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Modern Control Systems (MCS)
Dr. Imtiaz HussainAssistant Professor
email: [email protected] :http://imtiazhussainkalwar.weebly.com/
Lecture-25-26Modeling of Digital Control Systems
2
Lecture Outline
β’ Sampling Theorem
β’ ADC Model
β’ DAC Model
β’ Combined Models
3
Sampling Theoremβ’ Sampling is necessary for the processing of analog data
using digital elements.
β’ Successful digital data processing requires that the samples reflect the nature of the analog signal and that analog signals be recoverable from a sequence of samples.
4
Sampling Theoremβ’ Following figure shows two distinct waveforms with
identical samples.
β’ Obviously, faster sampling of the two waveforms would produce distinguishable sequences.
5
Sampling Theoremβ’ Thus, it is obvious that sufficiently fast sampling is a
prerequisite for successful digital data processing.
β’ The sampling theorem gives a lower bound on the sampling rate necessary for a given band-limited signal (i.e., a signal with a known finite bandwidth)
6
Sampling Theoremβ’ The band limited signal with
β’ can be reconstructed from the discrete-time waveform
β’ if and only if the sampling angular frequency satisfies the condition
π β (π‘ )= βπ=ββ
β
π (π‘ )πΏ(π‘βππ )
ππ >2ππ
7
Selection of Sampling Frequency
β’ A given signal often has a finite βeffective bandwidthβ beyond which its spectral components are negligible.
β’ This allows us to treat physical signals as band limited and choose a suitable sampling rate for them based on the sampling theorem.
β’ In practice, the sampling rate chosen is often larger than the lower bound specified in the sampling theorem.
β’ A rule of thumb is to choose asππ =πππ ,5β€πβ€10
8
Selection of Sampling Frequency
β’ The choice of depends on the application.
β’ In many applications, the upper bound on the sampling frequency is well below the capabilities of state-of-the-art hardware.
β’ A closed-loop control system cannot have a sampling period below the minimum time required for the output measurement; that is, the sampling frequency is upper-bounded by the sensor delay.
ππ =πππ ,5β€πβ€10
9
Selection of Sampling Frequency
β’ For example, oxygen sensors used in automotive air/fuel ratio control have a sensor delay of about 20 ms, which corresponds to a sampling frequency upper bound of 50 Hz.
β’ Another limitation is the computational time needed to update the control.
β’ This is becoming less restrictive with the availability of faster microprocessors but must be considered in sampling rate selection.
10
Selection of Sampling Frequency
β’ For a linear system, the output of the system has a spectrum given by the product of the frequency response and input spectrum.
β’ Because the input is not known a priori, we must base our choice of sampling frequency on the frequency response.
11
Selection of Sampling Frequency (1st Order Systems)
β’ The frequency response of first order system is
β’ where K is the DC gain and is the system bandwidth.
β’ Time constant and 3db bandwidth relationship
π» ( ππ )= πΎππ /ππ+1
ππ=1π
π 3ππ=1
2ππ
12
Selection of Sampling Frequency (1st Order Systems)
-30
-25
-20
-15
-10
-5
0
Mag
nitu
de (dB
)
System: sysFrequency (rad/s): 0.33Magnitude (dB): -2.97
System: sysFrequency (rad/s): 2.31Magnitude (dB): -16.9
10-2
10-1
100
101
-90
-45
0
Pha
se (de
g)
Bode Diagram
Frequency (rad/s)
πΊ (π )= 13π +1
13
Selection of Sampling Frequency (1st Order Systems)
β’ The frequency response amplitude drops below the DC level by a factor of about 10 at the frequency .
β’ If we consider , the sampling frequency is chosen as
ππ =πππ ,35 β€πβ€70
14
Selection of Sampling Frequency (2nd Order Systems)
β’ The frequency response of second order system is
β’ The bandwidth of the system is approximated by the damped natural frequency
β’ Using a frequency of as the maximum significant frequency, we choose the sampling frequency as
ππ =πππ ,35β€πβ€70
2)/(1/2)(
nnj
KjH
21 nd
15
Example-1β’ Given a first-order system of bandwidth 10 rad/s, select a
suitable sampling frequency and find the corresponding sampling period.
