Finite-Control-Set Model Predictive Control With ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 9, NO. 2, MAY 2013

Finite-Control-Set Model Predictive Control With Improved Steady-State Performance Ricardo P. Aguilera, Member, IEEE, Pablo Lezana, Member, IEEE, and Daniel E. Quevedo, Member, IEEE

Abstract—Finite-control-set model predictive control (FCSMPC) is a novel and promising control scheme for power converters and drives. Many practical and theoretical issues have been presented in the literature, showing good performance of this technique. The present work deals with one of the most relevant aspects of any controller, namely, the steady-state operation. As will be shown, basic FCS-MPC formulations can be enhanced to achieve a reduced average steady-state error. As an illustrative example, we apply our proposal to a simple H-Bridge power converter. Experimental results were carried out to verify the performance obtained by the proposed predictive strategies. Index Terms—Emerging control issues, power electronics, predictive control.

I. INTRODUCTION

P

OWER converters have been in constant development not only to obtain new advantageous topologies, but also to improve control techniques used. In this latter field, model predictive control (MPC) has emerged as a promising control technique for power converters [1]–[4]. Different predictive control approaches have been proposed to handle power converters, showing that these methods hold many advantages when compared with traditional PWM control schemes [5], [6]. In general, MPC or receding horizon control [7]–[9] is a control technique which calculates the control action by solving, at each sampling instant, an optimal control problem which forecasts, over a finite horizon, the future system behavior from the current system state. This generates an optimal control sequence. The control action to be applied to the plant is the first element of this sequence. The main advantage of MPC is that system constraints (e.g., voltage and current limitations) and nonlinearities can be explicitly taken into account. In particular, one of the most attractive predictive strategies for power converters and drives is the so-called finite-control-set model predictive control (FCS-MPC) [5], [10] (also known as direct MPC [3]). An important feature of FCS-MPC is the fact that this strategy directly takes into account the power switches as control input of the system. Since power switches

Manuscript received December 15, 2011; revised May 15, 2012; accepted July 09, 2012. Date of publication August 01, 2012; date of current version January 09, 2013. This work was supported in part by the Chilean Research Council (CONICYT) under Grant FONDECYT 1100697 and by the Australian Research Council’s Discovery Projects funding scheme under Project DP110103074. Paper no. TII-11-1024. R. P. Aguilera and D. E. Quevedo are with the School of Electrical Engineering and Computer Science, The University of Newcastle, Australia (e-mail: [email protected]; [email protected]). P. Lezana is with the Departamento de Ingeniería Eléctrica, Universidad Técnica Federico Santa María, Valparaíso, Chile (e-mail: [email protected]). Digital Object Identifier 10.1109/TII.2012.2211027

can adopt only two states, namely 1 or 0 (ON or OFF), the input is restricted to belong to a finite set of feasible switch combinations. Therefore, one obtains a finite number of predictions for the different electrical variables at each sampling period. To track references, a cost function, which considers the tracking error at each sampling instant is commonly used. This cost function is evaluated for each possible switch combination. Thus, the optimal switching action to be applied to the converter is the one which minimizes the cost function. The optimization procedure is carried out at each sampling instant. It is important to remark that this procedure can consider variables of different nature, as electrical, mechanical, or of any other kind [11]–[18]. In this way, FCS-MPC is an easy-to-understand strategy that can be implemented to several kind of power converters and covering a wide range of objectives [10]. However, practical issues must be considered, which include: • computational effort, such as the number of switching elements is increased (e.g., multilevel converters [19]); • design of cost function especially when more than one target is desired [20]; • measurement noise; • model parameter mismatch. Most of these problems have been studied in previous works showing that FCS-MPC is able to deal with them [12], [13], [21]. The present work is focused in one of the most important aspects of any controller, namely, the steady-state error. In [22], extensive simulations were carried out, showing that existing FCS-MPC algorithms give, in general, a nonzero steady-state error. Moreover, this error is more relevant when lower switching frequency and/or lower magnitude references are used. In the present paper, we extend [22] by performing a mathematical analysis to determine the steady-state error. To reduce the steady-state error, two different approaches are proposed. The first one is based on the idea of using different sampling and actuation instants, resembling classical PWM techniques [23]. The second one takes into account not only the errors at the sampling instant, but also the error generated during the intersampling [9]. To facilitate the understanding of the problem and the proposed solutions, a simple architecture consisting in current control for a four-quadrant chopper converter will be used. The remainder of this paper is organized as follows. Section II gives a brief review of FCS-MPC. In Section III, an analysis of the steady-state error generated by FCS-MPC is presented. Sections IV and V present our proposals which are aimed at reducing the steady-state error. Section VI illustrates the results obtained when our proposal is applied to a four quadrant chopper. Section VII draws conclusions.

