Block
Diagrams
What Is A Block
Diagram?
What
Is A Block Diagram?
Block diagrams are ways of representing relationships between signals in
a system. Here is a block diagram of a typical control system.
Each block in the block diagram establishs a relationship between signals.
Here are the relationships
for this particular system. (Click
here for a review of signal relationships in a block diagram.)

E(s) = U(s)  Y(s)

This relationship is for
the summer/subtractor (shown with a green circle)

W(s) = K(s)E(s)

This shows how W(s) 
the control effort that drives the system being controlled, G(s)  is related
to the error. The controller is probably an amplifier  probably
a power amplifier  that provides an output to drive the plant, G(s).

Y(s) = G(s)W(s)

This shows how the output,
Y(s), is related to the control effort that drives the plant (system being
controlled ) with a transfer function, G(s).
Next, you can combine all of those relationships and get an overall relationship
between the input and the output in the system. Here is the process.

Note that Y(s) = G(s)W(s)

Note that W(s) = KE(s),
and use that in the equation for Y(s). That gives you:

Y(s) = G(s)W(s) = G(s)KE(s)

Note that the error is
given by E(s) = U(s) = Y(s), and use that in the equation for Y(s).

Y(s) = G(s)W(s) = G(s)KE(s)
= G(s)K[U(s) = Y(s)]

Now, solve for Y(s), and
you get:

Y(s) = U(s)KG(s)/[1 +
KG(s)]
That's what you need to know, and the final relationship will allow you
to compute the output given knowledge of the system components and the
input.
What if you have a more complex system? Here is a block diagram of
a slightly more complex system.
A description of this
system is as follows.

The plant being controlled
includes a pump motor. The output is the height of a liquid in a
tank.

It takes some threshold
voltage on the pump to get it started. After the voltage exceeds
the threshold, the flow rate into the pump depends upon the amount by which
the threshold is exceeded.

In the block diagram model
above, the threshold voltage (V_{T})and attendant effects
are modelled using another summer.

The controller has a transfer
function, G_{C}(s).

The sensor has a transfer
function, G_{S}(s).

We can write the mathematical
relationships that exist in this block diagram.

Y(s) = G_{P}(s)[W(s)
 V_{T}(s)]

Y(s) = G_{P}(s)[G_{C}(s)E(s)
 V_{T}(s)]

Y(s) = G_{P}(s)[G_{C}(s)(U(s)
 G_{S}(s)Y(s))  V_{T}(s)]

Now, solve for Y(s), and
you get:

Y(s) = U(s)G_{P}(s)G_{C}(s)[1
+ G_{P}(s)G_{C}(s)G_{S}(s)]
 V_{T}(s)G_{P}(s)[1
+ G_{P}(s)G_{C}(s)G_{S}(s)]
Now,
notice that the output has two components. One of those components
is due to the input  something we know about. The other component
of the output is due to the threshold voltage  something we might not
have expected.
What do we make of all this? Actually, representing offsets and thresholds
like this is a particularly good way to incorporate some simple nonlinearities
into our block diagram algebra even though the block diagram representation
was originally used only for linear systems. It's not hard to incorporate
those offsets into your analysis. Here's what you can do.

Generate a complete block
diagram for the system and be sure that you incorporate all of the offsets
in your block diagram model.

Using your block diagram
model write out the algebraic equations for each block.

Solve the equations you
have written to determine the output of the system (or the error if that
is what you are interested in). Note that the output will probably
depend upon the input and all of the offset quantities you added.

Use the solution to determine
numerical values for the output. Remember, you are often interested
in steady state solutions (DC solutions) and you can get that by using
DC gains with s = 0 in your transfer functions.