model predictive control method of simulated moving bed...

Research ArticleModel Predictive Control Method of Simulated MovingBed Chromatographic Separation Process Based onSubspace System Identification

Zhen Yan1 Jie-Sheng Wang 1 Shao-Yan Wang2 Shou-Jiang Li2 Dan Wang1

and Wei-Zhen Sun3

1School of Electronic and Information Engineering University of Science and Technology Liaoning Anshan CityLiaoning Province China2School of College of Chemical Engineering University of Science and Technology Liaoning Anshan CityLiaoning Province China3Fujian Institute of Research on the Structure Fuzhou Fujian Province China

Correspondence should be addressed to Jie-Sheng Wang wang_jiesheng126com

Received 10 June 2019 Revised 30 August 2019 Accepted 1 October 2019 Published 22 October 2019

Academic Editor Laurent Dewasme

Copyright copy 2019 Zhen Yan et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Simulated moving bed (SMB) chromatographic separation is a new type of separation technology based on traditional fixed bedadsorption operation and true moving bed (TMB) chromatographic separation technology which includes inlet-outlet liquidliquid circulation and feed liquid separation e input-output data matrices were constructed based on SMB chromatographicseparation process data e SMB chromatographic separation process was modeled by utilizing two subspace system identi-fication algorithms multivariable output-error state-space (MOESP) identification algorithm and numerical algorithms forsubspace state-space system identification (N4SID) so as to obtain the 3rd-order and 4th-order state-space yield models of theSMB chromatographic separation process respectivelyemodel predictive control method based on the established state-spacemodels is used in the SMB chromatographic separation process e influence of different control indicators on the predictivecontrol system response performance is discussed e output response curves of the yield models were obtained by changing therelated parameters so that the yield model parameters are optimized set meanwhile Finally the simulation results showed that theyield models are successfully controlled based on the each control period and given yield range

1 Introduction

Due to the complexity of simulated moving bed (SMB)chromatographic separation mechanism and many pa-rameters affecting the separation effect [1 2] especially thesensitivity to operating parameters and disturbances theSMB chromatographic separation process is difficult to runat the optimal operating point for a long time e sepa-ration zones of SMB chromatographic separation processwere analyzed [3] A method to optimize the separation wasproposed so as to improve yield purity and economic effi-ciency [4] It is an effective method to set up the effective andaccurate model on the separation process and then

implement advanced process control methods in order tosolve the existing problems of SMB chromatographic sep-aration process ere are many modeling methods for theSMB model such as extracting the device features andsimplifing mathematical calculation Many researchers havestudied the modeling methods of SMB e general ratemodel (GRM) is the most accurate but the actual model isdifficult to solve e analytical solutions of GRM wereobtained by using the Laplace transform by using differentboundaries of linear adsorption isotherms [2] e modeloptimization sequence and the impact on the separationefficiency under different influencing factors were analyzed[5] An alternative model method was proposed to obtain a

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2391891 24 pageshttpsdoiorg10115520192391891

reduced-order model of SMB so as to reduce the compu-tational complexity of the model and achieve the desiredoptimization results [6] e front velocity method wasproposed for SMB process modeling in order to remove themethod that adsorption isotherm was obtained by means ofequilibrium experiments in the traditional SMB modelingprocess en the feasibility of the model was verified byexperiments [7] Aiming at the existing problems of thediscretization model by utilizing the finite-dimensionalpartial differential equations two Krylov-type reduced-or-der models were proposed to optimize the computationalaccuracy and computational efficiency of the model [8] emain control objective of the SMB chromatographic sepa-ration process is to increase the target yield of the SMBseparation process Many papers have made relevant re-search on process control methods e genetic algorithmwas adopted to optimize the parameters in the SMB sepa-ration process [9] e standing wave design method wasproposed to improve the purity and yield of the separatedcomponents while reducing the consumption of the ana-lytical solution [10] e flow rate control strategy wasadopted where two operational variables were added in eachswitching cycle and the SMB separation performance wasimproved by the SimCon operation [11] Because the tra-ditional numerical optimization method has many un-certainties the system parameters of size exclusionsimulated moving bed (SEC-SMB) were studied by batchchromatography [12] e impact of the fault column on theSMB separation process was analyzed through experimentsand theoretical analysis [13] An adaptive nonlinear modelpredictive control method was proposed for an SMB deviceto realize the praziquantel enantiomer separation Under theset purity levels the controller can respond quickly [14]Some scholars have carried out research and application onthe control methods of MPC Markus Bambach andMichaelHerty had studied to control the strain rate during theprocess using model predictive control [15] Saeed Sha-maghdari andMohammadHaeri had discussed the design ofa new robust model predictive control algorithm for non-linear systems [16] Wahid A et al study investigates amultivariable model predictive control (MPC) based on two-point temperature control strategy [17] Some articles on thestudy of process control methods are also discussed YunjianHu et al proposed an optimal tension and thickness controlmethod based on the receding horizon control (RHC)strategy to improve control performance and enhancesystem robustness [18] Ren et al studied a reduced-orderextended state observer- (ESO-) based sliding mode controlscheme for friction compensation of a three-wheeled om-nidirectional mobile robot [19]

In this paper a predictive control method based on thestate-space yield models of SMB chromatographic separationprocess obtained by utilizing the subspace system identifi-cation method is proposed e subspace system identifica-tion is an identification algorithm based on the input-outputdata matrix by means of mathematical tools such as ge-ometry matrix analysis and probability statistics so as toobtain the state-space matrix in state-space equations Manypapers have proposed many model identification methods for

complex process models van Overschee and de Moor elab-orated on the multivariable output-error state-space(MOESP) and the numerical algorithms for subspace state-space system identification (N4SID) [20] Hachicha et alcompared the morbid conditions of MOESP and N4SID andconfirmed that they are robust in the parameter estimationprocess of the controlled object [21] e discrete state-spaceequation was vectorized and scaled to solve the bilinear low-rank minimization problem [22] e subspace identificationmethod was applied to the fault diagnosis of the wind turbineshaft [22] Model predictive control is an advanced optimi-zation control method composed of the predictive modelrolling optimization and feedback correction Clarke de-veloped a controlled autoregressive integral moving averagemodel (CARIMA) [23] Garic and Morari proposed the in-ternal model control (IMC) algorithm [24] Ding made arelated study on the robustness ofmultiple time-delay systems[25] Kayacan et al successfully solved the trajectory trackingproblem of the self-made tractor trailer system by using thefast distributed nonlinear model predictive control algorithm[26] Li et al designed a distributed predictive controller tosatisfy the stability conditions by using a distributed imple-mentation [27] Farina and Scattolini proposed a new dis-tributed predictive control algorithm for linear discrete-timesystems [28] Some newly published papers also propose newmethods for model identification Hou et al designed asubspace identification method which is proposed for theHammerstein-type nonlinear systems subject to periodicdisturbances with unknown waveforms [29] Gradient-basedand least-squares-based iterative identification algorithms aredeveloped for output-error (OE) and output-error movingaverage (OEMA) systems and the simulation results confirmtheoretical findings [30] Ding et al proposed an algorithmabout a least-squares-based and a gradient-based iterativealgorithm for the Hammerstein nonlinear systems using thehierarchical identification principle [31] Ding used an iter-ative least-squares algorithm to estimate the parameters ofoutput-error systems and enhance computational efficiencies[32] Xu and Ding discussed the parameter estimationproblems for dynamical systems by means of the impulseresponse measurement data [33] Xu and Ding used thehierarchical identification principle and the iterative search tostudy the modeling of multifrequency signals based onmeasured data [34] e signal modeling methods based onstatistical identification are proposed by means of the dy-namical window discrete measured data [35] Ding proposedamethod for identifying the systemmodel parameters and thenoise model parameters for stochastic systems described bythe CARARMA models [36] Xu et al used the hierarchicalprinciple and the parameter decomposition strategy to de-velop a parameter estimation algorithm for linear continuous-time systems [37]

e structure of the paper is organized as followsSection 2 introduces the technique of SMB chromatogra-phy separation process Section 3 introduces the twosubspace system identification algorithms (MOESP andN4SID) Section 4 introduces the structure of the modelpredictive control algorithm and the predictive controlmethod based on the state-space model Section 5 shows

