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Journal of Process Control 23 (2013) 1455 1464

Contents lists available at ScienceDirect

Journal of Process Control

jo u r nal homep age: www.elsev ier .com/ locate / jprocont

daptive speed control based on just-in-time learning technique forermanent magnet synchronous linear motor

haowu Lua,, Shiqi Zhengb, Xiaoqi Tangb, Bao Songb

Wuhan University of Science and Technology, 947 Heping Road, Wuhan, ChinaHuazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, China

r t i c l e i n f o

rticle history:eceived 12 June 2012eceived in revised form 6 September 2013ccepted 23 September 2013

a b s t r a c t

In this paper, an adaptive two degrees of freedom (2Dof) PI controller based on a just-in-time learning(JITL) method is proposed for predictive speed control of permanent magnet synchronous linear motor(PMSLM). Firstly, to guarantee the high identification accuracy and high real-time performance simul-taneously, an improved JITL method is proposed to estimate the controlled model parameters of speed

vailable online 26 October 2013

eywords:eneralized predictive controlwo degrees of freedom controllerinusoid-type

control system. Then, based on the dynamic controlled model, a simplified generalized predictive control(GPC) supplies a 2Dof proportional integral (PI) controller with suitable control parameters to follow asinusoid-type speed command in operating conditions. The main motivation of this paper is the exten-sion of the predictive controller to replace traditional PI controller in industrial applications. Finally, theefficacy and usefulness of the proposed controller are verified through the experimental results.

ust-in-time learning

. Introduction

Among linear electrical actuators, permanent magnet syn-hronous linear motor (PMSLM) is particularly suitable forpplications where high speed and high precision are required [1].ue to its structural simplicity, PMSLM owns high thrust density

o ensure high level of dynamic performance, but it is more sen-itive to various disturbances than rotational actuators. Moreover,n industrial application, servo drive of PMSLM generally uses theraditional proportional integral (PI) controller with fixed parame-ers as the control core to complete the speed adjustment. Althoughhe control system using PI controller presents simplicity and fea-ibility to a certain extent, it is not robust enough to accommodatehe uncertainties, which usually are composed of model parameterariations, external disturbances and model nonlinear dynamics2]. In the last decade, two degrees of freedom (2Dof) controllersave been widely investigated [3,4], which consider the dynamicerformance and the robustness of control system together. Alfaro5] proposed a 2Dof PI controller to replace the traditional PI con-roller, but more control parameters need to be tuned. To guaranteetable and robust performance of PMSLM servo system, it is thusesirable to have an adaptive 2Dof PI controller that can self-tune its

ontrol parameters according to the uncertainties under differentperating conditions.

Corresponding author. Tel.: +86 13476040180.E-mail address: guoting518@163.com (S. Lu).

959-1524/$ see front matter 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.jprocont.2013.09.018

2013 Elsevier Ltd. All rights reserved.

A considerable amount of research work has been made inthe area of self-tuning control in order to improve the traditionalcontrol performance. Generally speaking, there are two groups ofthoughts to autotune the PID control parameters so that the con-trollers can adapt to the varying conditions: model-based methodsand rule-based methods [6]. Fuzzy theory [7] and neural networktechniques [8] were developed to provide rule-based methods thatincorporated self-tuning capability using nonlinear algorithms.However, the learning cost of these methods is considerably large,and these PID parameters cannot be adequately adjusted due tothe nonlinear properties [9]. For the industrial feasibility of self-tuning PID strategies, model-based methods appear to be moresuitable in real-time system. Generalized predictive control (GPC)[10,11] based on the linear model has been widely accepted as anadvanced control strategy. Many PID controllers based on a GPClaw have been proposed to improve the performance of controlsystem [1214]. Therefore, applying GPC into 2Dof PI controllernot only guarantees the stability of the speed control performancebut also tunes the control parameters automatically according tothe characteristic changes of controlled model.

