Design of set-point weighting PIλ + Dμ controller for vertical magnetic flux controller in Damavand tokamak.

Design of set-point weighting PIλ + Dμ controller for vertical magnetic flux controller in Damavand tokamak H. Rasouli and A. Fatehi Citation: Review of Scientific Instruments 85, 123508 (2014); doi: 10.1063/1.4904737 View online: http://dx.doi.org/10.1063/1.4904737 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Identification and control of plasma vertical position using neural network in Damavand tokamak Rev. Sci. Instrum. 84, 023504 (2013); 10.1063/1.4791925 Parameter Tuning of Fractional PI λ D μ Controllers With Integral Performance Criterion AIP Conf. Proc. 1019, 186 (2008); 10.1063/1.2952976 RealTime Plasma Control Tools for Advanced Tokamak Operation AIP Conf. Proc. 875, 385 (2006); 10.1063/1.2405971 Model-based dynamic resistive wall mode identification and feedback control in the DIII-D tokamak Phys. Plasmas 13, 062512 (2006); 10.1063/1.2214637 Modeling of stochastic magnetic flux loss from the edge of a poloidally diverted tokamak Phys. Plasmas 9, 4957 (2002); 10.1063/1.1521125

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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 123508 (2014)

Design of set-point weighting PIλ + Dμ controller for vertical magnetic flux controller in Damavand tokamak H. Rasouli and A. Fatehi Advanced Process Automation and Control (APAC) Research Group, Industrial Control Center of Excellence, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Seyed Khandan, P. O. Box 16315-1355 Tehran, Iran

(Received 5 July 2014; accepted 8 December 2014; published online 24 December 2014) In this paper, a simple method is presented for tuning weighted PIλ + Dμ controller parameters based on the pole placement controller of pseudo-second-order fractional systems. One of the advantages of this controller is capability of reducing the disturbance effects and improving response to input, simultaneously. In the following sections, the performance of this controller is evaluated experimentally to control the vertical magnetic flux in Damavand tokamak. For this work, at first a fractional order model is identified using output-error technique in time domain. For various practical experiments, having desired time responses for magnetic flux in Damavand tokamak, is vital. To approach this, at first the desired closed loop reference models are obtained based on generalized characteristic ratio assignment method in fractional order systems. After that, for the identified model, a set-point weighting PIλ + Dμ controller is designed and simulated. Finally, this controller is implemented on digital signal processor control system of the plant to fast/slow control of magnetic flux. The practical results show appropriate performance of this controller. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904737] I. INTRODUCTION

Recent developments in analysis of fractional order systems have inspired a lot of applications in controller, modeling and identification of physical phenomena.1 In this way, development of classical PID controllers to fractional order PID controllers as FOPID2 (PIλ Dμ ) and CRONE3 have been more accounted. Podlubny2 suggests using PIλ Dμ controller in fractional order processes and then he calculates tracking error for a unit step by unknown parameters. Parameters of fractional order controller are obtained by gradient method, to optimize a specified performance in time domain. Monje in Ref. 4 presents a numerical optimal algorithm to regulate PIλ Dμ controller based on a series of constraints in the frequency domain. In this algorithm, solutions are related to the initial estimation of unknown parameters. Then, he presents in Ref. 5 a method of tuning parameters of PIλ Dμ in integer order systems. This method is based on solving 5 nonlinear equations with 5 unknown variables. It offers robust performance in the presence of noise and variable gain. However, it depends on initial condition of parameters. Also, in Ref. 5 the self-tuning method is studied for closed loop relay feedback. Extending Ziegler-Nichols PID tuning rules in fractional order systems is presented in Ref. 6. This work is based on numerical minimization algorithms for a first order plus dead time (FOPDT) model of process. In Ref. 7, tuning of PIλ Dμ controller parameters is studied based on phase and gain margin for a special class of fractional order systems. In Ref. 8, an optimal PIλ Dμ controller has been designed with ISE cost function based on special gain and phase margin. In Ref. 9, for a special class of fractional order systems, parameters of PIλ and PDμ controllers are calculated with the method of 0034-6748/2014/85(12)/123508/13/$30.00

