Post on 31-Dec-2015
description
Prof. Wahied Gharieb Ali Abdelaal
CSE 502: Control Systems(1)
Topic#2
Mathematical Tools for Analysis
Faculty of EngineeringComputer and Systems Engineering Department
Master and Diploma Students
2
Outline
• Ordinary Differential Equations
(ODE)
• Laplace Transform and Its Inverse
• Laplace Transform Properties
• Sampling and Digital Systems
• Selection of Sampling Frequency
• Z-Transform and Its Properties
• Summary
3
Ordinary Differential Equations (ODE)
• An ordinary differential equation of order n is given by:
)()(y)(ydt
d...)(y
dt
d)(y
dt
d011-n
1-n
1n
n
tftatatat n
• Example:
Second order linear ordinary differential equation.
It is convenient to define the differential operators:
and
1)(y6)(ydt
d3)(y
dt
d2
2
ttt
dt
dD
n
n
dt
dnD
4
Characteristic Polynomial & Characteristic Equation
• The polynomial: is called the Characteristic Polynomial and the equation :
is called the characteristic equation
0
1
1
1
1 ... aDaDaD n
n
n
0... 0
1
1
1
1 aDaDaD n
n
n
The characteristic polynomial is D2 +3D +2.The characteristic equation is D2+3D+2=0(D+2)(D+1)=0. The roots are D1=-2 & D2=-1
u y2ydt
d3y
dt
d2
2
Example:
Ordinary Differential Equations (ODE)
5
Solution of Linear Ordinary Differential Equation
The solution of a differential equation contains two parts:• Free response• Forced response
The free response; is the solution of the differential equation when the input is zero.
The forced response ; is the solution of the differential equation when all initial conditions are zero.
The total response is the sum of the free response and the forced response.
Ordinary Differential Equations (ODE)
6
Example
The free response (y1)D2 + 3D +2 =0 (D+2)(D+1)=0
D=-1 ; D=-2
)(y2ydt
d3y
dt
d2
2
tf 0y(0) 1dt
dy(0)
-2t2
-t11 eA eA(t)y
The forced response (y2) depends on the forcing function f(t).If f(t)= cos t y2(t) = A1 cos t + A2 sin t
f(t)= t2 y2(t) = A1 + A2t + A3t2
f(t)= te-t y2(t) = A1 e-t + A2 te-t
f(t)= et y2(t) = Aet
The above forms will usually work if the forcing function is not a part of the free response !
Ordinary Differential Equations (ODE)
7
• In this example, f(t) = -4e-3t y2(t)= Ae-3t because
te 32
2
4 y2ydt
d3y
dt
d
• The total response y(t)= y1(t) + y2(t)
• We have to find A , A1 and A2 satisfying the differential equation and the initial conditions
9Ae-3t –9Ae-3t +2Ae-3t = -4e-3t A=-2
y2(t)=-2e-3t
Ordinary Differential Equations (ODE)
8
• Using initial conditions to find A1 and A2 y(0)=0 A1+A2-2=0 A1=2-A2
ý(0)= 1 -A1-2A2+6=1 A1=-1 and A2=3
y(t)= -e-t + 3e-2t - 2e-3t
• The total response y(t)= -2e-3t + A1e-t
+ A2e-2t
Ordinary Differential Equations (ODE)
Example
The free response (y1)
D2 + D +1/2 =0 Complex poles
D1,2= -1/2 j1/2
t2
1y
2
1y
dt
dy
dt
d2
2
1y(0) 2
1dt
dy(0)
tAtAety
t
2
1sin
2
1cos)( 21
2
1
1
9
Ordinary Differential Equations (ODE)
Forced response
Since f(t)=1/2t y2(t)=B1+B2t
0+B2 +1/2B1 +1/2B2t =1/2t B1=-2 and B2=1
y2(t)=-2+t
ty
2
1y
2
1
dt
dy
dt
d2
2
2
2
2
10
Ordinary Differential Equations (ODE)
11
Laplace Transform
Given a real function f(t) that satisfies
Laplace Transform of f(t) is defined as :
Where F(s) = L[f(t)] or f(t) = L-1[F(s)]
0
)( dtetf t
0
)]([)()( tfLdtetfsF st
Definition
12
Laplace Transform
1. f(t) = (t) = impulse signal defined by:
where for t0,
otherwise
s
edt
etL
sst
1)]([
0
11
lim)]([0
s
etL
s
)(lim)()(0
tttf
1)( t
0)( t
Examples
1)]([ tL
13
2. f(t) = u(t) = Unit Step
01)( ttu
00)( ttu
ss
edtetuLsF
st
st 1][)]([)( 00
stuL
1)]([
s
ettuL
stuL
st0
)]([1
)]([ 0
A shift in the time domain is equivalent to an exponential term in the s-plane.
