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Control Systems – Block Diagram Reduction
Parallel Form
Feedback Form
Moving Blocks
Superposition
Example
Control Systems
MEC-C305Block Diagram Reduction1
Block Diagram Reduction
Parallel Form
Feedback Form
Moving Blocks
Superposition
ExampleComponents of a Block Diagram for a
Linear Time Invariant System
System components are alternatively
called elements of the system.
Block diagram has four components:
Signals
System/ block
Summing junction
Pick-off/ Take-off pointSubsystems are represented in block diagrams as
blocks, each representing a transfer function. In
this unit we will consider how to combine the
blocks corresponding to individual subsystems so
that we can represent a whole system as a single
block, and therefore a single transfer function.
Here is an example of this reduction:
Reduced Form:
Block Diagram ReductionFirst we summarize the elements of block
diagrams:
We now consider the forms in which blocks are typically
connected and how these forms can be reduced to single
When multiple subsystems are connected such
that the output of one subsystem serves as the
input to the next, these subsystems are said to
The algebraic form of the final output clearly shows the
equivalent system TF—the product of the cascaded
subsystem TF’s.When reducing subsystems in cascade form we
make the assumption that adjacent subsystems
do not load each other. That is, a subsystem’s
output remains the same no matter what the
output is connected to. If another subsystem
connected to the output modifies that output,
we say that it loads the first system.
Consider interconnecting the circuits (a) and (b)
below:When reducing subsystems in cascade form we make
each other. That is, a subsystem’s output remains the
same no matter what the output is connected to. If
another subsystem connected to the output modifies
that output, we say that it loads the first system.
Consider interconnecting the circuits (a) and (b)
below:
The overall TF is not the product of the individual TF’s!We can prevent loading by inserting an amplifier. This
amplifier should have a high input impedance so it
does not load its source, and low output impedance so
it appears as a pure voltage source to the subsystem
it feeds into.
If no actual gain is desired then K = 1 and the
“amplifier” is referred to as a buffer.Parallel Form Graphs
Parallel subsystems have a common input and their
outputs are summed together.
The equivalent TF is the sum of parallel TF’s (with matched
signs at summing junction).Feedback Form
Systems with feedback typically have the following form:
Noticing the cascade form within the feedforward and
feedback paths wecan simplify:We can easily establish the following two facts:
E(s) = R(s) ∓ C(s)H(s)
C(s) = E(s)G(s)
We can now eliminate E(s) to obtain,
G(s)
Ge(s) =
1± G(s)H(s)Moving Blocks
A system’s block diagram may require some modification
before the reductions discussed above can be applied.
We may need to move blocks either to the left or right
of a summing junction:Or we may need to move blocks to the left or right of
a pickoff point:Canonical Form of a Feedback Control System
The system is said to have negative feedback if the sign at the summing
junction is negative and positive feedback if the sign is positive.1. Open loop transfer function

1. Feed Forward Transfer function
2. control ratio
3. feedback ratio
4. error ratio
5. closed loop transfer function
6. characteristic equation
7. closed loop poles and zeros if K=10.
H(s)Characteristic EquationUnity Feedback SystemCharacteristic Equation
• The control ratio is the closed loop transfer function of
the system.
• The denominator of closed loop transfer function
determines the characteristic equation of the system.
• Which is usually determined as:
1 G(s)H(s) = 0Reduction techniques
G1 G2 G1G2
9. Combining blocks in parallel5. Moving a pickoff point ahead of a block
10. Moving a summing point ahead of a block
11. Moving a pickoff point behind a block
Reduction techniques6. Eliminating a feedback loop
G H
12. Swap with two neighboring summing points
Reduction techniques
1 ∓ 𝐻𝐺Block Diagram Transformation Theorems
The letter P is used to represent any transfer function,
and W, X , Y, Z denote any transformed signals.Transformation Theorems Continue:Transformation Theorems Continue:Reduction of Complicated Block Diagrams:Example 1: Reduce the Following Block Diagram.Example 1 : Continue.
However in this example step-4 does not apply.
However in this example step-6 does not apply.Example 2: Simplify the Block Diagram.Example 2: Continue.Example 3: Reduce the Block Diagram.Example 3: Continue.Example 4: Reduce the Block Diagram.Example 4: Continue.Example 8: For the system represented by the
following block diagram determine:
13. Open loop transfer function
14. Feed Forward Transfer function
15. control ratio
16. feedback ratio
17. error ratio
18. closed loop transfer function
19. characteristic equation
20. closed loop poles and zeros if K=10.First we will reduce the given block diagram to
canonical form
+Example 9: For the system represented by the
following block diagram determine:
21. Open loop transfer function
22. Feed Forward Transfer function
23. control ratio
24. feedback ratio
25. error ratio
26. closed loop transfer function
27. characteristic equation
28. closed loop poles and zeros if K=100.Example-10: Reduce the system to a single transfer
function.Example 10: Continue.Example 10: Continue.Example
Reduce the following system to a single TF:
First we can combine the three summing
junctions together…We can now recognize the parallel form in the feedback
path:
We now have G1 cascaded with a feedback subsystem:Example 2
Reduce the following more complicated block
diagram:
Steps:
Rightmost feedback loop can be reduced Create
parallel form by moving G2 leftReduce parallel form involving 1/G2 and unity
Push G1to the right past the summing junction to create a
parallel form in the feedback pathReduce parallel form on left Recognize cascade
form on rightReduce feedback form on leftSuperposition of Multiple InputsExample: Multiple Input System. Determine the output
C due to inputs R and U using the Superposition
Method.Example: Continue.Example: Continue.Example 15: Multiple-Input System. Determine the
output C due to inputs R, U1 and U2 using the
Superposition Method.Example-15: Continue.Example 15: Continue.Example 16: Multi-Input Multi-Output System.
Determine C1 and C2 due to R1 and R2.Example 16: Continue.Example:
Continue.
When R1 = 0,
When R2 = 0,

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