A Multilevel Inverter Bridge Control Structure With Energy Storage Using Model Predictive Control...

8/12/2019 A Multilevel Inverter Bridge Control Structure With Energy Storage Using Model Predictive Control for Flat Systems

1/16

Hindawi Publishing CorporationJournal o EngineeringVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleA Multilevel Inverter Bridge Control Structure with EnergyStorage Using Model Predictive Control for Flat Systems

Paolo Mercorelli

Leuphana University of Lueneburg, Institute of Product and Process Innovation, Volgershall , Lueneburg, Germany

Correspondence should be addressed to Paolo Mercorelli; [email protected]

Received September ; Accepted December

Academic Editor: Claudio Mazzotti

Copyright Paolo Mercorelli. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Te paper presents a novel technique to control the current o an electromagnetic linear actuator ed by a multilevel IGB voltageinverter with dynamic energy storage. Te technique uses a cascade model predictive control (MPC), which consists o two MPCs.A predictive control o the trajectory position predicts the optimal current, which is considered to be the desired current or thesecond MPC controller in which a hysteresis control technique is also integrated. Energy is stored in a capacitor using energyrecovery. Te current MPC can handle a capacitor voltage higher than the source voltage to guarantee high dynamic current anddisturbance compensation. Te main contribution o this paper is the design o an optimal control structure that guarantees acapacitor recharge. In this context, the approach is quite new and can represent a general emerging approach allowing to reducethe complexity o the new generation o inverters and, in the meantime, to guarantee precision and acceptable switching requency.

Te proposed technique shows very promising results through simulations with real actuator data in an innovative transportationtechnology.

1. Introduction and Motivation

Te perormance o a multilevel inverter is better than thato a classical inverter. Te total harmonic distortion o aclassical inverter is very high; that is, the total harmonicdistortion or a multilevel inverter is low. Tis topic hasbecome the ocus o a signicant amount o research. Onestudy has simulated and implemented a multilevel inverter-

ed induction motor drive []. Te output harmonic contentis reduced by using a multilevel inverter. In a symmetricalcircuit, the voltage andpower increase as the number o levelsin the inverter increases. Te switching angle or the pulse isselected to reduce the harmonic distortion. Tis drive systemhas several advantages, including reduced total harmonic dis-tortion and higher torque. A normal neutral point potentialstabilization technique using the output current polarity hasbeen proposed []. Te neutral point potential balancingalgorithm or three-level neutral point clamped invertersusing an analytically injected zero-sequence voltage has beendeveloped []. Modulation schemes to eliminate commonmode voltage in multilevel inverter topology have been

suggested[]. A generalized multilevelinvertertopology withsel-voltage balancing hasalso beensuggested [].Asurveyotopologies, control, and applications o multilevel invertershas been published []. A digital modulation techniqueor dual three-phase alternating current (AC) machines hasbeen presented []. A space vector pulse-width modulation(PWM) technique or a dual three-phase AC machine and itsdigital signal processor (DSP) implementation has been pre-

sented []. Practical medium voltage converter topologies orhigh power applications have been described previously []. Industrial topologies have been presented []. Differentpulse-width modulation techniques or symmetrical cascadeinverters with high and undamental switching requencyhave been shown, or instance, in [, ]. An improvement interms o efficiency o the converter is presented in [] wherethe authors use different DC sources to reduce the switchinglosses. More recently in [], the authors proposed a novelH-bridge multilevel pulse modulation converter topology inwhich the structure is based on a series connection o high-

voltage diode-clamped inverter and low-voltage conventionalinverter. Te present work implements a controlled multilevel


2/16

Journal o Engineering

inverter eeding an electromagnetic actuator. In particular,two MPCs in cascade structure are presented to control ve-level inverter electromagnetic actuators which are becom-ing increasingly important in many industrial applications.Tanks to recent progress in permanent-magnet technology,

