Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller

doi:10.21629/JSEE.2018.01.17

Journal of Systems Engineering and Electronics ›› 2018, Vol. 29 ›› Issue (1): 166-175.doi: 10.21629/JSEE.2018.01.17

• Control Theory and Application • Previous Articles Next Articles

Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller

Lu LIU¹(), Liang SHAN^1,*(), Yuewei DAI¹(), Chenglin LIU²(), Zhidong QI¹()

¹ School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
² Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China

Received:2017-02-22 Online:2018-02-26 Published:2018-02-23
Contact: Liang SHAN E-mail:18551843710@163.com;shanliang@njust.edu.cn;daiywei@163.com;liucl@jiangnan.edu.cn;qizhidong@sina.com.cn
About author:LIU Lu was born in 1990. He received his B.S. degree in electrical engineering and automation from University of Jinan in 2013. He is currently a Ph.D. degree candidate in control science and engineering at Nanjing University of Science and Technology. His research interests include servo control system and intelligence control algorithm. E-mail: 18551843710@163.com|SHAN Liang was born in 1979. He received his B.S. degree in electrical engineering in 2002 and Ph.D. degree in control science and control engineering in 2007 both from Nanjing University of Science and Technology. Now he works as an associate professor in Nanjing University of Science and Technology. His research interests include intelligence control algorithm, nonlinear system, and control methods of motor servo system. E-mail: shanliang@njust.edu.cn|DAI Yuewei was born in 1962. He received his B.S. and M.S. degrees in system engineering from East China Engineering Institute in 1984 and 1987, respectively, and his Ph.D. degree in control science and engineering from Nanjing University of Science and Technology in 2002. Now he works as a professor in Nanjing University of Science and Technology. His research interests include multimedia security, system engineering theory and network security. E-mail: daiywei@163.com|LIU Chenglin was born in 1981. He received his B.S. degree in electrical engineering and automation from Nanjing University of Science and Technology in 2003, and his Ph.D. degree in control theory and control engineering from Southeast University. Since 2008, he has been with Jiangnan University, where he is currently a professor at Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of Internet of Things Engineering. His current research interests include cyber-physical systems, sensor networks and coordination control of multi-agent systems. E-mail: liucl@jiangnan.edu.cn|QI Zhidong was born in 1976. He received his B.S. degree in industrial automation from North China Electric Power University in 1999, M.S. degree in electrical engineering from Jiangsu University in 2002, and Ph.D. degree in control science and control engineering from Shanghai Jiao Tong University in 2006. Now he works as an associate professor in Nanjing University of Science and Technology, China. His research interests include fractional control algorithm, nonlinear system, modeling and control methods of fuel cell. E-mail: qizhidong@sina.com.cn
Supported by:
the National Natural Science Foundation of China(61374153);the National Natural Science Foundation of China(61473138);Natural Science Foundation of Jiangsu Province(BK20151130);Six Talent Peaks Project in Jiangsu Province(2015-DZXX-011);China Scholarship Council Fund(201606845005);This work was supported by the National Natural Science Foundation of China (61374153; 61473138), Natural Science Foundation of Jiangsu Province (BK20151130), Six Talent Peaks Project in Jiangsu Province (2015-DZXX-011), and China Scholarship Council Fund (201606845005)

Abstract

Abstract:

The quantum bacterial foraging optimization (QBFO) algorithm has the characteristics of strong robustness and global searching ability. In the classical QBFO algorithm, the rotation angle updated by the rotation gate is discrete and constant, which cannot affect the situation of the solution space and limit the diversity of bacterial population. In this paper, an improved QBFO (IQBFO) algorithm is proposed, which can adaptively make the quantum rotation angle continuously updated and enhance the global search ability. In the initialization process, the modified probability of the optimal rotation angle is introduced to avoid the existence of invariant solutions. The modified operator of probability amplitude is adopted to further increase the population diversity. The tests based on benchmark functions verify the effectiveness of the proposed algorithm. Moreover, compared with the integerorder PID controller, the fractional-order proportion integration differentiation (PID) controller increases the complexity of the system with better flexibility and robustness. Thus the fractional-order PID controller is applied to the servo system. The tuning results of PID parameters of the fractional-order servo system show that the proposed algorithm has a good performance in tuning the PID parameters of the fractional-order servo system.

Key words: bacterial foraging algorithm, fractional-order, quantum rotation gate, proportion integration differentiation (PID), servo system

Lu LIU, Liang SHAN, Yuewei DAI, Chenglin LIU, Zhidong QI. Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller[J]. Journal of Systems Engineering and Electronics, 2018, 29(1): 166-175.