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AGUILERA et al.: FCS-MPC WITH IMPROVED STEADY-STATE PERFORMANCE

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of the variables to be controlled at the instant but also to know their values at the instant . To do this, we take advantage of the fact that the previous optimal switching remains constant until the next sampling instant action . Thus, considering the system model presented in (1), , can be estimated by the system state, at the instant

Fig. 1. FCS-MPC: temporal scheme.

II. FINITE-CONTROL-SET MODEL PREDICTIVE CONTROL Here, a brief review of FCS-MPC is presented. Further theoretical analysis can be found in [24]–[26]. For some examples of applications in power electronics, see [3], [5], [10], [27]. A. Basic Principles FCS-MPC operates in discrete time with fixed sampling freand uses a discrete-time model of the plant quency (1) where represents the system states and stands for the control inputs. In this case, FCS-MPC directly considers the converter power switches as the control input of the system. Thus, each control input can adopt only two values, remaining con, for all stant during the sampling period, , i.e., . Consequently, the input belongs to a finite control set of switch combinations . To predict the future system behaviour over a horizon of , length , FCS-MPC uses a cost function, namely where is a feasible input sequence. Thus, the optimal switching sequence is obtained by minimizing this cost function

Since the input belongs to a finite set, the minimization process is, in general, carried out by evaluating all of the possible switch combinations. Finally, the optimal switching input to be applied is the first element of the optimal switching . Thus, this predictive strategy directly sequence, provides the optimal switching actions to be applied to the converter, and no intermediate modulation stages are required [10].

(2) 3) Prediction: In order to obtain the optimal input, FCS-MPC forecasts the system state behavior at the instant by evaluating all the possible switch combinations. This is expressed via (3) . These predictions are where compared to the reference by evaluating a cost function, , i.e.,

where is a positive definite matrix (weighting matrix). is that one which Thus, the optimal switching input produces the minimum value in the cost function. III. STEADY-STATE ISSUES Here, we investigate the steady-state performance of FCS-MPC in terms of the average value. A. Theoretical Background As already stated, the control target of FCS-MPC is to minimize the tracking error produced at each sampling instant . Consequently, the predictive controller will optimize the at sample response of the system. However, since the system states exist in continuous time, e.g., converter output current, in some cases, the system will give a poor intersample performance. Therefore, it is important to analyze the continuous-time response when using the discrete-time control law given by FCS-MPC. For that purpose, we propose to consider the deviation from the average of the controlled variable during the intersampling times. This is expressed via

B. Implementation In practical implementations of FCS-MPC with prediction , there exist three main stages: measurement, horizon estimation and prediction. A temporal scheme of these stages is depicted in Fig. 1. 1) Measurement: Since this predictive strategy works in discrete time, it is required to take a measurement of the state at each sampling instant . (in some cases, variables some of these variables can be estimated from other measurements [28]). These current state values are used by the controller to decide upon the optimal switching action to be applied. 2) Estimation: To account for computational delays, in standard formulations, the optimal switching action is applied , . Thus, the effect of this action will be at time . This delay of one samobserved only at the instant pling period results in the need to predict not only the value

(4) where

is the average value of the controlled variable

As will be shown later, with standard FCS-MPC, the average will not always be the same value of the controlled variable as the desired reference . This issue constitutes the main motivation of the present work. B. Study Case: H-Bridge As an illustrative example, we will analyze the continuoustime performance of a four-quadrant chopper converter under FCS-MPC. This topology is very popular in audio applications

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Fig. 2. Four-quadrant chopper converter.

