Cascade Sliding Mode-PID Controller for a Coupled-Inductor BoostConverter

Niliana Carrero, Carles Batlle and Enric Fossas,Member, IEEE,

Abstract— In this paper, a coupled-inductor Boost converteris modelled as a piece-wise complementarity system and con-trolled by means of two loops: a sliding mode control inner loopand an experimentally tuned PID outer loop control. The aimof the closed loop system is to regulate the output voltage ofthe coupled-inductor Boost converter. The control is carried outusing the piece-wise complementarity model of the converter,which takes into account its hybrid dynamic. In addition, theperformance and the effectiveness of the feedback control isvalidated through computer simulations using MATLAB andPsim.

I. I NTRODUCTION

In some emerging applications like fuel cells [1], windpower generation [2], photovoltaic systems [3], or lightingapplications [4], the use of switching power converters allowsto modify electrical power efficiently. However, power con-verter analysis and control design is not an easy task, becausethe key elements are active and passive switches (diodes),which possess piecewise-linear current/voltage characteris-tics [5] (voltages in diodes may be discontinuous).

Recently these switching networks have been modelled aslinear complementarity systems [6]. Voltages at capacitorsand currents through inductances are usually taken as statevariables, while diode currents and voltages act as comple-mentarity variables. The coupled-inductor boost converterconsidered in this paper can be described as a piece-wiselinear complementarity system. It is locally modelled bytwo switching structures, which correspond to the convertermodels when the switch is in ON and OFF positions.

Several feedback techniques have been used to solveregulation or tracking problems in power converters. Linearcontrol techniques are the most common, due to their sim-plicity. Continuous controllers are synthesized for averagedmodels in a continuous-time framework and implementedthrough pulse width modulation (PWM). Since the averagedmodels result, in general, in non-linear dynamical systems,they are replaced by linearized systems at specific operatingpoint regardless of any restrictions related to the controlimplementation. As a result, the controller design is valid

Work partially supported by Spanish CICYT projects DPI2010-15110 andDPI2011-25649

Niliana Carrero and Enric Fossas are with the Instituteof Industrial and Control Engineering, Universitat Polit`ecnicade Catalunya, ETSEIB, Av. Diagonal 647, 08028 Barcelona,Spain (niliana.andreina.carrero@upc.edu,enric.fossas@upc.edu)

Carles Batlle is with Department of Applied Mathematics 4 and In-stitute of Industrial and Control Engineering, Universitat Politecnica deCatalunya, EPSEVG, Av. V. Balaguer s/n 08800 Vilanova i la Geltru, Spain(carles.batlle@upc.edu)

in a neighbourhood of the operating point only, whichleads to low performance during transients [7]. Moreover,performance and stability of such controllers are guaranteedonly for one converter operation mode, even though thesystem may function in continuous (CCM) and discontinuous(DCM) conduction modes over the entire operating range.

Model predictive control (MPC) [8], passivity based con-trol [9], [10], linear quadratic regulators (LQR) [11], slidingmode control (SMC) [12], [13] may be found among the non-linear control techniques applied to regulation or trackingproblems in power converters. Sliding mode control (SMC)is particularly appropriate to control such converters sincethey intrinsically are discontinuous systems because of theswitch [14]. V. Utkin [15] popularized SMC in Westerncountries in the late seventies. Since then, SMC has beenapplied, among others, to mechanical, electrical and chemicalplants. The success of this discontinuous feedback controltechnique is due to its robustness in presence of systemparameters variations and model uncertainties, good perfor-mance, a reduced order dynamics designed by the engineerand because it is easy to implement. However, there arealso some disadvantages such as chattering and the need forsome sophisticated mathematical formulations. For details,the reader is referred to [16], [17]. Nevertheless, from theengineer’s side, the choice of a suitable sliding surface isnot an easy task.

