Comparison of Fuzzy, PI and Fixed Frequency Sliding Mode Controller

For DC-DC Converters

Hamit Erdem Department of Electrical and Electronics Engineering,

The Baskent university, Turkey herdem@baskent.edu.tr

Abstract- This paper presents a comprehensive study about the application of three control methods in DC/DC converters. Fuzzy, proportional–integral and fixed frequency sliding mode controllers are applied to DC/DC buck converter. Fixed frequency sliding mode controller is studied with two different design approaches. Advantages, disadvantages, similarities and design procedures of controllers are studied. The dynamic performance of these controllers under input voltage change and load current variations are presented.

I. INTRODUCTION

Since DC–DC converters are nonlinear systems, they represent a big challenge for control design. Traditional controller has been focused on the application of linear control theory based on linearised models [1-3]. These methods are based on small signal models of the DC-DC converters and describe the system behavior very well around operating point and offers simple solutions of feedback controllers. The small signal models are linear approximations; thus, they are only valid around an operating point and change due to changes in the system parameters. Since classical control methods are designed at one nominal operating point, they are not able to respond satisfactorily to operating point variations and load disturbance. They often fail to perform satisfactorily under large parameter or load variations [4]. Nonlinear controllers which are more robust and have faster dynamic response are applied to power converters to solve this problem. Many nonlinear control schemes, such as one-cycle control, current-mode control and sliding-mode control (SMC), have been proposed for DC/DC converters [5-8]. Between these controllers, SMC and fuzzy logic control (FLC) has advantages such as simple and model free implementation. These two methods are applied to power converters with satisfactory results. FLC have been developed and implemented for DC/DC converters in many works [9-12]. SMC is a powerful method that is able to yield a very robust closed-loop system under plant uncertainties and external disturbances. The system can be entirely independent of effects due to modeling uncertainties, parameter fluctuations and disturbances [13]. Sliding mode(SM) controllers are well known for their robustness and stability. This controller

ideally operates at an infinite switching frequency such that the controlled variables can track a certain reference path to achieve the desired dynamic response and steady-state operation [14]. There are two different methods for application of SMC scheme to DC/DC converters. The first scheme which is called direct SMC is the conventional method. In this method the result of control action or hysteresis circuit is applied directly to power stage. In this method, converter works at infinite switching frequency. This causes high speed switching in converters which results in increasing power losses [15]. For reduction of switching frequency or applying fixed switching frequency possible solutions which include the use of constant timer circuits into the SMC [16-17] or implementing PWM based SMC. This approach is called indirect SMC or Quasi SMC [18-19]. By this way the result of control action is applied to the PWM module. In this control scheme, the output of the SM control is compared with a triangular wave. Then the resultant signal is applied to switch [20]. Recent studies on application SM control on DC/DC converters focused on fixed frequency sliding mode (FFSM) controller [18][21]. Ref [4] is the first work which comprises FLC, PI and standard SMC for power converter. Last works for application of SMC in power electronics focuses on fixed frequency controller [18][21]. This paper comprises the application of FLC, PI and FFSM control methods for DC/DC converters. This work is organized as follows, second section studies control of power converters and comprises suggested controller in last Works. Section three presents performance evaluation of controller under load disturbances. Last section is about discussion of results.

II. Control of DC/DC Converters

The DC/DC converters are one of main parts of many industrial applications. The control of these converters is a subject of many research projects which are looking for the best control [22-23]. These converters are basically non-linear plant due to switching elements behavior. In closed loop control of power converters, the function of the control loop is to regulate the variation of the line voltage and change of the output current. Voltage-mode control (VMC) and current-

