Hybrid Posicast Controller for a DC-DC Buck Converter

Transcription

1 SERBIAN JOURNAL OF ELETRIAL ENGINEERING Vol. 5, No. 1, May, Hybri Posicast ontroller for a D-D Buck onverter Kaithamalai Uhayakumar 1, Ponnusamy Lakshmi, Kanasamy Boobal Abstract: A new Posicast compensate hybri controller for the D-D Buck converter is investigate. Posicast is a feeforwar compensator, which eliminates the overshoot in the step response of a lightly ampe system. However, the traitional metho is sensitive to variations in natural frequency. The new metho escribe here reuces this unesirable sensitivity by using Posicast within the feeback loop. Design of the Posicast function is inepenent of computational elay. The new controller results in a lower noise in the control signal, when compare to a conventional PID controller. Keywors: D-D Buck converter, Posicast, PID controller. 1 Introuction One of the challenges in controlling the D-D converter is compensation for the converter s nonlinear ampe ynamics which are a function of loa parameters [1]. Avances in signal processing technology have spurre research in new control techniques to improve converter control [3]. In this work, a new control technique base on the Posicast principle is employe. lassical Posicast is a fee forwar control metho for lightly ampe systems []. It has the potential to eliminate the oscillatory response of a lightly ampe system, but the rawback is sensitivity to moel uncertainty [1]. The Posicast approach can be more useful if the parametric sensitivity can be reuce. The rawback is eliminate by using the Posicast in a feeback loop; this reuces the sensitivity to parameter variations [5]. Research has shown that PID-base control of the power converter may require aitional algorithm moifications to achieve a combination of goo transient an steay-state performance []. These moifications inclue the use of a control ea zone, an averaging igital filter to counteract switching noise, an the use of two sets of gains (one set for the transient perio an the secon set for steay-state performance).the application of PID-type controllers for buck converters is further complicate in a igital implementation. 1 Department of Electrical an Electronics Engineering, ollege of Engineering Guiny, Anna University, hennai 5, Inia; k_uhayakumar@annauniv.eu Department of Electrical an Electronics Engineering, ollege of Engineering Guiny, Anna University, hennai 5, Inia. 11

2 K. Uhayakumar, P. Lakshmi, K. Boobal The Posicast-base controller escribe in this work, prouces many of the beneficial close loop effects of the PID-type controller, such as goo steaystate performance an goo amping of resonant behavior. ontroller gain parameters are very easy to etermine. The control metho prouces a very goo response that is preictable by the small-signal average continuous time moel. The key element of the Posicast controller structure is especially easy to implement in iscrete-time harware. The frequency response of the Posicast element inherently reuces high-frequency noise [5], an the amping effect of Posicast eliminates the nee for multiple sets of controller gains. The propose metho oes not require any of the aitional moifications escribe earlier for PID control (eazone, aitional filtering, gain scheuling). The state space moeling of a converter [-11] an Small signal Moel with parasitic elements is also erive for the converter [3-]. In summary, a classical Posicast has superior amping qualities, reuce sensitivity to parametric uncertainty an loa change through feeback. So, the propose control metho is a significant eparture from the classical posicast. The organization of this paper is as follows. Section provies mathematical moeling of a D-D buck converter. Section 3 evelops the controller esign base on the control objectives. In Section, the controller moeling is escribe base on a small signal moel of the converter in the transfer function form. Section 5 shows the obtaine simulation results of a Posicast compensate D-D buck converter for the output voltage, uty cycle an loa current for ifferent values of input voltage, loa resistances an Posicast parameters. The simulation results for a PID controller are also given for comparison purposes. Finally, some conclusions are rawn. Moeling.1 State space analysis State Space variables: Let, x 1 ( i L ) an x ( V ) be the two state variables corresponing to the inuctor current an the capacitor voltage [7]. Where, V is the input ( u ), an V o is the output voltage ( y ). Fig. 1 D-D buck converter. 1