In this paper, a sliding mode control for a coupled-inductorBoost converter is presented. To the authors’ knowledge, itis the first reference in the literature to a piece-wise linearcomplementarity converter controlled by means of SMC.The switching action is governed through an hysteresis.This converter is especially interesting because it achieveshigh efficiency without requiring an extreme duty ratio [18].An indirect feedback control with inner and outer loops issynthesized. The inner loop is a sliding mode current controlloop, while the outer one is a PID controller that tunes thecurrent reference to regulate the output voltage. This two-loop scheme is applied to the power converter, which ismodelled as a piece-wise complementarity system [19].

Although Utkin’s mathematical analysis for obtaining theideal sliding dynamics cannot be strictly applied to thecomplementarity formulation of our system, the simulationsin closed-loop demonstrate the effectiveness of the approach.

The paper is organized as follows. The dynamic model ofthe coupled-inductor Boost converter using the complemen-tarity framework is described in Section II and the controllaw is designed in Section III. The simulation results arein Section IV. Finally, some conclusions and considerations

L1

L2

D1

D2

C2

C1V

in

VR

S+

-

+

+

+

-

-

-

-

-+

+

Fig. 1. Coupled-inductor Boost Converter

about further research are presented in Section V.

II. COUPLED-INDUCTOR BOOST CONVERTER

The circuit topology of the coupled-inductor Boost con-verter is shown in Fig 1. According to [6] the dynamics ofthe converter using linear complementarity systems (LCS) isgiven by

• S=Off (is = 0)

x = A1x+B1ω +E1Vin

z =C1x+D1ω +F1Vin

06 z1⊥ω1 > 0

06 z2⊥ω2 > 0

(1)

and the complementary variables are given by the twopairs

ω1 =−vD1 → z1 = iD1 = x1,

ω2 =−vD2 → z2 = iD2 = x2,(2)

with

A1 =

0 0 −L2 M0 0 M −L1

1 0 −∆ −∆0 1 −∆ −∆

, B1 =

−L2 MM −L1

0 00 0

,

E1 =

L2

−M00

,C1 =

(

1 0 0 00 1 0 0

)

,

D1 =

(

0 00 0

)

,F1 =

(

00

)

(3)• S=On(vs = 0)

x = A2x+B2ω +E2Vin

z = C2x+D2ω +F2Vin

0 ≤ z1⊥ω1 ≥ 0

0 ≤ z2⊥ω2 ≥ 0 (4)

and the pairs of complementary variables are given by

ω1 = iD1 → z1 =−vD1 = x3

ω2 =−vD2 → z2 = iD2 = x2(5)

with

A2 =

0 0 0 M0 0 0 −L1

0 0 −∆ −∆0 1 −∆ −∆

, B2 =

0 M0 −L1

−1 00 0

,

E2 = E1, C2 =

(

0 0 1 00 0 0 1

)

,

D2 = D1, F2 = F1(6)

We use normalized continuous state variablesx1 = ΓVin

iL1,x2 = Γ

ViniL2, x3 = 1

VinvC1 and x4 = 1

VinvC2. The variablesiL1

and iL2 denote the inductors currents throughL1 and L2

respectively, and the voltages across the capacitorC1 andC2

are represented byvC1 and vC2. The diodesD1 and D2 aremodelled as ideal diodes, and their current-voltage character-istics are expressed by the pair of complementary variablesz and ω . This set of constraints is called “complementaryconditions” (CC) [20]. These inequalities hold component-wise, and⊥ denotes the orthogonality between both vectors,so that z can be positive only ifω = 0, and vice versa.The normalization of the state and time variables yields thefollowing normalized parametersk = L1L2−M2, τ = t√

kC1,

∆ = ΓR , Γ =

√

kC1

. Note that, with this normalization, thevoltage of the input sourceVin is equal to 1.

In compact form, the dynamics of the converter can thenbe rewritten as

x = A1x+B1ω +E1Vin +(Ax+Bω +EVin)uz =C1x+D1ω +F1Vin +(Cx+Dω +FVin)u

06 ω⊥z > 0

u =

1 → S = On0 → S = Off

(7)

where A= A2−A1, B= B2−B1, C=C2−C1, D= D2−D1,E= E2−E1 and F= F2−F1.