2 Mathematical Problems in Engineering

the experimental simulation and results analysis Finallythe conclusion is summarized in Section 6

2 Technique of Simulated Moving BedChromatographic Separation Process

e devices in the SMB chromatographic separation processare composed of a circulation pump filter constant tem-perature system pressure sensor flow rate meter rotaryvalve pipeline control part and a plurality of separationcolumns [38 39] According to the specific requirements ofthe separation target compound the number of separationcolumns is changed from 4 to 24 and the number of pumpsis 3 to 5 e SMB system is mainly divided into the mobilephase circulation system and the stationary phase circulation

system e mobile phase circulation system makes thecompounds of mixed components circulate in the columnsunder the action of the pressure pump through the valve ofthe entrance port e corresponding target separationsubstance is obtained at the valve outlet of extraction so-lution e stationary phase circulation system simulates theflow of the stationary phase by the cooperation of two valvese stationary phase flows in the opposite direction com-pared to the direction of the mobile phase

21 Import and Export Liquid of Chromatographic SeparationProcess e chromatographic separation process is dividedinto different zones depending on the function namely thesolid phase regeneration zone the two separation zones and

Eluent

Extract liquid

Entrance liquidRaffinate

Valve switching direction

Section I

Strong adsorption component AWeak adsorption component B

Section III

Section IV Section II

Figure 1 Distribution map of SMB zones

A

B

Switching direction

Section III

Section I

Sect

ion

II

Sect

ion

IV

EluentExtract liquid

A + B

Entranceliquid

Raffinate

(a)

B

Switching direction

Section III

Section ISe

ctio

n II

Sect

ion

IV

Eluent

A

Extractliquid

A + B

Entranceliquid

Raffinate

(b)

Figure 2 Switching state schematic of SMB (a) Switching State 1 (b) Switching State 2

Mathematical Problems in Engineering 3

the eluent regeneration zone e zones of chromatographicseparation process are shown in Figure 1 [22]

e first zone is located between the mobile phase inletand the extract liquid outlet the second zone is locatedbetween the extract liquid outlet and the entrance liquidinlet the third zone is located between the entrance liquid

inlet and the raffinate outlet and the fourth zone is locatedbetween the raffinate outlet and the mobile phase inlet InSection I the mobile phase detaches the adsorbent and theseparation material regeneration that is the solid phaseregeneration zone In Section II the strongly adsorbedcomponent moves relatively slowly in the column due to

R

M

E

UV-Vis

UV-Vis

UV-Vis

LR

LM

LE

Oi

Hi

i

Ii

Pi Di Fi

Ti

Pump 1

Pump 2

Pump 3

LF

LD

LP

F

D

PA

PB

Ri Mi Ei

Figure 3 SMB chromatographic separation unit

Table 1 Variable unit and range

Flow rate(F pump)

Flow rate(D pump)

Switchingtime

Targetsubstance purity

Impuritypurity

Targetsubstance yield

Impurityyield

Unit mlmin mlmin min mgml mgml Range 0-1 0-1 0-1 11ndash20 0-1 0ndash100 0ndash100


adsorption and is concentrated in the second region InSection III the weakly adsorbed component moves from thesecond region to the third region at a faster velocity and isaggregated in the third region In Section IV the mixture isseparated in the second zone and the third zone and themobile phase is regenerated Before the mixture is subjectedto a new cycle the mixed components must be completelydesorbed to allow the stationary phase to be purified eweak components in the separation zone are desorbed as

the liquid phase of the mobile phase moves and the strongcomponents are adsorbed and move with the stationaryphase

22 Cycle of Material Liquid in Chromatographic ColumnIn the SMB chromatographic separation process only themobile phase in the column is circulating and the stationaryphase does not move By controlling the valves the position

0 500 1000 150001

02

03

04

05m

lmin

0 500 1000 150005

1

15

2

mlm

in

0 500 1000 1500

min

10

15

20

0 500 1000 1500

Yiel

d

0

02

04

06

08

1

Yiel

d

0 500 1000 15000

02

04

06

08

1

Figure 4 Input-output data sets (a) u1 F pump flow rate (b) u2 D pump flow rate (c) u3 switching time (d) y1 Target substance yield(e) y2 impurity yield


of the liquid in each zone is switched to simulate the flow ofthe stationary phasee state switching of chromatographicseparation is shown in Figure 2 [22] At the initial momentthe positions of the eluent extract liquid entrance liquidand raffinate are shown in Switching State 1 in Figure 2After t time through the valve switching the position ofeach inlet and outlet is shown in Switching State 2 in Fig-ure 2 After the position switching the result is that thestationary phase forms a relative movement with the mobilephase at a speed of Lt so as to simulate the countercurrentprocess of the moving bed

23 Separation of Mixture By carrying out switching thelocation of inlet-outlet continuously after several cycles themixed compound is under the flushing of the eluent so thatthe sample components are gradually separated After theseparation for a period of time the mixture in the columncan be separated to some extent by adjusting and optimizingthe interval of valve switching the moving speed of the inlet-outlet and in-out of double inlet-outlet solutions per unittime Meanwhile the concentration separation curve in thecolumn is always maintained in a stable distribution stateUnder cyclic operation of the SMB chromatographic systemthe mixed compound is separated continuously

e core structure of the SMB device used in this paper isshown in Figure 3 e universal SMB system is built byseven self-designed chromatographic dedicated two-way

solenoid valves in each chromatographic column ere arefour channels in the mobile phase inlet of each chromato-graphic column which are respectively introduced into thecirculating liquid of the former root columne pumps 1 2and 3 are used to deliver the raw material liquid F eluent Dand eluent Pere are four pipes in the mobile phase outletwhich respectively output the current chromatographiccolumn circulating liquid the weak adsorption raffinate Rthe strong adsorption extract liquid E and the ordinaryadsorption substance or the feedback liquid M e sevenpipes are controlled by solenoid valves Each pipe of outlet isequipped with a detector e microprocessor controls theworking state of the solenoid valves (ldquoonrdquo or ldquooffrdquo) Underthe control of the solenoid valves the SMB system can workon multiple working modes as required

24 Selection of SMB Process Variables e variable in-formation of the SMB chromatographic separation processis shown in Table 1 F pump flow rate D pump flow rate andswitching time are chosen as input variables and targetsubstance yield and impurity yield are chosen as outputvariables Using the subspace identification algorithms thecorresponding state-space model can be established eactual operational data of the SMB chromatographic sepa-ration process are collected whose input data are pump flowrate flushing pump flow rate and switching time data andwhose output data are target substance purity impurity

Smooth

PlantOptimizationmin J(k)

Model

Currentcontrol

Future control

Past controlInput and output

Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)

Figure 5 Structure of the predictive control process

SMB state-space model

y(t)u(t)

Impulse output response

MPC parameters

Reference trajectory

Calculation of control action

Set point

MPC controller

Figure 6 MPC controller structure of SMB chromatographic separation process


100

y1 actual data3rd-order MOESP y13rd-order N4SID y1

ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued


y1 actual data4th-order MOESP y14th-order N4SID y1

100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)

Figure 7 Comparison of the output values between MOESP and N4SID


purity target substance yield and impurity yield enumber of data is more than 1400 groups e whole input-output data set is shown in Figure 4