3 Hybri Posicast ontroller for a D-D Buck onverter. onverter moel uring switch ON When the switch is ON, the ioe will not conuct an the resulting circuit is shown in Fig.. Fig. Buck converter uring switch ON. Now, the system governing equations are, V + Lx 1 + rlx1+ R( x1 x ) = (1) x r x + R( x1 x ) = () il t il = A Bu V + V t il Vo = V rl r R R ( r ) ( ) 1 + R r + R x 1 L L L x1 L u x = R 1 x +. (3) ( r ) ( ) + R r + R.3 onverter moel uring switch OFF When the switch is OFF, the ioe will conuct an the resulting circuit is shown in Fig. 3. Fig. 3 Buck converter uring switch OFF. 13

4 K. Uhayakumar, P. Lakshmi, K. Boobal The state equations uring switch OFF time ( V = ), when only the ioe is conucting is given by il t il = A V V () t il Vo = V (5) rl r R R ( r + R) ( r + R) x 1 L L L x1 x = R 1 x. () ( r + R ) ( r + R ) The converter moel inclues parasitic elements an the buck converter parameters are liste in Table 1. Table 1 Parameters of the Buck onverter. Symbo1 Parameter Value Units R Loa Resistance 1 Ω L Filter Inuctance 15 μ H Filter apacitance 1 μ F r L Effective series resistance of the Inuctor 1 mω r Effective series resistance of the capacitor 3 mω T Dampe natural perio. ms δ Overshoot. f s Switching frequency khz V Input voltage V. Open loop response For an input of V the output is 1V, the response is shown in Fig.. In the open loop response, the overshoot is high, the settling time is very high, an the response is oscillatory. The propose control strategy is able to eliminate the peak overshoot an reuce the settling time. 1

5 Hybri Posicast ontroller for a D-D Buck onverter 5 Open Loop Response ontroller Design 3.1 Posicast principle Fig. Open loop response. Fig. 5 Step response of a lightly ampe system. The step response is shown in Fig. 5. The response is characterize by the overshoot δ an the ampe time response perio T. The block iagram shown in Fig. escribes the structure of a classical half-cycle Posicast. A classical Posicast is esigne using knowlege of the step response overshoot δ an ampe time response perio T. Accurate knowlege of the step response parameters yiels a compensator whose lowest 15

6 K. Uhayakumar, P. Lakshmi, K. Boobal frequency zeros cancel the ominant plant pair. The controller is calle halfcycle because the embee time elay is one-half of the time perio ( T ). Fig. The classical half-cycle Posicast. The new hybri control system structure is illustrate in Fig. 7. Fig. 7 Propose Hybri Feeback control using Posicast. The Posicast function is given by T δ Ps () = e s 1. 1+δ Parameters of function Ps ( ) are the step response overshoot an the ampe response perio T. The Posicast essentially reshapes the reference signal into two parts, initially the controller subtracts the scales amount from the reference signal, so that the peak of a lightly ampe response coincies with the esire final value of the system response. The time to the peak of the step response is one half the natural ampe perio. After this elay, the full value of the step reference is applie to the system Gs ( ) so that the output remains at the esire final value. Another interpretation is that the reference amount originally subtracte from the input now cancels any unesire overshoot because it is elaye by precisely one-half the ampe natural perio. 1

7 Hybri Posicast ontroller for a D-D Buck onverter Posicast ontroller Moelling The key elements of the function Ps () are the scaling factor parameterize by δ an the time elay element parameterize by T. The esign metho for the propose control system has two steps. First, the function Ps () is esigne for the buck converter using (3-). Next, the controller s () is esigne to compensate the combine moel [1 + Ps ( )] Gp ( s). lassical frequency omain techniques are use. To counteract steay state isturbances, a pure integratortype compensator has been foun suitable for the buck converter K s () =. (7) s The gain K is chosen as large as possible to minimize the settling time, but not so large that the overshoot is excessive. The complete hybri controller transfer function is escribe by combining the compensator s ( ) an the Posicast transfer function as,.1 Posicast controller s + Ps = + e s 1+δ T K δ s ()(1 ()) () The key elements of function Ps ( ) are the scaling factor parameterize by δ an the time elay element parameterize by T. Small signal moel transfer function Vˆ o () s 1 + sr = V ˆ( ) rl L R r. s + 1+ s r + R + + s L R+ rl R+ r R+ r L Transfer function of converter 1 s + Gs () = s s+ 1 Unampe natural frequency 1 ω n =. (9) L

