E(s) is the difference of the input and the feedback and for a feedback that is positive, E(s) is the sum of the input and the feedback.

Now,

Hence, the above loop can be replaced by,

4) Moving a take off point to the left of the block.

When we need to move a take off point to the left of a block, we introduce a block with the same transfer function in that branch of a take off point. The diagram below will make it clear.

5) Moving a take off point to the right of the block.

Similar to the previous one, when we have to move a take off point to the right of a block, we introduce a block with the reciprocal of the transfer function in that branch of the take off point.

6) Moving a summing point to the right of block.

When a summing point has to be moved to the right of the summing block, the following modifications are to be made.

7) Moving a summing point to the left of a block.

When a summing point has to be moved to the left of the summing block, the following modifications are to be made.

8) Interchanging summing points.

The summing points can be interchanged without any modifications.

9) Moving a take off point to the right of the summing point.

When we move the take off point to the right of the summing point, we need to compensate for the arithmetic changes so the value of the branch of the take off point as well as the output doesn’t change.

10) Moving a take off point to the left of the summing point.

Similar to the previous rule, when we move the take off point to the left of the summing point, we need to compensate for the arithmetic changes as shown.

Okay, now we are done! The idea behind these rules are just to keep the resulting values the same by slightly modifying the block diagram which shall compensate for the changes. Take your time and go through this once again.

It’s time for us to try out these rules and reduce a slightly complicated block diagram into a single block.

As we can see, blocks G₃ and G₄ are in cascade, so we can combine them according to the first rule we learnt.

Now, the circled section forms a closed loop and we can reduce it using the third rule that we learnt.

You see that take off point there, we can move that to the right of the block with the help of the 5th rule.

Next, we can see that G₂ and the block next to it are in cascade, and hence, they can be combined easily.

The circled portion is a closed loop which can be reduced using the 3rd rule.

Again we can see that G₁ and the block next to it are in cascade and hence can be combined.

And again, this is a simple closed loop which we can reduce using the 3rd rule having a little patience while calculating.

And voila!! We have reduced that big block diagram into a single block. It is to be noted that reducing a block diagram to a single block is not always required and we reduce block diagrams as for our convenience of understanding.

To summarize, in this tutorial we learnt what block diagrams are, the basic terminologies involved. Then we learnt how to draw block diagrams for a system followed by the rules for reducing complex block diagrams. In the next tutorial, we shall learn about signal flow graphs. Till then, try reducing this block diagram into a single block and verify it with the given answer.