- Email: [email protected]
A nonlinear adaptive backstepping approach applied to a three phase PWM AC–DC converter feeding induction heating
Available online at www.sciencedirect.com
Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525 www.elsevier.com/locate/cnsns
A nonlinear adaptive backstepping approach applied to 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 b
a Laboratory of Electrotechnics of Constantine, University of Mentouri, 25000 Constantine, Algeria MSE Laboratory, Department of Electrical Engineering, University of Biskra, BP 145, 07000 Biskra, Algeria c GESC Department, University of Technologies Belfort-Montbe´liard, Belfort, Cedex 90010, France
Received 24 May 2006; received in revised form 10 February 2008; accepted 10 February 2008 Available online 26 February 2008
Abstract This paper presents a new control strategy for a three phase PWM converter, which consists of applying an adaptive nonlinear 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 is derived with state variables of the input current and the output voltage by using power balance of the input and output, the nonlinear 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 updates the system parameters and provide an efficient control design both for tracking and regulation in order to improve the power factor. Ó 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/DC converters using resonant link schemes which utilize high speed devices such as fast recovery transistors and GTOs. These new converters not only have high power density but also possess very low switching losses since switching of the devices are made at zero-voltage instants and thus enable the total system to operate at *
Corresponding author. Tel./fax: +213 33 74 27 33. E-mail address: [email protected] (A. Hadri-Hamida).
1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.02.005
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 resonant link converters are intended to operate at high power density, almost all the systems presented in the past require self commutated transistors and have some difficulty performing conversion at very high power levels because of the relatively low voltage and current margins that self commutated devices such as transistors typically have. These new converters with high frequency and high power density are necessary in induction heating application which leads to increase the switching frequency. However, increasing the switching frequency leads to significant switching losses, which is deteriorated overall system efficiency [1]. In recent year, three phase voltage source PWM converters are increasingly used for applications such as UPS systems, electric traction and induction heating. The attractive features of them are constant DC bus voltage, 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 by changing the original nonlinear dynamics into linear one [6–8], but this technique do not take into account the uncertainties of system parameters. Adaptive backstepping control is a newly developed technique for the control of uncertain nonlinear systems [9–12]. This paper presents the implementation of the adaptive backstepping control law to the three phase PWM converter. First, the nonlinear model of the system is introduced in the following section, where the exact nonlinear multiple-input multiple-output state space model was obtained in ðd; q; 0Þ reference frame using the power balance between the input and output sides. State feedback linearization control for a three phase PWM converter is given in Section 2. In Section 3, the uncertainty in the system is the resistance of the load. The uncertain nonlinear model of PWM converter is partially linearized through an input–output feedback linearization method when the parameter uncertainty is considered. To compensate the uncertainty, a nonlinear adaptive control technique is adopted to derive the control algorithm, and the uncertainty adaptation laws can also be derived systematically based on the backstepping control technique [13]. In Section 4, simulation results were obtained; these results clearly show how the adaptive backstepping law updates 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 of the quasi resonant DC link converter [14] feeding induction heating is introduced in Fig. 1. This circuit could be modelised and an equivalent circuit is derived (Fig. 2) [4]. It is assumed that a resistive load RL is connected to the output terminal. A voltage equation is derived from Fig. 2 as es ¼ Ris þ L
dis þ vF dt
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 axis converter input voltage. R and L represent the line resistance and the input inductance, respectively. Considering the terms of three phases, if one transforms (1) into a synchronous reference frame, then:
es R
L1
idc iload
PWM converter
L3
iC A
B
C
VC
i1
QRDCL converter
D1 L2
C
i2 D3 i3
S2
Induction Heating
S1 iC C’
VC’
D2 Inductor + Load
Fig. 1. A power circuit of PWM converter associated with a power circuit of the QRDCL converter.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525 iload
i dc
es
R
L
1517
iC A
B
C
VC
C
Rload
Fig. 2. Power circuit of PWM converter.