β’ We know
β’ Choosing k=60
Solution
ππ=10πππ /π ππ
ππ =πππ ,35 β€πβ€70
ππ =60ππ=600πππ /π ππ
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Example-1β’ Corresponding sapling period is calculated as
π=2πππ
=2Γ3141
600=0.01π ππ
17
Example-2β’ A closed-loop control system must be designed for a steady-
state error not to exceed 5 percent, a damping ratio of about 0.7, and an undamped natural frequency of 10 rad/s. Select a suitable sampling period for the system if the system has a sensor delay of 0.02 sec.
β’ Let the sampling frequency be
Solution
ππ β₯35ππ
2135 ns
27.011035 s
18
Example-2
β’ The corresponding sampling period is
β’ A suitable choice is T = 20 ms because this is equal to the sensor delay.
srads / 95.249
π β€2Γ3141249.95
19
Home Workβ’ A closed-loop control system must be designed for a steady-
state error not to exceed 5 percent, a damping ratio of about 0.7, and an undamped natural frequency of 10 rad/s. Select a suitable sampling period for the system if the system has a sensor delay of 0.03 sec.
20
Introductionβ’ A common configuration of digital control system is shown in
following figure.
21
ADC Modelβ’ Assume that
β ADC outputs are exactly equal in magnitude to their inputs (i.e., quantization errors are negligible)
β The ADC yields a digital output instantaneouslyβ Sampling is perfectly uniform (i.e., occur at a fixed rate)
β’ Then the ADC can be modeled as an ideal sampler with sampling period T.
T
t
u*(t)
0 t
u(t)
0
0 t
u*(t)
t
u(t)
0
T
t
Ξ΄T(t)
0
Γ οΌ
modulating pulse(carrier)
modulated wave
Modulation signal
u(t) u*(t)
0
* )()()(k
kTttutu
Sampling Process
23
DAC Modelβ’ Assume that
β DAC outputs are exactly equal in magnitude to their inputs.β The DAC yields an analog output instantaneously.β DAC outputs are constant over each sampling period.
β’ Then the input-output relationship of the DAC is given by
π’ (π )πππ»β
π’h (π‘ )=π’ (π ) ,ππ β€ π‘β€ (π+1 )π
u(k)u(t)
uh(t)
24
DAC Modelβ’ Unit impulse response of ZOH
β’ The transfer function can then be obtained by Laplace transformation of the impulse response.
25
DAC Modelβ’ As shown in figure the impulse response is a unit pulse of
width T.
β’ A pulse can be represented as a positive step at time zero followed by a negative step at time T.
β’ Using the Laplace transform of a unit step and the time delay theorem for Laplace transforms,
L {π’ (π‘)}=1π
L {βπ’(π‘βπ )}=β πβππ
π
26
DAC Model
β’ Thus, the transfer function of the ZOH is
L {π’ (π‘)}=1π
L {βπ’(π‘βπ )}=β πβππ
π
πΊπππ» (π )=1βπβππ
π
27
DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ The cascade of a DAC, analog subsystem, and ADC is shown in following figure.
β’ Because both the input and the output of the cascade are sampled, it is possible to obtain its z-domain transfer function in terms of the transfer functions of the individual subsystems.
28
DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ Using the DAC model, and assuming that the transfer function of the analog subsystem is G(s), the transfer function of the DAC and analog subsystem cascade is
πΊππ΄ (π )=πΊπππ»(π )πΊ(π )
πΊππ΄ (π )=1βπβππ
π πΊ(π )
29
DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ The corresponding impulse response is
β’ The impulse response is the analog system step response minus a second step response delayed by one sampling period.
πΊππ΄ (π )=1βπβππ
π πΊ(π )
πΊππ΄ (π )=πΊ(π ) βπΊ(π ) πβππ
π
πΊππ΄ (π )=πΊ(π )π
βπΊ(π ) πβππ
π
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DAC, Analog Subsystem, and ADC CombinationTransfer Function
πΊππ΄ (π )=πΊ(π )π
βπΊ(π ) πβππ
π
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DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ Inverse Laplace yields
β’ Where
πΊππ΄ (π )=πΊ(π )π
βπΊ(π ) πβππ
π
πππ΄ (π‘ )=ππ (π‘)βππ (π‘βπ )
32
DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ The analog response is sampled to give the sampled impulse response
β’ By z-transforming, we can obtain the z-transfer function of the DAC (zero-order hold), analog subsystem, and ADC (ideal sampler) cascade.
