Block diagrams
1 Simple inteconnection of blocks
We have already seen three basic forms of block diagrams:
We can use the basic rules above to simplify block diagrams. We provide a few examples below.
Example 1 Simplify the block diagram shown below.
Thus, the simplified TF is
\[ T(s) = \frac{G_1(s) G_2(s) G_3(s) } { 1 + G_1(s) G_2(s) G_3(s) H_1(s) H_2(s) }. \]
Example 2 Simplify the block diagram shown below.
In the first approach, we can view the system as three nested feedback loops. The first loop is highlighted above, which simplifies to
\[ \dfrac{G_2(s) G_3(s)}{1 + G_2(s) G_3(s) H_1(s)} \]
The second loop simplified to
\[ \dfrac{ \dfrac{G_2(s) G_3(s)}{1 + G_2(s) G_3(s) H_1(s)} } { 1 + \dfrac{G_2(s) G_3(s) H_2(s)}{1 + G_2(s) G_3(s) H_1(s)} } = \dfrac{ G_2(s) G_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) } \]
This is a simple feedback loop. Thus, we can simplify it to \[ \dfrac{ \dfrac{ G_1(s) G_2(s) G_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) }} { 1 + \dfrac{ G_1(s) G_2(s) G_3(s) H_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) } } = \dfrac{G_1(s) G_2(s) G_3(s) } {1 + G_2(s) G_3(s) \bigl[ H_1(s) + H_2(s) + H_3(s) \bigr] }. \]
This is a simple feedback circuit. Thus, the simplified transfer function is \[ \dfrac{G_1(s) G_2(s) G_3(s) } {1 + G_2(s) G_3(s) \bigl[ H_1(s) + H_2(s) + H_3(s) \bigr] }. \]
The above two examples show that we can simplify block diagrams using the simple rules for interconnected blocks. However, these simple ideas are unable to simplify more complicated block diagrams shown below.
Example 3 Simplify the block diagram shown below.
In order to simplify such diagrams, we need to develop rules for moving pick up points and adders around blocks.
2 Moving pickup points and adders around blocks
There are four basic rules for simplifying block diagrams.
Moving a pickup point after a block to before a block.
In both cases, \(Y(s) = X(s) G(s)\) and \(Z(s) = X(s)G(s)\).
Moving a pickup point before a block to after a block.
In both cases, \(Y(s) = X(s) G(s)\) and \(Z(s) = X(s)\).
Moving an adder after a block to before a block.
In both cases, \(Z(s) = X(s)G(s) + Y(s)\).
Moving an adder before a block to after a block.
In both cases, \(Z(s) = X(s)G(s) + Y(s)G(s)\).
3 Back to the example
We now see how we can use the rule above to simplify the block diagram of Example 3.
We look at the highlighted sub-block and move the pickup point from behind \(G_4(s)\) to after \(G_4(s)\).
The simplified diagram is of the form that we can identify the three basic blocks and iteratively simplify it. However, the method is tedious. We can make it much simpler by moving the adders around so that all of them are in the middle.
The final diagram now has three adders in feedback, similar to Example 2. Therefore, we can simplify the TF as
\[ \dfrac{G_1(s) G_2(s) G_3(s) G_4(s)} {1 + G_2(s)G_3(s)G_4(s) \Bigl[ \dfrac{H_1(s)}{G_2(s)} + \dfrac{H_2(s)}{G_4(s)} + H_3(s)G_1(s) \Bigr]}. \]