Predictive Current Control of Dual Three–phase Drives using Restrained Search Techniques and

Multi Level Voltage Source Inverters

Mario J. Durán Dept. of Electrical Engineering

University of Malaga Malaga, Spain

mjduran@uma.es

Federico Barrero, Joel Prieto, Sergio Toral Dept. of Electronic Engineering

University of Seville Seville, Spain

jprieto@esi.us.es Abstract– The interest in predictive control techniques and

multiphase drives has re–emerged in the last decade, and predictive control techniques have been recently presented as a viable alternative to conventional PI and carrier–based or Space Vector PWM techniques in the current regulation of power converters and drives. Proposed schemes have demonstrated good performance at the expenses of a high computational cost and an unknown switching frequency. However, little attention has been paid to the multi–level VSI application. A number of predictive techniques have been developed in recent times for six–phase 2–level voltage source inverters (VSI), like the restrained search predictive control technique or RSPC, which has been used to favor on–line implementation at the expense of achieving suboptimal solutions. This paper investigates the application of multi–level VSI converters in conjunction with the RSPC method in multiphase drive applications. Simulation results are provided to test the viability of a 3–level VSI power converter and the RSPC predictive control method for the current control of an asymmetrical six–phase drive.

I. INTRODUCTION

Both multiphase drives [1]–[2] and predictive control [3] have attracted much attention in the last years. Multiphase machines and drives are now playing an important role due to their specific advantages (low torque pulsations, low DC link current harmonics, high overall system reliability, power distribution per phase) in industrial applications [4]. One of the most frequently considered multiphase machines for these applications is the asymmetrical six–phase (or asymmetrical dual three–phase) machine, with two sets of three–phase stator windings spatially shifted by 30 electrical degrees and with two isolated neutral points. Predictive control strategies are interesting in the development of modern high–performance power electronics and drive systems because concepts are intuitive and easy to understand, constraints and nonlinearities are easily included, multivariable case can be considered, and the controller is easy to implement and redefine. Predictive control techniques are characterized by the use of the model of the system for the prediction of the future behavior of the controlled variables. This information is then used by the controller to obtain the actuation according to a predefined optimization criterion. In power converters and electrical drives, the optimization problem is simplified considering the discrete nature of the power converter which provides a finite set of possible switching states of the converter. The

number of the finite set possibilities depends on the possible combinations of the on/off switching states of the power switches. The optimization problem is then reduced to the evaluation of all possible states and the selection of the one which minimizes the cost function. However, the possible states to be used in the control algorithm can be somewhat high for an n–phase machine, obtaining 2n available vectors in a 2–level VSI and 3n in a 3–level VSI (64=26 and 36=729 respectively in the asymmetrical six–phase machine).

Studies aimed at combining the predictive control technique and multiphase drives have been explored together in [5]–[12], using the asymmetrical six–phase drive. The obtained results prove that predictive control can provide enhanced performance for multiphase drives using a high level design. Conventional 2–level VSI were used in addition with the asymmetrical six–phase machine. However, the applicability of the predictive control techniques is hindered due to the use of intensive computations [5]–[8], and special operation modes (quasi–balanced operation and sinusoidal output voltage) must be assumed to reduce the number of considered voltage vectors. Since the main shortcoming of predictive control methods is the high computational cost of the optimal solutions, other proposals have been recently presented to restrict the available voltage vectors at each switching period, aiming at reducing the computing effort [9]–[12]. Such control methods have been termed RSPC methods and they reduced the subset of available vectors at each instant but they do not discard a priori any vector, as it occurs in previous simplified methods [6]–[9]. Results presented in [9]–[12] show that RSPC methods provide good performance both in steady and transient state using 2–level VSI. However, although the computing time is reduced using RSPC methods, the control reference using these techniques is never achieved.

This work shows the performance of a RSPC method with a multi–level VSI. Using a 3–level VSI, more available voltage vectors are obtained, and the reference is more achievable using predictive control techniques. Real–time implementation of the predictive current control method is favored due to the considered suboptimal solutions. Simulations are provided, comparing the obtained results with standard predictive current control methods and other RSPC methods with 2–level power converters. The paper is structured so that the fundamentals about the asymmetrical six–phase drive and the predictive control method are described in section II. The introduced RSPC scheme is presented in section III. Section IV shows the simulation 978-1-4244-6392-3/10/$26.00 ©2010 IEEE 3171

results to compare the effectiveness and viability of the introduced control technique. Finally, section V summarizes the conclusions.

II. SIX–PHASE DRIVE AND PREDICTIVE CURRENT CONTROL

A squirrel cage induction motor with two isolated neutral points is considered in this work. A detailed scheme of the drive is provided in Fig. 1. The analyzed six–phase machine is a continuous system which can be described by a set of differential equations. Machine model, according to the vector space decomposition (VSD) approach [13], consists of three stator–rotor pairs of windings in mutually orthogonal subspaces. One stator–rotor pair engages in electromechanical energy conversion (α–β subspace), while the others do not. The α–β subspace represents the fundamental supply component plus supply harmonics of the order 12n±1 (n=1,2,3,…). The second stator–rotor pair (x–y subspace) represents supply harmonics of the order 6n±1. The zero sequence harmonic components do not exist due to the isolated neutral points. Consequently, the third subspace is not considered in what follows. The current components in the x−y subspace do not contribute to the air–gap flux and torque. They only produce losses and are limited by the stator resistance and stator leakage inductance, which is usually small. The model of the electromagnetic part of the considered multiphase machine in stationary reference frame is as follows:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅−⋅−⋅⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

βriαriβsiαsi

p

rLmLrLmL

mLsLmLsL

βriαriβsiαsi

rRrLrωmLrωrLrωrRmLrω

sRsR

=βsuαsu

0000

0000

00

000000

00

(1)

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

ys

xs

ls

ls

ys

xs

s

s

ys

xs

ii

pL

Lii

RR

uu

00

00

(2)

where p is the time derivative operator, ωr the rotor angular electrical speed, and Rs, Ls=Lls+Lm, Rr, Lr=Llr+Lm and Lm the electrical parameters of the machine, with index l denoting leakage inductances.

The 6–phase VSI is a 3–level inverter, and it has a total number of 36=729 switching states defined by six switching functions, which correspond to the six inverter legs [Sa, Sb, Sc, Sd, Se, Sf], where Siœ{0,0.5,1}. Different switching states and the voltage of the DC link (VDC) define the phase voltages which can in turn be mapped into the α–β and x–y subspaces according to the VSD approach. Consequently, the 729 different on/off combinations of the six VSI legs lead to 729 switching states in the α–β and x–y subspaces. Figure 2 shows the active vectors in the α–β and x–y subspaces, where each switching state is identified by a number [SaSbScSdSeSf]. It must be noted that the 729 possible switching states imply only 361 different voltage space vectors in the α–β and x–y subspaces. Nevertheless, redundant states have to be considered because they have a different impact on the switching patterns.

+

+

-

-

a

b c

d

e

far

br

cr

xr

yr

zr

30º

6 E.M.

adbecf

6 6Inverter

6 E.M.

Si

r

is

DC_LinkVdc

a

b

c

d

e

fCONTROL SIGNAL

(switching state)

DCV

Fig. 1. A general scheme of an asymmetrical six−phase (dual three–phase) AC drive. A 3–level VSI is considered.

Fig. 2. Voltage space vectors and switching states in the α–β (upper side) and x–y (lower side) subspaces for a 6–phase asymmetrical drive using a 3–level VSI.

The predictive control techniques overcome the difficulties

in the generalization of other current control methods for multiphase drives, guaranteeing fast current and torque responses. A block diagram of a standard predictive current control (SPC) technique for the asymmetrical dual three–phase induction machine is provided in Fig. 3.

3172

Fig. 3. Standard predictive current control method presented in [5] and [6]. The sequencer issues voltage vectors one at a time, while the minimizer chooses the one that provides the lower value of J.

The control actions are obtained by solving an

optimization problem aimed at minimizing a cost function. Different cost functions (J) can be used to express different control criteria [14]. In this case, the cost function is defined as J=|iαβs

*–îαβs|+λxy·|ixys*–îxys|, where is

* is the stator reference current, îs is the predicted stator current and λxy is a weighting factor. Other cost functions could be used like the square or integral current errors, the flux and torque errors, or terms to minimize the switching stress or the DC link voltage balancing. The machine model must be discretised in order to be of use as a predictive model. A forward Euler method with a sampling time Tm can be used, producing equations in the needed digital control form where predicted variables depend on past values and not on present values of variables. A detailed predictive model of the asymmetrical dual three–phase drive can be found in [5].

III. RSPC TECHNIQUES AND PROPOSED CONTROL METHOD

The SPC technique discards some voltage vectors a priori to allow real time implementation of the algorithm in the multiphase drive. This restriction is only available if sinusoidal output voltage and quasi–balanced operation of the drive is considered. However, RSPC methods use constraints to select the possible vectors, guaranteeing a reduced number of commutations in the VSI legs. These constrains are evaluated at each sampling time, and possible voltage vectors are not discarded a priori. With the proposed procedure, different subgroups of switching states are considered in each sampling time and the real time implementation is feasible because only the sampling time is decreased and the switching frequency is increased. Consequently, different voltage vectors are discarded at different times. The constraints proposed in [10] and [11] for a 2–level 6–phase VSI are:

a. The subset of available vectors must include those that imply one commutation or less.

b. Any VSI leg must not commutate in two consecutive switching periods.

The first restriction reduces the overall number of VSI

commutations, while the second one guarantees a low

number of commutations in every VSI leg. Condition b) is important because it allows halving the switching period compared to SPC which favours the drive performance. Condition a) helps to reduce the number of commutations but it is a somewhat restricting condition. Both conditions imply that only 6 voltage vectors, among the 64 possible in a 2–level VSI, are examined [10]. This method was termed in [10] as RSPC6v. A more sophisticated version of the RSPC6v was proposed in [12], redefining the condition a) to increase the possible voltage vectors considered by the control algorithm:

a. The subset of available vectors must include those

that imply two or less commutations.

According to this new condition, the control algorithm can change up to two VSI switching states. However, according to condition b), that it is preserved, if two power switches are changed in a sampling period, only the other four ones can be commutated during the next sampling time. The RSPC method combining these new conditions computes 11 voltage vectors if two legs were commutated, 16 voltage vectors if only one leg was switched in the previous switching period or 22 voltage vectors if no legs were commutated [12]. The method proved to provide higher performance than [10] but increasing the computational cost of the algorithm.

The applied RSPC method for the 3–level and multi–phase

drive is based on the RSPC6v method [10] to favour the real time implementation. In this case, the considered conditions imply that 11 voltage vectors, among the 729 possible, are taken into account by the predictive control algorithm and the method will be termer RSPC11v from now on. Let us illustrate this with an example where the vector [000000] is applied at k–1 sampling time, and the vector [100000] is then used (sampling period k). For the next sampling time k+1, the following vectors are possible:

1 → [1 0 0 0 0 0] 5 → [1 0 1 0 0 0] 9 → [1 0 0 0 1 0] 2 → [1 0.5 0 0 0 0] 6 → [1 0 0 0.5 0 0] 10 → [1 0 0 0 0 0.5] 3 → [1 1 0 0 0 0] 7 → [1 0 0 1 0 0] 11 → [1 0 0 0 0 1] 4 → [1 0 0.5 0 0 0] 8 → [1 0 0 0 0.5 0]

IV. SIMULATION RESULTS

This section examines the performance of the proposed RSPC11v technique. It will be compared with optimal SPC considering all 729 possible vectors (it will be termed as SPC729v) and with optimal SPC using 2–level VSIs (termed SPC64v). The steady state performance is firstly addressed by considering a current reference with amplitude of 0.67 A and an angular frequency of 300 rad/s (called test 1 from now on). Four performance parameters are considered for comparison purposes:

Mean square error of α current component (MSE iα). Mean square error of x current component (MSE ix). Total harmonic distortion (THD). Total number of commutations (TNC).

