The document discusses mathematical modeling of dynamic systems, including obtaining differential equations to represent system dynamics, different representations like transfer functions and impulse response functions, using block diagrams to visualize system components and signal flows, modeling various physical systems like mechanical, electrical, and thermal systems, and representing systems using signal flow graphs. It provides examples of obtaining transfer functions for different system types and using block diagram reduction techniques to find overall transfer functions.

1. Chapter 3
Mathematical Modeling of Dynamic System
3.1 Introduction
A mathematical model of a dynamic system is defined as a set of equations that represents
the dynamics of the system accurately, or at least fairly well.
Note that a mathematical model is not unique to a given system. A system may be
represented in many different ways and, therefore, may have many mathematical models,
depending on one’s perspective.
The dynamics of many systems, whether they are mechanical, electrical, thermal, economic,
biological, and so on, may be described in terms of differential equations.
Such differential equations may be obtained by using physical laws governing a particular
system—for example,
Newton’s laws for mechanical systems and
Kirchhoff’s laws for electrical systems.
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2. Cont..
Mathematical models may assume many different forms. Depending on the
particular system and the particular circumstances, one mathematical model
may be better suited than other models.
For example,
in optimal control problems, it is advantageous to use state-space representations.
for the transient-response or frequency-response analysis of single-input, single-output,
linear, time-invariant systems, the transfer-function representation may be more
convenient than any other.
Once a mathematical model of a system is obtained, various analytical and
computer tools can be used for analysis and synthesis purposes
Simplicity Versus Accuracy
In obtaining a mathematical model, we must make a compromise between
the simplicity of the model and the accuracy of the results of the analysis
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3. Cont..
Linear Systems
A system is called linear if the principle of superposition applies.
The principle of superposition states that the response produced by the
simultaneous application of two different forcing functions is the sum of the
two individual responses.
Hence, for the linear system, the response to several inputs can be calculated
by treating one input at a time and adding the results.
In an experimental investigation of a dynamic system, if cause and effect are
proportional, thus implying that the principle of superposition holds, then the
system can be considered linear
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4. Cont..
Linear Time-Invariant Systems and Linear Time-Varying Systems
A differential equation is linear if the coefficients are constants or functions
only of the independent variable.
Dynamic systems that are composed of linear time-invariant lumped-
parameter components may be described by linear time-invariant differential
equations—that is, constant-coefficient differential equations.
Systems that are represented by differential equations whose coefficients are
functions of time are called linear time-varying systems.
An example of a time-varying control system is a spacecraft control system.
(The mass of a spacecraft changes due to fuel consumption.)
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5. 3.2 TRANSFER FUNCTION AND IMPULSE RESPONSE FUNCTION
Transfer Function
The transfer function of a linear, time-invariant, differential equation system is defined as the
ratio of the Laplace transform of the output (response function) to the Laplace transform of
the input (driving function) under the assumption that all initial conditions are zero.
Consider the linear time-invariant system defined by the following differential equation:
where y is the output of the system and x is the input.
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6. Properties of transfer function
1. it is an operational method of expressing the differential equation that relates the output
variable to the input variable.
2. The transfer function is a property of a system itself, independent of the magnitude and
nature of the input or driving function.
3. It includes the units necessary information to relate the input to the output; however, it
does not provide any information concerning the physical structure of the system.
(The transfer functions of many physically different systems can be identical.)
4. If the transfer function of a system is known, the output or response can be studied for
various forms of inputs with a view toward understanding the nature of the system.
5. If the transfer function of a system is unknown, it may be established experimentally by
introducing known inputs and studying the output of the system.
Once established, a transfer function gives a full description of the dynamic characteristics
of the system, as distinct from its physical description.
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7. Cont..
For a linear, time-invariant system the transfer function G(s) is
Where X(s) is the Laplace transform of the input to the system and
Y(s) is the Laplace transform of the output of the system, where we assume that all initial
conditions involved are zero.