phase margin and gain margin constraint in the frequency domain. Prior to design a controller, it is necessary to have a model for plant. Some of the real systems have fractional dimension, so to achieve more accurate results, they have to be modeled by fractional order models. So far, there have been a lot of researches on identification of these types of systems. In a class of these methods, the fractional transfer function model of the plant is approximated by an integer order model using fractional order derivatives. Then, these parameters are identified by an estimation method that is sometimes nonlinear10, 11 and the fractional model is extracted from it. To avoid complexity, in some methods, a base fractional order (such as 0.5) is considered for system and then the multiples of this base are considered as fractional orders.12 Continuous order distribution is also used for the identification of fractional order systems at the frequency domain.13 Fractional order controllers are implemented in both digital and analog domains. Analog controllers of PIλ Dμ are constructed and evaluated in Ref. 14 to control the water level of a dual tank system. Also in Ref. 15 an analog fractional order controller is implemented to tune the voltage of a DC/DC converter. In Ref. 16, closed loop relay self-tuning method is tested practically for digital PIλ Dμ controller of a laboratory servo-motor system. In Ref. 17, the digital fractional PIλ controller is implemented by a digital FIR (Finite Impulse Response) filter as an approximation of fractional order controller to control the speed of a smart wheel. Tokamak is a machine to study fusion physics. It is based on electromagnetic dynamic system which is known to be one of the physical phenomena that can be stated by fractional order models. The use of fractional calculation is more considered in modeling the chaotic behavior of plasma and

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magnetohydrodynamic (MHD) instability of plasma in tokamak.18 Also, simulation of PIλ Dμ controller so as to control the plasma position is reported in STOR-M tokamak machine.19 In this paper, a PIλ + Dμ set-point weighted controller is suggested to control the process with special fractional order structure. For this purpose, the controller parameters are set by the pole placement method. Our goal is to make the closed loop system work as a reference model with fractional order transfer function. The desired polynomial system coefficients are chosen based on the transient time response of fractional order system generated by generalized characteristic ratio assignment (CRA) method.20 Damavand tokamak21, 22 is a practical machine for case study in this work. Some advanced control systems have been implemented in this machine.22 Because of the lack of mathematic model of magnetic flux in Damavand tokamak, it is identified by the experimental data of the plant. We use output error method10, 12 to identify the vertical magnetic flux. After that, a PIλ + Dμ controller is designed and simulated for the identified model. Finally, this controller is implemented on the plant to control the magnetic flux by means of a digital signal processor (DSP) control system. This paper is organized as follows: In Sec. II, the structure of PIλ + Dμ with a set-point weighted controller and its specifications are presented. Section III reviews the determination of coefficients of the desired commensurate fractional order transfer function by CRA method. After that, the regulation of PIλ + Dμ controller parameters is presented. In Sec. IV, for a case study, the vertical magnetic flux of Damavand tokamak is identified in continuous time domain. In Sec. V, PIλ + Dμ controller is designed and simulated. Then in Sec. VI, PIλ + Dμ controller is implemented and its control performance results are investigated. Then, its results are compared with previous PD digital controller implemented in the same tokamak machine. Finally, the paper is concluded in Sec. VII.

II. DESIGN OF SET-POINT WEIGHTED PIλ + Dμ CONTROLLER

One of the main challenges in using both classical integer and fractional order PID controllers is simultaneous reduction of the effect of disturbance (d), noise (n), and amplification of input reference signal effect. Due to the closeness of the main frequency band of the disturbance rejection and

set-point tracking, this is more crucial in the response to disturbance. Manipulation of set-point weighting is an efficient approach to deal with this problem. A. Structure of set-point weighted PIλ + Dμ controller

The structure of PIλ + Dμ with set-point weighting controller is a suitable structure to solve this problem. The structure of PI + D with set-point weighting is known for integer order systems.23 This structure can also be considered for fractional order systems. Figure 1 shows the structure of PIλ + Dμ controller with set-point weighting. According to this structure, the set-point is weighted for the proportional part of the controller. However, set-point does not have any effect on the derivative part of the controller to prevent abrupt change in the control signal while it appears with ordinary weight of one in integral part to reject the steady state error. According to Fig. 1, we can define feedback and feedforward controllers of a pole placement controller, as shown in Fig. 2; in which Gff =

T k = akp + λi , R s

Gc =

S k = kp + λi + kd s μ . R s (1)

By means of PIλ + Dμ controller we can attain disturbance rejection goal. Also, utilizing parameter a improves the effect of reference input on the system output, without any concern about changing the effect of noise and disturbance.

III. DESIGN OF DESIRED FRACTIONAL ORDER CLOSED LOOP MODEL

In integer order systems, different methods are presented to obtain coefficients of characteristic polynomial of a transfer function based on transient response. One of these methods is employing the CRA method in which an all-pole system is designed to obtain a non-overshooting step response.24 In Ref. 20, the CRA concept is generalized for commensurate fractional order systems. For commensurate fractional order transfer function shown in (2), the generalized time constantτ and characteristic ratios γ i are defined as follows: M (s) =

d0 , dn s nλ + dn−1 s (n−1)λ + . . . + d1 s λ + d0

(2)

FIG. 1. The structure of PIλ + Dμ with set-point weighting controller.


