Control Systems - Block Diagram Reduction

The concepts discussed in the previous chapter are helpful for reducing (simppfying) the block diagrams.

Block Diagram Reduction Rules

Follow these rules for simppfying (reducing) the block diagram, which is having many blocks, summing points and take-off points.

Rule 1 − Check for the blocks connected in series and simppfy.

Rule 2 − Check for the blocks connected in parallel and simppfy.

Rule 3 − Check for the blocks connected in feedback loop and simppfy.

Rule 4 − If there is difficulty with take-off point while simppfying, shift it towards right.

Rule 5 − If there is difficulty with summing point while simppfying, shift it towards left.

Rule 6 − Repeat the above steps till you get the simppfied form, i.e., single block.

Note − The transfer function present in this single block is the transfer function of the overall block diagram.

Example

Consider the block diagram shown in the following figure. Let us simppfy (reduce) this block diagram using the block diagram reduction rules.

Step 1 − Use Rule 1 for blocks $G_1$ and $G_2$. Use Rule 2 for blocks $G_3$ and $G_4$. The modified block diagram is shown in the following figure.

Step 2 − Use Rule 3 for blocks $G_1G_2$ and $H_1$. Use Rule 4 for shifting take-off point after the block $G_5$. The modified block diagram is shown in the following figure.

Step 3 − Use Rule 1 for blocks $(G_3 + G_4)$ and $G_5$. The modified block diagram is shown in the following figure.

Step 4 − Use Rule 3 for blocks $(G_3 + G_4)G_5$ and $H_3$. The modified block diagram is shown in the following figure.

Step 5 − Use Rule 1 for blocks connected in series. The modified block diagram is shown in the following figure.

Step 6 − Use Rule 3 for blocks connected in feedback loop. The modified block diagram is shown in the following figure. This is the simppfied block diagram.

Therefore, the transfer function of the system is

$$frac{Y(s)}{R(s)}=frac{G_1G_2G_5^2(G_3+G_4)}{(1+G_1G_2H_1)lbrace 1+(G_3+G_4)G_5H_3 brace G_5-G_1G_2G_5(G_3+G_4)H_2}$$

Note − Follow these steps in order to calculate the transfer function of the block diagram having multiple inputs.

Step 1 − Find the transfer function of block diagram by considering one input at a time and make the remaining inputs as zero.

Step 2 − Repeat step 1 for remaining inputs.

Step 3 − Get the overall transfer function by adding all those transfer functions.

The block diagram reduction process takes more time for comppcated systems. Because, we have to draw the (partially simppfied) block diagram after each step. So, to overcome this drawback, use signal flow graphs (representation).

In the next two chapters, we will discuss about the concepts related to signal flow graphs, i.e., how to represent signal flow graph from a given block diagram and calculation of transfer function just by using a gain formula without doing any reduction process.