III. C ONTROL DESING

In general, the main objective of control in DC-DCconverters is to regulate the output dc-voltage (VR) to agiven reference value (Vre f ). This is to be achieved by theapplication of an appropriate feedback control lawu, whichwill control the switched state. One major concern in thedesigning process is that regulation must be carried out in thepresence of load disturbances and input voltage variations.In addition, it must take also into account the restrictionsover the manipulated variable (duty cycle). In this context,the general control scheme for our coupled-inductor Boostconverter is shown in Fig. 2. For the inner loop, a non-linearcontrol strategy similar to sliding mode control is proposed,with the goal of controlling the fast dynamics of the converter(the inductors currents), and a PID controller is designedfor the outer loop. This second loop will set the internalreference (Ire f ) of the current loop and at the same timewill regulate the output voltageVR of the converter. In thiscontrol all state variables of the converter (inductor currentsand capacitor voltages) are sensed.

DC-DC

Converter+

+

-

-V refIref s u

VR

VR

iL1,2

PIDe (x)

Fig. 2. Control scheme for the considered coupled-inductorBoost converter

A. Current control analysis

Assuming that the inductors currents in steady state havean average value, the purpose of the current control loop isto drive the average value of the inductors currentsx1 andx2

to a reference valueIre f , and the sliding surface is defined asa linear combination of the state variables of the inductorscurrents given by

s(x) = L1x1+Mx2− Ire f , (8)

whereIre f is defined by the outer loop PID controller. Thecontrol action is designed as

u =

0 i f s(x)> 0,1 i f s(x)< 0,

(9)

and, furthermore, a hysteretic function is added to the slidingfunction in order to maintain this near zero. Thus, usingthis control law, both existence and reaching conditions arebeing imposed to ensure the stability condition, at leastin a neighborhood around the system equilibrium point.In the literature such condition is known as the reachingcondition [15]. If the trajectories of the converter reach thevicinity of the sliding surface in which there is a slidingmode then they cannot leave that surface, except possiblyat the boundaries of the sliding zone. This condition can beexpressed mathematically as lim

s→0+s< 0 and lim

s→0−s> 0. These

inequalities imply that in a neighborhood ofs(x), the functions(x) evaluated on the trajectories of the system solution (7)is increasing fors(x)< 0 and decreasing fors(x)> 0. Thus,during the interval of time in which the switch is closed(On), one gets

s = m1 = L1L2−M2. (10)

Since in a coupled inductor the mutual inductanceM hasto be positive and lower than the geometric mean of theinductances of the coils,i.e. 06 M 6

√L1L2, equation (10)

implies that the derivative ofs(x) evaluated on the vectorfield (4) is positive. On the other hand, during the intervalof time in which the switch is open (Off), one gets

s = (L1L2−M2)(1+ω1− x3). (11)

In this case the derivative ofs(x) depends on the comple-mentarity variableω1, which is defined in (2). Whenω1 = 0one has

s ≡ m2 = (L1L2−M2)(1− x3). (12)

Since the output voltagex3 must be greater than one, one hasthat, during the period of time in whichω1 = 0, the derivativeof s(x) is negative. In addition, it must also be negative whenω1 6= 0 in order to guarantee sliding mode control. Indeed,

in this case the inductor current is identically zero,z1 =iD1 = x1 = 0, so its derivative is also zero during this periodof time. Then, by replacing ˙x1 = 0 in the expression ˙s =α1x1+α2x2 = 0 and using ˙x2 from (1) one gets

ω1 =−L2+Mx4−L2x3

L2. (13)

Using this in (11) yields finally

s ≡ m3 =−M(L1L2−M2)x4

L2. (14)

Since the output voltage isx4 > 0, the derivative during thisperiod of time is also negative. Putting all the results together,one concludes that the reaching condition is satisfied.

Figure 3 provides the simulation results using the linearcomplementarity model (7) and under sliding mode control.As it can be observed from the surface trajectorys in Fig.3(d), the existence and reaching conditions for having slidingmodes ons(x) = 0 are satisfied. To the authors’ knowledge,there is no definition of an equivalent control and its corre-sponding ideal sliding dynamics for general complementaritysystems and, at present, we can only provide simulationresults to validate the correctness of this modelling andcontrol approach.

B. Voltage control analysis

The main goal of the voltage control loop is to regulate theoutput dc-voltage to a reference value. Thus, a discrete-timePID controller of the following form has been designed

C(z) = Kp +KiTs

2z+1z−1

+Kd

Ts

z−1z

, (15)

whereKp, Ki, Kd are the gain values of the controller andTs is the sampling period. The parameters of this controllerhave been designed through experimental tuning accordingto different performance criteria such as minimal overshootand shorter settling time. The values of the coefficients areKp = 1.4e− 8, Ki = 1.1e− 6 and Kd = 8e− 12 with Ts =50e−6.

IV. SIMULATION RESULTS

The main purpose of this section is to show, throughcomputer simulation, the effectiveness of the control lawdesigned using the piece-wise complementarity model of theconverter. In addition, simulations of the converter applyingthe control law designed using Psim where carried out.The simulations of the complementarity model given byequation (7) were performed using Matlab. The back-Eulermethod plus a specific solution of the linear complementarityproblem (LCP) was used for the solution of the ordinarydifferential equations.

Two different scenarios of the closed loop system arepresented. The first one takes into account a perturbationload and the second introduces step changes in the referencevoltage.

• Scenario one: The nominal values of the converterparameters areVin = 12V , L1 = 75µH, L2 = 525µH,M = 196µH, C1 = 22µF, C2 = 22µF, R = 400Ω.

0.0148 0.0149 0.0149 0.015 0.015 0.0151 0.0151 0.0152 0.0152

0

2

4

6

8

10

12

14

t

i L1

LCS

(a) Zoom of inductor current trajectoryiL1

0.017 0.017 0.0171 0.0171 0.0172 0.0172 0.0173 0.0173 0.0174

0

0.5

1

1.5

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2.5

3

3.5

4

t

i L2

LCS

(b) Zoom of inductor current trajectoryiL2

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0

20

40

60

80

100

120

140

t

VR

LCS

(c) Output voltage trajectoryvR

0.0109 0.0109 0.011 0.011 0.0111 0.0111 0.0112 0.0112

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x 10-7

t

s(x

)

LCS

(d) Zoom of surface trajectorys

Fig. 3. Simulation result: current control

The controller gains areKp = 1.4e− 8, Ki = 1.1e− 6,Kd = 8e − 12, and the hysteresis thresholds arehmax = 6.20e−8 andhmin =−6.20e−8. The controllergoal is to regulate the output dc-voltage to a valueVre f = 120V . The initial conditions arex(0) = [0,0,0,0].The waveforms of the inductors currents, output voltageand sliding surface trajectories are shown in Fig. 4. Theleft column corresponds to the complementarity modeland the right one to the Psim model. As it can beobserved from the simulation results, the output voltageVR follows the reference. The robust tracking of thecontrol law designed is made manifest when a changein the nominal value of the load fromR = 400Ωto R = 200Ω is applied during the period of time0.1 < t < 0.2. In Fig. 5(a) and Fig. 5(b) are shown azoom of the inductors currents when a load perturbationhas been applied. The left column corresponds to thetime (t = 0.1) at which the load perturbation is applied,while the right column corresponds to the final time(t = 0.2).

