Optimization of coal blending process control based on fuzzy self-tuning

Abstract

Due to the continuous improvement of industrial production requirements and green manufacturing demand, manufacturing enterprises and factories need to constantly optimize and improve the system structure. Based on the actual problems in the actual coal blending process, this paper analyzes the reason of the coal blending error in coal blending process and the optimization scheme of intelligent control of coal blending process. We optimized the overall structure of the coal blending system, improved the coal blending system which use the fuzzy control system. At the same time, we used MATLAB to simulate and analyse the simulation results. This paper gives an illustration to the optimal structure and system of the coal blending system, effectively improves the accuracy and stability of the coal blending process control system, reduces the coal blending error and improves the quality of coal blending in general.

Keywords

Green manufacturing PID intelligent control fuzzy self-tuning coal blending

1. Introduction

In order to save the production cost and increase the efficiency of coal blending, coking enterprises need to perform a series of coal blending tests such as coke oven tests or semi-industrial tests in the best coal blending program. The cost of the experiment is very expensive. It needs to be tested repeatedly and repeated experiments, but there is still a certain difference in order to meet the requirements of the enterprise [1 –4]. Maximizing the rational use of resources and optimizing the allocation of coke output are the problems that coking enterprises need to pay more attention to it. This requires optimizing the design on the basis of the original and optimizing the design of the coal blending process, so as to use the minimum resources to achieve the maximum optimization process of coal output. This is an arduous task that modern enterprises must complete [5 –8]. This is the arduous task that the modern enterprise must complete and continuously achieve the optimal allocation of the resources and realize the optimization and docking of the coal blending. It is the biggest goal of modern enterprises[9, 10].

In the early 1960s, many scholars studied the theoretical scheme of coking coal in large steel plants very profoundly. And most of the research was mainly to reduce costs and reduce environmental pollution. Because of the uncertainty of artificial coal blending, it is difficult to design a control system to optimize the allocation of coal blending process. The traditional control mode is insufficient to meet the demand for the coal blending system, and the requirements of its control are also very rare to meet the requirements [11 –14]. However, the overall structure control still needs to be improved. In the process of intelligent coal blending, human operation still needs to be carried out. The research on the guiding strategy of coking coal blending needs further optimization and promotion [15 –18].

Although some of the coal blending methods in foreign countries are much better than the domestic coal blending methods in terms of optimization control. However, high-precision requirements mean that high-precision infrastructure is required to match it, and professional personnel are required to monitor coal quality parameters [19, 20]. Based on the above research status, the current research focuses on the prediction of coke quality, and there is no systematic optimization of the system for the intelligent control of the coal blending process [21 –23]. In view of the characteristics and actual operation process of coal blending control in China’s large enterprises, this paper puts forward a coal blending process control and intelligent optimization system to optimize the coal blending control through the analysis of the control process of the coal blending in the actual steel plant. This article mainly studies the process of coal blending in coking plants of major companies. Based on the actual problems in the actual process of coal blending discovered through actual inspection, we analyzed the process of coal blending and the intelligent configuration scheme of coal blending. The artificial coal blending method determines the initial proportion of coal blending through experience, which obviously has great human factors and has great influence on the quality of coal blending. In view of the above shortcomings, we have established an intelligent control system for coking coal blending process to optimize the design[24 –26].

The rest of this paper is organized as follows. First, in Section 2 shows some related work to current situation of research on theoretical scheme of coking coal blending. Section 3 improves the structure of existing coal blending equipment. After that, Section 4 combines conventional PID control with fuzzy control to form a fuzzy self-tuning PID controller, and simulates the optimized controller with MATLAB. And finally, in Section 5 the optimization results of coal blending system is presented.

