8/6/2019 mpi boost

1/6

Abstract--This paper presents a comparative study

between two control strategies based on the sliding mode

control theory applied to a switched DC-DC Boost

converter. A master-slave control structure is applied to the

converter such that the inner control loop is a sliding mode

current controller and the outer voltage controller is

designed according to what follows. First a Proportional-

Integral compensator (PI) is proposed and second, a Model

Predictive Controller (MPC) is used aiming an improved

performance. The control strategies are validated on the

nonlinear hybrid model in a simulation study. Tracking and

disturbance rejection tests are performed in order to show

the robustness of the controllers.

Index Terms Boost converter, hybrid systems, sliding

mode control, model predictive control.

I. INTRODUCTION

Switched power converters require few componentsand, from a theoretical point of view, are simplistic tooperate. All DC-DC converters require control circuitry

in order to account for load variations, componenttolerance, system aging and source voltage variations.

The controllers used in practical implementations are

frequently of analogue nature and have classic linearstructures such as Proportional-Integral-Derivative (PID)compensator. These compensators normally involve

suboptimal design for specifications such as fast responseand stability. Hence, there is a need of advanced controlmethods, which can now be implemented in practice

thanks to the latest advances in digital signal processors(DSP) [1].

From a control engineering point of view, DC-DC

converters are a traditionally benchmark for testing

(advanced) nonlinear controllers. However, apart fromtheir nonlinear characteristics, DC-DC converters pose

another interesting feature: they have unstable zerodynamics yielding to nonminimum phase behaviorsystems. The control of nonminimum phase systems is

significantly more difficult than control of systems withstable zero-dynamics since this feature restricts theclosed-loop performance.

Due to these difficulties, a number of nonlinearcontrollers have been reported in literature, such as:sliding mode control strategies [2], nonlinear PI

This paper is funded by the ALFA project in the frame ofLaBioProC programme project II-0407-FA and partly by the Belgian

Science Policy Office in the frame of the STEREO II programme -

project SR/00/100.

controllers based on the method of extended linearization[3] and a predictive controller [4,5] using algorithms suchas the Extended Prediction Self-Adaptive Control

(EPSAC) [6].The results of an experimental comparison of five

control algorithms on a boost converter are presented in

[7]: linear averaged controller, feedback linearizing

controller, passivity-based controller, sliding modecontroller, and sliding mode plus passivity-basedcontroller.

The control laws derived for such systems can be

classified in two groups, depending on whether theygenerate directly the switching signal a hybrid systemapproach- or whether they require an auxiliary pulse

width modulation (PWM) circuit to determine the switchposition.

Important advantages of sliding mode control overconventional PWM control are stability, even for large

supply and load variations, robustness, good dynamicresponse and simple implementation [8]. However, some

drawbacks appear. First, the switching frequency dependson the operating point, due to the hysteretic nature of thecontrol method. Second, steady-state errors can appear in

the output response and a significant overshoot in thestate variables might arise during the transient regime.

This paper deals with a classical Sliding Mode Control

(SMC) strategy used as a inner-control for the currentloop in a master-slave scheme. The voltage outer loop isimplemented as a Proportional-Integral compensator in

order to later compare it with a novel control structurebased on a Model Predictive Controller. The closed-loopbehavior is analyzed with respect to response time, load

disturbances, input-source voltage disturbances and

robustness. A comparison between the two controlstrategies is given and advantages and limitations of each

approach are discussed.The paper is organized as follows: A brief description

of the nonlinear system is presented in section II. The

controllers are designed in section III and comparisonsare treated in section IV. Finally, a conclusion sectionsummarizes the main outcome of this work, pointing to

some future steps for a model based predictive controldesign.

II. CIRCUIT DESCRIPTION

The boost converter considered throughout this paperis illustrated in Fig. 1 and the differential equationsdescribing the dynamic of the circuit are presented.

Model Predictive and Sliding Mode Control of

a Boost Converter

D. Plaza*, R. De Keyser**, and J. Bonilla****Ghent University, LHWM Laboratory of Hydrology and Water Management, Coupure Links 653, B-9000 Gent, Belgium**Ghent University, EeSA department of Electrical energy, Systems and Automation,

Technologiepark 913, B-9052 Gent, Belgium

***Katholieke Universiteit Leuven, ESAT/SCD & CIT/BioTeC, W. de Croylaan 46, B-3001 Leuven, Belgium

SPEEDAM 2008International Symposium on Power Electronics,Electrical Drives, Automation and Motion

37978-1-4244-1664-6/08/$25.00 2008 IEEE

8/6/2019 mpi boost

2/6

Fig. 1. Boost schematic circuit.