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FIG. 2. The structure of pole placement for PIλ + Dμ controller.

d τ = 1, d0 λ

γi =

di2 di−1 di+1

,

1 ≤ i < n − 1,

(3)

τλ di = di−1 i−1 k=1

γk

,

b B = 2λ , A s + a1 s λ + a2

(7)

Bm ωn 2 = 2λ . Am s + 2ξ ωn s λ + ωn 2

(8)

Gp = γn = γ0 = ∞,

(4)

where λ is the commensurate order. According to (3) and (4), the following recursive relation can be used to determine the system parameters:20 d1 = τ λ d0 ,

tive parts.Gp and Gm are process model transfer function and desired fractional order system, respectively:

i = 2, 3, . . . , n. (5)

The characteristic ratios could specify the damping and overshoot in the time response. The time response speed could be tuned by the generalized time constant independently. This implies that in the systems with the same characteristic ratios, the generalized time constant can scale up the step responses proportionally. For the two models with identical characteristic ratios and different generalized time constants τ 1 and τ 2 , the settling times ts1 and ts2 and rise times tr1 and tr2 of these two models are related by the following relations: ts1 τ τ tr1 = 1, = 1. (6) ts2 τ2 tr2 τ2 According to (6), it is obvious that we can obtain the desired fractional order polynomial in two steps:20 Step 1: Having initial choice of τ 1 and characteristic ratios and determining the fractional order polynomial coefficients. The step response of this model is plotted and its settling time ts1 or its rise time tr1 is measured. Step 2: Considering the desired tr2 or ts2 , τ 2 is obtained using (6). Then, the coefficient of the desired fractional order polynomial is computed using (3) and (4), considering the same characteristic ratio for initial and final model. A. Tuning the controller parameters

In this subsection, a simple method is presented for tuning the parameters of PIλ + Dμ controller to follow the commensurate fractional pseudo-second order closed loop transfer function. We assume that the controller has the same integral and derivative order as process (μ = λ). Consider Fig. 2, where Gc is in the form of PIλ Dμ and Gff is a PIλ controller with the same fractional order integral and deriva-

Gm =

The Diophantine equation for closed loop systems is as follows: AR + BS = Ac = Am A0 ⇒ (s 2λ + a1 s λ + a2 )s λ + b(kd s 2λ + kp s λ + ki ) = (s 2λ + 2ξ ωn s λ + ωn 2 )(s λ + a0 ).

(9)

In this equation, Ac is the closed loop characteristic polynomial, Am is the polynomial of the desired closed loop poles A0 is the observer characteristic polynomial and B is the zero polynomial of the plant. Polynomials R and S are obtained from solving the Diophantine equation (9). Therefore, the controller parameters are obtained as follows: kp =

ωn 2 + 2ξ ωn a0 − a2 , b

(10)

2ξ ωn + a0 − a1 , b

(11)

kd =

ωn 2 a0 . (12) b The observer pole a0 is selected to be approximately 10 times greater than the dominant pole. A proper choice of a1 for tracking the reference input is that the zero in Gff rejects the observer pole. Then, it is necessary that ki =

T = (s λ + a0 )ωn 2 = b(akp s λ + ki ) ⇒ a =

ki . a0 kp

(13)

Using the above technique, the controller parameters of PIλ + Dμ are derived for the second order fractional order plant with an integer order. IV. IDENTIFICATION OF VERTICAL MAGNETIC FLUX AND POWER CONVERTER

In tokamak, the plasma magnetic control action varies the current in the poloidal field coils. These currents


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FIG. 3. The structure of magnetic fields control system. (a) The arrangement of poloidal field coil and saddle loop sensors for producing vertical magnetic field in Damavand tokamak. (b) Schematic diagram of power converter.

generate magnetic fields, which interact with the position and shape of plasma. In order to maximize the performance of machine, it is necessary to use plasmas with vertically elongated cross-sections. Damavand tokamak has an elongated plasma cross-section. Unfortunately, this elongation leads to instability of the plasma vertical movements. Indeed, to stabilize the plasma position, an active feedback system is needed. The feedback system used to control plasma position in Damavand tokamak is presented in Ref. 25. In this machine, hydrogen the plasma is sustained for 21 ms with a current peak about 35 kA.22 In the Damavand tokamak the magnetic flux (ψ) is manipulated to the control position and plasma shape in the tokamak. This magnetic flux itself is controlled by applying current into the poloidal field coils to obtain desired time response, which is one of the technical requirements for plasma physics researches, albeit without any comprehensive precise mathematical model for the system. So, we developed a model using experimental data of the plant and identification techniques.

