Sub Code
Sub Name
Credits
Semester
: 19ECE301
: Control Theory (Pre-Requisite(s): Signals and Systems)
:4
:V
Branch
Faculty
: ECE
: Ms. Bhavana V
Assistant Professor, Dept. of ECE
Lecture 15-18
Reduction of Multiple Subsystems
Signal Flow Graph
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Signal-flow graphs
• Signal-flow graphs are an alternative to block diagrams. This approach was coined by S.J
Mason
• Block diagrams, consists of blocks, signals, summing junctions, and pickoff points, etc
• A signal-flow graph consists only of branches, which represent systems, and nodes, which
represent signals.
• The main advantage of SFG, it doesn't require any reduction process because of
availability of gain formula which gives the transfer function of control system without
any transformation.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Signal-flow graphs
• SFG is a graphical representation of the block diagram or representation of the relationships
between the variables of a set of linear algebraic equations.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Signal-flow graphs
1. The SFG is applicable only to linear time invariant systems
2. The signal flow along the branch is the direction of arrow associated with the branch
3. The signal gets multiplied by the branch gain or branch transmittance when it travels
through branch.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Signal-flow graphs
4. The value of the variable represented by the node is given by the algebraic sum of
all the signals entering that node.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Signal-flow graphs
5. The value of the variable represented by any node is available to all the branches
leaving that node.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Numerical
A system is described by the following set of equations . Construct the signal flow
graph of the system.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution
Note : The SFG should
contain five nodes, since
there are five variables.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Terminologies used in SGF
•
Source Node: The node having only outgoing branches is known as the Source Node or
Input Node
In Fig A, the source node is 𝑥1
•
Sink Node/ Output Node: The node having only incoming branches is known as sink node
or output node.
In Fig A, the source node is 𝑥5 = output node, after transmitting 𝑥5 through a
branch of unity gain .
•
Chain Node: A node having incoming and outgoing branches is known as chain node.
In Fig A, the chain nodes = 𝑥2 , 𝑥3 , 𝑥4
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Terminologies used in SGF
• Forward Path : It is a path from input node to the output node.
---- In forward path no node should be traversed more than once.
---- In Fig A,
•
Feedback Loop: A path which originates from a particular node and terminating on
the same node , travelling through at least one other node without tracing any
other node twice is called feedback loop or simply the loop.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Terminologies used in SGF
• Self Loop: A loop which originates and terminates on the same node is called the self
loop.
• Path Gain: The product of all the branch gains in a forward path is called the path
gain.
• Loop Gain: It is the path gain of a loop
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Terminologies used in SGF
Non Touching Loops: Two loops are said to be non touching if they don’t share a common
node.
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Masons Gain Formula
The gain formula which gives the relationship between input variable and output variable
of SFG is known as Mason’s gain formula.
Where
𝐶(𝑆) σ𝑁
𝐾=1 𝑀𝐾 ∆𝐾
𝑇 𝑆 =
=
𝑅(𝑆)
∆
N= Number of forward paths between R(S) and C(S)
𝑀𝐾 =gain of the 𝐾 𝑡ℎ forward path
∆ = Determinant of the SFG , given by
∆ = 1 – [ sum of all the individual feedback loop gains including the self loop] + [ sum of the gain products of all
combinations of two non touching loops ] – [ sum of the gain products of all combination of three non touching
loops ] + …………
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Masons Gain Formula
∆𝐾 = 1 – [ sum of all the individual feedback loop gains including the self loop which are not present in the
𝐾 𝑡ℎ forward path] + [ sum of the gain products of all combinations of two non touching loops which are not
present in 𝐾 𝑡ℎ forward path ] – [ sum of the gain products of all combination of three non touching loops which
are not present in the 𝐾 𝑡ℎ forward path ] + …………
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Numerical
For the signal flow graph shown below ,obtain the transfer function
𝑥5
𝑥1
using masons gain
formula
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
∆ = 1 – [ sum of all the individual feedback loop gains including the self loop] + [ sum of the gain products of all combinations of
two non touching loops ] – [ sum of the gain products of all combination of three non touching loops ] + …………
∆𝟏 = 1 – [ sum of all the individual feedback loop gains including the self loop which are
not present in the 1st forward path] +
[ sum of the gain products of all combinations of two non touching loops which are not present in 1st forward path ] – [ sum of
the gain products of all combination of three non touching loops which are not present in the 1st forward path ] + …………
∆𝟐 = 1 – [ sum of all the individual feedback loop gains including the self loop which are not present in the 2nd forward path] + [ sum
of the gain products of all combinations of two non touching loops which are not present in 2 nd forward path ]
– [ sum of the gain products of all combination of three non touching loops which are not present in the 2 nd forward path ] + …………
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Numerical
For the signal flow graph shown below ,obtain the transfer function
𝑦7
𝑦1
and
𝑦6
𝑦1
using
masons gain formula
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Numerical
For the block diagram shown below, construct the SFG. Find the transfer function
𝐶(𝑆)
𝑅(𝑆)
using
Masons Gain Formula
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Numerical
For the network shown below, construct the SFG. Find the transfer function
𝑉0 (𝑆)
𝑉𝑖 (𝑆)
using
Masons Gain Formula
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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Solution (Contd…)
𝑉0 (𝑆)
10
=
𝑉𝑖 (𝑆) 10𝑆 2 + 21𝑆 + 10
Ms. Bhavana V,Assistant Professor, Dept. of ECE
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