did xLiq þ Rid ¼ ed vd dt diq þ xLid þ Riq ¼ eq vq L dt
L
where x is the angular frequency of the source voltage. So, we have: pffiffiffi V s ðtÞ ¼ V s 2 sin xt pffiffiffi iload ðtÞ ¼ iload 2 sinðxt uÞ
ð2Þ ð3Þ
ð4Þ 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 charging rate of the capacitor energy. The average rate of change of energy associated with AC link and DC link is given by 3 P ¼ ðed id þ eq iq Þ ¼ V dc idc 2
ð6Þ
where the input resistance loss and switching device loss are neglected. On the output side, we have: idc ¼ C
dV dc V dc þ dt RL
ð7Þ
From (6) and (7), 3 dV dc V 2dc þ ðed id þ eq iq Þ ¼ C dt RL 2
ð8Þ
Eq. (8) lead to nonlinear system with regard to Vdc. Combination of Eqs. (2), (3) and (8) describes a nonlinear model as 2 3 2 3 21 3 RL id þ xiq 0 i_d L e d vd 6 _ 7 6 7 6 R 17 i xi ð9Þ d 4 iq 5 ¼ 4 5þ 40 L 5 L q e q vq V dc 3 _V dc ðe i þ e i Þ 0 0 d d q q RL C 2CV dc 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 g1 ðxÞud þ g2 ðxÞuq
ð10Þ
1518
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
The system parameters may deviate from the nominal values. Let b ¼ R1L , then b = bn + Db, where bn represent the nominal value of parameter b, while Db represent the error of the nominal value. Taking into account these uncertainties, the system (10) can be changed as x_ ¼ f ðxÞ þ Df ðxÞ þ g1 ud þ g2 uq
ð11Þ
where f ðxÞ and Df(x) are the nominal and uncertain matrices of f(x), respectively. Such that: 2 3 2 3 RL id þ wiq 0 6 7 6 7 RL iq wid f ðxÞ ¼ 4 5 and Df ðxÞ ¼ 4 0 5: DbV dc 2 ðed id þ eq iq Þ bnCV dc C 3CV dc The control objective is to make DC output voltage track the desired reference output voltage command. We should select the DC output voltage as one of the output variable. Therefore, we choose the outputs variable of the PWM converter drive system as 8 > < y 1 ¼ h1 ðxÞ ¼ V dc ð12Þ y 2 ¼ Lf h1 ðxÞ > : y 3 ¼ id Then the dynamic model of PWM rectifier in the new coordinate is given by 2 3 2 3 Lf h1 þ LDf h1 þ Lg1 h1 ud þ Lg2 h1 uq y_ 1 6 7 6 L2 h þ L L h þ L L h u þ L L h u 7 4 y_ 2 5 ¼ 4 f 1 Df f 1 g1 f 1 d g2 f 1 q 5 Lf h2 þ LDf h2 þ Lg1 h2 ud þ Lg2 h2 uq y_ 3
ð13Þ
where Lf h1 ðxÞ ¼
L2f h1 ðxÞ
2 b V dc ; ðed id þ eq iq Þ n C 3CV dc
2ðed f1 ðxÞ þ eq f2 ðxÞÞ ¼ 3CV dc
Lg1 h1 ðxÞ ¼ Lg2 h1 ðxÞ ¼ 0; Lg2 Lf h1 ðxÞ ¼
(
R Lf h2 ðxÞ ¼ id þ wiq ; L
) 2ðed id þ eq iq Þ bn f3 ; þ C 3CV 2dc
1 Lg1 h2 ðxÞ ¼ ; L
Lg2 h2 ðxÞ ¼ 0;
LDf h1 ðxÞ ¼
LDf Lf h1 ðxÞ ¼
Lg1 Lf h1 ðxÞ ¼
DbV dc ; C
2ðed id þ eq iq Þ bn V dc Db þ 3C 2 V dc C2
2ed 3LCV dc
and
2eq 3LCV dc
In order to decouple the two control inputs, we construct the new control inputs as follows: ud Lg1 Lf h1 ud þ Lg2 Lf h1 uq ¼ Lg1 h2 ud þ Lg2 h2 uq uq In the new coordinate, the system can be written as follows: 3 3 2 3 2 3 2 2 y2 LDf h1 0 0 y_ 1 ud 7 7 6 7 6 L2 h 7 6 6 þ 4 LDf Lf h1 5 4 y_ 2 5 ¼ 4 f 1 5 þ 4 1 0 5 uq Lf h2 LDf h2 0 1 y_ 3 where
LDf h2 ðxÞ ¼ 0;
ð14Þ
ð15Þ
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
V dc ¼ hu1 ðxÞ LDf h1 ðxÞ ¼ Db C 2ðed id þ eq iq Þ bn V dc ¼ hu2 ðxÞ þ LDf Lf h1 ðxÞ ¼ Db 3C 2 V dc C2 LDf h2 ðxÞ ¼ 0
1519
ð16Þ ð17Þ ð18Þ
The structural property of the new system (15) contains two decoupled subsystems. The first subsystem is ( y_ 1 ¼ y 2 ð19Þ y_ 2 ¼ L2f h1 þ ud The second subsystem is uq y_ 3 ¼ Lf h2 þ
ð20Þ
where ud and uq are the input control of the subsystem (9) and (10), respectively. This structure allows us to conveniently use adaptive backstepping design technique to obtain the desired controller. The uncertainties of the system now are reflected by unknown parameter h. Thus, we obtain the compact form of the error-tracking model as follows: y_ ¼ AðxÞ þ DAðxÞ þ BðxÞU