A. Technical structure of vertical controlled magnetic field

The physical structure of control system of vertical magnetic field in the Damavand tokamak includes two poloidal field coils each with five turns. The two coils are installed symmetrically with respect to mid-tokamak plan. The coils are set up in series and their spatial position are R = 61.5 cm and Z = ±16.5 cm. These coils have inductance Lcr = 280 μH and resistance Rcr = 67.3 m.22 The vertical magnetic field is created by conducting the electrical current through coils. The current is produced by a power DC/AC converter with IGBT elements with the maximum current, voltage and frequency of 1.6 kA, 1.2 kV, and 10 kHz, respectively. Also, the energy source is a capacitive bank of 20 mF with maximum voltage of 1 kV. The coils are located out of the vacuum vessel. The saddle loop sensors26 for measuring vertical magnetic flux are

installed inside the vacuum vessel. Figure 3 shows the technical structure of the power converter plant. Figure 4 shows the block diagram of the closed loop system with unit feedback. In this Figure, Gp (s) and Ga (s) are the magnetic field process and the DC/AC power converter, respectively. ψ denotes the vertical magnetic flux and Ic denotes the converter output, known as the controller current. The manipulated signal u is the input command for IGBTs. A practical delay of 10 μs in electrical boards is added to protect IGBT elements. B. Experimental test to produce identification signals

According to the block diagram of Fig. 4, identification procedure constructs the transfer function model of the plant from control signal (u) to output flux (ψ). Direct and offline identification in continuous time domain using experimental data of the plant is employed to construct the model. In this research, the vertical magnetic flux is controlled by manipulating the current of poloidal coil without any plasma in the chamber. So, the physical process is stable. The first step in the system identification is designing a set of suitable experimental tests. The important issue is the selection of an input signal which could excite all modes of the plant. The input frequency should be around the system bandwidth. Because of the constraints in performance of the power converter, the maximum excitation frequency is 10 kHz. Therefore, the input reference signal has pulses with different amplitudes ±1 and pulse width from 0.5 ms to

FIG. 4. The block diagram of control system of the vertical magnetic flux in Damavand tokamak.


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FIG. 5. The complete collection of experimental signals of plant for identification. (a) Experimental signals Ref and ψ. (b) Experimental signals u and coil current Ic .

15 ms. Also the sampling time is 10 μs. In practical experiment, there are some other physical constraints. For example, the maximum time duration of any electrical discharge to magnetic flux control, known as “shot,” is about 45 ms. Therefore, it is necessary to divide the identification signal and to gather data from various shots. Figure 5 illustrates the experimental data gathered from 15 shots.

C. Fractional identification algorithm in time domain

Several techniques of identification are available for fractional order systems, including equation-error and outputerror approaches.10, 11 In this paper, the model of the system is in continuous time representation, thus it is preferable to use an output-error technique to estimate the derivative orders and model parameters. The model parameters are estimated using a Levenberg Marquardt nonlinear optimization algorithm.11 The following four different model structures are constructed and evaluated: G1 = G2 = G4 =

b2 s 2α + b1 s α + b0 , a3 s 3α + a2 s 2α + a1 s α + a0 b1 s α + b0 , + a1 s α + a0

a2

s 2α

a2

s 2α

b0 , + a1 s α + a0

(14) G3 =

b0 . a1 + a0 sα

Two different approaches are tested. In the first method all the model parameters are simultaneously estimated for minimization of the sum of the mean squared error. In the second method, the coefficient parameters of the models are estimated for different values of α from 0.05 to 1.5 with steps of 0.05.

The first method (denoted by n) is evaluated for the 4 above models with sampling time 10 μs. The second method (denoted by e) is used for 4 above models with different sampling times (10, 20, 50, and 100 μs). Table I shows the overall identification results. According to Table I, it is obvious that the best models with less error are G3n10 , G3e10 , and G1e10 all with sampling time of 10 μs. Figure 6 shows the RMSE error, model output, and experimental output of the test and train data for each of the above models. According to Fig. 6(b1), the commensurate fractional order is selected as 1.3 for model G3 , based on RMSE error. Also, in Fig. 6(c1), the commensurate fractional order as 0.4 is selected for model G1 . Figure 7 shows the 3 models errors, for comparison. It is obvious that between 3 above models, G1e10 has more error in high frequencies amplitude in comparison to the two other models, G3e10 and G3n10 . On the other hand, G3e10 Shows more error for low frequencies and large variants in amplitude than G1e10 and G3n10 . Nevertheless, these three models have fairly good performances. The above evaluation was based on direct identification from control signal u to magnetic flux output ψ. An alternative method for evaluation of the obtained models is based on the comparison of the simulated closed loop system with the real one. For this purpose, the block diagram of Fig. 8 is simulated with reference input of the real plant. In Fig. 8, Gp is the identified model. In both experimental and simulation test, the unit controller Gc = 1 has been used. Figure 13 shows the comparison of the practical plant output and models output and also the error of each model while tested in closed loop system. The RMSE of 3 models of closed loop system, G3e10 , G1e10 , and G3n10 are 0.1282, 0.1070, and 0.1118, respectively. According to Fig. 9, it is


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TABLE I. Evaluation result of fractional order models for vertical magnetic flux in Damavand tokamak. Model

ST (μs)

RMSE_test

%MAX_error

FOTF model 3.3339 0.0003s 1.257 −0.5775s 0.342 +1.18s 0.25 5.1196 G3n10 = 0.0002s 1.334 +0.515s 0.052

4

n

10

0.3039

47.73

3

n

10

0.09

15

2

n

10

0.1211

23

15.3944s 1.345 −0.7416s 0.417 0.0006s 2.669 +1.5925s 1.396 +1

1

n

10

0.1206

26

0.0143s 1.655 +11.3018s 0.921 −2.3114s 0.66 0.0001s 2.465 +0.0369s 1.448 +0.7236s 0.993 +10.2892

4

e

10

0.4840

100.08

3

e

10

0.09660

18.6

2

e

10

0.1141

22.7

−0.0286s 0.55 +8.3437 0.0019s 1.1 −0.0262s 0.55 +1.2769

1

e

10

0.08905

16.4

G1e10 =

4

e

20

0.4546

81.82

3

e

20

0.1695

22

2

e

20

0.1632

30

1

e

20

0.1278

23.2

4

e

50

0.4065

77.27

3

e

50

0.2170

49

2

e

50

0.1508

37

−0.0065s 0.6 +3.988 0.0004s 1.2 −0.0013s 0.6 +0.5609

1

e

50

0.1804

42

0.0046s 0.7 −0.1999s 0.35 +5.528 0.0024s 1.05 −0.029s 0.7 +0.2133s 0.35 +0.03

4

e

100

0.4821

89.55

3

e

100

0.5928

92

2

e

100

0.5694

78

1

e

100

0.5687

79.5

5.5531 0.5074s 0.5 −3.689s 0.25 +8.1974 G3e10 = 0.0004s7.748 1.3 +0.9887

0.4388 0.0161s 0.6 −0.1705s 0.3 +0.6055 9.1556 0.0003s 1.35 +1.3511 −0.0068s 0.6 +5.2121 0.0005s 1.2 −0.0051s 0.6 +0.7365 0.0023s 0.8 −0.1607s 0.4 +7.8235 0.0006s 1.2 +0.0002s 0.8 −0.0177s 0.4 +0.9951 1.6829 0.0221s 0.7 −0.2946s 0.35 +1.5085 −8.9424 −0.0009s 1.2 −1.0664

0.8641 0.0437s 0.6 −0.5172s 0.3 +2.0417 −8.7542 −0.001s 1.2 −2.105 −0.0054s 0.55 +2.3616 0.0008s 1.1 −0.023s 0.55 +1.0424 −0.0049s 0.9 +0.6704s 0.45 +4.5514 0.0016s 1.35 −0.0371s 0.9 +0.5911s 0.45 −0.0423

clear that the identified models have appropriate performance in the closed loop test.

For various practical physics fields of study, it is necessary to have different time responses in the control system of the magnetic flux. To provide this facility, first two desired closed loop reference models are presented based on necessary requirements in plant using generalized CRA method. Then, for the constructed model G3e10 of the plant, two PIλ + Dμ controllers are designed and simulated. This model is the simplest one with the appropriate response. After that, in Sec. VI, the controllers are compared practically through experimental implementation. A. Slow response PIλ + Dμ controller

According to identification of Sec. IV, open loop model of plant is presented as (15):

γi =

G3e10 =

b 7.748 = λ + a1 s + a2 0.0004s 1.3 + 0.9887

19370 , (15) s 1.3 + 2471.8 where coefficient a1 is equal to zero. We want the desired closed loop system to have a rise time of 10 ms and a maximum over shoot of 2%. Using derivation provided in Sec. III, the desired fractional order transfer function can be obtained. In our study, the magnetic flux plant is modeled by the pseudo-second commensurate fractional order system. Gm =

if : i = 2k,

ωn 2 . s 2λ + 2ξ ωn s λ + ωn 2

(16)

Considering the observer polynomial in (sλ + 700), it is finalized by the third order closed loop model. Therefore, it is sufficient to select an initial value for τ 1 and calculate the characteristic ratio of γ . To have a step response with maximum overshoot of 2%, the characteristic ratios are calculated using the following relations:20

−2β cos (π λ) , if : i = 2k + 1, −2 , βcos(πλ)

s 2λ

=

V. DESIGN OF SET-POINT WEIGHTING PIλ + Dμ CONTROLLER

0.0014s 0.8 −0.0945s 0.4 +5.3411 0.0005s 1.2 −0.0022s 0.8 +0.016s 0.4 +0.5578

β = 1.254λ4.717 − 0.05652.

(17)


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FIG. 6. (a) Model output and experimental output of plant for model of G3n10 . (b1) RMSE index of test and train data relevant with commensurate fractional order (α) for G3 . (b2) Model output and experimental output of plant for model of G3e10 . (c1) RMSE index of test and train data relevant with commensurate fractional order (α) for G1 . (c2) Model output and experimental output of plant for model of G1e10 .


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FIG. 7. Comparison of errors of 3 selected models G3n10 , G3e10 , and G1e10 for test shot.

The characteristic ratios determined for this case are γ = [0.0979 40.8491]. By the initial selection of a0 = 1 and τ 1 = 5 ms, the initial desired polynomial coefficients are calculated as a1 = 0.0319 and a2 = 0.0104. Plotting step response of the initial design system, the rise time of initial model is tr1 = 68.46 ms. Considering tr2 = 10 ms and considering (6), the final time constant is calculated as τ 2 = 0.73 ms. So, the desired system transfer function is computed as follows: Gm_slow =

1170.75 . s 1.3 + 10.71s 0.65 + 1170.75

(18)

A PIλ + Dμ controller is the designed to provide such a desired transient response. As a result, the equivalent feedback and feedforward transfer functions of PIλ + Dμ controller in (1) are computed as Gff

42.309 = 0.0604 + 0.65 , s

(19) 42.309 0.65 Gc = 0.3199 + 0.65 + 0.0367s . s The step response of the desired fractional order model and the simulation result of the closed loop control system of G3e10 with the PIλ + Dμ controller are presented in Fig. 12. The rise time of the step responses of the desired model and the closed loop simulation are approximately 9.8 ms and 9.2 ms, respectively, which are close to the desired value of 10 ms.

B. Fast response PIλ + Dμ controller

An alternative controller is designed with the goal of fast speed closed loop control of the plant. The desired closed loop system is designed for the rise time of 0.5 ms and maximum overshoot of 2%. Since the same characteristic of the response is demanded but with different speed, the characteristic ratios determined for this case are also γ = [0.0979 40.8491]. By the initial selection of a0 = 1000 and τ 1 = 2 ms, the initial desired polynomial coefficients are calculated as a1 = 17.606 and a2 = 3.166. Plotting step response of the initial design system, the rise time of the initial model is tr1 = 27.53 ms. Considering tr2 = 0.5 ms and considering (6), the final time constant is calculated as τ 2 = 0.03632 ms. Finally, the transfer function of the desired system is obtained as follows: 57908.8 . (20) + 75.28s 0.65 + 57908.8 As a result, the feedback and feedforward functions of the PIλ + Dμ controller are computed as follows: Gm_f sat =

s 1.3

Gff = 2.989 +

1195.85 , s 0.65

(21) 1195.85 0.65 + 4.4166 + 0.02454s . s 0.65 The step response of the desired fractional order model and the simulation result of the closed loop control system of G3e10 with the PIλ + Dμ controller are presented in Fig. 14. It Gc =

FIG. 8. The block diagram of closed loop system to control the magnetic flux in tokamak.


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FIG. 9. The performance of models in closed loop test. (a) Model output and experimental output of plant. (b) Error of models.

is obvious that the rise time of step response in desired model and closed loop simulation are approximately 0.4978 ms and 0.43 ms, respectively. It shows an appropriate accuracy in the following desired vertical magnetic flux.

VI. EXPERIMENTAL RESULT

The first step of digital implementation of a fractional order system (FOC) is discretizing the derivative of the fractional order s±λ . There are two general methods to discretize a fractional order: direct and indirect discretization.27 In indirect method, the continuous fractional order transfer function is approximated by an integer order continuous transfer function, which is then converted to a discrete transfer function. In the direct method, the fractional order transfer function is directly discretized using some algorithms such as power series expansion (PSE), continued fraction expansion (CFE) with the Tustin operator and alternative methods based on numerical integrating. In this study, the controllers are implemented through indirect discretization method.

For implementation of the fractional order controllers Gc and Gf , at first the fractional order transfer function is approximated with an integer order system, by Oustaloup filter28 of order N = 3 and frequency range of ω ∈ [10−2 10−5 rad/s]. Then, the resulted integer order system is discretized by the Tustin method in prewarp frequency of 3 × 105 rad/s and sample time of T = 10 μs. The result is a high order IIR filter. These transfer functions are implemented after reducing its order and adaptation of DC gain. Controllers are implemented in a hardware based on DSP series 5000 of Texas Instrument Co. (LTD).29 The hardware has a resolution of 16 bit which is appropriate for our study. The data are sampled every 10 μs. The control signal is converted to On/Off commands using pulse width modulation (PWM) technique, and sent to the power converter of IGBTs. Figure 10 displays the implemented hardware. This hardware is designed and implemented to execute digital control. This includes two separated boards mounted on each other; the first level includes 6 input channels that are responsible for integrating and amplifying the analogue signal, as well as analogue/digital conversion procedure. The second level is a

FIG. 10. The hardware of the control system, consist of DSP board, and its interface circuits. (a) Level 1: Integrator, amplifier, and A/D electronic board. (b) Level 2: DSP processor and interface electronic board.


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FIG. 11. The implemented visual program to tune controller parameters and the results of its implementation for slow growing of the magnetic flux. (a) Experimental signals of Ref and ψ. (b) Experimental signals of Ref and Ic .

general purpose DSP board that includes the kernel processor of DSP series 5000 of Texas Instrument Co. that handles the procedure of the real time processing, communicating with PC, and data storing. The control parameters and other safety parameters are sent to the hardware through a visual C++ program. This software has the capability of regulating sampling frequency and command frequency. It also tunes the start and stop time of the control procedure and assigns the type of the controller, tunes the hysteresis dead band, amplitude, the type of reference signal and the allowed controller current boundary. This software is responsible for data gathering from the hardware, storing and monitoring them, at the end of the process. Figures 11 and 13 show screenshots of this software. All processes must be developed in the sampling time interval of 10 μs. These processes consist of data gathering from analog channels, digital processing of Gc and Gff transfer functions, fulfilling the safety boundary of the con-

Gc (z−1 ) = 386.159 ∗

troller current and control signal and sending controller commands. This was one of the challenges in the implementation of the digital control system in this hardware. To overcome this problem, the transfer function of Gff , which is out of the closed loop system, was realized offline and its result was sent as a numerical array to the DSP controller hardware. As a result, the time limitation of the processor is fulfilled, while the transfer function Gff is also calculated. Therefore, the Gc controller is realized in DSP processor in the real time without violating the sample time limit. A. Experimental results of slow control system

In this section, the experimental results of the closed loop control system are presented. The fractional order transfer function of Gc (19) is approximated by the following discrete transfer function using Oustaloup approximation and discretization method:

1 − 5.364z−1 + 11.921z−2 − 14.036z−3 + 9.226z−4 − 3.207z−5 + 0.4598z−6 . 1 − 3.048z−1 + 2.375z−2 + 1.447z−3 + −3.100z−4 1.601z−5 + −0.275z−6

Figure 11 shows a screenshot of the practical implementation for slow rate growth of the magnetic flux in the chamber of tokamak. In Figures 11 and 13, the vertical magnetic flux ψ is the output of control system of the plant. The control current Ic is produced by power converter. The oscillation of out-

put signal is because of the performance of relay power converter. Its commands are resulted by hysteresis block and it can vibrate up to frequency of 10 kHz; however, the amplitude and frequency of these oscillations depend on hysteresis dead band; the more the value of hysteresis dead band, the more the


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FIG. 12. (a) Desired reference model, simulation, and experimental output for slow growth of the vertical magnetic flux in tokamak. (b) Control signals of simulation and experiment, controlled current in experiment.

oscillation amplitude and the less the oscillation frequency and vice versa. So, the hysteresis dead band can be used to tune the frequency and amplitude of the signals oscillations. Figure 12 shows the step response for the desired reference model, closed loop simulation and experimental implementation for slow growth of magnetic flux. The rise time of the step response in the desired model, closed loop simulation, and experimental implementation are approximately 9.8 ms, 9.2 ms, and 11 ms, respectively. According

to these results, the control system provides the required performance. B. Experimental results of the fast control system

In this section, the experimental results of the closed loop control system are presented. The fractional order transfer function of Gc (21) is approximated by the following discrete transfer function using Oustaloup approximation and

FIG. 13. The implemented visual program to tune control parameters and the results of its implementation for the fast growing of the magnetic flux. (a) Experimental signals of Ref and ψ. (b) Experimental signals of Ic , u, and ψ.