Because of the high real-time requirement of PSMLM servosystem, some problems for using GPC need to be solved in thefollowing.

Firstly, the main disadvantage of GPC is the calculation of con-trol action requires a large computation load since it requires the

solution of optimization problem at each sampling time. The real-time requirement of the online optimization restricts the industrialapplications of GPC to some extent. However, because of theadvanced development of computing hardware, some cheeringdx.doi.org/10.1016/j.jprocont.2013.09.018http://www.sciencedirect.com/science/journal/09591524http://www.elsevier.com/locate/jproconthttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jprocont.2013.09.018&domain=pdfmailto:guoting518@163.comdx.doi.org/10.1016/j.jprocont.2013.09.018

1 s Control 23 (2013) 1455 1464

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456 S. Lu et al. / Journal of Proces

esearch reports on the application of predictive control for servoystem have been presented [1517]. An important reason for therowing application is the linear controlled model of servo systeman be obtained by both analytical means and identification tech-iques. In [17], to apply predictive control into the speed controlystem, a first-order linear model is required for simplifying thenline implementation. Thus, in this paper, a dynamic first-orderinear model is also used to represent the controlled object of speedontrol system, and GPC can be straightforwardly replaced by 2DofI controller. Furthermore, since Diophantine equations based on arst-order controlled model will be simplified obviously, real-timeerformance of speed control system can be ensured due to theeduction of computational burden.

Secondly, in GPC, a dynamic controlled model is developed toredict the future outputs in the prediction horizon by minimiz-

ng a pre-specified cost function [10]. Therefore, the effectivenessf GPC relies heavily on the availability of a reasonably accurateontrolled model. Sato [18] used a recursive least squares (RLS)ethod to update the local model in a nonlinear GPC strategy.owever, the RLS method focuses on the global modeling, its com-utational cost is low but the identification results cannot reflecthe uncertainties satisfactorily. To solve the problem, Kansha andhiu [19] proposed an adaptive GPC algorithm based on the just-

n-time learning (JITL) technique. Though the JITL technique hastrong prediction capability for nonlinear systems and inherentdaptive nature [2023], the computational expense on searchingatabase for modeling is related directly to the number of dataamples in database, so conventional JITL method is difficult toatisfy the real-time requirement of servo system. In fact, at eachampling time, the identification of a first-order controlled modelust needs several input and output data samples. Thus, once theatabase is updated online by the available query data, the most

rrelevant data with the query data will be discarded for preventingrom deteriorating the identification process. Therefore, despite thexpense of the identification accuracy of conventional JITL method,he improved JITL method is still superior to RLS method, and itsomputational burden is significantly reduced.

Thirdly, to reduce overshoot and avoid saturation in the con-rol system, the conventional GPC-based PID controllers use theuture reference trajectory to obtain control performance as goods that of GPC, and have been designed to follow a step-type or

ramp-type reference command [18,24]. However, a step-typer a ramp-type speed command will lead to flexible impact onlectrical and mechanical agencies of servo system. Meanwhile,peed command should be suitable for the limited travel rangef PMSLM. In this paper, because of the features of continuouspeed curve and acceleration curve, a sinusoid-type signal is cho-en as a speed command, and can guarantee the input signal ofhe controlled model is persistently exciting, so an adaptive 2DofI controller will be designed to follow a sinusoid-type speed com-and.Our contribution in this paper: (1) a first-order controlled model

s chosen for a simplified GPC to satisfy the requirement of the real-ime performance for speed control system; (2) to enhance thedentification accuracy and reduce computational burden simul-aneously, an improved JITL method is proposed as an attractivelternative for modeling the nonlinear system; (3) based on theynamic controlled model, the simplified GPC will supply a 2Dof PIontroller with suitable controller parameters to ensure good con-rol performance, and high robustness of speed control system isntroduced to follow a sinusoid-type speed command.