very compact and high-power electromagnetic actuators are

now available. Particularly in automotive systems, these actu-ators have ofen replaced conventional mechanical compo-nents due to their high efficiency, excellent dynamic behavior,and control exibility; thus, mechatronics is one o the mostnotable innovative elds in the automobile industry. A recentdesign o such an actuator (e.g., []) considers optimizingwith respect to dimensions and orces. Nevertheless, practicaltests show that this actuator structure is not always effectiveor a large variety o applications because a complex controlstructure is needed to control such a valve []. Recently,or instance, a new type o electromagnetic valve drivesystem has been proposed and described []. Tis paperpresents the control design or a novel permanent-magnetlinear valve actuator or use in a variable engine control toallow short-stroke and high-dynamic motions. In a linearactuator, the electromagnetic orce is totally independento the constrained motion. Specically, the electromagneticorce is parallel to the magnetic axis, and thus it is orthogonalto the constrained motion. Tese perormance character-istics make this type o actuator an excellent candidate inapplications requiring both high precision and high-loadcapacity. In order to obtain an actuator structure that couldbe easily controlled, various linear designs with differentpermanent magnet actuator topologies using NdFeB magnetswere considered. In particular, the objective o this paper is todesign a controller structure consisting o

(i) a atness-based eedorward control,

(ii) a multilevel inverter conguration,

(iii) a cascade MPC structure based on optimized ener-getic unctions to control the proposed multilevelinverter with storage energy.

An important aspect o this approach is the use o geometricsystem properties, such as the differential atness neededto track the desired trajectories. In this context, recentlypublished research results have been considered (e.g., [])in which the author proposed an interesting method tointegrate MPC anda eedorward atness control. Te second

part o the paper shows the structure o the cascade MPCdened above. For trajectory-based motion cycles, there isusually a need to control the actuator current separately. Tereerence value or the actuator current is normally generatedby the speed or position controller in an outer control loopbased on the desired speed or position proles. Te currentcontrol is used to provide the orce or torque required or thedesired motion quickly and precisely. Generally, three maintechniques are employed or current control o a voltage-source inverter (VSI):

(i) closed loop control (e.g., PID control) using pulse-width modulated terminal voltages [],

(ii) hysteresis control techniques [,],

(iii) predictive current control [].

Te PWM technique is currently the most popular methodor an inverter current control. Te main advantage oPWM techniques is that the inverter switches operate at a

xed requency, and hardware- or sofware-based standardmodulators are available as industrial products (e.g., on-board microcontrollers). A staggered space vector modula-tion technique applicable to three-phase cascaded voltage-source inverter topologies has also been demonstrated witha single-phase cascaded voltage-source inverter that uses aseries connection o insulated gate bipolar transistor (IGB)H-bridge modules with isolated DC buses []. Among

various modulation techniques or a multilevel inverter, thespace vector pulse width modulation (SVPWM) is widelyused. However, implementing the SVPWM or a multilevelinverter is complicated because it is difficult to determinethe location o the reerence vector, calculate on times, and

determine and select switching states. A previous paperhas proposed a general SVPWM algorithm or multilevelinverters based on standard two-level SVPWM. However, thesystem response is affected by the stability requirements othe eedback loop, which also depend on load parameters;moreover, appreciable phase lag may arise even in the steadystate. Recently, many studies have ocused on solving theseproblems (e.g., [,]) o multilevel inverter modules withindependent control o the phase angle and magnitude o theoutput voltage []. Control andnew power bridge structureshave been studied. Very recent works have presented a novelbridge structure [, ]. In both structures, capacitors areused but not recharged. In particular, an MPC strategy isproposed or a three-phase power bridge []. Hysteresiscurrent control has a ast response and a good accuracy.It can be implemented with a simple hardware structure,and, in many cases, it does not require any knowledgeo load parameters. However, it can sometimes cause veryhigh switching requencies or, i the maximum switchingrequency is limited, the current waveorm may vary widelyand the current peaks may appreciably exceed the hystere-sis band depending on the operation conditions and loadparameters. Conventional predictive current control uses asimple gradient model to predict load current in vector spaceand determine the proper switching voltage vector basedon the one-step prediction []. However, no eedbackloop is applied to compensate or model uncertainties. o

overcome the disadvantages o the mentioned methods, thisstudy develops a novel approach that combines hysteresiscontrol with a model predictive control (MPC) strategy. Teproposed control system integrates atness and two cascadeMPCs. Te rst MPC generates an optimal desired currentby minimizing position error. Te second MPC considersthis optimal desired current to be the reerence signal tond an optimal switching law in combination with thehysteresis control. A special energy storage and chargingcircuit was designed to provide multilevel voltages or aneffective current control. By applying the proper voltagelevel, which was determined by the MPC strategy, to theactuator, dynamic current changes and small current ripples


3/16


Coil

Valve

Lower spring

Iron pole

Permanent magnet

Upper spring

F : Cross-section o the perpendicular linear actuator.

are possible in spite o relatively low switching requencies.A recent work [] proposes ast algorithms based oncomputing the ormulated problems in parallel. In act, the

optimization technique uses subsets o the state subspaceoffline and then optimizes them in parallel. In this paper, acascade structure o MPCs is utilized to control a multilevelinverter. wo different optimization techniques are presentedas well. Te main contribution o the paper is the optimalcontrol structure, which also guarantees that the capacitorrecharges. In this context, the approach is quite new. Tepaper is organized in the ollowing way.