Laplace Transform
14
3. f(t) = t u(t) = Ramp Function
0tt
00 t
2000
11][
1)]([)(
sdte
ste
sdttettuLsF ststst
Laplace Transform
se
sdteettuLsF tsstt 1
][1
)]([)( 0)(
0
4. Exponential Function f(t)=e-t u(t)
15
][sin][cos][)( tLjtLeLsF tj
5. Sinusoidal Functions f(t) =
ejωtu(t)
222222
1][
s
js
s
s
js
jseL tj
22][cos
s
stL
22][sin
s
tL
Laplace Transform
16
Laplace Transform
17
Laplace Transform
18
Laplace Transform
19
Laplace Transform
20
If the limit
exist
Initial Value Theorem
1.
F(s) has a pole of order 2 at zero and theorems can not be
applied.
)()( limlims0t
ssFtf
Provided that sF(s) does not have any
poles on the j axis and in the right half s-
plane
Final Value Theorem)()( limlim
0st
ssFtf
Examples
)3(2s
2sF
2
ss
2. ; No poles in the right half s-plane
(Stable).
)6(
3sF
2
sss
2
1
6
3)()(
20s0st
limlimlim
ss
ssFtf
Initial and Final Value Theorems
21
Inverse Laplace Transform using Partial Fraction Expansion
The inverse Laplace Transform does not relay on the use of the inversion Integral. Rather the inverse Laplace transform operation involving rational functions can be carried out using a Laplace Transform table and Partial fraction expansion. f(t)=L-1[F(s)]
Suppose that all poles of the transfer function are simple
)2()1()1)(2(
3s
23
3ssF
2
s
B
s
A
ssss
2)2(
)3()()1(A
1
1
s
s s
ssFs
1)1(
)3()()2(
2
2
s
s s
ssFsB
)2(
1
)1(
2sF
ss
From the Laplace Transform
Table,
tt ee 22tf
Inverse Laplace Transform
22
Example
Consider
:
)2()1()1)(2(
52
)23(
52sF
2
s
C
s
B
s
A
sss
ss
sss
ss
The first term is the steady-state solution; the last two
terms represent the transient solution. Unlike the
classical method, which requires separate steps to give the
transient and the steady state solutions, the Laplace
transform method gives the entire response.
02
35
2
5tf 2 tee tt
2
3;5;
2
5 CBA
By tacking the inverse Laplace transform of this
equation, we get the complete solution as:
Inverse Laplace Transform
23
Example
Inverse Laplace Transform
24
Example
Inverse Laplace Transform
25
Inverse Laplace Transform
26
f(t) F(s)
1. δ(t) 1
2. u(t)
3. t u(t)
4. tn u(t)
5. e-at u(t)
6. sin t u(t)
7. cos t u(t)
s
1
2
1
s
1
!ns
n
as 1
22 s
22 s
s
Laplace Transform of Basic Functions
27
Laplace Transform Properties
28
Laplace Transform Properties
29
Laplace Transform Properties
30
Sampling and Digital Systems
31
Advantages of digital control
Hardware is replaced by software, which is costly-effective
Complex function can be implemented in software so easily
rather than hardware
Reliability in implementation, that means, you can simply
modify the control function in software without extra cost.
Computers can be used in data logging (monitoring),
supervisory control and can control multiple loop
simultaneously as the computers are well fast.
Sampling and Digital Systems
32
Sampling and Digital Systems
33
Sampling and Digital Systems
34
Sampling and Digital Systems
35
Sampling and Digital Systems
36
Digital controllers could take one of the forms:• A computer or simply microprocessor board. Once they have developed and started to be manufactured commercially, digital controllers are developed.• Microcontroller is a microprocessor system on chip as a single integrated circuit. It is used in embedded control applications such as TV, mobile phones, Air conditioner, Video Camera, Hard disk controllers, Robots, Smart car manufacturing, ...etc. It is used for a limited number of I/O signals in real time applications.• Programmable logic controller (PLCs). PLC can handle a very large number of I/O signals (as hundreds or thousands) in industrial control applications. It has a standard interfaces with the field measurements. The PLC technology replaces the old hardwired control (relay logic control) cabinets in the industry.
Sampling and Digital Systems
37
The analog signal is a continuous representation of a signal, that it takes different values with time. Digital signals have two values only or two level corresponding to logic 1 and logic zero
Sampling and Digital Systems
38
The ADC requires three operations in sequence:
1- Sampling, we need to sample the analog signal at a constant rate. The sampler could be an electronic switch. The critical question is how to select the sampling frequency.