The paper is organized as follows. In Section 2, the system

odel of PMSLM is built in detail. An improved JITL method and an

daptive 2Dof PI controller are proposed in Section 3, while experi-ental results are presented in Section 4 and conclusions are drawn

n the final section.

Fig. 1. Average model of PMSLM.

2. System model of PMSLM

2.1. Speed control system and disturbances of PMSLM

In PMSLM, excitation flux is set-up by magnets; subsequentlyno magnetizing current is needed from the supply under vectorcontrol. As a result, the electromagnetic torque will be directlyproportional to the torque current, so the nonlinear dynamic per-formance of PMSLM can be significantly improved [25]. Therefore,an average model of speed loop for PMSLM is shown in Fig. 1. Inpractical servo control, the current loop is necessary to ensure accu-rate current tracking, and the dynamics of current loop is usuallymuch faster than that of speed loop, so it can be equivalently actedas a current source amplifier within the current loop bandwidth.Accordingly, the electromechanical dynamics can be reasonablygiven in Laplace domain as

r(s) =kf i

q(s) fdistms + B (1)

where r is linear speed; iq is expected force current; kf is theforce coefficient related with flux linkage of PMLSM; m is the massof carriage; fdist is the external disturbances; B is viscous frictioncoefficient.

The popularity of GPC comes in great part from the fact thata suitable model can be obtained, thus GPC can be easily imple-mented with a direct physically understandable model. In thispaper, a dynamic first-order model is employed to represent thecontrolled object of speed control system. With the zero-order-hold(ZOH) conversion, the controlled model can be described with thediscrete difference equation:

A(z1)r(k) = B(z1)iq(k 1) +(k)

(2)

A(z1) = 1 + ak1z1 (3)

B(z1) = bk0 (4)

where ak1 and bk0 are the estimated model parameters; (k) is white

noise; is the differential operator, = 1 z1.

2.2. 2Dof PI controller

The control structure of a 2Dof PI controller is shown in Fig. 2.In order to use GPC to tune the control parameters, the 2Dof PIcontroller can be described as

iq(k) = (kpv1 + kiv1)r (k) (kpv2 + kiv2)r(k) (5)

iq(k) = iq(k 1) + iq(k) (6)

where kpv1 and kpv2 are proportional gain parameters; kiv1 and kiv2are integral gain parameters; r (k) is speed command; iq(k) usu-ally need to be limited; iq(k) is used for GPC to replace 2Dof PIcontroller.

S. Lu et al. / Journal of Process Cont

+

+

z1

+

+

-

+

Sum

Sum

Limit

Limit

(k)r*

(k)r

(k)iq*

+

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z1

+

+

Sum

Sum

Limit

iv1k

iv2k

pv2k

pv1k

3

cton2s

3

mdimmWbntwtwwpwT

Fig. 2. two degrees of freedom PI controller.

. JITL-based predictive 2Dof PI controller design

GPC is a model-based control algorithm, when the structure ofontrolled model is fixed, the identification of its model parame-ers is the sole task for GPC. To enhance the identification accuracyf model parameters and reduce computational burden simulta-eously, an improved JITL is used. Meanwhile, a GPC will supplyDof PI controller with suitable control parameters online. Thetructure of the adaptive controller is presented in Fig. 3.

.1. JITL modeling technique

In this paper, there are three main steps in the traditional JITLethod to computer model parameters corresponding to the query

ata: (1) by some nearest neighborhood criterion, all data samplesn the database are searched to match the query data; (2) a local

odel is built based on the rational use of all data samples; (3) theodel output is calculated based on local model and the query data.hen the next query data comes, a new local model will be built

ased on the aforementioned procedure. The traditional JITL tech-ique has the following problems: (1) the traditional JITL techniqueakes so much time to search relevant data samples in the database,hich cannot meet the high-real time requirement for speed con-

rol system, and moreover, the traditional database update strategyill add more and more data samples into the database, which

ill increase the complexity of the search, (2) in the calculatingrocess of the local model parameters, matrix inverse operationill increase the computational burden on the online identification.

hese problems will be discussed at length in the following.