8 K. Uhayakumar, P. Lakshmi, K. Boobal Dampe time response perio π T =. (1) ω n 1 ζ Overshoot ζπ 1 ζ δ= e. (11) ontroller parameters are given below T =. ms, δ=.. The gain K is chosen as large as possible to minimize the settling time, but not so large that the overshoot is excessive an it is chosen to be 35. The resulting transfer function of the hybri controller is 35 1.s s ()(1 + Ps ()) = 1+.55( e 1) s. (1) The frequency response of the Posicast-compensate open-loop function, s ()[1 + Ps ()] Gp () s is shown in Fig.. The phase margin is about 7 an the gain margin is approximately 1B. Mag in DB phase eg freq in ra freq in ra Fig. Frequency response of the Posicast- ompensate system.. PID compensator ontrollers base on the PID approach are commonly use for D D converter applications. Power converters have relatively low orer ynamics that can be well controlle by the PID metho. In fact, the PID controller is ifficult to outperform for several reasons. The integrator increases the system type 1

9 Hybri Posicast ontroller for a D-D Buck onverter number, thus minimizing the steay-state error. Two zeros in the controller enable ampening of resonant characteristics an improving of the transient response. ompare to many other control philosophies, the PID controller structure is fairly easy to explain an unerstan. Yet, there are some limitations for PID controllers. The integrator introuces aitional phase elay an tens to slow the system response. The phase elay can be countere, however, by the two zeros of the PID controller. A minimum of three gain parameters must be esigne for controllers of the PID variety. Two parameters etermine the compensators zero characteristics, an the thir parameter is use to ajust the overall loop gain. One approach for tuning is to use the two zeros to cancel unesirable lightly ampe poles in the converter moel. The basic PID transfer function is not strictly proper, however, an amplification of high-frequency noise is a serious rawback in switching converter applications. Practical applications of the PID controller inclue one or more aitional instrumentation poles to reuce the high frequency gain. As such, typical PID compensators have four to five gain coefficients to esign. The pure integrator in the PID controller also evelops problems if the control signal becomes saturate. This is the well-known integrator winup problem; many practical controllers substitute a lag compensator for the pure integrator. Alternatively, some steps to minimize winup effects must be esigne an implemente. Mag in DB phase eg freq in ra freq in ra Fig. 9 Frequency response of the PID compensate system. 19

10 K. Uhayakumar, P. Lakshmi, K. Boobal The PID compensator is esigne for the buck converter for comparison; the frequency response of the PID-compensate open-loop function is shown in Fig Gc ( s) = s. (13) s.3 omparison with PID control omparing the propose Posicast-base control to classical PID control yiels useful insights. For the same phase margin, however, the Posicastcompensate magnitue response is significantly suppresse at higher frequencies compare to that of the PID-compensate system. Therefore, the Posicast-compensate system suppresses high frequency noise much better than the PID approach. For ientical phase margins, the Posicast-compensate approach yiels a larger gain margin. 5 Simulation Results an Discussion 5.1 lose loop response of a Posicast controller For the input of V the output is 1V, the settling time is 1s. The overshoot eliminate response is shown below for the output voltage an uty cycle Fig. 1 Output voltage (V in V). From the close loop response it is inferre that the measurement noise is highly suppresse by the Posicast controller, the overshoot is eliminate an the regulate voltage of 1V is obtaine. 13

11 Hybri Posicast ontroller for a D-D Buck onverter.7 Duty ycle..5 Duty ycle Fig. 11 Duty ycle (V in V). 5. Posicast responses for various values of the Input Voltage Simulate output voltage an uty cycle for ifferent input voltages an loa resistances are plotte in Figs The output settling time is 1 ms. 1 Output Voltage Fig. 1 Output Voltage (V in 15V). 131

