Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong Fixed Structure Robut Loop Shaping Controller for a Buck-Boot Converter uing Genetic Algorithm Somyot aitwanidvilai, Piyapong Olranthichachat and anukid Parnichkun Abtract Robut controller deigned by H loop haping i complicated and it order i much higher than that of the plant. It i not eay to implement thi controller in practical engineering application. To overcome thi problem, we propoe an algorithm, GA baed fixed-tructure H loop haping control, to deign a robut controller. Genetic algorithm i ued to olve the H loop haping deign problem under a tructure pecified controller. The performance and robutne of the propoed controller are invetigated in a buck-boot converter in comparion with the controller deigned by conventional H loop haping method. Reult of imulation demontrate the advantage of imple tructure and robutne againt plant perturbation and diturbance of the propoed controller. Experiment are performed to verify the effectivene of the propoed technique. Index Term H loop haping, genetic algorithm, buck boot converter I. INTRODUCTION DC-DC converter have been widely ued in computer hardware and indutrial application. Controlling of thee converter i a challenging field becaue of their intrinic nature of nonlinear, time-variant ytem []. In previou reearch work, the linear model of thee converter were derived by uing linearization method [-3]. Some linear control technique were applied to thee converter baed on the linear model [, 4-5]. NAI, R., et.al.[4], applied the H control to a boot converter. Three controller; voltage mode, feed-forward and current mode control were invetigated and compared the performance. G.C. loannidi and S.N. ania [5] applied the H loop haping control cheme for a buck converter. In their paper, the µ-analyi wa ued to examine the robut feature of the deigned controller. Simone Buo [] aded the robut µ-ynthei to deign a robut voltage controller for a buck-boot converter with current mode control. The parameter variation in the converter tranfer function were decribed in term of perturbation of linear fraction tranformation (LFT) cla. In DC to DC converter, normally, the controller i deigned by uing analog circuit. Although the higher control anucript received January 3, 008. Thi work wa upported in part by Faculty of Engineering, ing ongkut' Intitute of Technology Ladkrabang, Bangkok, Thailand. Piyapong and Somyot are with the Department of Electrical Engineering, Faculty of Engineering, ing ongkut' Intitute of Technology Ladkrabang, Bangkok 050, Thailand. Email : kkomyot@kmitl.ac.th anukid i with the School of Engineering and Technology, Aian Intitute of Technology, P.O. Box 4, long Luang, Pathumthani 0, Thailand.. technique mentioned above are powerful technique for deigning the high performance and robut controller; however, the tructure of thee controller i complicated with a high order. It i not eay to implement thee controller in the converter. Neverthele, the deign of analog circuit for thee controller i not feaible. To overcome thi problem, fixed-tructure controller i invetigated. Fixed-tructure robut controller have become an intereting area of reearch becaue of their imple tructure and acceptable controller order. However, the deign of thi controller by uing the analytical method remain difficult. To implify thi problem, the earching algorithm uch a genetic algorithm, particle warm imization technique, gradient method, etc., can be employed. Several approache to deign a robut control for tructure pecified controller were propoed in [6-8]. In [6], a robut H imal control problem with tructure pecified controller wa olved by uing genetic algorithm (GA). A concluded in [6], genetic algorithm i a imple and efficient tool to deign a tructure pecified H imal controller. Bor-Sen.Chen. et. al.[7], propoed a PID deign algorithm for mixed H /H control. In their paper, PID controller parameter were tuned in the tability domain to achieve mixed H /H imal control. A imilar work wa propoed in [8] by uing the intelligent genetic algorithm to olve the mixed H /H imal control problem. The technique in [6-8] are baed on the concept of H imal control which two appropriate weight for both the uncertainty of the model and the performance are eentially choen. A difficulty with the H imal control approach i that the appropriate election of cloe-loop objective and weight i not traightforward. In robut control, H loop haping which i a imple and efficient technique for deigning a robut controller can be alternatively ued to deign