 It follows that the output Y(s) can be written as the product of G(s) and
X(s), or
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8. Impulse-Response Function
Consider the output (response) of a linear time-invariant system to a unit-impulse input when the
initial conditions are zero
Since the Laplace transform of the unit-impulse function is unity, the Laplace transform of the
output of the system is
The inverse Laplace transform of the output given by Equation above gives the impulse response of
the system
The inverse Laplace transform of G(s), or is called the impulse-response function. This function g(t)
is also called the weighting function of the system
Therefore, the transfer function and impulse-response function of a linear, time-invariant system contain
the same information about the system dynamics
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9. 3.3 Block Diagrams
A block diagram of a system is a pictorial representation of the functions performed by each
component and of the flow of signals
• Such a diagram depicts the interrelationships that exist among the various components.
• Differing from a purely abstract mathematical representation, a block diagram has the
advantage of indicating more realistically the signal flows of the actual system.
• In a block diagram all system variables are linked to each other through functional blocks.
• The functional block or simply block is a symbol for the mathematical operation on the input
signal to the block that produces the output.
• The transfer functions of the components are usually entered in the corresponding blocks,
which are connected by arrows to indicate the direction of the flow of signals.
• Note that the signal can pass only in the direction of the arrows. Thus a block diagram of a
control system explicitly shows a unilateral property.
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10. Note that the dimension of the output signal from the block is the dimension of the input
signal multiplied by the dimension of the transfer function in the block.
Summing Point
• a circle with a cross is the symbol that indicates a summing operation.
• The plus or minus sign at each arrowhead indicates whether that signal is to be added or
subtracted.
• It is important that the quantities being added or subtracted have the same dimensions and
the same units.
Branch Point
• A branch point is a point from which the signal from a block goes concurrently to other blocks
or summing points
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11. Cont...
The advantages
Block diagram representation of a system are that it is easy to form the overall block
diagram for the entire system
The functional operation of the system can be visualized more readily by examining the
block diagram than by examining the physical system itself.
The disadvantage
A block diagram contains information concerning dynamic behavior, but it does not
include any information on the physical construction of the system.
Consequently, many dissimilar and unrelated systems can be represented by the same
block diagram.
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12. Block diagram of closed loop system
Consider the simple feedback control system
the ratio of the feedback signal B(s) to the
actuating error signal E(s) is called the open-
loop transfer function. That is,
The ratio of the output C(s) to the actuating
error signal E(s) is called the feed-forward
transfer function, so that
If the feedback transfer function H(s) is unity,
then the open-loop transfer function and the
feed-forward transfer function are the same.
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13. Note : If the feedback signal is added to the reference input the feedback is said to be positive feedback. On the
other hand if the feedback signal is subtracted from the reference input, the feedback is said to be negative
feedback).
• The block diagram of a complex system can be constructed by finding the transfer functions of simple
subsystems. The overall transfer function can be found by reducing the block diagram, using block
diagram reduction techniques
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14. Block Diagram Reduction Techniques
• A block diagram with several summers and pick off points can be reduced to a
single block, by using block diagram algebra, consisting of the following rules
Step 1:Two blocks G1 and G2 in cascade can be replaced by a single block as shown,
 Step 2 : A summing point can be moved from right side of the block to left side of the block as shown,
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15. Step 3: A summing point can be moved from left side of the block to right side of the block as shown,
 Step 4: A Pick off point can be moved from the right side of the block to left side of the block as shown
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16. Step 5: A Pick off point can be moved from left side of the block to the right side of the block as shown
 Step 6: A feedback loop can be replaced by a single block as shown
 Using these six rules any complex block diagram can be simplified to a single block with an input and output.
The summing points and pick off points are moved to the left or right of the blocks so that we have either two blocks
in cascade, which can be combined, or a simple feedback loop results, which can be replaced by a single block
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17. Summary Rules of Block diagram algebra
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18. Example
• Find the overall transfer function of the system shown in Figure below using block diagram
reduction technique.
