Abstract—This paper presents a model predictive control (MPC)approach for dc–dc boost converters. A discrete-time switchednonlinear (hybrid) model of the converter is derived, which cap-tures both the continuous and the discontinuous conduction mode.The controller synthesis is achieved by formulating an objectivefunction that is to be minimized subject to the model dynamics.The proposed MPC strategy, utilized as a voltage-mode controller,achieves regulation of the output voltage to its reference, withoutrequiring a subsequent current control loop. Furthermore, a stateestimation scheme is implemented that addresses load uncertain-ties and model mismatches. Simulation and experimental resultsare provided to demonstrate the merits of the proposed controlmethodology, which include a fast transient response and a highdegree of robustness.
Index Terms—DC–DC boost converter, hybrid system, modelpredictive control (MPC), optimal control, voltage control.
I. INTRODUCTION
OVER the past decades dc–dc conversion has matured intoa ubiquitous technology, which is used in a wide vari-
ety of applications, including dc power supplies and dc motordrives [1]. DC–DC converters are intrinsically difficult to con-trol due to their switching behavior, constituting a (continuous-time) switched linear or hybrid system. To date, a plethora ofcontrol schemes has been proposed to address these difficul-ties. These control techniques range from linear techniques,such as proportional-integral (PI) controllers based on aver-age models [2], [3] to fuzzy logic [4], [5], and from nonlineartechniques [6], [7] and feedforward control [8], [9] to slidingmode [10], [11] and H∞ methods [12].
Although existing control approaches have been shown tobe reasonably effective, several challenges have not been fullyaddressed yet, such as ease of controller design and tuning, aswell as robustness to load parameter variations. Furthermore,the computational power available today and the recent theoret-ical advances with regards to controlling hybrid systems allow
Manuscript received October 22, 2012; revised January 7, 2013; acceptedMarch 25, 2013. Date of current version August 20, 2013. Recommended forpublication by Associate Editor B. Wang.
P. Karamanakos and S. Manias are with the Department of Electricaland Computer Engineering, National Technical University of Athens, 15780Zografou, Athens, Greece (e-mail: [email protected]; [email protected]).
T. Geyer is with the ABB Corporate Research, 5405 Baden-Dattwil,Switzerland (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2013.2256370
one to tackle these problems in a novel way. The aim is notonly to improve the performance of the closed-loop system, butto also enable a systematic design and implementation proce-dure. Model predictive control (MPC) is a particularly promis-ing candidate to achieve this [13], [14], since it allows one todirectly include constraints in the design phase and to addressthe switching or hybrid nature of dc–dc converters. MPC wasdeveloped in the 1970s in the process control industry, and hasrecently been introduced to the field of power electronics, in-cluding three-phase dc–ac and ac–dc systems [15]–[20], as wellas dc–dc converters [21]–[30].
In MPC, the control action is obtained by solving on-lineat each time-step an optimization problem with a given ob-jective function over a finite prediction horizon, subject to thediscrete-time model of the system and constraints. The optimalsequence of control inputs is the one that minimizes the objec-tive function. To provide feedback, allowing one to cope withmodel uncertainties and disturbances, only the first element ofthe sequence of control inputs is applied to the converter. Atthe next time-step, the optimization problem is repeated withupdated measurements or estimates. This procedure is knownas the receding horizon policy [14].
In this paper, MPC is employed as a voltage-mode controllerfor the dc–dc boost converter. The main control objective is theregulation of the output voltage to a commanded value, while re-jecting the impact of variations in the input voltage and the load.The discrete-time model of the converter used by the controlleris designed such that it accurately predicts the plant behaviorboth when operating in continuous (CCM) as well as in discon-tinuous conduction mode (DCM). As a result, the formulatedcontroller is applicable to the whole operating regime, ratherthan just to a particular operating point. To address time-varyingand unknown loads, a Kalman filter is added that estimates theconverter states and provides offset-free tracking of the out-put voltage due to its integrating action, despite changes in theload [31]. In that way, the robustness of the controller is ensuredeven when the converter operates under nonnominal conditions.
