INTELLIGENT CONTROL OF DC/DC SWITCHING BUCK CONVERTER

Transcription

1 Helsinki University of Technology Control Engineering Laboratory Espoo 2000 Report 122 INTELLIGENT CONTROL OF DC/DC SWITCHING BUCK CONVERTER Zekeriya Uykan TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI

2 Helsinki University of Technology Control Engineering Laboratory Espoo December 2000 Report 122 INTELLIGENT CONTROL OF DC/DC SWITCHING BUCK CONVERTER Zekeriya Uykan Abstract: In this report, Neural-Fuzzy and Sliding-Mode control based Intelligent methods have been applied to DC/DC Converter. The focus is on DC/DC buck converter with equivalent series resistors operating in voltage mode. The usual method in controller design for this kind of converter is based on linear theory: The dynamics of the inherently nonlinear converter is approximated by linear systems e.g. using small-signal models, and then classical linear control methods have been used to design a controller. The goal of this work is to obtain intelligent and robust controllers using information only from output voltage of the converter (under line and load disturbances). The methods used are i) Neural-Fuzzy Control using Radial Basis Function Network and ii) Sliding Mode Control. Our investigations resulted in a robust controller structure consisting of Sliding Mode Controller and Integral Controller being developed for the converter. The proposed controller gave the best performance being more robust for model inaccuracies and disturbances in comparison with neural-fuzzy controller and linear PID controller. On the other hand, the results obtained by Neural-Fuzzy control technique were not satisfactory, and therefore are not presented here. All simulations in this work use parameters from a real DC/DC buck converter. Accordingly, the algorithms can readily be implemented on a real DSP setup. Keywords: DC/DC buck converter, radial basis function networks, neural-fuzzy systems, sliding mode control, PID control Helsinki University of Technology Department of Automation and Systems Technology Control Engineering Laboratory

3 Distribution: Helsinki University of Technology Control Engineering Laboratory P.O. Box 5400 FIN HUT, Finland Tel Fax ISBN ISSN Picaset Oy Helsinki 2001

4 Contents 1 Introduction 3 2 Buck Converter model 5 3 Intelligent Control of DC/DC Buck Converter Neural-Fuzzy + Integral Controller design SMC + Integral Design PID Design 12 5 Computer Simulations 14 6 Conclusions 20 1

5 2

6 Chapter 1 Introduction The performance of switched mode power supplies under line and load disturbances highly depends on the design of the controller. The usual method in controller design for DC/DC buck converter with equivalent series resistors operating in voltage mode is based on linear theory: The dynamics of the inherently nonlinear converter is approximated by linear systems e.g. using small-signal models [11], [12], and then classical linear control methods have been used to design a controller. However, as control theory has developed, there has been a growing interest to use nonlinear controllers, which could lead to better performance and robustness of the closed loop system [16]. The goal of this work is to obtain intelligent and robust controllers using information only from output voltage of the converter (under line and load disturbances). DC/DC converters are nonlinear systems due to their swithching characteristics and have intensively been studied in the literature using dierent control strategies. PID controller (e.g. [4]), Sliding Mode Controller (e.g. [1], [7]), Fuzzy controller (e.g. [8]) have been applied to various DC/DC converters in last decades. This report presents an intelligent and robust controller structure for DC/DC Buck converter with equivalent series resistors operating in voltage mode. The methods used are i) Neural-Fuzzy Control using Radial Basis Function Network and ii) Sliding Mode Control. The main purpose of sliding mode control is to reject disturbances and make the system insensitive to unknown parametric perturbations [5]. Our investigations resulted in a robust controller structure consisting of Sliding Mode Controller (SMC) and Integral (I) controller developed for the converter. The proposed controller gave the best performance being more robust for model inaccuracies and disturbances in comparison with Neural-Fuzzy (NF) controller and linear PID controller. One of the motivations of this study is the need for new control solutions in power supplies is to replace analog control structures with processor based solutions. 3

7 S L r l i o i l + DC v in C v c R o V r c - v o d PWM v c G c (s) e - + v ref Figure 1.1: System controlled in voltage mode [16]. The DSP technology today is capable of using discrete algorithms meeting the high sampling frequencies needed in the control of DC-DC converters [14]. This report is organized as follows: The buck converter model used is shown in Chapter 2. The intelligent control design for the converter in voltage mode is presented in Chapter 3. Chapter 4 explains the PID design. Simulation results are presented in Chapter 5 followed by Conclusions in Chapter 6. 4