Lowpa

filter

Optimal controlparameters calculati

Control parameters

mapping

2Dof PI controller)(k

r

* )(kiq

*

Fig. 3. Proposed JITL-based pred

rol 23 (2013) 1455 1464 1457

Define regression vector for the controlled model given in Eq.(2) as

xk = [r(k 1), iq(k 1)] (7)The database (yi, xi)i=1N is given by

[r(k), r(k 1), iq(k 1)]k=1N , where r(k) and iq(k) are theoutput and input data samples collected at the kth sampling instantin the identification test. Similarly, the query data xq at the (k 1)thsampling instant is arranged in the form of [r(k 1), iq(k 1)] asthe input to the JITL method. The evaluation of similarity betweentwo data samples is the first step. So, the following similaritymeasure is defined:

si =

ed2(xq,xi) + (1 ) cos(i) if cos(i)0 (8)where is a weight parameter and is constrained between 0 and 1,and i is the angle between xi and xq, where xi = xi xi1 andxq = xq xq1. The value of si is bounded between 0 and 1, whensi approaches to 1, it shows xi resembles closely to xq. It is notedthat if cos(i) is negative, the query data xq will be discarded, andthe model parameters will keep the last values.

Based on the similarity measure, all D(xi, xq) can be computedand will be reordered as follows:

nM = {(X(1), x1), . . ., (X(N), xN)}, D(x1, xq) > > D(xN, xq) (9)In the traditional JITL method, the data samples in database will

be more and more in the running of speed control system, whichwill lead to the increasing computational burden of reordering pro-cedure. If the number of data samples in database is fixed, thecomputational burden of reordering procedure will be also fixed.So, the traditional database update strategy will be improved as fol-lows: (1) when D(xN, xq) is not less than a pre-specified threshold, itindicates the model parameters is not enough changed greatly, andthe query data xq will be discarded; (2) when D(xN, xq) is smallerthan the pre-specified threshold, the data sample xN will be discard,and the query data xq is considered as new data and the databasewill be updated. The improved database update strategy can beillustrated in Fig. 4.

After the database is updated, the database should be rationallyused for building a local optimal model. A nearest neighborhoodshould be firstly constructed. At sampling time k, the size of theneighborhood is specified to be n [nm, nM](nm < nM N). So thelocal model parameters are calculated in the following recursive

form without the matrix inverse operation:

Vm+1 = Vm Vmxm+1D(xm+1, xq)xm+1T Vm1 + D(xm+1, xq)xm+1T Vmxm+1

(10)

Controlled

model

Lowpass

filter

JITL

modeling

ss

Performance

predictionon

GPC

)(kr

ictive 2Dof PI controller.

1458 S. Lu et al. / Journal of Process Control 23 (2013) 1455 1464

......1x 2x 3x 4x kx ......

.....

......

.....

1x 2x 3x 4x kx ......)1(X

)2(X)3(X)4(X

)(kX

database ..........

)1(X

)2(X)3(X)4(X

)(kX

of tra

r

wv

oetrpcp

3

dwdp

J

wNNtr

wgc

a

1

Fig. 4. Database update strategy

m+1 = Vm+1xm+1D(xm+1, xq) (11)

m+1 = Xm+1 xm+1T m (12)

m+1 = m + m+1rm+1 (13)here m ranges from 1 to n 1; Vm, m and rm are the intermediate

ariables used to calculate n and 0 = 0, V0 = I( = 104106).Finally, in order to verify the local model effectively, the leave-

ne-out cross validation test [21] is conducted and the validationrror is calculated. Once the validation error starts to become large,he local model has the trend of deterioration, so further recur-ence optimization will be interrupted and the local optimal modelarameters can be obtained. Upon the completion of the above pro-edure, the model output is calculated based on the optimal modelarameters and the query data.

