Welcome to elearning session on CONTROL ENGINEERING ME 55

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Welcome to e-learning session onCONTROL
ENGINEERING (ME 55)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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ByDr. B.K. Sridhara HeadDepartment of
Mechanical EngineeringThe National Institute of
EngineeringMysore 570 008
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Session 15 CHAPTER II MATHEMATICAL
MODELING (continued) CHAPTER III SYSTEM
RESPONSE
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Recap of Session VII Chapter II Mathematical
Modeling

• Mathematical Modeling of Mechanical systems
• Mathematical Modeling of Electrical systems
• Models of Hydraulic Systems
• Liquid Level System
• Fluid Power System

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Modeling Thermal Systems
Tov
Tamb
qin heat inflow rate Tov Temperature of the
oven Tamb Ambient Temperature T Rise in
Temperature (Tov - Tamb)
Oven
qout
Parts
qin
Example Heat treatment oven
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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From Law of Conservation Energy qin heat
inflow rate qin qs qout --- (1) qout
heat loss through the walls of the oven qs
Rate at which heat is stored (Rate at which heat
is absorbed by the parts)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Thermal Resistance R
--- (a)
Thermal Capacitance C Q/T
Heat stored
--- (b)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Substitute (a) and (b) in (1)
qin qs qout --- (1)
Model
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Chapter III System Response

• Prediction of the performance of control systems
  requires
• Obtaining the differential equations
• Solutions
• System behaviour can be expressed as a function
  of time
• Such a study System response or system analysis
  in time domain

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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System Response in Time Domain

• System Response The output obtained
  corresponding to a given Input.
• Total response Two parts
• Transient Response (yt)
• Steady state response (yss)
• Total response is the sum of steady state
  response and transient response
• y yt yss

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
11
Transient Response (yt)

• Initial state of response and has some specific
  characteristics which are functions of time.
• Continues until the output becomes steady.
• Usually dies out after a short interval of time.
• Tends to zero as time tends to 8

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Steady State Response (yss)

• Ultimate Response obtained after some interval
  of time
• Response obtained after all the transients die
  out
• It is not independent of time
• As time approaches to infinity system response
  attains a fixed pattern

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• When the weight is added the deflection abruptly
  increases
• System oscillates violently for some time
  (Transient)
• Settles down to a steady value (Steady state)

Transient
SS
Transient and Steady-state Response of a spring
system
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Steady State Error

• Steady State Response may not agree with Input
• Difference is called steady state error
• Steady state error Input Steady state
  response

Input
Input or Response
Steady state error
Response
t 8
Time
t 0
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Test Input Signals

• Systems are subjected to a variety of input
  signals (working conditions)
• Most cases it is very difficult to predict the
  type of input signal
• Impossible to express the signals by means of
  Mathematical Models

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Common Input Signals
• - Step Input
• - Ramp Input
• - Sinusoidal
• - Parabolic
• - Impulse functions, etc.,

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• In system analysis one of the standard input
  signal is applied and the response produced is
  compared with input
• Performance is evaluated and Performance index
  is specified
• When a control system is designed based on
  standard input signals generally, the
  performance is found satisfactory

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Common System Input Signals
a) Step Input
K
i (t)
t 0
time

• Input is zero until t 0
• Then takes on value K which remains constant
  for t gt 0
• Signal changes from zero level to K
  instantaneously

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematically i (t) K for t gt 0 0 for t
lt 0 for t 0, step function is not
defined When a system is subjected to sudden
disturbance step input can be used as a test
signal
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Examples
• Angular rotation of the Shaft when it starts
  from rest
• Change in fluid flow in a hydraulic system due
  to sudden opening of a valve
• Voltage applied on an electrical network when it
  is suddenly connected to a power source

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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b) Ramp Input
Input
i (t)
Kt

• Signals is linear function of time
• Increases with time
• Mathematically i (t) Kt for t gt 0
• 0 for t lt 0
• Example Constant rate heat input in thermal
  system

t 0
time
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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c) Sinusoidal Input
Input
k Sin ?t

• Mathematically
• i (t) k Sin ?t
• System response in frequency domain
• Frequency is varied over a range
• Example Voltage, Displacement, Force etc.,

i (t)
ime
i (t) k Sin ?t
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Order of the System

• The responses of systems of a particular order
  are Strikingly similar for a given input
• Order of the system It is the order of the
  highest derivative in the ordinary linear
  differential equation with constant coefficients,
  which represents the physical system
  mathematically.

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Illustration First order system
.
Cy ky kx
x (t) i/p
K
y (t) o/p
C
Order Order of the highest derivative 1 First
order system
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Illustration Second order system
.
..
x (t)
K
y (t)
m
C
Order Order of the highest derivative 2 Second
order system
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Response of First Order Mechanical Systems to
Step Input
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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THANK YOU
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore