Multi-model fractional predictive functional control design with

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Multi-model fractional predictive functional control design with application on an industrial heating furnace xiaomin hu, hongbo zou, jili tao, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03741 • Publication Date (Web): 26 Sep 2018 Downloaded from http://pubs.acs.org on September 28, 2018

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Industrial & Engineering Chemistry Research

Multi-model fractional predictive functional control design with application on an industrial heating furnace

Xiaomin Hua, Hongbo Zoua, Jili Taob,*, Furong Gaoc

a

School of Science, Hangzhou Dianzi University, Hangzhou 310018, P R China

b

Ningbo Institute of technology, Zhejiang University, Ningbo 315100, P R China

c

Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay,

Kowloon, Hong Kong

Corresponding author: Jili Tao Emails: [email protected]

Abstract: This paper mainly targets the nonlinear characteristics in the industrial heating furnace control and uses the multi-model method to decompose the global dynamics of the process into a series of local model sets. The corresponding predictive functional controller is designed using the local fractional order models of the set within their working range. The weight coefficients of the different models in the model sets are obtained based on the error of the different model sets at current moment and the system performance at current moment is obtained by weighting between many models using the weight coefficient. Since the accuracy of the local models is increased, the error between the local model set and the process output is reduced, which decreases the influence of the system model inaccuracy on the system performance. Finally, the strategy is verified through the experiment on the temperature of an heating furnace. Keywords: Multi-model set; Fractional order model; Predictive functional control

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1. Introduction With the increasing complexity of industrial processes and people’s demands on control precision, reduced raw material cost, resources saving and so on, it is more difficult to obtain satisfactory control performance based on control technologies using integer order theory, while in the realistic world, the actual industrial processes are often of the fractional order and it is difficult for integer order system models to better describe the dynamics of such types of systems [1-2]. Generally, it is difficult to achieve good control performance on fractional order systems using traditional integer order control methods [3]. The emergence of fractional order calculus theory is an effective way to solve this problem. Fractional order models can better describe a system’s characteristics. Fractional order calculus can provide a new mathematical tool for natural science and an effective method to solve issues in practice [4]. Recently, the continuous development of its theory has enabled it to be gradually used in the control field. In terms of fractional order control strategies, in addition to the representative CRONE controller [5], P‫ ܫ‬ఒ ‫ܦ‬ఓ controller [6], TID controller [7], etc., the sliding mode control and other advanced control strategies [8-11] also reflect the advantages that integer order control methods do not have. With the development of theory and the rise of real industrial demand, it has also aroused more and more scholars' attention and in-depth research to apply fractional order calculus theory to model predictive control (MPC) is also a hot topic. As a branch of advanced control technologies with rich theoretical results and practical applications, the development of MPC has experienced model algorithmic control using impulse response models [12], dynamic matrix control based on step response models [13] and generalized predictive control using parametric models [14], etc. MPC can deal with constraints explicitly, predict future dynamics of a system and incorporate the constraints into the future input, output or


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the state variables to convert the optimization into the quadratic programming [15-16]. But since nonlinear systems are more common in practice, it will become more difficult to find effective solutions of MPC for such systems. This has also attracted some researchers and relevant results have emerged, such as the nonlinear MPC with dead-zone compensator for distributed solar collectors proposed in [17]. Ref. [18] proposed the input and output feedback linearization scheme of a dual induction generator. The authors of [19] studied the MPC scheme of the reactive distillation columns for hydrogenation of benzene. In [20], the application of MPC in the precise tracking control in the electromechanical servo system and the constraint dealing are discussed. In [21], the constrained MPC for the industrial batch processes was proposed. Although MPC has achieved some results in industrial process control, lack of accuracy in the off-line recognition model of nonlinear systems is still a problem to be considered. Therefore, the delay, large inertia and parameters changing with time should be paid attention to for both the control algorithm and the process models. In a nonlinear system, the system response will ultimately differ from a linear process model. In addition, to improve the accuracy of process modeling, some optimization efforts must be made [22-23]. Multi-model control is an effective way for modeling nonlinear processes with strong nonlinearity and large range of operating conditions [24]. In [25], the authors proposed a number of models to model a nonlinear hybrid system using Bayesian theory. In [26], a multi-model MPC for the component content in the CePr/Nd counter current extraction process was discussed. In [27], decomposing of the nonlinear system into a series of linear models and the nonlinear MPC are discussed. Fractional order systems can improve the accuracy of process models and improve control performance may be anticipated. By introducing the fractional order system into the model set, the



local fractional order models can be built to improve modeling accuracy and reduce the number of models established in the local range due to the low modeling accuracy. The fractional order model and predictive functional control will also improve the poor control performance and other adverse effects caused by the low accuracy of traditional models. There have been some results in this field and good performance can be anticipated. It is known that we can combine MPC with fractional order models and multiple models to deal with nonlinear problems in industrial processes [28-34]. Aiming at the factors mentioned above, this paper proposes the predictive functional control method using the multi-model fractional order modeling, which is mainly used to reduce the requirement of the predictive functional control on the model accuracy and the control performance is improved by decomposing the nonlinear dynamics of the system into several local linear models and establishing the fractional order models in the local working range. Finally, the proposed method is tested on the temperature in a SXF4-10 electric heating furnace. This paper is presented as follows. （1）First, the inner PID controller and the furnace is treated as a generalized process and the PID parameters are set through the internal model control principle, which serves as the internal process model for the outer fractional PFC design. （2）The local fractional order models are established by dividing the working range into different regions and are transformed into the high order integer models using Outsaloup approximation method. （3）The local fractional PFC is systematically designed and implemented on the heating furnace, where the control performance at the current moment is obtained by the weighting among various local models. 2. Inner Model Establishment and PID Controller Tuning


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The response parameters of the process can be obtained by analyzing the time domain process response. A first-order plus dead time (FOPDT) model is often used as:

g% p (s) = where

y(s) K −τ s e = u(s) Ts +1

y(s) ，u(s) are the Laplace transforms of the input u(t )

model gain，T is the time constant,

and the output

(1)

y(t ) , K

is the

τis the delay. The response of the FOPDT model to a step input

is described as follows:

, t 0

(−γ )

, and ωj

= 0 for

j < 0.

The optimal control law based on Eq. (22) can be obtained as U = −(ψ TWψ ) −1ψ TW [ L ( y (k ) − r (k )) + G ∆x(k ) − Su (k − 1) − Q ∆R]

(23)

where

 CAP1 −1B + CAP1 −2 B + L + CB  P1 P1 −1   S =  CA B + CA B + L + CB  M  P2 −1  P2 −2 CA B CA B + L + CB  +   CA P1 + CA P1 −1 + L + CA   P1 +1 + CA P1 + L + CA  G =  CA  M  P2  P2 −1 + L + CA   CA + CA

L = [1 1 L 1]T P1 −1 P1 −1  j CBf ( P 1) CA Bf ( P 1 l ) CBf ( P 1) − + − − − + ∑ ∑ CA j Bf2 ( P1 − 1 − l )  1 1 1 1 2 2 l =1 l =1  P1 P1 j  CBf1 ( P1 ) + ∑ CA Bf1 ( P1 − l ) CBf 2 ( P1 ) + ∑ CA j Bf 2 ( P1 − l ) ψ = l =1 l =1  M M P2 −1 P2 −1  j j CBf1 ( P2 − 1) + ∑ CA Bf1 ( P2 − 1 − l ) CBf 2 ( P2 − 1) + ∑ CA Bf 2 ( P2 − 1 − l ) l =1 l =1 

1 1 L 1 0 L Q = 1 1 O O O O M M O O O O 1 1 L 1 L L

0 M 0 1


P1 −1



L CBf M ( P1 − 1) + ∑ CA j Bf M ( P1 − 1 − l )     l =1  L M P2 −1  j L CBf M ( P2 − 1) + ∑ CA Bf M ( P2 − 1 − l )  l =1  l =1

L

P1

CBf M ( P1 ) + ∑ CA j Bf M ( P1 − l )

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U = [µ1, µ2 ,L, µM ]T

∆R = [∆r(k +1) ∆r(k + 2) L ∆r(k + P)]T and Q is the matrix of (P2 − P1 +1) × P2 dimension. Denote

µ1 = − (1, 0, L , 0) （ψ T Wψ ) −1ψ T W [ L ( y ( k ) − r ( k )) + G ∆x ( k ) − Su ( k − 1) − Q ∆R ] = − h1[ y ( k ) − r ( k )] − g1∆x ( k ) + v1u ( k − 1) − q1∆R （ψ T Wψ ) −1ψ T W [ L ( y ( k ) − r ( k )) + G ∆x ( k ) − Su ( k − 1) − Q ∆R ] µ 2 = − (0,1,L , 0) = − h2 [ y ( k ) − r ( k )] − g 2 ∆x ( k ) + v2 u ( k − 1) − q2 ∆R

(24)