mode control (CMC) are two methods which are used in regulation of output voltage in DC/DC converters. From view point of linearity the controller is linear or nonlinear controller. Control of DC/DC converters, which are most common in industrial application, have been extensively investigated for their serious non-linearity properties. [22-23]. There are several methods for control of these converters. Between these controller FLC, PI control and SMC method are chosen for this investigation. The suggested controllers can be used in VCM and CMC. PI control is based on linearised system model but FLC and SLM are nonlinear controllers. There are some similarity between FLC and SMC. These controllers can be designed based on state of system in phase plan depended on selected output variable. Phase plane shows output variable error and change of the error. These controllers do not need mathematical model of the system but require the parameter variation ranges [4]. A simple DC/DC buck converter which is shown in Fig. 1 is considered as a plant for applying control methods. Three control methods, FLC, PI, FFSMC issued for control of this converter. In comparison of controllers, FFSM is considered with two different design approaches which are studied in [18][21]. The parameters of converter are shown in Table. 1. In each controller the resultant of control action is applied to power switch via PWM module which is common in controllers. The calculation of duty cycle variation in each controller is presented. In PWM based controller a fixed frequency ramp is compared with continuous voltage and the result of comparison is applied to power switch driver. Frequency of the ramp provides that the frequency of the output switching signal will be constant, regardless of how the duty cycle varies with the variation of the control signal.

Fig. 1. Step down buck converter

TABLE I Buck converter parameters

Parameter Symbol Value Input voltage Vin 50 v Output voltage Vo 20v Capacitor C 22µf Load R 5Ω Switching frequency f 100kHz

A. Proportional-Integral Controller PI control is one of the most commonly used controllers in industry and can be analyzed mathematically. PI control is based on linear model of system which describes the system behavior very well around any fixed operating point and gives

simple solutions of feedback controllers. Parameters of controller affect the controller performance. In PI control, the parameter selection is a trade off between robustness and response speed. Increase of robustness will imply decrease of response speed. PI controller is very sensitive to the system parameters variation. PID family of controllers uses only output voltage as the feedback. However, their performances are not satisfactory under parameter variation, nonlinearity, large supply and load disturbances [4]. For design of PI control, it is essential to define the loop transfer function and the closed loop transfer functions for the output voltage as a function of the reference voltage. Bode Plots is the one of the tools is used in the design of the controller for the power electronic. By using the Bode plots one can easily design the controller that will result in a desired performance around working point. Bode plots provides possibility of stability measurement of system. This can be done by imposing an adequate value for the Phase Margin (PM) for the loop transfer function which is defined as the value given by the difference between the phase of the loop transfer function at the crossover frequency (ωx) and -180°. The Gain Margin (GM) is defined as the difference between 0 dB and the gain of the loop transfer function at the frequency where the phase becomes -180°. A stable system presents positive phase and gain margins. In PI controller, the result of PI control action based on error voltage can be applied directly to PWM module. B. Fuzzy Logic Controller FLC, as a nonlinear control technique, has been applied to control of the DC-DC converters in many papers [9-12]. The design of FLC is primarily based on an expert knowledge, trial and error procedure. Number of linguistic variables, the base width of linguistic variables, the number of rules and slope of membership function (MF) affect the performance of controller [23]. The design and tuning of a fuzzy system can be formulated as a search problem in a high-dimensional space where each point in the space represents a rule set, membership function and the corresponding system performance, that is, the performance of the system forms a hyper-surface in the space according to given performance criteria. Thus, finding the optimal location of this hyper-surface is a search problem, which is equivalent to developing the optimal fuzzy system design [23]. On the other hand, the design procedure of FLC can be reduced by using other soft computing techniques like Neural Networks and genetic algorithms. Advantages of FLC are:

• FLC work with less precise inputs; • FLC does not need accurate mathematical model and

can handle nonlinearity, • It is possible to take account for local nonlinearities

in FLC. Disadvantages of fuzzy logic:

• Controllers traditionally designed by trial and error • Difficult to get stability and performance analysis

Vin

outreferror VVV −=

)1()(_ −−= kVkVV errorerrorchangeerror

)()1()( kPWMkPWMkPWMd ∆+−==

• Difficult to get transfer functions and small signal analysis

In FL based feedback controller, error between output voltage and reference voltage, and change of error are inputs of FLC. The output of controller is change of duty cycle. Accumulating change of duty cycle is applied directly to PWM module. K is the sampling time of controller. Equation (1-3) show the relation between inputs (error and error rate) and outputs (duty cycle change) The calculated duty cycle signal is then sent to a pulse width modulator which generates the appropriate switching pattern for the switch in the DC/ DC converter.

(1)

(2)

(3) Control algorithm of FLC in VMC is as follow; 1. Measure the output voltage variable, 2. Calculate the error and error change 3. Normalize the input variables and fuzzify the inputs using the rule base table. 4. Transform the result value into fuzzy inference 5. Defuzzify the data using the center of gravity method to convert to fuzzy control signal to produce the real value. In this work, based on equation above, the output voltage is considered as system variable. For every input and output variables seven MF is selected. Design of controller is based on congenital FLC method and is a simple application. Triangular membership is selected for input and output variables. The triangular membership is considered because of its simplicity of implementation and because less computational intensity is required. Table 1 shows the rule bases. In simulation work, center of gravity is considered for defuzzification.

TABLE I Rules table of FLC

e/de nb nm ns zero ps pm pb nb nb nb nb nb nm ns Ze nm nb nb nb nm ns ze Ps ns nb nb nm ns ze ps Pm zero nb nm ns ze ps pm Pb ps nm ns ze ps Pm Pb Pb pm ns ze ps pm pb pb Pb pb ze ps pm pb pb pb pb

C. Fixed Frequency Sliding Mode Controller SMC controllers which are well known for their robustness and stability were introduced for variable structure system and work with variable frequency. Originally SM controller operates at an infinite switching frequency such that the controlled variables can track a certain reference path in phase plane to reach zero error in output variable. The application and implementation of SM controller in power converter is investigated in many papers [4][8][10]. In power converter, the control action forces the output variable to track a certain

trajectory and remain on sliding surface, against load current and input voltage variation. The main problem with using of this controller in control of DC/DC converters is variable and high switching frequency which increases losses. The switching frequency changes with load current and input voltage variations. Variable switching complicates L and C selections and filter design. On the other hand in practical application the switching frequency is limited by property of electronic devices. Operation of converter in fixed switching frequency prevents these problems but decreases the system robustness. In recent years, FFSM control has been investigated as a better control then conventional SM control method for DC-to-DC. In last years in many works some techniques are suggested and applicated for reducing and fixing the switching frequency of SM control. In literature, there are three main methods to obtain a constant switching frequency in SMC which are as follow;

• Adding a constant ramp or timing function into controller[24]

• Applying adaptive control into the hysteresis modulation based SMC controller [25-26]

• PWM based SMC controller[18] First method does not show good dynamic behavior, but transient response of the second method is better. The hardware implementation of the second method is complicated and needs additional components. The first two methods are complex and increase the implementation cost of the controller [18]. On the other hand the switching frequency is not absolutely constant [26]. Among three methods which mentioned, PWM based converter shows better Performance than others and discussed in some works [18- 19]. For implementation of PWM based SM control, the relation between SM control and duty cycle must be obtained. For implementation of PWM –based SM control, two main approaches investigated in recent studies are as follows;

• Using sate space averaging method in controller modeling[21]

• Mapping equivalent control function onto the duty cycle function [19]

Reference [21] describes first method in details. This method is based on sate space averaging method which describes circuit behavior over a switching period with consideration of ON/OFF conditions. By using state space averaging, a first order path is selected for output voltage trajectory on phase plane. Phase plane is defined based on output voltage error and error variation. Equation (4) describes voltage error and (5) shows selected path in phase plane. Where λ, is convergence factor which effects system dynamic performance. The larger the convergence factor the faster the system will reach its steady state. However, due to limits on the system parameters such as duty cycle, it is not possible to increase the convergence factor beyond a certain value. From stability view point, increase the very high values for λ; the converter behaves unstable [21].

Fig. 2. Block diagram of SMC1 in Simulink

(4)

(5) With combining the (5) with buck converter state space model, the duty cycle (d) can be provided based on (6) (6) The result of (6) can be applied directly to PWM module. The coefficient ‘a’ depended on L, R, C and λ values. Equation (6) shows that the duty cycle can be change with variation of output and input voltage. The simulink model of block of control section is shown in Fig. 2. Design steps of state space averaging based first order SMC (SMC1) control in VCM are as follow;

• State space averaging of converter • Define the sliding surface • Inserting the λ in state space equation of converter • Selection of value for α which effects dynamic

response of system. The second method is investigated in [18-19]. This method offers mapping the equivalent control onto the duty cycle function of the pulse width modulator [71]. This method is analyzed with simulation and practical implementation in [18-19]. These works show the theoretical analysis and practical approaches for PWM based SMC. The advantages of this approach are that additional hardware circuits are not needed as the switching function is performed by the PWM modulator, and that the transient response is not deteriorated. However, the implementation is non-trivial in order to preserve the original SM control law [18]. The second method suggests sliding surface with three coefficients α1, α2 and α3 and results second order response for output variable response as in (7).

(7) Equation (8) shows the result of SMC as Vc, which is considered equal as d (duty cycle). This value can be calculated as in (8) [19].

(8) The constant gain parameters, Kp1 and kp2 parameters are depended on values of L, C and sliding coefficients αi. The

dynamic performance of controller can be changed depended on values of kpi [18]. These coefficients behave like PI coefficients. Reference [19] surveys design steps of PWM based SM control in second method as follow;

• Selection of desired settling time and damping • Calculation of sliding coefficient • Calculation of kp1 and kp2

III. Simulation Results

Performance analysis of four controllers is evaluated in simulation. FLC, PI, SMC1 and SMC2 developed based on last descriptions. Performance analysis of converters is evaluated under input voltage variation and load current disturbances. SMC1 indicate controller with first order response and SMC2 response is a second order controller based on (8). Fig’s 2-6 show the start up behavior of the controllers. FLC behaves like second order system and has the lowest settling time. FLC and PI controller has the same overshoot for output voltage. The maximum overshoot is in SMC2. SMC1 shows order reduction which is one of SMC properties. The response of current disturbances is evaluated in three steps. Fig. 7 shows comparison of controller’s response for %20, %40, and % 50 of load increases. Overshot and undershot of output voltage in SMC and FLC is better than other controllers. Settling time of FLC is better then SMC2. Maximum undershot of voltage is in SMC1.Voltage reduction is 5 volt for the %50 of load increase. Fig.8 shows controller performance under load decrease for %40 of nominal load. Simulation started with nominal load of 5Ω then applied a load change to 7 Ω at 2ms. Against load reduction, Voltage overshot is nearly same in PI, FLC and SMC2 but settling time in FLC is better then others. PI and SMC2 behave in underdemped response. The time for reaching to steady state in SMC2 is better then PI. SMC1 has the maximum overshoot which is nearly 3.5 volt. The performance analyses of converter under input voltage variation is shown in Fig. 9. Feedforward control is not included in regulation of output voltage. It means that the amplitude of ramp signal of PWM modulator is constant in selected controllers. 2ms after start up %50 input voltage reduction is applied for testing line disturbances. With voltage reduction, FLC and SMC2 with minimum change in output voltage shows better performance then the other controllers and have the lowest deviation from reference value. There is an increase in output voltage and decrease in ripple in SMC1. Output ripple increases in SMC2 against input voltage reduction. With consideration the results, the dynamic performance of FLC and SMC2 is better than others. Instantaneous voltage change in SMC2 is better than first order SMC1.

oxrefxe −=

)(ee λ−=o

( ) ( )inV

k2xaktd −+=

321S α+α+α=

ovovrefv2pkCi1pkcV β+β−+−= )(

Fig. 3. Responses FL controller in start up and load change

Fig. 4. PI controller reposes in start up and load change

Fig. 5. Responses of PWM based SMC in start up and load change

Fig. 6. Responses of first order PWM based SMC in start up and load change

Fig. 7. Response of controllers under increase of load

Fig. 8. Response of controllers under decrease of load

Fig. 9. Response of controllers under input voltage change

FLC SMC1

SMC2 PI

FLC SMC1

SMC2 PI

FLC SMC1

SMC2 PI

IV. Conclusions

Application of three control method based on design procedure and dynamic responses for DC/DC converters are comprised. The relationship between duty cycle of PWM and output of each controller presented based on recent studies. Simulation results showed that the PWM based sliding mode controller (SMC2) with second order response and FLC has the acceptable performance under load and line disturbances. Steady state performances of controllers are acceptable. State space averaging based SMC1 has maximum overshot and undershot in dynamic response. Steady state performances of controller are acceptable. With consideration of design procedure, dynamic response and hardware implementation, FLC is better then SMC2. FLC can be implemented by using fuzzy based microcontroller which causes decreasing cost of practical implementation and increase speed of responses. Despite of robustness and dynamic performances of second order PWM based SMC (SMC2), the hardware implementation of this controller needs more devices and more mathematical procedures then FLC. FLC and PI controller can be applied on low cost microcontrollers. Recent works shows that SMC has better performance with use of analog component but digital implementation of this controller is subject of research. With solution of complexity of design procedure and hardware implementation of PWM based SMC; this controller will be preferred in control of dc/dc converters. With expanding user friendly soft computing tools for tuning of FL controller, these controller can be used in many industrial applications.

REFERENCES [1] Middlebrook, R.D., and Cuk, S.: ‘Advances in switched-mode power conversion: Volumes I and II’ TESLAco, Pasadena, CA, USA, 1983 [2] Ioannidis, G., Kandianis, A., ‘Novel control design for the buck converter’, IEE Proc., Electr. Power Appl., 145, (1), pp. 39–47 1998 [3] Himmelstoss, F.A., and Zach, F.C.: ‘State space control for a step-up converter’. Proc. Telecom. Energy Conf., pp. 275–282, November 1991 [4] V.S.C. Raviraj and P.C. Sen, “Comparative study of proportional-integral, sliding mode, and FLC for power converters,” IEEE Transactions on Industry Applications, vol. 33 no. 2, pp. 518–524, March/April 1997. [5 ] Smedley, K.M., and Cuk, S.: ‘One-cycle control of switching converters’, IEEE Trans. Power Electron., 10, (6), pp. 625–633,1995 [6] Sanders, S.R., Verghese, G.C., and Cameron, D.E.: ‘Nonlinear control of switching power converters’, Control Theory Adv. Tech., 5, (4), pp. 601–627,1989 [7] Liu, Y.F., and Sen, P.C.: ‘A general unified large signal model of current programmed DC-to-DC Converters’, IEEE Trans. Power Electron., 9, (4), pp. 414–424, 1994 [8] M. Ahmed, M. Kuisma, K. Tolsa, and P. Silventoinen, “Implementing sliding mode control for buck converter,” in IEEE Power Electronics Specialists Conference Record (PESC), vol. 2, pp. 634–637, June 2003. [9] A. Balestrino, A. Landi, and L. Sani, “CUK converter global control via fuzzy logic and scaling factor,” IEEE Trans. Ind. Appl., vol. 38, no. 2, pp. 406–413, Mar./Apr. 2002. [10] E. Vidal, L. Martine, F. Guinjoan, J. Calvente, and S. Gomariz, “Sliding and fuzzy control of a boost converter using an 8-bit microcontroller,” Proc. Inst. Elect. Eng., Electr. Power Appl., vol. 151, no. 1, pp. 5–11, Jan. 2004.

[11] T. Gupta and R. R. Boudreaux, “Implementation of a fuzzy controller for DC–DC converters using an inexpensive 8-b micro-controller,” IEEE Trans. Ind. Electron., vol. 44, no. 5, pp. 661–669, Oct. 1997. [12] W. So, C. K. Tse, and Y. Lee, “Development of a FLC for DC/DC converters.,” IEEE Trans. Power Electron., vol. 11, no. 1, pp. 24–32, Jan. 1996. [13] J. Y. Hung, W. Gao and J. C. Hung, “Variable Structure Control: A Survey”, IEEE Transaction on Industrial Electronics, Vol. 40, No. 1, Feb 1993 [14] V. Utkin, J. Guldner, and J. X. Shi, Sliding Mode Control in Electromechanical Systems. London, UK: Taylor and Francis, 1999. [15] H.W. Whittington, B.W. Flynn, Switched Mode Power Supplies: Design and Construction, New York: Wiley, 1997. [16] M. Castilla, L. C. de Vicuna, M. Lopez, O. Lopez, and J. Matas, “On the design of sliding mode control schemes for quantum resonant converters,” IEEE Trans. Power Electron., vol. 15, no. 6, pp. 960–973, Nov. 2000. [17] B. J. Cardoso, A. F. Moreira, B. R. Menezes, and P. C. Cortizo, “Analysis of switching frequency reduction methods applied to sliding mode controlled DC-DC converters,” in Proc. APEC, pp. 403–410.Feb. 1992, [18] S. C. Tan, Y. M. Lai, C. K. Tse, and M. K. H. Cheung, “A fixed-frequency pulse-width-modulation based quasi sliding mode controller for buck converters,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1379–1392, Nov. 2005. [19] S.C. Tan, Y.M. Lai, and Chi K. Tse, “A unified approach to the design of PWM based sliding mode voltage controller for basic DC–DC converters in continuous conduction mode”, IEEE Trans. on Circuits and Systems. Volume 53, no.8, pp 1816 – 1827. Aug. 2006 [20] Shiau, L.G, Lin. J.L.,Direct and indirect SMC control schemes for DC-DC switching converters, SICE '97., P.1289 – 1294 July 1997 [21] J. Mahdavi, A. Emadi, and H.A. Toliyat, “Application of state space averaging method to sliding mode control of PWM DC/DC converters,” in Proceedings, IEEE Conference on Industry Applications (IAS), vol. 2, [22] G. Ioannidis, A. Kandianis, and S.N. Manias, “Novel control design for the buck converter,” IEE Proc., Electr. Power Appl., vol.145, no.1, pp.39–47, Jan. 1998. [23] W.-D. Xiao and W.G. Dunford, “Fuzzy logic auto-tuning applied on DC-DC converter,” The 30th Annual Conference of the IEEE Ind. Electron. Society, vol.3, pp.2661–2666, Nov. 2004. [24] B.J. Cardoso, A.F. Moreira, B.R. Menezes, and P.C. Cortizo, “Analysis of switching frequency reduction methods applied to sliding mode controlled DC–DC converters,” in Proceedings, IEEE Applied Power Electronics Conference and Exposition (APEC), pp. 403–410, Feb 1992. [25] S.C. Tan, Y.M. Lai, C. K. Tse, and M.K.H. Cheung, “Adaptive feedforward and feedback control schemes for sliding mode controlled power converters”, IEEE Transactions on Power Electronics, vol. 21, no. 1, pp. 182–192, Jan. 2006. [26] V.M. Nguyen and C.Q. Lee, “Tracking control of buck converter using sliding-mode with adaptive hysteresis,” in IEEE Power Electronics Specialists Conference Record (PESC), vol. 2, pp. 1086–1093, June 1995.