3 Yield Model Identification of SMBChromatographic Separation ProcessBased on Subspace SystemIdentification Methods

e structure of the typical discrete state-space model isshown in Figure 4 e state-space model can be describedas

xk+1 Axk + Buk

yk Cxk + Duk(1)

where uk isin Rm yk isin Rl A isin Rntimesn B isin Rntimesm C isin Rltimesn andD isin Rltimesm

e constructed input Hankel matrices are as follows

Up def

Uo|iminus 1

u0 u1 middot middot middot ujminus 1

u1 u2 middot middot middot uj

⋮ ⋮ ⋮ ⋮

uiminus 1 ui middot middot middot ui+jminus 2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1

ui ui+1 middot middot middot ui+jminus 1

ui+1 ui+2 middot middot middot ui+j

⋮ ⋮ ⋮ ⋮

u2iminus 1 u2i middot middot middot u2i+jminus 2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)

where i denotes the current moment o|i minus 1 denotes fromfirst row to ith row p denotes past and f denote future

Similarly the output Hankel matrices can be defined asfollows

Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)

e status sequence is defined as

Xi xi xi+1 xi+jminus 11113872 1113873 (4)

We obtain the following after iterating Formula (1)

Yf ΓiXf + HiUf + V (5)

where V is the noise term Γi is the extended observabilitymatrix and Hi is the lower triangular Toeplitz matrix whichare expressed as

Γi C CA middot middot middot CAiminus 1( 1113857T

Hi

D 0 middot middot middot 0

CB D middot middot middot 0

⋮ ⋮ ⋱ ⋮

CAiminus 2B CAiminus 3B middot middot middot D

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)

and the oblique projections Oi is expressed as

Oi def Yf

Wp

Uf1113890 1113891

(7)

In [1] the oblique projections has been explained Fi-nally we get

Oi ΓiXfWp

Uf1113890 1113891

(8)

Singular value decomposition for Oi is expressed as

Oi USVΤ (9)

Get the extended observability matrix Γi and the Kalmanestimator of Xi as 1113954Xi At the same time in the process ofsingular value decomposition the order of the model isobtained from the singular value of S

In [2] system matrices A B C and D are determined byestimating the sequence of states erefore the state-spacemodel of the system is obtained

emathematical expression of the MOESP algorithm isgiven below

LQ decomposition of system matrices is constructed byusing the Hankel matrices

Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)

and singular value decomposition on L32 is as follows

L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)

e extended observability matrix is equal to

Γi Um1 middot S

m1( 1113857

12 (12)

Multivariable output-error state-space (MOESP) isbased on the LQ decomposition of the Hankel matrixformed from input-output data where L is the lower tri-angular and Q is the orthogonal A singular value de-composition (SVD) can be performed on a block from the L(L22) matrix to obtain the system order and the extendedobservability matrix From this matrix it is possible to es-timate the matrices C and A of the state-space model efinal step is to solve overdetermined linear equations usingthe least-squares method to calculate matrices B and D e


Time (s)

ndash6ndash4ndash2

02

m = 1

3rd-order MOESP y13rd-order MOESP y2

4th-order MOESP y14th-order MOESP y2







ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued


MOESP identification algorithm has been described in detailin [40 41]

e mathematical expression of the N4SID algorithm is

Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)

Given two matrices W1 isin Rlitimesli andW2 isin Rjtimesj

W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)


Γi C CA middot middot middot CAiminus 1( 1113857Τ (15)

e method of numerical algorithms for subspace state-space system identification (N4SID) starts with the obliqueprojection of the future outputs to past inputs and outputs intothe future inputsrsquo directione second step is to apply the LQdecomposition and then the state vector can be computed bythe SVD Finally it is possible to compute thematricesA BCand D of the state-space model by using the least-squaresmethod N4SID has been described in detail in [42]

4 Model Predictive Control Algorithm

41 Fundamental Principle of Predictive Control

411 Predictive Model e predictive model is used in thepredictive control algorithm e predictive model justemphasizes on the model function rather than depending on

3rd-order N4SID y13rd-order N4SID y2

4th-order N4SID y14th-order N4SID y2




4th-order N4SID y14th-orderN4SID y2



ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)

Figure 8 Output response curves of yield models under different control time domains (a) Output response curves of the MOESP yieldmodel (b) Output response curves of the N4SID yield model


ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)







4th-order MOESP y13rd-order MOESP y2


(a)

Figure 9 Continued


the model structure e traditional models can be used aspredictive models in predictive control such as transferfunction model and state-space model e pulse or stepresponse model more easily available in actual industrialprocesses can also be used as the predictive models inpredictive controle unsteady system online identificationmodels can also be used as the predictive models in pre-dictive control such as the CARMA model and the CAR-IMAmodel In addition the nonlinear systemmodel and thedistributed parameter model can also be used as the pre-diction models

412 Rolling Optimization e predictive control is anoptimization control algorithm that the future output will beoptimized and controlled by setting optimal performanceindicators e control optimization process uses a rolling

finite time-domain optimization strategy to repeatedly op-timize online the control objective e general adoptedperformance indicator can be described as

J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)

where y(k + j) and yr(k + j) are the actual output andexpected output of the system at the time of k + j N1 is theminimum output length N2 is the predicted length Nu isthe control length and cj is the control weighting coefficient

is optimization performance index acts on the futurelimited time range at every sampling moment At the nextsampling time the optimization control time domain alsomoves backward e predictive process control has cor-responding optimization indicators at each moment e

ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)

Figure 9 Output response curves of yield models under different predicted time domains (a) Output response curves of the MOESP yieldmodel (b) Output response curves of the N4SID yield model


ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued


control system is able to update future control sequencesbased on the latest real-time data so this process is known asthe rolling optimization process

413 Feedback Correction In order to solve the controlfailure problem caused by the uncertain factors such astime-varying nonlinear model mismatch and interference inactual process control the predictive control system in-troduces the feedback correction control link e rollingoptimization can highlight the advantages of predictiveprocess control algorithms by means of feedback correctionrough the above optimization indicators a set of futurecontrol sequences [u(k) u(k + Nu minus 1)] are calculated ateach control cycle of the prediction process control In orderto avoid the control deviation caused by external noises andmodel mismatch only the first item of the given control

sequence u(k) is applied to the output and the others areignored without actual effect At the next sampling instant theactual output of the controlled object is obtained and theoutput is used to correct the prediction process

e predictive process control uses the actual output tocontinuously optimize the predictive control process so as toachieve a more accurate prediction for the future systembehavior e optimization of predictive process control isbased on the model and the feedback information in order tobuild a closed-loop optimized control strategye structureof the adopted predictive process control is shown inFigure 5

42 Predictive Control Based on State-Space ModelConsider the following linear discrete system with state-space model

ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)

Figure 10 Output response curves of yield models under control weight matrices (a) Output response curves of the MOESP yield model(b) Output response curves of the N4SID yield model


ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued


x(k + 1) Ax(k) + Buu(k) + Bdd(k)

yc(k) Ccx(k)(17)

where x(k) isin Rnx is the state variable u(k) isin Rnu is thecontrol input variable yc(k) isin Rnc is the controlled outputvariable and d(k) isin Rnd is the noise variable

It is assumed that all state information of the systemcan be measured In order to predict the system futureoutput the integration is adopted to reduce or eliminatethe static error e incremental model of the above lineardiscrete system with state-space model can be described asfollows

Δx(k + 1) AΔx(k) + BuΔu(k) + BdΔd(k)

yc(k) CcΔx(k) + yc(k minus 1)(18)

where Δx(k) x(k) minus x(k minus 1) Δu(k) u(k) minus u(k minus 1)and Δ d(k) d(k) minus d(k minus 1)

From the basic principle of predictive control it can beseen that the initial condition is the latest measured value pis the setting model prediction time domain m(m lep) isthe control time domainemeasurement value at currenttime is x(k) e state increment at each moment can bepredicted by the incremental model which can be de-scribed as

ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)

Figure 11 Output response curves of yield models under different output-error weighting matrices (a) Output response curves of theMOESP yield model (b) Output response curves of the N4SID yield model


4th-order MOESP4th-order N4SIDForecast target value

ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)

Figure 12 Output prediction control curves of 4th-order yield model under different inputs (a) Output prediction control curves of the4th-order model without noises (b) Output prediction control curves of the 4th-order model with noises


Δx(k + 1 | k) AΔx(k) + BuΔu(k) + BdΔd(k)