2. Related work

There are many corresponding studies on the design and control system scheme of coal blending process. Z Zhong [27] uses the design control system scheme of coal blending process to carry out serial optimization design process, and uses fuzzy controller to control coal blending. By establishing a stable model of coal blending, the accuracy of coal blending is improved, which provides a theoretical reference for coal blending process. Li Yonghua [28] studied the combustion mode of blended coal, tested 640MJ/H grinding system and exhaust gas online analysis system, and explained the control treatment methods of exhaust gas, cinder and some blended coal. By reducing the sulfur content of coal, the emission of sulfur oxide can be reduced, which make a theoretical foundation for other distribution modes. Ruhul A. Sarker [29] has applied coal blending modeling to coal blending schemes in a variety of ways in response to the complex problems of coal blending in the real world. These different approaches are to add or subtract some of the coal blending processes in reality. It explains the use of algebraic formula to solve the problem of coal mixing in different practical situations, and improves the practical significance of coal blending research. Shameek Ganguly [30] uses conventional PID position controller, empirical PAM model and internal pressure regulating circuit to realize the trajectory tracking extension of single-degree-of-freedom manipulator. The results of pressure regulation and position control confirm the validity of the method adopted, and reduce the complexity of control and the total cost of setting up in the actual experimental results. It provides theoretical support for coal blending control system. In order to improve the management and control level of coke ovens, Gongfa Li [31] and others studied the intelligent integrated control system and introduced the integrated management and control system of the coke oven including system model, production planning and management, heating control system and model. And method coke oven temperature, intelligent combustion control and pressure control air trap, pointed out the management and control integrated development of the coke oven control system positioning method, proposed the use of heating gas flow adjustment method, established the mixing intelligence of flue gas temperature Control model.

\enlargethispage 12pt Based on the above research status, the current research mainly focuses on the prediction of coke quality, and there is no systematic optimization of the system for intelligent control of coal blending process. The actual situation is often very complicated. In the coal blending process, it is not possible to rely entirely on the workers for experienced on-site coal blending. It is difficult for them to improve the accuracy of blending. This situation is neither economic nor reasonable.

3. Structure optimization of coal blending equipment

3.1. Structure of coal blending equipment

Coal blending is a closed process and complicated process. Therefore, coal blending error is easily affected by unsteady state interference. When the coal is taken, the manual operation heap is used to pick up the coal machine, and a certain amount of coal is put on the transmission device according to the demand. The transport equipment is transported through the conveyor belt to the coal trough through the conveyer belt, and the quality of the coal before the electronic scale is weighed. Then, a variety of coal is mixed according to a certain coal blending ratio, and the blended coal is transported to the coke oven by the conveyor belt quantitatively. In this process, it is necessary to sample the mixed coal on the conveyor belt regularly, carry out the test and error analysis of the sample after the sampling and determine whether the amount of coal is in the allowable range of error [32 –36]. If the error is exceeded, the corresponding parameters of the coal blending system need to be tested and adjusted. Due to the technical means of manual sampling and weighing detection errors, this will guarantee detection accuracy and accuracy, but also cause waste of time and labor costs.

3.2. Structure improvement of coal blending equipment

The actual situation shows that all kinds of external factors and the restriction of current technological conditions lead to greater errors in coal blending process. The coal in the coal tank still contains some impurities and moisture. The error of single quantitative coal extraction in coal blending plant is about 5%, which is hard to avoid and reduce at present. After processing, the error of coal weighing before mixing is about 1%, because the relative disturbance factors of coal in this part are few, so it is easy to control and change [37 –40]. At present, the intelligent control system of coal blending is also used to correct and deal with the current error. The system can control and detect the ratio of each coal in real time which could adjust the set value of the system and detect the measured value but it could not reflect and detect actual errors and proportions. Therefore, in actual work, people need regular sampling and testing for the quality ratio of the current blended coal, which will cost more time and manpower [41 –44].

Taking into account the practical application and technical cost, the coal treatment base is large, the environment is bad and the human factors could not be estimated. This three methods need to be further research. This article only studies the two technical methods. One is to optimize the structure of coal blending equipment, the other is to optimize the coal blending control system. The structure optimization method studied in this paper is to add the high precision electronic scale in the conveyor belt structure after the coal mixing, measure the quality of the mixed coal according to the system demand at any time, and compare the quality error before and after the system comparison and analysis, in order to judge the cause of the error and modify it. The purpose of adding electronic scale is to replace the technical means of previous artificial sampling weighing error, which can not only improve the accuracy and accuracy, but also save time and manpower cost. The structure of the improved coal blending process is shown in Fig. 1.

Fig. 1

Process flow chart of coal blending after structure improvement.

After the above structural improvement, the comparison of the actual design values of the 7 coals after the coal blending process was performed and the resulting error comparison table is shown in Table 1.

Table 1

Comparison of error values before and after structural improvement

	Species 1	Species 2	Species 3	Species 4	Species 5	Species 6	Species 7
Pre-improvement error	0.912%	0.635%	0.799%	0.953%	0.877%	0.679%	0.811%
Improved error	0.863%	0.612%	0.745%	0.879%	0.805%	0.656%	0.786%

According to the comparison of the data in the table, the larger the structural error after improvement, the more obvious the effect is. Although the overall effect is not particularly obvious, the results are of great value for practical applications. So it can be seen that after adding the high-precision electronic scale to the conveyor belt, the improved structure can reduce the error in coal blending process.

4. Optimization of intelligent control system

4.1. Conventional PID control system

At present, the control of coal blending process in domestic coking plant is still controlled by conventional PID. PID controller is a controller composed of three control models. The three control models are proportional element (P), integral unit (I) and differential unit (D) [45, 46]. The control of linear and stable system is achieved through setting three parameters of each unit. PID control theory can be regarded as a negative feedback control. When the designed loop is under control, the value returned by the feedback is compared with the target value set by the system, and the output of the system is adjusted according to the obtained error so as to control the object to be stable [47 –50]. Because the PID controller has formed a typical fixed structure, the hardware is easy to implement, the structure is changed flexibly, and its operation reliability is high, so it is widely used in production control. PID control schematic shown in Fig. 2.

Fig. 2

PID control schematic.

The relationship between PID controller input x_i (t), output x_o (t) and deviation ɛ (t) can be expressed as: $ɛ (t) = x_{i} (t) - x_{o} (t)$ (1)

The PID control law is formulated as: $m (t) = K_{p} [ɛ (t) + \frac{1}{T_{i}} \int_{0}^{t} ɛ (t) dt + T_{d} \frac{d ɛ (t)}{dt}]$ (2)

In the formula: K_Pɛ (t) is the proportional control item and K_p is the proportional coefficient; $\frac{1}{T_{i}} \int_{0}^{1} ɛ (t) dt$ is the integral control term, T_i is the integral time constant $T_{d} \frac{d ɛ (t)}{dt}$ is the derivative control term and T_d is the derivative time constant.

The proportionality coefficient should not be too large. When it is too large, the dynamic quality will become worse, causing the change of control volume or even the instability of the system.

After adding integral links, the dynamic process slows down, and the excessive integration will make the overshoot larger and cause the system to be unstable.

Differential control is a little bit more, but differential action amplifies the noise signal and reduces the performance of the system.

The fuzzy self-tuning PID controller based on fuzzy inference is one of them, which has the characteristics of simple structure, high stability and high control accuracy.

4.2. Fuzzy control system

Fuzzy control theory is a bionic intelligent control method. In the process, experts form the expert system to establish the control rules in the fuzzy rule controller through the experience formed by the technical personnel. After relevant processing, the input is obfuscated, the input fuzzy quantity is adapted to the adaptive control rule, the logical reasoning is completed, and the output quantity is obtained. The amount of blur is processed by the sharpening process, and it is converted into the actual quantity as the input of the actuator to control the controlled object. Fuzzy control can be regarded as a fuzzy controller represented by multi-dimensional functions. It is assumed that these several functions are multiple variable functions whose domain is [0, 1] and each function represents a fuzzy control rule. System input and input membership functions are blurred to obtain a fuzzy input. According to the relevant laws, an input is obtained through fuzzy inference, and then the fuzzy output is subjected to defuzzification processing according to the output membership function to obtain a system output that can be used for actual control. Fuzzy control can be summarized as the Fig. 3 [41 –43].

Fig. 3

Control theory block diagram.

(1) Establishment of Coal Blending Control Model

The PID process controller of the traditional coal blending system could not meet the actual needs of the current progress. Based on the characteristics and superior performance of fuzzy control, this paper combines fuzzy control and PID control to optimize the system and study the fuzzy self-tuning PID controller. After research and comparison, it has better performance characteristics than traditional PID controllers, and it is applied to the coal blending process control to optimize the coal blending system, so that the new coal blending system is more stable, more practical and more Precise performance.

In actual industrial production, the domestic coking plant coal blending system sets various coal blending ratios according to the amount of coal required by technical experts for actual demand, then the system is configured according to the set value. In the coal blending process of the system, errors are unavoidable due to various factors. In order to reduce the generation of errors, this system has introduced feedback signals to make the system form a closed-loop control and adopt fuzzy self-tuning PID control algorithm to achieve precise control. The system closed-loop control structure diagram is shown in Fig. 4. According to the above figure, combined with the coal blending system of the coking plant, the system control block diagram is obtained, as shown in Fig. 5. In the figure, r (s) is the cumulative input of coal and c (s) is the cumulative output of coal. Therefore, the system is a first-order inertial link system, and the transfer function of system control can be obtained as follows: $G (s) = \frac{1}{TS + 1}$ (3)

Fig. 4

System closed-loop control structure diagram.

Fig. 5

Closed-loop control structure diagram of the system.

In formula: time constant $T = \frac{1}{K_{0} K_{1} K_{2}} \approx 200$

Since various external factors affect the time constant, it is necessary to approximate the above transfer function to meet the requirements of the actual control system. The approximate transfer function is: $G_{1} (s) = \frac{1}{TS + 1} e^{- ts}$ (4)

According to the characteristics of coal blending control system, the lag time is set to t = 20s. Conventional PID control could not meet the actual requirements of coal blending because the parameters are set to a fixed value, so it needs to consider a better controller. However, the fuzzy self-tuning PID control can meet the relevant requirements and ensure the stability and accuracy of the control.

(2) Controller design

The controller designed in this paper is fuzzy self-tuning controller. It takes the error and its variation as the input value. In the working process, the system continuously detects the error is e and the error change is Δe, and returns the detection value back, thus can change the PID ratio coefficient K_p, the integral time constant T_i and the differential time constant T_d three parameters, in order to ensure the stability of the control quantity. The control theory block diagram is shown in Fig. 6.

Fig. 6

Control theory block diagram.

When setting a given target value of 40, the design of the system in conventional PID and fuzzy parameter self-tuning PID is shown in Fig. 7.

Fig. 7

Fuzzy parameter self-tuning PID.

The fuzzy controller is loaded and the corresponding parameters are set up, and the step response curve of time constant changes shown in Figs. 8–10.

Fig. 8

Response curve of time constant T = 200.

Fig. 9

Response curve of time constant T = 180.

Fig. 10

Response curve of time constant T = 160.

5. Conclusion

In view of the shortcomings of the overall structure of the existing coal blending control system in the domestic coking plant, the overall structure and control system of the coal blending system are improved and optimized. In terms of equipment structure, electronic scales connected to the system are added and data is fed back to the system. The test shows that the improved structure effectively reduces the error of coal blending. In terms of the controller, this paper optimizes the conventional PID controller, combines conventional PID control and fuzzy control to form a fuzzy self-tuning PID controller, and performs MATLAB simulation on the optimized controller. The results show that the control effect of fuzzy self-tuning PID is obviously better than that of conventional self-tuning PID controller. The stability, rapidity, and accuracy of the fuzzy self-tuning PID control are obviously improved. Comparing with the above data, when T = 200, the system is in the optimal working condition. This paper optimizes the structure and system of the coal blending system, improves the precision and stability of the control system of the coal blending process. Combined with the complex situation of coking coal blending process control system, the fuzzy self-tuning PID parameter controller of this system was designed and simulated by MATLAB and its conventional PID controller. Finally, the simulation results are analyzed and it is concluded that the fuzzy self-tuning PID parameter control can meet some design requirements of the coking coal blending process control system. The control effect is much better than the conventional PID controller.

Footnotes

Acknowledgments

The research reported in the paper was supported by Foundation of Wuhan Polytechnic University (No.2017y02). This research was also supported by Wuhan Polytechnic University Research Grant (Grant No.2014RZ37) and Educational Commission of Hubei Province of China (D20161706). These support are greatly acknowledged.

References

Li ,

Zhu ,

Zhang , et al., Effect of coal blending and ashing temperature on ash sintering and fusion characteristics during combustion of Zhundong lignite, Fuel195 (2017), 131–142.

Wang ,

Zhao ,

Chu , et al., Effects of coal and iron ore blending on metallurgical properties of iron coke hot briquette, Powder Technology (2018), 328:318–328.

Li ,

Miao ,

Jiang , et al., Intelligent control model and its simulation of flue temperature in coke oven, Discrete and Continuous Dynamical Systems - Series S6 (2015), 1223–1237.

Sun and

Gao , Application of support vector machine optimized by improved ant colony optimization algorithm in power coal blending prediction, International Journal of Applied Mathematics & Statistic6 (2014), 96–106.

Tian ,

Li ,

Shi , et al., A method of various alternatives for coal blending optimization with integrated performance index, Journal of Yunnan University (2016).

Duan ,

Lu ,

Xiong , et al., Evaluation method of the energy conversion efficiency of coal gasification and related applications, International Journal of Energy Research2 (2016), 620–631.

Li ,

Qu ,

Kong , et al., Influence of working lining parameters on temperature and stress field of ladle, Applied Mathematics & Information Sciences2 (2013), 439–448.

Wen ,

Deng and

Y.C.

Co , Practical exploration of coking with large proportion high sulfur coal using the existing equipment, Coal Chemical Industry (2017).

Li ,

Liu ,

Jiang , et al., Numerical simulation of the influence factors for rotary kiln in temperature field and stress field and the structure optimization, Advances in Mechanical Engineering6 (2015), 1–15.

10.

Yörükoğlu , Coal blending for thermal power stations, Madencilik3 (2017), 109–116.

11.

Wang ,

Liang ,

Gong , et al., Influence of coal blending on ash fusion property and viscosity, Fuel189 (2017), 15–22.

12.

Wei and

Wang , Preparation of commercially applicable slurry fuels from rapid hydrogasification char by blending with coal, Fuel Processing Technology143 (2016), 18–26.

13.

Li ,

Qu and

Kong , et al., Coke oven intelligent integrated control system, Applied Mathematics & Information Sciences3 (2013), 1043–1050.

14.

Lv ,

Xu ,

Xie , et al., Equilibrium strategy based coal blending method for combined carbon and pm 10 pm 10 mathcontainer loading mathjax, emissions reductions, Applied Energy183 (2016), 1035–1052.

15.

Li ,

Liu ,

Jiang , et al., Numerical simulation of temperature field and thermal stress field in the new type of ladle with the nanometer adiabatic material, Advances in Mechanical Engineering4 (2015), 1–13.

16.

Li ,

Gu ,

Kong , et al., Intelligent control of air compressor production process, Applied Mathematics & Information Sciences3 (2013), 1051–1058.

17.

Mahto and

Mukherjee , Fractional order fuzzy PID controller for wind energy-based hybrid power system using quasi-oppositional harmony search algorithm, Iet Generation Transmission & Distribution13 (2017), 3299–3309.

18.

Shah and

Agashe , Review of fractional PID controller, Mechatronics38 (2016), 29–41.

19.

Li ,

Kong ,

Jiang , et al., Air-fuel ratio intelligent control in coke oven combustion process, Information-An International Interdisciplinary Journal11 (2012), 4487–4494.

20.

Hu ,

Yin ,

Li , et al., Trajectory planning algorithm and simulation of 6-DOF manipulator, International Journal of Wireless & Mobile Computing2 (2018), 138.

21.

Smolka ,

Slupik ,

Fic , et al., 3-D coupled CFD model of a periodic operation of a heating flue and coke ovens in a coke oven battery, Fuel165 (2016), 94–104.

22.

Gamrat ,

Poraj ,

Bodys , et al., Influence of external flue gas recirculation on gas combustion in a coke oven heating system, Fuel Processing Technology152 (2016), 430–437.

23.

Buczynski ,

Weber ,

Kim , et al., One-dimensional model of heat-recovery, non-recovery coke ovens Part V: Coking-bed sub-model using an inverse procedure, Fuel225 (2018), 443–459.

24.

Zhang ,

Zhao ,

Xu , et al., Characteristics of pressure drop of charred layer in coke dry quenching over coke oven gas, Energy & Fuels4 (2017).

25.

Wang ,

Chu ,

Guo , et al., Mathematical simulation on blast furnace operation of coke oven gas injection in combination with top gas recycling, Steel Research International5 (2016), 539–549.

26.

T.K.

Das ,

K.N.

Ghosh ,

Misra , et al., Studies on the effect of usage of inert-rich coal in coal blend on coke properties through pilot oven carbonization tests, Transactions of the Indian Institute of Metals6 (2016), 1209–1216.

27.

Zhong ,

J.Y.

Ooi and

J.M.

Rotter , Predicting the handlability of a coal blend from measurements on the source coals, Fuel17 (2005), 2267–2274.

28.

