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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 5, OCTOBER 1992 429 Large-Signal Feedback Control of a Bidirectional Coupled-Inductor Cuk Converter 1 Joan Maj6, Luis Martinez, Albert0 Poveda, Luis Garcia de Vicuiia, Francisco Guinjoan, Antonio F. Shnchez, Max Valentin, and Jean Claude Marpinard Abstract-Under conditions of order reduction, a nonlinear control of a bidirectional coupled-inductor Cuk converter suitable for large-signal applications is presented. The converter is accurately modeled as a second-order bilinear system and the conditions established for local controllability. The integration of converter state equations and the subsequent introduction of a linear recurrence between the output variable and an external reference signal lead to a nonlinear control law that is implemented by means of an analog divider, standard operational amplifiers, and a pulse-width modulator. As a result, the output variable follows proportionally the reference signal, this allowing the obtention of different types of power waveforms in the converter output. Experimental results verify the theoretical predictions. designing a switching converter-based servosystem using a variable reference or when, in a conventional regulator, a load perturbation appears. The dynamical analysis becomes also rather complex in the discontinuous conduction mode where the instant in which the switch current becomes zero depends on the converter state and not on its duty cycle. Therefore, two constraints expressing the current cancellation must be added to the three piecewise-linear vector differential equations. First, the switch current takes the same value at both the beginning of the first subinterval and the end of the second one; second, the switch current is zero during the third subinterval of the switching period. Moreover, the existence of an additional nonlinearity nonrelated to the control variables but I. INTRODUCTION to the converter state variables makes the design of a WITCHING converters can be represented taking part feedback loop suitable for large-signal operation particuof a class of feedback systems as illustrated in Fig. 1. larly difficult. In order to avoid the discontinuous conducThe basic operation of the regulator loop consists in tion mode and, at the same time, allow the existence of comparing the output variable with a reference input to light load levels, the bidirectional switch was introduced, generate an error signal, which, after being appropriately this having led to a bilinear description of the switching filtered and amplified, results in a continuous control converter [l], [2]. signal. This is eventually transformed by means of a The transformation of the canonical buck, boost, or modulator into a discrete variable defining the switch duty buck-boost structures into bidirectional cells has extended cycle. The main difficulties in the analysis of switching the dynamic performances of the power converter as the regulators are due to the highly nonlinear nature of time constant of the output filter is strongly affected by dc-to-dc converter systems. the low-impedance input network. On one hand, the To solve this problem, different linearizing modeling well-known linear control strategies cannot be used in techniques were proposed in the past, their validity limits such bidirectional converters since large-signal transients being located in a small neighborhood of the steady-state are often present in the current flow reversal. Therefore, operating point. However, the analysis becomes more in such converters, large-signal control becomes mandadifficult beyond these limits, when large-signal behavior is tory, this having to take into account the nonlinearities required. This case is particularly important either when introduced by the sampling process as well as the addiManuscript received August 3, 1990; revised March 20, 1992. This tional constraints that are needed to obtain the desired work has been sponsored by Grant Acci6n Integrada Hispano-Francesa