Download In The Name of Allah The Most Beneficent The

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In The Name of Allah
The Most Beneficent The Most Merciful
ECE4545:
Control Systems
Lecture:
Mathematical
Modeling of SystemsFrequency Domain
Engr. Ijlal Haider
UoL, Lahore
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Control Talk
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The 2003 Jaguar features a forward alert system from
Delphi that advises a driver to brake in the approach of
slow moving traffic ahead [Sharke 2003]. If the driver does
not adequately respond to warnings, collision avoidance
systems might take control of the throttle, brakes, or
steering, to maneuver the vehicle back to a safe state.
Nissan’s new gas pedal will lift itself to alert the driver of a
possible collision, and the brakes are automatically
applied if the radar sensor detects a possible collision
ahead and when the driver’s foot is off the gas
[Kageyama 2006]
Taking a step further, the 2006 Honda Acura RL will even
actively brake the car if an imminent rear-end collision is
sensed [Honda 2005]
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Control Talk
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The so-called Honda Collision Mitigation Braking System TM
(CMBSTM) predicts potential collision situations with a
millimeter wave radar unit, which monitors the distance
between the RL and objects in front of the car, as well as
closing rates. If CMBS determines the closing rate between
the RL and the vehicle directly in front has increased
beyond an acceptable level, visual and audible warnings
prompt the driver to take preventive action. If the
distance further diminishes, the system provides a tactile
warning by gently retracting the seatbelt and then applies
light braking. If an accident is determined to be
unavoidable, the system applies strong braking and strong
retraction of the front seatbelt to reduce the speed of
impact and to mitigate the damage of a collision.
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Control Talk
 With
funding from DARPA, researchers at
Princeton have developed "a new type of
sensor [that] can identify substances as
small as a molecule by examining the
light they reflect, potentially leading to
sensors for a wide range of substances,
from explosives to cancer”.
ASEE Weekly Newsletter
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Plate Dispenser
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Mathematical ModelingDefinition
 Determining
equations and relationships
governing the behavior of system
©copyright: M Junaid Khan
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Mathematical ModelingSignificance
 The
power of a mathematical model lies
in the fact that it can be simulated in
hypothetical situations, be subject to
states would be dangerous in reality and
it can be used as a basis for synthesizing
controllers
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Model Development
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Essential step of the “Design Procedure”
Three different choices for modeling:
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Analysis: Mathematical models based on physical
principles.
Grey-Box: Model is developed and then parameters are
inferred from experiments.
Black-Box: Input / Output data is used to infer a dynamic
relationship. (System Identification)
Nonlinear models can be linearized.
Linear models are most often used in analysis and
design.
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On Building Models
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High order models can only be justified where
there is little uncertainty
Control-relevant models are often quite simple
compared to the true system and generally
combine physical reasoning with experimental
data
Actuators should be included as they often are
nonlinear and have their own dynamic behavior
A model is never perfect, hence modeling errors
are a fact of life
Simplicity versus Accuracy
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Mathematical Models
A model is any representation of a real system.
 May involve words, diagrams, mathematical notation,
physical structure
 A mathematical model may be as simple as a single
equation relating a single dependent variable (y) to
another independent variable (x) such as:
y = ax + b
 May be multi-component involving the interaction of many
equations having several mutually dependent variables
a11x1  a12 x2  ...a1n xn  b1
a21x1  a22 x2  ...a2 n xn  b2
an1 x1  an 2 x2  ...ann xn  bn
dy1
 f 1 (t , y1 , y 2 ,..., y n ); y1 (t 0 )  y1, 0
dt
dy 2
 f 2 (t , y1 , y 2 ,..., y n ); y 2 (t 0 )  y 2, 0
dt
dy n
 f n (t , y1 , y 2 ,..., y n ); y n (t 0 )  y n , 0
dt
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System
 Engineering
process
(continuous)
–
chemical plants, aircraft, traffic system,
Electrical, Mechanical, Thermal systems
 Information process (computer-based or
digital) – computer network, telephone
system,
 Economic process – Pakistan economy,
world economy, software industry
 Social systems – suburban culture, welfare
system
©copyright: M Junaid Khan
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Purpose of Modeling
 To
analyze systems
 To analyze each element in the system
 Mathematical Models are equations
representing input/output relationships of a
system
©copyright: M Junaid Khan
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Types of Models Under Consideration
Transfer Function Models---Output/Input ratio Models
Differential Equation Models
State Space Models-A set of first Order Differential
Equations
Flow Diagram Models
Block Diagram Models----Simulink Models
©copyright: M Junaid Khan
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Transfer Function
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Engineering Systems Under Consideration
 Mechanical
 Electrical
 Thermal
 Fluid
(Hydraulic and Pneumatic)
©copyright: M Junaid Khan
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Ingredients of a Model
Building Blocks
 Each block has a single property or function
 Example
 Electrical Circuit
 The building blocks are resistors,
capacitors and inductors.
 Mechanical System
 The building blocks are Spring,
Mass and dampers
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Note: There may be similarities in the building
blocks used for different type of systems
(analogous systems)
©copyright: M Junaid Khan
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Building Blocks of Electrical Systems
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Building Blocks of Mechanical Systems
(Translational)
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Building Blocks of Mechanical Systems
(Rotational)
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Physical Laws Governing any System

Law of Conservation of Energy
 KVL
 KCL
 Newton’s
2nd Law
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Thank You!