[29], [30] as well as power drives [19]. This converter is comprised of two pairs of complementary power switches ( , , , as depicted in Fig. 2. and , ) and a dc-voltage source The load voltage can be easily obtained from

Fig. 3. Steady-state pattern for an input sequence

Since

.

, we finally obtain (9)

where . Thus, the continuous-time dynamic model for the load current is given by

Since finite set

,

, the control input,

(5)

A similar analysis can be carried out to obtain the upper bound . First, we note from Fig. 3 that . Then, considering (8) one can derive the following expression:

, belongs to the

(10) Consequently, the average value can be determined as follows:

produced by this pattern

Therefore, can take only one of three different values, namely , 0 and . Due to the fact that, with FCS-MPC, the power switches remain constant during the intersampling, the output current in (5) can be exactly represented, in continuous time via

(6)

(11)

for all , where is the system time constant. Consequently, the value of the load current at the next sampling instant is

We next analyze under which conditions FCS-MPC will pro. To obtain the optimal vide the sequence , (see Fig. 3), the input to be applied at the instant predictive controller will compare the error at the next sampling instant , produced by the different possible inputs, i.e.,

(7) To analyze the steady-state performance of FCS-MPC when applied to this converter, we will focus on the particular peri. Under this situaodical input sequence tion, the system will reach a steady-state behavior as depicted in Fig. 3. To properly describe this pattern, it is necessary to oband of the state trajectory. tain the steady-state bounds From Fig. 3, we can notice that the lower bound satisfies that . Thus, from (7) and considering , we have that (8) Then, taking into account (7) with

, we have that

. Considering that , the input will generate the tracking error represented via

while the input expressed by

will produce the tracking error

where, from (4), . , an To maintain this periodical pattern, at the instant input should be applied to the converter. Due to the optimization carried out by FCS-MPC, this will occur whenever . Thus, a critical average error can be obtained by equalizing both predicted errors , yielding (12)


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Fig. 5. Intermediate sampling model predictive control temporal scheme.

Fig. 4. Synchronous current sampling with PWM.

Therefore, if , then the minimum forecast error will be . Consequently, will be applied. It is important to emphasize that the input sequence pattern, , will prevail provided This lower limit can be determined via

.

Then, similar to the standard FCS-MPC algorithm, predictions are calculated by considering the measurement and the system state estimation, . For IS-MPC, however, . predictions are obtained for the instant This gives a better approximation to the average of the output , which, when controlled, can be expected to get variable, closer to the reference. With this modifications, the proper expression for the prediction is

This expression is used to evaluate the IS-MPC cost function (13) Assuming that , it is possible to approximate . Then, from (13), the critical reference can be approximated as (14) , in A similar result can be obtained for the case when which case the critical limit becomes . The preceding analysis shows that for any reference which , FCS-MPC will produce the belongs to the range same average value . This steady-state analysis can be extended for different steady-state patterns. In Sections IV–VII, we will propose two modifications of FCS-MPC which are aimed at reducing the steady-state error. IV. INTERMEDIATE SAMPLING (IS-MPC) A. Basic Principle The use of synchronous current sampling with sinusoidal PWM schemes is widespread by sampling either at the maximum or minimum of the triangular carrier [23]. Thus, a good approximation of the average value of the controlled variable can be obtained, as shown in Fig. 4. This idea can be easily adapted for use in the FCS-MPC . As strategy in order to approximate the average value of previously stated, FCS-MPC can only change the value of the converter switches at the beginning of each sampling period . Then, the measurement of is taken at each instant , which will be denoted by . Due to this temporal shift of , the final estimate of the system state can be calculated as