The proposed scheme carries several benefits. The very fastdynamics achieved by MPC, combined with its inherent robust-ness properties, are some of its key beneficial characteristics.Furthermore, thanks to the fact that the control objectives areexpressed in the objective function in a straightforward manner,the design process is simple and laborious tuning is avoided.The inherent computational complexity is the most prominentdrawback—the computational power required increases expo-nentially as the prediction horizon is extended. To address thisissue, a move blocking strategy is adopted [32], which results
KARAMANAKOS et al.: DIRECT VOLTAGE CONTROL OF DC–DC BOOST CONVERTERS USING MPC 969
Fig. 1. Topology of the dc–dc boost converter.
in a significant reduction of the computations required and fa-cilitates the real-time implementation of the controller. Finally,the absence of a modulator and the direct manipulation of theconverter switches imply a variable switching frequency.
This paper is organized as follows: In Section II, thecontinuous-time model of the converter, suitable for both CCMand DCM, is presented. Furthermore, the discrete-time modelthat will be used as the prediction model is derived. The controlproblem is stated in Section III. In Section IV, the MPC prob-lem is formulated and solved, using a move blocking schemeand enumeration, and a Kalman filter is added to address loadvariations. Section V presents simulation results illustrating theperformance of the proposed control approach. In Section VI,the experimental validation of the introduced control strategyis provided. The paper is summarized in Section VII, whereconclusions are drawn.
II. MODEL OF THE BOOST CONVERTER
A. Continuous-Time Model
The dc–dc boost converter shown in Fig. 1 is a converterthat increases the (typically uncontrolled) dc input voltage vs(t)to a higher (controlled) dc output voltage vo(t). The converterconsists of two power semiconductors—the controllable switchS, and the diode D. The inductor L with the internal resistor RL
is used to store and deliver energy depending on the operatingmode of the converter, while the filter capacitor Co is connectedin parallel with the load resistor R so as to ensure a constantoutput voltage during steady-state operation of the converter.
Three different linear dynamics are associated with the switchpositions. When the switch S is ON (S = 1), energy is stored inthe inductor L and the inductor current iL (t) increases. Whenthe switch S is OFF (S = 0), the inductor is connected to theoutput and energy is released through it to the load, resultingin a decreasing iL (t). Furthermore, when the switch S remainsOFF and iL (t) = 0, then both S and D are OFF; the topologyis reduced to the mesh formed by the capacitor Co and the load.In this case, the converter operates in DCM.
The state-space representation of the converter in thecontinuous-time domain is given by the following equations[33]:
dx(t)dt
=(A1 + A2u(t)
)x(t) + Bvs(t) (1a)
y(t) = Cx(t), (1b)
Fig. 2. DC–DC converter presented as a continuous-time automaton.
where
x(t) = [iL (t) vo(t)]T (2)
is the state vector, encompassing the inductor current and theoutput voltage across the output capacitor. The output y = vo(t)is given by the output voltage. The system matrices are
A1 =
⎡
⎢⎢⎣
−dauxRL
L−daux
L
daux
Co− 1
CoR
⎤
⎥⎥⎦ , A2 =
⎡
⎢⎢⎣
01L
− 1Co
0
⎤
⎥⎥⎦ ,
B =[daux
L0]T
, and C = [0 1].
The variable u denotes the switch position, with u = 1 implyingthat the switch S is ON, and u = 0 referring to the case where theswitch S is OFF. Finally, daux is an auxiliary binary variable [34]that is daux = 1 when the converter operates in CCM, i.e., eitheru = 1 or u = 0 and iL (t) > 0. When the converter operates inDCM, i.e., u = 0 and iL (t) = 0, then daux = 0 holds
daux(t) ={
1 if u(t) = 1, or u(t) = 0 and iL (t) > 0
0 if u(t) = 0 and iL (t) = 0.(3)
For a graphical summary, representing the boost converter asan automaton, see Fig. 2.