Δx(k + 2 | k) AΔx(k + 1 | k) + BuΔu(k + 1) + BdΔd(k + 1)

A2Δx(k) + ABuΔu(k) + Buu(k + 1) + ABdΔd(k)


A3Δx(k) + A

2BuΔu(k) + ABuΔu(k + 1) + BuΔu(k + 2) + A

2BdΔd(k)

Δx(k + m | k) AΔx(k + m minus 1 | k) + BuΔu(k + m minus 1) + BdΔd(k + m minus 1)

AmΔx(k) + A

mminus 1BuΔu(k) + A

mminus 2BuΔu(k + 1) + middot middot middot + BuΔu(k + m minus 1) + A

mminus 1BdΔd(k)

⋮

Δx(k + p) AΔx(k + p minus 1 | k) + BuΔu(k + p minus 1) + BdΔd(k + p minus 1)

ApΔx(k) + A

pminus 1BuΔu(k) + A

pminus 2BuΔu(k + 1) + middot middot middot + A

pminus mBuΔu(k + m minus 1) + A

pminus 1BdΔd(k)

(19)

where k + 1 | k is the prediction of k + 1 time at time k econtrolled outputs from k + 1 time to k + p time can bedescribed as

yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)

CcAΔx(k) + CcBuΔu(k) + CcBdΔd(k) + yc(k)

yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2

+ CcA1113872 1113873Δx(k) + CcABu + CcBu( 1113857Δu(k) + CcBuΔu(k + 1)

+ CcABd + CcBd( 1113857Δd(k) + yc(k)

yc(k + m | k) CcΔx(k + m | k) + yc(k + m minus 1 | k)

1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA

iminus 1BuΔu(k) + 1113944

mminus 1

i1CcA

iminus 1BuΔu(k + 1)

+ middot middot middot + CcBuΔu(k + m minus 1) + 1113944m

i1CcA

iminus 1BdΔd(k) + yc(k)

⋮

yc(k + p | k) CcΔx(k + p | k) + yc(k + p minus 1 | k)

1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA


+ middot middot middot + 1113944

pminus m+1

i1CcA

iminus 1BuΔu(k + m minus 1) + 1113944

p

i1CcA


(20)

Define the p step output vector as

Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮

yc(k + m minus 1 | k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)

Define the m step input vector as

ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)


e equation of p step predicted output can be defined as

Yp(k + 1 | k) SxΔx(k) + Iyc(k) + SdΔd(k) + SuΔu(k)

(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su

CcBu 0 0 middot middot middot 0

1113944

2

i1CcA

iminus 1Bu CcBu 0 middot middot middot 0

⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA

iminus 1Bu middot middot middot middot middot middot CcBu

⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA

iminus 1Bu middot middot middot middot middot middot 1113944

pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

e given control optimization indicator is described asfollows

J 1113944

p

i11113944

nc

j1Γyj

ycj(k + i | k) minus rj(k + i)1113872 11138731113876 11138772 (25)

Equation (21) can be rewritten in the matrix vector form

J(x(k)ΔU(k) m p) Γy Yp(k + i | k) minus R(k + 1)1113872 1113873

2

+ ΓuΔU(k)

2

(26)

where Yp(k + i | k) SxΔx(k) + Iyc(k) + SdΔd(k) + SuΔu(k) Γy diag(Γy1 Γy2 Γyp) and Γu diag(Γu1

Γu2 Γum)e reference input sequence is defined as

R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)

en the optimal control sequence at time k can beobtained by

ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where

Ep(k + 1 | k) R(k + 1) minus SxΔx(k) minus Iyc(k) minus SdΔd(k)

(29)

From the fundamental principle of predictive controlonly the first element of the open loop control sequence actson the control system that is to say

Δu(k) Inutimesnu0 middot middot middot 01113960 1113961

ItimesmΔUlowast(k)

Inutimesnu0 middot middot middot 01113960 1113961

ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let

Kmpc Inutimesnu0 middot middot middot 01113960 1113961

ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)

en the control increment can be rewritten as

Δu(k) KmpcEp(k + 1 | k) (32)

43 Control Design Method for SMB ChromatographicSeparation e control design steps for SMB chromato-graphic separation are as follows

Step 1 Obtain optimized control parameters for theMPC controller e optimal control parameters of theMPC controller are obtained by using the impulseoutput response of the yield model under differentparametersStep 2 Import the SMB state-space model and give thecontrol target and MPC parameters calculating thecontrol action that satisfied the control needs Finallythe control amount is applied to the SMB model to getthe corresponding output

e MPC controller structure of SMB chromatographicseparation process is shown in Figure 6

5 Simulation Experiment and Result Analysis

51 State-Space Model Based on Subspace Algorithm

511 Select Input-Output Variables e yield model can beidentified by the subspace identification algorithms by


utilizing the input-output data e model input and outputvector are described as U (u1 u2 u3) and Y (y1 y2)where u1 u2 and u3 are respectively flow rate of F pumpflow rate of D pump and switching time y1 is the targetsubstance yield and y2 is the impurity yield

512 State-Space Model of SMB Chromatographic Separa-tion System e input and output matrices are identified byusing the MOESP algorithm and N4SID algorithm re-spectively and the yield model of the SMB chromatographicseparation process is obtained which are the 3rd-orderMOESP yield model the 4th-order MOESP yield model the3rd-order N4SID yield model and the 4th-order N4SIDyield model e parameters of four models are listed asfollows

e parameters for the obtained 3rd-order MOESP yieldmodel are described as follows

A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

minus 189315 minus 00393

36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦

D minus 00760 minus 00283

minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)

e parameters for the obtained 4th-order MOESP yieldmodel are described as follows

A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800

minus 00003 minus 00067 minus 00618 02460

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)

e parameters for the obtained 3rd-order N4SID yieldmodel are described as follows

A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)

e parameters for the obtained 4th-order N4SID yieldmodel are described as follows

A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527

minus 0008868 minus 03551 08102 minus 01679

006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 1123 6627 minus 1731 minus 2195

minus 5739 4897 minus 1088 minus 28181113890 1113891

D 0 0

0 01113890 1113891

(36)

513 SMB Yield Model Analysis and Verification Select thefirst 320 data in Figure 4 as the test sets and bring in the 3rd-order and 4th-order yield models respectively Calculate therespective outputs and compare the differences between themodel output values and the actual values e result isshown in Figure 7

It can be seen from Figure 7 that the yield state-spacemodels obtained by the subspace identification methodhave higher identification accuracy and can accurately fitthe true value of the actual SMB yielde results verify thatthe yield models are basically consistent with the actualoutput of the process and the identification method caneffectively identify the yield model of the SMB chro-matographic separation process e SMB yield modelidentified by the subspace system provides a reliable modelfor further using model predictive control methods tocontrol the yield of different components

52 Prediction Responses of Chromatographic SeparationYield Model Under the MOESP and N4SID chromato-graphic separation yield models the system predictionperformance indicators (control time domain m predicted


time domain p control weight matrix uw and output-errorweighting matrix yw) were changed respectively e im-pulse output response curves of the different yield modelsunder different performance indicators are compared

521 Change the Control Time Domain m e control timedomain parameter is set as 1 3 5 and 10 In order to reducefluctuations in the control time-domain parameter theremaining system parameters are set to relatively stablevalues and the model system response time is 30 s esystem output response curves of the MOESP and N4SIDyield models under different control time domains m areshown in Figures 8(a) and 8(b) It can be seen from thesystem output response curves in Figure 8 that the differentcontrol time domain parameters have a different effect onthe 3rd-order model and 4th-order model Even though thecontrol time domain of the 3rd-order and 4th-order modelsis changed the obtained output curves y1 in the MOESP andN4SID output response curves are coincident When m 13 5 although the output curve y2 has quite difference in therise period between the 3rd-order and 4th-order responsecurves the final curves can still coincide and reach the samesteady state When m 10 the output responses of MOESPand N4SID in the 3rd-order and 4th-order yield models arecompletely coincident that is to say the output response isalso consistent