Section describes the model. InSection , the atnessproperty o the system is shown. In Section , the generalMPC goal is dened and the main idea o using a cascadepositional MPC ollowed by a current MPC is explained. InSection , a positional MPCaroundthe desired trajectory isproposed.Section is devoted to analyzing and optimizingthe current MPC. Te simulation results and some conclud-ing remarks end the paper.

2. Description of the Physical Systems

Te electromagnetic actuator is depicted in Figure . Testructure o this actuator is a moving coil; thus, the coilsare mounted on the upper part o the stem o the valve.Te moving valve was connected to the power system toeed the coil with two normal cables with x contacts. Tisarrangement is possible because o the short stroke to becovered ( mm). Te valve can be modeled mathematically

in the ollowing way:

coil() = coilcoil coil()+in() coil() , ()coil , ()

() = () , () () =

coil() , () coil()+ () ()+ 0 ()

,

()

where

coil() , () = coil() , () coilcoil , coil() , () = coil() , () () .

()

It is important to note that(coil(), ()) > 0or allcoil()and or all().coil andcoil are the resistance and theinductance o the coil windings;in() is the input voltage;() is the induced back voltage; is the magnetic uxpenetrating the coil;coil()is the coil current;coilis the coillength.(),(), andare the position, velocity, and masso the actuator, respectively, while(),(), and0()represent the viscose riction, the total spring orce, and thedisturbing orce acting on the valve. Equation () representsthe electrical system o the actuator. Both ()and () describethe mechanical behavior o the actuator, and () and () alsorepresent the magnetic system. In particular, the ollowingexpression

()= coil() , () coil() ()describes the Lorentz orce generated by the actuator. Essen-tially, the magnetic ux, generated by the permanent mag-nets, has two components in the air gap: () the main ux(coil()), which does not depend on the displacemento the mover and is responsible or the Lorentz orceand the induced back voltage, and () the leakage ux(coil(),()), which disperses around the coil and doesnot contribute to the electromagnetic orce and induced back

voltage:

coil() + coil() , () = PMcoil() . ()

However, due to the special actuator design, the leakage uxis almost equal to zero. Tus, it is possible to conclude

coil() , () coil() =PMcoil() coilcoil . ()3. Differential Flatness of the System

Roughly speaking, a system is differentially at i it is possibleto nd a set o outputs equal in number to the number oinputs such that all states and inputs are expressed in terms othose outputs and their derivatives. o be more precise, i thesystem has state variablesx R and inputsu R, thenthe system is at i the outputs y R have the ollowingorm:

y=yx, u,u, . . . ,u() , ()x=xy, y, . . . ,y() ; u=u y, y, . . . ,y(+1) . ()

Differentially at systems are especially interesting insituations in which explicit trajectory tracking is required.Because the behavior o the at system is given by the atoutput, it is possible to plan trajectories in output space andthen map them to the appropriate inputs. Concerning the


4/16


PM

PM

F : Permanent-magnet demagnetization curve.

at output, or dating neither the necessary and sufficientcondition nor a general method or the determination othe at output has been provided. ypically, a at output is

guessed, and Denitions () and () are used to veriy thatthe chosen output is at. Once the system is proven to bedifferentially at, the basic approach o two degrees o ree-dom controller design consists o two steps: rst, separatingthe nonlinear controller synthesis problem into designing aeasible eedorward controlled trajectory or the nominalmodel o the system and second regulating that trajectoryusing controllers, that guarantee robust perormance in thepresence o uncertainties and disturbances. o explain themeaning o atness intuitively, it is adequate to consider asystem to be at when the dynamic o at least one output isvisible rom the state andthe inputo the system; thus, thereis at least oneoutput that canbe unctionally controlled by theinputs as well by the states.

Now the rst step is to show that our model representedin (), (), and () is at. I the positiono the moving parto the actuator is chosen as guessed output, then

()= () ()= () ()= () ()= () , ()

coil() coil() = ()+ ()+ () ,

()

in

()= coil

coil() +

coil

coil

()+

coil

() () .()o veriy the atness property it is necessary to look at(). As evident rom (),(coil())is proportional to theux coupled with the permanent magnets which is againproportional to the ux densityPM . Te operating point(PM , PM)o the permanent magnets is mainly determinedby the magnet geometry and the demagnetization curve(Figure ). Te coil current only changes the eld strengtharound the operating point() algebraically. An exactanalytical expression betweenandcoil()is very difficultto be derived because it also depends on the saturation level othe iron parts. However, a numerical solution exists. In other

Linear motorInverter bridgeEnergy storePower source

Us

Uc

D1

CZ1

V2

V3

V1V5

V4

R

L

uq+

F : Multilevel inverter bridge structure.

words, there is always a (locally) unique solution orcoil()that depends oncoil(). Tus, using (), the coil currentcan always be determined uniquely by the actuator positionand its derivatives. Considering additional system equations,it can be easily seen that the atness conditions dened in() are satised. Relationships () and () dene the inversesystem. Based on the atness o the system, the linearizingtrajectory o the model () and () is as ollows:

()= () ()+

()+ () , ()where

()is the desired trajectory and

() is the corre-

sponding eed orward control current. As it was explainedbeore, the ux does not depend on the position o thearmature because the leakage ux is almost equal to zero.Tis means that the ux is a unction o the current. Function(coil()) is never equal to zero, at least or our conceivedstructure, because o controllability. Te actuator is conceivedanddesigned in a wayin order to guarantee the controllabilityat any point o its movement. For (), the inverse systemthat determines the desired eedorward control input()is dened as ollows:

()= coil () + coil ()

+ () , () () (). ()

Based on the eed orward control presented previously, thenext step is to build a eedback control law that, in thepresence o external disturbances and uncertainties in theparameters, takes the system around the desired trajectory.

4. General MPC Problem Formulation

Figure shows the power electronic circuit eeding thepermanent magnet linear actuator. Te power supply is a


5/16


Current

FuturePast

Time

Desired currentMeasured current

Predicted currentNot used

0

2 + 2 +

F : Principle o the complete applied MPC strategy.

constant voltage source (battery). However, to realize acurrent change as ast as possible to ensure the requireddynamic, it is desirable to have a higher voltage appliedto the actuator. Tereore, a capacitor, charged with aninitial voltage higher than the battery level, is used. omaintain the high voltage level o the capacitor during the

valve operation, an energy storage block consisting o thecapacitor, the diode D, and the IGB switch V is used.Besides the current regulation, another task o the controlstrategy is to recharge the capacitor appropriately by utilizingthe braking energy. On the right, the IGB inverter bridgeand the linear actuator are represented. Model predictivecontrol has been a widely used control concept or over years, especially in the process industry. Applications oMPC in the eld o electrical drives are quite rare becauseo a high computational complexity. Te central point oa model predictive controller is a process model, which iscapable to predict uture output signals based on utureinput signals and initial values. Using this process model,the uture dynamic behavior o the real plant is predictedwithin a prediction horizon. Tese predicted output sig-nals can be used to minimize an open loop perormancecriterion (e.g., the sum o squared control errors withinthe prediction horizon) and to calculate the input signals()or a control horizon. Outside the control horizon, theinput remains constant. Te calculated input signals eedthe plant until a new measurement becomes available. Tisprocedure is repeated with a receding prediction and control

horizon. Te receding horizon strategy makes a closed loopcontrol law rom the originally open loop minimization. Teminimization step can easily include constraints, such asinput, output, or state constraints, that may be taken intoaccount already in the controller design. Te principle othe predictive control design or the current control problemconsidered in this paper is depicted in Figure with thedesired current (dashed-point line)thatresults rom the outerposition control loop and the measured current (solid line).At the present time, all possible currents or the uture twotime steps + + 2 are calculated. Tis calculationis perormed based on a multimode model and assessedaferwards with the help o a dened cost unction (details

ollow). Te optimum switching conguration is also shown(bold-dashed line).

Figure shows the whole proposed control structure.Te presented technique can be interpreted in the ollowingpoints.

(i) Te positional MPC structure generates an optimal

current trajectory with the desired optimal horizonthrough the electrical model.

(ii) Te optimal current trajectory is used as a desiredtrajectory or the current MPC.

(iii) Te current MPC makes it possible to calculate theswitch control system structure.

Figure shows the control system structure in which it ispossible to recognize the desired current()which is usedto linearize the system, the predicted reerence current opt(),and currentcoil().

5. Solving a Linear Position MPCOptimization Problem

o obtain the desired current trajectory, a position modelpredictive control is implemented. In this case, the sampletime is relatively short, s ( kHz), to make the algorithmas ast as possible. Considering the model in (), (), and ()in which0() = 0and an Euler discretization with = , , whereis the sampling time, the ollowing system isobtained:

coil( + 1)= coil()+ coil coilcoil()+ mpc()

+ () () ( + 1)= ()+ ()

( + 1)= ()+ coil() coil()+ () () ,

()

wherempc() is the model predictive control input to becalculated and () and () are the discretized voltages o()and() already dened in (), (), and (). I () isdiscretized, then

()= ( 1) ()+

()+ ( + 1) () .()

Considering the Euler discretization o (), which allows usto express the uture value o an output as a unction o the


6/16


Feedforwardcontrol

CurrentMPC

MultilevelIGBTbridge

DC-motor

systemPositionalMPC

Current

Switchingsignal

Electricalmodel

Desiredtrajectories

Desiredcurrent Obtained

trajectories

Obtainedtrajectories

Optimalcurrent

mpc

in

in

F : Te whole control scheme.

pass inputs, which is the required orm o the MPC approach,the ollowing equation is obtained:

()= coil ()+ coil ( + 1) () + () .

()

I () and () are inserted into the discretized equations (),(), and (), then the ollowing linear system is obtained:

coil( + 1)= coil() ()+ ( + 1) coilcoil coil() ()

mpc()coil ( + 1)= ()+ ()

( + 1)= ()+ ()+ ()

()+ () .

()

Impc() = mpc() mpc( 1)is assumed, the systemdescribed in () becomes

x( + 1)= Ax()+ B mpc()+ mpc( 1)+ Ed ()

()=Hx() ,()

where matrix H= 0 1 0 is the output matrix whichdetermines the position, and the previous notation means

A=

1 coilcoil 0 00 1

0 1

,

B=

coil00,

E=

1 coilcoil 1 0 0

0 0 0 00 0

,

d ()= ( + 1) () () ()

.

()

In the considered representation, vector d() is the knowninput. In the model approach, just two samples are consid-ered:

( + 1)= HAx()+HBmpc()+HBmpc( 1)

+HEd

() ,


7/16


( + 2)= HA2x()+HABmpc()+HABmpc( 1)+HBmpc( + 1)+HBmpc()+HAEd ()+HEd () .

()

I the ollowing perormance criterion is assumed,

=12=1

+ + Q + +

+

=1mpc

+

Rmpc + ,

()

where( + ), = 1,2 , . . . ,, is the position reerencetrajectory, the prediction horizon, and Q and R arenonnegative denite matrices, then the solution minimizingperormance index () may be then obtained by solving

mpc = 0. ()I only two steps or the prediction horizon are considered,then

G=HA

HA2

, F1= HB 0HAB+HB HB ,F2= HBHAB+HB ,

F3= HE 0HAE+HE HE .()

A direct computation may be obtained explicitly as

mpc()= F

1QF1+ R1

F1Q Y () G () F2mpc( 1) F3d () ,

()

whereY()is the desired output column vector. Once thepredicted optimized voltage is obtained, then it is possibleto obtain the predicted optimized current as the reerencecurrent or the current MPC based on the model o thesystem.

1

1

2

2

3

3

4

4

F : Working phases one and two (ed by).

1

1

2

2

3

3

4

4

F : Working phases three and our (ed by).

Remark . Itshould be noted that, because o the linearizationaround a trajectory, the model depends on the value o thetrajectory, which is ormalized in vector d().

6. Inverter and Motor Electrical Model inCurrent MPC Structure

Given a desired current reerence, the goal is to controlthe real actuator current ollowing this signal by a switched

voltage. Direct control o the switching states has some


8/16


1

1

2

2

3

3

4

4

1

2

3

4

1

2

3

4

F : Phase ve: capacitor charging phase.

1

1

2

2

3

3

4

4

F : Working phase six (reewheeling).

advantages over the conventional pulse-width modulationmethod: the reachable dynamic is higher, and no pulse-pattern generation is needed. However, a proper switchingstrategy is necessary, and chattering phenomena can occurdue to measurement noise. o provide a good reerence, the

Optimum

0

Time

+ 2 +

F : Tree o the combinations in two-step prediction.

Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

opt

coil

F : Scheme o the multimodel predictive control.

ollowing control problem is stated: given the system depictedinFigure and the ollowing linear system or the actuator:

coil() = coilcoil coil+Uin ()coil , ()

where Uin 1, 2, 3, 4, 5 = I is the available set oinput voltages applied to the actuator and() is the inducedback voltage o the linear actuator. Find a sequence U oelements I such that

= minUin

12=1

coil + opt + Q coil + opt +

+ =1

1

2 + R + ,

()


9/16


PI controller

PWMand

bridge Motor

Currentreference

Motorposition

Motorcurrent

structure

(a)

0

Output from the

PI controller

0

(b)

F : (a) Control scheme using PI controller in the loop. (b)PWM generation.

where coil()is the predicted current, opt()is an optimizedcurrent which is assumed to be a reerence current (thiscurrent is calculated by the positional MPC structure), and is the prediction horizon (equal to the control horizonin our case).Q is the selection matrix o the weights o thecurrent error, andRis a selection matrix o the weights o theinput voltage.

It is worthwhile to remark that the matrices Q provideadaptive weights depending on the length o the predictionhorizon. Te ormulation in () and () is a well-knownsystem with switching command variables. In our case, ia short prediction horizon ( samples) is considered, theinduced back voltage could, in this interval, be treatedas constant. Observing Figure , ve working phases areidentied. Tese working phases are summarized as ollows.

(i) Working phase one: IGB and IGB on; IGB and IGB off; IGB off (Figure ).

(ii) Working phase two: IGB and IGB off; IGB and IGB on; IGB off (Figure ).

(iii) Working phase three: IGB and IGB on; IGB and IGB off; IGB on (Figure ).

(iv) Working phase our: IGB and IGB off; IGB and IGB on; IGB on (Figure ).

(v) Working phase ve: IGB and IGB off; IGB and IGB off; IGB on (Figure ).

MPC Representation.In general, dependence on the switch-ing conguration the discrete-time dynamic model o theconsidered system can be ormulated in one o the ollowingtwo ways:

coil

( + 1)=A1

coil

()+B1

() , ()

where the pair (A1,B1) represents the rst order R-L systemdepicted inFigure or by dening

x()=coil() ()

. ()Te dynamic o phases three and our is

x( + 1)=A2 ()+ B2 () ,

()=C2x

()= coil

() ,

()

where the term(2, 2, 2)represents the second order R-L-C system depicted inFigure .

(i) During the commutation rom phases one and twoto phases three and our, the predicted current witha prediction horizon equal to is

coil( + 2)= 22 [1coil()+ 1 () () ]+ 22 () ,

()

where()is the measured capacitor voltage,coil()is themeasured coil current, and() is induced voltage (backvoltage), which can be estimated by a state observer.

(ii) During commutation rom phases three and ourto phases one and two, the predicted current witha prediction horizon equal to samples is

coil( + 2)= 122 coil() ()

+ 22 ()

+ 1

1

() .

()

(iii) Phase ve is the phase in which the capacitor ischarged, and it could be represented as

coil( + 2)= 22 [1coil()+ 1 () () ] 22 () .

()

Te voltage o the capacitor,(), is measurable, and thevoltage() is assumed to be constant in the predictionperiod.


10/16


0 1 2 3

Current tracking using PI controller

Time (ms)

0.5 1 1.5 2Time (ms)

0

5

10

15

20

Current(A

)

Reference currentMotor current

Reference currentMotor current

18

16

14

12

10

8

6

20

15

10

5

F : Current tracking obtained with the control scheme oFigure .

HysteresisBridge

Motor

Currentreference

Motorposition

Motorcurrent

Error

structure

F : Control scheme using hysteresis controller in the loop.

.. Optimization and Current Control Technique. Generally,MPCs suffer rom high computational complexity in onlineapplications. In the presented case, there are theoretically possible congurations o the IBG switching states, but only6o them are used in practice; most o them cannot be usedbecause they generateshort circuits, and others arepractically

redundant. I a global optimum is wanted in a one-stepprediction horizon, then there are at most5possible voltagecombinations. In general, or-step prediction, there are5 possible combinations. Tis problem is known as an NP-complete problem (the solution time grows exponentiallywith the problem size). Tus, in a general solution o the

optimization problem, all5 possible states must be takeninto account, which means that the calculation time isexpected to be relatively high. However, or short predictionhorizons, using this method to determine the controller isstill reasonable. InFigure , this situation is depicted. Treelogical inputs1(),2(),3() {0, 1}which identiy thecongurations are dened. Besides the general optimization

solution described previously, a different optimization pro-cedure, which is actually the well-known branch and boundmethod, was used. In this case - combinations through abinary tree are explored. Te easible region is partitionedin subdomains systematically, and valid upper and lower

bounds are generated at different levels o the binary tree.Te main advantage o the branch and bound method isthat it can be interrupted at any intermediate step to obtaina suboptimal solution, although, in this case, the trackingperormance deteriorates. o build a procedure or nding alocal optimum, the ollowing denition is introduced.

Denition . Given a voltage and inverter bridge cong-uration at time that corresponds to the minimum othe unction dened in () and its logical code, a nearpermutation at time + 1 is dened as the combinationcode, which identies the two nearest possible voltages to theminimum at time

.

I the near permutation at time is considered, then theollowing procedure is proposed.Step . Let u be an initial optimum o logic inputs that isound by a total search.

Step . Consider u and its near permutations in anprediction horizonbranch phase.

Step . Check the cost unction dened in () around u tobound the branch.

Step . Find the minimum in the branch.


11/16


0 1 2 3 4

Time (ms)

Current tracking using hysteresis controller with ve states

0.5 1 1.5 2Time (ms)

0

5

10

15

20

Current(A

)

Reference currentMotor current Reference current

Motor current

18

16

14

12

10

8

6

20

15

10

5

F : Current tracking obtained with the control scheme oFigure .

0 1 2 3 4

0

5

10

15

20Current tracking using MPC 2-step prediction

Time (ms)

Current(A)

0.5 1 1.5 2Time (ms)Reference current

Motor current

Reference current

Motor current

20

15

10

20

5

18

16

14

12

10

8

F : Current tracking obtained using an MPC (-step prediction) with the control scheme oFigure .

Step . Branch around the new minimum candidate andcheck the cost unction () tobound the branch.

Step . Go toStep until the speciedtime out.

Te procedure described previously is suitable or currentreerencevalues that do notcontain steps or very ast changes.In these cases, it is possible to assume that the optimum in

() changes slowly and that the computational advantagesare known. At the rst step control horizon, all the switchingpossibilities are considered to manage abrupt changes thatmay arise. Te ollowing step horizons only consider theadjacent switch possibility. In general, independently othe optimal search technique,Figure shows the structureo the proposed controller. As shown previously [], thistype o approach could also be useul in nonlinear control


12/16


0 1 2 3 4 5

0

3

6

MPC compared with PI controller

Time (ms)

Reference currentMPC 2-step controlPI control

Curr

ent(A)

6

3

F : Step test: current tracking obtained using MPC (-stepprediction) and PI controller.

systems. In act, as proposed elsewhere [], it is possibleto describe a nonlinear system with this technique, whichnormally works around known working points, with a seto corresponding linear systems. In our presented case the

set has5 models. As already discussed, current control isparticularly important or positioning a mechatronic system.It is also possible to build an MPC outer loop, as depictedinFigure . In other words, the desired current is calculatedto minimize the quadratic error between the desired positionand the predicted position or the current control loop.

7. Robust Stability Considerations andSimulation Results

In the presented case the perormances are tested throughnumerical simulations. Te numerical simulations are per-ormed considering an extension o a sufficient conditionstated in [, Lemma .]. I the systems dened previouslyare characterized by the ollowing dynamic matrices, A, A1,and A2are stable, and i or all matrices Qand Qas in ()and () there exist matrices P, P1, and P2such that

APA

P

= Q

,A1P1A1 P1= Q,A

2P2A2 P2= Q,

()

then one has the perturbed systems:

x( + 1)=Ax()+mAx() ,x( + 1)=A1x()+m1A1x() ,x( + 1)=A2x()+m2A2x() ,

()

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

1

2

3

Time (s)

Position(mm)

Desired positionObtained position

103

5

4

3

2

1

(a)

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

5

10

15

20

Time (s)

Current(A)

Predicted reference currentMeasured current

15

10

5

(b)

0.01 0.02 0.03 0.04 0.05 0.06 0.07

45

50

55

60

65

70

75

80

85

90

Time (s)

Capacitorvoltage(V)

(c)

F : From the top: three cycles or position, current tracking,

and capacity charging using MPC.


