1. Chapter 3: Mathematical Modeling
Outline
 Introduction
 Types of Models
• Theoretical Models
• Empirical Models
• Semi-empirical Models
 LTI Systems
• State variables Models
• Transfer function Models
 Block diagram algebra
 Signal flow graph and Mason’s gain formula
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2.  Introduction
• A model is a mathematical representation of a
physical , biological or information system.
Models allow us to reason about a system and
make predictions about how a system will
behave.
Roughly speaking, a dynamic system is one in
which the effect of actions do not occur
immediately.
A model should capture the essence of the
reality that we like to investigate
Depending on the questions asked and
operational ranges, a physical system may have
different models. 2

3.  Types of Models
• Models can be classified based on how they are
obtained.
[A] Theoretical (or White Box) Models
• Are developed using the physical and chemical
laws of conservation, such as mass balance ,
component balance, moment balance and
energy balance.
Advantages:
 provide physical insight into process behavior.
 applicable over wide ranges of operating
conditions
Disadvantage(s): 3

4.  Types of Models…
[B] Empirical (or Black Box) Models
• Are obtained by fitting experimental data.
Advantages:
 easier to develop than theoretical models.
 applicable over wide ranges of operating
conditions
Disadvantage(s):
Typically don‟t extrapolate well!
Caution!
Empirical models should be used with caution for
operating conditions that were not included in the
experimental data used to fit the model
4

5.  Types of Models…
[C] Semi-empirical (or Gray Box) Models
• Are a combination of the models in categories (a)
& (b).
• Used in situation where much physical insight is
available but certain information( parameter) or
understanding is lacking.
• Those unknown parameter(s) in a theoretical
model are calculated from experimental data.
Advantages:
They incorporate theoretical knowledge
They can be extrapolated over a wide range of
operating conditions.
Require less development effort
5

6.  Theoretical ( White Box) Models
• In this chapter we will be dealing models that are
generated as a set of linear differential equations
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3

7.  Theoretical ( White Box) Models
7
3

8.  Theoretical ( White Box) Models
8
3

9.  Theoretical ( White Box) Models
Example 3.1: Cruise Control for a car
• Goals - maintain the speed of a car at a
prescribed value in the presence of external
disturbances (external forces such as wind
gusts, gravitational forces on a incline, etc).
Assume two forces show in the fig.
• Fp(t) – the propulsive force from the engine
• Fd(t) – a “ disturbance” force from the
9

10.  Theoretical ( White Box) Models
Example 3.1:Cruise Control for a car…
Also assume where is the gas-
pedal depression and is a constant.
Then from a simple force balance:

10

11.  Theoretical ( White Box) Models
Example 3.1:Cruise Control for a car…
• The corresponding block diagram ( for
simulation)
• Closed-loop control for the same(using P
controller)
11

12.  Theoretical ( White Box) Models
Example 3.2(a) Armature Controlled DC Motor
12
w
va
TL

13.  Theoretical ( White Box) Models
Example 3.2: Armature Controlled DC Motor…
• the back emf (refer the previous schematic) is
given by
(3.1a)
• Applying KVL to the armature circuit
(3.1b)
• Because of const. flux, the torque produced at
the shaft by the armature current is
(3.1c)
• Assuming J and B for the motor, and TL load
coupled to the shaft of the motor, the equation
becomes
(3.1d) 13

14.  Theoretical ( White Box) Models
Example 3.2: Armature Controlled DC Motor…
• By substitution of eqns. (3.1c) & (3.1d), we have

• Substituting the above result into the armature
eqn.(and assuming constant TL).
14

15.  Theoretical ( White Box) Models
Example 3.2 :Armature Controlled DC Motor…
• or equivalently,
Example 3.2(b): Field Controlled DC Motor 15
+
--
+
--
1/L
m
k
T
B
R
m
kb
DC Motor Block

16.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
Ans: Find the relation between 1 & 3, i.e. , the
Model equation relating the two waveforms.
16

17.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
Typical waveforms in the circuit are:
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18.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
•A linearized model of the transformer/rectifier
circuit , with a voltage source (with a waveform as
at (2) above) and a series resistor R ( a Thévenin
Source)
Objective: Find 18

19.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?...
Solution:
• combine series and parallel impedances to
simplify the structure. Draw as an impedance graph
Where
19

20.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
Solution…
• The system output is
• Choose mesh loops to contact all branches as
shown.
The loop equations are:
20

21.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
Solution…
• Or
• solving for i2(using Cramer‟s Rule)
21

22.  Theoretical ( White Box) Models
Example 3.3: How does the circuit shown below
work?
Solution…
• And since
•Substituting the impedances of the branches
22

23.  Theoretical ( White Box) Models
Example 3.4: A Bus Suspension System(1/4
Bus):
Assumptions:
• 1/4 model ( one of the four wheels) is used to
simplify the problem to 1D multiple spring-damper
system.
23

24.  Theoretical ( White Box) Models
Example 3.4: A Bus Suspension System(1/4
Bus)…
From the free-body diagram, the dynamic equations
become
24
M1
M2

25.  LTI Systems
• The set of ODEs drived so far are not suitable for
analysis and design, hence rearranged to a more
suitable form, i.e., State Space Model & TF
Model
[1] SS-Model:
Def: State Variable
A set of characterizing variables which give the
total information about the system under study at
any time provided the initial state & the external 25
Set of ODEs
State Space
Model
Transfer Function
Model

26.  LTI Systems
[1] SS-Model…
Where
A: system or dynamic matrix,
B: input matrix ,
C: output matrix,
D: direct transfer matrix, 26

27.  LTI Systems
Differential equation SS-Model
Example3.5:
Derive the state-space model (i.e. find the A,B,C
and D matrices) for each of the following differential
equations. Take u(t) to be the input and y(t) to be
the output.
(1)
(2)
Solution:
(1) Define
So we have the state equations:
27

28.  LTI Systems
Differential equation SS-Model
Example3.5…
And the output equation:
The state-space model is then:
28

29.  LTI Systems
[2] TF-Model
In general Transfer function is expressed as (
)
Differential equation TF model
Example 3.6:
Find the transfer function for the system given in
example 3.5
Solution:
Taking LT on both sides of the equation gives (
assuming zero initial conditions)
29

30.  LTI Systems
Differential equation TF-Model
Example 3.6….
Exercise3.1: Find the TF-model for part (2) in
example 3.5
Ans:
Exercise3.2: Find TF for a bus suspension
discussed (refer pp 23-24) using „Matlab Symbolic
Toolbox „ 30

31.  Block Diagram Algebra (Interconnection Rules)
[1] Series (Cascade) connection:
Note: This is only true if the connection of H2(s) to
H1(s) doesn‟t alter the output of H1(s)-known as the
“no-loading” condition
[2] Parallel Connection
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+
+

32.  Block Diagram Algebra (Interconnection Rules)
[3] Associative Rule:
[4] Commutative Rule:
32
+
+
+
+

33.  Block Diagram Algebra (Interconnection Rules)
[5] The “closed-loop” TF:
[5.a] Unity Feedback:
From the block diagram:
and
or
Rearranging:
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+
-Reference
input
error
Controller Plant
Feedback
path

34.  Block Diagram Algebra (Interconnection Rules)
[5.b] Feedback with sensor dynamic:
Similarly, in this case:
But now E(s) is the “indicated error” ( as opposed to
the actual error):
So
34
+
-Reference
input
error
Controller Plant
Indicated
Output
Actual
output