Solution:
Step 1 : Using rule 6, the feedback loop with G2 and H2 is replaced by a single block as shown
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19. Step 2: The two blocks in the forward path are in cascade and can be replaced using rule 1 as shown
 Step 3: Finally, the loop in Figure above is replaced by a single block using rule 6
Exercise: Obtain the overall transfer function of
the system shown
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20. 3.4 Mechanical, Electrical, Liquid –level and thermal Systems
Electrical system
• Most of the electrical systems can be modelled by three basic elements :
• Resistor,
• inductor, and
• capacitor.
• Circuits consisting of these three elements are analyzed by using Kirchhoff's
Voltage law and Current law.
• The transfer function can be obtained by taking Laplace transform the
differential equation and rearranging them as a ratio of output to input.
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21. Current-voltage relationship of R,L&C
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22. Example
• Consider the parallel RLC network excited by a current source. Find the (a) differential
equation representation and (b) transfer function representation of the system.
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23. Mechanical Systems
• Mechanical systems can be divided into two basic systems.
1. Translational systems and
2. Rotational systems
• We will consider these two systems separately and describe these systems in terms of three
fundamental linear elements.
• Translational systems:
1. Mass: represents an element which resists the motion due to inertia. According to Newton's
second law of motion, the inertia force is equal to mass times acceleration.
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24. • 2. Dash pot: This is an element which opposes motion due to friction.
• If the friction is viscous friction, the frictional force is proportional to velocity. This force is also
known as dampling force.
• Thus we can write as,
• Where B is the damping coefficient.
• 3. Spring: The third element which opposes motion is the spring.
• The restoring force of a spring is proportional to the displacement.
Thus,
• Where K is known as the stiffness of the spring or simply spring constant
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25. Rotational systems:
• Corresponding to the three basic elements of translation systems, there are three basic elements
representing rotational systems.
• 1. Moment of Inertia: This element opposes the rotational motion due to Moment of inertia. The
opposing inertia torque is given by,
• Where α, ω, and θ are the angular acceleration, angular velocity and angular displacement
respectively. J is known as the moment of inertia of the body.
• 2. Friction: The damping or frictional torque which opposes the rotational motion is given by,
• Where B is the rotational frictional coefficient.
• 3. Spring: The restoring torque of a spring is proportional to the angular displacement θ and is
given by,
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26. • If three elements of rotational systems are similar in nature to those of translational systems
no separate symbols are necessary to represent these elements.
• For basic elements of mechanical systems, we able to write differential equations for the
system, when these mechanical systems are subjected to external forces.
• This is done by using the D' Alembert's principle which is similar to the Kirchhoff's laws in
Electrical Networks. Also, this principle is a modified version of Newton's second law of
motion.
• The D' Alembert's principle states that,
• "For any body, the algebraic sum of externally applied forces and the forces opposing the motion
in any given direction is zero".
• To apply this principle to any body, a reference direction of motion is first chosen.
• All forces acting in this direction are taken positive and those against this direction are taken as
negative.
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27. For example, consider a translational mechanical system
• Let us take a reference direction of motion of the body from left
to right.
• Let the displacement of the mass be x.
• We assume that the mass is a rigid body, i.e., every particle in the
body has the same displacement, x.
• Let us enumerate the forces acting on the body as,
1. external force = f
2. resisting forces :
• This is the differential equation
governing the motion of the mechanical
translation system.
• The transfer function can be easily
obtained by taking Laplace transform.
Thus,
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28. • consider a rotational mechanical system
• Similarly, the differential equation governing the
motion of rotational system can also be obtained.
Torque:
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29. Analogous system
• The system have the same type of equations even though they have different
physical appearance.
• Mechanical systems, fluid systems, temperature systems etc. may be governed
by the same types of equations as that of electrical circuits.