(15) . The optimal for all possible combinations of , which minimizes (15), is then chosen. switching action The temporal scheme of the IS-MPC algorithm is shown in Fig. 5. Remark 1: Notice that, since the measurements are taken in the middle of the sampling period, the obtained samples are less affected by commutation noises. Additionally, IS-MPC obtains predictions of the system state for one sampling period ahead after measurements are taken (see Fig. 5) while standard FCS-MPC carries out the state predictions for two sampling periods ahead, as described in Section II. Thus, our proposal exhibits smaller estimation and prediction errors. B. Application to H-Bridge Here, we apply the proposed IS-MPC algorithm to the four quadrant chopper presented in Section III-B. At the middle of , a measurement of the load each sampling period, . Then, from (7), the load current, at current is taken, the instant , is estimated by (16) Afterwards, the predictions are obtained considering from (16), as the initial state

,

(17) Finally, the load current prediction given by (17) is compared with the desired reference given at the beginning of the next sampling period . Thus, the cost function for the four quadrant chopper under IS-MPC is expressed via

(18)

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B. H-Bridge for the four-quadrant chopper is obThe integral term tained by replacing (6) in (19) to yield Fig. 6. Temporal scheme of FCS-MPC with integral minimization.

(21)

V. INTEGRAL ERROR TERM A. Basic Principle As mentioned in Section III-A, FCS-MPC optimizes the at sample response of the system. However, since the system states exist in continuous time, a poor intersample response can be obtained in some cases. To overcome this situation, we propose to take into account the intersample performance by adding an extra term in the cost function . This term will be chosen as , and the prethe integral of the error between the reference, dicted continuous behaviour of the controlled variable within a . Taking into acsampling period , count the one-sampling-period delay, this integral error is represented via (19) This integral term gives us information about the average error generated by each switch combination. In addition, we propose to include -previous integral error terms to improve the continuous-time performance of the predictive controller. Thus, the proposed cost function for this strategy is given by

(20) is a weighting factor which gives us a higher/lower where represents preimportance to the integral term, while vious integral errors. In the sequel, we will denote this strategy , where the subindex stands for the number of as previous integral terms included in the cost function. The temporal scheme for this predictive algorithm is depicted in Fig. 6. Remark 2: It is important to emphasize that this cost function can be associated with a discrete Proportional-Integral (PI) controller, which is comprised of a proportional part, which acts over the instantaneous error, , providing a fast response, and an integral part which is related with the accumulated error. In (20), the accumulated error considers the continuous behaviour of the control variable achieving a better intersample perforcan be adjusted in a similar mance. Therefore, the factor fashion than in a PI design. Remark 3: The pure integrator in the cost function (20) may be replaced by a low-pass filter. This may improve the system behavior due to the fact that the cost function does not depend on the exact value of the previous -states. On the other hand, a low pass filter will introduce errors in magnitude and phase angle which will depend on the chosen cutoff frequency. This issue remains a topic for future work, see [31].

previous inteFinally, according to (20) and considering gral errors, the final cost function to be evaluated for this particular case becomes (22) Remark 4: Notice that only the last term of (21) depends on depends the switching input while the entire term on previously calculated errors. Thus, most of this expression does not need to be recalculated when predictions are carried out. Consequently, the online implementation of this cost function does not rely upon extensive additional computations when compared to standard FCS-MPC, as revised in Section II. VI. RESULTS To verify the performance of the prosed solutions, simulation and experimental studies were carried out on an H-Bridge converter depicted in Fig. 2. A main dc-link voltage source of 150 V was considered. The electrical load parameters are 15 and 10 mH. The proposed solutions as well as the standard FCS-MPC strategy were implemented using a 200 s. In this case, the load current sampling period of is the variable to be controlled and the control target is to maintain its average value near to the reference. Since we are interested in controlling only one variable, standard FCS-MPC and IS-MPC, as shown in previous sections, do not require any tuning. On the other hand, according to (22), strategy requires to select the weight factor and the number of previous integral errors . In this case, these and . tuning parameters where chosen as A. Average Error The steady-state average errors, for different current reference, produced by the standard FCS-MPC and the proposed solutions IS-MPC and are presented in Fig. 7. Here, only positive values of the load current are considered due to the fact that the average error pattern is symmetric with respect to zero. It can be observed that standard FCS-MPC generates higher errors when the current reference is near to zero and when it is close to the maximum current limit. This is due to the fact whenthat this predictive strategy applies a null voltage ever the current reference satisfies (0, 0.8) A, and a max, whenever the current reference is imum voltage, (9.2, 10) A. For these particular references, it can be noreduce ticed that the proposed solutions IS-MPC and these average errors.


