# Control Systems Block Diagram Reduction - Control Systems

## How to simplify the block diagrams?

The concepts discussed as of now are very useful for reducing (simplifying) the block diagrams.

## Block Diagram Reduction Rules

Following rules are used for simplifying (reducing) the block diagram, which includes many blocks, summing points and take-off points.
• Rule 1 − Check for the blocks connected in series and simplify.
• Rule 2 − Check for the blocks connected in parallel and simplify.
• Rule 3 − Check for the blocks connected in feedback loop and simplify.
• Rule 4 − If there is difficulty with take-off point while simplifying, shift it towards right.
• Rule 5 − If there is difficulty with summing point while simplifying, shift it towards left.
• Rule 6 − Repeat the above steps till you get the simplified form, i.e., single block.
Note − The transfer function present in this single block is the transfer function of the overall block diagram.

### Example

Let’s see following block diagram as mentioned in the diagram. You can simplify (reduce) this block diagram by using the block diagram reduction rules.
Step 1 – Here apply Rule 1 for blocks${G}_{1}$andG2. After that, apply Rule 2 for blocksG3andG4
G4
4. Now see the simplified block diagram as mentioned below.
Step 2 – Now apply Rule 3 for blocks ${G}_{1}{G}_{2}$ and ${H}_{1}$. Then use Rule 4 for shifting take-off point after the block ${G}_{5}$ . See the following modified block diagram as mentioned in the following figure.
Step 3 – Apply Rule 1 for blocks $\left({G}_{3}+{G}_{4}\right)$ and ${G}_{5}$. Let’s see the simplified block diagram as mentioned in the following figure.
Step 4 – Now use Rule 3 for blocks $\left({G}_{3}+{G}_{4}\right){G}_{5}$ and ${H}_{3}$. The modified block diagram is shown in the following figure.
Step 5 – Let’s apply the Rule 1 for blocks connected in series. The modified block diagram is shown in the following figure.
Step 6 – You can use Rule 3 for blocks ${G}_{1}{G}_{2}$connected in feedback loop. The modified block diagram is mentioned below in the following figure. This is the simplified block diagram.
Therefore, the transfer function of the system is $\frac{Y\left(s\right)}{R\left(s\right)}=\frac{{G}_{1}{G}_{2}{G}_{5}^{2}\left({G}_{3}+{G}_{4}\right)}{\left(1+{G}_{1}{G}_{2}{H}_{1}\right)\left\{1+\left({G}_{3}+{G}_{4}\right){G}_{5}{H}_{3}\right\}{G}_{5}-{G}_{1}{G}_{2}{G}_{5}\left({G}_{3}+{G}_{4}\right){H}_{2}}$
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 uses more time for complicated systems. The reasons might be need to draw the (partially simplified) block diagram after each step. To overcome this you need use signal flow graphs (representation).
Coming two chapters will discuss about the concepts related to signal flow graphs which means how to represent signal flow graph from a given block diagram and how to calculate the transfer function with a formula without doing any reduction process.