12 K. Uhayakumar, P. Lakshmi, K. Boobal.9 Duty ycle..7. Duty ycle Fig. 13 Duty ycle (V in 15V). 1 Output Voltage Duty Ratio Fig. 1 Output Voltage (V in V) Fig. 15 Duty ycle (V in - V). 5.3 Response for various values of Loa Resistances (R) For the change in the loa resistance the regulate voltage of 1V is obtaine, the loa current is change accoringly. The Posicast compensate buck converter maintains goo transient response throughout a :1 range of loa resistance (Fig. 1). 13

13 Hybri Posicast ontroller for a D-D Buck onverter 1 Output Voltage 1 Output Voltage R=ohms 1 1 R=1 ohms 1 1 Output Voltage Output Voltage Fig. 1 Output voltage (R 5 ohms) Fig. 17 Output voltage for ifferent loa resistances. Also, the Posicast controller proves to be effective in withstaning the perturbations in terms of loa resistance changes. The propose metho has been confirme not to require any of the aitional moifications for PID control. The results iscusse are achieve using a single controller gain setting. Fig. 1 Loa current for ifferent loa resistances. 133

14 K. Uhayakumar, P. Lakshmi, K. Boobal 5. Response for changes in controller parameter values Figs. 19 an show that the changes in the Posicast parameters, the Overshoot an the Time Delay ( T /) have very little effect on the steay state response of the controller. Fig. 19 Response for change in a Posicast Parameter (Overshoot). Fig. Response for change in a Posicast Parameter (Time Delay). 5.5 Results of the PID ontroller A classical PID type feeback controller is also esigne for the purposes of comparison. Zeros of the PID controller are esigne to cancel the lightly ampe ominant poles of Go(s). An instrumentation pole is ae to the PID 13

15 Hybri Posicast ontroller for a D-D Buck onverter controller so that the transfer function woul be physically realizable. The instrumentation pole is place corresponing to a corner frequency that is one ecae higher than the ampe natural frequency of the PID zeros. The simulate responses are plotte in Fig. 1; the output exhibits a small overshoot, noise suppression is less when compare to the Posicast controller. The Posicast controlle response cannot be tune, however, since the function 1 + Ps ( ) is uniquely etermine by the system overshoot an ampe natural perio. The propose hybri controller gain K an PID base controller gain are tune to yiel similar settling times an close loop banwith. The PID controller yiels a slightly faster rise time than the propose controller, an can be tune to give even faster response. Both feeback systems exhibit some eviation from steay state ue to the measurement noise that is inclue in the simulation. The input resulting from PID control exhibits a very high initial rate of change ue to the erivative action. The propose control is much less aggressive. The PID controller s noise sensitivity can be reuce, however, by aing more instrumentation poles to reuce the high frequency gain. Both of these responses exhibit oscillations at the natural frequency of the secon pair of non-moele lightly ampe poles. The propose controller with a Posicast insie the feeback loop performs much better PID ontroller omparison of Posicast an PID controllers Posicast ontroller Fig. 1 Output voltage (V in = V, R = 1 Ω). 5. Responses of a PID ontroller for various values of the input Voltage Figs. -5 show that the controller is robust enough to ensure the regulate output voltage at 1 V, for ifferent input voltage changes. 135

16 K. Uhayakumar, P. Lakshmi, K. Boobal 15 ontrol Signal 1 PID ontroller Response(Vin 15 V) 1 1 Duty ycle Fig. ontrol Signal (with noise) Fig. 3 Output Voltage (V in 15V). 15 ontrol Signal 1 Output Voltage Fig. ontrol Signal (with noise) Fig. 5 Output Voltage (V in V). 5.7 Responses of a PID ontroller for various values of Loa Resistance (R) Figs. - show the responses of a PID controller for loa current an output voltages for ifferent loa resistances. From the response it is clear that the PID controller is ineffective in eliminating the overshoot an high frequency noise suppression. This happens because of several reasons. The integrator increases the system type number, thus minimizing the steay-state error. Two zeros in the controller make it possible to ampen resonant characteristics an improve transient response. The aitional phase elay introuce by the integrator tens to slow own the response. PID controllers help amplification of high frequency noise which is a serious rawback in switching converter applications. 13