522 Change the Prediction Time Domain p e predictiontime-domain parameter is set as 5 10 20 and 30 In order toreduce fluctuations in the prediction time-domain param-eter the remaining system parameters are set to relativelystable values and the model system response time is 150 se system output response curves of the MOESP andN4SID yield models under different predicted time domainsp are shown in Figures 9(a) and 9(b)

It can be seen from the system output response curves inFigure 9 that the output curves of the 3rd-order and 4th-orderMOESP yield models are all coincident under differentprediction time domain parameters that is to say the re-sponse characteristics are the same When p 5 the outputresponse curves of the 3rd-order and 4th-order models ofMOESP and N4SID are same and progressively stable Whenp 20 and 30 the 4th-order y2 output response curve of theN4SIDmodel has a smaller amplitude and better rapidity thanother 3rd-order output curves

523 Change the Control Weight Matrix uw e controlweight matrix parameter is set as 05 1 5 and 10 In orderto reduce fluctuations in the control weight matrix pa-rameter the remaining system parameters are set to rel-atively stable values and the model system response time is20 s e system output response curves of the MOESP andN4SID yield models under different control weight ma-trices uw are shown in Figures 10(a) and 10(b) It can beseen from the system output response curves in Figure 10that under the different control weight matrices althoughy2 response curves of 4th-order MOESP and N4SID have

some fluctuation y2 response curves of 4th-order MOESPand N4SID fastly reaches the steady state than other 3rd-order systems under the same algorithm

524 Change the Output-Error Weighting Matrix ywe output-error weighting matrix parameter is set as [1 0][20 0] [40 0] and [60 0] In order to reduce fluctuations inthe output-error weighting matrix parameter the remainingsystem parameters are set to relatively stable values and themodel system response time is 400 s e system outputresponse curves of the MOESP and N4SID yield modelsunder different output-error weighting matrices yw areshown in Figures 11(a) and 11(b) It can be seen from thesystem output response curves in Figure 11 that whenyw 1 01113858 1113859 the 4th-order systems of MOESP and N4SIDhas faster response and stronger stability than the 3rd-ordersystems When yw 20 01113858 1113859 40 01113858 1113859 60 01113858 1113859 the 3rd-order and 4th-order output response curves of MOESP andN4SID yield models all coincide

53 Yield Model Predictive Control In the simulation ex-periments on the yield model predictive control the pre-diction range is 600 the model control time domain is 10and the prediction time domain is 20e target yield was setto 5 yield regions with reference to actual process data [004] [04 05] [05 07] [07 08] [08 09] and [09 099]e 4th-order SMB yield models of MOESP and N4SID arerespectively predicted within a given area and the outputcurves of each order yield models under different input areobtained by using the model prediction control algorithmwhich are shown in Figures 12(a) and 12(b) It can be seenfrom the simulation results that the prediction control under4th-order N4SID yield models is much better than thatunder 4th-order MOESP models e 4th-order N4SID canbring the output value closest to the actual control targetvalue e optimal control is not only realized but also theactual demand of the model predictive control is achievede control performance of the 4th-order MOESP yieldmodels has a certain degree of deviation without the effect ofnoises e main reason is that the MOESP model has someidentification error which will lead to a large control outputdeviation e 4th-order N4SID yield model system has asmall identification error which can fully fit the outputexpected value on the actual output and obtain an idealcontrol performance

6 Conclusions

In this paper the state-space yield models of the SMBchromatographic separation process are established basedon the subspace system identification algorithms e op-timization parameters are obtained by comparing theirimpact on system output under themodel prediction controlalgorithm e yield models are subjected to correspondingpredictive control using those optimized parameters e3rd-order and 4th-order yield models can be determined byusing the MOESP and N4SID identification algorithms esimulation experiments are carried out by changing the


control parameters whose results show the impact of outputresponse under the change of different control parametersen the optimized parameters are applied to the predictivecontrol method to realize the ideal predictive control effecte simulation results show that the subspace identificationalgorithm has significance for the model identification ofcomplex process systems e simulation experiments provethat the established yield models can achieve the precisecontrol by using the predictive control algorithm

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

Authorsrsquo Contributions

Zhen Yan performed the main algorithm simulation and thefull draft writing Jie-Sheng Wang developed the conceptdesign and critical revision of this paper Shao-Yan Wangparticipated in the interpretation and commented on themanuscript Shou-Jiang Li carried out the SMB chromato-graphic separation process technique Dan Wang contrib-uted the data collection and analysis Wei-Zhen Sun gavesome suggestions for the simulation methods

Acknowledgments

is work was partially supported by the project by NationalNatural Science Foundation of China (Grant No 21576127)the Basic Scientific Research Project of Institution of HigherLearning of Liaoning Province (Grant No 2017FWDF10)and the Project by Liaoning Provincial Natural ScienceFoundation of China (Grant No 20180550700)

References

[1] A Puttmann S Schnittert U Naumann and E von LieresldquoFast and accurate parameter sensitivities for the general ratemodel of column liquid chromatographyrdquo Computers ampChemical Engineering vol 56 pp 46ndash57 2013

[2] S Qamar J Nawaz Abbasi S Javeed and A Seidel-Mor-genstern ldquoAnalytical solutions and moment analysis ofgeneral rate model for linear liquid chromatographyrdquoChemical Engineering Science vol 107 pp 192ndash205 2014

[3] N S Graccedila L S Pais and A E Rodrigues ldquoSeparation ofternary mixtures by pseudo-simulated moving-bed chroma-tography separation region analysisrdquo Chemical Engineeringamp Technology vol 38 no 12 pp 2316ndash2326 2015

[4] S Mun and N H L Wang ldquoImprovement of the perfor-mances of a tandem simulated moving bed chromatographyby controlling the yield level of a key product of the firstsimulated moving bed unitrdquo Journal of Chromatography Avol 1488 pp 104ndash112 2017

[5] X Wu H Arellano-Garcia W Hong and G Wozny ldquoIm-proving the operating conditions of gradient ion-exchangesimulated moving bed for protein separationrdquo Industrial ampEngineering Chemistry Research vol 52 no 15 pp 5407ndash5417 2013

[6] S Li L Feng P Benner and A Seidel-Morgenstern ldquoUsingsurrogate models for efficient optimization of simulatedmoving bed chromatographyrdquo Computers amp Chemical En-gineering vol 67 pp 121ndash132 2014

[7] A Bihain A S Neto and L D T Camara ldquoe front velocityapproach in the modelling of simulated moving bed process(SMB)rdquo in Proceedings of the 4th International Conference onSimulation and Modeling Methodologies Technologies andApplications August 2014

[8] S Li Y Yue L Feng P Benner and A Seidel-MorgensternldquoModel reduction for linear simulated moving bed chroma-tography systems using Krylov-subspace methodsrdquo AIChEJournal vol 60 no 11 pp 3773ndash3783 2014

[9] Z Zhang K Hidajat A K Ray and M Morbidelli ldquoMul-tiobjective optimization of SMB and varicol process for chiralseparationrdquo AIChE Journal vol 48 no 12 pp 2800ndash28162010

[10] B J Hritzko Y Xie R J Wooley and N-H L WangldquoStanding-wave design of tandem SMB for linear multi-component systemsrdquo AIChE Journal vol 48 no 12pp 2769ndash2787 2010

[11] J Y Song K M Kim and C H Lee ldquoHigh-performancestrategy of a simulated moving bed chromatography by si-multaneous control of product and feed streams undermaximum allowable pressure droprdquo Journal of Chromatog-raphy A vol 1471 pp 102ndash117 2016

[12] G S Weeden and N-H L Wang ldquoSpeedy standing wavedesign of size-exclusion simulated moving bed solventconsumption and sorbent productivity related to materialproperties and design parametersrdquo Journal of Chromatogra-phy A vol 1418 pp 54ndash76 2015