ð21Þ
where 3 e2 7 6 AðxÞ ¼ 4 L2f h1 5;
2
2
Lf h2
hu1 ðxÞ
2
3
7 6 DAðxÞ ¼ 4 hu2 ðxÞ 5 and 0
0
6 BðxÞ ¼ 4 1 0
0
3
7 0 5: 1
In order to obtain good transient performance, a linear reference model is defined as y_ m ¼ k m y m þ Bm uref 2 3 2 y_ m1 0 1 6 7 6 4 y_ m2 5 ¼ 4 k m1 k m2 y_ m3 0 0
0
32
y m1
3
2
0
76 7 6 0 54 y m2 5 þ 4 k m1 k m1 0 y m3
3 7 V dc 0 5 id k m3 0
ð22Þ
Using the reference model, the performance of the system can easily be evaluated, as the tracking problem could be changed to a regulation problem. Define the error variables as 2 3 2 3 y 1 y m1 e1 6 7 6 7 ð23Þ e ¼ 4 e2 5 ¼ 4 y 2 y m2 5 e3 y 3 y m3 and use the following transformation: ~ ud þ k m1 y m1 þ k m2 y m2 k m1 V dc ud e ¼ U ¼ ~ uq uq þ k m3 y m3 k m3 id
ð24Þ
Then the differential equations of the errors can be derived as follows: e e_ ¼ AðxÞ þ DAðxÞ þ BðxÞ U
ð25Þ
The uncertain parameter error is defined as: ~ h ¼ h ^h, where ^h is the estimation of h and ~h is the estimation error. For the first equation of (25), e2 is taken as the new control input according to backstepping control technique. It can be easily obtained that, if uncertainty h is known, the first equation of (25) is obviously stable by a Lyapunov function V ¼ 12 e2 and a virtual controller a0 as a0 ðxÞ ¼ k 1 e1 hu1
ð26Þ
1520
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
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 k 1 e1 ^ hu1 Step 1: Define new variables as z1 ¼ e 1 ;
z2 ¼ e2 aðxÞ;
ð27Þ z3 ¼ e 3
ð28Þ
The virtual control a for z2 stabilizes the first equation as follows: a ¼ k 1 z1 ^ hu1
ð29Þ
The derivatives of the new variables are written as hu1 z_ 1 ¼ z2 þ a þ hu1 ¼ k 1 z1 þ z2 þ ~
ð30Þ
_ ~d k 1 z_ 1 þ ^ hu1 z_ 2 ¼ e2 þ a_ ¼ L2f h1 ðxÞ þ hu2 þ u
ð31Þ
Step 2: The first control Lyapunov function V 1 ðz1 ; z2 ; ~hÞ is written as 1 1 1 2 hÞ ¼ z21 þ z22 þ ~ h V 1 ðz1 ; z2 ; ~ 2 2 2c
ð32Þ
where c is the adaptation gains. The derivative of V 1 ðz1 ; z2 ; ~hÞ is 1 ~_ hÞ ¼ z1 z_ 1 þ z1 z_ 1 þ ~ V_ 1 ðz1 ; z2 ; ~ hh c 1_ _ h z2 u2 þ z1 u1 þ k 1 z2 u1 þ ~h þ z2 fz1 þ L2f h1 þ ^hu2 þ ud þ k 1 ðz2 k 1 z1 Þ þ ^hu1 g ¼ k 1 z21 þ ~ c ð33Þ Step 3: We can write the third equations of (22) as uq z_ 3 ¼ Lf h2 ðxÞ þ ~
ð34Þ
Define the Lyapunov function V2(z3) for the third equation of (22) as 1 V 2 ðz3 Þ ¼ z23 2
ð35Þ
The derivative of V2(z3) is as follows: ud g V_ 1 ðz3 Þ ¼ z3 z_ 3 ¼ z3 fLf h2 ðxÞ þ ~
ð36Þ
Step 4: To design the final adaptive backstepping nonlinear control for the system, we define the augmented hÞ as Lyapunov function V ðz1 ; z2 ; z3 ; ~ hÞ ¼ V 1 ðz1 ; z2 ; ~ hÞ þ V 2 ðz3 Þ V ðz1 ; z2 ; z3 ; ~ hÞ is computed as The derivative of V ðz1 ; z2 ; z3 ; ~ 1_ hÞ ¼ k 1 z21 þ ~ h z2 u2 þ z1 u1 þ k 1 z2 u1 þ ~h þ z2 fz1 þ L2f h1 þ ^hu2 þ ud þ k 1 ðz2 k 1 z1 Þ V_ ðz1 ; z2 ; z3 ; ~ c _^ þ hu1 g þ z3 ðLf h2 þ ~ uq Þ
ð37Þ
ð38Þ
~ 6 0, the simplest way is to make the items in the square brackets in the second, third To make V_ ðz1 ; z2 ; z3 ; hÞ and fourth items equal to zero to cancel the uncertainties and make the fifth term equal to k 2 z22 , the sixth term equal to k 3 z23 and V_ ¼ 0 if, and only if, z = 0. Then the following results can be obtained:
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1521
Fig. 3. Adaptive backstepping control block diagram of PWM converter.