The converter is a hybrid system due to the interaction

between continuous and discrete signals, vS(t), vO(t), iL(t)and vC(t) are continuous variables while q represents the position of the switch (logical variable). When q=0 the

switch is opened and when q=1 the switch is closed. Forsimplicity in the mathematical expressions, the ideal casefor the circuit is considered (without losses).

Consequently,RL andRCare equal to zero, yielding to:

( )( )( ) ( ) ( )

( )( )( ) ( ) ( )tv

RCti

Ctq

dt

tdv

tvL

tvL

tqdt

tdi

CLC

SCL

111

111

=

+=

(1)

It is assumed that the source voltage is constant

(vS(t)=VS). Thus, the dynamic system is represented with

the set of equations:

SVLx

x

RCC

uL

u

x

x

+

=

0

1

11

10

2

1

2

1

(2)

wherex1, x2 and u correspond to the current, voltage andlogical variable q, respectively. It is well known, that the

system presents a nonminimum phase and highnonlinearity, as reported in [9]. Therefore, due to thenonlinearity of the system, it is important that the

designed controllers perform reasonably on a wideoperational range of the boost converter.

The system is modelled using the approach presented

in [10], where neither linearization nor averaging of the

states is performed. The model for simulation is built inMATLAB/SIMULINKTM as a block structure and

presented in Fig. 2. The controllers that are presented inthis work are implemented in Matlab/SIMULINK

TMas

well. SIMULINK is a graphical dynamics modeling

software package built on the top of the Matlab numericalworkspace.

The advantage of using this interface is that models are

entered as block diagrams when the correspondingmathematical descriptions are available for the targetsystems. Extensive documentation and internal functions

of the software make easier the implementation of modelsand controllers. On the other hand, the graphical

environment increases the computational load duringsimulations, which is an important feature to take intoaccount during simulations.

Fig. 2. Boost converter modelled in SIMULINK

The nominal values of the circuit parameters and the

source voltage are: Vs= 230V, R= 200, L=1mH,

C=100uF. The disturbances applied to the system undercontrol are presented in table I.

TABLEI

SIGNALS FORTESTING THE CONTROLLERS

I Steady State

II Set-point increase of 20 V

III Input Voltage Vs decrease of 50 V

IV Load resistorR reduction in 25%

III. CONTROLLERDESIGN

Switching converters are a particular case of variable

structure system (VSS), research on sliding mode controltechniques in VSS report good performance regarding toparameters variations and external disturbances.

A. Sliding Mode Control applied to the Boost converter

The design of a sliding mode controller is basically

performed in two steps [11]. In the first step the slidingsurface is selected according to:

*11 xxs = (3)

where s is the sliding surface and x1*

is the desiredcurrent. In the second step, the corresponding control

signal u is chosen. Since u is a discrete signal with values0 and 1, the control law can be designed based on the signfunction:

))(1(2

1ssignu = (4)

Expression (4) represents the case of an ideal switch.Additionally, hysteresis of 1 ampere is set to reduce the

chattering phenomena with a switching frequency of 50

kHz. The closed loop with the SMC policy exhibits aslow time voltage response and steady-state error.

38

8/6/2019 mpi boost

3/6

B. Cascade control of the converter with sliding modecontrol (PI+SMC)

Fig. 3 shows the block diagram of a cascade control,the inner control loop corresponds to the sliding modecurrent controller and the outer loop is a Proportional-Integral compensator.

Fig. 3. Cascade control scheme using SMC and PI (PI+SMC).

A generalized approach for designing compensatingnetworks in sliding mode controlled DC-DC switching

converters and its validation is well known. The approachpresented in [12] is used here for this purpose.

Fig. 4 is the generalized block diagram for a two states

switching DC-DC converter. The describing function1/m, can represent the linearized transfer function of apulse width modulator PWM (m0) or the ideal control

law (4) in sliding mode (m=0).Hereafter, a State Space Average Model (SSA) [12] is

obtained from (2) and a linearization procedure is

performed. In order to design a PI controller, the input tostates (U(s) to X1(s), X2(s)) transfer functions G1(s),G2(s) are considered to be known.

Fig. 4. Block diagram for a switching DC-DC converter with two states.

