Analog Lecture: and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Analog and Digital Implementations of Fractional Order Operators
J. A. Tenreiro Machado Dept. of Electrotechnical Engineering Institute of Engineering of Porto PORTUGAL Email: [e-mail]
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Goal
Time domain based implementations of fractional order operators
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
2
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Discrete-Time Conversion Schemes (1)
Method Euler, Letnikov, First backward difference s z conversion Taylor series
s
1 1 - z -1 T
2 1 - z -1 T 1 + z -1
(
)
1 -1 ( - 1) - 2 z L 1 - z + 2! T
Tustin
s
Simpson
2 2 -2 -1 1 - 2z + 2 z L T
[
]
3 1 + z -1 1 - z -1 s -1 -2 T 1 + 4z + z
(
)(
)
3 -1 -2 1 - 4z + 2(4 + 3)z L T
[
]
3
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Discrete-Time Conversion Schemes (2)
Method Euler, Letnikov, First backward difference Pad (m=n=1)
-1 1 2 - ( + 1) z -1 T 2 + ( - 1) z
Pad (m=n=2)
-1 -2 2 1 12 - 6( + 2) z + ( + 3 + 2) z 2 -1 -2 T 12 + 6( - 2) z + ( - 3 + 2) z
Tustin
-1 2 1 - z -1 T 1 + z
2 -1 -2 2 3 - 3z + ( - 1) z 2 -1 -2 T 3 + 3z + ( - 1) z
Simpson
-1 3 2 - ( 4 - 3) z -1 T 2 + (4 + 3) z
a)
a)
-1 -2 2 3 2 4 3 2 3 3(16 - 25) - 6(16 - 24 - 25 + 33) z + (64 - 144 - 92 + 171 + 28) z -1 -2 2 3 2 4 3 2 T 3(16 - 25) + 6(16 + 24 - 25 - 33) z + (64 + 144 - 92 - 171 + 28) z
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
4
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
D1/2 - Amplitude of the Series Coefficients
10 10 10 term amplitude 10 10 10 10 10 10
4 3
S impson
2
1
0
-1
Tustin
-2
Letnikov
-3
Geometric
-4
10
0
10 te rm orde r
1
10
2
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
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Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Considerations
An integer order derivative implies a finite series A fractional order derivative requires an infinite number of terms fractional derivatives are `global' operators The Simpson approach shows convergence problems (not discussed further!) The Tustin and the Letnikov aproximations are more attractive from control pointview and will be adressed in next slides
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail] 6
integer derivatives are `local' operators while
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
D (0<<1) - Amplitude of Series
10
0
Letnikov
10
1
Tus tin
10
-1
10 10
-2
0
=0.9
10
-3
=0.1
10
-1
10
-4
10
-2
=0.9 10
-5
10
0
10 k
1
10
2
10
-3
=0.1
0
10
10 k
1
10
2
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
7
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
(0<<1) I
0
- Amplitude of Series
=0.9 10
1
10
Letnikov
Tus tin
10 10
-1
0
=0.9
10
-2
-1
10
10 =0.1 10
-3 0 1 2
-2
10
-3
=0.1
0
10
10 k
10
10
10 k
1
10
2
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
8
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Discrete Approximation Methods (1)
Series expansion:
Letnikov:
1 - z -1 1 = PN ( z -1 ) D ( z -1 ) = series T N T
Tustin:
2 1 - z -1 2 -1 = PN ( z -1 ) D ( z ) = series T 1 + z -1 N T
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
9
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Discrete Approximation Methods (2)
Pad (fraction) approximation:
Letnikov:
1 - z -1 1 Pm ( z -1 ) = D ( z -1 ) = pad -1 T T Qn ( z ) m ,n
Tustin:
-1 2 1 - z -1 2 Pm ( z ) D ( z -1 ) = pad T 1 + z -1 = T Q ( z -1 ) m ,n n
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
10
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Elemental Control System
R(s) + -
G (s ) = K s
C(s)
We will show:
Unit step response (open-loop and closed-loop) Frequency response
Comparison of series and Pad approximations versus the ideal case for =1/2 System parameters: K=1, T=0.001
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail] 11
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Open-loop Ideal Caracteristics
20 log10 |G(j)| -20 dB/dec
arg {G(j)} -/2 - /2 -
log
log
Transfer function: Go=K/s Amplitude of constant slope of -20 dB/dec Constant phase of -/2 rad Step response: y(t)=Kt/ (+1)
12
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Closed-loop Ideal Caracteristics
20 log10 |G(j)|
0 dB
arg {G(j)} -/2 - /2 -
Transfer function: Gc=K/(s+K) -20 dB/dec Infinite gain margin Constant PM=(1-/2) rad log Step response: y(t)=1-E(-Kt) where E(x) is the MittagLeffler function defined as: PM=(1-/2)
log
E ( x ) =
n =0
xn
( n + 1)
, >0
13
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Letnikov - Open Loop Unit Step Response
1.4 1.2 1 10 0.8 y(t) 0.6 10 0.4 0.2 0 10
-2 -1
Lin e ar plot
serie s, N=100 Pad, m=n=15 Idea l
10
1
Log -log p lot
series, N=100 Pad, m=n=15 Idea l
0
0
0.2
0.4 time [s]
0.6
0.8
1
y(t)
10
-3
10
-2
10 time [s]
-1
10
0
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
14
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Letnikov - Closed Loop Unit Step Response
0.7 0.6 0.5 0.4 y(t) y(t) 10
-1
Lin ear plot
series, N=100 Pad, m=n=15 Idea l
10
0
Log -log p lot
series, N=100 Pad, m=n=15 Idea l
0.3 0.2 0.1 0 10
-2
0
0.2
0.4 time [s]
0.6
0.8
1
10
-3
10
-2
10 time [s]
-1
10
0
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
15
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Tustin - Open Loop Unit Step Response
1.4 1.2 1 10 0.8 y(t) 0.6 10 0.4 0.2 0 10
-2 -1
Lin ear plot
series, N=100 Pad, m=n=15 Idea l
10
1
Log -log p lot
series, N=100 Pad, m=n=15 Idea l
0
0
0.2
0.4 time [s]
0.6
0.8
1
y(t)
10
-3
10
-2
10 time [s]
-1
10
0
J. A. Tenreiro Machado, of Institute Engineering of Porto, Portugal, [e-mail]
16
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Tustin - Closed Loop Unit Step Response
0.7 0.6 0.5 0.4 y(t) y(t) 10
-1
Lin ear plot
series, N=100 Pad, m=n=15 Idea l
10
0
Log -log p lot
series, N=100 Pad, m=n=15 Idea l
0.3 0.2 0.1 0 10
-2
0
0.2
0.4 time [s]
0.6
0.8
1
10
-3
10
-2
10 time [s]
-1
10
0
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
17
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Letnikov - Frequency Response, I1/2
20 Amplitude (dB) 0 -20 -40 -2 10 100 Phase (degrees ) 50 0 -50 -100 -2 10
-1 0 1 2 3 4
Op en Loop
series, N=100 Pad, m=n=15 Ideal
Closed Loop
0 -10 -20 -30 -40 series , N=100 P ad, m=n=15 Ideal 10
-1
10
-1
10
0
10
1
10
2
10
3
10
4
-50 -2 10 50 0 -50 -100 -2 10
10
0
10
1
10
2
10
3
10
4
10
10
10 [ra d/s]
10
10
10
10
-1
10
0
10 [ra d/s]
1
10
2
10
3
10
4
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
18
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Tustin - Frequency Response, I1/2
20 Amplitude (dB) 0 -20 -40 -60 -2 10 100 Pha se (degrees ) 50 0 -50 -100 -2 10
-1 0 1 2 3 4
Op en Loop
0 -20
Closed Loop
series , N=100 P ad, m=n=15 Ideal 10
-1
-40 -60 -2 10 100 50 0 -50 -100 -2 10
s erie s, N=100 P ad , m=n=15 Ideal 10
-1
10
0
10
1
10
2
10
3
10
4
10
0
10
1
10
2
10
3
10
4
10
10
10 [ra d/s]
10
10
10
10
-1
10
0
10 [ra d/s]
1
10
2
10
3
10
4
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
19
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
Application Example
r + - e KD u SH e Ms2
-sTD
y
We will show:
Root-locus :
TD=0, ={1/2, -1/2}, for ideal, series and Pad approximations of the Letnikov and Tustin conversion schemes
Time response:
TD={0, 0.1}, =1/2, for ideal, series and Pad approximations of the Letnikov and Tustin conversion schemes
Parameters : System: TD=0.1, M=1; Controller: K=10*T1/2, T=0.1
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J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
D1/2 - Root Locus of Infinite Series
2 1.5 1 0.5 Im(z) 0 -0.5 -1 -1.5 -2 -5 Im(z) K=0.58
Letnikov
2 1.5
Tus tin
K=1.42 1 0.5 0 -0.5 -1 -1.5 -2 -5
-4
-3
-2 Re(z)
-1
0
1
2
-4
-3
-2 Re(z)
-1
0
1
2
D1 / 2 ( z -1 ) = T -1 / 2 (1 - z -1 )1 / 2
-1 -1 -1 / 2 1 - z 1/ 2 D (z ) = T 2 1 + z -1
1/ 2
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
21
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
I1/2 - Root Locus of Infinite Series
3
Letnikov
3
Tus tin
2
2
1 Im(z) Im(z) -4 -3 -2 Re(z) -1 0 1 2
1
0
0
-1
-1
-2
-2
-3 -5
-3 -5
-4
-3
-2 Re(z)
-1
0
1
2
I 1 / 2 ( z -1 ) = D -1 / 2 ( z -1 ) = T 1 / 2 (1 - z -1 ) -1 / 2
-1 -1 -1 / 2 -1 1/ 2 1/ 2 1 - z I (z ) = D (z ) = T 2 1 + z -1
-1 / 2
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
22
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
D1/2 - Root Locus of Series (N=5)
2 1.5 1 0.5 Im(z) 0 -0.5 -1 -1.5 -2 -5 Im(z) Letnikov 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -5 Tustin
-4
-3
-2 Re(z)
-1
0
1
2
-4
-3
-2 Re(z)
-1
0
1
2
D1 / 2 ( z -1 ) =
1 1 -1 1 -2 1 -3 5 -4 7 -5 z - z 1 - z - z - z - 2 8 16 128 256 T
D1 / 2 ( z -1 ) =
2 1 - 2 1 - 3 3 - 4 3 -5 -1 1 - z + z - z + z - z 2 2 8 8 T
J. A. Tenreiro Machado, Institute of Engineering of Porto, Portugal, [e-mail]
23
Lecture: Analog and Digital Implementations of Fractional Order Operators, Las Vegas, CDC 2002
1/2 I
3
- Root Locus of Series (N=5)
Letnikov 3 Tustin 2
2
1 Im(z)
1
0
Im(z) -4 -3 -2 Re(z) -1 0 1 2
0
-1
-1
-2
-2
-3 -5
-3 -5
-4
-3
-2 Re(z)
-1
0
1
2
3 5 35 -4 63 -5 1 I 1 / 2 ( z -1 ) = T 1 + z -1 + z -2 + z -3 + z + z 8 16 128 256 2
I 1 / 2 ( z -1 ) =
1 - 2 1 - 3 3 - 4 3 ...