2- Holding, that holds the sample in during the sampling period until a new sample is captured. This is necessary to convert a constant value into digital word.
3- Conversion, it is often sequential circuit that takes a considerable time to convert the holding sample into digital word.
Sampling and Digital Systems
39
Sampling and holding process
A/D Converters
n
VV
VVN 2
)(
)(
minmax
min
Digitized Value
Sampling and Digital Systems
40
Sampling and Digital Systems
41
DAC requires two operations in sequence:
1- DAC, in general is faster than ADC ones and easier in
implementation.
2- Holding, it is very difficult to apply the discrete signal that
outputs from DAC directly to an analog process. It will excite
the system and fatigue the actuator. Therefore, holding these
samples makes them in a continuous form (stepping levels).
Sampling and Digital Systems
42
Sampling and Digital Systems
43
It is imperative that an ADC's sample time is fast enough to capture
essential changes in the analog waveform. In data acquisition
terminology, the highest-frequency waveform that an ADC can
theoretically capture is called Nyquist frequency, which equals to
one-half of the ADC's sample frequency. Therefore, if an ADC circuit
has a sample frequency of 5000 Hz, the highest frequency waveform
will be the Nyquist frequency of 2500 Hz. If an ADC is subjected to
an analog input signal whose frequency exceeds the Nyquist
frequency for that ADC, the converter will output a digitized signal of
falsely low frequency.This phenomenon is known as aliasing effect.
Selection of Sampling Frequency
44
Selection of Sampling Frequency
45
Selection of Sampling Frequency
46
Aliasing Phenomenon
In practice, the sampling frequency = 10 *frequency bandwidth of the analog signal.
Selection of Sampling Frequency
47
The system bandwidth frequency is not the only limit to select the sampling frequency, there is also other constraints due to time considerations in ADC, DAC, and microprocessor to execute the control program. In general, the sampling period Ts to control a single loop can be selected using the following relationship:
1)/2 fB > (Ts > (TADC + Tμp +TDAC)
Where fB = frequency bandwidth of the analog signalTADC = conversion time of ADCTDAC = conversion time of DAC(can be ignored)Tμp = Execution time of the control program in
microprocessor, it depends the speed of microprocessor
Selection of Sampling Frequency
48
0 1 2 3 4 5 6 7 8 9 10-5
-4
-3
-2
-1
0
1
2
3
4
5
FourierTransform
F()
0-0
0 1 2 3 4 5 6 7 8 9 10
-5
-4
-3
-2
-1
0
1
2
3
4
5
Fs()
0-0-s s 2s-2s
0 1 2 3 4 5 6 7 8 9 100.5
1
1.5
2
2.5
3
3.5
4
Fs()
0-0-s s 2s-2s
Aliasing effect
Selection of Sampling Frequency
49
Preventing aliases Make sure your sampling frequency is greater than twice
of the highest frequency component of the signal. In practice, take it ten times the highest frequency component.
Pre-filtering of the analog signal
Set your sampling frequency to the maximum if possible
Selection of Sampling Frequency
50
Selection of Sampling Frequencyw
ith
ou
t p
re-
filt
erin
g
Wit
h p
re- fi
lter
ing
51
Z-Transform and Its Properties
52
Z-Transform and Its Properties
53
Z-Transform and Its Properties
54
Z-Transform and Its Properties
55
Z-Transform and Its Properties
56
Z-Transform and Its Properties
57
Z-Transform and Its Properties
58
Z-Transform and Its Properties
59
Z-Transform and Its Properties
60
Z-Transform and Its Properties
61
Inverse of Z-Transform
Z-Transform and Its Properties
62
Z-Transform and Its Properties
63
Z-Transform and Its Properties
64
Z-Transform and Its Properties
65
Inverse of Z-Transform
For simple poles of X(z) zk-1 at z=zi the corresponding residue K is given by
If X(z) zk-1 contains multiple pole of order q at z=zj the corresponding residue K is given by
Z-Transform and Its Properties
66
Examples
Z-Transform and Its Properties
67
Z-Transform and Its Properties
68
Z-Transform and Its Properties
69
Laplace transform is a necessary mathematical tool for analysis and design of continuous control systems
Z-Transform is a necessary mathematical tool for analysis and design of sampled and digital control systems
MATLAB/SIMULINK software is a powerful tool for analysis, simulation, and design for both continuous and digital control systems
Summary
70
In real time digital control applications, the sampling period is bounded by lower and upper limits
The lower sampling limit depends on the computational and conversion delays
The upper sampling limit depends on the frequency bandwidth of the sampled signals
Noise rejection in the measured signals by using low pass filter is necessary to avoid aliasing effect in the frequency spectrum
Summary