closed-loop regulation. On the other hand, a large-signal 183. control technique in bidirectional cells has provided excelJ. Maj6, L. Garcia de Vicufia, F. Guinjoan, and A. F. Sinchez are with the Escuela Universitaria PolitCcnica de Vilanova i la Geltrh, 08800 lent performances in the elementary converters as reVilanova i la Geltrh, Barcelona, Spain. ported in [l], [2]. This approach is now applied to control L. Martinez and A. Poveda are with the Escuela TCcnica Superior de a complFx converter such as the bidirectional coupled-inIngenieros de Telecomunicaci6n, Departamento de Ingenieria ductor Cuk converter, shown in Fig. 2, under conditions of Electrbnica, 08034 Barcelona, Spain. J. C. Mapinard and M. Valentin are with the Laboratoire d’Automa- order reduction. tique et d’halyse des Systkmes (L.A.A.S.1-(Centre National de la In such a converter, a reduction of a fourth two Recherche Scientifique), UniversitC Paul Sabatier, 31077 Toulouse second-order can be achieved if two conditions are fulCedex, France. IEEE Log Number 9202498. filled. First, if the magnetic coupling satisfies the matching S 0278-0046/92$03.00 0 1992 IEEE IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 5, OCTOBER 1992 430 ERROR AYPUFIER AND 41- YODUUTOR J 134 d(t) e b POWER 5l'ACE 8 L- .__I Fig. 1. Block diagram of a one-loop regulator system using a convertor as power stage. 20 1 Fig. 2. Bidirectional coupled-inductor duk converter. L,= L,= L z condition n = k , zero current ripple will be obtained at of a coupled inductor Cuk converter with the the output and the need for output capacity C2 will be Fig. 3. Equivalent circuitcondition n = k = 1. completely eliminated as it was reported by Cuk in [41, [51. The elimination of this capacitor results in a simplified Equations (1) and (2) can be combined in only one and extremely favorable loop-gain dynamics when this converter is used in push-pull configuration, thus imple- bilinear expression: menting a push-pull switching audio power amplifier x = (Ax + a ) + (Bx b)u (3) [4]-[6]. Second, an additional order reduction can be obtained if the magnetic coupling is perfect, i.e., k = 1, where which results in only one equivalent inductor. Thus, this U = 1 for t I To, paper covers the analysis and design of a nonlinearly controlled bidirectional second-order converter suitable u =0 for TonI t I T (4) for large-signal applications. + 11. BILINEAR REPRESENTATION AND LOCAL x= CONTROLLABILITY Fig. 3 shows a? equivalent circuit of a bidirectional coupled inductor Cuk converter with the condition n = k = 1. Transistors and diodes have been assumed to be ideal, therefore no parasitics or storage-time modulation effects have been considered. Because of the bidirectional current capability of the switch, the discontinuous conduction mode does not take place and therefore the switching converter can be represented by two piecewise-linear vector differential equations: i =A,x + blVg for .i = A,x + b2Vg for t I Ton TonI t I T where X is the power-stage state vector and input. ['; I and matrices A , a, B , and b are given, respectively, by (7) (9) (1) (2) where 0 is the null matrix. 5 is the dc (5) The bilinear system described by (3) is locally controllable since the dimension of the Lie algebra generated by vectors Ax + a and Bx is 2. To prove that [see Appendix], it can be shown after tedious calculations that the Lie 43 1 MAJO et al.: LARGE-SIGNAL FEEDBACK CONTROL algebra contains the vectors I W, = Bx W, = [AX + a, Bx] w,= [ A x + a , ( R x + a , Bx)] which are related among them as follows. I) W, and W, are linearly independent in the complementary of conic. vg F c,, = {(im,ul)I - 2 ~ , i , u , + L,V,i, -RC,I/,u, 11) 0) * W, and W, are linearly independent in the complementary of conic. cI3= {(im,u1)~z~(2 ~V ;, () ~ R C , ) +i,(R2ClV, 111) = + L M V , )+ U1(3RC11/,) = 0) C,, n C,, represents a point in R2 where W, and W, are linearly independent. 