unit-v_modelling-and-analysis-of-mechatronic-system.pdf

Model

1. Mechatronics
UNIT –V
Modelling and Analysis of
Mechatronic System

2. Syllabus
Modelling and Analysis of Mechatronic System
•System modeling (Mechanical, Thermal and Fluid),
•Stability Analysis via identification of poles and zeros,
•Time Domain Analysis of System and estimation of Transient
characteristics: % Overshoot, Damping factor, Damping frequency, Rise
time,
•Frequency Domain Analysis of System and Estimation of frequency
domain parameters such as Natural Frequency, Damping Frequency and
Damping Factor.

3. Objectives
1.Understand key elements of Mechatronics system,
representation into block diagram
2.Understand concept of transfer function, reduction and
analysis
3.Understand principles of sensors, its characteristics, interfacing
with DAQ microcontroller
4.Understand the concept of PLC system and its ladder
programming, and significance of PLC systems in industrial
application
5.Understand the system modeling and analysis in time
domain and frequency domain.
6.Understand control actions such as Proportional, derivative and
integral and study its significance in industrial applications.

4. Outcomes
1.Identification of key elements of mechatronics system and its
representation in terms of block diagram
2.Understanding the concept of signal processing and use of
interfacing systems such as ADC, DAC, digital I/O
3.Interfacing of Sensors, Actuators using appropriate DAQ micro
-controller
4.Time and Frequency domain analysis of system model (for
control application)
5.PID control implementation on real time systems
6.Development of PLC ladder programming and implementation
of real life system

5. System Modelling
￿To understand and control complex systems, one must obtain
quantitative mathematical models of these systems to analyze
the relationships between the system variables.
￿Because the systems under consideration are dynamic in nature,
the descriptive equations are usually differential equations
obtained by utilizing the physical laws of the process

6. System Modelling
￿The Laplace transform can be used to obtain a solution
describing the operation of the system.
￿In practice, the complexity of systems and our ignorance of all
the relevant factors necessitate the introduction of assumptions
concerning the system operation.

7. System Modelling
Transfer Function based Approach
￿Why Transfer Function?
￿Because it is easier / better to assess some things using classical
techniques, such as gain and phase margin.
￿How to determine TF?
￿Derive the Governing Differential Equation
￿Assume I.C=Zero and
￿Take Laplace transform of output
￿Take Laplace transform of input
￿Transfer function = L (output) / L (input)

8. System Modelling: Mechanical System

9. System Modelling: Thermal System
￿A closed, insulated vessel filled with liquid and contains an electrical heater
immersed in liquid. Heating element is contained within metal jacket that has
a thermal resistance of R
HL
. Thermal resistance of vessel and its insulation is
R
La
. Heater has a thermal capacitance of C
H
, and liquid has a thermal
capacitance of C
L
. Heater temperature is T
H
and that of liquid is T
L
(assumed
to be uniform due to the mixer). Rate at which energy is supplied to the
heating element is q
i
.

10. System Modelling: Thermal System
￿Take Laplace of Eq. (1) and determine the equation for T
H
(s) .
￿Take Laplace of Eq. (2), determine the equation for T
L
(s) and
substitute the equation for T
H
(s) in it.
￿Transfer Function is: T
L
(s)/q
i
(s)

11. System Modelling: Hydraulic System

12. System Modelling: Hydraulic System

13. System Modelling: Hydraulic System

14. Poles and Zeros
▪Poles and zeros are properties of transfer function, which characterize
the differential equation, and provide a complete description of the
behavior of the system .
▪The poles of a transfer function are the roots of the characteristic
equation in the denominator of the transfer function.
▪The zeros of a transfer function are the roots of the characteristic
equation in the numerator of the transfer function.

15. Relation: Poles of System and Damping and Natural
Frequency
Relation between Pole Location, Damping (ζ) and Natural
Frequency (ω
n
)

16. Relation: Pole Location and Response of System
Pole Location for
Different Systems
Response of
Systems

17. Example: Identification of Poles

18. Example: Identification of Zeros

19. Stability Analysis based on Location of Poles
▪For all initial conditions, if the response of a system decays to
equilibrium, the system is presumed to be stable, at large.
▪Absolute Stability: Whether stable or not
▪Relative Stability: Stable under what conditions
▪The real part of the poles of a given system must be on the left
side of the s-plane for the system to be stable at large.

20. Stability Analysis based on Location of Poles
▪In case the poles are a complex conjugate pair, their real part
must be negative for the system to be stable.
▪If any of the poles have a value, zero, the system is deemed to
be marginally stable
▪If the poles are positive / have a positive real part, the system is
deemed to be un-stable.

21. Time Domain Analysis of System
▪Time is used as an independent variable in most systems.
▪It is usually of interest to evaluate the output response of the system
with respect to time or, simply, the time response.
▪Time response is divided into: Transient Response, Steady State
Response.
▪In the time domain analysis problem, a reference input signal is
applied to a system, and the performance of the system is evaluated in
the form of the time domain specifications w.r.t to the transient as
well as the steady state response.

