This document provides an overview of mathematical modeling and linear time-invariant (LTI) systems. It discusses different types of models including theoretical, empirical, and semi-empirical models. It also describes state-space models, transfer function models, and how to derive these from differential equations. Block diagram algebra and signal flow graphs are introduced as ways to represent LTI systems. Mason's gain formula is presented as a method for calculating transfer functions using signal flow graphs. Several examples are worked through to illustrate these modeling concepts and techniques.

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
1

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
6
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:
17

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
31
+
+

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:
33
+
-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
41

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