.2. Performance prediction

Based on the least error control in a broad sense, GPC intro-uces the idea of multi-step prediction during the optimizationhich significantly improves the ability of anti-disturbance, ran-om noise, and delay changes of controlled system [10]. The controlerformance evaluation is introduced [26]:

= E

N2

j=N1

[r(k + j) (k + j)]2 + Nuj=1

iq(k + j 1)2 (14)

here N1 is minimum predictive horizon, usually choose N1 = 1;2 is maximum predictive horizon; Nu is control horizon, generallyu N2; is the control increment weighting factor. In the paper,

he speed command is a sinusoid-type signal. Hence, the futureeference trajectory is defined as

1(k) = r(k) (15)

(k + j) = (1 )r (k + j) + 1(k + j 1) (16)

r (k) = sin(0kTs) (17)here is design parameter adjusting the reference trajectory,

enerally [0, 1]; Ts is sampling time; 0 is frequency of the speedommand.

Because B(z1) is a constant, simplified Diophantine Eqs. (18)nd (19) are solved to obtain r(k + j)

= Ej(z1)A(z1) + zjFj(z1) (18)

ditional JITL and improved JITL.

Ej(z1)B(z1) = Gj(z1) (19)

Ej(z1) = e0 + e1z1 + + ejzj+1 (20)

Fj(z1) = f j0 + f

j1z

1 (21)

Gj(z1) = g0 + g1z1 + + gjzj+1 (22)

where j = 1, . . ., N2. The z1 operator on the left hand side of (20),(21) and (22) is omitted, The j steps ahead predictive output is givenas

r(k + j) = Gjiq(k + j 1) + Fjr(k) + Ej(k + j) (23)Obviously, on the right of (23), the first two terms and the third

term are irrelevant. If the first two terms are regarded as the optimalprediction, then the third term is the prediction error, that is, theoptimal prediction outputs can be expressed as:

r(k + j) = Gj(z1)iq(k + j 1) + Fj(z1)r(k) (24)The vector form of (24) is expressed as:

r = Giq + Fr(k) + E (25)where

Tr = [r(k + 1), . . ., r(k + N2)] (26)

iTq = [iq(k), . . ., iq(k + Nu 1)] (27)

FT = [F1(z1), . . ., FN2 (z1)] (28)

ET = [E1(k + 1), . . ., EN2 (k + N2)] (29)

G =

g0

g1 g0

...

gNu1 gNu1 g0...

gN21 gN21 gN2Nu

(30)

So the vector form of control performance evaluation (14) canbe obtained:

J = E[(r )T (r ) + iTq iq] (31)

s Control 23 (2013) 1455 1464 1459

w

S

i

T

w

[

P

F

3

st

wg

t

P

p

p

p

e

k

k

k

k

mand and control parameters to servo drive, and receives expectedforce current and actual speed from servo drive simultaneously. Theimproved JITL algorithm and the GPC algorithm are implementedevery 5 ms by commercial Servo Self-adapted Turning Tool (SSTT)

Table 1Specification of the PMSLM.

Rating

Mass of carriage 110 (kg)Pole pitch 48 (mm)Force coefficient 103 (N/A)

S. Lu et al. / Journal of Proces

here

T = [(k + 1), . . ., (k + N2)] (32)

o the optimal solution with respect to iq is obtained as

q = (GT G + I)

1GT [ Fr] (33)

he use of Receding Horizon gives the following control law:

iq(k) = P(z1)(k + N2) F(z1)r(k) (34)

here

pN1 pN2 ] = [10 0](GT G + I)1

GT (35)

(z1) = pN2 + pN21z1 + + pN1 z(N2N1) (36)

(z1) = pN1 FN1 (z1) + pN1+1FN1+1(z1) + + pN2 FN2 (z1)

= f0 + f1z1 (37)

.3. Optimal control parameters calculation

In the 2Dof PI controller, the future reference input cannot betraightforwardly achieved. To compare the 2Dof PI controller withhe GPC law, the future reference input (15)(17) is expressed as

(k + j) = f (j)r (k) + g(j)r (k) + jr(k) (38)

here f(j) and g(j) are related to j. Detailed expressions of (38) areiven in Appendix A.