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References 20

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Algorithm	$\theta _0 $	Optimal value	Worst value	Average value	Variance
IQBFO	0.05π	1.136 9E?09	5.308 6E+01	9.783 2E+00	5.335 8E+02
	0.1π	1.136 9E?09	2.293 7E?01	1.595 1E?02	2.738 1E?03
	0.3π	1.136 9E?09	1.648 5E?07	9.549 7E?09	1.337 1E?15
	0.5π	1.136 9E?09	4.286 8E?02	1.042 9E?02	1.857 9E?04
	0.7π	1.136 9E?09	2.082 4E?01	6.827 4E?02	5.795 8E?03
	0.9π	1.136 9E?09	4.102 7E?01	7.175 8E?02	1.348 1E?02
	0.11π	1.136 9E?09	3.754 5E?01	7.623 3E?02	1.519 1E?02

Function	Mathematical representation	Range of search	Optimal value
Griewank	$f_1 (x) = { }\sum\limits_{i = 1}^D {\frac{x_i^2 }{4\; 000}}-\prod\limits_{i = 1}^D {\cos (\frac{x_i }{\sqrt i })} +1$	$[-100, 100]^D$	0
Ackley	$f_2 (x) = 20+{{{\rm{e}}}}-{ }20{{{\rm{e}}}}^{-\frac{1}{5}\sqrt{\frac{1}{D}\sum\limits_{i = 1}^D {x_i^2 } }}-{{{\rm{e}}}}^{-\frac{1}{D}\sum\limits_{i = 1}^D {\cos (2 \pi x_i)} }$	$[-100, 100]^D$	0
Rastrigrin	$f_3 (x) = { }\sum\limits_{i = 1}^D {[x_i^2-10\cos (2 \pi x_i)+10]} $	$[-100, 100]^D$	0
Schewefel	${ } f_4 (x) = 418.982\; 9\ast D+\sum\limits_{i = 1}^D {(-x_i \sin(\sqrt {x_i }))} $	$[-500, 500]^D$	0
Rosenbrock	$f_5 (x) = { }\sum\limits_{i = 1}^{D-1} {[100(x_{i+1}-x_i^2)^2+(x_i-1)^2]} $	$[-10, 10]^D$	0
Sum squares	$f_6 (x) = { }\sum\limits_{i = 1}^D {ix_i^2 } $ & $[-10, 10]^D$	$[-10, 10]^D$	0
Dixon-price	$f_7 (x) = (x_1 -1)^2+{ }\sum\limits_{i = 2}^D {i(2x_i^2-x_{i-1})^2} $	$[-10, 10]^D$	0

Function	IQBFO		QBFO		QGA		CPSO
Function	Iter	Average value	Iter	Average value	Iter	Average value	Iter	Average value
Griewank	308	1.025 2E?01	372	1.126 9E+00	338	4.7286E+00	166	1.557 2E+00
Ackley	315	1.585 7E?01	389	1.107 8E+01	328	5.458 2E+01	181	2.597 8E+01
Rastrigrin	290	2.096 1E+01	387	1.022 9E+02	369	3.839 4E+02	147	3.201 5E+03
Schewefel	156	1.735 5E+03	116	1.331 3E+04	153	1.314 8E+04	96	1.842 4E+04
Rosenbrock	249	2.608 6E+01	265	1.209 8E+02	255	1.146 1E+02	107	1.810 5E+02
Sum squares	271	2.149 3E?01	402	6.425 5E+00	368	8.617E+00	168	1.030 6E+01
Dixon-price	237	2.016 2E+02	249	5.439 2E+03	239	4.750 8E+03	153	8.374 8E+03

$K_{p2} $	$K_{i2} $	$K_{d2} $	$K$	$\alpha $	$T_{on} $	$K_{AP}$	$T_Y $	$K_g $	$\theta '$	$G_{com} $
0.8	10	0	480	19.1	0.08	23/440	0.08	1	sin(0.1t)	$ {\left({1/25} \right)^2}{s^2} + 2 \times \left({1/25} \right) \times {\rm{ }}0.06s + 1 $
0.8	10	0	480	19.1	0.08	23/440	0.08	1	sin(0.1t)	$ {\left({1/40} \right)^2}{s^2} + 2 \times {\rm{ }}\left({1/40} \right) \times {\rm{ }}0.09s + 1 $

Parameter	Algorithm
Parameter	IQBFO	QBFO	QGA	CPSO
$K_p $	0.253 6	0.181 2	0.140 5	0.235 5
$K_i $	0.001 3	0.001 3	0.001 5	0.001 6
$K_d $	0.102 2	0.073 4	0.051 5	0.070 2
$K_{fe} $	19.010 4	18.988 2	19.104 7	19.021 4
$\lambda $	0.000 3	0.000 1	0.000 4	0.000 3
$\mu $	0.827	0.778 9	0.681	0.701 1

Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller

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Criterion	Algorithm
Criterion	IQBFO	QBFO	QGA	CPSO
Rise time/s	0.151	0.175	0.682	0.148
Setting time/s	0.805	1.002	1.123	1.001
Overshoot/%	0	0	0	0
Steady-state error	4.52E?02	8.98E?02	1.46E?01	8.56E?01

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