[13] J Y Song D Oh and C H Lee ldquoEffects of a malfunctionalcolumn on conventional and FeedCol-simulated moving bedchromatography performancerdquo Journal of ChromatographyA vol 1403 pp 104ndash117 2015

[14] A S A Neto A R Secchi M B Souza et al ldquoNonlinearmodel predictive control applied to the separation of prazi-quantel in simulatedmoving bed chromatographyrdquo Journal ofChromatography A vol 1470 pp 42ndash49 2015

[15] M Bambach and M Herty ldquoModel predictive control of thepunch speed for damage reduction in isothermal hot form-ingrdquo Materials Science Forum vol 949 pp 1ndash6 2019

[16] S Shamaghdari and M Haeri ldquoModel predictive control ofnonlinear discrete time systems with guaranteed stabilityrdquoAsian Journal of Control vol 6 2018

[17] A Wahid and I G E P Putra ldquoMultivariable model pre-dictive control design of reactive distillation column forDimethyl Ether productionrdquo IOP Conference Series MaterialsScience and Engineering vol 334 Article ID 012018 2018

[18] Y Hu J Sun S Z Chen X Zhang W Peng and D ZhangldquoOptimal control of tension and thickness for tandem coldrolling process based on receding horizon controlrdquo Iron-making amp Steelmaking pp 1ndash11 2019

[19] C Ren X Li X Yang X Zhu and S Ma ldquoExtended stateobserver based sliding mode control of an omnidirectionalmobile robot with friction compensationrdquo IEEE Transactionson Industrial Electronics vol 141 no 10 2019

[20] P Van Overschee and B De Moor ldquoTwo subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo in Proceedings of the 31st IEEE Conference on De-cision and Control Tucson AZ USA December 1992

[21] S Hachicha M Kharrat and A Chaari ldquoN4SID and MOESPalgorithms to highlight the ill-conditioning into subspace


identificationrdquo International Journal of Automation andComputing vol 11 no 1 pp 30ndash38 2014

[22] S Baptiste and V Michel ldquoK4SID large-scale subspaceidentification with kronecker modelingrdquo IEEE Transactionson Automatic Control vol 64 no 3 pp 960ndash975 2018

[23] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralizedpredictive control-part 1 e basic algorithmrdquo Automaticavol 23 no 2 pp 137ndash148 1987

[24] C E Garcia and M Morari ldquoInternal model control Aunifying review and some new resultsrdquo Industrial amp Engi-neering Chemistry Process Design and Development vol 21no 2 pp 308ndash323 1982

[25] B Ding ldquoRobust model predictive control for multiple timedelay systems with polytopic uncertainty descriptionrdquo In-ternational Journal of Control vol 83 no 9 pp 1844ndash18572010

[26] E Kayacan E Kayacan H Ramon and W Saeys ldquoDis-tributed nonlinear model predictive control of an autono-mous tractor-trailer systemrdquo Mechatronics vol 24 no 8pp 926ndash933 2014

[27] H Li Y Shi and W Yan ldquoOn neighbor information utili-zation in distributed receding horizon control for consensus-seekingrdquo IEEE Transactions on Cybernetics vol 46 no 9pp 2019ndash2027 2016

[28] M Farina and R Scattolini ldquoDistributed predictive control anon-cooperative algorithm with neighbor-to-neighbor com-munication for linear systemsrdquo Automatica vol 48 no 6pp 1088ndash1096 2012

[29] J Hou T Liu and Q-G Wang ldquoSubspace identification ofHammerstein-type nonlinear systems subject to unknownperiodic disturbancerdquo International Journal of Control no 10pp 1ndash11 2019

[30] F Ding P X Liu and G Liu ldquoGradient based and least-squares based iterative identification methods for OE andOEMA systemsrdquo Digital Signal Processing vol 20 no 3pp 664ndash677 2010

[31] F Ding X Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET Control9eory amp Applications vol 7 no 2 pp 176ndash184 2013

[32] F Ding ldquoDecomposition based fast least squares algorithmfor output error systemsrdquo Signal Processing vol 93 no 5pp 1235ndash1242 2013

[33] L Xu and F Ding ldquoParameter estimation for control systemsbased on impulse responsesrdquo International Journal of ControlAutomation and Systems vol 15 no 6 pp 2471ndash2479 2017

[34] L Xu and F Ding ldquoIterative parameter estimation for signalmodels based on measured datardquo Circuits Systems and SignalProcessing vol 37 no 7 pp 3046ndash3069 2017

[35] L Xu W Xiong A Alsaedi and T Hayat ldquoHierarchicalparameter estimation for the frequency response based on thedynamical window datardquo International Journal of ControlAutomation and Systems vol 16 no 4 pp 1756ndash1764 2018

[36] F Ding ldquoTwo-stage least squares based iterative estimationalgorithm for CARARMA system modelingrdquo AppliedMathematical Modelling vol 37 no 7 pp 4798ndash4808 2013

[37] L Xu F Ding and Q Zhu ldquoHierarchical Newton and leastsquares iterative estimation algorithm for dynamic systems bytransfer functions based on the impulse responsesrdquo In-ternational Journal of Systems Science vol 50 no 1pp 141ndash151 2018

[38] M Amanullah C Grossmann M Mazzotti M Morari andM Morbidelli ldquoExperimental implementation of automaticldquocycle to cyclerdquo control of a chiral simulated moving bed

separationrdquo Journal of Chromatography A vol 1165 no 1-2pp 100ndash108 2007

[39] A Rajendran G Paredes and M Mazzotti ldquoSimulatedmoving bed chromatography for the separation of enantio-mersrdquo Journal of Chromatography A vol 1216 no 4pp 709ndash738 2009

[40] P V Overschee and B D Moor Subspace Identification forLinear Systems9eoryndashImplementationndashApplications KluwerAcademic Publishers Dordrecht Netherlands 1996

[41] T Katayama ldquoSubspace methods for system identificationrdquoin Communications amp Control Engineering vol 34pp 1507ndash1519 no 12 Springer Berlin Germany 2010

[42] T Katayama and G PICCI ldquoRealization of stochastic systemswith exogenous inputs and subspace identification method-sfication methodsrdquo Automatica vol 35 no 10 pp 1635ndash1652 1999


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reduced-order model of SMB so as to reduce the compu-tational complexity of the model and achieve the desiredoptimization results [6] e front velocity method wasproposed for SMB process modeling in order to remove themethod that adsorption isotherm was obtained by means ofequilibrium experiments in the traditional SMB modelingprocess en the feasibility of the model was verified byexperiments [7] Aiming at the existing problems of thediscretization model by utilizing the finite-dimensionalpartial differential equations two Krylov-type reduced-or-der models were proposed to optimize the computationalaccuracy and computational efficiency of the model [8] emain control objective of the SMB chromatographic sepa-ration process is to increase the target yield of the SMBseparation process Many papers have made relevant re-search on process control methods e genetic algorithmwas adopted to optimize the parameters in the SMB sepa-ration process [9] e standing wave design method wasproposed to improve the purity and yield of the separatedcomponents while reducing the consumption of the ana-lytical solution [10] e flow rate control strategy wasadopted where two operational variables were added in eachswitching cycle and the SMB separation performance wasimproved by the SimCon operation [11] Because the tra-ditional numerical optimization method has many un-certainties the system parameters of size exclusionsimulated moving bed (SEC-SMB) were studied by batchchromatography [12] e impact of the fault column on theSMB separation process was analyzed through experimentsand theoretical analysis [13] An adaptive nonlinear modelpredictive control method was proposed for an SMB deviceto realize the praziquantel enantiomer separation Under theset purity levels the controller can respond quickly [14]Some scholars have carried out research and application onthe control methods of MPC Markus Bambach andMichaelHerty had studied to control the strain rate during theprocess using model predictive control [15] Saeed Sha-maghdari andMohammadHaeri had discussed the design ofa new robust model predictive control algorithm for non-linear systems [16] Wahid A et al study investigates amultivariable model predictive control (MPC) based on two-point temperature control strategy [17] Some articles on thestudy of process control methods are also discussed YunjianHu et al proposed an optimal tension and thickness controlmethod based on the receding horizon control (RHC)strategy to improve control performance and enhancesystem robustness [18] Ren et al studied a reduced-orderextended state observer- (ESO-) based sliding mode controlscheme for friction compensation of a three-wheeled om-nidirectional mobile robot [19]

