21 nonlinear adaptive backstepping

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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525

www.elsevier.com/locate/cnsns

A nonlinear adaptive backstepping approach appliedto a three phase PWM AC–DC converter

feeding induction heating

A. Hadri-Hamida a,*, A. Allag b, M.Y. Hammoudi b, S.M. Mimoune b, S. Zerouali b,M.Y. Ayad c, M. Becherif c, E. Miliani c, A. Miraoui c

a Laboratory of Electrotechnics of Constantine, University of Mentouri, 25000 Constantine, Algeriab MSE Laboratory, Department of Electrical Engineering, University of Biskra, BP 145, 07000 Biskra, Algeria

c GESC Department, University of Technologies Belfort-Montbeliard, Belfort, Cedex 90010, France

Received 24 May 2006; received in revised form 10 February 2008; accepted 10 February 2008Available online 26 February 2008

Abstract

This paper presents a new control strategy for a three phase PWM converter, which consists of applying an adaptivenonlinear control. The input–output feedback linearization approach is based on the exact cancellation of the nonlinearity,for this reason, this technique is not efficient, because system parameters can vary. First a nonlinear system modelling isderived with state variables of the input current and the output voltage by using power balance of the input and output, thenonlinear adaptive backstepping control can compensate the nonlinearities in the nominal system and the uncertainties.Simulation results are obtained using Matlab/Simulink. These results show how the adaptive backstepping law updatesthe system parameters and provide an efficient control design both for tracking and regulation in order to improve thepower factor.� 2008 Elsevier B.V. All rights reserved.

PACS: 05.45.�a; 02.30.Yy; 07.50.Ek; 07.05.Dz

Keywords: Adaptive nonlinear control; AC–DC converter; PWM converter; Induction heating

1. Introduction

In the past few years remarkable progress has been made in development of high power density DC/DCconverters using resonant link schemes which utilize high speed devices such as fast recovery transistorsand GTOs. These new converters not only have high power density but also possess very low switching lossessince switching of the devices are made at zero-voltage instants and thus enable the total system to operate at

1007-5704/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2008.02.005

* Corresponding author. Tel./fax: +213 33 74 27 33.E-mail address: [email protected] (A. Hadri-Hamida).

mailto:[email protected]

1516 A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525

very high frequency compared to conventional DC link transistorized converters. Although these resonantlink converters are intended to operate at high power density, almost all the systems presented in the pastrequire self commutated transistors and have some difficulty performing conversion at very high power levelsbecause of the relatively low voltage and current margins that self commutated devices such as transistors typ-ically have.

These new converters with high frequency and high power density are necessary in induction heating appli-cation which leads to increase the switching frequency. However, increasing the switching frequency leads tosignificant switching losses, which is deteriorated overall system efficiency [1].

In recent year, three phase voltage source PWM converters are increasingly used for applications such asUPS systems, electric traction and induction heating. The attractive features of them are constant DC busvoltage, low harmonic distortion of the utility currents, bidirectional power flow and controllable power factor[2–5].

A linearization technique using input–output feedback have been used to design the nonlinear controller bychanging the original nonlinear dynamics into linear one [6–8], but this technique do not take into account theuncertainties of system parameters. Adaptive backstepping control is a newly developed technique for the con-trol of uncertain nonlinear systems [9–12].

This paper presents the implementation of the adaptive backstepping control law to the three phase PWMconverter. First, the nonlinear model of the system is introduced in the following section, where the exact non-linear multiple-input multiple-output state space model was obtained in ðd; q; 0Þ reference frame using thepower balance between the input and output sides. State feedback linearization control for a three phasePWM converter is given in Section 2.

In Section 3, the uncertainty in the system is the resistance of the load. The uncertain nonlinear model ofPWM converter is partially linearized through an input–output feedback linearization method when theparameter uncertainty is considered. To compensate the uncertainty, a nonlinear adaptive control techniqueis adopted to derive the control algorithm, and the uncertainty adaptation laws can also be derived systemat-ically based on the backstepping control technique [13].

In Section 4, simulation results were obtained; these results clearly show how the adaptive backstepping lawupdates the system parameters of PWM converter and gave a good performance.

2. Modelling of the proposed converter

A power circuit of PWM three phase voltage source AC–DC converter associated with a power circuit ofthe quasi resonant DC link converter [14] feeding induction heating is introduced in Fig. 1. This circuit couldbe modelised and an equivalent circuit is derived (Fig. 2) [4]. It is assumed that a resistive load RL is connectedto the output terminal. A voltage equation is derived from Fig. 2 as

es ¼ Ris þ Ldis

dtþ vF ð1Þ

where ed, and eq are the d–q axis source voltage, id, and iq the d–q axis source current, vd, and vq are the d–q axisconverter input voltage. R and L represent the line resistance and the input inductance, respectively. Consid-ering the terms of three phases, if one transforms (1) into a synchronous reference frame, then:

Inductor + Load

PWM converter

C

iload

iC

VC

idc

A B C

R es

i3

L1

L3

D3

i1

C’

D1

D2S2

S1

L2

i2

iC

VC’

QRDCL converter Induction Heating

Fig. 1. A power circuit of PWM converter associated with a power circuit of the QRDCL converter.