13/16


where m, m1, and m2are matrices with appropriate dimen-sions and are open loop stable i

mAmin min Q2 AP+P ,1,

m1A1min min Q2 A1P1+P1 ,1,m1A2min min Q2 A2P2+P2 ,1.

()

o stabilize the open loop structure robustly, Q()= I. Asdescribed in [, Teorem .] states that it is possible tostabilize the positional MPC in a closed loop. In act, thetheorem mentioned previously states a sufficient condition:

F1QF1+R1F1Q


14/16


15/16


[] A. Bellini, S. Biaretti, and S. Costantini, Implementationon a microcontroller o a space vector modulation techniqueor NPC inverters, in Proceedings of the International IEEESymposium on Industrial Electronics (IEEE-ISlE ), pp. , LAquila, Italy, May .

[] J. F. Martins, A. J. Pires, and J. F. Silva, A Novel and simple cur-rent controller or three-phase IGB PWM power invertersa comparative study, inProceedings of the IEEE InternationalSymposium on Industrial Electronics (IEEE-ISIE ), pp. , July .

[] J. Holtz and S. Stadteld, A predictive controller or the statorcurrent vector o ac machines ed rom a switched voltagesource, in Proceedings of the International Power ElectronicsConference (IPEC ), pp. , okio, Japan, .

[] H. le Huy and L. A. Dessaint, An adaptive current control orpwm inverters, inProceedings of the of IEEE Power ElectronicsSpecialists Conference (PESC ), .

[] R. Kennel and D. Schrder., Predictive control strategy orconverters, in Proceedings of the rd IFAC Symposium onControl in Power Electronics and Electrical Drives, pp. ,

Losanne, Switzerland, .[] R. eodorescu, F. Blaabjerg,J. K. Pedersen, E. Cengelci, and P. N.

Enjeti, Multilevel inverter by cascading industrial VSI, IEEETransactions on Industrial Electronics, vol. , no. , pp. ,.

[] Z. Ye, P. K. Jain, and P. C. Sen, A ull-bridge resonant inverterwith modied phase-shif modulation or high-requency ACpower distribution systems, IEEE Transactions on IndustrialElectronics, vol. , no. , pp. , .

[] Z. Ye, P. K. Jain, and P. C. Sen, A two-stage resonant inverterwith control o the phase angle and magnitude o the outputvoltage, IEEE Transactions on Industrial Electronics, vol., no., pp. , .

[] P. Cortes, A. Wilson, S. Kouro, J. Rodriguez, and H. Abu-

Rub, Model predictive control o multilevel cascaded H-bridgeinverters,IEEE Transactions on Industrial Electronics, vol. ,no. , pp. , .

[] K. A. ehrani, H. Andriatsioharana, I. Rasoanarivo, and F. M.Sargos, A novel multilevel inverter model, in Proceedings ofthe th IEEE Annual Power Electronics Specialists Conference(PESC ), pp. , June .

[] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, Teexplicit linear quadratic regulator or constrained systems,Automatica, vol. , no. , pp. , .

[] A. Bemporad and M. Morari, Control o systems integratinglogic, dynamics, and constraints,Automatica, vol. , no. , pp., .

[] C. Pedret, A. Poncet, K. Stadler et al., Model-varying predictivecontrol o a nonlinear system, Internal Report in ComputerScience Department, ESE de la Universitat Autonoma deBarcelona, .

[] H. Sunan, . K. Kiong, and L. . Heng, Applied PredictiveControl, Springer, London, UK, .

[] P. Mercorelli, N. Kubasiak, and S. Liu, Multilevel bridgegovernor by using model predictive control in wavelet packetsor tracking trajectories, in Proceedingsof the IEEE InternationalConference on Robotics and Automation, vol. , pp. ,May .

[] N. Kubasiak, P. Mercorelli, and S. Liu, Model predictive controlo transistor pulse converter or eeding electromagnetic valveactuator with energy storage, in Proceedings of the th IEEE

Conference on Decision and Control, and the European ControlConference (CDC-ECC ), pp. , December .


16/16

Submit your manuscripts at

http://www.hindawi.com

Date post:	03-Jun-2018
Category:	Documents
Upload:	ospino
View:	221 times
Download:	0 times

A Multilevel Inverter Bridge Control Structure With Energy Storage Using Model Predictive Control...

Documents