Reduction of Multiple Subsystems
reduce a block diagram of multiple
subsystems to a signal block representing
the transfer function from input to output
Before this we only worked with individual
subsystems represented by a block with
its input and output.
Complex systems are represented by the
interconnection of many subsystems.
In order to analyze our system, we want
to represent multiple subsystems as a
single transfer function.
A subsystems is represented as a block
with an input and output and a transfer
Many systems are composed of multiple
subsystems. So, we need to add a few
more schematic elements to the block
Output signal, C(s), is the algebraic sum of the
input signals, R1(s), R2(s) and R3(s).
Distributes the input signals, R(s),
undiminished, to several output points.
There are three topologies that can be
used to reduce a complicated system to a
a. cascaded subsystem
b. equivalent transfer function
Equivalent transfer function is the output
divided by the input.
Parallel subsystems have a common input and
output formed by the algebraic sum of the
outputs from all of the subsystems.
It is the same as the closed loop system that
we learn in Chapter 1.
a. closed loop system
b. closed loop, G(s)H(s) is open loop transfer
Moving blocks to create familiar forms
Cascade, parallel and feedback topologies are
not always apparent in a block diagram.
You will learn block moves that can be made in
order to establish familiar forms when they
almost exist. I.e. move blocks left and right
past summing junctions and pickoff points.
algebra for summing
equivalent forms for
moving a block
a. to the left past a
b. to the right past a
algebra for pickoff
for moving a
a. to the left past
a pickoff point;
b. to the right
past a pickoff
Block diagram reduction via familiar forms
Reduce the block diagram to a single
Steps in solving
a. collapse summing
b. form equivalent
in the forward path
parallel system in the
c. form equivalent
feedback system and
multiply by cascaded
Block diagram reduction by moving blocks
Reduce the system shown to a single
First, move G2(s) to the left past the pickoff point
to create parallel subsystems, and reduce the
feedback system consisting of G3(s) and H3(s).
Second, reduce the parallel pair consisting of
1/g2(s) and unity and push G1(s) to the right past
the summing junction, creating parallel subsystems
in the feedback.
Third, collapse the summing junctions, add the two
feedback elements together, and combined the last
two cascaded blocks.
Fourth, use the feedback formula to obtain
Finally multiply the two cascaded blocks and
obtain the final result.
Find the equivalent transfer function,
Combine the parallel blocks in the forward path. Then, push 1/s to
the left past the pickoff point.
Combine the parallel feedback paths and get 2s. Apply the
feedback formula and simplify
A technique for reducing signal-flow graphs to
single transfer function that relate the output
of system to its input.
We must understand some components
before using Mason’s rule
Product of branch gains found by going through a path
that starts at a node and ends at the same node,
following the direction of the signal flow, without passing
through any other node more than once.
Product of gains found by going through a path from the
input node of the signal-flow graph in the direction of
Loops that do not have any nodes in common.
Loop G2(s)H1(s) does not touch loops G4(s)H2(s),
G4(s)G5(s)H3(s) and G4(s)G6(s)H3(s)
Product of gains form nontouching loops taken
two, three, four, or more at a time.
The transfer function, C(s)/R(s), of a system
represented by a signal-flow graph is
G (s) =
k = number of forward path
Tk = the kth forward - path gain
∆ k = formed by eliminating from ∆
those loop gains that touch the kth forward path.
∆ = 1 - Σ loop gains
+ Σ nontouching loop gains taken two at a time
− Σ nontouching loop gains taken three at a time
+ Σ nontouching loop gains taken four at a time