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Fig. 7. Average error as a function of the current reference for the MPC strategies examined.

Fig. 9. Steady-state performance with reference . (b) IS-MPC. (c)

0.6 A. (a) FCS-MPC.

B. Experimental Results Fig. 8. Normalized average error.

TABLE I EXECUTION TIME OF EACH PREDICTIVE ALGORITHM

Another interesting case is related to the error produced around 5 A (half of the maximum load current). This error is . generated by the particular input sequence, It follows from the analysis in Section III-B that FCS-MPC applies this input sequence whenever the current reference lies (4.7, 5.3) A. Regarding the proposed solution, in the range for this reference range, IS-MPC presents a similar average reduces this error error to standard FCS-MPC while considerably. Fig. 8 shows a normalized average error as a function of the current reference value. In this way a better sense of the importance of the average error can be observed. From this viewpoint, the main problem of the predictive techniques are around , and even zero. In this zone the benefits of the use of IS-MPC, become apparent.

Here, experimental results of standard FCS-MPC and our proposals IS-MPC and are presented. The predictive strategies were implemented in a standard TMS320C6713 DSP. Then, the optimal input was applied to the converter by using an XC3S400 FPGA. This is a standard digital configuration used to control power converters [32]. Table I summarizes the execution times taken by the predictive algorithms. Here, it is possible to see that IS-MPC takes the same time than standard FCS-MPC to obtain the optimal input, due to the fact that no extra calculations are needed. On the other hand, IS-MPC takes about twice the time to obtain the optimal values. Fig. 9 shows the steady-state performance achieved by the 0.6 A. different predictive strategies for a load current of Here, it is possible to observe that the converter applies a null 0 V to the load when governed by standard FCSvoltage MPC. This behavior is improved when IS-MPC and are implemented. An average value of 0.67 A is obtained when IS-MPC is used while 0.71 A is achieved with . 4.8 A A different situation occurs when a reference of is desired. Here, FCS-MPC generates the optimal sequence , which was previously analyzed in Section III-B. This situation is depicted in Fig. 10 and an

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4.8 A. (a) FCS-MPC.

average reference of 4.99 A is obtained. For the same reference, IS-MPC produces the same optimal pattern as standard FCS-MPC. Therefore, the same average value, for the load current, is achieved. On the other hand, when the converter is , a different pattern is obtained. This governed by results in a decrease in the average error, achieving an average 4.73 A. Nevertheless, as a value for the load current of side effect, an increment in the load current ripple is obtained. produces a more It is important to observe that variable input pattern when compared with FCS-MPC and is the only IS-MPC. This is due to the fact that predictive strategy among them that explicitly considers the continuous-time behavior of the system state. FCS-MPC and IS-MPC only optimize the at-sample response. With an improved inter-sample performance is obtained though more random-like switching. Consequently, there is a tradeoff between the average steady-state error and the ripple achieved in the load current. In Fig. 11, the steady-state behavior for a load current refer7.4 A is presented. For this reference, the three ence of predictive strategies produce a similar average value, as anticipated by simulations documented in Fig. 7. One of the most important benefits of FCS-MPC is the high dynamic response achieved [10]. Fig. 12 shows the step response for a load current reference from

2 A to


7.4 A. (a) FCS-MPC.

7 A. Here, it is possible to observe that the proposed modifications to standard FCS-MPC do not deteriorate the dynamic performance. Consequently, these modifications only improve the steady-state performance of this predictive strategy. C. Tracking of Sinusoidal References The steady-state analysis of FCS-MPC presented in this work as well as the proposed modifications are based on constant references. Nevertheless, in a wide range of applications, it is required to operate power converters in order to track sinusoidal references. Here, we study the continuous-time behavior of the four-quadrant chopper governed by FCS-MPC and the two proposed modifications when tracking sinusoidal references. Table II summarizes the fundamental component and THD of the load current obtained by FCS-MPC and the proposed mod, for different magnitude refifications, IS-MPC and erences considering a 50-Hz frequency. As can be observed, for reference magnitudes between 4 and 8 A, the three predictive control methods present a similar behavior obtaining a small tracking error in terms of fundamental component. Nevertheless, for lower and higher references, FCS-MPC produces a significant deviation from the desired reference, while IS-MPC achieve better performances from a fundamental and


Fig. 12. Step response. (a) FCS-MPC. (b) IS-MPC. (c)

THD

AND

TABLE II FUNDAMENTAL COMPONENT MAGNITUDE SINUSOIDAL REFERENCE

-

FOR A

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.

50-HZ

Fig. 13. Experimental results for a 50-Hz sinusoidal reference of 9 A. (a) FCSMPC. (b) IS-MPC. (c) IE-MPC.

tween the fundamental component tracking error and the THD obtained. In particular, Fig. 13 shows the current waveforms obtained for the three predictive control strategies previously discussed when a 50-Hz sinusoidal reference with an amplitude of 9 A is required. It can be observed that the load current behavior obtained with FCS-MPC presents a lower switching frequency near to zero and the extreme values, which is consistent with the results depicted in Fig. 7. This leads to a fundamental component of 9.66 A, which is a 7.3% higher than the desired reference of 9 A. Regarding the proposed modifications, IS-MPC and present a higher switching frequency, which leads to higher THD values (as shown in Table II), but with a fundamental component value closer to the desired reference, 8.76 A . (2.7% error) for IS-MPC and 8.79 A (2.4% error) for VII. CONCLUSION component viewpoint. To do so, however, the proposed solutions increase the current THD. Therefore, there is a tradeoff be-

In this paper, we have studied the steady-state performance of power converters when governed by FCS-MPC.

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We have shown, analytically and experimentally, that existing FCS-MPC gives, in general, a nonzero steady-state error, even when models and parameter values are exactly known. To deal with this issue, two different modifications to FCS-MPC strategy have been proposed. In the first one, IS-MPC, the measurements and the optimal switch implementation are temporarily shifted by half of the sampling period. This allows one to approximate the average value of the controlled variable by the sample value. On the other hand, the second modification reduces the average tracking error by using an analytical expression of the intersampling integral error. Thus, the continuous-time trajectory during the inter-sampling times of the controlled variable is considered. Moreover, past integral errors can be also included to improve the continuos-time response of the average tracking. An important characteristic of our proposals is that, for the cases examined, they improve steady-state performance, without sacrificing the dynamic response achieved with standard FCS-MPC. According to the presented results, IS-MPC reaches a low average steady-state error only for low reference values, while algorithm, considering two previous integral errors, gives in general the best steady-state performance. However, a higher ripple in the load current is obtained. Consequently, there is a tradeoff between the ripple in the variable to be controlled and its average steady-state error. The benefits of using our proposals are also apparent when tracking of sinusoidal references is desired. In this case, IS-MPC and IE-MPC achieve better performance in terms of fundamental component when compared with standard FCS-MPC. Nevertheless, an increment of the THD is produced. Therefore, similar to the constant reference case, there is a tradeoff between the tracking error of the fundamental component and the THD obtained. REFERENCES [1] G. C. Goodwin, D. Q. Mayne, T. Chen, C. Coates, G. Mirzaeva, and D. E. Quevedo, “Opportunities and challenges in the application of advanced control to power electronics and drives,” in Proc. IEEE Int. Conf. Ind. Technol., Mar. 2010, pp. 27–39. [2] G. C. Goodwin, D. Q. Mayne, K. Chen, C. Coates, G. Mirzaeva, and D. E. Quevedo, “An introduction to the control of switching electronic systems,” Annu. Rev. Control, vol. 34, no. 2, pp. 209–220, Dec. 2010. [3] A. Linder, R. Kanchan, R. Kennel, and P. Stolze, Model-Based Predictive Control of Electrical Drives. Göttingen, Germany: Cuvillier Verlag Göttingen, 2010. [4] J. Rodríguez and P. Cortés, Predictive Control of Power Converters and Electrical Drives, 1st ed. New York: Wiley-IEEE, 2012. [5] P. Cortés, M. P. Kazmierkowski, R. M. Kennel, D. E. Quevedo, and J. Rodríguez, “Predictive control in power electronics and drives,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, Dec. 2008. [6] T. Geyer, “A comparison of control and modulation schemes for medium-voltage drives: Emerging predictive control concepts versus PWM-based schemes,” IEEE Trans. Ind. Appl., vol. 47, no. 3, pp. 1380–1389, May–Jun. 2011. [7] G. C. Goodwin, M. Serón, and J. De Doná, Constrained Control and Estimation: An Optimization Approach. Berlin, Germany: SpringerVerlag, 2005. [8] J. Rawlings and D. Mayne, Model Predictive Control: Theory and Design. Madison, WI: Nob Hill, 2009. [9] L. Grüne and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, ser. Communications and Control Engineering. Berlin, Germany: Springer-Verlag, 2011.