B. Discrete-Time Model
The derivation of an adequate model of the boost converterto serve as an internal prediction model for MPC is of fun-damental importance. As can be seen in Fig. 3, after the dis-cretization of the model in time, the converter can operate infour different modes, depending on the shape of the inductorcurrent. In order to precisely describe the operating modes ofthe converter, the matrices Γ1 = A1 for daux = 1, Γ2 = A1 fordaux = 0, Γ3 = A2 , and Δ = B for daux = 1 are introduced.Then, based on the continuous-time state-space model (1) andusing the forward Euler approximation approach, the following
970 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Fig. 3. Operation modes used in the mathematical model to describe the boostconverter. Depending on the shape of the current four different modes are used.(a) Current. (b) Switch position.
Fig. 4. Discrete-time mathematical model of the dc–dc converter representedas a discrete-time automation.
discrete-time model of the converter is derived
x(k + 1) =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
E1x(k) + F1vs(k) Mode “1”
E2x(k) + F2vs(k) Mode “2”
E3x(k) + F3vs(k) Mode “3”
E4x(k) Mode “4”
(4a)
y(k) = Gx(k) (4b)
where the matrices are E1 = 1 + (Γ1 + Γ3)Ts , E2 = 1 +Γ1Ts , E3 = 1
Ts(τ1E2 + τ2E4), E4 = 1 + Γ2Ts , F1 = ΔTs ,
F2 = F1 , F3 = Δτ1 , and G = C. Furthermore, τ1 denotes thetime-instant within the sampling interval, when the inductor cur-rent reaches zero, i.e., iL (k + τ1/Ts) = 0, and τ1 + τ2 = Ts .Finally, 1 is the identity matrix and Ts is the sampling interval.Note that E3 is derived by averaging over modes “2” and “4”.
The four different operating modes of the converter’s mathe-matical model are illustrated in Fig. 4. The transitions from one
mode to another are specified by conditions, such as the switchposition and the value of the current.
III. CONTROL PROBLEM
For the dc–dc converter, the main control objective is for theoutput voltage to accurately track its given reference—or equiv-alently to minimize the output voltage error—by appropriatelymanipulating the switch. This is to be achieved despite changesin the input voltage and load. During transients, the output volt-age is to be regulated to its new reference value as fast and withas little overshoot as possible.
IV. MODEL PREDICTIVE CONTROL
In this section, an MPC scheme for dc–dc boost convertersis introduced, which directly controls the output voltage bymanipulating the switch S. Using an enumeration technique,the user-defined objective function is minimized subject to theconverter dynamics.
A. Objective Function
The objective function is chosen as
J(k) =k+N −1∑
�=k
(|vo,err(� + 1|k)| + λ|Δu(�|k)|) (5)
which penalizes the absolute values of the variables of concernover the prediction horizon N , which is of finite length. The firstterm penalizes the absolute value of the output voltage error
vo,err(k) = vo,ref − vo(k). (6)
By penalizing the difference between two consecutive switch-ing states, the second term aims at decreasing the switchingfrequency and avoiding excessive switching
Δu(k) = u(k) − u(k − 1). (7)
The weighting factor λ > 0 sets the tradeoff between outputvoltage error and switching frequency, fsw . Note that the sam-pling interval Ts implicitly imposes an upper bound on theswitching frequency, i.e., fsw < 1/(2Ts). This value corre-sponds to the case when λ = 0, the output voltage is twicethe input voltage, i.e., vo = 2vs , and when the inductor is idealwith RL = 0.
B. Optimization Problem
The optimization problem underlying MPC at time-step kamounts to minimizing the objective function (5) subject to theconverter model dynamics
U ∗(k) = arg min J(k)
subject to eq. (4).(8)
The optimization variable is the sequence of switching statesover the horizon, which is U(k) = [u(k) u(k + 1) . . . u(k +N − 1)]T . Minimizing (8) yields the optimal switching se-quence U ∗(k). Out of this sequence, the first element u∗(k)
KARAMANAKOS et al.: DIRECT VOLTAGE CONTROL OF DC–DC BOOST CONVERTERS USING MPC 971
is applied to the converter. The procedure is repeated at k + 1,based on new measurements acquired at the following samplinginstance.