Using (38), the first term on the right hand side of (34) is rewrit-en as

(z1)(k + N2) = prr (k) + psr (k) + pyr(k) (39)

r =N2

j=N1

pjf (j) (40)

s =N2

j=N1

pjg(j) (41)

y =N2

j=N1

pjj (42)

So, the GPC law (34) is rewritten as

iq(k) = (pr + ps)r (k) (f0 + f1 py f1)r(k) (43)

Then comparing (5) with (38), the 2Dof PI controller gain param-ters are calculated by the following equations

pv1 = ps (44)

iv1 = pr (45)

pv2 = f1 (46)

iv2 = f0 + f1 py (47)

Fig. 5. Experimental apparatus for PMSLM system.

With aforementioned discussion, the implementation of JITL-based predictive 2Dof PI controller design is summarized asfollows:

Step 1: Given the related parameters for JITL and GPC.Step 2: Calculate the similarity measure and update the databaseadopting the method discussed in Section 3.1.Step 3: With the database, the current controlled model can beobtained by using the leave-one-out cross validation test.Step 4: Based on the controlled model, iq(k) is obtained by (38),and the control parameters of the 2Dof PI controller can be tunedby (39)(42).Step 5: Set k = k + 1 and go to step 2.

4. Experiment

The apparatus for the experiments contains three major partsand some data transferring buses, as shown in Fig. 5. These threemajor parts are: (1) a PC and a PCI with sampling time equal to 5 ms;(2) a servo drive using a DSP plus a FPGA, where DSP TMS320F2812mainly accomplishes position, velocity and torque control, andFPGA EP2C8Q208C8N is responsible for the analysis and realiza-tion of absolute ruler and NCUC-Bus protocols; (3) PMSLM with theparameters described in Table 1, and absolute ruler fixed in PMSLMis LC183, the resolution of which is 10 nm.

In the experimental tests, PC using the PCI sends speed com-

Winding resistance 0.38 ()Continuous thrust 3090 (N)Continuous current 30 (A)

Table 2Computational efficiency comparison.

Time (ms) N = 10 N = 20 N = 40 N = 80

RLS 0.0181 0.0181 0.0181 0.0181JITL 0.979 1.857 3.526 6.781

1460 S. Lu et al. / Journal of Process Control 23 (2013) 1455 1464

Table 3Validation error for different values of .

4

sq5u

4

ipCtthJots

= 0.1 = 0.15 = 0.2

MSE 4.72 103 4.70 103 4.69 103

oftware based on the environments of VC. In this paper, the fre-uency of speed command is set to 5 Hz and its amplitude is set to0 m/min. Finally, all experimental data from SSTT are plotted bysing Matlab 6.5.

.1. The identification of the controlled model

The computational efficiency of the RLS algorithm and themproved JITL algorithm is compared and tested on the datarocessing software. The processor for the PC is Intel(R) Pentium(R)PU E5300@ 2.60 GHz and the memory is 1.98 GHz. Table 2 recordshe running time based on the different number of data samples inhe database. It is clear from Table 2 that the RLS algorithm owns theigh real-time performance, and the time spent on the improved

ITL will gradually increase according to the increasing numberf data samples in the database. To ensure the signal integrity ofhe send and receive operations in every 5 ms, the number of dataamples is chosen to be 10.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20

40

60

Time (s)

Line

ar s

peed

(m

/min

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (s)

Iden

tific

atio

n er

ror

base

d R

LS (

m/m

in)

Fig. 6. Data used to update the mod

= 0.25 = 0.3 = 0.35 = 0.4

.68 103 4.70 103 4.71 103 4.72 103

To obtain the effective weight parameter , a validation test fordifferent values of is conducted by the mean-squared-error (MSE)of the model identification error. The experimental results of thevalidation error are shown in Table 3, when increases from 0.1to 0.25, the validation error gradually decreases; when furtherincreases from 0.25, the validation error begins to increase. So thereasonable is chosen to be 0.25.