3173

1) Influence of weighting factor λxy.

Both SPC and RSPC11v methods are dependent on the weighting factor λxy and a suitable value must be achieved to obtain good α–β current tracking but with low values of x–y current components (note that x–y components do not contribute to the overall torque generation but only generate losses). Table I and Fig. 4 shows the performance parameters against λxy when the switching frequency of the PWM is 5 kHz. Please note that the switching period of the different methods is characterized by the maximum achievable switching frequency since the switching frequency is inherently variable in predictive methods. In the case of Table I and Fig. 4 the maximum achievable switching frequency is set to 10 kHz and the switching periods of the SPC and RSPC methods are 0.05 ms and 0.025 ms respectively. It can be noted in Fig. 5 that higher values of the weighting factor λxy leads to higher MSE iα and lower MSE ix for both SCP and RSPC11v methods as expected. The THD is reduced as λxy increases and the TNC slightly increases. The value of λxy is set to 0.4 in the subsequent results to obtain a good compromise between the four performance parameters. It can be observed that the proposed RSPC11v method achieves better performance in general than the SPC729v even though 98.5% of the available voltage vectors are discarded each switching period. This fact shows the potential of restrained search predictive control (RSPC) methods to both reduce the computational cost and to improve the current control performance. In the considered 3–level six–phase VSI the benefits of the RSPC approach over optimal solutions is more evident than in the previous 2–level six–phase VSIs [10]–[12] because the number of possible vectors is much higher in the former case.

Table I: Performance of SCP729v and RSPC11v for test 1 with variable value of the weighting factor and maximum switching frequency of 10 kHz.

Test 1 TNC MSE iα MSE ix THD λxy=0.1 SCP729v 7.686⋅103 4.4475 0.0454 8.4631 RSPC11v 4.637⋅103 2.8263 0.0887 19.0759 λxy=0.2 SCP729v 7.232⋅103 4.5041 0.0418 7.6246 RSPC11v 4.240⋅103 2.8101 0.0523 10.8767 λxy=0.3 SCP6729v 7.163⋅103 4.5453 0.0374 7.1367 RSPC11v 4.749⋅103 3.2228 0.0498 10.3785 λxy=0.4 SCP729v 9.065⋅103 4.1765 0.0327 6.0311 RSPC11v 4.718⋅103 3.3688 0.0462 9.9334

2) Influence of switching frequency fsw.

Since the performance of predictive methods is highly dependent on the switching frequency, Table II and Fig. 5 show the performance parameters as a function of the maximum achievable switching frequency. It can be observed that the predictive control performance (of both SCP729v and RSPC11v) greatly improves as the switching period is decreased (higher switching frequency, fswitch in Fig.

5). The RSPC11v reduces the switching stress and improves the error in α current tracking compared to SCP729v while the MSE ix and the THD are slightly worse. Nevertheless the global performance is at least similar to optimal methods while the computational cost is extremely lower.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.452.5

3

3.5

4

4.5

5

5.5

λxy

MS

E i α

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450.02

0.04

0.06

0.08

0.1

λxy

MS

E i x

SPC729v

RSPC11v

SPC729v

RSPC11v

Fig. 4. MSE iα, MSE ix, as a function of the weighting factor λxy with maximum switching frequency of 10 kHz for SPC729v and RSPC11v methods (test 1).

2 3 4 5 6 7 8 9 100

5

10

15

20

f sw itchmax

MS

E i α

RSPC11v

SPC729v

2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

f sw itchmax

MS

E i x

RSPC11v

SPC729v

Fig. 5. MSE iα, MSE ix, as a function of the maximum switching frequency for SPC729v and RSPC11v methods with λxy=0.4 (test 1).

Table II: Performance of SCP729v and RSPC11v for test 1 with variable value of the maximum switching frequency and weighting factor λxy=0.4.