In this paper a predictive control method based on thestate-space yield models of SMB chromatographic separationprocess obtained by utilizing the subspace system identifi-cation method is proposed e subspace system identifica-tion is an identification algorithm based on the input-outputdata matrix by means of mathematical tools such as ge-ometry matrix analysis and probability statistics so as toobtain the state-space matrix in state-space equations Manypapers have proposed many model identification methods for

complex process models van Overschee and de Moor elab-orated on the multivariable output-error state-space(MOESP) and the numerical algorithms for subspace state-space system identification (N4SID) [20] Hachicha et alcompared the morbid conditions of MOESP and N4SID andconfirmed that they are robust in the parameter estimationprocess of the controlled object [21] e discrete state-spaceequation was vectorized and scaled to solve the bilinear low-rank minimization problem [22] e subspace identificationmethod was applied to the fault diagnosis of the wind turbineshaft [22] Model predictive control is an advanced optimi-zation control method composed of the predictive modelrolling optimization and feedback correction Clarke de-veloped a controlled autoregressive integral moving averagemodel (CARIMA) [23] Garic and Morari proposed the in-ternal model control (IMC) algorithm [24] Ding made arelated study on the robustness ofmultiple time-delay systems[25] Kayacan et al successfully solved the trajectory trackingproblem of the self-made tractor trailer system by using thefast distributed nonlinear model predictive control algorithm[26] Li et al designed a distributed predictive controller tosatisfy the stability conditions by using a distributed imple-mentation [27] Farina and Scattolini proposed a new dis-tributed predictive control algorithm for linear discrete-timesystems [28] Some newly published papers also propose newmethods for model identification Hou et al designed asubspace identification method which is proposed for theHammerstein-type nonlinear systems subject to periodicdisturbances with unknown waveforms [29] Gradient-basedand least-squares-based iterative identification algorithms aredeveloped for output-error (OE) and output-error movingaverage (OEMA) systems and the simulation results confirmtheoretical findings [30] Ding et al proposed an algorithmabout a least-squares-based and a gradient-based iterativealgorithm for the Hammerstein nonlinear systems using thehierarchical identification principle [31] Ding used an iter-ative least-squares algorithm to estimate the parameters ofoutput-error systems and enhance computational efficiencies[32] Xu and Ding discussed the parameter estimationproblems for dynamical systems by means of the impulseresponse measurement data [33] Xu and Ding used thehierarchical identification principle and the iterative search tostudy the modeling of multifrequency signals based onmeasured data [34] e signal modeling methods based onstatistical identification are proposed by means of the dy-namical window discrete measured data [35] Ding proposedamethod for identifying the systemmodel parameters and thenoise model parameters for stochastic systems described bythe CARARMA models [36] Xu et al used the hierarchicalprinciple and the parameter decomposition strategy to de-velop a parameter estimation algorithm for linear continuous-time systems [37]

e structure of the paper is organized as followsSection 2 introduces the technique of SMB chromatogra-phy separation process Section 3 introduces the twosubspace system identification algorithms (MOESP andN4SID) Section 4 introduces the structure of the modelpredictive control algorithm and the predictive controlmethod based on the state-space model Section 5 shows







Eluent

Extract liquid



Section I


Section III



A

B

Switching direction

Section III

Section I

Sect

ion

II

Sect

ion

IV


A + B

Entranceliquid

Raffinate

(a)

B

Switching direction

Section III

Section ISe

ctio

n II

Sect

ion

IV

Eluent

A

Extractliquid

A + B

Entranceliquid

Raffinate

(b)






R

M

E

UV-Vis

UV-Vis

UV-Vis

LR

LM

LE

Oi

Hi

i

Ii

Pi Di Fi

Ti

Pump 1

Pump 2

Pump 3

LF

LD

LP

F

D

PA

PB

Ri Mi Ei



Flow rate(F pump)

Flow rate(D pump)

Switchingtime


Impuritypurity


Impurityyield






0 500 1000 150001

02

03

04

05m

lmin

0 500 1000 150005

1

15

2

mlm

in

0 500 1000 1500

min

10

15

20

0 500 1000 1500

Yiel

d

0

02

04

06

08

1

Yiel

d

0 500 1000 15000

02

04

06

08

1








Smooth


Model

Currentcontrol

Future control


Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)



y(t)u(t)


MPC parameters



Set point

MPC controller



100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018













Eluent

Extract liquid



Section I


Section III



A

B

Switching direction

Section III

Section I

Sect

ion

II

Sect

ion

IV


A + B

Entranceliquid

Raffinate

(a)

B

Switching direction

Section III

Section ISe

ctio

n II

Sect

ion

IV

Eluent

A

Extractliquid

A + B

Entranceliquid

Raffinate

(b)






R

M

E

UV-Vis

UV-Vis

UV-Vis

LR

LM

LE

Oi

Hi

i

Ii

Pi Di Fi

Ti

Pump 1

Pump 2

Pump 3

LF

LD

LP

F

D

PA

PB

Ri Mi Ei



Flow rate(F pump)

Flow rate(D pump)

Switchingtime


Impuritypurity


Impurityyield






0 500 1000 150001

02

03

04

05m

lmin

0 500 1000 150005

1

15

2

mlm

in

0 500 1000 1500

min

10

15

20

0 500 1000 1500

Yiel

d

0

02

04

06

08

1

Yiel

d

0 500 1000 15000

02

04

06

08

1








Smooth


Model

Currentcontrol

Future control


Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)



y(t)u(t)


MPC parameters



Set point

MPC controller



100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018











R

M

E

UV-Vis

UV-Vis

UV-Vis

LR

LM

LE

Oi

Hi

i

Ii

Pi Di Fi

Ti

Pump 1

Pump 2

Pump 3

LF

LD

LP

F

D

PA

PB

Ri Mi Ei



Flow rate(F pump)

Flow rate(D pump)

Switchingtime


Impuritypurity


Impurityyield






0 500 1000 150001

02

03

04

05m

lmin

0 500 1000 150005

1

15

2

mlm

in

0 500 1000 1500

min

10

15

20

0 500 1000 1500

Yiel

d

0

02

04

06

08

1

Yiel

d

0 500 1000 15000

02

04

06

08

1








Smooth


Model

Currentcontrol

Future control


Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)



y(t)u(t)


MPC parameters



Set point

MPC controller



100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018











0 500 1000 150001

02

03

04

05m

lmin

0 500 1000 150005

1

15

2

mlm

in

0 500 1000 1500

min

10

15

20

0 500 1000 1500

Yiel

d

0

02

04

06

08

1

Yiel

d

0 500 1000 15000

02

04

06

08

1








Smooth


Model

Currentcontrol

Future control


Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)



y(t)u(t)


MPC parameters



Set point

MPC controller



100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018













Smooth


Model

Currentcontrol

Future control


Future output

u(k + 1)

y(k + 1)

y(k + N)

u(k + Nu ndash 1)

y(k)

ω(k + d)

yr(k + 1)

yr(k + N)



y(t)u(t)


MPC parameters



Set point

MPC controller



100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018








100


ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

150 200 250 300 3500 50

(a)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(b)

Figure 7 Continued



100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018









100 150 200 250 300 3500 50ndash02

0

02

04

06

08

1

12Ta

rget

subs

tanc

e yie

ld

(c)


100 150 200 250 300 3500 5001

02

03

04

05

06

07

08

09

1

11

Impu

rity

yiel

d

(d)






xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018











xk+1 Axk + Buk

yk Cxk + Duk(1)



Up def

Uo|iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Uf def

Ui|2iminus 1



⋮ ⋮ ⋮ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)