• Designing and constructing a model is easier in electrical systems.
• The system can be built up with cheap elements,
• the values of the elements can be changed with ease and
• experimentation is easy with electrical circuits.
• Once a circuit is designed with the required characteristics, it can be readily
translated into a mechanical system.
• It is not only true for mechanical systems but also several other systems like
acoustical, thermal, fluid and even economic systems.
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30. Analogous quantities based on force voltage analogy
Analogous quantities based on Force - Current analogy
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31. Example
• Write the equations describing the motion of the mechanical system shown in Figure below.
• Also find the transfer function X1(s)/F(s)
Solution:
• The first step is to identify the displacements of masses M1 and M2 as X1
and X2 in the direction of the applied external force!
• Next we write the equilibrium equation for each of the masses by
identifying the forces acting on them.
Let us first find out the forces acting on mass M 1
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33. • From eqns. (1) & (2), force voltage analogous electrical circuits can be drawn as shown in fig.
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34. Assignment 1
1. Briefly explain about Block Diagrams of Liquid –level and thermal
Systems
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35. 3.4. Signal Flow graph
• Another useful way of representing a system is by a signal flow graph.
• A signal flow graph describes how a signal gets modified as it travels from
input to output and the overall transfer function can be obtained very easily
by using Mason's gain formula.
• Signal flow graph: It is a graphical representation of the relationships
between the variables of a system.
• Node: Every variable in a system is represented by a node.
• The value of the variable is equal to the sum of the signals coming towards
the node.
• Its value is unaffected by the signals which are going away from the node.
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36. • Branch : A signal travels along a branch from one node to another node in the direction
indicated on the branch. Every branch is associated with a gain constant or transmittance
• The signal gets multiplied by this gain as it travels from one node to another.
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Branches are: 𝑎12, 𝑎23, 𝑎42
Nodes are:𝑥1, 𝑥2, 𝑥3, 𝑥4
Input node: It is a node at which only outgoing branches are present. In Figure below node X1 is an input
node. It is also called as source node.

37. • Output node: It is a node at which only incoming signals are present. Node X4 is an output
node.
• Path: It is the traversal from one node to another node through the branches in the direction
of the branches such that no node is traversed twice
• Forward path: It is a path from input node to output node.
• In figure above, X1-X2-X3-X4-X5 and
• X1-X2-X5 is also a forward path
• Loop: It is a path starting and ending on the same node.
• x3 - x 4 - x3 is a loop.
• Similarly x 2 - x3 - x 4 - x 2 is also a loop.
• Non touching loops: Loops which have no common node, are said to be non touching loops.
• Forward path gain: The gain product of the branches in the forward path is called forward
path gain.
• Loop gain : The product of gains of branches in the loop is called as loop gain.
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38. Construction of a Signal Flow Graph for a System
• A signal flow graph for a given system can be constructed by writing down the equations governing
the variables.
• Consider the following circuits:
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• The variables I1(s), I2(s), V1(s), V2(s) and Vo(s) are represented by nodes and these nodes are
interconnected with branches to satisfy the relationships between them.
• The resulting signal flow graph is shown

39. Mason's Gain Formula
• The transfer function (gain) of the given signal flow graph can be easily obtained by using Mason's
gain formula.
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where Pmr is the product of the gains of mth possible combination of r non touching loops.
Or
Δ= 1 - (sum of gains of individual loops) + sum of gain products of possible combinations
of 2 non touching loops) - (sum of gain products of all possible combinations of 3 non touching
loops) + .....
and Δk is the value of Δ for that part of the graph which is non touching with kth forward path.

40. Example
To explain the use of this formula, let us consider the signal flow graph shown in
Figure below, and find the transfer function C(s)/R(s)
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41. 8/6/2019 41

42. Exercise
• Find the transfer function for the block diagram using Mason’s gain
formula.
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