111. NONFINEAR CONTROL OF THE COUPLED INDUCTOR CUKCONVERTER WITH n = k = l i i = 1 = 2 for for to, = 0 t; CONDITION Fig. 4. Block diagram of a large-signal feedback control of a bidirectional coupled inductor Cuk converter with n = k = I. t I Ton Ton4 t I T t", = Ton Considering the subinterval (10) becomes THE < t < t; (lo) Expression (12) can be also written as a bilinear recurrence for x and 7 : +7 x(t;+') = Hx(t;) + FX(t;)7 + G (13) 432 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 5, OCTOBER 1992 sc3524 lOOK 4K7 ,, K 1 2 = I I -1.5" TIE 4K7 + LH74 1 +v Fig. 5. Practical implementation of a coupled inductor Cuk converter: C, = 2.2 p F ; VR= 10 V; L , f, = 80 KHz; ( D )= 0.3; 0.1 < D < 0.5. where -- L T T - 1-- c F= [ 1 - - LC L T 1 f21 T -E+= T g11 = [gl, z u1(t;") RC T fll Lc T2 ( L -?E)& T2 LC + - -T- ) & RC = L, = llK9 3.25 mH; Finally, introducing in (15) the following control recurrence: T 1 Ct llK9 T2 R2C2 - c, = W [ u , ( t ; )- C,] (17) where C, represents an external reference and W is a positive arbitrary constant less than 1, results in T(t;> - -g12 - h2lim(t;) + Ul(t;)(W- f 2 1 i,(t;> h22) + CA1 - w > ff22Udt;) ( 18) This nonlinear control law is represented by the block diagram depicted in Fig. 4 and has been implemented as shown in Fig. 5, where the complete bidirectional converter and its control circuit are depicted in detail. MAJO et 433 LARGE-SIGNAL FEEDBACK CONTROL : o u t p u t Voltage (V) -3:: -EiEEEEl -5.5 -6 8 8.5 9 9.5 10.5 10 11 11.5 12 12.5 13 13.5 14 Input Voltage (V) (a) - 10 _~ - a _____ __- - 4 - - 2 0 I 1 I 4 I 100 00 GO 40 20 0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.0 0.8 Output C u r r e n t (A) Fig. 7. Efficiency. 0 Measured value. --Interpolation curve. Fig. 6(a) and (b) shows the regulation up and down of the prototype, respectively. Fig. 7, in turn, represents the obtained efficiency versus the output current. On the other hand, Fig. 8 illustrates the excellent dynamic behavior of the regulation by following different types of variable reference. The measured bandwidth is 714 Hz. Finally, Fig. 9 shows the output voltage behavior when using a sinusoidal reference and the load has pulsating characteristics, this being implemented as represented in Fig. 10. The output voltage exhibits in this case a fast recovery, i.e., only 5% of a half a cycle time and negligible overshoot 10%. IV. CONCLUSIONS The obtained results show the feasibility of a large-signal control of PWM complex converters. The technique presented in the paper can be used to implement highperformance sinusoidal inverters or power amplifiers. It 434 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 5, OCTOBER 1992 r Fig. 10. Practical implementation of the pulsating load. must be remarked that the developed prototype has been designed for low-power applications since our main interest was to verify the validity of the proposed control loop. The control procedure can also be easily applied to highqrder converters such as the fourth-order bidirectional Cuk converter as reported in reference [13]. In all cases, the system performances are basically limited by the characteristics of the analog multiplier. In that sense, further research is needed in order to simplify the number of elements in the feedback path by implementing some of the analog functions by means of digital circuits. ACKNOWLEDGMENT The authors want to express their acknowledgment to Dr. E. Fossas for his helpful suggestions in the controllability analysis. APPENDIX (b) Fig. 8. Output voltage under different types of control references (2 V/div. and 5 ms/div.). (a) Sinusoidal type. (b) Triangular type. Upper waveform corresponds to reference voltage. Lower waveform corresponds to output voltage. Considering the vector fields V and W whose coordinates are V(x)= Ax + a and W ( x ) = Bx, respectively, in the usual base of R",the Lie derivate of the field W(x) along V(x)is defined as L;(X) = 6W [ V , W ] = -6VX- 6V -W 6X (Al) or, in coordinates, [ V , W ]= B ( A x + U ) - A ( B x ) . The bilinear system described by (3) will be controllable if and only if the Lie algebra generated by operators V and W is of full rank in any operating point of the converter [12]. Since, in this case, we analyze a secondorder system, the Lie algebra will be of full rank if its dimension is 2. In order to investigate a base of this algebra, we define the set of vectors { W,> W, 9 w, ... 