8 Chapter 2 Buck Converter model The basic model of the controlled buck converter in voltage mode is given in Fig.1.1. The state space equations of the system can easily be derived to be " _vc _i L # = " ; 1 C(Ro+r c) ; Ro + " 0 1 L # L(R o+r c) R o C(R o+r c) ; r CR o+r L R o+r C r L L(R o+r c) #" vc i L # v in s w (2.1) # (2.2) i L v o = h 1 1+r c=r o r c 1+r c=r o i " v c S 2f0 1g and i L 0 (2.3) where v c (t) and i L (t) are capacitor voltage and inductor current respectively r c and r L are ESRs of the capacitor and the inductor respectively v in is the input voltage s w represents the switch (s w 2f0 1g), R o is the resistive load and v o is the output voltage. The parameters in the system corresponding those of a real converter are: V in = 140 V, V o = 54 V, L = 100 H, C = 1000 F, R o = 11 (resistive load), f s = 100 khz, r L = 0.015, r c = The converter is controlled in voltage mode conguration and in continuous conduction mode. In order to maintain the stability of real converters (operating e.g. in current programmed mode) the duty cycle is limited to the values between 0 and 0.5. Under these conditions, the aim here is to design a robust controller keeping the output voltage v o (t) equal to a reference voltage v ref (t) under line and load disturbances. 5

9 Chapter 3 Intelligent Control of DC/DC Buck Converter The proposed control structure, which is a hierarchical one, of the DC/DC buck converter is shown in Fig Figure 3.1: The proposed control structure of DC/DC buck converter. Dierent controllers may be applied in the hierarchical control structure in Fig.3.1 as long as the coordinator assures the stability as switching among the controllers. In this study, we will design Neural-Fuzzy (NF) controller (using Radial Basis Function Network (RBFN), Sliding Mode Contoller (SMC), and Integral (I) Controller, and will consider the following 2 cases: a) Neural Fuzzy + Integral control, b) Sliding Mode + Integral control, The role of the Integral controller is to get rid of steady-state errors in output voltage. 6

10 In order to compare the simulatoin results, a well-tuned PID controller is designed in Chapter 4 as a reference case Neural-Fuzzy + Integral Controller design The NF+I design is shown in Fig.3.2. Figure 3.2: Neural-Fuzzy + Integral control of DC/DC buck converter. Radial Basis Function Network (RBFN) is used as a Neural-Fuzzy (NF) system whose input-output relation is given in (3.1) F ( C x) = NX j=1 j e( kx ; c jk 2 ) (3.1) j 2 where N is the number of neurons, x 2 < M is the input, kkdenotes a norm, C = [c 1 c N ] is a matrix whose columns are the centers of RBFN and = [ 1 N ]isthevariance vector and =[ 1 N ] is the linear weight vector. Figure 3.3: Radial Basis Function Network. 7

11 In [10], it is shown that the functional behaviours of RBFN and a fuzzy system are equivalent. This equivalance enables us to combine the advantages of fuzzy systems and neural networks as presented below. Using the information of output error and its derivative, an initial set of fuzzy rules for controlling the DC/DC converter are set up without using any priori information of the converter. These fuzzy rules are optimized during a training set of dierent line and load conditions using neural networks techniques thanks to the neural-fuzzy behaviour of the RBFN. In a similar way as presented in [2], [15], the following cost function is used to optimize the RBFN parameters, i.e. C. E( C ) = SX s=1 n N s X n=1 (e(t)+ _e(t)) 2 o s (3.2) where e(t) = v ref ; v o (t), S is the number of line and load conditions in the training set and for each load/line condition N s shows the number of steps to reach the zero error. We minimize the RBFN parameters by a gradient type algorithm and is set to 0. and j (n +1)= j (n)+a C j j =1 ::: N (3.3) and c ij (n +1)=c ij (n)+a C ij i =1 ::: M j =1 ::: N (3.4) j (n +1)= j C j j =1 ::: N (3.5) where a l a c a > 0 and N is the number of neurons, each of which stand for a fuzzy rule, and M is the number of inputs to RBFN. The initial fuzzy rules for the controller are in the form of (3.6). Rule 1 : IF e(t) is... AND _e(t) is..., THEN du is.... Rule N : IF e(t) is... AND _e(t) is..., THEN du is... (3.6) where du shows the increment to the control signal, not the actual control signal. As explained in [2] and [15], after minimizing the RBFN parameters by a gradient type algorithm, we obtain optimized Fuzzy Rules thanks to the functional equivalance of RBFN and a fuzzy system (Fig.3.4). 8

