Parameter Setting Method of Fractional Order PI λ D μ Controller Based on Bode s Ideal Transfer Function and its Application in Buck Converter

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1 Parameter Setting Method of Fractional Order PI λ D μ Controller Baed on Bode Ideal Tranfer Function and it Application in Buck Converter Yiwen He, a, Zhen Li,b School of Mechanical and Automotive Engineering, Nanyang Intitute of Technology, Nanyang, Henan 4736, China a hyw_66@ina.com.cn, b lizhenhexu@63.com Keyword: Bode' ideal tranfer function, PI λ D μ controller, parameter etting, Buck converter Abtract. The Bode ideal tranfer function i applied to the parameter etting of fractional order PI λ D μ controller in thi paper. The cloed loop ytem compoed of Bode ideal tranfer i ued to be a reference model in thi optimized ytem, where the controller parameter are earched by genetic algorithm and the imulation oftware. The optimized ytem i imilar to the reference model a much a poible and poee it excellent propertie. The validity of the method i proved by imulation reult. In addition, the above method i applied to the BUCK converter with contant power load. The imulation reult how that the optimal controller i uperior to the effect of the integer order PID controller. Introduction The order of calculu i promoted to fraction even to complex field by fractional calculu. But becaue of it computation complexity and lacking of definite phyical meaning, the theoretical reearch ha jut been applied to mathematic. With the rapid development of computer cience, the application of fractional calculu ha become poible[,2]. The theory of fractional order control i the ummary and upplement of the traditional integer order control theory, which would be more imulate and robut. At preent, everal important fractional order controller, uch a PI λ D μ controller[3], CRONE controller[4] and TID controller[5] have been promoted. Among them, the PI λ D μ controller ha greatet influence and ha been widely ued. Becaue the fractional order PI λ D μ controller ha two adjutable parameter more than the integer order PID controller, it could adjut the five parameter more flexibly and reaonably to achieve better reult. But a a reult of the λ, u, Kp, KI, KD parameter need to deign, it make the deign harder. Therefore, parameter etting and optimization of fractional PI λ D μ controller i one of the focu on the reearch topic. The Bode ideal tranfer function i applied to the parameter etting of fractional order PI λ D μ controller in thi paper. It can not only atify the ytem requirement of the cut-off frequency and phae margin, but alo make the phae frequency characteritic curve of Bode diagram of compenated ytem ha a level area near the cut-off frequency. That i to ay, the cloed-loop ytem i robut to the change of gain. In addition, thi method i applied to control of the Buck converter and the good control effect ha been achieved. Fractional Order Calculu and Fractional Order PI λ D μ Controller n n Fractional Order Calculu. Generally, the d y dx i n order differentiation of y for x. But what i the meaning if n i equal to the /2? Thi problem ha been put forward by the French mathematician Guillaume more than 3 year ago. Although the fractional order calculu theory had been raied more than 3 year ago, it mainly focued on theoretical reearch in the early day. However, with the development of the computer in recent year, the application field of fractional order calculu i more and more wide. For example, fractional order control ha emerged in the field of automatic control theory. In term of modeling, ome ytem are difficult to accurately decribe by uing an 26. The author Publihed by Atlanti Pre 293

2 integer order differential equation. Introducing fractional differential operator, the fractional order differential equation can be ued to decribe the ytem, and it would be more cloe to the nature of the ytem characteritic. In addition, becaue of the characteritic of fractional order calculu operator, fractional order controller alo ha a lot of uperiority that can't be realized by integer order controller [6]. In the proce of the development of fractional order calculu, there have been many definition. The Riemann Liouville calculu i mot widely ued among them. The definition i a follow: [( ta)/ h] j adt f() t lim ( ) f( t jh) h h j j. () According to the definition, it ha an aociation with all the value from the initial to that point, which i different from the limit of integer order differential. A a reult, it ha memory that make the information of the pat ha an influence on the preent and the future[7]. Fractional Order PI λ D μ Controller. With the development of fractional order calculu, TID, CRONE and PI λ D μ controller appeared. Fractional order differential equation of PI λ D μ controller i a follow: u() t KPe() t KID e() t KDD e() t. (2) The tranfer function i: KI G () K K,. (3) C P D The appropriate value u and λ of PI λ D μ controller can be elected according to the order, characteritic and deign requirement of the control ytem. The dynamic and tatic characteritic of ytem would be improved to get better control effect. Parameter Setting of Fractional Order PI λ D μ Controller Baed on Bode Ideal Tranfer Function Bode Ideal Tranfer Function. An ideal form of a kind of open loop tranfer function wa put forward by Bode in the deign of feedback amplifier: c G (), R. (4) Where ωc i the cut-off frequency of the ytem, that i to ay G j c. Alpha can be fraction or integer. Subtantially, G () i a fractional order differentiator when alpha i le than zero and a fractional integrator when alpha i greater than zero. The Bode diagram of Bode ideal tranfer function i very imple when <α< 2. The amplitude-frequency characteritic curve i a traight line that the lope i -2αdB/dec, and the phae frequency characteritic curve i a level traight line with the value of -απ/2 [7]. Figure i the Bode diagram of ideal tranfer function when ωc=,α=.2. Bode Diagram 5 Magnitude (db) -5 Phae (deg) R() + _ G( ) c Y() Frequency (rad/ec) Fig. Bode plot of Bode ideal tranfer function Fig.2Fractional-order control ytem with ideal tranfer function Figure 2 i the block diagram of unit feedback ytem including the ideal tranfer function. The ideal tranfer function make the cloed-loop ytem ha the expected characteritic that i not enitive to gain change. The cut-off frequency ωc will change when the gain change, while the 26. The author Publihed by Atlanti Pre 294