Yp def

Yo|iminus 1

Yf def

Yi|2iminus 1

Wp def Yp

Up

⎡⎣ ⎤⎦

(3)


Xi xi xi+1 xi+jminus 11113872 1113873 (4)





Hi



⋮ ⋮ ⋱ ⋮


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)


Oi def Yf

Wp

Uf1113890 1113891

(7)


Oi ΓiXfWp

Uf1113890 1113891

(8)


Oi USVΤ (9)





Uf

Wp

Yf

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

L11 0 0

L21 L22 0

L31 L32 L33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

QΤ1

QΤ2

QΤ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (10)


L32 Um1 Um

2( 1113857Sm1 0

0 01113888 1113889

Vm1( 1113857Τ

Vm2( 1113857Τ

⎛⎝ ⎞⎠ (11)


Γi Um1 middot S

m1( 1113857

12 (12)



Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018








Time (s)

ndash6ndash4ndash2

02

m = 1









ndash6ndash4ndash2

02

m = 3

ndash4

ndash2

0

ndash4

ndash2

0

m = 5

m =10

0 5 10 15 20 25 30

15 20 305 25100Time (s)

15 20 305 25100Time (s)

5 10 250 15 3020Time (s)

(a)

Figure 8 Continued




Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018










Xf AiXp + ΔiUp

Yp ΓiXp + HiUp

Yf ΓiXf + HiUf

(13)


W1OiW2 U1 U21113858 1113859S1 0

0 01113890 1113891

VΤ1

VΤ2

⎡⎣ ⎤⎦ U1S1VΤ1 (14)















ndash4

ndash2

0

ndash6ndash4ndash2

02

ndash6ndash4ndash2

02

m = 1

m = 3

m = 5

ndash4

ndash2

0

m = 10

5 10 15 20 25 300Time (s)

15 20 305 25100Time (s)

5 100 20 25 3015Time (s)

5 10 15 20 25 300Time (s)

(b)



ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018








ndash4

ndash2

0

P = 52

ndash4ndash2

02 P = 10

ndash4ndash2

02

ndash4ndash2

02

P = 20

P = 30

100 150500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

454015 20 3530 500 105 25Time (s)









(a)

Figure 9 Continued





J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018











J E 1113944

N2

jN1

y(k + j) minus yr(k + j)1113858 11138592

+ 1113944

Nu

j1cjΔu(k + j minus 1)1113960 1113961

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(16)



ndash4

ndash2

0

2

ndash4

ndash2

0

2

ndash4ndash2

02

ndash4ndash2

02

P = 5

P = 10

P = 20

P = 30

50 100 1500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

105 15 20 25 30 35 40 45 500Time (s)









(b)



ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018








ndash10ndash5

0

ndash4ndash2

0

ndash20ndash10

0uw = 05

uw = 1

uw = 5

ndash2ndash1

01

uw = 10

0 2 16 20184 146 10 128Time (s)

184 10 122 6 14 16 200 8Time (s)

2 166 14 200 108 184 12Time (s)

104 8 12 182 14 166 200Time (s)









(a)

Figure 10 Continued







ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018













ndash4

ndash2

0

ndash20

ndash10

0

uw = 05

ndash10

ndash5

0uw = 1

uw = 5

ndash2ndash1

01

uw = 10

2 4 6 8 10 12 14 16 18 200Time (s)

14 166 8 10 122 4 18 200Time (s)

642 8 10 12 14 16 18 200Time (s)

2 4 6 8 10 12 14 16 18 200Time (s)









(b)



ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018








ndash2

0

2yw = 10

ndash6ndash4ndash2

0yw = 200

ndash10ndash5

0yw = 400

ndash20

ndash10

0yw = 600

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)









(a)

Figure 11 Continued



yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018









yc(k) Ccx(k)(17)







ndash2

0

2

ndash6ndash4ndash2

0

ndash10

ndash5

0

ndash20

ndash10

0

yw = 10

yw = 200

yw = 400

yw = 600

350100 150 200 250 300 4000 50Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)

50 100 150 200 250 300 350 4000Time (s)









(b)




ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018









ndash04

ndash02

0

02

04

06

08

1

12O

bjec

tive y

ield

400 500300 6000 200100

(a)


100 200 300 400 500 6000ndash04

ndash02

0

02

04

06

08

1

12

Obj

ectiv

e yie

ld

(b)







A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018












A3Δx(k) + A


2BdΔd(k)


AmΔx(k) + A



mminus 1BdΔd(k)

⋮


ApΔx(k) + A




pminus 1BdΔd(k)

(19)


yc k + 1 | k) CcΔx(k + 1 | k( 1113857 + yc(k)


yc(k + 2 | k) CcΔx(k + 2| k) + yc(k + 1 | k)

CcA2




1113944m

i1CcA

iΔx(k) + 1113944m

i1CcA


mminus 1

i1CcA



i1CcA


⋮


1113944

p

i1CcA

iΔx(k) + 1113944

p

i1CcA


pminus 1

i1CcA



pminus m+1

i1CcA


p

i1CcA


(20)


Yp(k + 1) def

yc(k + 1 | k)

yc(k + 2 | k)

⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(21)


ΔU(k) def

Δu(k)

Δu(k + 1)

⋮

Δu(k + m minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)




(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018










(23)

where

Sx

CcA

1113944

2

i1CcA

i

⋮

1113944

p

i1CcA

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

I

Inctimesnc

Inctimesnc

⋮

Inctimesnc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Sd

CcBd

1113944

2

i1CcA

iminus 1Bd

⋮

1113944

p

i1CcA

iminus 1Bd

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Su


1113944

2

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

m

i1CcA

iminus 1Bu 1113944

mminus 1

i1CcA


⋮ ⋮ ⋮ ⋮ ⋮

1113944

p

i1CcA

iminus 1Bu 1113944

pminus 1

i1CcA


pminus m+1

i1CcA

iminus 1Bu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)


J 1113944

p

i11113944

nc

j1Γyj




2

+ ΓuΔU(k)

2

(26)



R(k + 1)

r(k + 1)

r(k + 2)

⋮

r(k + 1)p

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimes1

(27)


ΔUlowast(k) STuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓyEp(k + 1 | k)

(28)

where


(29)



ItimesmΔUlowast(k)


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1

middot STuΓ

TyΓyEp(k + 1 | k)

(30)

Let


ItimesmS

TuΓ

TyΓySu + ΓTuΓu1113872 1113873

minus 1S

TuΓ

TyΓy

(31)


Δu(k) KmpcEp(k + 1 | k) (32)











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018











A

09759 03370 02459

minus 00955 06190 12348

00263 minus 04534 06184

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


36586 00572

162554 00406

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729

minus 02218 02143 minus 00772⎡⎢⎣ ⎤⎥⎦


minus 38530 minus 00090⎡⎢⎣ ⎤⎥⎦

(33)


A

09758 03362 02386 03890

minus 00956 06173 12188 07120

00262 minus 04553 06012 12800


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B


minus 192690 00527

minus 78163 00170

130644 00242

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C minus 03750 02532 minus 01729 00095

minus 02218 02143 minus 00772 01473⎡⎢⎣ ⎤⎥⎦

D 07706 minus 00190

minus 37574 minus 00035⎡⎢⎣ ⎤⎥⎦

(34)


A

09797 minus 01485 minus 005132

003325 0715 minus 01554

minus 0001656 01462 09776

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

02536 00003627

0616 0000211


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C 1231 5497 minus 1402

6563 4715 minus 19651113890 1113891

D 0 0

0 01113890 1113891

(35)


A

0993 003495 003158 minus 007182

minus 002315 07808 06058 minus 03527


006837 02852 01935 08489

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

01058 minus 0000617

1681 0001119

07611 minus 000108


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



D 0 0

0 01113890 1113891

(36)













6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018
















6 Conclusions




Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018









Data Availability






Acknowledgments


References






















































Journal of












Journal of







Volume 2018






Volume 2018







































Journal of












Journal of







Volume 2018






Volume 2018








model predictive control method of simulated moving bed...

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