1 (A31 where W, = W Fig. 9. Output voltage with sinusoidal reference and pulsating load. Upper waveform represents the pulsating behavior of the load. Lower waveform corresponds to output voltage. = BX (A41 + U , Bx] W, = [ V ,W ] = [ A x W, = [V,[V,W]= ] [Ax (W + u , [ h + u , B x ] ] (A6) 435 MAJO et ai.: LARGE-SIGNAL FEEDBACK CONTROL If two vectors of this set are linearly independent, the model will be controllable. On the other hand, the model Will be uncontrollable if the following determinants are zero: det (W,, W,) = 0 (A7) det(W,,W,) = 0 (‘48) det(W2,W,) = 0 (A91 nal modelling and control in bidirectional switching converters,” in Proc. IEEE Int. Symp. on Circuits and Systems, ISCAS’91, Sin@ pore, June 11-14, 1991, pp. 1081-1084. [14] A. Monin, “Contributions algebriques and topologiques B la representation des systemes non-lineaires,” Thkse de Doctorat. Universit6 Paul Sabatier, Toulouse, France, Dec. 1987. Joan Maj6 graduated as Ingeniero Industrial in 1978, and received the Ph.D. degree in 1990, both at the Universidad PolitCcnica de Cataluiia. Since October 1978, he has been teaching at the Escuela Universitaria PolitCcnica de Vilanova i la Geltri, where he is an Assistant Professor. His research interests are in the field of control in power electronics, variable structure systems, and modern control theory applied to drive systems. The analysis of this set of equations reveals that we can always find two linearly independent vectors since W,, W,, and W, are related among them as follows: I> W, and W, are linearly independent in the complementary of conic. c,, 2L,imu, RC,I/,U, = 0) W, and W, are linearly independent in the complementary of conic. = {(im,Ul)l +I,&& 11) - - Luis Martinez graduated as Ingeniero de Telecomunicaci6n in 1978, and received the Ph.D. degree in 1984, both at the Univeraidad PolitCcnica de Catalufia. Since October 1978, he has been teaching at the Escuela TCcnica Superior de Ingenieros de Telecomunicacidn of Barcelona, where he is an Assistant Professor. His research interests are in the field of analog circuit design and structure and control of power conditioning systems. c,, = { ( i , , , , u I ) ~ i ~ ( 2 ~ , ) u : ( ~ R c , ) - +im(R2C& 111) + I,/&) + U1(3RC,V,)= 0) C , , n C , , represents a point in R 2 where W, and W, are linearly independent. REFERENCES [11 M. Valentin, “Contribution a I’analyse des convertisseurs statiques et leur commande B fort signal,” Thtse d’Etat, UniversitC Paul Sabatier, Toulouse, France, Dec. 1984. 121 G. Salut, J. C. Marpinard, and M. Valentin, “Large signal feedback control for power switching conversion,” in Proc. 1985 IEEE PESC Conf. Rec., pp. 741-750. 131 S. Cuk, “Switching dc-to-dc converter with zero input or output current ripple,” in Proc. IEEE Industry Applications Society Ann. Mtg, Toronto, Canada, 1978, pp. 1131-1146. 141 S. Cuk and R. Erickson, “A conceptually new high-frequency switched-mode amplifier technique eliminates current ripple,” in Proc. 5th National Solid-state Power Conversion Conf., San Francisco, CA, 1978, pp. G.3.1-G.3.22. [51 S. duk and R. D. Middlebrook. “Advances in switched-mode power conversion. Part 11,” IEEE Trans. Industrial Electron., vol. IE-30, pp. 19-29, Feb. 1983. J. Fortuiio Mata and L. Salamero, Martinez, “Simplified models for push-pull switching power amplifiers,” Electron. Lett., vol. 24, no. 1, pp. 49-50, Jan. 1988. A. Capel, J. C. Marpinard, G. Salut, M. Valentin, D. O’Sullivan, A. Weinberg, N. Limbourg, and J. C. Rym, “A bidirectional highpower cell using large-signal feedback control with maximum current control for space applications,” ESA J., vol. 10, pp. 387-402, 1986. M. Clique and A. J. Fossard, “Design of feedback laws for dc/dc converters using modern control theory,” in Proc. 1978 IEEE PESC Conf. Rec., pp. 2-11. A. Poveda, J. Maj6, M. Valentin, L. Martinez, L. Garcia de Vicufia, and F. Guinjoan, “Linear and nonlinear control of high order converters after reducing the order by appropriate techniques,’’ in Proc. European Space Power Conf., Madrid, 1989, pp. 375-380. J. C. Marpinard, P. Bidan, and J. M. Hernandez, “High speed digital control of high frequency PWM converters,” in Proc. European Space Power Conf., Madrid, 1989, pp. 399-402. 