Test 1 TNC MSE iα MSE ix THD fswitchmax=2.5kHz SCP729v 1.922⋅103 18.9630 0.1352 23.7389 RSPC11v 0.996⋅103 11.4884 0.1465 31.1920 fswitchmax=5kHz SCP729v 3.621⋅103 8.8575 0.0648 12.9612 RSPC11v 1.993⋅103 5.5764 0.0718 15.8092 fswitchmax=7.5kHz SCP6729v 6.163⋅103 5.8346 0.0433 8.5951 RSPC11v 3.107⋅103 3.9709 0.0530 11.6258 fswitchmax=10kHz SCP729v 9.065⋅103 4.1765 0.0327 6.0311 RSPC11v 4.718⋅103 3.3688 0.0462 9.9334

3174

3) Current tracking capability and voltage waveform.

The performance of the different methods has been explored so far through the calculation of four performance parameters (MSE iα, MSE ix, THD and TNC).

Figure 6 shows directly the current tracking of the α reference current components when a switching frequency of 5 kHz is used. The RSPC11v proves to provide slightly enhanced performance compared to SPC729v and better performance than 2–level optimal solution SPC64v in spite of using only 11 vectors and involving less commutations.

4) Performance with different amplitudes and frequencies.

All results reported so far are related to test 1, using a reference current with an amplitude (ipeak) of 0.67 A and an angular frequency (ω) of 300 rad/s. Table I summarizes the results for another two test, where different amplitudes and angular frequencies for the reference current are considered:

Test 2: ipeak = 4.50 A and ω = 300 rad/s. Test 3: ipeak = 5.85 A and ω = 30 rad/s.

From Table III it can be observed that the proposed RSPC

method obtains a lower performance in terms of current error than SPC, especially in test 3.

0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25-1

0

1

Time (s)

i α (

A)

0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25-1

0

1

Time (s)

i α (A

)

0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25-1

0

1

Time (s)

i α (A

)

Fig. 6. Current tracking of the α current component for RSPC11v (upper), SPC729v (middle) and 2–level VSI SPC64v (bottom) methods (test 1) when the maximum switching frequency is set to 5 kHz.

Nevertheless the total number of commutations is highly reduced, especially in test 2, and the computational cost is reduced in a 98.5 %. From the previous results it can be observed that in all cases there is a great reduction in terms of computational effort (98.5%) and a high reduction of the switching stress (up to 7 times lower) while the current control performance is variable depending on the reference current. For some references the constraints are more adequate and the RSPC achieves even better performance while for other references the subset of voltage vectors is not optimum and the performance can be lower. Better performance can be achieved using RSPC if more vectors are considered, as in [12]. This would reduce the performance improvement assuming a higher computational effort (but still much lower than using all 729 vectors).

Table III: Performance of SCP729v and RSPC11v for tests 2 and 3 with weighting factor λxy=0.4 and maximum switching frequency of 5 and 10

kHz.

Test 2 TNC MSE iα MSE ix THD fswitchmax = 5 kHz SPC729v 0.936⋅104 3.6327 0.0122 0.9286 RSPC11v 1.473⋅103 5.5088 0.0318 7.1561 fswitchmax = 10 kHz SPC729v 2.390⋅104 1.4775 0.0408 1.7028 RSPC11v 3.495⋅103 3.3419 0.0183 4.3892 Test 3 fswitchmax = 5 kHz SPC729v 4.513⋅103 1.2902 0.0222 10.9252 RSPC11v 1.4072⋅103 7.7690 0.0307 15.3343 fswitchmax = 10 kHz SPC729v 0.8441⋅104 0.666 0.0114 10.3444 RSPC11v 3.2983⋅103 5.0451 0.0172 13.2415

V. CONCLUSION

A modification of the SPC scheme for the current control of the asymmetrical 6–phase induction machine has been proposed. The introduced scheme is based on a restrained search which limits the available voltage vectors in the optimization process and forces the corresponding inverter switching states to comply with a series of restrictions. A suboptimal predictive control is then obtained, which reduces the required control computing time. A 3–level VSI is considered to have a more achievable reference using the predictive control techniques. The introduced technique provides better performance than standard predictive current control methods in steady and transient states at similar switching frequencies, and that RSPC with conventional 2–level VSI converters. Simulation results are provided to examine the potential of the control method, showing the obtained benefits. The presented technique also favours the real time implementation compared to optimal predictive control techniques, and it can be easily extended to other multiphase drives.