17 Hybri Posicast ontroller for a D-D Buck onverter 3 Loa urrent for ifferent loa resistances.5 R=5 Ohms Loa urrent(a) R =1 Ohms R= Ohms Fig. Loa urrent for ifferent loa resistances. Output Voltage 1 Output Voltage Fig. 7 Output Voltage (R 5 Ω) Fig. Output Voltage (R Ω). 5. Avantages of the propose controller The controller gain parameters are very easy to etermine. This metho prouces a very goo response that is preictable by the small-signal average continuous time moel. The key element of the Posicast controller structure is especially easy to implement in igital harware also. The frequency response of the Posicast element inherently reuces high-frequency noise, an the amping effect of the Posicast eliminates the nee for multiple sets of controller gains. onclusion A Posicast base control metho enabling regulate output voltage for a buck converter has been presente in this paper. The control metho uses a Posicast element within the feeback control system to take avantage of superior amping qualities, while reucing the sensitivity of the classical,

18 K. Uhayakumar, P. Lakshmi, K. Boobal feeforwar Posicast. Posicast element parameters T an δ can be irectly compute from the analytical, ieal metho of D-D converters. Different variations of a classical Posicast were originally use in a fee forwar manner, by reshaping the system input to minimize the oscillatory behaviour of the lightly ampe system that followe the Posicast. The peak overshoot is eliminate an the settling time is much lower with the new control strategy. The measurement noise is highly suppresse an is much better than the PID controller since it has a lower gain at a higher cross over frequency. An integral compensator with a single gain K is use with the Posicast element to ensure the proper steay state response. In contrast to the PID controller, the propose metho only nees to tune the gain K, an the compensate system has improve the gain margin an phase margin an, its narrow open loop banwith ensures suppression of high frequency noise. The effect of the lightly ampe poles of the buck converter moel in the close loop response is cancelle by the Posicast function. ontroller parameters are easy to etermine. With the propose new control strategy the parametric an the loa sensitivity is much reuce. In summary, the results obtaine inicate that the hybri Posicast-base controller is an effective approach for D-D converter output voltage regulation. 7 References [1] O.J.M. Smith: Posicast ontrol of Dampe Oscillatory Systems, Proc. IRE, Vol. 5, No. 9, 1957, pp [] G. ook: An Application of Half-cycle Posicast, IEEE Trans. Autom. ontrol, Vol. A-11, No. 3, July 19, pp , [3] R.P. Severns, G.E. Bloom: Moern D-to-D Switchmoe Power onverter ircuits. New York: Van Nostran Reinhol, 195. [] N. Mohan, T.M. Unelan,W.P. Robbins: Power Electronics: onverters, Applications an Design, New York, John Wiley & Sons, Inc., [5] J.Y. Hung: Feeback ontrol with Posicast, IEEE Trans. In. Electron., Vol. 5, No. 1, 3, pp [] L. Guo, J.Y. Hung, R.M. Nelms: PID ontroller Moifications to Improve Steay-State Performance of Digital ontrollers for Buck an Boost onverters, in Proc. 17th Annu. IEEE Appl. Power Electron. onf. Expo., Dallas, TX, Feb., pp [7] M.H. Rashi: Power Electronics Hanbook, Acaemic Press Series in Engineering,. [].T. Rim, G.B. Joung, G.H. ho: A State Space Moeling of Non-ieal D-D onverters, IEEE Power Electronics Specialists onf. Rec., 19, pp [9] R.D. Milebrook, S. uk: A General Unifie Approach to Moeling Switching onverter Stages, IEEE Power Electronics Specialists onf. Rec., 197, pp [1] P. Woo: General Theory of Switching Power onverters, IEEE Power Electronics Specialists onf. Rec., 1979, pp [11] V.F. Pires, J.F.A. Silva: Teaching Nonlinear Moeling, Simulation, an ontrol of Electronic Power onverters Using MATLAB/SIMULINK, IEEE Trans. Eucation, Vol. 5, No. 3, Aug., pp