Fig. 4. Block diagram of adaptive backstepping controller.
Control outputs: _ ~ud ¼ e1 k 2 e2 L2f h1 hu2 þ k 1 ðk 1 z1 z2 Þ ^hu1
ð39Þ
~ uq ¼ k 3 e3 Lf h2 ðxÞ
ð40Þ
Parameter adaptation laws: _ ^_ ¼ ~ h h ¼ cðz1 u1 þ k 1 z2 u1 þ z2 u2 Þ
ð41Þ
The final control inputs ud and uq can be easily derived through (23) and (26). The block diagram of the nonlinear 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 performance. 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 shows the parameter values used in the ensuing simulations.
1522
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
Table 1 Parameter of the PWM converter Supply’s voltage and frequency Line’s inductor and resistance DC link resistance Output capacitors PWM carrier frequency
220 V(rms), 50 Hz 0.1 mH, 2 mX 20 X 370 lF 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.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1523
The output DC link voltage of the PWM converter, the supply voltage and the input line current responses of 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 reference resulting from the reference model perfectly. The ideal input–output feedback linearization is based on the
b l ðtÞ. Fig. 8. Estimation of the parameter R
Fig. 9. Output DC link voltage response with adaptive law.
Fig. 10. Supply current and supply voltage with adaptive law.
1524
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
Normalised amplitude
1.5
1
0.5
0
0
1000
2000 3000 4000 Frequency [Hz]
5000
Fig. 11. Harmonic spectra of line current.
exact cancellation of the nonlinearity, for this reason, the ideal nonlinear control cannot deal with the system uncertainties. Fig. 7 presents the output DC link voltage of the system without an adaptation law. Figs. 8–10 give the simulation results of the proposed backstepping controller with adaptation laws. Fig. 11 shows the harmonic spectra of line current. In order to test the robustness of the controller to the change of the system parameters, the resistance of the load is changed to RL = 2RL nom at t = 0.15 s. An output DC voltage drop is observed, but this voltage drop is successfully rejected due to the effectiveness of the adaptation laws. The proposed control scheme gives satisfactory 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 the block system is guaranteed. Simulation results obtained were in good performance as it is expected. The strategy control was very robust to uncertain parameters and gave a very high power factor and small ripple in the current line supply. References [1] Kim ES, Lee DY, Hyun DS. A novel partial series resonant DC/DC converter with zero-voltage/zero-current switching. In: Proceedings of the 15th annual IEEE applied power electronics conference and exhibition, APEC; 2000. [2] Hase S, Konishi T, Okui A. Control methods and characteristics of power converter with large capacity for electric railway system. In: Proceedings of the power conversion conference, vol. 3; 2002. p. 1039–44. [3] Dong-Choon L, G-Myoung L, Ki-Do L. DC-bus voltage control of three-phase AC/DC PWM converters using feedback linearization. IEEE Trans 2000;36(3):826–33. [4] Canales F, Barbosa P, Lee FC. A zero-voltage and zero-current switching three-level DC/DC converter. IEEE Trans Power Electron 2002;17(6):898–904. [5] Chen Y, Qiu S, Wu Y. Extension of characteristic equation method to stability analysis of equilibrium points for closed-loop PWM power switching converters. Commun Nonlinear Sci Numer Simul 1999;4:276–80. [6] Slotine J, LI W. Applied nonlinear control. New Jersey: Prentice-Hall; 1991. [7] Isidori A. Nonlinear control systems. 3rd ed. Berlin: Springer; 1995. [8] Harband AM, Zaher AA. Nonlinear control of permanent magnet stepper motors. Commun Nonlinear Sci Numer Simul 2004;9:443–58. [9] Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: Wiley; 1995. [10] Su CY, Stepanenko Y, Svoboda J, Leung TP. Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis. IEEE Trans Automat Control 2000;45:2427–32. [11] Ezal K, Pan Z, Kokotovic P. Locally optimal and robust backstepping design. IEEE Trans Automat Control 2000;45(2). [12] Li D, Jiang X, Li L, Xie M, Guo J. The inverse system method applied to the derivation of power system nonlinear control laws. Commun Nonlinear Sci Numer Simul 1997;2:120–5. [13] Nayfeh AH, Harb A, Chin C. Bifurcations in a power system model. Commun Nonlinear Sci Numer Simul 1996;6(3):497–512.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1525
[14] Hadri Hamida A, Allag A, Mimoune SM, Zerouali S, Srairi K. Analysis and design of a passively clamped two switch quasi resonant DC link converter for induction heating application. In: Proceedings of the fourth electrical engineering conference; 2004. p. 45–9. [15] Valderrama GE, Mattavelli P, Stankovic AM. Reactive power and unbalance compensation using STATCOM with dissipative based control. IEEE Trans Control Syst Technol 2001;9(5):718–27.
Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525 www.elsevier.com/locate/cnsns
A nonlinear adaptive backstepping approach applied to 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 b
a Laboratory of Electrotechnics of Constantine, University of Mentouri, 25000 Constantine, Algeria MSE Laboratory, Department of Electrical Engineering, University of Biskra, BP 145, 07000 Biskra, Algeria c GESC Department, University of Technologies Belfort-Montbe´liard, Belfort, Cedex 90010, France
Received 24 May 2006; received in revised form 10 February 2008; accepted 10 February 2008 Available online 26 February 2008
Abstract This paper presents a new control strategy for a three phase PWM converter, which consists of applying an adaptive nonlinear 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 is derived with state variables of the input current and the output voltage by using power balance of the input and output, the nonlinear 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 updates the system parameters and provide an efficient control design both for tracking and regulation in order to improve the power factor. Ó 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/DC converters using resonant link schemes which utilize high speed devices such as fast recovery transistors and GTOs. These new converters not only have high power density but also possess very low switching losses since switching of the devices are made at zero-voltage instants and thus enable the total system to operate at *
Corresponding author. Tel./fax: +213 33 74 27 33. E-mail address: [email protected] (A. Hadri-Hamida).
1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.02.005
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 resonant link converters are intended to operate at high power density, almost all the systems presented in the past require self commutated transistors and have some difficulty performing conversion at very high power levels because of the relatively low voltage and current margins that self commutated devices such as transistors typically have. These new converters with high frequency and high power density are necessary in induction heating application which leads to increase the switching frequency. However, increasing the switching frequency leads to significant switching losses, which is deteriorated overall system efficiency [1]. In recent year, three phase voltage source PWM converters are increasingly used for applications such as UPS systems, electric traction and induction heating. The attractive features of them are constant DC bus voltage, 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 by changing the original nonlinear dynamics into linear one [6–8], but this technique do not take into account the uncertainties of system parameters. Adaptive backstepping control is a newly developed technique for the control of uncertain nonlinear systems [9–12]. This paper presents the implementation of the adaptive backstepping control law to the three phase PWM converter. First, the nonlinear model of the system is introduced in the following section, where the exact nonlinear multiple-input multiple-output state space model was obtained in ðd; q; 0Þ reference frame using the power balance between the input and output sides. State feedback linearization control for a three phase PWM converter is given in Section 2. In Section 3, the uncertainty in the system is the resistance of the load. The uncertain nonlinear model of PWM converter is partially linearized through an input–output feedback linearization method when the parameter uncertainty is considered. To compensate the uncertainty, a nonlinear adaptive control technique is adopted to derive the control algorithm, and the uncertainty adaptation laws can also be derived systematically based on the backstepping control technique [13]. In Section 4, simulation results were obtained; these results clearly show how the adaptive backstepping law updates 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 of the quasi resonant DC link converter [14] feeding induction heating is introduced in Fig. 1. This circuit could be modelised and an equivalent circuit is derived (Fig. 2) [4]. It is assumed that a resistive load RL is connected to the output terminal. A voltage equation is derived from Fig. 2 as es ¼ Ris þ L
dis þ vF dt
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 axis converter input voltage. R and L represent the line resistance and the input inductance, respectively. Considering the terms of three phases, if one transforms (1) into a synchronous reference frame, then:
es R
L1
idc iload
PWM converter
L3
iC A
B
C
VC
i1
QRDCL converter
D1 L2
C
i2 D3 i3
S2
Induction Heating
S1 iC C’
VC’
D2 Inductor + Load
Fig. 1. A power circuit of PWM converter associated with a power circuit of the QRDCL converter.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525 iload
i dc
es
R
L
1517
iC A
B
C
VC
C
Rload
Fig. 2. Power circuit of PWM converter.