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discretization method: Gc (z−1 ) = 38.884 ∗

1 − 5.354z−1 + 11.891z−2 − 14.018z−3 + 9.245z−4 − 3.232z−5 + 0.468z−6 . 1 − 5.001z−1 + 10.301z−2 − 11.172z−3 + 6.717z−4 − 2.119z−5 + 0.274z−6

Figure 13 shows a screenshot of an experimental implementation for the fast rate growth of the magnetic flux in the chamber of tokamak. Figure 14 shows the step response for the desired reference model, closed loop simulation and experimental implementation for the fast growth of magnetic flux. The rise time of the step response in the desired model, closed loop simulation, and experimental implementation are approximately 0.52 ms, 0.46 ms, and 0.48 ms, respectively. According to these results, the control system provides the required performance. The ability of the control system to reject disturbances is a crucial objective of the tokamak control system. A step disturbance is introduced to both the simulation and the experimental plants in t = 3.65 ms with the amplitude of 0.5 V. Figure 15 shows that the fractional order controller rejects the disturbance in 0.5 ms with a reasonable control signal.

C. Comparison of fractional order controller and optimal integer order PD controller

In this section, we experimentally compare the previous optimal PD controller25 of Damavand tokamak with the present the PIλ + Dμ controller. Figure 16 shows the result of experimental implementation of digital PD controller and PIλ + Dμ controller for both step reference input and disturbance. The rise times of PD controller and PIλ + Dμ controllers are equal to 2.07 ms and 0.48 ms, respectively. The amplitude of the steady state output oscillation for PIλ + Dμ controller is

FIG. 14. (a) Desired reference model, simulation, and experimental output for fast growth of the vertical magnetic flux in tokamak. (b) Control signals of simulation and experiment, controlled current in experiment.

FIG. 15. The results of simulation and experimental for fast growth of the vertical magnetic flux in tokamak with disturbance. (a) Desired reference model, simulation, and experimental output. (b) Control signals of simulation and experiment, controlled current in experiment.

less than PD controller. Also, the settling time of disturbance rejection by PD and PIλ + Dμ controllers are equal to 0.6 ms and 0.5 ms, respectively, while the control signal overshoot of the PD controller is lower than that of the fractional order controller. As a conclusion, the presented fractional order controller has better performance versus PD in the experimental study.

FIG. 16. The results of experimental implementation of PD and PIλ + Dμ controllers (WFOPI+D) for step input with disturbance. (a) Desired reference model, experimental output with PD and PIλ + Dμ controllers. (b) Experimental control signals with PD and PIλ + Dμ controllers. (c) Experimental controlled current with PD and PIλ + Dμ controllers.


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VII. CONCLUSION

In this paper, a set-point weighting PIλ + Dμ controller was developed for the commensurate fractional order systems. Tuning of the parameters of this controller was presented based on the pole placement for the pseudo-secondorder fractional order systems. In this research, several experiments were designed and implemented for identification of vertical magnetic flux in Damavand tokamak. Using the experimental data, various fractional order models were constructed by applying the output-error model and Levenberg Marquardt nonlinear optimization algorithm. Evaluation of models with practical outputs confirms the accuracy of the models. One of the requirements in Damavand tokamak plant for plasma research activities is the control of the magnetic flux position with an alternative time response. This requirement was met by designing a desired fractional order closed loop model using CRA extended method for commensurate fractional order systems. In this paper, two desired reference models were designed with the above mentioned method for the fast and slow time responses of the vertical magnetic flux. Then, the set-point weighting PIλ + Dμ controllers were designed to build up these response speeds. At last, these controllers were converted to the integer order systems using Oustaloup approximation method. Then, these controllers were quantized as discrete transfer functions and implemented on DSP processor for the first time in the tokamak plants. The results of the experimental implementation of this controller confirm that its performance is compatible with simulation results. Also, the results of experimental implementation of PIλ + Dμ controller were compared with PD controller. The comparison shows that fractional order controller has better performance compared with the integer order PD controller on both set-point tracking and disturbance rejection. By using the fractional order identification in the tokamak, the fractional models are resulted with less orders versus integer order ones. So, the order of model and related controller are reduced as an advantage of fractional order identification in this machine. The control structure mentioned in this paper includes two free parameters: fractional order (λ) and weighted input reference (a). These parameters increase the design degree of freedom, which offers the possibility of decreasing the disturbance effect and noise of closed loop system, simultaneously. Therefore, by using the PIλ + Dμ controller on the tokamak, it is possible to reject the effect of the disturbance and the noise from the closed loop system simultaneously, with an appropriate tracking of the reference input. This causes to generate a fast, accurate, and bumpless magnetic field and as a result, reduces the energy consumption of the tokamak. On the other hand, it results

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in a desired magnetic field with alternative speeds. This control structure can be viewed as a pioneer to accurate the design of shape and position of desired plasma in a colossal tokamak. 1 D.

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