research article a multilevel inverter bridge control...

Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 750190 15 pageshttpdxdoiorg1011552013750190

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 1 21339 Lueneburg Germany

Correspondence should be addressed to Paolo Mercorelli mercorelliunileuphanade

Received 5 September 2012 Accepted 30 December 2012

Academic Editor Claudio Mazzotti

Copyright copy 2013 Paolo Mercorelli This 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

The paper presents a novel technique to control the current of an electromagnetic linear actuator fed by a multilevel IGBT voltageinverter with dynamic energy storageThe technique uses a ldquocascademodel predictive control (MPC)rdquo which consists of twoMPCsA predictive control of the trajectory position predicts the optimal current which is considered to be the desired current for thesecond MPC controller in which a hysteresis control technique is also integrated Energy is stored in a capacitor using energyrecovery The current MPC can handle a capacitor voltage higher than the source voltage to guarantee high dynamic current anddisturbance compensation The main contribution of this paper is the design of 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 of the new generation of inverters and in the meantime to guarantee precision and acceptable switching frequencyThe proposed technique shows very promising results through simulations with real actuator data in an innovative transportationtechnology

1 Introduction and Motivation

The performance of a multilevel inverter is better than thatof a classical inverter The total harmonic distortion of aclassical inverter is very high that is the total harmonicdistortion for a multilevel inverter is low This topic hasbecome the focus of a significant amount of research Onestudy has simulated and implemented a multilevel inverter-fed induction motor drive [1] The output harmonic contentis reduced by using a multilevel inverter In a symmetricalcircuit the voltage and power increase as the number of levelsin the inverter increases The switching angle for the pulse isselected to reduce the harmonic distortionThis 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 [2] The neutral point potential balancingalgorithm for three-level neutral point clamped invertersusing an analytically injected zero-sequence voltage has beendeveloped [3] Modulation schemes to eliminate commonmode voltage in multilevel inverter topology have been

suggested [4] A generalizedmultilevel inverter topologywithself-voltage balancing has also been suggested [5] A survey oftopologies control and applications of multilevel invertershas been published [6] A digital modulation techniquefor dual three-phase alternating current (AC) machines hasbeen presented [7] A space vector pulse-width modulation(PWM) technique for a dual three-phase ACmachine and itsdigital signal processor (DSP) implementation has been pre-sented [8] Practical medium voltage converter topologies forhigh power applications have been described previously [9ndash11] Industrial topologies have been presented [12] Differentpulse-width modulation techniques for symmetrical cascadeinverters with high and fundamental switching frequencyhave been shown for instance in [13 14] An improvement interms of efficiency of the converter is presented in [15] wherethe authors use different DC sources to reduce the switchinglosses More recently in [16] the authors proposed a novelH-bridge multilevel pulse modulation converter topology inwhich the structure is based on a series connection of high-voltage diode-clamped inverter and low-voltage conventionalinverterThepresentwork implements a controlledmultilevel

2 Journal of Engineering

inverter feeding an electromagnetic actuator In particulartwo MPCs in cascade structure are presented to control five-level inverter electromagnetic actuators which are becom-ing increasingly important in many industrial applicationsThanks to recent progress in permanent-magnet technologyvery compact and high-power electromagnetic actuators arenow available Particularly in automotive systems these actu-ators have often replaced conventional mechanical compo-nents due to their high efficiency excellent dynamic behaviorand control flexibility thus mechatronics is one of the mostnotable innovative fields in the automobile industry A recentdesign of such an actuator (eg [17]) considers optimizingwith respect to dimensions and forces Nevertheless practicaltests show that this actuator structure is not always effectivefor a large variety of applications because a complex controlstructure is needed to control such a valve [18] Recentlyfor instance a new type of electromagnetic valve drivesystem has been proposed and described [19] This paperpresents the control design for a novel permanent-magnetlinear valve actuator for use in a variable engine control toallow short-stroke and high-dynamic motions In a linearactuator the electromagnetic force is totally independentof the constrained motion Specifically the electromagneticforce is parallel to the magnetic axis and thus it is orthogonalto the constrained motion These performance character-istics make this type of 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 differentpermanentmagnet actuator topologies usingNdFeBmagnetswere considered In particular the objective of this paper is todesign a controller structure consisting of

(i) a flatness-based feedforward control

(ii) a multilevel inverter configuration

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

An important aspect of this approach is the use of geometricsystem properties such as the differential flatness neededto track the desired trajectories In this context recentlypublished research results have been considered (eg [20])in which the author proposed an interesting method tointegrateMPC and a feedforward flatness controlThe secondpart of the paper shows the structure of the cascade MPCdefined above For trajectory-based motion cycles there isusually a need to control the actuator current separately Thereference value for the actuator current is normally generatedby the speed or position controller in an outer control loopbased on the desired speed or position profiles The currentcontrol is used to provide the force or torque required for thedesired motion quickly and precisely Generally three maintechniques are employed for current control of a voltage-source inverter (VSI)

(i) closed loop control (eg PID control) using pulse-width modulated terminal voltages [21]

(ii) hysteresis control techniques [22 23](iii) predictive current control [24ndash26]

The PWM technique is currently the most popular methodfor an inverter current control The main advantage ofPWM techniques is that the inverter switches operate at afixed frequency and hardware- or software-based standardmodulators are available as industrial products (eg 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 of insulated gate bipolar transistor (IGBT)H-bridge modules with isolated DC buses [27] Amongvarious modulation techniques for a multilevel inverter thespace vector pulse width modulation (SVPWM) is widelyused However implementing the SVPWM for a multilevelinverter is complicated because it is difficult to determinethe location of the reference vector calculate on times anddetermine and select switching states A previous paperhas proposed a general SVPWM algorithm for multilevelinverters based on standard two-level SVPWMHowever thesystem response is affected by the stability requirements ofthe feedback loop which also depend on load parametersmoreover appreciable phase lag may arise even in the steadystate Recently many studies have focused on solving theseproblems (eg [28 29]) of multilevel inverter modules withindependent control of the phase angle and magnitude of theoutput voltage [29] Control and new power bridge structureshave been studied Very recent works have presented a novelbridge structure [30 31] In both structures capacitors areused but not recharged In particular an MPC strategy isproposed for a three-phase power bridge [30] Hysteresiscurrent control has a fast response and a good accuracyIt can be implemented with a simple hardware structureand in many cases it does not require any knowledgeof load parameters However it can sometimes cause veryhigh switching frequencies or if the maximum switchingfrequency is limited the current waveform 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 [24ndash26] However no feedbackloop is applied to compensate for model uncertainties Toovercome the disadvantages of the mentioned methods thisstudy develops a novel approach that combines hysteresiscontrol with a model predictive control (MPC) strategy Theproposed control system integrates flatness and two cascadeMPCs The first MPC generates an optimal desired currentby minimizing position error The second MPC considersthis optimal desired current to be the reference signal tofind an optimal switching law in combination with thehysteresis control A special energy storage and chargingcircuit was designed to provide multilevel voltages for aneffective current control By applying the proper voltagelevel which was determined by the MPC strategy to theactuator dynamic current changes and small current ripples

Journal of Engineering 3

Coil

Valve

Lower spring

Iron pole

Permanent magnet

Upper spring

Figure 1 Cross-section of the perpendicular linear actuator

are possible in spite of relatively low switching frequenciesA recent work [32] proposes ldquofastrdquo algorithms based oncomputing the formulated problems in parallel In fact theoptimization technique uses subsets of the state subspaceoffline and then optimizes them in parallel In this paper acascade structure of MPCs is utilized to control a multilevelinverter Two different optimization techniques are presentedas well The main contribution of the paper is the optimalcontrol structure which also guarantees that the capacitorrecharges In this context the approach is quite new Thepaper is organized in the following way

Section 2 describes the model In Section 3 the flatnessproperty of the system is shown In Section 4 the generalMPC goal is defined and the main idea of using a cascadepositional MPC followed by a current MPC is explained InSection 5 a positional MPC around the desired trajectory isproposed Section 6 is devoted to analyzing and optimizingthe current MPC The simulation results and some conclud-ing remarks end the paper

2 Description of the Physical Systems

The electromagnetic actuator is depicted in Figure 1 Thestructure of this actuator is a moving coil thus the coilsare mounted on the upper part of the stem of the valveThe moving valve was connected to the power system tofeed the coil with two normal cables with fix contacts Thisarrangement is possible because of the short stroke to becovered (8mm) The valve can be modeled mathematicallyin the following way

120597119894coil (119905)

120597119905= minus

119877coil119871coil

119894coil (119905) +119880in (119905) minus 119906119902 (119894coil (119905) 119904 (119905))

119871coil (1)

120597119904 (119905)

120597119905= 119907 (119905) (2)

120597119907 (119905)

120597119905=119863119892 (119894coil (119905) 119904 (119905))

119898119894coil (119905)

+minus119896119889119907 (119905) minus 119896119891119904 (119905) + 1198650 (119905)

119898

(3)

where

119863119892 (119894coil (119905) 119904 (119905)) =Φℎ (119894coil (119905) 119904 (119905)) 119897coil

119860coil

119906119902 (119894coil (119905) 119904 (119905)) = 119863119892 (119894coil (119905) 119904 (119905)) 119907 (119905)

(4)

It is important to note that119863119892(119894coil(119905) 119904(119905)) gt 0 for all 119894coil(119905)and for all 119904(119905) 119877coil and 119871coil are the resistance and theinductance of the coil windings 119880in(119905) is the input voltage119906119902(119905) is the induced back voltage Φℎ is the magnetic fluxpenetrating the coil 119894coil(119905) is the coil current 119897coil is the coillength 119904(119905) 119907(119905) and 119898 are the position velocity and massof the actuator respectively while 119896119889119907(119905) 119896119891119904(119905) and 1198650(119905)represent the viscose friction the total spring force and thedisturbing force acting on the valve Equation (1) representsthe electrical system of the actuator Both (2) and (3) describethe mechanical behavior of the actuator and (2) and (4) alsorepresent the magnetic system In particular the followingexpression

119865119871 (119905) = 119863119892 (119894coil (119905) 119904 (119905)) 119894coil (119905) (5)

describes the Lorentz force generated by the actuator Essen-tially the magnetic flux generated by the permanent mag-nets has two components in the air gap (1) the main fluxΦℎ(119894coil(119905)) which does not depend on the displacementof the mover and is responsible for the Lorentz forceand the induced back voltage and (2) the leakage fluxΦ120590(119894coil(119905) 119904(119905)) which disperses around the coil and doesnot contribute to the electromagnetic force and induced backvoltage

Φℎ (119894coil (119905)) + Φ120590 (119894coil (119905) 119904 (119905)) = ΦPM (119894coil (119905)) (6)

However due to the special actuator design the leakage fluxis almost equal to zero Thus it is possible to conclude

119863119892 (119894coil (119905) 119904 (119905)) asymp 119863119892 (119894coil (119905)) =ΦPM (119894coil (119905)) 119897coil

119860coil (7)

3 Differential Flatness of the System

Roughly speaking a system is differentially flat if it is possibleto find a set of outputs equal in number to the number ofinputs such that all states and inputs are expressed in terms ofthose outputs and their derivatives To be more precise if thesystem has state variables x isin R119899 and inputs u isin R119898 thenthe system is flat if the outputs y isin R119898 have the followingform

y = y (x u u u(119901)) (8)

x = x (y y y(119902)) u = u (y y y(119902+1)) (9)

Differentially flat systems are especially interesting insituations in which explicit trajectory tracking is requiredBecause the behavior of the flat system is given by the flatoutput it is possible to plan trajectories in output space andthen map them to the appropriate inputs Concerning the


119867 minus119867119888

119888

Δ119867

119867PM

119861119903

119861

119861PM

Figure 2 Permanent-magnet demagnetization curve

flat output for dating neither the necessary and sufficientcondition nor a general method for the determination ofthe flat output has been provided Typically a flat output isguessed and Definitions (8) and (9) are used to verify thatthe chosen output is flat Once the system is proven to bedifferentially flat the basic approach of two degrees of free-dom controller design consists of two steps first separatingthe nonlinear controller synthesis problem into designing afeasible feedforward controlled trajectory for the nominalmodel of the system and second regulating that trajectoryusing controllers that guarantee robust performance in thepresence of uncertainties and disturbances To explain themeaning of flatness intuitively it is adequate to consider asystem to be flat when the dynamic of at least one output isldquovisiblerdquo from the state and the input of the system thus thereis at least one output that can be functionally controlled by theinputs as well by the states

Now the first step is to show that our model representedin (1) (2) and (3) is flat If the position 119904 of the moving partof the actuator is chosen as ldquoguessed outputrdquo then

119910 (119905) = 119904 (119905) 997904rArr 119904 (119905) = 119910 (119905)

119904 (119905) = 119910 (119905) 997904rArr 119907 (119905) = 119910 (119905)

(10)

119894coil (119905) 119863119892 (119894coil (119905)) = 119898(119896119889

119898119907 (119905) +

119896119891

119898119904 (119905) +

120597119907 (119905)

120597119905)

(11)

119880in (119905) = 119871coil120597119894coil (119905)

120597119905+ 119877coil119894coil (119905) + 119863119892 (119894coil (119905)) 119907 (119905)

(12)

To verify the flatness property it is necessary to look at(13) As evident from (7) 119863119892(119894coil(119905)) is proportional to theflux coupled with the permanent magnets which is againproportional to the flux density 119861PM The operating point(119861PM 119867PM) of the permanent magnets is mainly determinedby the magnet geometry and the demagnetization curve(Figure 2) The coil current only changes the field strengtharound the operating point (Δ119867) algebraically An exactanalytical expression between Δ119867 and 119894coil(119905) is very difficultto be derived because it also depends on the saturation level ofthe 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+

Figure 3 Multilevel inverter bridge structure

words there is always a (locally) unique solution for 119894coil(119905)that depends on 119863119892119894coil(119905) Thus using (13) the coil currentcan always be determined uniquely by the actuator positionand its derivatives Considering additional system equationsit can be easily seen that the flatness conditions defined in(9) are satisfied Relationships (13) and (12) define the inversesystem Based on the flatness of the system the linearizingtrajectory of the model (3) and (2) is as follows

119894119889 (119905) =119898

119863119892 (119894119889 (119905))(119896119889

119898119907119889 (119905) +

119896119891

119898119904119889 (119905) +

120597119907119889 (119905)

120597119905) (13)

where 119904119889(119905) is the desired trajectory and 119894119889(119905) is the corre-sponding feed forward control current As it was explainedbefore the flux does not depend on the position of thearmature because the leakage flux is almost equal to zeroThis means that the flux is a function of the current Function119863119892(119894coil(119905)) is never equal to zero at least for our conceivedstructure because of controllabilityThe actuator is conceivedand designed in away in order to guarantee the controllabilityat any point of its movement For (1) the inverse systemthat determines the desired feedforward control input 119906119889(119905)is defined as follows

119906119889 (119905) = 119871coil120597119894119889 (119905)

120597119905+ 119877coil119894119889 (119905)

+ 119863119892 (119894119889 (119905) 119904 (119905))120597119904119889 (119905)

120597 (119905)

(14)

Based on the feed forward control presented previously thenext step is to build a feedback control law that in thepresence of external disturbances and uncertainties in theparameters takes the system around the desired trajectory

4 General MPC Problem Formulation

Figure 3 shows the power electronic circuit feeding thepermanent magnet linear actuator The power supply is a


Current

FuturePast

Time

Desired currentMeasured current

Predicted currentNot used

0

119905 119905minus 2119879119904 119905 minus 119879119904 119905 + 2119879119904 119905 + 119879119904

119880119888

119880119904

minus119880119888

Figure 4 Principle of the complete applied MPC strategy

constant voltage source 119880119878 (battery) However to realize acurrent change as fast as possible to ensure the requireddynamic it is desirable to have a higher voltage appliedto the actuator Therefore a capacitor 119862 charged with aninitial voltage 119880119888 higher than the battery level is used Tomaintain the high voltage level of the capacitor during thevalve operation an energy storage block consisting of thecapacitor 119862 the diode D5 and the IGBT switch V5 is usedBesides the current regulation another task of the controlstrategy is to recharge the capacitor appropriately by utilizingthe braking energy On the right the IGBT inverter bridgeand the linear actuator are represented Model predictivecontrol has been a widely used control concept for over15 years especially in the process industry Applications ofMPC in the field of electrical drives are quite rare becauseof a high computational complexity The central point ofa model predictive controller is a process model which iscapable to predict future output signals based on futureinput signals and initial values Using this process modelthe future dynamic behavior of the real plant is predictedwithin a prediction horizon These predicted output sig-nals can be used to minimize an open loop performancecriterion (eg the sum of squared control errors withinthe prediction horizon) and to calculate the input signals119906(119896) for a control horizon Outside the control horizon theinput remains constant The calculated input signals feedthe plant until a new measurement becomes available Thisprocedure is repeated with a receding prediction and controlhorizon The receding horizon strategy makes a closed loopcontrol law from the originally open loop minimization Theminimization step can easily include constraints such asinput output or state constraints that may be taken intoaccount already in the controller design The principle ofthe predictive control design for the current control problemconsidered in this paper is depicted in Figure 4 with thedesired current (dashed-point line) that results from the outerposition control loop and the measured current (solid line)At the present time 119905 all possible currents for the future twotime steps 119905 + 119905119904 sdot sdot sdot 119905 + 2119905119904 are calculated This calculationis performed based on a multimode model and assessedafterwards with the help of a defined cost function (details

follow) The optimum switching configuration is also shown(bold-dashed line)

Figure 5 shows the whole proposed control structureThe presented technique can be interpreted in the followingpoints

(i) The positional MPC structure generates an optimalcurrent trajectory with the desired optimal horizonthrough the electrical model

(ii) The optimal current trajectory is used as a desiredtrajectory for the current MPC

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

Figure 5 shows the control system structure in which it ispossible to recognize the desired current 119894119889(119905) which is usedto linearize the system the predicted reference current 119894opt(119905)and current 119894coil(119905)

5 Solving a Linear Position MPCOptimization Problem

To obtain the desired current trajectory a position modelpredictive control is implemented In this case the sampletime is relatively short 40 120583s (25 kHz) to make the algorithmas fast as possible Considering the model in (1) (2) and (3)in which 1198650(119905) = 0 and an Euler discretization with 119896 = 119899119905119904119899 isin 119873 where 119905119904 is the sampling time the following system isobtained

119894coil (119896 + 1)

= 119894coil (119896) +119905119904

119871coil(minus119877coil119894coil (119896) + 119906mpc (119896)

+119906119889 (119896) minus 119906119902 (119896))

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119863119892 (119894coil (119896))

119898119894coil (119896)

+minus119896119889119907 (119896) minus 119896119891119904 (119896)

119898)

(15)

where 119906mpc(119896) is the model predictive control input to becalculated and 119906119889(119896) and 119906119902(119896) are the discretized voltages of119906119889(119905) and 119906119902(119905) already defined in (1) (2) and (3) If (13) isdiscretized then

119894119889 (119896) =119898

119863119892 (119894119889 (119896 minus 1))

sdot (119896119889

119898119907119889 (119896) +

119896119891

119898119904119889 (119896) +

119907119889 (119896 + 1) minus 119907119889 (119896)

119905119904

)

(16)

Considering the Euler discretization of (14) which allows usto express the future value of an output as a function of the


Feedforwardcontrol

CurrentMPC

MultilevelIGBTbridge

DC-motorsystemPositional

MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories

Desiredcurrent Obtained

trajectories

Obtainedtrajectories

Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889

Figure 5 The whole control scheme

pass inputs which is the required form of theMPC approachthe following equation is obtained

119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)

If (16) and (17) are inserted into the discretized equations (1)(2) and (3) then the following linear system is obtained

119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904


(119894coil (119896) minus 119894119889 (119896) minus119906mpc (119896)

119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)

If Δ119906mpc(119896) = 119906mpc(119896) minus 119906mpc(119896 minus 1) is assumed the systemdescribed in (18) becomes

x (119896 + 1) = A119896x (119896) + B119896 (Δ119906mpc (119896) + 119906mpc (119896 minus 1))

+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)

where matrix H119896 = [0 1 0] is the output matrix whichdetermines the position and the previous notation means

A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)

In the considered representation vector d(119896) is the knowninput In the model approach just two samples are consid-ered

119904 (119896 + 1) = H119896A119896x (119896)

+H119896B119896Δ119906mpc (119896) +H119896B119896Δ119906mpc (119896 minus 1)

+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)

If the following performance criterion is assumed

119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)

where 119904119889(119896 + 119895) 119895 = 1 2 119873 is the position referencetrajectory 119873 the prediction horizon and Q119901 and R119901 arenonnegative definite matrices then the solution minimizingperformance index (22) may be then obtained by solving

120597119869

120597Δ119906mpc= 0 (23)

If only two steps for the prediction horizon are consideredthen

G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)

A direct computation may be obtained explicitly as

Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)

minus F2119901119906mpc (119896 minus 1) minus F3119901d (119896) ))

(25)

where Y119889119901(119896) is the desired output column vector Once thepredicted optimized voltage is obtained then it is possibleto obtain the predicted optimized current as the referencecurrent for the current MPC based on the model of thesystem

119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

Figure 6 Working phases one and two (fed by 119880119904)

119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

Figure 7 Working phases three and four (fed by 119880119862)

Remark 1 It should be noted that because of the linearizationaround a trajectory the model depends on the value of thetrajectory which is formalized in vector d(119896)

6 Inverter and Motor Electrical Model inCurrent MPC Structure

Given a desired current reference the goal is to controlthe real actuator current following this signal by a switchedvoltage Direct control of the switching states has some


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634

Figure 8 Phase five capacitor charging phase

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

Figure 9 Working phase six (freewheeling)

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 To provide a good reference the

Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904

Figure 10 Three of the combinations in two-step prediction

Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil

Figure 11 Scheme of the multimodel predictive control

following control problem is stated given the systemdepictedin Figure 3 and the following linear system for the actuator

120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)

where Uin isin [1199061 1199062 1199063 1199064 1199065] = I is the available set ofinput voltages applied to the actuator and 119906119902(119896) is the inducedback voltage of the linear actuator Find a sequence Ulowast of 119901elements 119906 isin I such that

119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894

sdot (coil (119896 + 119895) minus 119894opt (119896 + 119895))

+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)


PI controllerPWM and

bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0

Output from thePI controller

119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)

Figure 12 (a) Control scheme using PI controller in the loop (b)PWM generation

where coil(119896) is the predicted current 119894opt(119896) is an optimizedcurrent which is assumed to be a reference current (thiscurrent is calculated by the positional MPC structure) and119873119901 is the prediction horizon (equal to the control horizonin our case) Q119894 is the selection matrix of the weights of thecurrent error andR119894 is a selectionmatrix of the weights of theinput voltage

It is worthwhile to remark that the matrices Q119894 provideadaptive weights depending on the length of the predictionhorizon The formulation in (26) and (27) is a well-knownsystem with switching command variables In our case ifa short prediction horizon (2ndash4 samples) is considered theinduced back voltage 119906119902 could in this interval be treatedas constant Observing Figure 3 five working phases areidentified These working phases are summarized as follows

(i) Working phase one IGBT 2 and IGBT 3 on IGBT 1and IGBT 4 off IGBT 5 off (Figure 6)

(ii) Working phase two IGBT 2 and IGBT 3 off IGBT 1and IGBT 4 on IGBT 5 off (Figure 6)

(iii) Working phase three IGBT 1 and IGBT 4 on IGBT 2and IGBT 3 off IGBT 5 on (Figure 7)

(iv) Working phase four IGBT 1 and IGBT 4 off IGBT 2and IGBT 3 on IGBT 5 on (Figure 7)

(v) Working phase five IGBT 2 and IGBT 3 off IGBT 1and IGBT 4 off IGBT 5 on (Figure 8)

MPC Representation In general dependence on the switch-ing configuration the discrete-time dynamic model of theconsidered system can be formulated in one of the followingtwo ways

119894coil (119896 + 1) = A1119894coil (119896) + B1 (119880119904 minus 119906119902 (119896)) (28)

where the pair (A1B1) represents the first order ldquoR-Lrdquo systemdepicted in Figure 6 or by defining

x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)

The dynamic of phases three and four is

x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)

where the term (1198602 1198612 1198622) represents the second order ldquoR-L-Crdquo system depicted in Figure 8

(i) During the commutation from phases ldquoone and twordquoto phases ldquothree and fourrdquo the predicted current witha prediction horizon equal to 2 is

119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)

where 119880119888(119896) is the measured capacitor voltage 119894coil(119896) is themeasured coil current and 119906119902(119896) is induced voltage (backvoltage) which can be estimated by a state observer

(ii) During commutation from phases ldquothree and fourrdquoto phases ldquoone and twordquo the predicted current witha prediction horizon equal to 2 samples is

119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)

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

119894coil (119896 + 2) = minus 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)

The voltage of the capacitor 119880119888(119896) is measurable and thevoltage 119906119902(119896) is assumed to be constant in the predictionperiod


0 1 2 3

Current tracking using PI controller

Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)

Reference currentMotor current


minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5

Figure 13 Current tracking obtained with the control scheme of Figure 12

Hysteresis Bridge Motor

Current reference

Motorposition

Motorcurrent

Errorstructureminus

Figure 14 Control scheme using hysteresis controller in the loop

61 Optimization and Current Control Technique GenerallyMPCs suffer from high computational complexity in onlineapplications In the presented case there are theoretically 32possible configurations of the IBGT switching states but only6 of them are used in practice most of them cannot be usedbecause they generate short circuits and others are practicallyredundant If a global optimum is wanted in a ldquoone-steprdquoprediction horizon then there are at most 5 possible voltagecombinations In general for 119873-step prediction there are5119873 possible combinations This problem is known as an NP-complete problem (the solution time grows exponentiallywith the problem size) Thus in a general solution of theoptimization problem all 5119873 possible states must be takeninto account which means that the calculation time isexpected to be relatively high However for short predictionhorizons using this method to determine the controller isstill reasonable In Figure 10 this situation is depicted Threelogical inputs 1199061198971(119905) 1199061198972(119905) 1199061198973(119905) isin 0 1 which identify theconfigurations are defined 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 0-1 combinations through abinary tree are explored The feasible region is partitionedin subdomains systematically and valid upper and lowerbounds are generated at different levels of the binary treeThe main advantage of the branch and bound method isthat it can be interrupted at any intermediate step to obtaina suboptimal solution although in this case the trackingperformance deteriorates To build a procedure for finding alocal optimum the following definition is introduced

Definition 2 Given a voltage and inverter bridge config-uration at time 119896 that corresponds to the minimum ofthe function defined in (27) and its logical code a ldquonearpermutationrdquo at time 119896 + 1 is defined as ldquothe combinationcoderdquo which identifies the two nearest possible voltages to theminimum at time 119896

If the ldquonear permutationrdquo at time 119896 is considered then thefollowing procedure is proposed

Step 1 Let u⋆119897be an initial optimum of logic inputs that is

found by a total search

Step 2 Consider u⋆119897and its ldquonear permutationsrdquo in an 119873119901

prediction horizon branch phase

Step 3 Check the cost function defined in (27) around u⋆119897to

bound the branch

Step 4 Find the minimum in the branch


0 1 2 3 4Time (ms)

Current tracking using hysteresis controller with five states

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)

Reference currentMotor current Reference current

Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15

20Current tracking using MPC 2-step prediction

Time (ms)

Curr

ent (

A)

05 1 15 2Time (ms)Reference current

Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8

Figure 16 Current tracking obtained using an MPC (2-step prediction) with the control scheme of Figure 11

Step 5 Branch around the new minimum candidate andcheck the cost function (27) to bound the branch

Step 6 Go to Step 4 until the specified time out

The procedure described previously is suitable for currentreference values that do not contain steps or very fast changesIn these cases it is possible to assume that the optimum in

(27) changes slowly and that the computational advantagesare known At the first step control horizon all the switchingpossibilities are considered to manage abrupt changes thatmay arise The following step horizons only consider theldquoadjacent switch possibilityrdquo In general independently ofthe optimal search technique Figure 11 shows the structureof the proposed controller As shown previously [33] thistype of approach could also be useful in nonlinear control


0 1 2 3 4 5

0

3

6

MPC compared with PI controller

Time (ms)

Reference currentMPC 2-step control PI control

Curr

ent (

A)

minus6

minus3

Figure 17 Step test current tracking obtained using MPC (2-stepprediction) and PI controller

systems In fact as proposed elsewhere [34] it is possibleto describe a nonlinear system with this technique whichnormally works around known working points with a setof corresponding linear systems In our presented case theset has 5119873 models As already discussed current control isparticularly important for positioning a mechatronic systemIt is also possible to build an MPC outer loop as depictedin Figure 5 In other words the desired current is calculatedto minimize the quadratic error between the desired positionand the predicted position for the current control loop

7 Robust Stability Considerations andSimulation Results

In the presented case the performances are tested throughnumerical simulations The numerical simulations are per-formed considering an extension of a sufficient conditionstated in [35 Lemma 31] If the systems defined previouslyare characterized by the following dynamic matrices A119896 A1and A2 are stable and if for all matricesQ119901 and Q119894 as in (22)and (27) there exist matrices P119901 P1119894 and P2119894 such that

A119879119896P119901A119896 minus P119901 = minusQ119901



(34)

then one has the perturbed systems

x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)

Desired positionObtained position

times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)

Predicted reference currentMeasured current

minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)

Figure 18 From the top three cycles for position current trackingand capacity charging using MPC


wherem119901m1 andm2 are matrices with appropriate dimen-sions and are open loop stable if

10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)

To stabilize the open loop structure robustly Q(sdot) = I Asdescribed in [35 Theorem 31] states that it is possible tostabilize the positional MPC in a closed loop In fact thetheorem mentioned previously states a sufficient condition

100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)

In our case the control structure is a cascade of con-trollers No stability conditions are given in the literatureThe heuristic procedure to obtain the stability of the wholesystem which was adopted in this paper consists of findingthe gain of the positional MPC according to relation (37) andtesting the stability of the whole loop control structure If theloop is not stable then the value of matrix Q119894 is reducedand the method is repeated This trial-and-error techniquecauses the system to achieve stability It is always possibleto consider a number of models as in [34] such that theuncertainties of the system respect the sufficient conditionstated in [35 Lemma 31] To justify the presented approachthe following comparison results are reported that trackthe current control These results provided some essentialinformation regarding the presented research direction Tocompare the performance of a possible control strategy thesecond MPC was considered The results shown in Figures13 15 and 16 are obtained through the control structurepresented in Figures 11 12 and 14 using a current sinusoidaltest signal In Table 1 the results are summarized From theseresults a majority of the approximations of the MPC aresuperior if a minimum number of models are consideredThis set of models was used to determine the best value ofthe 119906119902(119896) voltage at the moving working point as proposedin [34] A sinusoidal test function was chosen based onthe fundamental harmonic of the current in a real workingsystemMoreover to complete the comparison a step test theresults of which are summarized in Figure 17 was done Alsoin this case the superiority of the MPC seems to be evidentsee Table 2 With both the hysteresis controller and the PIcontroller it is not possible to recharge the capacitor at leastnot with our proposed bridge structure After determiningthe superiority of the MPC for our task [36] an MPC thatfollows a given current profile was presented and in [37] asimilar control strategy was shown but with a bridge withthe capacitor recharging structure and a new control law In[37] no trajectory tracking was analyzed Nevertheless PI

Table 1 Comparison of the results in the sinusoidal test current

119867 MPC Hysteresis regulator PI regulator

Energyerror(int 1198902119889119905)

023 (039) 035 125

Calculationcomplexity 30 (5) models mdash mdash

Table 2 Comparison of the results in the step test current


Energyerror(int 1198902119889119905)

24 (42) 40 45


controllers and hysteresis controller are very often used inthese kinds of applications Even thoughwemust renounce torecharge the capacitor PI and hysteresis controllers are veryoften used because of their simplicity to be realized Based onthe real pressure acting in a valve actuator motion with anexponential disturbance with an initial value equal to 400Nis simulated even though in the control law 1198650(119905) = 0 Specialattention was paid to the applicability of the control systemIn fact the sample time was 40 120583s In this time just a limitednumber of arithmetical operations can be performed andfor this reason the simulation was performed with 119873 = 2

(prediction horizon)

71 Capacitor Recharging Along with the tracking taskanother interesting aspect is the following a self-rechargingvoltage capacitor control A capacitor can be charged inworking phase 5 where the motor inductance either allowsthe charging current or themechanical braking energy can befed backThus appropriately partitioning the working phasescan make both tracking and recharging possibleThe balancebetween these two tasks is achieved by the cost functiondefined in (27) Depending on the motion cycle dynamicthe weights Q119895 and R119895 were set to maintain the level ofthe voltage capacitor within an acceptable tolerance Thusin particular Δ119880119888(119896) = (119880119888(119896) minus 119880

lowast

cd) where 119880119888(119896) is themeasured voltage of the capacitor and119880lowastcd is its desired levelwhich should be kept within a tolerance band The capacitorvoltage is used to enable high dynamic changes in the actuatorcurrent which in turn cause large variations in the capacitorvoltageThus the system could not be operated continuouslyif the capacitor control was not included in the optimizationprocedure Furthermore efficiencywas improved by properlyutilizing the mechanical braking energy Simulation resultswere achieved using real data and they show promisingresults In this phase some different control strategies weretested using real data from the already-built valveThe resultsencourage proceeding in the prototyping phase in whichthe rest of the structure which basically consists of a DSPmust have a power bridge according to the simulated data


Some interesting aspects are worth identifying The lowerpart of Figure 18 shows how the self-charging phase can startduring the following phase In this case an initial error occursas long as the capacitor does not achieve the full voltageIn the presented simulated case the initial voltage value inthe capacitor is 48V It is possible to start from 0V in thiscase the charging phase lasts 6 cycles more if the sameweighting matrices are considered in the cost function Toachieve a higher charging velocity it is necessary to increasethe matrix R119895 in the index (27) Finally it is interesting tonote that throughMPC it is possible to obtain a self-balancedswitch of the IBGT In fact Figure 9 shows two possiblefreewheeling circuits It is acceptable to balance the switchingphase on these two circuits to obtain a balance frequencyA compromise between good tracking and stability of thesupply voltage is achieved In particular good capacitorvoltage stability provides the necessary condition for goodfollow-up results it is always necessary to identify the trade-off between capacitor voltage stability and followup

8 Conclusions and Outlook

Thepaper implements a multilevel inverter control system byintegrating flatness and two cascaded MPCs The first MPCgenerates an optimal desired current byminimizing the posi-tion error The second MPC considers this optimal desiredcurrent as a reference signal to find an optimal switchinglaw for the proposed multilevel inverter The current MPCconsiders energy storage optimization using a capacitor inthe proposed inverters Finally simulations using real data areshown

Acknowledgment

Theauthor thanks the Institute forAutomation and Informat-ics (IAI) in Wernigerode Germany for its collaboration

References

[1] G Pandian and S Rama Reddy ldquoImplementation of multilevelinverter-fed induction motor driverdquo Journal of Industrial Tech-nology vol 24 no 1 pp 2ndash6 2008

[2] K Yamanaka AM Hava H Kirino Y Tanaka N Koga and TKume ldquoA novel neutral point potential stabilization techniqueusing the information of output current polarities and voltagevectorrdquo IEEE Transactions on Industry Applications vol 38 no6 pp 1572ndash1580 2002

[3] Q Song W Liu Q Yu X Xie and Z Wang ldquoA neutral-pointpotential balancing algorithm for three-level NPC invertersusing analytically injected zero-sequence voltagerdquo in Proceed-ings of the 18th Annual IEEE Applied Power Electronics Con-ference and Exposition pp 228ndash233 Miami Beach Fla USAFebruary 2003

[4] H Zhang A von Jouanne S Dai A K Wallace and F WangldquoMultilevel invertermodulation schemes to eliminate common-mode voltagesrdquo IEEE Transactions on Industry Applications vol36 no 6 pp 1645ndash1653 2000

[5] F Z Peng ldquoA generalized multilevel inverter topology with selfvoltage balancingrdquo IEEE Transactions on Industry Applicationsvol 37 no 2 pp 611ndash618 2001

[6] J Rodrıguez J S Lai and F Z Peng ldquoMultilevel inverters a sur-vey of topologies controls and applicationsrdquo IEEE Transactionson Industrial Electronics vol 49 no 4 pp 724ndash738 2002

[7] R Bojoi A Tenconi F Profumo G Griva and D MartinelloldquoComplete analysis and comparative study of digital modu-lation techniques for dual three-phase AC motor drivesrdquo inProceedings of the 33rdAnnual IEEEPower Electronics SpecialistsConference (PESC rsquo02) vol 2 pp 851ndash857 Cairns AustraliaJune 2002

[8] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 p 1112 2006

[9] J Dixon and LMoran ldquoA clean four-quadrant sinusoidal powerrectifier using multistage converters for subway applicationsrdquoIEEE Transactions on Industrial Electronics vol 52 no 3 pp653ndash661 2005

[10] J Dixon and L Moran ldquoHigh-level multistep inverter opti-mization using a minimum number of power transistorsrdquo IEEETransactions on Power Electronics vol 21 no 2 pp 330ndash3372006

[11] P K Steimer and M D Manjrekar ldquoPractical medium voltageconverter topologies for high power applicationsrdquo in Proceed-ings of the IEEE Industrial Applications Society Annual Meetingvol 3 pp 1723ndash1730 October 2001

[12] J Rodrıguez S Bernet B Wu J O Pontt and S KouroldquoMultilevel voltage-source-converter topologies for industrialmedium-voltage drivesrdquo IEEE Transactions on Industrial Elec-tronics vol 54 no 6 pp 2930ndash2945 2007

[13] R Naderi and A Rahmati ldquoPhase-shifted carrier PWM tech-nique for general cascaded invertersrdquo IEEE Transactions onPower Electronics vol 23 no 3 pp 1257ndash1269 2008

[14] M S A Dahidah and V G Agelidis ldquoSelective harmonic elim-ination PWM control for cascaded multilevel voltage sourceconverters a generalized formulardquo IEEE Transactions on PowerElectronics vol 23 no 4 pp 1620ndash1630 2008

[15] B Ozpineci Z Du L M Tolbert and J N Chiasson ldquoFunda-mental frequency switching strategies of a seven-level hybridcascaded H-bridge multilevel inverterrdquo IEEE Transactions onPower Electronics vol 24 no 1 pp 25ndash33 2009

[16] A Nami F Zare A Ghosh and F Blaabjerg ldquoA hybridcascade converter topology with series-connected symmetricaland asymmetrical diode-clamped H-bridge cellsrdquo IEEE Trans-actions on Power Electronics vol 26 no 1 pp 51ndash65 2011

[17] A di Gaeta L Glielmo V Giglio and G Police ldquoModelingof an electromechanical engine valve actuator based on ahybrid analyticalmdashFEM approachrdquo IEEEASME Transactionson Mechatronics vol 13 no 6 pp 625ndash637 2008

[18] J Tsai C R Koch and M Saif ldquoCamless enginerdquo in Proceedingof the 47th IEEE Conference on Decision and Control pp 5689ndash5703 Cancun Mexico 2008

[19] T A Parlikar W S Chang Y H Qiu et al ldquoDesign andexperimental implementation of an electromagnetic enginevalve driverdquo IEEEASME Transactions on Mechatronics vol 10no 5 pp 482ndash494 2005

[20] V Hagenmeyer Robust Nonlinear Tracking Control Based onDifferential Flatness Reihe 8 Nr 978 Fortschritt-Berichte VDIDusseldorf Germany 2003

[21] T M Rowan and D W Kerkman ldquoA new synchronous currentregulator and an analysis of current regulated pwm invertersrdquoin Proceedings of the IEEE Industry Applications Society AnnualMeeting 1985


[22] A Bellini S Bifaretti and S Costantini ldquoImplementationon a microcontroller of a space vector modulation techniquefor NPC invertersrdquo in Proceedings of the International IEEESymposium on Industrial Electronics (IEEE-ISlE rsquo04) pp 935ndash940 LrsquoAquila Italy May 2004

[23] J F Martins A J Pires and J F Silva ldquoA Novel and simple cur-rent controller for three-phase IGBT PWM power invertersmdasha comparative studyrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (IEEE-ISIE rsquo97) pp 241ndash246 July 1997

[24] J Holtz and S Stadtfeld ldquoA predictive controller for the statorcurrent vector of ac machines fed from a switched voltagesourcerdquo in Proceedings of the International Power ElectronicsConference (IPEC rsquo83) pp 1665ndash1675 Tokio Japan 1983

[25] H le Huy and L A Dessaint ldquoAn adaptive current control forpwm invertersrdquo in Proceedings of the of IEEE Power ElectronicsSpecialists Conference (PESC rsquo86) 1986

[26] R Kennel and D Schrder ldquoPredictive control strategy forconvertersrdquo in Proceedings of the 3rd IFAC Symposium onControl in Power Electronics and Electrical Drives pp 415ndash422Losanne Switzerland 1983

[27] R Teodorescu F Blaabjerg J K Pedersen E Cengelci and PNEnjeti ldquoMultilevel inverter by cascading industrial VSIrdquo IEEETransactions on Industrial Electronics vol 49 no 4 pp 832ndash8382002

[28] Z Ye P K Jain and P C Sen ldquoA full-bridge resonant inverterwith modified phase-shift modulation for high-frequency ACpower distribution systemsrdquo IEEE Transactions on IndustrialElectronics vol 54 no 5 pp 2831ndash2845 2007

[29] Z Ye P K Jain and P C Sen ldquoA two-stage resonant inverterwith control of the phase angle and magnitude of the outputvoltagerdquo IEEE Transactions on Industrial Electronics vol 54 no5 pp 2797ndash2812 2007

[30] P Cortes A Wilson S Kouro J Rodriguez and H Abu-Rub ldquoModel predictive control of multilevel cascaded H-bridgeinvertersrdquo IEEE Transactions on Industrial Electronics vol 57no 8 pp 2691ndash2699 2010

[31] K A Tehrani H Andriatsioharana I Rasoanarivo and F MSargos ldquoA novel multilevel inverter modelrdquo in Proceedings ofthe 39th IEEE Annual Power Electronics Specialists Conference(PESC rsquo08) pp 1688ndash1693 June 2008

[32] A Bemporad MMorari V Dua and E N Pistikopoulos ldquoTheexplicit linear quadratic regulator for constrained systemsrdquoAutomatica vol 38 no 1 pp 3ndash20 2002

[33] A Bemporad and M Morari ldquoControl of systems integratinglogic dynamics and constraintsrdquo Automatica vol 35 no 3 pp407ndash427 1999

[34] C Pedret A Poncet K Stadler et al ldquoModel-varying predictivecontrol of a nonlinear systemrdquo Internal Report in ComputerScience Department ETSE de la Universitat Autonoma deBarcelona 2000

[35] H Sunan T K Kiong and L T Heng Applied PredictiveControl Springer London UK 2002

[36] P Mercorelli N Kubasiak and S Liu ldquoMultilevel bridgegovernor by using model predictive control in wavelet packetsfor tracking trajectoriesrdquo inProceedingsof the IEEE InternationalConference on Robotics and Automation vol 4 pp 4079ndash4084May 2004

[37] N Kubasiak PMercorelli and S Liu ldquoModel predictive controlof transistor pulse converter for feeding electromagnetic valveactuator with energy storagerdquo in Proceedings of the 44th IEEE

Conference on Decision and Control and the European ControlConference (CDC-ECC rsquo05) pp 6794ndash6799 December 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



inverter feeding an electromagnetic actuator In particulartwo MPCs in cascade structure are presented to control five-level inverter electromagnetic actuators which are becom-ing increasingly important in many industrial applicationsThanks to recent progress in permanent-magnet technologyvery compact and high-power electromagnetic actuators arenow available Particularly in automotive systems these actu-ators have often replaced conventional mechanical compo-nents due to their high efficiency excellent dynamic behaviorand control flexibility thus mechatronics is one of the mostnotable innovative fields in the automobile industry A recentdesign of such an actuator (eg [17]) considers optimizingwith respect to dimensions and forces Nevertheless practicaltests show that this actuator structure is not always effectivefor a large variety of applications because a complex controlstructure is needed to control such a valve [18] Recentlyfor instance a new type of electromagnetic valve drivesystem has been proposed and described [19] This paperpresents the control design for a novel permanent-magnetlinear valve actuator for use in a variable engine control toallow short-stroke and high-dynamic motions In a linearactuator the electromagnetic force is totally independentof the constrained motion Specifically the electromagneticforce is parallel to the magnetic axis and thus it is orthogonalto the constrained motion These performance character-istics make this type of 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 differentpermanentmagnet actuator topologies usingNdFeBmagnetswere considered In particular the objective of this paper is todesign a controller structure consisting of

(i) a flatness-based feedforward control

(ii) a multilevel inverter configuration

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

An important aspect of this approach is the use of geometricsystem properties such as the differential flatness neededto track the desired trajectories In this context recentlypublished research results have been considered (eg [20])in which the author proposed an interesting method tointegrateMPC and a feedforward flatness controlThe secondpart of the paper shows the structure of the cascade MPCdefined above For trajectory-based motion cycles there isusually a need to control the actuator current separately Thereference value for the actuator current is normally generatedby the speed or position controller in an outer control loopbased on the desired speed or position profiles The currentcontrol is used to provide the force or torque required for thedesired motion quickly and precisely Generally three maintechniques are employed for current control of a voltage-source inverter (VSI)

(i) closed loop control (eg PID control) using pulse-width modulated terminal voltages [21]

(ii) hysteresis control techniques [22 23](iii) predictive current control [24ndash26]

The PWM technique is currently the most popular methodfor an inverter current control The main advantage ofPWM techniques is that the inverter switches operate at afixed frequency and hardware- or software-based standardmodulators are available as industrial products (eg 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 of insulated gate bipolar transistor (IGBT)H-bridge modules with isolated DC buses [27] Amongvarious modulation techniques for a multilevel inverter thespace vector pulse width modulation (SVPWM) is widelyused However implementing the SVPWM for a multilevelinverter is complicated because it is difficult to determinethe location of the reference vector calculate on times anddetermine and select switching states A previous paperhas proposed a general SVPWM algorithm for multilevelinverters based on standard two-level SVPWMHowever thesystem response is affected by the stability requirements ofthe feedback loop which also depend on load parametersmoreover appreciable phase lag may arise even in the steadystate Recently many studies have focused on solving theseproblems (eg [28 29]) of multilevel inverter modules withindependent control of the phase angle and magnitude of theoutput voltage [29] Control and new power bridge structureshave been studied Very recent works have presented a novelbridge structure [30 31] In both structures capacitors areused but not recharged In particular an MPC strategy isproposed for a three-phase power bridge [30] Hysteresiscurrent control has a fast response and a good accuracyIt can be implemented with a simple hardware structureand in many cases it does not require any knowledgeof load parameters However it can sometimes cause veryhigh switching frequencies or if the maximum switchingfrequency is limited the current waveform 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 [24ndash26] However no feedbackloop is applied to compensate for model uncertainties Toovercome the disadvantages of the mentioned methods thisstudy develops a novel approach that combines hysteresiscontrol with a model predictive control (MPC) strategy Theproposed control system integrates flatness and two cascadeMPCs The first MPC generates an optimal desired currentby minimizing position error The second MPC considersthis optimal desired current to be the reference signal tofind an optimal switching law in combination with thehysteresis control A special energy storage and chargingcircuit was designed to provide multilevel voltages for aneffective current control By applying the proper voltagelevel which was determined by the MPC strategy to theactuator dynamic current changes and small current ripples


Coil

Valve

Lower spring

Iron pole

Permanent magnet

Upper spring






120597119894coil (119905)

120597119905= minus



119871coil (1)

120597119904 (119905)

120597119905= 119907 (119905) (2)

120597119907 (119905)

120597119905=119863119892 (119894coil (119905) 119904 (119905))

119898119894coil (119905)

+minus119896119889119907 (119905) minus 119896119891119904 (119905) + 1198650 (119905)

119898

(3)

where

119863119892 (119894coil (119905) 119904 (119905)) =Φℎ (119894coil (119905) 119904 (119905)) 119897coil

119860coil

119906119902 (119894coil (119905) 119904 (119905)) = 119863119892 (119894coil (119905) 119904 (119905)) 119907 (119905)

(4)


119865119871 (119905) = 119863119892 (119894coil (119905) 119904 (119905)) 119894coil (119905) (5)





119860coil (7)



y = y (x u u u(119901)) (8)

x = x (y y y(119902)) u = u (y y y(119902+1)) (9)



119867 minus119867119888

119888

Δ119867

119867PM

119861119903

119861

119861PM




119910 (119905) = 119904 (119905) 997904rArr 119904 (119905) = 119910 (119905)

119904 (119905) = 119910 (119905) 997904rArr 119907 (119905) = 119910 (119905)

(10)

119894coil (119905) 119863119892 (119894coil (119905)) = 119898(119896119889

119898119907 (119905) +

119896119891

119898119904 (119905) +

120597119907 (119905)

120597119905)

(11)

119880in (119905) = 119871coil120597119894coil (119905)

120597119905+ 119877coil119894coil (119905) + 119863119892 (119894coil (119905)) 119907 (119905)

(12)



Us

Uc

D1

CZ1

V2

V3

V1V5

V4

R

L

uq+



119894119889 (119905) =119898

119863119892 (119894119889 (119905))(119896119889

119898119907119889 (119905) +

119896119891

119898119904119889 (119905) +

120597119907119889 (119905)

120597119905) (13)


119906119889 (119905) = 119871coil120597119894119889 (119905)

120597119905+ 119877coil119894119889 (119905)

+ 119863119892 (119894119889 (119905) 119904 (119905))120597119904119889 (119905)

120597 (119905)

(14)





Current

FuturePast

Time



0

119905 119905minus 2119879119904 119905 minus 119879119904 119905 + 2119879119904 119905 + 119879119904

119880119888

119880119904

minus119880119888











119894coil (119896 + 1)

= 119894coil (119896) +119905119904


+119906119889 (119896) minus 119906119902 (119896))

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119863119892 (119894coil (119896))

119898119894coil (119896)

+minus119896119889119907 (119896) minus 119896119891119904 (119896)

119898)

(15)


119894119889 (119896) =119898

119863119892 (119894119889 (119896 minus 1))

sdot (119896119889

119898119907119889 (119896) +

119896119891

119898119904119889 (119896) +

119907119889 (119896 + 1) minus 119907119889 (119896)

119905119904

)

(16)



Feedforwardcontrol

CurrentMPC



MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories


trajectories


Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889



119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)


119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904



119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)



+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)


A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)


119904 (119896 + 1) = H119896A119896x (119896)


+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Coil

Valve

Lower spring

Iron pole

Permanent magnet

Upper spring






120597119894coil (119905)

120597119905= minus



119871coil (1)

120597119904 (119905)

120597119905= 119907 (119905) (2)

120597119907 (119905)

120597119905=119863119892 (119894coil (119905) 119904 (119905))

119898119894coil (119905)

+minus119896119889119907 (119905) minus 119896119891119904 (119905) + 1198650 (119905)

119898

(3)

where

119863119892 (119894coil (119905) 119904 (119905)) =Φℎ (119894coil (119905) 119904 (119905)) 119897coil

119860coil

119906119902 (119894coil (119905) 119904 (119905)) = 119863119892 (119894coil (119905) 119904 (119905)) 119907 (119905)

(4)


119865119871 (119905) = 119863119892 (119894coil (119905) 119904 (119905)) 119894coil (119905) (5)





119860coil (7)



y = y (x u u u(119901)) (8)

x = x (y y y(119902)) u = u (y y y(119902+1)) (9)



119867 minus119867119888

119888

Δ119867

119867PM

119861119903

119861

119861PM




119910 (119905) = 119904 (119905) 997904rArr 119904 (119905) = 119910 (119905)

119904 (119905) = 119910 (119905) 997904rArr 119907 (119905) = 119910 (119905)

(10)

119894coil (119905) 119863119892 (119894coil (119905)) = 119898(119896119889

119898119907 (119905) +

119896119891

119898119904 (119905) +

120597119907 (119905)

120597119905)

(11)

119880in (119905) = 119871coil120597119894coil (119905)

120597119905+ 119877coil119894coil (119905) + 119863119892 (119894coil (119905)) 119907 (119905)

(12)



Us

Uc

D1

CZ1

V2

V3

V1V5

V4

R

L

uq+



119894119889 (119905) =119898

119863119892 (119894119889 (119905))(119896119889

119898119907119889 (119905) +

119896119891

119898119904119889 (119905) +

120597119907119889 (119905)

120597119905) (13)


119906119889 (119905) = 119871coil120597119894119889 (119905)

120597119905+ 119877coil119894119889 (119905)

+ 119863119892 (119894119889 (119905) 119904 (119905))120597119904119889 (119905)

120597 (119905)

(14)





Current

FuturePast

Time



0

119905 119905minus 2119879119904 119905 minus 119879119904 119905 + 2119879119904 119905 + 119879119904

119880119888

119880119904

minus119880119888











119894coil (119896 + 1)

= 119894coil (119896) +119905119904


+119906119889 (119896) minus 119906119902 (119896))

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119863119892 (119894coil (119896))

119898119894coil (119896)

+minus119896119889119907 (119896) minus 119896119891119904 (119896)

119898)

(15)


119894119889 (119896) =119898

119863119892 (119894119889 (119896 minus 1))

sdot (119896119889

119898119907119889 (119896) +

119896119891

119898119904119889 (119896) +

119907119889 (119896 + 1) minus 119907119889 (119896)

119905119904

)

(16)



Feedforwardcontrol

CurrentMPC



MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories


trajectories


Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889



119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)


119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904



119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)



+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)


A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)


119904 (119896 + 1) = H119896A119896x (119896)


+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119867 minus119867119888

119888

Δ119867

119867PM

119861119903

119861

119861PM




119910 (119905) = 119904 (119905) 997904rArr 119904 (119905) = 119910 (119905)

119904 (119905) = 119910 (119905) 997904rArr 119907 (119905) = 119910 (119905)

(10)

119894coil (119905) 119863119892 (119894coil (119905)) = 119898(119896119889

119898119907 (119905) +

119896119891

119898119904 (119905) +

120597119907 (119905)

120597119905)

(11)

119880in (119905) = 119871coil120597119894coil (119905)

120597119905+ 119877coil119894coil (119905) + 119863119892 (119894coil (119905)) 119907 (119905)

(12)



Us

Uc

D1

CZ1

V2

V3

V1V5

V4

R

L

uq+



119894119889 (119905) =119898

119863119892 (119894119889 (119905))(119896119889

119898119907119889 (119905) +

119896119891

119898119904119889 (119905) +

120597119907119889 (119905)

120597119905) (13)


119906119889 (119905) = 119871coil120597119894119889 (119905)

120597119905+ 119877coil119894119889 (119905)

+ 119863119892 (119894119889 (119905) 119904 (119905))120597119904119889 (119905)

120597 (119905)

(14)





Current

FuturePast

Time



0

119905 119905minus 2119879119904 119905 minus 119879119904 119905 + 2119879119904 119905 + 119879119904

119880119888

119880119904

minus119880119888











119894coil (119896 + 1)

= 119894coil (119896) +119905119904


+119906119889 (119896) minus 119906119902 (119896))

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119863119892 (119894coil (119896))

119898119894coil (119896)

+minus119896119889119907 (119896) minus 119896119891119904 (119896)

119898)

(15)


119894119889 (119896) =119898

119863119892 (119894119889 (119896 minus 1))

sdot (119896119889

119898119907119889 (119896) +

119896119891

119898119904119889 (119896) +

119907119889 (119896 + 1) minus 119907119889 (119896)

119905119904

)

(16)



Feedforwardcontrol

CurrentMPC



MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories


trajectories


Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889



119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)


119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904



119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)



+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)


A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)


119904 (119896 + 1) = H119896A119896x (119896)


+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Current

FuturePast

Time



0

119905 119905minus 2119879119904 119905 minus 119879119904 119905 + 2119879119904 119905 + 119879119904

119880119888

119880119904

minus119880119888











119894coil (119896 + 1)

= 119894coil (119896) +119905119904


+119906119889 (119896) minus 119906119902 (119896))

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119863119892 (119894coil (119896))

119898119894coil (119896)

+minus119896119889119907 (119896) minus 119896119891119904 (119896)

119898)

(15)


119894119889 (119896) =119898

119863119892 (119894119889 (119896 minus 1))

sdot (119896119889

119898119907119889 (119896) +

119896119891

119898119904119889 (119896) +

119907119889 (119896 + 1) minus 119907119889 (119896)

119905119904

)

(16)



Feedforwardcontrol

CurrentMPC



MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories


trajectories


Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889



119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)


119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904



119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)



+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)


A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)


119904 (119896 + 1) = H119896A119896x (119896)


+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Feedforwardcontrol

CurrentMPC



MPC

Current

Switchingsignal

Electrical model

Desiredtrajectories


trajectories


Optimal current

119906119902

119880119888

119906mpc

119880in

119880in

119904119889



119906119889 (119896) = 119877coil119894119889 (119896) + 119871coil (119894119889 (119896 + 1) minus 119894119889 (119896)

119905119904

)

+ 119906119902 (119896)

(17)


119894coil (119896 + 1) = 119894coil (119896) minus 119894119889 (119896) + 119894119889 (119896 + 1)

minus 119905119904



119877coil)

119904 (119896 + 1) = 119904 (119896) + 119905119904119907 (119896)

119907 (119896 + 1) = 119907 (119896) + 119905119904 (119896119889119907119889 (119896) + 119896119891119904119889 (119896)

119898

minus119896119889119907 (119896) + 119896119891119904 (119896)

119898)

(18)



+ E119896d (119896)

119910 (119896) = H119896x (119896)

(19)


A119896 =

[[[[[[[[

[

(1 minus119905119904119877coil119871coil

) 0 0

0 1 119905119904

0 minus119905119904119896119891

119898(1 minus

119905119904119896119889

119898)

]]]]]]]]

]

B119896=[[[

[

119905119904

119871coil

0

0

]]]

]

E119896 =[[[[[[

[

1 (119905119904119877coil119871coil

minus 1)

0 0

0 0 0 0

0 0119905119904119896119891

119898

119905119904119896119889

119898

]]]]]]

]

d (119896) =[[[

[

119894119889 (119896 + 1)

119894119889 (119896)

119904119889 (119896)

119907119889 (119896)

]]]

]

(20)


119904 (119896 + 1) = H119896A119896x (119896)


+H119896E119896d (119896)


119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119904 (119896 + 2) = H119896A2

119896x (119896)

+H119896A119896B119896Δ119906mpc (119896)

+H119896A119896B119896Δ119906mpc (119896 minus 1)

+H119896B119896Δ119906mpc (119896 + 1)

+H119896B119896Δ119906mpc (119896) +H119896A119896E119896d (119896)

+H119896E119896d (119896)

(21)


119869 =1

2

119873

sum

119895=1

(119904119889 (119896 + 119895) minus 119904 (119896 + 119895))119879

sdotQ119901 (119904119889 (119896 + 119895) minus 119904 (119896 + 119895))

+

119873

sum

119895=1

(Δ119906mpc (119896 + 119895))119879

sdot R119901Δ119906mpc (119896 + 119895)

(22)


120597119869

120597Δ119906mpc= 0 (23)


G119901 = [[

H119896A119896H119896A2119896

]

]

F1119901 = [H119896B119896 0

H119896A119896B119896 +H119896B119896 H119896B119896]

F2119901 = [H119896B119896

H119896A119896B119896 +H119896B119896]

F3119901 = [H119896E119896 0

H119896A119896E119896 +H119896E119896 H119896E119896]

(24)


Δ119906mpc (119896) = (F119879

1119901Q119901F1119901 + R119901)

minus1

sdot (F1198791119901Q119901 (Y119889119901 (119896) minus G119901119909 (119896)


(25)


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871


119880119904

119880119904

1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862






1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871

119862

119862

1198631

1198632

1198633

1198634

1198631

1198632

1198633

1198634


1198811

1198811

1198812

1198812

1198813

1198813

1198814

1198814

119872

119872

119868119871

119868119871



Optimum

0

Time

119905119905 + 2119879119904119905 + 119879119904

119880119888

minus119880119888

minus119880119904

119880119904


Optimizer

Switchstates

Current MPC

Multimodel

Switchstates

119890

119906119902119880119888

119998opt

119998coil



120597119894coil (119905)

120597119905= minus


119894coil +Uin minus 119906119902 (119896)

119871coil (26)


119869 = minUin

1

2

119873119901

sum

119895=1

(coil (119896 + 119895) minus 119894opt (119896 + 119895))Q119894


+

119873119901

sum

119896=1

1

2(Δ119880119888 (119896 + 119895))R119894 sdot (Δ119880119888 (119896 + 119895))

(27)



bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











bridge Motor

Current reference

Motorposition

Motorcurrent

structureminus

(a)

0


119880119888

minus119880119888

minus119880119904

119880119904

0

119880119888

minus119880119888

minus119880119904

119880119904

(b)












x (119896) = [[

119894coil (119896)

119880119888 (119896)]

]

(29)


x (119896 + 1) = A2119909 (119896) + B2 (minus119906119902 (119896))

119910 (119896) = C2x (119896) = 119894coil (119896) (30)



119894coil (119896 + 2) = 11986221198602 [1198601119894coil (119896) + 1198611 (119880119904 minus 119906119902 (119896))

119880119888 (119896)]

+ 11986221198612 (minus119906119902 (119896))

(31)



119894coil (119896 + 2) = 119860111986221198602[

[

119894coil (119896)

119880119888 (119896)]

]

+ 11986221198612 (minus119906119902 (119896))

+ 11986211198611 (119880119904 minus 119906119902 (119896))

(32)



119880119888 (119896)]

minus 11986221198612 (minus119906119902 (119896))

(33)



0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3


Time (ms)

05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)



minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5



Current reference

Motorposition

Motorcurrent

Errorstructureminus











bound the branch



0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4Time (ms)


05 1 15 2Time (ms)

0

5

10

15

20

Curr

ent (

A)


Motor current

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus20

minus15

minus10

minus5


0 1 2 3 4

0

5

10

15


Time (ms)

Curr

ent (

A)


Motor current


minus20

minus15

minus10

minus

minus20

5

minus18

minus16

minus14

minus12

minus10

minus8







0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5

0

3

6


Time (ms)


Curr

ent (

A)

minus6

minus3








(34)


x (119896 + 1) = A119896x (119896) +m119901A119896x (119896)

x (119896 + 1) = A1x (119896) +m1A1x (119896)

x (119896 + 1) = A2x (119896) +m2A2x (119896) (35)

001 002 003 004 005 006 007

0

1

2

3

Time (s)

Posit

ion

(mm

)


times10minus3

minus5

minus4

minus3

minus2

minus1

(a)

001 002 003 004 005 006 007

0

5

10

15

20

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

(b)

001 002 003 004 005 006 007

45

50

55

60

65

70

75

80

85

90

Time (s)

Capa

cito

r vol

tage

(V)

(c)




10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











10038171003817100381710038171003817m119901A119896

10038171003817100381710038171003817le min

120582min (Q119901)

210038171003817100381710038171003817A119879119896P11990110038171003817100381710038171003817+10038171003817100381710038171003817P11990110038171003817100381710038171003817

1

1003817100381710038171003817m1A11003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198791P1

1003817100381710038171003817 +1003817100381710038171003817P1

1003817100381710038171003817

1

1003817100381710038171003817m1A21003817100381710038171003817 le min

120582min (Q119894)21003817100381710038171003817A1198792P2

1003817100381710038171003817 +1003817100381710038171003817P2

1003817100381710038171003817

1

(36)


100381710038171003817100381710038171003817(F1198791119901Q119901F1119901+R119901)

minus1

(F1198791119901Q119901)

100381710038171003817100381710038171003817ltmin 1

210038171003817100381710038171003817A119879P119901+P119901

10038171003817100381710038171003817

1

(37)




Energyerror(int 1198902119889119905)

023 (039) 035 125




Energyerror(int 1198902119889119905)

24 (42) 40 45





lowast






Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgment


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a multilevel inverter bridge control...

Documents