did xLiq þ Rid ¼ ed vd dt diq þ xLid þ Riq ¼ eq vq L dt
L
where x is the angular frequency of the source voltage. So, we have: pffiffiffi V s ðtÞ ¼ V s 2 sin xt pffiffiffi iload ðtÞ ¼ iload 2 sinðxt uÞ
ð2Þ ð3Þ
ð4Þ 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 charging rate of the capacitor energy. The average rate of change of energy associated with AC link and DC link is given by 3 P ¼ ðed id þ eq iq Þ ¼ V dc idc 2
ð6Þ
where the input resistance loss and switching device loss are neglected. On the output side, we have: idc ¼ C
dV dc V dc þ dt RL
ð7Þ
From (6) and (7), 3 dV dc V 2dc þ ðed id þ eq iq Þ ¼ C dt RL 2
ð8Þ
Eq. (8) lead to nonlinear system with regard to Vdc. Combination of Eqs. (2), (3) and (8) describes a nonlinear model as 2 3 2 3 21 3 RL id þ xiq 0 i_d L e d vd 6 _ 7 6 7 6 R 17 i xi ð9Þ d 4 iq 5 ¼ 4 5þ 40 L 5 L q e q vq V dc 3 _V dc ðe i þ e i Þ 0 0 d d q q RL C 2CV dc 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 g1 ðxÞud þ g2 ðxÞuq
ð10Þ
1518
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
The system parameters may deviate from the nominal values. Let b ¼ R1L , then b = bn + Db, where bn represent the nominal value of parameter b, while Db represent the error of the nominal value. Taking into account these uncertainties, the system (10) can be changed as x_ ¼ f ðxÞ þ Df ðxÞ þ g1 ud þ g2 uq
ð11Þ
where f ðxÞ and Df(x) are the nominal and uncertain matrices of f(x), respectively. Such that: 2 3 2 3 RL id þ wiq 0 6 7 6 7 RL iq wid f ðxÞ ¼ 4 5 and Df ðxÞ ¼ 4 0 5: DbV dc 2 ðed id þ eq iq Þ bnCV dc C 3CV dc The control objective is to make DC output voltage track the desired reference output voltage command. We should select the DC output voltage as one of the output variable. Therefore, we choose the outputs variable of the PWM converter drive system as 8 > < y 1 ¼ h1 ðxÞ ¼ V dc ð12Þ y 2 ¼ Lf h1 ðxÞ > : y 3 ¼ id Then the dynamic model of PWM rectifier in the new coordinate is given by 2 3 2 3 Lf h1 þ LDf h1 þ Lg1 h1 ud þ Lg2 h1 uq y_ 1 6 7 6 L2 h þ L L h þ L L h u þ L L h u 7 4 y_ 2 5 ¼ 4 f 1 Df f 1 g1 f 1 d g2 f 1 q 5 Lf h2 þ LDf h2 þ Lg1 h2 ud þ Lg2 h2 uq y_ 3
ð13Þ
where Lf h1 ðxÞ ¼
L2f h1 ðxÞ
2 b V dc ; ðed id þ eq iq Þ n C 3CV dc
2ðed f1 ðxÞ þ eq f2 ðxÞÞ ¼ 3CV dc
Lg1 h1 ðxÞ ¼ Lg2 h1 ðxÞ ¼ 0; Lg2 Lf h1 ðxÞ ¼
(
R Lf h2 ðxÞ ¼ id þ wiq ; L
) 2ðed id þ eq iq Þ bn f3 ; þ C 3CV 2dc
1 Lg1 h2 ðxÞ ¼ ; L
Lg2 h2 ðxÞ ¼ 0;
LDf h1 ðxÞ ¼
LDf Lf h1 ðxÞ ¼
Lg1 Lf h1 ðxÞ ¼
DbV dc ; C
2ðed id þ eq iq Þ bn V dc Db þ 3C 2 V dc C2
2ed 3LCV dc
and
2eq 3LCV dc
In order to decouple the two control inputs, we construct the new control inputs as follows: ud Lg1 Lf h1 ud þ Lg2 Lf h1 uq ¼ Lg1 h2 ud þ Lg2 h2 uq uq In the new coordinate, the system can be written as follows: 3 3 2 3 2 3 2 2 y2 LDf h1 0 0 y_ 1 ud 7 7 6 7 6 L2 h 7 6 6 þ 4 LDf Lf h1 5 4 y_ 2 5 ¼ 4 f 1 5 þ 4 1 0 5 uq Lf h2 LDf h2 0 1 y_ 3 where
LDf h2 ðxÞ ¼ 0;
ð14Þ
ð15Þ
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
V dc ¼ hu1 ðxÞ LDf h1 ðxÞ ¼ Db C 2ðed id þ eq iq Þ bn V dc ¼ hu2 ðxÞ þ LDf Lf h1 ðxÞ ¼ Db 3C 2 V dc C2 LDf h2 ðxÞ ¼ 0
1519
ð16Þ ð17Þ ð18Þ
The structural property of the new system (15) contains two decoupled subsystems. The first subsystem is ( y_ 1 ¼ y 2 ð19Þ y_ 2 ¼ L2f h1 þ ud The second subsystem is uq y_ 3 ¼ Lf h2 þ
ð20Þ
where ud and uq are the input control of the subsystem (9) and (10), respectively. This structure allows us to conveniently use adaptive backstepping design technique to obtain the desired controller. The uncertainties of the system now are reflected by unknown parameter h. Thus, we obtain the compact form of the error-tracking model as follows: y_ ¼ AðxÞ þ DAðxÞ þ BðxÞU