RloadC

Res

iload

iC

VC

idc

LA

B C

Fig. 2. Power circuit of PWM converter.

A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525 1517

Ldid

dt� xLiq þ Rid ¼ ed � vd ð2Þ

Ldiq

dtþ xLid þ Riq ¼ eq � vq ð3Þ

where x is the angular frequency of the source voltage.So, we have:

V sðtÞ ¼ V s

ffiffiffi2p

sin xt ð4ÞiloadðtÞ ¼ iload

ffiffiffi2p

sinðxt � uÞ ð5Þ

where u is the phase between the voltage and the load current.For fast voltage control, the input power should supply instantaneously the sum of load power and charg-

ing rate of the capacitor energy.The average rate of change of energy associated with AC link and DC link is given by

P ¼ 3

2ðedid þ eqiqÞ ¼ V dcidc ð6Þ

where the input resistance loss and switching device loss are neglected. On the output side, we have:

idc ¼ CdV dc

dtþ V dc

RLð7Þ

From (6) and (7),

3

2ðedid þ eqiqÞ ¼ C

dV dc

dtþ V 2

dc

RLð8Þ

Eq. (8) lead to nonlinear system with regard to Vdc. Combination of Eqs. (2), (3) and (8) describes a nonlinearmodel as

_id

_iq

_V dc

264375 ¼ � R

L id þ xiq

� RL iq � xid

32CV dcðedid þ eqiqÞ � V dc

RLC

264375þ

1L 0

0 1L

0 0

264375 ed � vd

eq � vq

� �ð9Þ

The system is of the third order which has a two control inputs.

3. Input–output feedback linearization

The input–output feedback linearization control for a PWM converters is introduced in [15].We can write the system (9) as follows:

_x ¼ f ðxÞ þ g1ðxÞud þ g2ðxÞuq ð10Þ


The system parameters may deviate from the nominal values. Let b ¼ 1RL

, then b = bn + Db, where bn representthe nominal value of parameter b, while Db represent the error of the nominal value. Taking into accountthese uncertainties, the system (10) can be changed as

_x ¼ �f ðxÞ þ Df ðxÞ þ g1ud þ g2uq ð11Þ

where �f ðxÞ and Df(x) are the nominal and uncertain matrices of f(x), respectively. Such that:

�f ðxÞ ¼� R

L id þ wiq

� RL iq � wid

23CV dcðedid þ eqiqÞ � bnV dc

C

264375 and Df ðxÞ ¼

0

0�DbV dc

C

264375:

The control objective is to make DC output voltage track the desired reference output voltage command. Weshould select the DC output voltage as one of the output variable. Therefore, we choose the outputs variableof the PWM converter drive system as

y1 ¼ h1ðxÞ ¼ V dc

y2 ¼ Lf h1ðxÞy3 ¼ id

8><>: ð12Þ

Then the dynamic model of PWM rectifier in the new coordinate is given by

_y1

_y2

_y3

264375 ¼ L�f h1 þ LDf h1 þ Lg1h1ud þ Lg2h1uq

L2�f h1 þ LDf L�f h1 þ Lg1L�f h1ud þ Lg2L�f h1uq

L�f h2 þ LDf h2 þ Lg1h2ud þ Lg2h2uq

264375 ð13Þ

where

L�f h1ðxÞ ¼2

3CV dc

ðedid þ eqiqÞ �bnV dc

C; L�f h2ðxÞ ¼ �

RL

id þ wiq; LDf h1ðxÞ ¼ �DbV dc

C; LDf h2ðxÞ ¼ 0;

L2�f h1ðxÞ ¼

2ðedf1ðxÞ þ eqf2ðxÞÞ3CV dc

� 2ðedid þ eqiqÞ3CV 2

dc

þ bn

C

( )f3; LDf L�f h1ðxÞ ¼

2ðedid þ eqiqÞ3C2V dc

þ bnV dc

C2

� �Db

Lg1h1ðxÞ ¼ Lg2h1ðxÞ ¼ 0; Lg1h2ðxÞ ¼1

L; Lg2h2ðxÞ ¼ 0; Lg1L�f h1ðxÞ ¼

2ed

3LCV dc

and

Lg2L�f h1ðxÞ ¼2eq

3LCV dc

In order to decouple the two control inputs, we construct the new control inputs as follows:

�ud

�uq

� �¼

Lg1L�f h1ud þ Lg2L�f h1uq

Lg1h2ud þ Lg2h2uq

� �ð14Þ

In the new coordinate, the system can be written as follows:

_y1

_y2

_y3

264375 ¼ y2

L2�f h1

L�f h2

264375þ 0 0

1 0

0 1

264375 �ud

�uq

� �þ

LDf h1

LDf L�f h1

LDf h2

264375 ð15Þ

where


LDf h1ðxÞ ¼ Db � V dc

C

� �¼ hu1ðxÞ ð16Þ

LDf L�f h1ðxÞ ¼ Db2ðedid þ eqiqÞ

3C2V dc

þ bnV dc

C2

� �¼ hu2ðxÞ ð17Þ

LDf h2ðxÞ ¼ 0 ð18Þ

The structural property of the new system (15) contains two decoupled subsystems. The first subsystem is

_y1 ¼ y2

_y2 ¼ L2�f h1 þ �ud

(ð19Þ

The second subsystem is

_y3 ¼ L�f h2 þ �uq ð20Þ

where ud and uq are the input control of the subsystem (9) and (10), respectively. This structure allows us toconveniently use adaptive backstepping design technique to obtain the desired controller. The uncertainties ofthe system now are reflected by unknown parameter h. Thus, we obtain the compact form of the error-trackingmodel as follows:

_y ¼ AðxÞ þ DAðxÞ þ BðxÞ �U ð21Þ

where

AðxÞ ¼e2

L2�f h1

L�f h2

264375; DAðxÞ ¼

hu1ðxÞhu2ðxÞ

0

264375 and BðxÞ ¼

0 0

1 0

0 1

264375:

In order to obtain good transient performance, a linear reference model is defined as

_ym ¼ kmym þ Bmuref

_ym1

_ym2

_ym3

264375 ¼ 0 1 0

�km1 km2 0

0 0 km1

264375 ym1

ym2

ym3

264375þ 0 0

km1 0

0 km3

264375 V �dc

i�d

� � ð22Þ

Using the reference model, the performance of the system can easily be evaluated, as the tracking problemcould be changed to a regulation problem. Define the error variables as

e ¼e1

e2

e3

264375 ¼ y1 � ym1

y2 � ym2

y3 � ym3

264375 ð23Þ

and use the following transformation:

eU ¼ ~ud

~uq

� �¼

�ud þ km1ym1 þ km2ym2 � km1V �dc

�uq þ km3ym3 � km3i�d

� �ð24Þ

Then the differential equations of the errors can be derived as follows:

_e ¼ AðxÞ þ DAðxÞ þ BðxÞ eU ð25Þ

The uncertain parameter error is defined as: ~h ¼ h� h, where h is the estimation of h and ~h is the estimationerror. For the first equation of (25), e2 is taken as the new control input according to backstepping controltechnique. It can be easily obtained that, if uncertainty h is known, the first equation of (25) is obviously stableby a Lyapunov function V ¼ 1

2e2 and a virtual controller a0 as

a0ðxÞ ¼ �k1e1 � hu1 ð26Þ


However, h is actually unknown and e2 is not the real control. Hence, an estimate h is used to replace h in (26).Define the new virtual control a for e2 as

aðxÞ ¼ �k1e1 � hu1 ð27Þ
Step 1: Define new variables as
z1 ¼ e1; z2 ¼ e2 � aðxÞ; z3 ¼ e3 ð28Þ

The virtual control a for z2 stabilizes the first equation as follows:

a ¼ �k1z1 � hu1 ð29Þ

The derivatives of the new variables are written as

_z1 ¼ z2 þ aþ hu1 ¼ �k1z1 þ z2 þ ~hu1 ð30Þ

_z2 ¼ e2 þ _a ¼ L2�f h1ðxÞ þ hu2 þ ~ud � k1 _z1 þ _hu1 ð31Þ

Step 2: The first control Lyapunov function V 1ðz1; z2; ~hÞ is written as

V 1ðz1; z2; ~hÞ ¼1

2z2

1 þ1

2z2

2 þ1

2c~h2 ð32Þ

where c is the adaptation gains. The derivative of V 1ðz1; z2; ~hÞ is

_V 1ðz1; z2; ~hÞ ¼ z1 _z1 þ z1 _z1 þ1

c~h _~h

¼ �k1z21 þ ~h z2u2 þ z1u1 þ k1z2u1 þ

1

c_~h

� �þ z2fz1 þ L2

�f h1 þ hu2 þ �ud þ k1ðz2 � k1z1Þ þ _hu1g

ð33Þ

Step 3: We can write the third equations of (22) as

_z3 ¼ L�f h2ðxÞ þ ~uq ð34Þ

Define the Lyapunov function V2(z3) for the third equation of (22) as

V 2ðz3Þ ¼1

2z2

3 ð35Þ

The derivative of V2(z3) is as follows:

_V 1ðz3Þ ¼ z3 _z3 ¼ z3fL�f h2ðxÞ þ ~udg ð36Þ

Step 4: To design the final adaptive backstepping nonlinear control for the system, we define the augmentedLyapunov function V ðz1; z2; z3; ~hÞ as

V ðz1; z2; z3; ~hÞ ¼ V 1ðz1; z2; ~hÞ þ V 2ðz3Þ ð37Þ

The derivative of V ðz1; z2; z3; ~hÞ is computed as

_V ðz1; z2; z3; ~hÞ ¼ �k1z21 þ ~h z2u2 þ z1u1 þ k1z2u1 þ

1

c_~h

� �þ z2fz1 þ L2

�f h1 þ hu2 þ �ud þ k1ðz2 � k1z1Þ

þ _hu1g þ z3ðL�f h2 þ ~uqÞ ð38Þ

To make _V ðz1; z2; z3; ~hÞ 6 0, the simplest way is to make the items in the square brackets in the second, thirdand fourth items equal to zero to cancel the uncertainties and make the fifth term equal to �k2z2

2, the sixthterm equal to �k3z2

3 and _V ¼ 0 if, and only if, z = 0. Then the following results can be obtained:

Fig. 3. Adaptive backstepping control block diagram of PWM converter.

Fig. 4. Block diagram of adaptive backstepping controller.


Control outputs:

~ud ¼ e1 � k2e2 � L2�f h1 � hu2 þ k1ðk1z1 � z2Þ � _hu1 ð39Þ

~uq ¼ �k3e3 � L�f h2ðxÞ ð40Þ

Parameter adaptation laws:

_h ¼ � _~h ¼ cðz1u1 þ k1z2u1 þ z2u2Þ ð41Þ
The final control inputs ud and uq can be easily derived through (23) and (26). The block diagram of the non-linear controller and adaptation laws is shown in Figs. 3 and 4, respectively.
4. Simulation results

We implemented the controller in Matlab/Simulink to verify the stability and asymptotic tracking perfor-mance. The overall block diagram for proposed control scheme is shown in Fig. 3.

The simulation of the steady state operation was performed at nominal power Pn = 10 kW. Table 1 showsthe parameter values used in the ensuing simulations.

Table 1Parameter of the PWM converter

Supply’s voltage and frequency 220 V(rms), 50 HzLine’s inductor and resistance 0.1 mH, 2 mXDC link resistance 20 XOutput capacitors 370 lFPWM carrier frequency 1 kHz

Fig. 5. Output DC link voltage response.

Fig. 6. Supply current and supply voltage without uncertain.

Fig. 7. Output DC link voltage response without adaptation law.



The output DC link voltage of the PWM converter, the supply voltage and the input line current responsesof the ideal input–output feedback linearization, are presented in Figs. 5 and 6, respectively.

If there is no uncertainty, the actual output DC link voltage response can track the DC voltage referenceresulting from the reference model perfectly. The ideal input–output feedback linearization is based on the

Fig. 8. Estimation of the parameter bRlðtÞ.

Fig. 9. Output DC link voltage response with adaptive law.

Fig. 10. Supply current and supply voltage with adaptive law.

0 1000 2000 3000 4000 50000

0.5

1

1.5

Nor

mal

ised

am

plitu

de

Frequency [Hz]

Fig. 11. Harmonic spectra of line current.


exact cancellation of the nonlinearity, for this reason, the ideal nonlinear control cannot deal with the systemuncertainties.

Fig. 7 presents the output DC link voltage of the system without an adaptation law. Figs. 8–10 give thesimulation results of the proposed backstepping controller with adaptation laws. Fig. 11 shows the harmonicspectra of line current.

In order to test the robustness of the controller to the change of the system parameters, the resistance of theload is changed to RL = 2RL nom at t = 0.15 s. An output DC voltage drop is observed, but this voltage drop issuccessfully rejected due to the effectiveness of the adaptation laws. The proposed control scheme gives satis-factory simulation results.

5. Conclusion

We have implemented and simulated the adaptive backstepping control for an uncertain PWM converter,which provides an efficient control design for both tracking and regulation. Global asymptotic stability of theblock system is guaranteed. Simulation results obtained were in good performance as it is expected. The strat-egy control was very robust to uncertain parameters and gave a very high power factor and small ripple in thecurrent line supply.

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21 nonlinear adaptive backstepping

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