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2 Calculation of {varvec K}_{{varvec p}} ,{varvec K}_{{varvec I}} ,{varvec K}_{{varvec D}} for Cart Position of CIP

2 Calculation of {varvec K}_{{varvec p}} ,{varvec K}_{{varvec I}} ,{varvec K}_{{varvec D}} for Cart Position of CIP

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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

features. IEEE Trans. Pattern Anal. Mach. Intell. 27(11), 1698–1709 (2005)

3. Sun, Z., Tan, T., Wang, Y., Li, S.Z.: Ordinal palm print representation for personal

identification. In: Proceedings of CVPR 2005, vol. 1, no. 1, pp. 279–284 (2005)

4. Jain, A.K., Ross, A., Pankanti,S.: A prototype hand geometry based verification system. In:

Proceedings of AVBPA. Washington, DC, vol. 1, no. 1, pp. 166–171 (1999)

5. Kumar, A., Zhang, D.: Improving biometric authentication performance from the user quality.

IEEE Trans. Instrum. Measur. 59(3), 730–735 (2010)

6. Zhang, L., Zhang, L., Zhang, D.: Finger-knuckle-print: a new biometric identifier. In:

Proceedings of IEEE International Conference on Image Processing, Cairo, Egypt, vol. 1, no.

1, pp. 76–82 (2009)

7. Woodard, D.L., Flynn, P.J.: Finger surface as a biometric identifier. Comput. Vis. Image

Underst. 100(1), 357–384 (2005)

8. Kumar, A., Ravikanth, Ch.: Personal authentication using finger knuckle surface. IEEE Trans.

Inf. Secur. 4(1), 98–110 (2009)

9. Kumar, A., Venkataprathyusha, K.: Personal authentication using hand vein triangulation and

knuckle shape. IEEE Trans. Image Process. 18(9), 640–645 (2009)

10. Zhang, L., Zhang, L., Zhang, D.: Finger-knuckle-print verification based on band-limited

phase-only correlation. LNCS 5702, vol. 1. No. 1, pp. 141–148. Springer, Berlin (2009)

11. Zhang, L., Zhang, L., Zhang, D.: MonogenicCode: a novel fast feature coding algorithm with

applications to finger-knuckle-print recognition. In: IEEE International Workshop on

Emerging Techniques and Challenges (ETCHB), vol. 1. No. 1, pp. 222–231 (2010)

12. Meraoumia, A., Chitroub, S., Bouridane, A.: Fusion of finger-knuckle-print and palm print for

an efficient multi-biometric system of person recognition. In: Proceedings of IEEE

International Conferences on communications (ICC), vol. 1, No. 1, pp. 1–5 (2011)

13. Hegde, C., Phanindra, J., Deepa Shenoy, P., Patnaik, L.M.: Human Authentication using

finger knuckle print. In: Proceedings of COMPUTE’11 ACM, Bangalore, Karnataka, India,

vol. 1. No. 1, pp. 124–131 (2011)

14. Hegde, C., Deepa Shenoy, P., Venugopal, K.R., Patnaik, L.M.: Authentication using finger

knuckle prints, signal, image and video processing, vol. 7. No. 4, pp. 633–645. Springer,

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)

16. Yang, L., Guo, B.L., Ni, W.: Multimodality medical image fusion based on multiscale

geometric analysis of contourlet transform. Neurocomputing 72(1–3), 203–211 (2008)

17. Lu, Y., Do, M.N.: A new contourlet transform with shape frequency localization. IEEE Int.

Conf. Image Process. 1(1), 1629–1632 (2009)

18. Hu, H., Yu, S.: An image compression scheme based on modified contourlet transform.

Comput. Eng. Appl. 41(1), 40–43 (2005)

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





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



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



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.



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