ð21Þ
where 3 e2 7 6 AðxÞ ¼ 4 L2f h1 5;
2
2
Lf h2
hu1 ðxÞ
2
3
7 6 DAðxÞ ¼ 4 hu2 ðxÞ 5 and 0
0
6 BðxÞ ¼ 4 1 0
0
3
7 0 5: 1
In order to obtain good transient performance, a linear reference model is defined as y_ m ¼ k m y m þ Bm uref 2 3 2 y_ m1 0 1 6 7 6 4 y_ m2 5 ¼ 4 k m1 k m2 y_ m3 0 0
0
32
y m1
3
2
0
76 7 6 0 54 y m2 5 þ 4 k m1 k m1 0 y m3
3 7 V dc 0 5 id k m3 0
ð22Þ
Using the reference model, the performance of the system can easily be evaluated, as the tracking problem could be changed to a regulation problem. Define the error variables as 2 3 2 3 y 1 y m1 e1 6 7 6 7 ð23Þ e ¼ 4 e2 5 ¼ 4 y 2 y m2 5 e3 y 3 y m3 and use the following transformation: ~ ud þ k m1 y m1 þ k m2 y m2 k m1 V dc ud e ¼ U ¼ ~ uq uq þ k m3 y m3 k m3 id
ð24Þ
Then the differential equations of the errors can be derived as follows: e e_ ¼ AðxÞ þ DAðxÞ þ BðxÞ U
ð25Þ
The uncertain parameter error is defined as: ~ h ¼ h ^h, where ^h is the estimation of h and ~h is the estimation error. For the first equation of (25), e2 is taken as the new control input according to backstepping control technique. It can be easily obtained that, if uncertainty h is known, the first equation of (25) is obviously stable by a Lyapunov function V ¼ 12 e2 and a virtual controller a0 as a0 ðxÞ ¼ k 1 e1 hu1
ð26Þ
1520
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
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 k 1 e1 ^ hu1 Step 1: Define new variables as z1 ¼ e 1 ;
z2 ¼ e2 aðxÞ;
ð27Þ z3 ¼ e 3
ð28Þ
The virtual control a for z2 stabilizes the first equation as follows: a ¼ k 1 z1 ^ hu1
ð29Þ
The derivatives of the new variables are written as hu1 z_ 1 ¼ z2 þ a þ hu1 ¼ k 1 z1 þ z2 þ ~
ð30Þ
_ ~d k 1 z_ 1 þ ^ hu1 z_ 2 ¼ e2 þ a_ ¼ L2f h1 ðxÞ þ hu2 þ u
ð31Þ
Step 2: The first control Lyapunov function V 1 ðz1 ; z2 ; ~hÞ is written as 1 1 1 2 hÞ ¼ z21 þ z22 þ ~ h V 1 ðz1 ; z2 ; ~ 2 2 2c
ð32Þ
where c is the adaptation gains. The derivative of V 1 ðz1 ; z2 ; ~hÞ is 1 ~_ hÞ ¼ z1 z_ 1 þ z1 z_ 1 þ ~ V_ 1 ðz1 ; z2 ; ~ hh c 1_ _ h z2 u2 þ z1 u1 þ k 1 z2 u1 þ ~h þ z2 fz1 þ L2f h1 þ ^hu2 þ ud þ k 1 ðz2 k 1 z1 Þ þ ^hu1 g ¼ k 1 z21 þ ~ c ð33Þ Step 3: We can write the third equations of (22) as uq z_ 3 ¼ Lf h2 ðxÞ þ ~
ð34Þ
Define the Lyapunov function V2(z3) for the third equation of (22) as 1 V 2 ðz3 Þ ¼ z23 2
ð35Þ
The derivative of V2(z3) is as follows: ud g V_ 1 ðz3 Þ ¼ z3 z_ 3 ¼ z3 fLf h2 ðxÞ þ ~
ð36Þ
Step 4: To design the final adaptive backstepping nonlinear control for the system, we define the augmented hÞ as Lyapunov function V ðz1 ; z2 ; z3 ; ~ hÞ ¼ V 1 ðz1 ; z2 ; ~ hÞ þ V 2 ðz3 Þ V ðz1 ; z2 ; z3 ; ~ hÞ is computed as The derivative of V ðz1 ; z2 ; z3 ; ~ 1_ hÞ ¼ k 1 z21 þ ~ h z2 u2 þ z1 u1 þ k 1 z2 u1 þ ~h þ z2 fz1 þ L2f h1 þ ^hu2 þ ud þ k 1 ðz2 k 1 z1 Þ V_ ðz1 ; z2 ; z3 ; ~ c _^ þ hu1 g þ z3 ðLf h2 þ ~ uq Þ
ð37Þ
ð38Þ
~ 6 0, the simplest way is to make the items in the square brackets in the second, third To make V_ ðz1 ; z2 ; z3 ; hÞ and fourth items equal to zero to cancel the uncertainties and make the fifth term equal to k 2 z22 , the sixth term equal to k 3 z23 and V_ ¼ 0 if, and only if, z = 0. Then the following results can be obtained:
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1521
Fig. 3. Adaptive backstepping control block diagram of PWM converter.