22. Time Domain Analysis of System
▪Stable systems exhibit transient phenomena to some extent before the steady
state is reached.
▪e.g. in Mechanical systems Inertia is unavoidable and, thus, response
cannot follow sudden changes in input, instantaneously, and transients
are usually observed.
▪Control of transient response, which leads to deviation between output
response and the input response, before the steady state is reached, must be
closely controlled.
▪Control of the steady state response is also very important since it determines
the accuracy of the system
▪More so W.R.T position control system

23. Time Domain Specifications
Unit Step Response of Second Order System

24. Time Domain Specifications
▪Percentage Overshoot (% O.S): It is the amount that the response
overshoots the steady state, or final, value at the peak time, expressed
as a percentage of the steady-state value.
▪Rise Time (T
r
): Time required for the step response to rise from 10%
to 90% of its final value.
▪Delay Time (T
d
): Time required for the step response to reach 50%
of final value

25. Time Domain Specifications
▪2% Settling Time (T
s
): Time required for the step response to
decrease and stay within ±2% of its final value
▪Steady State Error (e
ss): It is the difference between the output
and the reference input after the steady state has reached

26. Example: Time Domain Specifications
▪Using the values of the natural frequency= =1.414 and the
damping factor=ζ=0.177, determine the values for overshoot,
rise time and 2% settling time

27. Effect of Damping and Natural Frequency in Time Domain
￿Response of system depends on damping ζ and natural
frequency ω
n
￿Settling time and rise time of the system reduces with increase in
the natural frequency, ω
n
￿As damping decreases below 1, the response overshoots and
oscillates about final value
￿Smaller the value of damping: larger the overshoot and
longer it takes for the oscillations to die

28. Effect of Damping on System Response

29. Effect of Natural Frequency on System Response

30. Frequency Domain Analysis of System
▪The frequency domain analysis of a system is defined as the steady-
state response of the system to a sinusoidal input signal.
▪The sinusoid is a unique input signal, and the resulting output signal
for a linear system, as well as signals throughout the system, is
sinusoidal in the steady state.
▪Output differs from the input only in amplitude and phase.

31. Frequency Domain Analysis of System
▪Frequency domain analysis is a better option w.r.t to higher order
system
▪Time response of a higher order system is difficult to determine,
analytically.
▪Frequency domain analysis is better suited when it comes to
determining sensitivity of system to uncertainty (parameter/process
variation, mechanical / electrical noise)
▪Frequency domain analysis is better suited when it comes to accessing
relative stability of a system.

32. Frequency Domain Analysis of System
▪Input applied is some form of sine wave.
▪ The output of the system is also some form of sine wave given by:
▪In above: A is the amplitude of the sine wave, ω is the frequency of
the sine wave, Y is the magnitude of output sine wave and Ø is the
phase shift.

33. Frequency Domain Analysis using Bode Plot
▪Steady state performance can be characterised in the form of
magnitude and phase shift w.r.t to frequency (ω)

34. Rules for Drawing Bode Plot: Effect of Gain

35. Rules for Drawing Bode Plot: Effect of Real Pole

36. Rules for Drawing Bode Plot: Real Zero

37. Rules for Drawing Bode Plot: Effect of Pole at Origin

38. Rules for Drawing Bode Plot: Effect of Zero at Origin

39. Example: Bode Plot for Typical Transfer Function with
single Pole at origin

40. Example: Bode Plot for Typical Transfer Function with two
poles at origin

41. Example: Bode Plot for Typical Transfer Function with one
Pole on negative real axis

42. Example: Bode Plot for Typical Transfer Function with one
Pole at origin and one Pole on negative real axis

43. Example: Bode Plot for Typical Transfer Function with both
poles on negative real axis

44. Frequency Domain Specifications
▪Resonant Peak (M
r
): It is the maximum value of the magnitude.
▪M
r
gives indication on the relative stability of a stable closed-loop
system. Normally, a large M
r
corresponds to a large maximum overshoot
of the step response. For most control systems, it is generally accepted in
practice that the desirable value of M
r
should be between 1.1 and 1.5.
▪Resonant Frequency (ω
r
): It is the frequency at with peak resonance, M
r
,
occurs.
▪Bandwidth (BW): It is the frequency range over which the magnitude drops
3 decibels (dB) from its zero frequency value.
▪BW gives indications of the transient response properties in time domain.
A large bandwidth corresponds to a faster rise time.

45. Frequency Domain Specifications

46. Gain & Phase Margin from Bode Plot
▪Simply knowing that system is stable is not enough
▪Important to access relative stability
▪Stability Margins help accessing the relative stability
▪Gain Margin: It is the factor by which system gain can be
increased before the system becomes un-stable in closed loop.
▪Gain Margin should be > 1 for system to be stable in closed
loop.
▪Gain Margin is determined at phase cross over
▪Phase Margin: It is the amount by which the phase exceeds -180
0
▪Phase Margin should be > 0 for system to be stable in closed
loop
▪Phase Margin is determined at gain cross over

47. Closed Loop Stability of System based on Gain &
Phase Margin

48. Effect of Damping on Bode Plot

49. Estimation of Transfer Function from Bode Plot
▪Step 1-Order of the system: The final value of the phase speaks about the
order
▪If the final value of the phase is -90 deg it means there is one extra pole
compared to the zero
▪Step 2-Number of Poles and Zeros: From the Magnitude Plot, determine the
shift in slope.
▪If the initial slope is -20dB/dec and final slope is -40 dB/dec it means there
are two poles and no zeros
▪Step 3-Location of Poles and Zeros: If the poles / zeros are at origin, they will
pass 0 dB at 1 rad/s. If the poles / zeros are real, determine the location
(frequency ) at which the plot breaks away.
▪Step 4-Gain: From the max gain, determined from magnitude plot, subtract
gain of poles/zero. Then equating this value to the formula 20log10(K),
determine K.

50. Example: Estimation of Transfer Function from Bode Plot

51. Example: Estimation of Transfer Function from Bode Plot
▪Step 1-Order of the system:
▪The final value of the phase is -90 deg it means there is one extra pole
compared to the zero; could be 2 poles and 1 zero OR 1 pole only
▪Step 2-Number of Poles and Zeros:
▪One Pole
▪Step 3-Location of Poles and Zeros:
▪The pole is at 1 rad/s.
▪Step 4-Gain:
▪20*log10(k)=20; Thus k=20
▪Transfer Function: