2 Calculation of {varvec K}_{{varvec p}} ,{varvec K}_{{varvec I}} ,{varvec K}_{{varvec D}} for Cart Position of CIP
Tải bản đầy đủ - 0trang
Contourlet Transform Based Feature Extraction Method …
415
Table 2 Comparative analysis of proposed method with some of the existing feature extraction
methods in hand based biometrics
Reference
Data set
Feature extraction methodology
Results
obtained
(EER %)
Meraoumia
et al. [12]
PolyU database for palm
print. PolyU database
for knuckle print
PolyU FKP Database
with 165 subjects and
7,440 images
Generation of phase correlation
function based on discrete fourier
transform
Measurement of average gray scale
pixel values and frequency values of
the block subjected to 2D DCT and
matching using correlation method
Computation of Eigen values by
subjecting the FKP image to random
transform and matching by calculating the minimal distance value
Contourlet transform based feature
extraction method
1.35
Saigaa et al.
[15]
Hedge et al.
[14]
PolyU FKP Database
with 165 subjects and
7,440 images
This paper
PolyU FKP Database
with 165 subjects and
7,440 images
1.35
1.28
0.82
The proposed ﬁnger knuckle recognition system is compared with the existing
personal recognition systems based on texture analysis methods which has been
implemented on various hand based biometric traits such as ﬁnger knuckle print,
palm print, hand vein structure and ﬁnger prints. The following Table 2 illustrates
the summary of reported results of existing systems and comparative analysis of
those results with the performance of proposed system.
In the comparative analysis as shown in the Table 2, it has been found that, the
existing ﬁnger knuckle print authentication system based on coding method and
appearance based method produces accuracy which is purely dependent upon the
correctness of the segmentation techniques and quality of the image captured
respectively. But in the case of the proposed Contourlet Transform based Feature
Extraction Method which is based on texture analysis produces the lower equal
error rate (EER) of 0.82 % with less computational complexity.
5 Conclusion
This paper has presented a robust approach for feature extraction from ﬁnger
knuckle print using Contourlet transform. The proposed CTFEM approach extracts
reliable feature information from ﬁnger knuckle print images is very effective in
achieving high accuracy rate of 99.12 %. From the results analysis presented in the
paper, it is obvious that the ﬁnger back knuckle print offers more features for
personal authentication. In addition, it requires less processing steps as compared to
the other hand traits used for personal authentication and hence it is suitable for all
types of access control applications. As a future work, ﬁrst we plan to incorporate
416
K. Usha and M. Ezhilarasan
shape oriented features of FKP along with this CTFEM texture feature extraction
method. Secondly, analyze the performance of the recognition methods using two
different ﬁnger knuckle datasets. Finally, derive the computational complexity of
the recognition methods in order to analyze its space and time requirements.
References
1. Rao, R.M., Bopardikar, A.S.: Hand-based biometrics. Biometric Technol. Today 11(7), 9–11
(2003)
2. Ribaric, S., Fratric, I.: A biometric identiﬁcation system based on Eigen palm and Eigen ﬁnger
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6. Zhang, L., Zhang, L., Zhang, D.: Finger-knuckle-print: a new biometric identiﬁer. In:
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1, pp. 76–82 (2009)
7. Woodard, D.L., Flynn, P.J.: Finger surface as a biometric identiﬁer. Comput. Vis. Image
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an efﬁcient multi-biometric system of person recognition. In: Proceedings of IEEE
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ﬁnger knuckle print. In: Proceedings of COMPUTE’11 ACM, Bangalore, Karnataka, India,
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14. Hegde, C., Deepa Shenoy, P., Venugopal, K.R., Patnaik, L.M.: Authentication using ﬁnger
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Berlin (2013)
15. Saigaa, M., Meraoumia, A., Chitroub, S.B.: A efﬁcient person recognition by ﬁnger-knuckleprint based on 2D discrete cosine transform. In: Proceedings of ICITeS, vol. 2. No. 1, pp. 1–6
(2012)
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geometric analysis of contourlet transform. Neurocomputing 72(1–3), 203–211 (2008)
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18. Hu, H., Yu, S.: An image compression scheme based on modiﬁed contourlet transform.
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19. PolyU Finger Knuckle Print Database. http://www.comp.polyu.edu.hk/biometrics/FKP.htm
CDM Controller Design for Non-minimum
Unstable Higher Order System
T.V. Dixit, Nivedita Rajak, Surekha Bhusnur and Shashwati Ray
Abstract The major problem in the ﬁeld of control system design is to develop a
control design procedure which is simple to implement and reliable in performance
for complex control problems. Yet, the design of an effective controller for highly
complicated, Non-minimum phase, higher order and unstable system is a challenging problem in the control community. The classical control procedure using
PID controller is only effective for ordinary control problems and fails, when it is
applied to some complex. In the present work, an algorithm is proposed to calculate
the PID parameters. Coefﬁcient Diagram Method (CDM) of controller design has
been proposed that keeps a good balance of stability, speed of response and
robustness. In this work, a CDM controller is designed and implemented to a fourth
order non-minimum unstable system. The results of proposed CDM method depict
a better disturbance rejection property, stability and speed of response as compared
to the PID controller.
Keywords Cart-inverted pendulum (CIP)
method (CDM) Stability indices
Á
Á PID controller Á Coefﬁcient diagram
T.V. Dixit (&)
Sarguja University, Ambikapur (C.G.), India
e-mail: tvdixit@gmail.com
N. Rajak Á S. Bhusnur Á S. Ray
Bhilai Institute of Technology, Durg (C.G.), India
e-mail: nivedita.5rajak@gmail.com
S. Bhusnur
e-mail: sbhusnur@yahoo.co.in
S. Ray
e-mail: shashwatiray@yahoo.com
© Springer India 2015
L.C. Jain et al. (eds.), Computational Intelligence in Data Mining - Volume 3,
Smart Innovation, Systems and Technologies 33, DOI 10.1007/978-81-322-2202-6_38
417
418
T.V. Dixit et al.
1 Introduction
In this paper, a classical problem of Cart Inverted Pendulum (CIP) is considered.
CIP is a fourth order unstable non-minimum system. Since, the CIP is inherently
unstable [1, 2] the pendulum will not remain upright without the external forces.
The problem involves a cart, which moves back and forth, and a pendulum, hinged
to the cart at the bottom of its length such that the pendulum can move in the same
plane as the cart [3]. Thus, the pendulum mounted on the cart is free to fall along
the cart’s axis of motion. The system is to be controlled so that the pendulum
remains balanced and upright. Like any other physical system, the CIP also exhibits
non-linear behaviour as the system parameters may change during operation [4, 5].
Therefore, designing a sufﬁciently robust controller to the aforesaid CIP model is a
challenging task [6, 7].
1.1 CIP: A Fourth Order System and Its Transfer Function
In the state model of CIP, the position, velocity of the cart, the angular position and
change of angular position of the pendulum are taken as a set of state variables.
This system state-space equation can be written as follows
2
x_
6 x::
6
6
4 u_
u
X_ ẳ Ax ỵ Bu; Y ẳ CX ỵ Du
3 2
0 1
7 6 0 I ỵ ml2 b=IM ỵ mị ỵ Mml2
7 6
7ẳ6
5 40 0
0 mlb=IM ỵ mị ỵ Mml2
2
3
0
6 I ỵ ml2 =IM ỵ mị ỵ Mml2 7
6
7
ỵ6
7
40
5
2
2
I þ ml =IðM þ mÞ þ Mml
0
m2 gl2 =IðM þ mị ỵ Mml2
0
mglM ỵ mị=IM ỵ mị ỵ Mml2
2 3
!
! x
0
1 0 0 0 6 x_ 7
x
u
ỵ
ẳ
Yẳ
4u 5
0
0010
u
u_
0
0
1
0
32
x
76
76 x_
74 u
5
u_
3
7
7
5
1ị
!
2ị
The state equations that describe the behavior of the inverted pendulum are
given in (2).
1.2 Parameters of the Cart-Inverted Pendulum
In this paper the parameters of the Cart-inverted pendulum for simulation are listed
in Table 1.
CDM Controller Design …
Table 1 Parameters of the
cart-inverted pendulum
419
M
b
l
m
I
g
Mass of the cart
Friction of the cart
Length of the pendulum
Mass of the pendulum
Inertia of the pendulum
Gravity
0.5 kg
0.1 N/m/s
0.3 m
0.2 kg
6E-4 kg m2
9.8 m2
By considering the parameters (from Table 1) the transfer functions for pendulum angle and cart position control are obtained as given in (3) and (4).The
transfer function for pendulum angle control is given by
usị
4:5455s2
ẳ 4
Usị s ỵ 0:18181s3 31:1818s2 4:4545s
3ị
Xsị
1:8182s2 44:5455
ẳ 4
Usị s ỵ 0:18181s3 31:1818s2 À 4:4545s
ð4Þ
2 PID Controller Design
The PID controller TF is given by
KD s2 ỵ Kp s ỵ KI
U sị
KI
ẳ Kp ỵ ỵ KD s ẳ
E sị
s
s
5ị
To ensure better performance of PID controller, poles other than the dominant
poles of the desired Characteristic Equation (CE) are selected 2, 3 and 5 times
farther than the dominant poles (Pendulum Angle and cart position). The dominant
poles are decided based on desired settling time ts and peak overshoot MP . Where,
the unknown PID parameters of controller are obtained by comparing it with
desired CE.
3 A Brief Review on CDM
The CDM is fairly new and its basic philosophy has been known in control
community since four decades [8]. The power of CDM lies in that it generates not
only non-minimum phase controllers but also unstable controllers when required.
Unstable controllers are shown to be very effective in controlling such unstable
plants as CIP [9, 10]. The detailed description of CDM design method can be found
in [11–13].
420
T.V. Dixit et al.
Fig. 1 Mathematical model of CDM control system
Block diagram of CDM is shown in Fig. 1. Where, r is the reference input, y is
the output, u is the control and d is the external disturbance signal. Further, Nc ðsÞ
and Dc ðsÞ are controller polynomials designed to meet the desired transient
response and the pre-ﬁlter FðsÞ is used to provide the steady-state gain. The output
of this closed loop system is
yẳ
Nc sịF sịr Dc sịNc sịd
ỵ
Psị
Psị
6ị
where P(s) is the characteristic polynomial and is given by.
Psị ẳ Dp sịDc sị ỵ Np sịNc sị
ẳ an sn ỵ ỵ a1 s1 ỵ a0 ẳ
n
X
7ị
ai s i
8ị
iẳ0
The mathematical relations involving coefﬁcients of characteristic equation and
stability indices, limits and time constant are detailed in [8, 14].
The characteristic polynomial is expressed using a0 ; s and ci as follows:
"(
a0
n
i1
X
Y
1
iẳ2
j
jẳ1 cij
!
ssị
)
i
#
ỵ ss ỵ 1
9ị
The details of basic stability conditions involved are discussed in detail in the
earlier papers on CDM [11, 12, 15].
3.1 Proposed Algorithm for CDM Controller
Let the degree of the numerator and denominator polynomial of the plant TF be
assumed as mp and np respectively. So, the TF of an LTI system can be expressed in
the form [4]:
CDM Controller Design …
421
Np sị Nmp smp ỵ Nmp1 smp1 ỵ N1 s ỵ N0
ẳ
Dp sị
Dnp snp ỵ Dnp1 snp1 ỵ D1 s ỵ D0
10ị
Let the degree of the denominator polynomial Dc ðsÞ of the controller TF be nc
and that of the numerator polynomial of the controller transfer function Nc ðsÞ be mc .
The controller polynomials are suitably chosen so that the effect of disturbance and
noise is decreased to the minimum. Let the polynomials be given by
Dc ðsÞ ¼
nc
X
li si ; Nc ðsÞ ¼
i¼0
mc
X
ki s i
ð11Þ
i¼0
Considering only input r, the pre-lter F sị is chosen as
Psịsẳ0
F sị ẳ
Np sị
12ị
In CDM, s is chosen as
sẳ
ts
or ts ẳ 2:5 $ 3Þs
2:5 $ 3
ð13Þ
where ts is user-speciﬁed settling time [5].
By using the standard values of stability indices and the value of equivalent time
constant s calculated from (13), the target characteristic polynomial and coefﬁcients
are calculated. The closed loop characteristic equation is obtained with the help of
(10)–(11) in terms of unknown controller parameters and plant parameters. From (7),
Dp sịDc sị ỵ Np sịNc sị ẳ Ptarget sị
14ị
The controller parameters are obtained by solving the Diophantine Eq. (14).
To control the selected design problem following steps are adopted in the
proposed design algorithm:
1. Deﬁne Np ðsÞ, Dp ðsÞ as in (10) and select. Nc ðsÞ, Dc ðsÞ of the controller as in
(11).
2. Derive the Closed Loop Characteristic Polynomial (CLCP) in terms of unknown
controller parameter and deﬁned polynomials of plant.
3. Derive the target Characteristic Polynomial (CP) from (9) (order of target CP is
selected on the basis of order of CLCP from (14).
4. To derive the target CP, select a0 ; s; ci . The value of s is obtained on the basis of
desired settling time from (13).
5. Draw the Coefﬁcient Diagram and compare the target characteristic polynomial
with closed loop characteristic polynomial by using Diophantine Eq. (14).
6. Derive the controller parameters and calculate Pre-ﬁlter from (12).
7. Simulate the plant with proposed controller.
422
T.V. Dixit et al.
Table 2 PID controller results of pendulum angle of CIP
ts ðsÞ
%Mp
f ðrad=sÞ
Dominant poles
Other poles
Parameters Kp ; KI ; KD
2
10
0.591
À2 Ỉ 2:73
–4, –6
23.456, 63.889, 3.039
2
10
0.591
À2 Æ 2:73
–4
12.896, 13.562, 1.719
(1st row) WOPZC-without, (2nd row) WPZC-with pole-zero cancelation
4 Results and Discussion
The simulation results of CIP control system are obtained using MATLAB/SIMULINK 7.1 (R2010a) environment.
4.1 Calculation of K p ; K I ; K D for Pendulum Angle of CIP
The location of dominant poles, other assumed poles as well as parameters of PID
Controller of pendulum angle for desired settling time and percentage overshoot are
listed in Table 2.
4.2 Calculation of K p ; K I ; K D for Cart Position of CIP
Substituting the plant and proposed controller transfer functions from (4) and (5),
the characteristic equation of the overall system is obtained as:
b5 s 5 ỵ b4 s 4 þ b3 s 3 þ b2 s 2 À b1 s b0 ẳ 0
15ị
where,
b5 ẳ 1; b4 ẳ 0:1818 þ 1:8182KD ; b3 ¼ 1:8182KP À 31:1818;
b2 ¼ 1:8182KI À 44:5455KD À 4:4545; b1 ¼ 44:5455KP ; b0 ¼ 44:5455KI :
The location of dominant poles, other assumed poles as well as parameters of
PID Controller of cart-position for desired settling time and percentage overshoot
are listed in Table 3.
Table 3 PID controller results of cart-position of CIP
ts ðsÞ
%Mp
f ðrad=sÞ
Dominant
poles
Other poles
Parameters
Kp ; KI ; KD
12.5
10
0.591
À0:32 Ỉ 0:44
15
10
0.591
À0:27 Ỉ 0:36
–0.64, –0.96,
–1.59
–0.54, –0.81,
–1.35
19.927, 52.088,
1.957
19.239, 44.349,
1.659
CDM Controller Design …
423
Table 4 CDM controller results of pendulum angle of CIP
Sys=FðsÞ
TOC
ts ðsÞ
s
P target
ref
WOPZC
WPZC
WPZC
Controller parameters
k2 ; k1 ; k0 ; l2 ; l1 ; l0
1
4:544s2
1
4:544s
2/2
5
2
(16)
0.032, 0.23, 0.36, 6E-4, 0.013, 0
2/2
5
2
(17)
0:4
4:544s
2/2
2
0.8
(18)
0.223, 0.1226, 0.560, 0.013,
0.126, 0
0.004, 0.031, 0.072, 1E-4, 12E-4,
0
4.3 Calculation of Controller Parameters for Proposed CDM
Controller
The procedure involved in the CDM controller design is described as follows: The
target polynomial (pendulum angle) WOPZC with ts ¼ 2 s is given in (16) and the
corresponding values of controller parameters are listed in Table 4. Similarly, the
target polynomials WPZC with ts ¼ 2 s as well as ts ¼ 5 s are presented in (17) and
(18) respectively.
Ptrg sị ẳ 0:0006s6 ỵ 0:0128s5 ỵ 0:128s4 ỵ 0:64s3 ỵ 1:6s2 ỵ 2s ỵ 1
Ptrg sị ẳ 0:0128s5 ỵ 0:128s4 ỵ 0:64s3 ỵ 1:6s2 ỵ 2s ỵ 1
Ptrg sị ẳ 0:0001s5 ỵ 0:0013s4 ỵ 0:0164s3 ỵ 0:1024s2 ỵ 0:32s ỵ 0:4
16ị
17ị
18ị
4.4 For Cart-Position of Cart-Inverted Pendulum
The seventh order target polynomial (cart-position) with ts ¼ 12:5 s and ts ¼ 15 s
are given in (19) and (20) respectively and their controller parameters are listed in
Table 5.
Table 5 CDM controller results of cart position of CIP
Sys=FðsÞ
TOC
ts ðsÞ
s
P trg
ref
ða0 ¼ 3Þ
3
1:82s2 À44:55
ða0 ¼ 2:5Þ
2:5
1:82 s2 À44:55
3/3
12.5
5
(19)
3/3
15
6
(20)
Controller parameters
k3 ; k2 ; k1 ; k0 ; l3 ; l2 ; l1 ; l0
14.263,80.541, 10.961, −0.067
0.029, 0.464, −21.357, −112.94
30.452, 167.068, 23, −0.056
0.088, 1.150, 45.073, 233.38
424
T.V. Dixit et al.
Ptrg sị ẳ 0:0293s7 ỵ 3:75s6 ỵ 0:0128s5 ỵ 15s4 ỵ 30s3 ỵ 30s2 ỵ 15s ỵ 3
19ị
Ptrg sị ẳ 0:0875s7 ỵ 1:1664s6 ỵ 7:7760s5 þ 25:92s4 þ 43:2s3 þ 36s2 þ 15s þ 2:5
ð20Þ
The Coefﬁcient diagrams for (16)–(20) are shown in Fig. 2.
4.5 Simulation Results of Proposed PID Controller
The calculated values of Kp ; KI ; KD for the proposed PID controller (pendulum
angle), from Table 2 are simulated with the plant and the result is shown in Fig. 3a.
Both achieve the reference condition of pendulum angle i.e. h ¼ 0 (vertically
upright position). But, the system WOPZC depicts better performance with respect
to both, disturbance rejection time and effect of step disturbance as compared to the
system WPZC.
The simulation results of cart position with PID controller parameters of Table 2
are shown in Fig. 3b. It is observed that for both the conditions viz., WOPZC and
WPZC the system shows unstable behavior i.e., the cart is unable to achieve the rest
position and it keeps on moving forward.
4.6 Simulation Results of Proposed CDM Controller
The calculated values of controller parameters of proposed CDM controller WOPZC and WPZC for pendulum angle are simulated with the plant and the results are
shown in Fig. 4a. The system WOPZC shows better performance in disturbance
rejection as compared to the system WPZC.
Coefficents of s
10
10
10
2
CTP WOPZC with ts=5 secs
CTP WPZC with ts=2 secs
CTP WPZC with ts=5 secs
a0=1 10
a2=1.6 a1=2
a0=1
a3=0.64 a2=1.6 a1=2
a3=0.64
a4=0.128
a0=0.4
a1=0.32
a4=0.128
a2=0.1024
a5=0.0128
0
-2
a5=0.0128
a6=0.0006
10
10
a3=0.0164
CTP with ts=15 secs
CTP with ts= 12.5 secs
2
a3=43.2
a4=25.92
a2=36
a3=30 a2=30
a5=7.7760 a4=15
a0=3
a0=2.5
a5=3.75
a6=1.1664
0
a1=15
a1=15
a6=0.4688
a7=0.0875
a4=0.0013
-4
a7=0.0293
a5=0.0001
10
6
5
4
3
2
For CIP Angle (Powers of s)
1
0
-2
8
6
4
2
For Cart Position (Powers of s)
Fig. 2 Coefﬁcient diagram for pendulum angle WOPZC, WPZC and for cart-position
0
CDM Controller Design …
425
Fig. 3 Response due to PID controller with a for pendulum angle b for cart position
Fig. 4 Response due to CDM controller for a h ¼ 00 ; with step disturbance for pendulum angle
b pendulum angle WPZC
The values of controller parameters of proposed controller obtained from Table 5
for pendulum angle of CIP WPZC are simulated with plant and the simulation result
is shown in Fig. 4b. The system shows more transient behavior when it is subjected
to impulse disturbance as compared to step disturbance.
In Fig. 5a, the cart is initially at rest position x = 0, the system is subjected to a
step disturbance and an impulse disturbance. The cart moves 1.6 m and takes
11.94 s to reject the step disturbance which is less than its desired disturbance
rejection time of 12.5 s and settles into rest position x = 0. The cart moves 0.65 m
and takes 11.93 s to reject the impulse disturbance which is less than its desired
disturbance rejection time of 12.5 s and gets into rest position x = 0. The cart moves
3.4 m and takes 17.5 s to reject the step disturbance which is greater than its desired
disturbance rejection time of 15 s and settles into rest position x = 0. The cart moves
1.35 m and takes 17 s to reject the impulse disturbance which is greater than its
desired disturbance rejection time of 15 s and gets into rest position x = 0. The
controller designed for ts ¼ 12:5 s gives better performance when the system is
given an impulse disturbance as compared to the case of a step disturbance. The
controller designed for ts ¼ 12:5 s also rejects the disturbances (step and impulse)
but shows more transient effect on the system as compared to the controller
designed for ts ¼ 12:5 s.