Fig. 4. Block diagram of adaptive backstepping controller.
Control outputs: _ ~ud ¼ e1 k 2 e2 L2f h1 hu2 þ k 1 ðk 1 z1 z2 Þ ^hu1
ð39Þ
~ uq ¼ k 3 e3 Lf h2 ðxÞ
ð40Þ
Parameter adaptation laws: _ ^_ ¼ ~ h h ¼ cðz1 u1 þ k 1 z2 u1 þ z2 u2 Þ
ð41Þ
The final control inputs ud and uq can be easily derived through (23) and (26). The block diagram of the nonlinear 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 performance. 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 shows the parameter values used in the ensuing simulations.
1522
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
Table 1 Parameter of the PWM converter Supply’s voltage and frequency Line’s inductor and resistance DC link resistance Output capacitors PWM carrier frequency
220 V(rms), 50 Hz 0.1 mH, 2 mX 20 X 370 lF 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.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1523
The output DC link voltage of the PWM converter, the supply voltage and the input line current responses of 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 reference resulting from the reference model perfectly. The ideal input–output feedback linearization is based on the
b l ðtÞ. Fig. 8. Estimation of the parameter R
Fig. 9. Output DC link voltage response with adaptive law.
Fig. 10. Supply current and supply voltage with adaptive law.
1524
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
Normalised amplitude
1.5
1
0.5
0
0
1000
2000 3000 4000 Frequency [Hz]
5000
Fig. 11. Harmonic spectra of line current.
exact cancellation of the nonlinearity, for this reason, the ideal nonlinear control cannot deal with the system uncertainties. Fig. 7 presents the output DC link voltage of the system without an adaptation law. Figs. 8–10 give the simulation results of the proposed backstepping controller with adaptation laws. Fig. 11 shows the harmonic spectra of line current. In order to test the robustness of the controller to the change of the system parameters, the resistance of the load is changed to RL = 2RL nom at t = 0.15 s. An output DC voltage drop is observed, but this voltage drop is successfully rejected due to the effectiveness of the adaptation laws. The proposed control scheme gives satisfactory 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 the block system is guaranteed. Simulation results obtained were in good performance as it is expected. The strategy control was very robust to uncertain parameters and gave a very high power factor and small ripple in the current line supply. References [1] Kim ES, Lee DY, Hyun DS. A novel partial series resonant DC/DC converter with zero-voltage/zero-current switching. In: Proceedings of the 15th annual IEEE applied power electronics conference and exhibition, APEC; 2000. [2] Hase S, Konishi T, Okui A. Control methods and characteristics of power converter with large capacity for electric railway system. In: Proceedings of the power conversion conference, vol. 3; 2002. p. 1039–44. [3] Dong-Choon L, G-Myoung L, Ki-Do L. DC-bus voltage control of three-phase AC/DC PWM converters using feedback linearization. IEEE Trans 2000;36(3):826–33. [4] Canales F, Barbosa P, Lee FC. A zero-voltage and zero-current switching three-level DC/DC converter. IEEE Trans Power Electron 2002;17(6):898–904. [5] Chen Y, Qiu S, Wu Y. Extension of characteristic equation method to stability analysis of equilibrium points for closed-loop PWM power switching converters. Commun Nonlinear Sci Numer Simul 1999;4:276–80. [6] Slotine J, LI W. Applied nonlinear control. New Jersey: Prentice-Hall; 1991. [7] Isidori A. Nonlinear control systems. 3rd ed. Berlin: Springer; 1995. [8] Harband AM, Zaher AA. Nonlinear control of permanent magnet stepper motors. Commun Nonlinear Sci Numer Simul 2004;9:443–58. [9] Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: Wiley; 1995. [10] Su CY, Stepanenko Y, Svoboda J, Leung TP. Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis. IEEE Trans Automat Control 2000;45:2427–32. [11] Ezal K, Pan Z, Kokotovic P. Locally optimal and robust backstepping design. IEEE Trans Automat Control 2000;45(2). [12] Li D, Jiang X, Li L, Xie M, Guo J. The inverse system method applied to the derivation of power system nonlinear control laws. Commun Nonlinear Sci Numer Simul 1997;2:120–5. [13] Nayfeh AH, Harb A, Chin C. Bifurcations in a power system model. Commun Nonlinear Sci Numer Simul 1996;6(3):497–512.
A. Hadri-Hamida et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1515–1525
1525
[14] Hadri Hamida A, Allag A, Mimoune SM, Zerouali S, Srairi K. Analysis and design of a passively clamped two switch quasi resonant DC link converter for induction heating application. In: Proceedings of the fourth electrical engineering conference; 2004. p. 45–9. [15] Valderrama GE, Mattavelli P, Stankovic AM. Reactive power and unbalance compensation using STATCOM with dissipative based control. IEEE Trans Control Syst Technol 2001;9(5):718–27.