[10] S. Kouro, P. Cortés, R. Vargas, U. Ammann, and J. Rodríguez, “Model predictive control—A simple and powerful method to control power converters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1826–1838, Jun. 2009. [11] S. Muller, U. Ammann, and S. Rees, “New time-discrete modulation scheme for matrix converters,” IEEE Trans. Ind. Electron., vol. 52, no. 6, pp. 1607–1615, Jun. 2005. [12] R. Vargas, U. Ammann, and J. Rodríguez, “Predictive approach to increase efficiency and reduce switching losses on matrix converters,” IEEE Trans. Power Electron., vol. 24, no. 4, pp. 894–902, Apr. 2009. [13] P. Lezana, R. Aguilera, and D. E. Quevedo, “Model predictive control of an asymmetric flying capacitor converter,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1839–1846, Jun. 2009. [14] T. Geyer, G. Papafotiou, and M. Morari, “Model predictive direct torque control—Part I: Concept, algorithm, and analysis,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1894–1905, Jun. 2009. [15] J. Beerten, J. Verveckken, and J. Driesen, “Predictive direct torque control for flux and torque ripple reduction,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 404–412, Jan. 2010. [16] E. Fuentes, C. Silva, and J. Yuz, “Predictive speed control of a twomass system driven by a permanent magnet synchronous motor,” IEEE Trans. Ind. Electron., vol. 59, no. 7, pp. 2840–2848, 2011. [17] K. Ahmed, A. Massoud, S. Finney, and B. W. Williams, “A modified stationary reference frame-based predictive current control with zero steady-state error for LCL coupled inverter-based distributed generation systems,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1359–1370, Apr. 2011. [18] C. Xia, M. Wang, Z. Song, and T. Liu, “Robust model predictive current control of three-phase voltage source PWM rectifier with online disturbance observation,” IEEE Trans. Ind. Inf., vol. 8, no. 3, pp. 459–471, Aug. 2012. [19] M. Malinowski, K. Gopakumar, J. Rodríguez, and M. Pérez, “A survey on cascaded multilevel inverters,” IEEE Trans. Ind. Electron., vol. 57, no. 7, pp. 2197–2206, Jul. 2010. [20] P. Cortés, S. Kouro, B. L. Rocca, R. Vargas, J. Rodríguez, J. Leon, S. Vazquez, and L. Franquelo, “Guidelines for weighting factors design in model predictive control of power converters and drives,” in Proc. IEEE Int. Conf. Ind. Technol., Feb. 2009, pp. 1–7. [21] D. E. Quevedo, R. P. Aguilera, M. A. Pérez, P. Cortés, and R. Lizana, “Model predictive control of an AFE rectifier with dynamic references,” IEEE Trans. Power Electron., vol. 27, no. 7, pp. 3128–3136, Jul. 2012. [22] P. Lezana, R. Aguilera, and D. Quevedo, “Steady-state issues with finite control set model predictive control,” in Proc. IEEE 35th Annu. Conf. Ind. Electron., 2009, pp. 1776–1781. [23] D. G. Holmes, T. A. Lipo, and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice. New York: Wiley-IEEE, 2003. [24] D. E. Quevedo, G. C. Goodwin, and J. A. D. Doná, “Finite constraint set receding horizon quadratic control,” Int. J. Robust Nonlinear Control, vol. 14, no. 4, pp. 355–377, Jan. 2004. [25] R. P. Aguilera and D. E. Quevedo, “On the stability of MPC with a finite input alphabet,” in Proc. 18th IFAC World Congress, Milan, Italy, Aug. 2011, pp. 7975–7980. [26] R. P. Aguilera and D. E. Quevedo, “On stability and performance of finite control set MPC for power converters,” in Proc. Workshop Predictive Control of Electrical Drives and Power Electron., Oct. 2011, pp. 55–62. [27] M. Rivera, J. Rodríguez, J. Espinoza, and H. Abu-Rub, “Instantaneous reactive power minimization and current control for an indirect matrix converter under a distorted AC-supply,” IEEE Trans. Ind. Inf., vol. 8, no. 3, pp. 482–490, Aug. 2012. [28] R. P. Aguilera and D. E. Quevedo, “Capacitor voltage estimation for predictive control algorithm of flying capacitor converters,” in Proc. IEEE Int. Conf. Ind. Technol., Melbourne, Australia, Feb. 2009, pp. 1–6. [29] J.-S. Hu and K.-Y. Chen, “Analysis and design of the receding horizon constrained optimization for class-D amplifier driving signals,” Digital Signal Process., vol. 20, no. 6, pp. 1511–1525, Dec. 2010. [30] S.-H. Yu and M.-H. Tseng, “Optimal control of a nine-level class-D audio amplifier using sliding-mode quantization,” IEEE Trans. Ind. Electron., vol. 58, no. 7, pp. 3069–3076, Jul. 2011. [31] P. Cortés, J. Rodríguez, D. E. Quevedo, and C. Silva, “Predictive current control strategy with imposed load current spectrum,” IEEE Trans. Power Electron., vol. 23, no. 2, pp. 612–618, Feb. 2008.