Using Masons rule, the closed loop transfer functionof the block diagram in fig. 4 is represented by

2211

22

*

2

2

Gs

zs

kGk

Gs

zsk

X

X

+

+

+

= (5)

In order to analyze the stability of the system, theRouth-Hurwitz criterion is used. The characteristic

polynomial is found for the closed loop transfer functionassuming k1=1 from the original sliding surface (3):

02)(

1 2 =+

+

++

sRC

V

VRs

V

VL

s

zsk O

S

S

O

(6)

where VO is the desired output voltage and stability isaccomplished when:

0,0,0 22 >>> LkV

VRCkz

O

S (7)

By setting k2=1 and using (6), the zero of the PI controlleris found based on the zero-pole cancellation method:

RCz

s

zsk

22 =

+(8)

The PI control loop modifies the original slidingsurface (3) according to

( ) ( ) ++= dtxxzkxxkxks*222

*22211 (9)

The addition of the integral term eliminates the steady-state error of the output voltage.

C. Model Predictive and Sliding control in a cascade

structure. (MPC+SMC)

The MPC strategy is applied as a compensatingnetwork to close the voltage loop while the current

controller is not modified. The MPC is based on thedynamical model of the process. However, in this work alinear SSA MPC approach is used.

A novel control structure is presented in this study.The novelty consists in the use of a MPC algorithm as the

voltage controller. The EPSAC algorithm is a well knownMPC variant, based on the input/output models of theprocess [6].

Fig. 5 shows a block diagram of the proposed controlscheme. The EPSAC controller sets the desired currentand the current is controlled by the SMC approach.

Fig. 5. MPC and SMC (MPC+SMC) applied to the boost converter.

The first step to apply the EPSAC as a controller is to

determine a proper model for prediction. This predictor isobtained from the linearized dynamic equations of the

converter and relates the converter input current with theoutput voltage. The model used for prediction is given by

2)()(

)(

+

+

=

sRC

V

VRs

V

VL

sI

sV O

S

S

O

O (10)

Additionally, the noise model employed in theControlled Auto-Regressive Integrated Moving Average(CARIMA) predictor:

)(

)(

1

1

)(

)(

11

1

te

te

qqD

qC C=

=

(11)

39

8/6/2019 mpi boost

4/6

is chosen as the disturbance process model in order to

account for model mismatch and achieve zero steady-state error. Here, eC(t) corresponds to a colored noisesignal while e(t) is assumed as white noise signal.

The objective of the model predictive control is to findthe input sequence that minimizes a given cost functionJ,

based on a desired output trajectory over a predictionhorizon (N1N2). The cost function considers the errors between the predicted model outputsy(t+kt) and thereference trajectory r(t+kt), formulated as:

[ ]=

+

++=

2

1

2

)1()...()|()|(min

N

NkNtutu

tktytktrJu

(12)

The degrees of freedom in the control law u can be

reduced defining a control horizon Nu, with Nu

8/6/2019 mpi boost

5/6

system leading to an overshoot of around 4 volts in the

voltage response. The fast response in the voltagegenerates a big peak in the current as can be seen in fig.10.

0 5 10 15 20 25 30 35 40 45 50 55 60455

460

465

470

475

480

485

Time (ms)

OutputVoltage(V)

I II III IV

Fig. 8. Output voltage around 460V: PI+SMC - (continuous line);

EPSAC+SMC - (dotted line).

0 5 10 15 20 25 30 35 40 45 50 55 60585

590

595

600

605

610

615

0 5 10 15 20 25 30 35 40 45 50 55 60

455

460

465

470

475

480

485

OutputVoltage(V)

0 5 10 15 20 25 30 35 40 45 50 55 60325

330

335

340

345

350

355

Time (ms)

II III IVI

III IV

II III IV

II

I

I

Fig. 9. Output voltage for three operating points PI+SMC - (continuous

line); EPSAC+SMC - (dotted line).

A test on disturbance rejection shows importantdifferences, while the PI+SMC structure recovers slowlyfrom disturbances; the EPSAC approach with the desiredcurrent estimation block presents a faster recovery. The

time response for disturbance rejection is slightlydifferent among the three operating points. Fig. 8 depictsmore clearly these times for 460 V. Interval III presents

the effects of an input voltage decrement. The approach based on the PI compensator shows a time response of20ms and the EPSAC around 3ms. On the other hand, a

decrement in the load resistance, described in interval IV,produces a time response for PI+SMC around 23ms whilefor the EPSAC+SMC controller is only 3ms.