πππ΄ (π‘ )=ππ (π‘)βππ (π‘βπ )
πππ΄ (ππ )=ππ (ππ )βππ (ππ βπ )
33
DAC, Analog Subsystem, and ADC CombinationTransfer Function
β’ Z-Transform is given as
β’ The * in above equation is to emphasize that sampling of a time function is necessary before z-transformation.
β’ Having made this point, the equation can be rewritten more concisely as
πππ΄ (ππ )=ππ (ππ )βππ (ππ βπ )
πΊππ΄π (π§ )=(1β π§β1)Z {ππ ββ(π‘) }
πΊππ΄π (π§ )=(1β π§β1)Z[Lβ1 {πΊ(π )π }
β]
πΊππ΄π (π§ )=(1β π§β1)Z[πΊ(π )π ]
34
Example-3β’ Find GZAS(z) for the cruise control system for the vehicle shown
in figure, where u is the input force, v is the velocity of the car, and b is the viscous friction coefficient.
β’ The transfer function of system is given as
β’ Re-writing transfer function in standard form
Solution
πΊ (π )=π (π )π (π )
= 1ππ +π
πΊ (π )= πΎπ π +1
=πΎ /ππ +1/π
35
Example-3
β’ Where and β’ Now we know
β’ Therefore,
β’ The corresponding partial fraction expansion is
πΊ (π )π
= πΎ /ππ (π +1/π )
πΊ (π )= πΎ /ππ +1/π
πΊππ΄π (π§ )=(1β π§β1)Z[πΊ(π )π ]
πΊ (π )π
=(πΎπ ) [ππ β ππ +1 /π ]
36
Example-3
β’ Using the z-transform table, the desired z-domain transfer function is
πΊππ΄π (π§ )=z(1βπ§β1)Z[(πΎπ ){ππ β ππ +1/π }]
πΊππ΄π (π§ )=(1β π§β1)Z[πΎ {1π β
1π +1/π }]
πΊππ΄π (π§ )= π§β1π§ [πΎ { π§
π§β1β
π§π§βπβπ /π }]
πΊππ΄π (π§ )=[πΎ {1β π§β1π§βπβπ /π }]
37
Example-3
πΊππ΄π (π§ )=[πΎ {1β π§β1π§βπβπ /π }]
πΊππ΄π (π§ )=πΎ π§βπβ ππ βπ§+1
π§βπβπ /π
πΊππ΄π (π§ )=πΎ 1βπβ ππ
π§βπβπ /π
38
Example-4β’ Find GZAS(z) for the vehicle position control system, where u is
the input force, y is the position of the car, and b is the viscous friction coefficient.
β’ The transfer function of system is given as
β’ Re-writing transfer function in standard form
Solution
πΊ (π )= π (π )π (π )
= 1π (ππ +π)
πΊ (π )= πΎπ (π π +1)
=πΎ /π
π (π +1π
)
π¦π οΏ½ΜοΏ½
39
Example-4
β’ Where and β’ Now we know
β’ Therefore,
β’ The corresponding partial fraction expansion is
πΊ (π )π
= πΎ /ππ 2(π +1/π )
πΊ (π )= πΎ /π
π (π +1π
)
πΊππ΄π (π§ )=(1β π§β1)Z[πΊ(π )π ]
πΊ (π )π
=πΎ [ 1
π 2βππ +
ππ +1/π ]
40
Example-4β’ The desired z-domain transfer function can be obtained as
πΊππ΄π (π§ )=(1β π§β1)ZπΎ [ 1
π 2βππ +
ππ +1/π ]
πΊππ΄π (π§ )= π§β1π§
πΎ [ π§(π§β1)2
βπ π§π§β1
+ π π§π§βπβπ /π ]
πΊππ΄π (π§ )=πΎ [ 1π§ β1
βπ+π (π§β1)π§βπβπ /π ]
πΊππ΄π (π§ )=πΎ [ (1βπ+ππβ ππ )π§+[πβπβππ (π+1)]
(π§β1)(π§βπβπ /π ) ]
41
Example-5β’ Find GZAS(z) for the series R-L circuit shown in Figure with
the inductor voltage as output.
42
END OF LECTURES-25-26
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