Minimizing (8) is a challenging task, since it is a mixed-integer nonlinear optimization problem.1 A straightforward al-ternative is to solve (8) using enumeration, which involves thefollowing three steps. First, by considering all possible com-binations of the switching states (u = 0 or u = 1) over theprediction horizon, the set of admissible switching sequencesis assembled. For each of the 2N sequences, the correspondingoutput voltage trajectory is predicted and the objective functionis evaluated. The optimal switching sequence is obtained bychoosing the one with the smallest associated cost.
C. Move Blocking
A fundamental difficulty associated with boost convertersarises when controlling their output voltage without an interme-diate current control loop, since the output voltage exhibits anonminimum phase behavior with respect to the switching ac-tion. For example, when increasing the output voltage, the dutycycle of switch S has to be ramped up, but initially the outputvoltage drops before increasing. This implies that the sign of thegain (from the duty cycle to the output voltage) is not alwayspositive.
To overcome this obstacle and to ensure closed-loop stability,a sufficiently long prediction interval NTs is required, so thatthe controller can “see” beyond the initial voltage drop whencontemplating to increase the duty cycle. On the one hand,increasing N leads to an exponential increase in the numberof switching sequences to be considered and thus dramaticallyincreases the number of calculations needed. On the other hand,long sampling intervals Ts reduce the resolution of the possibleswitching instants, since switching can only be performed at thesampling instants.
A long prediction interval NTs with a small N and a small Ts
can be achieved by employing a move blocking technique [32].For the first steps in the prediction horizon, the prediction modelis sampled with Ts , while for steps far in the future, the modelis sampled more coarsely with a multiple of Ts , i.e., nsTs , withns ∈ N+ [35]. As a result, different sampling intervals are usedwithin the prediction horizon, as illustrated in Fig. 5. We useN1 to denote the number of prediction steps in the first part ofthe horizon, which are sampled with Ts . Accordingly, N2 refersto the number of steps in the last part of the horizon, sampledwith nsTs . The total number of time-steps in the horizon isN = N1 + N2 .
An illustrative example of the effectiveness of the move block-ing strategy is depicted in Fig. 6. Assume that at time instantkTs the output voltage reference increases in a stepwise man-ner and the output voltage is to follow that change. However,as mentioned above, because of the nonminimum phase natureof the system, the output voltage initially tends to decrease. In
1It should be noted that the mathematical model of the converter given by(4a) for modes “1” and “2” is affine (linear plus offset), and for mode “4” islinear, while the expression for mode “3” is nonlinear.
Fig. 5. Prediction horizon with move blocking: a) output voltage, b) inductorcurrent, and c) control input. The prediction horizon has N = 10 time-steps,but the prediction interval is of length 19Ts , since ns = 4 is used for the lastN2 = 3 steps.
Fig. 6. Effect of the move blocking scheme. In (a), without move blocking,a prediction horizon of N = 20 steps of equal time-intervals is needed. In(b), with the move blocking strategy employed, an N = 11 prediction horizonis sufficient to achieve the same closed-loop result (N1 = 7, N2 = 4, andns = 4, total length 23Ts ).
972 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
order to ensure that MPC is able to predict the final voltage in-crease and will thus pick the corresponding switching sequencethat achieves this, in this example, a prediction interval of 20time-steps is required, i.e., NTs = 20.
By employing the move blocking scheme, the 11-step hori-zon N = 11, with N1 = 7, N2 = 4, and ns = 4 suffices, re-sulting in a prediction interval of a 23 time-steps. In this way,the computational cost is significantly reduced. Without moveblocking, the number of switching sequences to be examined is220 = 1048576, and the state evolution has to be predicted for20 steps into the future. In contrast to this, when using the moveblocking scheme, the total number of sequences is 211 = 2048,and the evolution of the state needs to be calculated only for 11steps. As a result, the computations required are decreased bythree orders of magnitude, or 99.9%.
It is important to point out that a high timing resolution isrequired only around the current time-step and the very nearfuture. Further ahead, a rough timing resolution suffices, dueto the receding horizon policy. The coarse plan of the secondpart of the prediction horizon is step by step shifted towards thebeginning of the prediction horizon and simultaneously refined.
D. Load Uncertainty
In most applications, the load is unknown and time varying.Thus, an external estimation loop should be added, which allowsthe elimination of the output voltage error in the presence of loaduncertainties. This additional loop is employed to provide stateestimates to the previously derived optimal controller, where theload was assumed to be known and constant. The output voltagereference will be adjusted so as to compensate for the deviationof the output voltage from its actual reference.
To achieve this, a discrete-time Kalman filter [36] is designedsimilar to [25]; thanks to its integrating nature the Kalman filterprovides a zero steady-state output voltage error. Two integratingdisturbance states, ie and ve , are introduced in order to modelthe effect of the load variations on the inductor current andoutput voltage, respectively. The measured state variables, iLand vo , together with the disturbance state variables form theaugmented state vector
xa = [iL vo ie ve ]T . (9)
The Kalman filter is used to estimate the state vector given by(9). Depending on the operating mode of the converter, as shownin Fig. 3, four different affine systems result. The respectivestochastic discrete-time state equations of the augmented modelare
xa(k + 1) = Ezaxa(k) + Fzavs(k) + ξ(k), (10)
where z = {1, 2, 3, 4} corresponds to the four operating modesof the converter.
The measured state vector is given by
x(k) =[
iL (k)
vo(k)
]= Gaxa(k) + ν(k) (11)
and the matrices are
Eza =[Ez 0
0 1
], F1a =
⎡
⎣F1
00
⎤
⎦= F2a =
⎡
⎣F2
00
⎤
⎦, F3a =
⎡
⎣F3
00
⎤
⎦
F4a = [ 0 0 0 0 ]T , and Ga = [1 1 ] (12)
where, 1 is the identity matrix of dimension two and 0 aresquare zero matrices of dimension two. The variables ξ ∈ R4
and ν ∈ R2 denote the process and the measurement noise,respectively. These terms represent zero-mean, white Gaussiannoise sequences with normal probability distributions. Theircovariances are given by E[ξξT ] = Q and E[ννT ] = R, andare positive semidefinite and positive definite, respectively.
A switched discrete-time Kalman filter is designed based onthe augmented model of the converter. The active mode of theKalman filter (one out of four) is determined by the switchingposition and the operating mode of the converter.
Due to the fact that the state-update for each operating modeis different, four Kalman gains Kz need to be calculated. Con-sequently, the equation for the estimated state xa(k) is
The noise covariance matrices Q and R are chosen such thathigh credibility is assigned to the measurements of the physicalstates (iL and vo ), whilst low credibility is assigned to the dy-namics of the disturbance states (ie and ve ). The Kalman gainsare calculated based on these matrices. The estimated distur-bances, provided by the resulting filter, can be used to removetheir influence from the output voltage. Hence, the disturbancestate ve is used to adjust the output voltage reference vo,ref
vo,ref = vo,ref − ve . (14)
To this end, the estimated states, iL and vo , are used as inputsto the controller, instead of the measured states, iL and vo .
E. Control Algorithm
The proposed control concept is summarized in Algorithm 1.The function f stands for the state-update given by (4), with thesubscripts 1 and 2 corresponding to the sampling interval beingused, i.e., Ts and nsTs , respectively. Fig. 7 depicts the flowchartof the introduced MPC algorithm, while the block diagram ofthe entire control scheme is shown in Fig. 8.
V. SIMULATION RESULTS
In this section, simulation results are presented to demonstratethe performance of the proposed controller under several operat-ing conditions. Specifically, the closed-loop converter behavioris examined in both CCM and DCM. The dynamic performanceis investigated during start-up. Moreover, the responses of theoutput voltage to step changes in the commanded voltage refer-ence, the input voltage and the load are illustrated.
The circuit parameters are L = 450 μH, RL = 0.3 Ω, andCo = 220 μF. The nominal load resistance is R = 73 Ω. If nototherwise stated, the input voltage is vs = 10 V and the refer-ence of the output voltage is vo,ref = 15 V.
KARAMANAKOS et al.: DIRECT VOLTAGE CONTROL OF DC–DC BOOST CONVERTERS USING MPC 973
Fig. 7. Flowchart of the MPC algorithm.
The weight in the objective function is λ = 0.1, the predic-tion horizon is N = 14 and the sampling interval is Ts = 2.5 μs.A move blocking scheme is used with N1 = 8, N2 = 6, andns = 4, i.e., the sampling interval for each of the last sixsteps in the prediction interval is Ts = 10 μs.2 Finally, the
2The length of the prediction horizon in time should be as long as possible. Ahorizon of about 80 μs is sufficient. The first part of the prediction horizon shouldbe finely sampled, since switching is possible only at the sampling instants. Assuch, the sampling interval Ts should be as small as possible. The number of
Fig. 8. Block diagram of the MPC scheme and Kalman filter.
Fig. 9. Simulation results for nominal start-up: a) output voltage (solid line)and output voltage reference (dashed line); and b) inductor current.
covariance matrices of the Kalman filter are chosen as Q =diag(0.1, 0.1, 50, 50) and R = diag(1, 1).
A. Start-Up
The first case to be examined is that of the start-up behav-ior under nominal conditions. As can be seen in Fig. 9, theinductor current is very quickly increased until the capacitor ischarged to the desired voltage level. The output voltage reachesits reference value in about t ≈ 1.8 ms, without any noticeableovershoot. Subsequently, the converter operates in DCM withthe inductor current reaching zero.
B. Step Changes in the Output Reference Voltage
Next, step changes in the reference of the output voltageare considered. First, a step-up change in the output reference
steps in the prediction horizon N = N1 + N2 determines the computationalcomplexity. To ensure that the control law can be computed within Ts , Nshould be relatively small, leading to the choice made above.
974 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Fig. 10. Simulation results for a step-up change in the output voltage reference:a) output voltage (solid line) and output voltage reference (dashed line); andb) inductor current.
voltage is examined: at time t = 2 ms the reference is doubledfrom vo,ref = 15 to 30 V. As can be seen in Fig. 10, the controllerincreases the current temporarily in order to quickly ramp upthe output voltage. Note that this favorable choice is made bythe controller thanks to its long prediction horizon and despitethe nonminimum phase behavior of the converter. Once theoutput voltage has reached its reference, the inductor current isdecreased to the level that corresponds to the steady state powerbalance. The controller exhibits an excellent behavior during thetransient, reaching the new output voltage in about t ≈ 1.8 ms,without any overshoot.
Furthermore, the behavior of the controller is tested undera step-down change in the output reference voltage. At timet = 2 ms, the output voltage reference changes from vo,ref = 20to 15 V; the segment of interest is depicted in Fig. 11. Sincethe proposed MPC strategy is formulated as a voltage-modecontroller effort is put into decreasing the voltage to its newdesired level as quickly as possible. To do so, the controllableswitch is turned OFF, the current instantaneously reaches zero,and the capacitor discharges through the load until it reaches itsnew demanded value in about t ≈ 1.2 ms.
C. Step Change in the Input Voltage
Operating at the steady-state operating point correspondingto vo,ref = 30 V, the input voltage is changed in a step-wisefashion. At time t = 0.4 ms the input voltage is increased fromvs = 10 to 15 V. The transient response of the converter is de-picted in Fig. 12. The output voltage remains practically unaf-fected, with no undershoot observed, while the controller settlesvery quickly at the new steady-state operating point.
Fig. 11. Simulation results for a step-down change in the output voltagereference: a) output voltage (solid line) and output voltage reference (dashedline); and b) inductor current.
Fig. 12. Simulation results for a step-up change in the input voltage: a) outputvoltage (solid line) and output voltage reference (dashed line); and b) inductorcurrent.
D. Load Step Change
The last case examined is that of a drop in the load resistance.As can be seen in Fig. 13, a step-down change in the loadfrom R = 73 to 36.5 Ω occurs at t = 1 ms (the input voltage isvs = 15 V, and the output voltage reference is vo,ref = 30 V).The Kalman filter adjusts the output voltage reference to its newvalue so as to avoid any steady-state tracking error. This can
KARAMANAKOS et al.: DIRECT VOLTAGE CONTROL OF DC–DC BOOST CONVERTERS USING MPC 975
Fig. 13. Simulation results for a step-down change in the load: a) outputvoltage (solid line) and output voltage reference (dashed line); and b) inductorcurrent.
be observed in Fig. 13(a); after the converter has settled at thenew operating point, the output voltage accurately follows itsreference.
VI. EXPERIMENTAL VALIDATION
To further investigate the potential advantages of the proposedalgorithm, the controller was implemented on a dSpace DS1104real-time system. A boost converter was built using an IRF60MOSFET and a MUR840 diode as active and passive switches,respectively. The values of the circuit elements are the same asin Section V. Moreover, the nominal input and output voltagesand the nominal load resistance are the same as previously. Thevoltage and current measurements were obtained using Halleffect transducers.
Due to computational restrictions imposed by the computa-tional platform, a six-step prediction horizon was implemented,i.e., N = 6 and the sampling interval was set to Ts = 10 μs.The prediction horizon was split into N1 = 4 and N2 = 2 withns = 2. The weight in the objective function was chosen asλ = 0.5. The covariance matrices of the Kalman filter are thesame as mentioned previously.
A. Start-up
In Fig. 14, the output voltage and the inductor current ofthe converter are depicted during start-up. The inductor currentrapidly increases to charge the output capacitor to the referencevoltage level as fast as possible. The output voltage reaches itsdesired value in about t ≈ 1.8 ms. Subsequently, the inductorcurrent reaches its nominal value and the converter operates inDCM.
Fig. 14. Experimental results for nominal start-up: a) output voltage andb) inductor current.
Fig. 15. Experimental results for a step-up change in the output voltage refer-ence: a) output voltage and b) inductor current.
B. Step Changes in the Output Reference Voltage
The second case to be analyzed is that of the transient be-havior during step changes in the output reference voltage. Astep-up change in the output reference voltage from vo,ref = 15to 30 V occurs at t ≈ 1.7 ms. The response of the converter isillustrated in Fig. 15. The inductor current instantaneously in-creases, enabling the output voltage to reach its new desired levelas fast as possible. This happens in about t ≈ 1.9 ms, without asignificant overshoot. Moreover, a step-down change, illustrated
976 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Fig. 16. Experimental results for a step-down change in the output voltagereference: a) output voltage and b) inductor current.
in Fig. 16, is investigated. The output reference voltage changesfrom vo,ref = 20 to 15 V at t ≈ 1.9 ms. As can be seen, thecontroller exhibits a favorable performance; the inductor cur-rent is instantly reduced to zero so as to allow the capacitor todischarge through the resistor, and the converter reaches the newsteady-state operating point in about t ≈ 1.2 ms.
C. Ramp Change in the Input Voltage
Subsequently, the input voltage is manually increased fromvs = 10 to 15 V (the output reference voltage is vo,ref = 30 V),resulting in a voltage ramp from t ≈ 16 ms until t ≈ 38 ms (seeFig. 17). During the transient, the inductor current changes ac-cordingly in a ramp-like manner down to its new steady-statevalue. It can be seen that the output voltage remains unaffectedand is kept equal to its reference value, implying that input volt-age disturbances are very effectively rejected by the controllerand the Kalman filter.
D. Load Step Change
The last case examined is that of a step-down change inthe load resistance occurring at t ≈ 1.2 ms. With the converteroperating at the previously attained operating point, the loadresistance is halved, i.e., from R = 73 to 36.5 Ω. As can beobserved in Fig. 18, the Kalman filter quickly adjusts the voltagereference accordingly, resulting in a zero steady-state error inthe output voltage, thanks to its integrating nature.