[32] C. Buccella, C. Cecati, and H. Latafat, “Digital control of power converters—A survey,” IEEE Trans. Ind. Inf., vol. 8, no. 3, pp. 437–447, Aug. 2012. Ricardo P. Aguilera (S’02–M’11) received the B.Sc. degree in electrical engineering from the Universidad de Antofagasta, Antofagasta, Chile, in 2002, the M.Sc. degree in electronic engineering from the Technical University Federico Santa María, Valparaíso, Chile, in 2007, and the Ph.D. degree from the School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia, in 2012. He is currently a Research Academic with the School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia. His main research interests include modern control strategies for power electronics and drives.

Pablo Lezana (S’06–M’07) was born in Temuco, Chile, in 1977. He received the M.Sc. and D.Eng. degrees in electronic engineering from the Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2005 and 2006, respectively. From 2005 to 2006, he was a Research Assistant with the Departamento de Electrónica, Universidad Técnica Federico Santa María (UTFSM), Valparaíso, Chile. From 2007 to 2009, he was a Researcher with the Departamento de Ingeniería Eléctrica, UTFSM, and since 2010 he has been a Lecturer with the same

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department. His research interests include power converters and modern digital control devices (DSPs and fiel programmable gate arrays). Dr. Lezana received the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Best Paper Award in 2007.

Daniel E. Quevedo (S’97–M’05) received the Ingeniero Civil Electrónico and Magister en Ingeniería Electrónica degrees from the Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2000, and the Ph.D. degree from The University of Newcastle, Newcastle, Australia, in 2005. He he is currently an Associate Professor with The University of Newcastle, Newcastle, Australia. He has been a Visiting Researcher with various institutions, including Uppsala University, Sweden, KTH Stockholm, Sweden, Aalborg University, Denmark, Kyoto University, Japan, and INRIA Grenoble, France. His research interests include several areas of automatic control, signal processing, and power electronics. Dr. Quevedo was supported by a full scholarship from the alumni association during his time at the Universidad Técnica Federico Santa María and received several university-wide prizes upon graduating. He received the IEEE Conference on Decision and Control Best Student Paper Award in 2003 and was also a finalist in 2002. In 2009, he was awarded an Australian Research Fellowship.