• Scenario two: In this scenario, the nominal parametersare the same than in the load perturbation scenario,except for the load, which takes the valueR = 500Ω.The simulation results for the step reference changesare shown in Fig. 6. Just like before, the left columncorresponds to the complementarity model and the rightcorresponds to the Psim model. The reference voltage isincreased toVre f = 120V from its initial valueV re f =80V . This step reference is applied during the periodof time 0.2 < t < 0.4. Fig. 7(a) and Fig. 7(b) displaya zoom of the inductor currents when the referencevoltage is increased. The left column corresponds to theinitial time (t =0.2), while the right column correspondsto the final time (t = 0.4). It can be seen that there isa discrepancy between the Psim and complementaritymodels during the transient, although it disappears in thestationary regime. The results show that the controlleris able to track this varying reference in an stable way.

V. CONCLUSIONS

A cascade control law has been applied to a coupled-inductor converter. It has been shown that excellent perfor-mance and stable tracking can be obtained with the proposedcontrol law, which was tested using a piece-wise comple-mentarity model of the converter, which takes into accountits hybrid dynamics. The whole system was simulated inMATLAB. On the other hand, the same control law obtainedwas applied to the model of the converter provided by Psim,showing negligible steady state discrepancies with the piece-wise complementarity model.

Besides being more faithful to the actual behaviour ofthe electronic components, the complementarity formulationallows for a systematic search of generalized discontinuousmodes [20]. Furthermore, we have observed that, in specificsimulations with the Psim model, increasing the switchingfrequency results in a loss of coupling inductances and,

0 0.05 0.1 0.15 0.2 0.25

5

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30

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40

45

t

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25

30

35

40

45

t

i L1

i L1

LCS Psim

(a) Inductor current trajectoryiL1

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

14

16

18

t

LCS

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14

16

18

t

Psim

i L2

i L2

(b) Inductor current trajectoryiL2

0 0.05 0.1 0.15 0.2 0.25 0.3110

115

120

125

t

VR

0 0.05 0.1 0.15 0.2 0.25 0.3110

115

120

125

t

VR

VrefLCS

VrefPsim

(c) Output voltage trajectoryvR

0 0.05 0.1 0.15 0.2 0.25 0.3-8

-6

-4

-2

0

2

4

6

8x 10

-5

t

S(x

)

0 0.05 0.1 0.15 0.2 0.25 0.3

-8

-6

-4

-2

0

2

4

6

8x 10

-5

t

S(x

)

LCS Psim

(d) Surface trajectorys

Fig. 4. Simulation results: load perturbation

consequently, in a decrease of efficiency, while this does nothappen with the complementarity-based Matlab model.

Future research will include experimental results for thespecific converter discussed in this paper, and a general anal-ysis of the ideal sliding dynamics for linear complementaritysystems.

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0.0997 0.0998 0.0998 0.0999 0.0999 0.1 0.1 0.1001 0.1001 0.1002 0.1002

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t

0.1997 0.1998 0.1999 0.2 0.2001 0.2002 0.2003

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t

PsimLCS

PsimLCS

i L1

i L1

(a) Inductor current trajectoryiL1

0.0997 0.0998 0.0999 0.1 0.1001 0.1002 0.1003

0

0.5

1

1.5

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2.5

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3.5

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4.5

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t

PsimLCS

PsimLCS

i L2

i L2

(b) Inductor current trajectoryiL2

Fig. 5. Zoom of inductor currents trajectories

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0 0.1 0.2 0.3 0.4 0.50

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t

LCSi L

1

i L1

Psim

(a) Inductor currentiL1

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LCS Psim

(b) Inductor currentiL2

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vR

VrefLCS

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vR

VrefPsim

t t

(c) Output voltage trajectoryvR

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

-1.5

-1

-0.5

0

0.5

1

x 10-3

s(x

)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

-1.5

-1

-0.5

0

0.5

1

x 10-3

s(x

)

t t

LCS Psim

(d) Surface trajectoryvR

Fig. 6. Simulation result: step change in reference voltage

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i L1

i L1

tt

PsimLCS

PsimLCS

(a) Inductor current trajectoryiL1

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-1

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0.5

1

1.5

i L2

i L2

tt

PsimLCS

PsimLCS

(b) Inductor current trajectoryiL2

Fig. 7. Zoom of inductor currents trajectories