VII. CONCLUSION
A model predictive control (MPC) approach based on enu-meration for dc–dc boost converter is proposed that directlyregulates the output voltage along its reference, without the use
Fig. 17. Experimental results for a ramp change in the input voltage: a) inputvoltage, b) output voltage, and c) inductor current.
Fig. 18. Experimental results for a step-down change in the load: a) outputvoltage and b) inductor current.
KARAMANAKOS et al.: DIRECT VOLTAGE CONTROL OF DC–DC BOOST CONVERTERS USING MPC 977
of an underlying current control loop. This enables very fast dy-namics during transients. Since the converter model is includedin the controller, the time-consuming tuning of controller gainsis avoided. The computational complexity is somewhat pro-nounced, but kept at bay by using a move blocking scheme.In addition to that, the switching frequency is variable. A loadestimation scheme, namely a discrete-time switched Kalmanfilter, is implemented to address load variations and to ensurerobustness to parameter variations. Simulation and experimen-tal results demonstrate the potential advantages of the proposedmethodology.
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Petros Karamanakos (S’10) received the Diplomadegree in electrical and computer engineering fromthe National Technical University of Athens, Athens,Greece, in 2007, where he is currently working to-ward the Ph.D. degree in the School of Electrical andComputer Engineering.
From 2010 to 2011, he was with the ABB Cor-porate Research Center, Baden-Dattwil, Switzerland.His main research interests lie at the intersection ofoptimal control theory and power electronics, includ-ing model predictive control for power electronics
converters and ac drives.Mr. Karamanakos received the First Prize Paper Award of the
Industrial Drives Committee at the 2012 IEEE Energy Conversion Congress andExposition.
978 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Tobias Geyer (M’08–SM’10) received the Dipl.-Ing.and Ph.D. degrees in electrical engineering from ETHZurich, Zurich, Switzerland, in 2000 and 2005, re-spectively.
From 2006 to 2008, he was with the High PowerElectronics Group of GE’s Global Research Centre,Munich, Germany, where he focused on control andmodulation schemes for large electrical drives. Sub-sequently, he spent three years at the Department ofElectrical and Computer Engineering, The Universityof Auckland, Auckland, New Zealand, where he de-
veloped model predictive control schemes for medium-voltage drives. In 2012,he joined ABB’s Corporate Research Centre, Baden-Dattwil, Switzerland. Hisresearch interests are at the intersection of power electronics, modern controltheory, and mathematical optimization. This includes model predictive controland medium-voltage electrical drives.
Dr. Geyer received the Second Prize Paper Award at the 2008 IEEE IndustryApplications Society Meeting and the First Prize Paper Award of the IndustrialDrives Committee at the 2012 IEEE Energy Conversion Congress and Exposi-tion. He serves as an Associate Editor of the Industrial Drives Committee of theIEEE TRANSACTIONS ON INDUSTRY APPLICATIONS and as an Associate Editorfor the IEEE TRANSACTIONS ON POWER ELECTRONICS.
Stefanos Manias (M’85–SM’92–F’05) received theB.Eng., M.Eng., and Ph.D. degrees in electrical en-gineering from Concordia University, Montreal, QC,Canada, in 1975, 1980, and 1984, respectively.
In 1975, he joined the Canadian Broadcasting Cor-poration where he was responsible for the design ofradio and television automation systems. In 1989, hejoined the Electrical and Computer Engineering De-partment, National Technical University of Athens,Athens, Greece. He is currently holding the positionof a Full Professor where he is teaching and contact-
ing research in the area of Power Electronics and Motor Drive Systems. He isthe author of more than 80 IEEE and IEE publications in power electronics andmotor drive systems.
Dr. Manias is the Chapter Chairman and founder of the IEEE Greece sectionIAS-PELS-IES and a member of the IEEE motor drives committee. He is aRegistered Professional Engineer in Canada and Europe.