Fig. 10 presents the current behavior for threedifferent operating points. The big overshoot in thecurrent is due to the fast voltage response for setpoint

variation described on interval II. This peak could bereduced through the proportional gain of the PI+SMC

controller and increasing the prediction horizon in theMPC approach.

The next step to improve the performance of the

structure based on MPC method is the addition of theinput penalization in the optimal control problem (12) toreduce or eliminate the overshoot in the current behavior.

However, including constraints on the control increasesthe complexity of the control law.

0 5 10 15 20 25 30 35 40 45 50 55 600

10

20

30

40

0 5 10 15 20 25 30 35 40 45 50 55 600

10

20

30

40

InductorCurrent(A)

0 5 10 15 20 25 30 35 40 45 50 55 600

10

20

30

40

Time (ms)

I II III IV

I II III IV

II III IVI

Fig. 10. Input Current at three operating points: PI+SMC - (continuous

line); EPSAC+SMC - (dotted line).

V. CONCLUSIONS

A comparative study over two different controltechniques on a DC-DC boost converter has been presented. Although the MPC approach shows better

performance than the classical PI technique fordisturbance rejection, there is still an expectedimprovement over the presented behavior in EPSACalgorithm by tuning appropriately the model for

prediction. On the other hand, a novel structure based onthe sliding mode control technique and MPC is developedand studied under simulation. This is an alternative to the

traditional control strategies for switching DC-DCconverters based on the use of a PWM unit.

Further research on this topic involves the use of a

better noise model in the EPSAC algorithm in order toavoid the use of the external filter, the penalization of theinput to eliminate the current peaks and the study of the

optimal parameter values in the EPSAC algorithm.

REFERENCES

[1] Special issue on digital control in power electronics. IEEETransaction Power Electronics, 18(1) Vol.II/II, 2003.

[2] H. Sira-Ramirez: Sliding motions in bilinear switchednetworks, IEEE Trans. Circuits Systems, vol. CAS-34,Aug.1987, p. 919-933.

[3] H. Sira-Ramirez: Design of PI controllers for DC-to-DC power supplies via extended linearization, Int. J. Control,

51(3), 1990, p.601-620.[4] M. Lazar, R. De Keyser:Non-linear predictive control of a

DC-to-DC converter. SPEEDAM04 Symposium onPower Electronics, Electrical Drives, Automation &Motion, 16-18 June, Capri, Italy; 5p paper nr. A206.

41

8/6/2019 mpi boost

6/6

[5] J. Bonilla, R. De Keyser, D. Plaza: Nonlinear PredictiveControl of a DC-DC Converter. A NEPSAC approach.Proc. of the EUCA - European Control Conference, July

2007.[6] R. De Keyser: Model Based Predictive Control. Invited

Chapter in UNESCO Encyclopedia of Life SupportSystems (EoLSS). EoLSS Publishers Co Ltd, Oxford,

2003.[7] G. Escobar, R. Ortega, H. Sira-Ramirez: J.P. Vilain and I.

Zein: An experimental comparison of several nonlinearcontrollers for power converters, IEEE Control Systems,19(1), 1999, p.66-82.

[8] Utkin, V. I., Sliding Modes and their applications inVariable Structure Systems. (Mir, Moscow, 1974).

[9] R. Erickson, D. Maksimovic, Fundamentals of PowerElectronics, (USA: Kluwer Academic Publishers), 2001.

[10]Juing-Huei Su, Jiann-Jong Chen, Dong-Shiuh Wu: Learning feedback controller design of switchingconverters via MATLAB/SIMULINK. IEEE Transactionson Education, 45, 2002, 307-315.

[11]V. I. Utkin, J. Guldner, and J. Shi, , Sliding Mode Controlin Electromechanical Systems. (London: Taylor &Francis), 1999.

[12]R. Giral, L. Martnez, J. Hernanz, J. Calvente, F. Guinjoan,A. Poveda, R. Leyva, Compensating networks for Sliding

Mode Control. IEEE International Symposium in Circuitsand Systems, 3, 1995, 2055-2058.

[13]R. De Keyser, C. Ionescu, M. Sbarciog. Advanced Controlof a Boost Converter. IEEE Symposium on PowerElectronics, Electrical Drives, Automation & Motion(Speedam2004), 2004, pp. 745-750.

42