1111 R. W. Brockett, “Nonlinear systems and differential geometry,” Proc. IEEE, vol. 64, pp. 61-72, Jan. 1976. “Lie algebras and Lie groups in control theory,” Geometric 1121 -, Methods in System Theory D. Q. Mayne and R. W. Brockett, Eds. Reidel, Dordrecht, 1973, pp. 43-82. L. Martinez, J. Maj6, A. Poveda, L. Garcia de Vicufia, F. Guinjoan, A. F. Sinchez, J. C. Marpinard, and M. Valentin, “Large-sig- Albert0 Poveda graduated as Ingeniero de Telecomunicaci6n in 1978, and received the Ph.D. degree in 1988, both at the Universidad Polittcnica de Catalufia. He is an Assistant Professor in the Departamento de Ingenieria Electrbnica at the Escuela Ttcnica Superior de Ingenieros de Telecomunicaci6n of Barcelona where he teaches analog circuits and microprocessors. His research interests include analog circuit design and power processing modeling, control, and simulation. Luis Garcia de Vicuiia graduated as Ingeniero de Telecomunicaci6n in 1980, and received the Ph.D. degree in 1990, both at the Universidad Polittcnica de Cataluiia. Docteur de I’Universitt Paul Sabatier de Toulouse in 1992. He is an Assistant Professor in the Departmento de Ingenieria Electr6nica at the Escuela Universitaria PolitCcnica de Vilanova i la Geltri where he teaches power electronics, analog circuits, and control theory. His research interests are power processing electronics, design-oriented circuit analysis, and measurement techniques. Francisco Guinjoan graduated as Ingeniero de Telecomunicaci6n in 1984 and received the Ph.D. degree in 1990, both at the Universidad PolitCcnica de Cataluiia, Docteur de I’UniversitC Paul Sabatier de Toulouse in 1992. He is an Assistant Professor in the Departamento de Ingenieria Electrdnica at the Escuela Universitaria PolitCcnica de Vilanova i la Geltri, where he teaches analog circuits and control theory. His research interests are power electronics modeling, nonlinear circuits analysis, and analog circuit design. I 436 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 5, OCTOBER 1992 Antonio F. Sanchez graduated as Ingeniero TCcnico de Telecomunicaci6n in 1988 at the Universidad Polittcnica de Cataluiia. He is an Assistant Professor in the Departamento de Ingenieria Electr6nica at the Escuela Universitaria PolitCcnica de Vilanova i la Geltri, where he teaches analog circuits and power electronics. His research interests are power electronics, variable structure systems, and mode m control theory applied to drive systems. Max Valentin was born in 1941. He took his degree in physics at the University of Toulouse and took his 3rd Cycle Thesis at the same University in 1974. He received the Ph.D. degree (‘‘Thkse d’Etat”) in 1984, whose subject was about PWM large signal control of power converters: classification of power cells and nonlinear feedback loop. Since 1977, he has been doing research at “Laboratoire d’Automatique et d h a l y s e des Systkmes, LAAS-CNRS” (Toulouse). His main interest is in power conditioning and in the mathematical modeling of converters. He teaches electrotechnics and automatic contriol at the University of Toulouse 111. Jean Claude Marpinard was born in Semblanqay, France, in 1938. He received the degree in physics at the University of Toulouse and the Ph.D. degree in the same university in 1967. He presently holds a chair as Professor of electrical engineering at the University Paul Sabatier (Toulouse), where he is doing research in the Laboratory of Automatic Control and Systems Analysis (LAAS) of National Scientific Research Center (CNRS) in Toulouse. His main interest is in power conditioning and control as well as space applications with the support of the European Space Agency and Alcatel Espace on electric/hybrid automotives control.