M （ψ T Wψ ) −1ψ T W [ L ( y ( k ) − r ( k )) + G ∆x ( k ) − Su ( k − 1) − Q ∆R ] µ M = −(0, 0,L ,1) = − hM [ y ( k ) − r ( k )] − g M ∆x ( k ) + vM u ( k − 1) − qM ∆R

The control input is derived as

u(k ) = −H y [ y(k ) − r(k )] − Gx ∆x(k ) + Vuu(k −1) − Qu ∆R

(25)

where M

H y = ∑ h j f j (0) j =1 M

Gx = ∑ g j f j (0) j =1 M

Vu = ∑ v j f j (0) j =1 M

Qu = ∑ q j f j (0) j =1

3.3 Weighted Control of Multi-model Sets Since the output of each model y j (t) deviates from the actual process output yout (t) at current time, the current model will concentrate the deviation values of each model, and this is expressed as: e j (t) = yout (t) − y j (t) ， j = 1, 2, L , i

Here,

(26)

yout (t) is the actual process output, y j (t) is the actual output of the channel j , and e j (t)

represented the deviation between the j th sub-model and the actual process output. Using the deviation values of the current model and the past-time models, the following factors



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can be selected to get the influence weight coefficient of each submodel on the system. 2

 1    ∑ k =0  ei (t − k )  j = 1, 2, L , i w j (t) = 2 ， i l  1  ∑∑   j =1 k =0  ei (t − k )  n

where, wj (t) represents the weighting coefficient of the j th submodel and

(27)

ei (t − k ) is the error of

past time instants. The control input at the current moment can then be expressed as: i

u (t) = ∑ w j u j

(28)

j =1

Remark. It will show that the fractional order model adopted in Eq.(9) will improve the modeling accuracy and the subsequent PFC controller design using the proposed state space model shown through Eq.(12) to Eq.(15) in terms of the fractional cost function shown in Eq.(21) together with the multi-model sets will improve the control performance.

4. Experimental Results 4.1 Experiment setup The experimental study in this paper uses SXF-4-10 electric heating furnace. Its overall diagram is described in Fig.1 and it consists of the temperature acquisition module, industrial computer (IPC) and the execution module. The functions of these main components will be discussed in the subsequent sections. Fig.2 is the overall structure diagram of this experimental device.


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Fig.1 The diagram of wiring connection and industrial personal computer 610L.

SG-3011 CR signal amplifier +

_ thermocouple

DN-37 CR terminal strip +

+

0 0

0

0

0

PCI-1802LU CR signal acquisition card

_

_ computer control algorithm

heating wire + + SSR-380D40 solid state relay _ _

+ PIO-D24U signal output card _ DN-37 CR terminal strip

IPC-610L _

SXF-4-10 heating furnace AC 220V +

Fig.2 Process flow chart of the electric heating furnace (SXF-4-10). 4.1.1 The Temperature Measurement Module



This module mainly consists of a K-type thermocouple and a voltage amplifier. The K-type thermocouple is a temperature detector commonly used in industrial control. Its main feature is that its temperature measurement range is large and it can accurately collect the data within a reasonable range no matter how bad the environment is and it has the simple structure and both ends of the couple can quickly generate the voltage signals when the temperature changes. The K-type thermocouple can generate a voltage value in the temperature difference at both ends of the sensor. Because the measured voltage at both ends of the thermocouple is very small, which are from only a few tens of millivolts to several hundred millivolts and the voltage needed to be processed by the processing module of the host computer is between 0-5v, a signal amplifier is added. The magnification of the signal can be adjusted by the connection to the amplifier, which can adjust the thermocouple output voltage to the range that meets the measurement range of the industrial control computer. 4.1.2 The Industrial Computer The industrial control computer is mainly composed of a PCI-1802LU converter, a PIO-D24U converter and the program processing part. The analog signal collected through the temperature can be converted into the digital signals that will be accepted by the industrial computers through the PCI-1802LU converter. The host computer collects the converted digital signal to get the temperature value of the current electric heating furnace in the specified sampling time through the program set and controls the temperature actuator to heat the electric heating furnace through the corresponding algorithm and converts the calculated control input to the duty ratio signal of the heating capacity of the temperature execution module. 4.1.3 The manipulated Variable


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The temperature execution module is primarily controlled by a relay to provide different amounts of heat to the heating time of the electrical heating furnace in a unit time. The current on-off time should be controlled to reach the heating effect. The heating state of the heating furnace is judged by the magnitude of the voltage value at the input end of the relay, such that the temperature of the electric heating furnace is increased. When the input voltage is 5V, the output end is at the closed state so that there is current going through. At this time, the electrical heating furnace is heated; otherwise, the output end is at the open state and the current is zero. The heating of the furnace is then stopped. The heating time can be controlled by the adjustment of the on-off time of the relay. 4.2 Real-time Application Results 4.2.1 Establishment of the Multi-model Set The temperature in this furnace is to be controlled to 300℃ when the experiment is carried out. First of all, the working section of the electric heating furnace is divided. The temperature of the laboratory is usually from 20 to 30℃. If the regional divisions of the temperature are too large, the calculation burden will be rather large. In view of this, this paper mainly divides the working range of the electric heating furnace into two sections of 0-150℃ and 150-300℃. The set-point values of the controller of the internal PI are therefore set to 150℃ and 300℃ respectively. The data of the temperature that is from the room temperature to the 150℃ in the electrical heating furnace is collected to establish the first model (model 1) and the second model (model 2) is established when the temperature rises from 150℃ to 300℃. The fractional order models are established respectively for the collected two groups of data as shown in Fig.3a-3b.



160

140

Temperature (°C)

120

100

80

60 Set point Pratical temperature curve fractional order model Oustaloup approximate model

40

20

0

0

2000

4000

6000 Time (s)

8000

10000

12000

Fig.3a Step response for temperature set-point 150℃ 320

300

280

260 Temperature (°C)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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240

220

Set point Pratical temperature curve fractional order model Oustaloup approximate model

200

180

160 0

2000

4000

6000 Time (s)

8000

10000

Fig.3b Step response for temperature set-point 300℃. The parameter of model 1 is obtained as:


12000

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g% p ( s ) =

1 e − 100 s 500 s 0.98 + 1

The parameter of model 2 is obtained as: g% p ( s ) =

1 e −100 s 400 s 0.88 + 1

Then the fractional order models mentioned above can be approximately transformed to the high-order integer model through Outsaloup approximation. In order to incorporate all the characteristics of the process as much as possible, the upper frequency limit of the model is selected −6

to be Wh = 10 and the lower limit of the frequency is selected to be Wb =10 . In order to calculate 6

easily and not influence the model accuracy, the order of the model is selected as 4. The high-order model after conversion is expressed as follows. The high-order integer model of model 1 is:

g% p (s) =

s4 + 6.6 ×105 s3 + 4.3×108 s2 + 2.8×108 s +1.9 e−100s 7 4 11 3 11 2 8 5 7.6 ×10 s +1.5×10 s +1.7 ×10 s + 5.5×10 s +1.9 ×10

The high-order integer model of model 2 is:

s4 + 9.3×105 s3 + 8.7 ×108 s2 + 8.1×108 s + 7.6 g% p (s) = e−100s 8 4 11 3 11 2 9 5 3.7 ×10 s + 4.1×10 s + 4.3×10 s + 1.3×10 s + 7.6 ×10 In order to test the accuracy of the models after their establishment, comparisons are done among the step responses of the established fractional order models, the integer higher order models and the actual process responses respectively. It is shown from Figs. 3a-3b that the model output responses are close to the actual process output. 4.2.2 Parameter Selection of Fractional PFC The predictive functional controller is designed for the high-order integer models in the different ranges, one coincidence point is selected to be 15 and the smoothing factor is 0.98 after many times of experiments under the condition that the process may not have significant overshoot



during the experiment process. 4.2.3 Weighting Control for the Multi-models Since the process has 10 unit times of delays, 10 units of the output errors in the past are chosen as the reference of the choice for the proportional coefficient of the weights on the model. So far, we have established the high-order integer models of different temperature sections. Using the two established models of the SXF-4-10 electric heating furnace, the corresponding predictive functional controller and the control structure will be designed as shown in Fig. 4. Finally, the control input of the current moment is obtained based on the weighted model among the multiple models.

Fig.4 The block of MMFPFC for heating furnace 4.2.4

Results

Fig.5 shows the comparison results of traditional PID, traditional predictive functional control, fractional order model based predictive functional control and the multi-model fractional order predictive functional control, where the temperature is to be controlled to 300℃. Fig. 6 shows the duty cycle of the solid state relays (indicating the heating time of the heating furnace) for every control method at each sampling time. By comparing the method of this paper with traditional PID, traditional predictive functional control and fractional order model based predictive functional


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control, the following conclusion is drawn in terms of system tracking performance, steady performance and the disturbance rejection performance. The tracking performance of the system is tested by forcing the temperature to 300℃. Under the condition that guarantees Shannon's theorem and in order to obtain high sampling precision, the sampling time is selected to be 10S. It can be seen from Figs.5-6 that the multi-model based fractional order predictive functional control takes about 42 minutes to force the temperature to reach the set-point, while the fractional order control using the single model (the fractional order model 1) and the PID control method both spend more than 100 minutes. It takes traditional predictive functional control about 48 minutes to force the temperature to reach the set-point with a small overshoot generated and it takes a long time to eliminate this overshoot. The predictive functional control based on the fractional order model has a very quick response speed at the early stage, but when the temperature is about to reach the set-point, its speed slows down and even has the trend to decrease. Thus the temperature reaches the set-point after a long time. Based on this comparison, it is shown that the tracking performance of the predictive functional control using the multi-model fractional order is better than those of the other control methods. The statistical results in Table 1 show the comparison in terms of the rising time (the first time the output reaches 90% of the output), and the overshoot. It shows that the proposed method improves the tracking performance.



350

300

Temperature (°C)

250

200

150

Set point PID PFC FPFC MMFPFC

100

50

0

0

100

200

300

400

500 Time (k)

600

700

800

900

1000

Fig.5 The performance of the set-point tracking. 100 Set point PID PFC FPFC MMFPFC

90 80 70 Duty-ratio (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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60 50 40 30 20 10 0

0

100

200

300

400

500 Time (k)

600

700

800

900

1000

Fig.6 The duty rate for the set-point tracking. Table 1．Results of tracking performance algorithm

MMFPFC

FPFC

PFC

Rising time (min)

12.8

13.3

23

Overshoot

0

0

6


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The system steady performance reflects the accuracy when the process output reaches the set-point. In order to compare this performance after the system reaches its steady operation point, 1000 sets of data are chosen and the result is shown in Fig.7. The statistical results in Table 2 show the comparison in terms of the mean, the maximum, the minimum values, the extreme deviation and the standard deviation. It shows through the analysis that the proposed method has the overall better control than the other methods.

340 330 320 Temperature (°C)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


310 300 290 Set point PFC FPFC MMFPFC

280 270 260

0

100

200

300

400

500 Time (k)

600

700

800

900

1000

Fig7. The steady state performance. Table 2．Results of steady state performance algorithm

MMFPFC

FPFC

PFC

Average

300.047

299.599

301.109

Maximum

302

302

305

Minimum

299

298

298

Extreme Deviation

3

4

7

Standard Deviation

0.86541

0.92718

0.96022



In addition, the system performance is not only valued based on the tracking performance and the steady performance, the disturbance rejection capability of the system is also a very important performance indicator of the system. When the system is steady, we open the furnace door with a fixed angle to generate a constant disturbance to the electric heating furnace, thus to test the disturbance rejection capability of the system. The control input is obtained by measuring the temperature deviation in the electric heating furnace. It is seen from Fig.8 and Fig. 9 that the multi-model fractional order control system can effectively suppress the external disturbance and can let the temperature rise to the previous state in a certain period of time compared with the other methods, which reflects its fast recovery ability. The statistical results in Table 3 show the comparison in terms of the recovering time (the first time the output reaches 90% of the output after the disturbance), and the overshoot. It shows that the proposed method improves the overall disturbance rejection performance.

320 310 300 290 Temperature (°C)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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280 270 Set point PFC FPFC MMFPFC

260 250 240 230

0

500

1000 Time (k)

Fig8. The disturbance rejection Performance.


1500

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60 Set point PFC FPFC MMFPFC

50

40 duty rate (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


30

20

10

0

0

500

1000

1500

Time (k)

Fig9. The duty rate for disturbance rejection. Table 3．Results of disturbance rejection performance

5.

algorithm

MMFPFC

FPFC

PFC

Recovering time (min)

12.8

12

12

Overshoot

4

8

8

Conclusion A multi-model fractional order predictive functional control is designed and applied to the

temperature of a heating furnace. By introducing the fractional order model into the predictive functional control system to improve the modeling accuracy and reduce the number of the model set, the calculation burden and the control performance are both considered and improved temperature control is achieved. References [1] West, B., Bologna, M., & Grigolini, P. (2012). Physics of fractal operators. Springer Science & Business Media. [2] Matychyn, I., Onyshchenko,V. (2018). On time-optimal control of fractional-order systems. Journal of Computational and Applied Mathematics, 339, 245-257 [3] Caponetto, R., Fortuna, L., & Porto, D. (2004). A new tuning strategy for a non integer order PID controller. In First IFAC workshop on fractional differentiation and its application (pp. 168-173). [4] Sarafnia, N., Malekzadeh, M., Askari, J. (2018). Fractional order PDD control of spacecraft rendezvous. Advances in Space Research, 62(7),1813-1825



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Multi-model fractional predictive functional control design with

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