Li ,

Wang and

Chen , Study on combustion characteristics of blended coals, Frontiers of Energy and Power Engineering in China2 (2007), 96–100.

29.

Ruhul , Sarker, alternative mathematical programming models: A case for a coal blending decision process, Optimization32 (2009), 383–399.

30.

Huang ,

Shan ,

Cheng and

Sun , TRec: An efficient recommendation system for hunting passengers with deep neural networks, Neural Computing and Applications (2018).

31.

Li ,

Xiao ,

Jiang , et al., Soft-sensing model of coke oven flue temperature, Sensors and Transducers12 (2013), 265–270.

32.

Wang ,

Liang ,

Gong , et al., Influence of coal blending on ash fusion property and viscosity, Fuel189 (2017), 15–22.

33.

H.B.

Vuthaluru and

D.K.

Zhang , Effect of coal blending on particle agglomeration and defluidisation during spouted-bed combustion of low-rank coals, Fuel Processing Technology1 (2001), 41–51.

34.

H.P.

Tiwari ,

Kumar ,

Bhattacharjee , et al., Prediction of operating parameters range for ammonia removal unit in coke making by-products, Metallurgical Research & Technology2 (2018), 211.

35.

Xiong ,

Fan ,

Jiang , et al., A simulation -based study of dispatching rules in a dynamic job shop scheduling problem with batch release and extended technical precedence constraints[J], European Journal of Operational Research1 (2017), 13–24.

36.

Liu ,

Xie ,

Sun , et al., Exergy analysis and optimization of coking process, Energy (2017), 139.

37.

Lv ,

Xu ,

Xie , et al., Equilibrium strategy based coal blending method for combined carbon and PM10 emissions reductions, Applied Energy183 (2016), 1035–1052.

38.

Xiong ,

Fan ,

Li , et al., Research on steady-state simulation in dynamic job shop scheduling problem, Advances in Mechanical Engineering9 (2015), 1–11.

39.

Li ,

Fan , et al., Understanding ash fusion and viscosity variation from coal blending based on mineral interaction, Energy & Fuels1 (2017).

40.

Cheng ,

Ning ,

Xia , et al., Modelling of coal trade process for the logistics enterprise and its optimisation with stochastic predictive control, International Journal of Production Research8 (2016), 1–19.

41.

Manamanni ,

Mansouri ,

Hamzaoui , et al., Relaxed conditions in tracking control design for a TS fuzzy model, Journal of Intelligent & Fuzzy Systems2 (2017), 185–210.

42.

Ramakrishnan and

Devaraju , Attack's feature selection-based network intrusion detection system using fuzzy control language, International Journal of Fuzzy Systems2 (2016), 1–13.

43.

M.H.

Khooban and

M.R.

Soltanpour , Swarm optimization tuned fuzzy sliding mode control design for a class of nonlinear systems in presence of uncertainties, Journal of Intelligent & Fuzzy Systems2 (2013), 383–394.

44.

H.D.

Zhang ,

S.L.

Liao and

W.Y.

Ma , Generalized multi-fuzzy soft set and its application in decision making, Journal of Intelligent & Fuzzy Systems4 (2016), 31–42.

45.

Zhou ,

Chen and

Fang , Fuzzy adaptive control design for collecting main pressure in coke-oven, International Conference on Fuzzy Systems & Knowledge Discovery, 2010.

46.

Jiang ,

Li ,

Kong , et al., Research on fuzzy intelligent control of coke oven temperature, IEEE (2010), 221–224.

47.

Huang ,

Chen and

Yu , Fuzzy PID control study and simulation of gas collector pressure of coke ovens, Electronic & Mechanical Engineering & Information Technology International Confere1 (2011), 169–172.

48.

Ahmadi ,

R.H.

Abdul and

R.R.

Abdul , Optimization of a self-tuning PID type fuzzy controller and a PID controller for an inverted pendulum, Journal of Intelligent & Fuzzy Systems4 (2014), 1987–1999.

49.

Parada ,

Sbarbaro ,

R.A.

Borges , et al., Robust PI and PID design for first-and second-order processes with zeros, time-delay and structured uncertainties, International Journal of Systems Science1 (2017), 95–106.

50.

Niu ,

Shen ,

Yang , et al., Analysis of Duffing oscillator with time-delayed fractional-order PID controller, International Journal of Non-Linear Mechanics92 (2017), 66–75.