To use the query data xq for the first-order linear model, thespecified threshold for updating database is set to be 0.9, nM is setto be 10 and nm is set to be 5. The initial database for the improvedJITL method is conducted by effectively sampling the force currentand the linear speed in an open-loop test. To verify the identificationaccuracy, the RLS algorithm is considered for the purpose of com-parison, the results of the two methods are shown in Fig. 6. Whenthe data samples of linear speed and force current are obtained

online for updating the controlled model, the maximum predictivespeed error based on the improved JITL method is only 0.23 m/min,it is evident that the improved JITL method with prediction capa-bility outperforms the RLS algorithm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6

Time (s)

For

ce c

urre

nt (

A)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (s)

Iden

tific

atio

n er

ror

base

d JI

TL

(m/m

in)

el and predictive speed error.

S. Lu et al. / Journal of Process Control 23 (2013) 1455 1464 1461

0 0.2 0.4 0.6 0.8 1-60

-40

-20

0

20

40

60

Time (s)

Line

ar s

peed

(m

/min

)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

Time (s)

Spe

ed e

rror

(m

/min

)

Fig. 7. Speed response and speed error of the RLS-based predictive 2Dof PI control design.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time (s)

Kpv

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time (s)

Kiv

1

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time (s)

Kpv

2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time (s)

Kiv

2

Fig. 8. Control parameters of the RLS-based predictive 2Dof PI controller.

1462 S. Lu et al. / Journal of Process Control 23 (2013) 1455 1464

0 0.2 0.4 0.6 0.8 1-60

-40

-20

0

20

40

60

Time (s)

Line

ar s

peed

(m

/min

)

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

6

8

Time (s)

Spe

ed e

rror

(m

/min

)

Fig. 9. Speed response and speed error of the JITL-based predictive 2Dof PI control design.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time (s)

Kpv

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time (s)

Kiv

1

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time (s)

Kpv

2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time (s)

Kiv

2

Fig. 10. Control parameters of the JITL-based predictive 2Dof PI controller.

S. Lu et al. / Journal of Process Control 23 (2013) 1455 1464 1463

0 0.2 0.4 0.6 0.8 1-60

-40

-20

0

20

40

60

Time (s)

Line

ar s

peed

(m

/min

)

0 0.2 0.4 0.6 0.8 1-15

-10

-5

0

5

10

Time (s)

Spe

ed e

rror

(m

/min

)

Fig. 11. Speed response and speed error of the RLS-based predictive 2Dof PI control design in the presence of modeling error.

0 0.2 0.4 0.6 0.8 1-8

-6

-4

-2

0

2

4

6

Time (s)

Spe

ed e

rror

(m

/min

)

0 0.2 0.4 0.6 0.8 1-60

-40

-20

0

20

40

60

Time (s)

Line

ar s

peed

(m

/min

)

edicti

4

m2ohTtfhbsttJ

Fig. 12. Speed response and speed error of the JITL-based pr

.2. Predictive 2Dof PI control

In the predictive 2Dof PI controller design, once the controlledodel is updated by the improved JITL method, GPC will supply the

Dof PI controller with suitable control parameters. In the designf the proposed GPC, maximum predictive horizon N2 = 10, controlorizon Nu = 2, weighting factor = 0.01, and design factor = 0.o evaluate the control performance, a predictive 2Dof PI con-roller based on the RLS algorithm is also considered. It is clearrom Figs. 710 that the JITL-based predictive 2Dof PI controlleras better tracking performance than that achieved by the RLS-ased predictive 2Dof PI controller. In particular, the controller with

lowly changed control parameters is difficult to satisfactorily tracehe sinusoid-type speed command. Meanwhile, the experimen-al results imply that the identification model using the improvedITL is better than those using RLS to present the actual controlled

ve 2Dof PI control design in the presence of modeling error.