35.  Block Diagram Algebra (Interconnection Rules)
[5.2] Feedback with sensor dynamic:
Or

35

36.  Signal flow graph
• is a diagram consisting of nodes that are connected
by several directed branches and is a graphical
representation of a set of linear relationships .
•The signal can flow only in the direction of the arrow
of the branch and it is multiplied by a factor indicated
along the branch, which happens to be the coefficient
of a model equation(s).
Terminologies:
Node: A node is a point representing a variable or
signal
Branch: A branch is a directed line segment between
two nodes. The transmittance is the gain of a branch.
Input node: An input node has only outgoing branches
and this represents an independent variable 36

37.  Signal flow graph…
Terminologies…
Output node: An output node has only incoming
branches representing a dependent variable
Mixed node: A mixed node is a node that has both
incoming and outgoing branches
Path: Any continuous unidirectional succession of
branches traversed in the indicated branch direction is
called a path.
Loop: A loop is a closed path
Loop gain: The loop gain is the product of the branch
transmittances of a loop
37

38.  Signal flow graph…
Terminologies…
Non-touching loops: Loops are non-touching if they do
not have any common node.
Forward path: A forward path is a path from an input
node to an output node along which no node is
encountered more than once.
Feedback path (loop): A path which originates and
terminates on the same node along which no node is
encountered more than once is called a feedback
path.
Path gain: The product of the branch gains
encountered in traversing the path is called the path
gain.
38

39.  Signal flow graph…
Illustrative example:
Q. Write the equations for the system described by
the signal flow graph above. 39
Mixed
nodes
input
node
input
node
output node
x3x1
x2
x4
x3
g12 g23
g43
g32
1
Fig. Signal flow graph

40.  Signal flow graph…
Properties of Signal flow graphs
1. A branch indicates the functional dependence of
one variable on another.
2. A node performs summing operation on all the
incoming signals and transmits this sum to all outgoing
branches
40
G(s)
R(s
)
Y(s
)
Y(s
)
G(s
)
R(s)
R(s
)
-
H(s)
Y(s
)
G(s
)
E(s)1
G(s)
H(s)
Y(s
)
E(s
)
R(s
)
+
-
Fig: Block diagrams and corresponding signal flow
graphs

41.  Manson’s gain formula
In a control system the transfer function between
any input and any output may be found by Mason‟s
Gain formula. Mason‟s gain formula is given by
Where
where
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42.  Manson’s gain formula…
42

43.  Manson’s gain formula
Example3.7: Find the closed loop transfer function
Y(s)/R(s) using gain Manson‟s formula.
43
R(s) G1(s) G2(s) G3(s) x1(s)x3(s)x4(s)
-H2(s)
-H1(s)
G4(s)
Y(s)
-1

44.  Manson’s gain formula
Example 3.7…
Here we have two forward paths with gains
,
And five individual loops with gains
Note for this example there are no non-touching
loops, so ∆ for this graph is
44

45.  Manson’s gain formula
Example 3.7…
The value of ∆1is computed in the same way as ∆ by
removing the loops that touch 1st forward path M1
• In this example, since path M1 touches all the five
loops, ∆1 is found as
• Proceeding the same way , we find
•Therefore, the closed loop transfer function
between the input R(s) and output Y(s) is given by,
45

46.  Manson’s gain formula
Example 3.8
Find Y(s)/R(s) for the system represented by the
signal flow graph shown below.
46
G4
X2X3
G2 G5 X1
G3
G1
X4
-H1
-H2
G6
G7
R(s) Y(s)

47.  Manson’s gain formula
Example 3.8…
Observe from the signal flow graph , there are three
forward paths between R(s) and Y(s)
• The respective forward path gains are:
There are four individual loops with gains:
47

48.  Manson’s gain formula
Example 3.8…
Since the loops L2 & L4 are the only non-touching
loops in the graph, the determinant ∆ will be given
by:
Computing ∆1,which is computed by removing the
loops that touch fist forward path M1
∆1=1
Similarly , ∆2=1 and
Thus, the closed-loop TF is given by Y(s)/R(s)
48