the robut controller for the ytem. Uncertaintie in thi approach are modeled a normalized co-prime factor; thi uncertainty model doe not repreent actual phyical uncertainty, which uually i unknown in real problem. Thi technique require only two pecified weight, pre-compenator and pot-compenator, for haping the nominal plant o that the deired open loop hape i achieved. Fortunately, the election of uch weight i baed on the concept of claical loop haping which i a well known technique in controller deign. By the reaon mentioned above, thi technique i impler and more intuitive than other robut control technique. However, the controller deigned by H loop haping i till complicated and ha high order. To overcome thi problem, in thi paper, we propoe a fixed-tructure H loop haping control to deign a robut controller for a buck boot converter. In the propoed technique, the controller tructure i firtly pecified and the genetic algorithm i then ISBN: 978-988-70--3 IECS 008
Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong ued to evaluate the control parameter. Simulation and experimental reult how the advantage of imple tructure, lower order and robutne of the propoed controller. The remainder of thi paper i organized a follow. Converter dynamic are decribed in ection II. H loop haping and the propoed technique are dicued in ection III. Section IV demontrate the deign example and reult. Finally, ection V conclude the paper with ome final remark. and denominator factor. Fig. how the coprime perturbed plant and robut tabilization ued in thi approach. - N N - II. CONVERTER ODELING A typical circuit of buck-boot converter with current mode control i hown in Fig.. The dynamic model of thi converter from the current reference (i r ) to output voltage (u 0 ) i given by [-3] L Vo Vo Vi du (- ) o Vi R L Vi Vo = R Vo V () L dir Vi Vo i ( C R L ) V V Where R L i the nominal load reitant, V o i the nominal output voltage, V i i the nominal input voltage, L i the inductance of an inductor ued in the circuit, C i the capacitance, f w i the witching frequency. Fig.. Buck boot converter with current mode control. The accuracy of thi model ha been proved to be accepted, at leat in frequency of interet in thi application [-3]. III. H LOOP SHAPING CONTROL AND PROPOSED TECHNIQUE Thi ection illutrate the concept of the tandard H loop haping control and the propoed technique. A. Standard H Loop Shaping H loop haping control [9] i an efficient method to deign a robut controller. Thi approach require only a deired open loop hape in frequency domain. Two weighting function, W (pre-compenator) and W (pot-compenator), are pecified to hape the original plant G o. In thi approach, the haped plant i formulated a normalized coprime factor, which eparate the plant G into normalized nominator N o i Fig.. Co-prime factor robut tabilization problem. If the haped plant G = W G W = N, then a perturbed plant i written a [9] ( o )( G = N N () Where N and are table, unknown repreenting the uncertainty atifying N, ε, ε i the uncertainty boundary, called tability margin. According to the tandard procedure of H loop haping, the following tep can be applied to deign the H loop haping controller. Step Shape the ingular value of the nominal plant G o by uing a pre-compenator W and/or a pot-compenator W to get the deired loop hape. W can be choen a an identity matrix, ince we can neglect the enor noie effect when the ue of good enor i aumed [0]. Weight election i very important for the deign. Typically, weight W and W are elected uch that the open loop of the haped plant ha the following conflict propertie: To achieve a good performance tracking, good diturbance rejection, large open loop gain (normally at low frequency range) i required. To achieve a good robut tability and enor noie rejection, mall open loop gain (normally at high frequency range) i required. There are ome guideline for the weight election in [0]. In SISO ytem, the weighting function W and W can be choen a W = Where W a b and W = (3), a and b are poitive value w Step inimize -norm of the tranfer matrix T zw over all tabilizing controller, to obtain an imal cot γ, a [0] I γ = ε = inf ( I G) (4) tab ε << indicate that W or W deigned in tep are incompatible with robut tability requirement. If ε i not atified (ε << ), then return to tep, adjut W. Step 3 Select ε < ε and then yntheize a controller that atifie ) ISBN: 978-988-70--3 IECS 008
Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong I T zw = ( ) I G ε (5) Controller i obtained by olving the imal control problem. See [] for more detail. Step 4 Final controller () follow = W W (6) B. Genetic Algorithm baed Fixed-Structure H Loop Shaping Optimization The controller, which i derived from H loop haping method, i complicated and high-order. It i difficult to apply thi controller in real work. Nowaday, the fixed-tructure robut controller become an intereting reearch area becaue of their advantage in imple tructure and acceptable controller order. In thi paper, the genetic earching algorithm i aded to olve thi problem. Although the propoed controller i tructured, it till retain the entire robutne and performance guarantee a long a a atifactory uncertainty boundary ε i achieved. The propoed algorithm i explained a following. Aume that the predefined tructure controller p) ha a atified parameter p. Baed on the concept of H infinity loop haping, imization goal i to find parameter p in controller p) that minimize infinity norm T. From (6), the controller p) can be written a p) = W W (7) Aume that W and W are invertible, then = W p) W (8) the weight W = I which implie that enor noie i negligible and not conidered [0]. Thu, = W - p) (9) By Subtitution of (9) into (5), then the -norm of the tranfer function matrix from diturbance to tate, T zw, which i ubjected to be minimized can be written a I J co t = γ = Tzw = ( I GW p)) W p) (0) In thi paper, GA i aded to find the imal control parameter p* in the tabilizing controller p) uch that the T zw i minimized. The imization problem can be written a inimize Subject to Where ( I G W p)) W p) I p < p < p i, min i i,max pi, min and pi, max value of the parameter zw () are the lower and upper bound pi in controller p), repectively. Genetic Algorithm Our propoed technique ue GA to olve the imization problem in (). GA i well known a a biologically inpired cla of algorithm that can be applied to any nonlinear imization problem. Thi algorithm applie the concept of chromoome, and the operation of croover, mutation and reproduction. At each tep, called generation, fitne value of all chromoome in population are calculated. Chromoome, which ha the maximum fitne value (minimum cot value), i kept a a olution in the current generation and paed to the next generation. The new population of the next generation i obtained by performing the genetic operator uch a croover, mutation, and reproduction. Croover randomly elect a ite along the length of two chromoome, and then plit the two chromoome into two piece by breaking them at the croover ite. The new chromoome are then formed by matching the headpiece of one chromoome with the tailpiece of the other. utation operation form a new chromoome by randomly changing value of a ingle bit in the chromoome. Reproduction operation form a new chromoome by jut copying the old chromoome. Chromoome election in genetic algorithm depend on the fitne value. High fitne value mean high chance to be elected. Operation type election; mutation, reproduction, or croover, depend on the pre-pecified operation probability. Chromoome in genetic population i coded a binary number. However, for the real number problem, decoding binary number to floating number i applied [].. Our propoed algorithm i ummarized a Step Shape the ingular value of the nominal plant G o by W and W. Then evaluate the ε uing (4). If ε < 0.5, then go to tep to adjut the weight W. Step Select a controller tructure p) and initialize everal et of parameter p a population in the t generation. Define the genetic parameter uch a initial population ize, croover and mutation probability, maximum generation, etc. Step 3 Evaluate the cot function J cot of each chromoome uing (0). Aign J cot = 00, or large number if p) doe not meet the contraint in our imization problem. The fitne value i aigned a. Select the chromoome with J cot minimum cot function a a olution in the current generation. For the firt generation, Gen =. Step 4 Increment the generation for a tep. Step 5 While the current generation i le than the maximum generation, create a new population uing genetic operator and go to tep 3. If the current generation i the maximum generation, then top. Step 6 Check performance in both frequency and time domain. If the performance i not atified, uch a too low ε (too low fitne function), then go to tep 3 to change the control tructure. Low ε indicate that the elected control tructure i not uitable for the problem. ISBN: 978-988-70--3 IECS 008
Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong Start Specify weighting function W and W, evaluate ε I ε atified? Initialize population of parameter (p), probability of genetic operation, max Gen, etc. Gen = ; Ye Evaluate fitne function (J cot) -, calculate the fitne value of each chromoome. I Gen = maxgen? No No Select the chromoome with maximum fitne value a the olution of current generation. Ye Gen = Gen ; Sampling new population by genetic operation Stop W i choen a ince we neglect the enor noie effect when the ue of good enor i aumed. Fig. 4 (a) how the plot of open loop hape of nominal plant and haped plant. A een in thi figure, the bandwidth of the nominal plant i about 600 rad/ec. With thee weighting function, bandwidth of the deired control ytem i increaed to 5,000 rad/ec. Significant performance and robutne improvement are carried out by thee weighting function. The haped plant i written a ( 30)(-0.004 605.6) G = L = WGW = 5 (4) ( 0)(0.9963 7) By applying the H loop haping method, the imal tability margin (ε ) i founded at 0.708 ( γ =. 43 ). Thi value indicate that the elected weighting function i compatible with the robut tability requirement. The ε = 0.663 (γ =.53), which i le than the imal tability margin, i choen to ynthei the controller. Baed on the conventional technique i ection II, the conventional H loop haping controller i yntheized a following ) = W W 6 8 9 4.599x0.763x0 4.5x0 = 3 5 6 7.845x0 7.386x0 5.54x0 (5) Check the performance. If it i not atified uch a too low fitne, then elect a new control tructure p) and then go to tep 4. Fig. 3. Flow chart of the propoed deign procedure. IV. SIULATION AND EXPERIENTAL RESULTS In thi paper, a buck-boot converter deigned for a photovoltaic ytem i tudied. Converter parameter and conidered variation range ued in thi paper are given in Table. Table Converter parameter and conidered variation range. Parameter Name Nominal Value R L Load Reitant 40 Ω V o Output Voltage 30 V V i Input Voltage V L Inductance 00 µh C Capacitor 470 µf f w Switching frequency 00 khz By (), the nominal tranfer function i found to be (-0.004 480) G = () o (0.7896 7) Both H loop haping control and our propoed technique are applied to thi converter. Firtly, we deign a controller by the conventional H loop haping procedure. In thi cae, W i elected a ( 30) W = 5 ( 0) (3) A hown in (5), the controller i 3 th order controller and complicated tructure. Next, PI controller i invetigated a a fixed-tructure controller. The controller tructure i expreed in (6). p and i are parameter that will be evaluated. i p) = p (6) Select the controller parameter, their range, and genetic algorithm parameter a following: p [0,00], i [0,000], population ize = 00, croover probability =0.7, mutation probability =0.5, and maximum generation = 30. An imal olution i obtained after 8 generation. The imal olution i hown in (7), which ha tability margin (ε) of 0.6598(γ =.57). ( p) * 989.7 =.88 (7) Fig. 5 how plot of convergence of cot function J cot veru generation by genetic algorithm. A een in thi figure, the imal fixed-tructure controller provide the atified tability margin at 0.6598(γ =.57). ISBN: 978-988-70--3 IECS 008
Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong (a) (b) Fig. 5 Cot function J cot veru iteration in genetic algorithm. The open loop bode diagram of the nominal and haped plant are hown in Fig. 4(a). A hown in thi figure, at low frequency, the open loop gain of haped plant i much larger than that of the nominal plant. Thi make the deigned ytem ha good performance tracking and good diturbance rejection. Open loop bode diagram are plotted in Fig. 4(b) to verify the propoed algorithm. It i clearly hown that the loop hape of H infinity control and PI are cloe to the deired loop hape. Fig. 4(c) how the tep repone of the imal olution from the propoed robut PI and the conventional H controller. A hown in thi figure, the ettling time of all repone i about 350 µec. To verify the robut performance, we change the converter parameter a: R L = 0 Ω, V i = 0.8 V, L=0 µh and C=6 µf. The deigned controller in (5) and (7) i aded to control thi perturbed plant. Obviouly, thi condition (increae the L and C and decreae the load and input voltage) i wore than the nominal condition. In thi cae, for imulation, the plant i changed to (-0.004896 9.6) G = (8) (0.938 70.8) Fig. 6 how the tep repone of all controller in the perturbed plant. The repone are almot the ame a the repone in the nominal plant with ome different in the etting time. The reult how that the deigned ytem from the propoed controller and H infinity loop haping ha a good performance and robutne. (c) Fig. 4. (a) Bode plot of the nominal plant and the haped plant (deired loop hape, L) (b) The deired loop hape and the loop hape by the conventional H loop haping and the propoed PI, (c) Step repone by the propoed PI and H loop haping controller. Fig. 6 Step repone in the perturbed plant.(r L = 0 Ω, V i = 0.8 V, L=0 µh and C=6 µf). ISBN: 978-988-70--3 IECS 008