model, control parameters are tuned more suitably according tothe variation of the speed command and model parameters. Lastly,to evaluate the robustness of two predictive 2Dof PI controller, 10%modeling error in the parameter b0 is assumed. As can be seen inFigs. 11 and 12, the JITL-based predictive 2Dof PI controller designstill maintains superior control performance.

5. Conclusions

Speed control system of PMSLM employed in various industriesis almost controlled by traditional PI controller. In order to meetthe development requirements of high performance linear motor,

this paper has proposed a JITL-based predictive 2Dof PI controller toguarantee regulatory control performance and system robustnesssimultaneously. The experimental results show that the JITL-basedpredictive 2Dof PI controller is able to handle the nonlinearities

1 s Cont

aT2b

A

da2

A

l

o

f

g

s

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[25] Y. A.-R.I. Mohamed, Adaptive self-tuning speed control for permanent-magnet

464 S. Lu et al. / Journal of Proces

nd the variations of disturbances and parameters satisfactorily.he comparative studies also reveal that the JITL-based predictiveDof PI controller exhibits superior control performance to the RLS-ased predictive 2Dof PI controller.

cknowledgements

The work is supported by the National Natural Science Foun-ation of China (NSFC) (Grant No. 61174106), the National Sciencend Technology Major Project (Grant Nos. 2012ZX04001012 and012ZX04001022).

ppendix A.

The speed command and the speed command difference in theearning process of GPC can be expressed as

r (k) = sin(0kTs)

r (k) = sin(0kTs) sin[0(k 1)Ts] = r (k)[1 cos(0Ts)]+ cos(0kTs) sin(0Ts) (A.1)

Making some manipulation, the future speed command can be

btained.

r (k + j) = sin[0(k + j)Ts] = sin(0kTs) cos(0jTs)

+ cos(0kTs) sin(0jTs)

= r (k)[

cos(0jTs) 1 cos(0Ts)

sin(0Ts)sin(0jTs)

]+ r (k)

sin(0jTs)sin(0Ts)

(A.2)

If /= 0, the future reference input can be expressed as

(k + j) =[

(1 )j1i=0

ir (k + j i)]

+ jr(k) (A.3)

Substituting (A.2) into (A.3), f(j) and g(j) can be obtained

(j)=j1i=0

(1 )i[

cos(0(j i)Ts) 1 cos(0Ts)

sin(0Ts)sin(0(j i)Ts)

](A.4)

(j) =j1i=0

(1 )i sin(0(j i)Ts)sin(0Ts)

(A.5)

If = 0, the expression of the future reference input can beimplified as

(k + j) = r (k + j) = r (k)[

cos(0jTs) 1 cos(0Ts) sin(0jTs)

]

sin(0Ts)

+r (k)sin(0jTs)sin(0Ts)

(A.6) [

rol 23 (2013) 1455 1464

According to (A.4) and (A.5), f(j) and g(j) can be obtained

f (j) = [cos(0jTs) 1 cos(0Ts)

sin(0Ts)sin(0jTs)] (A.7)

g(j) = sin(0jTs)sin(0Ts)

(A.8)

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speed control based on just-in-time learning technique for permanent magnet synchronous linear motor1 Introduction2 System model of PMSLM2.1 Speed control system and disturbances of PMSLM2.2 2Dof PI controller3 JITL-based predictive 2Dof PI controller design3.1 JITL modeling technique3.2 Performance prediction3.3 Optimal control parameters calculation4 Experiment4.1 The identification of the controlled model4.2 Predictive 2Dof PI control5 ConclusionsAcknowledgementsReferencesReferences