ACKNOWLEDGMENT The authors gratefully acknowledge the Spanish

Government for the economical support provided within the National Research, Development and Innovation Plan, DPI2009/07955 and PTI–Itaipu Binacional.

REFERENCES [1] E. Levi, R. Bojoi, F. Profumo, H.A. Toliyat, S. Williamson,

“Multiphase induction motor drives – A technology status review,” IET Electric Power Applications, vol. 1, no. 4, 2007, pp. 489–516.

[2] G.K. Singh, “Multi–phase induction machine drive research–a survey”, Electric Power Systems Research, 61, pp. 139–147, 2002.

[3] P. Cortés, M.P. Kazmierkowski, R.M. Kennel, D.E. Quevedo, J. Rodríguez, “Predictive Control in Power Electronics and Drives,” IEEE Transactions on Industrial Electronics, vol. 55, no. 12, pp. 4312–4324, 2008.

[4] E. Levi, “Multiphase electric machines for variable–speed applications,” IEEE Trans. on Ind. Electronics, vol. 55, no. 5, 2008, pp. 1893–1909.

3175

[5] M.R. Arahal, F. Barrero, S. Toral, M.J. Durán, R. Gregor, “Multi–phase current control using finite–state model–predictive control,” Control Engineering Practice, vol. 17, no. 5, pp. 579–587, 2009.

[6] F. Barrero, M.R. Arahal, R. Gregor, S. Toral, M.J. Durán, “A proof of concept study of predictive current control for VSI driven asymmetrical dual three–phase AC machines,” IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 1937–1954, 2009.

[7] F. Barrero, M.R. Arahal, R. Gregor, S. Toral, M.J. Durán, “One–step Modulation Predictive Current Control Method for the Asymmetrical Dual–three phase Induction Machine,” IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 1974–1983, 2009.

[8] R. Gregor, F. Barrero, S. Toral, M.J. Durán, M.R. Arahal, J. Prieto, J.L. Mora, “Predictive–SVPWM Current Control Method for Asymmetrical Dual–three phase Induction Motor Drives,” IET Electric Power Applications, doi: 10.1049/iet–epa.2008.0274.

[9] R. Gregor, F. Barrero, J. Prieto, M.R. Arahal, S.L. Toral, M.J. Durán, “Enhanced Predictive Current Control Method for the Asymmetrical Dual–three phase Induction Machine,” in Proc. of the IEEE International Electric Machines and Drives Conference (IEMDC09), Miami, USA, 2009.

[10] M.J. Durán, M.R. Arahal, F. Barrero, S.L. Toral, R. Gregor, “Restrained Search Predictive Control of Dual Three–phase Induction Motor Drives,” in Proc. of the International Conference on Renewable Energies and Power Quality (ICREPQ09), Valencia, Spain, 2009.

[11] M.J. Durán, M.R. Arahal, F. Barrero, S.L. Toral, R. Gregor, “Predictive Control of Asymmetrical Six–phase Induction Motor Drives with Submodulation and Restrained Search,” in Proc. of the European Conference on Power Electronics and Applications (EPE09), Barcelona, Spain, 2009.

[12] M.J. Durán, M.R. Arahal, F. Barrero, S.L. Toral, R. Gregor, “Improved Techniques of Restrained Search Predictive Control for Multiphase Drives,” in Proc. of the IEEE International Electric Machines and Drives Conference (IEMDC09), Miami, USA, 2009.

[13] Y. Zhao, T.A. Lipo, “Space vector PWM control of dual three–phase induction machine using vector space decomposition,” IEEE Trans. on Ind. Applications, vol. 31, no. 5, 1995, pp. 1100–1109.

[14] P. Cortés, S. Kouro, B. La Rocca, R. Vargas, J. Rodríguez, J.I. León, S. Vázquez, L.G. Franquelo, “Guidelines for Weighting Factors Design in Model Predictive Control of Power Converters and Drives,” in Proc. of the IEEE International Conference on Industrial Technology (ICIT09), Churchill, Australia, 2009.

3176