Proceeding of the International ulticonference of Engineer and Computer Scientit 008 Vol II IECS 008, 9- arch, 008, Hong ong Some experiment are performed to verify the effectivene of the propoed controller. The nominal value in Table are ued to deign a buck boot converter with current mode control. A propoed controller, robut PI controller in (7) i ued to control the converter. Fig. 7 how the experimental reult of tep repone of the propoed controller. The ettling time of the repone i about 350 µec. A een in Fig. 4(c) and Fig. 7, the repone of experimental reult i almot the ame a that of the imulation reult. V. CONCLUSION Both of H loop haping and the propoed technique can be applied to deign a robut controller for a buck boot converter. However, the propoed approach ignificantly improve in practical control viewpoint by implifying the controller tructure, reducing the controller order and retaining the robut performance. Although the propoed controller i tructured, it till retain the entire robutne and performance guarantee a long a a atifactory uncertainty boundary ε i achieved. Structure of controller in the propoed technique i electable. Thi i deirable, epecially in the DC-DC converter which analog circuit i normally ued to deign the controller. In concluion, by combining of the approache, genetic algorithm and H loop haping; fixed-tructure controller deign can be deigned. Implementation in buck-boot converter aure that the propoed technique i valid and flexible. ACNOWLEDGENT Thi reearch work i financially upported by the Thailand Reearch Fund (Project. No. RG4980087) and the reearch fund from the faculty of engineering, ing ongkut' Intitute of Technology Ladkrabang. Fig. 7 Step repone in the cloed loop in nominal condition for propoed PI controller. To verify the robut performance of the ytem, an experiment i performed. The component value and operating point of converter are changed to: R L = 0 Ω, V i = 0.8 V, L=0 µh and C=6 µf. The controller from the previou experiment i ued to control thi perturbed plant. The performance i verified by uing the tep repone. A hown in Fig. 8, the tep repone i almot the ame a the repone in nominal condition. Thi repone i over damp repone with a mall different in the ettling time. Experimental reult verify that the propoed controller can be applied for the buck-boot converter to achieve a good robut performance. REFERENCES [] Simone Buo, Deign of a Robut Voltage Controller for a Buck-Boot Converter Uing Synthei, IEEE Tran. On Control Sytem Technology, Vol. 7, No., pp. -9, arch 999. [] R. B. Ridley, A new continuou-time model for current-mode control, in Power Converion Intell. otion (PCI) Conf. Proc., 989, pp.455 464. [3] J. G. aakian,. F. Schlecht, and G. C. Verghee, Principle of Power Electronic. Reading, A: Addion-Weley, 99. [4] NAI, R., WEISS, G., and BEN-YAAOV, S., H control of boot converter: comparion to voltage mode, feed-forward and current mode control. PESC 95, Atlanta, USA, pp. 37-33, June 995. [5] G.C. loannidi, S.N.ania, H loop-haping control cheme for the buck converter and their evaluation uing µ-analyi, IEE Proc.-Electr. Power AppL, Vol. 46. No., arch 999. [6] Bor-Sen Chen and Yu-in Cheng, A Structure-Specified imal Control Deign for Practical Application: A Genetic Approach, IEEE Tran. on Control Sytem Technology, Vol. 6, No. 6, November 998. [7] Bor-Sen Chen, Yu-in Cheng and Ching-Hiang Lee, A Genetic Approach to ixed H / H Optimal PID Control, IEEE Tran. on Control Sytem, p. 5-60, 995. [8] Shinn-Jang Ho, Shinn-Ying Ho, ing-hao Hung, Li-Sun Shu, and Hui-Ling Huang. Deigning Structure-Specified ixed H / H Optimal Controller Uing an Intelligent Genetic Algorithm IGA. IEEE Tran. on Control Sytem 005; 3(6):9-4. [9] cfarlane, D.C. &. Glover, A loop haping deign procedure uing H ynthei, IEEE Tran. On Automatic Control AC-37 (6):759 769, 99. [0] emin Zhou, Jhon C. Doyle, 998. Eential of Robut Control. Prentice-Hall, pp 35-37. [] Siguard Skogetad, Ian Potlethwaite, ultivariable Feedback Control Analyi and Deign. John Wiley & Son, pp.8,376-380, 996. [] Chri Houck, Jeff Joine, and ike ay, "A Genetic Algorithm for Function Optimization: A ATLAB Implementation" by, NCSU-IE TR 95-09, 995. Fig. 8 Step repone in the cloed loop in perturbed condition for the propoed PI controller. ISBN: 978-988-70--3 IECS 008