12 Figure 3.4: Optimization of Fuzzy Rules for control of the converter using RBFN. The I controller is used to avoid steady-state errors in the output voltage and is evoked when e(t) =v ref ; v o (t) is below a threshold. where k I > 0. u I = k I s (3.7) So, the coordinator switches between NF and I controllers as follows ( unf if je(t)j u = u I if je(t)j < (3.8) where u NF and u I are the outputs of NF and I controller respectively and e(t) = v ref ; v o (t) and >0isathreshold SMC + Integral Design The SMC + I design is shown in Fig.3.5. Figure 3.5: SMC + Integral control of DC/DC buck converter. The main purpose of sliding mode control is to reject disturbances and make the system insensitive to unknown parametric perturbations [5]. The dymamics of the sliding surface [13] is chosen as in (3.9) s(t) =e(t)+ _e(t) (3.9) 9

13 where e(t) =v ref (t) ; v o (t) and _e(t) isthederivative ofe(t) with respect to time, and >0 is a design parameter. The Lyapunov function candidate is chosen as s 2 (t)=2 where s(t) is the sliding surface in (3.9). Let's denote the output of the SMC as u SMC and let it vary as in (3.10) u SMC = where > 0 is a design parameter. ( if s(t) 0 ; if s(t) < 0 (3.10) By omitting r c in the calculations for the sake of brevity (i.e., r c = 0), one may obtain the derivative of the sliding surface with respect to time after some calculations as follows in which i L 0. By selecting 1 _s = C (; + 1 R o C )+ r L i L LC ; v o R o C (; + 1 R o C )+(v o ; v SMC ) (3.11) 1 = R o C and u SMC = sign(s) (3.12) where >v o, we nally obtain the sliding condition s _s <0asfollows s _s <0 if i L ; v o r L (3.13) In other words, if = 1 R and i oc L ;vo r L, the control signal u SMC in (3.12) ensures the sliding condition s _s < 0, which also provides a sucient condition for the stability oftheclosed loop system during the time SMC is evoked. As will be seen from the simulation results, the obtained upper limit to i L is well satised for the converter parameters given in Table 5.1 corresponding a real converter. Chosing = 140, we obtain the following condition: \if i L < 5733A, then the system reaches sliding mode". Simulations show that i L is less than only 12 A even under line and load disturbances in the operational region of the converter. On the other hand, the I controller is used to avoid steady-state errors in the output voltage and is evoked when e(t) =v ref ; v o (t) is below a threshold. So, the coordinator switches between SMC and I controller as follows 10

14 ( usmc if js(t)j u = u I if js(t)j < (3.14) where u SMC and u I are the outputs of SMC in (3.12) and I controller in (3.7) respectively. Neural-Fuzzy Adaptation of SMC parameters Two design parameters in SMC, and, are usually constant parameters chosen by the designer in the case of DC/DC converter control (e.g., [1], [7] among many others). The parameters and mainly aect the time period during which the phase plane trajectory rst hits the sliding surface s(t) and the time period during which the trajectory then reaches zero point in sliding mode, respectively. Using the method in [9], [3], these two parameters are adapted for the SMC according to output error, sliding surface and their derivatives. These fuzzy rules (for changing the parameters of SMC) are based on expert knowledge and are in the form of (3.15) and (3.16). Rule 1 : IF e(t) is... AND _e(t) is..., THEN is.... Rule N c : IF e(t) is... AND _e(t) is..., THEN is... Rule 1 : IF s(t) is... AND _s(t) is..., THEN is.... Rule N d : IF s(t) is... AND _s(t) is..., THEN is... (3.15) (3.16) 11

15 Chapter 4 PID Design The performance of the intelligent control methods presented in Chapter 3 is compared with that of PID controller. A well-tuned PID contoller for the converter under investigation is borrowed from [16] and [4], which is obtained by the following classical frequency domain technique [4]. The frequency response of the open loop system is approximated from the corresponding transfer function using small signal model. From the ideal model, a lead-lag compensator of the form (4.1) was designed [4]. Figure 4.1:. PID control of DC/DC buck converter. 1+ s! G c (s) =G z 1+! l s cm 1+ s (4.1)! p From [16] and [4], the parameter values are G cm = 36.9, w z = 21998, w l = 3136, w p = The compensator was tuned by setting the bandwidth at about rad/s, which is large enough to achieve a fast control. A practical PID control is written as [6] which can be approximated by G PID (s) =K 1+ 1 st i + st d 1+ T d N s! (4.2) 12

16 G 0 PID(s) =K s(Td + Td st i 1+ T d N ) N s! (4.3) if T d =T i is "small". This formula has the same structure as the compensator above leading to a PID tuning K = 36.9, T i = 3:2 10 ;4, T d = 0:2 10 ;4, N 1 [16]. 13

17 Chapter 5 Computer Simulations The parameters of the implemented DC/DC Buck converter are listed in Table 5.1. The parameters of the SMC+I structure are as follows: = 1 R = 90:91 oc from (3.12), and, and k I are chosen as 0.05, 140 and respectively. Using these values, it is observed in the silmulations that i L < 12 A under line and load disturbances in the operational region of the converter (and during the initial transient, i.e., before reaching the steady state, i L is observed to be less than 190 A). Checking (3.13), the condition i L < 5733A is obtained, which iswell satised. This also provides a sucient condition for stability of the closed loop system during the time SMC is evoked. On the other hand, the results obtained by NF + I control structure in Fig.3.2 were not satisfactory, and therefore are not presented in this report. Table 5.1: The parameters of the converter. Input voltage V in 140 V Reference voltage V ref 54 V Inductance L 100 H Capacitance C 1000 F Resistive load R o 11 ESR of inductance r L ESR of capacitance r c 0.05 Switching frequency f s 100 khz The following two points concerning our simulations should be noted: i) before entering the PWM block, the control signal is multiplied with the inverse of PWM gain, ii)the design of PWM assures the condition that the duty cycle D is limited to 0 D 0:5 for the real converter considered. 14

18 The simulation results concerning line and load disturbance rejections are shown in Fig.5.1 and 5.2 respectively. The results as both disturbances enter the system are presented in Fig.5.3. The switching function for the SMC in (3.12) is shown in Fig.5.1.c. The superior performance of the SMC+I in comparison with PID is clear from Fig.5.1 to Fig.5.4. The phase plane for the SMC is shown in Fig

19 Input Voltage [Volt] time (a) time [s] - Input voltage [V] PID Converter Output [V] SMC + I time [s] (b) time [s] - Converter output [V] Switching function for SMC time [second] x 10 3 (c) time [s] - (t). Figure 5.1: (a) Line change with repect to time, (b) Line disturbance rejection, (c) Switching function for the SMC in (3.12). 16

20 Resistive Load [ohm] time (a) time [s] - Resistive load [] PID Converter Output [V] SMC+I time [s] (b) time [s] - Converter output [V] PID Converter Output [V] SMC + I time [s] (c) time [s] - Converter output [V]. Figure 5.2: (a) Load change with repect to time, (b) Load disturbance rejection, (c) closer look. 17

21 Input Voltage [Volt] time [second] Resistive Load [ohm] time [second] (a) Line and load changes with respect time PID 54.2 Converter Output [V] SMC+I time [s] (b) time [s] - Converter output [V] PID Converter Output [V] SMC + I time [s] (c) time [s] - Converter output [V]. Figure 5.3: (a) Line and load changes with repect to time, (b) Line and load disturbance rejection, (c) closer look. 18

22 54.08 PID. SMC Converter Output [Volt] time [second] Figure 5.4: Ripple for PID and SMC+I cases (no disturbance) derivative of (filtered) error error Figure 5.5: Phase plane. 19

23 Chapter 6 Conclusions This work presents an intelligent and hierarchical controller structure for DC/DC Buck converter with equivalent series resistors operating in voltage mode. The methods used are i) NF control using RBFN and ii) SMC, each of which is accompanied with I controller to avoid steady-state output error in voltage output. A robust controller structure consisting of SMC and I controller has been developed for DC/DC buck converter with ESRs operating in voltage mode. This controller gave the best performance in comparison with those of neural-fuzzy controller and linear PID controller. On the other hand, the results obtained by NF + I control structure were not satisfactory, and therefore are not presented here. All simulations in this work use parameters from a real DC/DC buck converter. Accordingly, the methods can readily be implemented on a real DSP setup. Acknowledgment The author would like to thank Imatran Voiman Saatio Finland for support. 20

24 Bibliography [1] Cardoso B.J., Moreira A.F., Menezes B.R., and P.C. Cortizo. Analysis of switching reduction methods applied to sliding mode controlled dc/dc converters. In APEC'92, pages 403{410, [2] Chen C.L and Chen W.C. Fuzzy controller design by using neural networks techniques. IEEE Trans. on Fuzzy Sytems, 2(3):235{244, [3] Erbatur K., Kaynak O., Sabanovic A., and Rudas I. Fuzzy parameter adaptation for a sliding mode controller as appliad to the control of an artculated arm. In IEEE Int. Conference on Robotics and Automation, pages 817{822, New Mexico, [4] Zenger K., Gadoura I., and Suntio T. Control engineering methods in analysis and design of dc-dc converters. In submitted to 2000 Nordic Workshop on Power and Industrial Electronics (Norpie/2000), [5] Young K.D., Utkin V.I., and Ozguner U. A control engineer's guide to sliding mode control. submitted to American Control Conference 2000, Chicago, USA, [6] Astrom K.J. and Hagglund T. PID Controllers: Theory, Design, and Tuning. Instrument Society of America, [7] Shiau L.G. and Lin J.L. Direct and indirect smc control schemes for dc-dc switching converters. In SICE'97, pages 1289{1294, [8] Mattavelli P., Rossetto L., Spiazzi G., and Tenti P. General-purpose fuzzy controller for dc-dc converters. IEEE Trans. Power Electronics, 12(1):79{86, January [9] Basbug R. Fuzzy Adaptive Sliding Mode Robot Control (in Turkish). PhD thesis, Istanbul Technical University, [10] Jang J. R. and Sun C. T. Functioanl equivalance between radial basis function networks and fuzzy inference systems. IEEE Trans. on Neural Networks, 4(1),

25 [11] Middlebrook R.D. Small-signal modeling of pulse-width modulated switched-mode power converters. Proc. of the IEEE, 76(4):343{354, April [12] Erickson R.W. Fundamentals of Power Electronics. Chapman & Hall, [13] Utkin V.I. Sliding Modes in Control and Optimization. Springer-Verlag, Berlin, [14] Duan Y. and Jin H. Digital controller design for switchmode power converters. In Proc. of APEC'99, pages 967{973, [15] Uykan Z. System identication and neuro-fuzzy control using radial basis function networks (in turkish). Master's thesis, Control and Computer Eng. Dep., Istanbul Technical University, [16] Uykan Z., Zenger K., and Koivo H. Sliding mode control for dc/dc buck converter in voltage mode. In IPEC-Tokyo 2000, volume 1, pages 102{107,

26 HELSINKI UNIVERSITY OF TECHNOLOGY CONTROL ENGINEERING LABORATORY Editor: H. Koivo Report 109 Report 110 Report 111 Report 112 Report 113 Report 114 Report 115 Report 116 Report 117 Niemi, A. J., Berndtson, J., Karine, S., Automatic Control of Paper Machine by Dry Line Measurement. December Ylén, J-P, Nissinen, A. S., Sumean logiikan sovelluksen kehistysprosessi ja sen soveltaminen fuzzytech-ohjelmiston evaluointiin. December Hyötyniemi, H., Mental Imagery: Unified Framework for Associative Representations. August Hyötyniemi, H., Koivo, H. (eds.), Multivariate Statistical Methods in Systems Engineering. December Robyr, S., FEM Modelling of a Bellows and a Bellows-Based Micromanipulator. February Hasu, V., Design of Experiments in Analysis of Flotation Froth Appearance. April Nissinen, A. S., Hyötyniemi, H., Analysis Of Evolutionary Self-Organizing Map. September Hätönen, J., Image Analysis in Mineral Flotation. September Hyötyniemi, H., GGHA Toolbox for Matlab. November Report 118 Nissinen, A. S. Neural and Evolutionary Computing in Modeling of Complex Systems. November Report 119 Gadoura, I. A. Design of Intelligent Controllers for Switching-Mode Power Supplies. November Report 120 Ylöstalo, T., Salonen, K., Siika-aho, M., Suni, S., Hyötyniemi, H., Rauhala, H., Koivo, H. Paperikoneen kiertovesien konsentroitumisen vaikutus mikrobien kasvuun. September Report 121 Cavazzutti, M. Fuzzy Gain Scheduling of Multivariable Processes. September Report 122 Uykan, Z. Intelligent Control of DC/DC Switching Buck Converter. December ISBN ISSN Picaset Oy, Helsinki 2001