3 phae margin remain the ame (φm=π-απ/2).thi mean that the ytem ha trong robutne to gain change. The overhoot of the ytem i only related with the alpha and ha nothing to do with gain. The exceive overhoot caued by gain change can be avoided. Becaue of the enor or environment change, the variation of the gain i difficult to be avoided in the proce of actual control. Therefore, the robut feature i one of expected feature of control ytem. Given the ideal tranfer function ha excellent propertie, it can be conidered a a goal to deign the controller. The controller i deigned to make the open loop tranfer function of the compenated ytem ame a the ideal tranfer function. A the ideal tranfer function i fractional order, thi viion can't be achieved uing an integer order controller. The development of the theory of fractional order calculu and fractional order control make it poible to thi idea. Parameter Setting of Fractional Order PI λ D μ Controller Baed on Bode Ideal Tranfer Function. In recent year, Bode ideal tranfer function ha gotten wide attention of the reearcher. The ideal tranfer function performed a a reference model for integer order control in literature [8]. It i ued for parameter etting of integer order controller in literature [7]. The fractional order PID controller ha two degree of freedom, order of integral and differential item, more than the conventional one. A a reult it ha a greater adjutment range, which can make the control ytem a better approximation of fractional order reference mode. Therefore, the fractional order controller i optimized uing cloed loop ytem contituted by Bode ideal tranfer function a a reference model in thi article. The repone of optimized ytem i made cloe to reference model a much a poible and poee it excellent propertie. The phae frequency repone curve of Bode diagram of the control ytem i not exactly the ame a that of the ideal tranfer function, but you can make it there i a "flat" area near the cut-off frequency. The cloed-loop ytem i robut to the change of gain. The principle of parameter etting of fractional order controller baed on the ideal tranfer function i hown in figure 3. The control object can be either integer order or fractional order. Setting tep are a follow: firt of all, Bode ideal tranfer function i calculated according to the ytem requirement in the time domain or frequency domain. Secondly, the cloed-loop tranfer function of the ideal ytem i ued a a reference model. The ITAE or ISE of y (t) - y r (t), that i to ay, the error of the ytem output and the reference model output, i ued a optimization indexe. Finally, the mot optimal fractional order controller i earched by the genetic algorithm baed on Simulink block diagram. r(t) + _ e(t) 分数阶 PID 控制器 控制对象 r(t) + _ e(t) 分数阶 PID 控制器 控制对象 + _ G () c yr () t 优化过程 优化过程 Fig.3 Sytem tructure for PI λ D μ controller tuning Fig.4 Sytem tructure for traditional PI λ D μ controller tuning The optimal deign method [9, ] baed on ITAE of ytem error for fractional order PID controller i an optimization deign method promoted from common integer order PID controller. To illutrate the difference and contact of the ideal tranfer function method and the optimal deign method baed on ITAE of ytem error and to facilitate imulation and compare, the brief introduction i given a follow. The principle block diagram i hown in figure 4. Except that it optimization indexe are different from the ideal tranfer function method, the optimization algorithm, the algorithm of fractional order module and imulation environment are all ame a ideal tranfer function method. A third-order object i ued a an example to tetify the above parameter etting method a follow. The tranfer function of the object i: 26. The author Publihed by Atlanti Pre 295

4 GP (). (5) 3 ( ) Selecting phae margin φm=π/4 and cutoff frequency ωc=, the correponding ideal tranfer function - 3/2 i get. The fractional order PID controller adjuted by the ideal tranfer function method i:.7.3 Gc ( ) (6) The integer order PID controller adjuted by the ideal tranfer function method i(order of calculu λ and μ are fixed of in the tuning proce ): Gc2 ( ) (7) The controller applying to equation (5) obtained according to the method illutrated in figure 4 i:.9 Gc3 ( ) (8) The Bode diagram of the ytem that the object of equation (5) are controlled by GC, GC2 and GC3 controller repectively are compared in Figure 5. It can be een from the diagram, there i a "flat" area near the cut-off frequency in the phae frequency characteritic curve of GC and GC2 control ytem adjuted uing the ideal tranfer function. Thi mean that the ytem ha robutne to the gain change. Thi area of fractional order controller i broader than that of integer order, o the robutne will be tronger. Although the phae margin of GC3 control ytem i greater than GC and GC2 control ytem (the deign value of phae margin of GC, GC2 ytem i π/4, but the phae margin of GC3 ytem i obtained from another optimization method), it doe not have the "flat" area. 5 Magnitude (db) Phae (deg) Integral PID control ytem(bode' ideal tranfer function method...fractional PID control ytem(bode' ideal tranfer function method -.-.Fractionl PID control ytem(traditional method) Fig5.Bode plot of G C,G C2 and G C3 control ytem In order to verify the robutne of GC control ytem for gain change, the gain of the object i et to.5, and.5. The tep repone curve are compared in Figure 6. It can be een from the diagram, the repone peed i affected when the gain i changed, while the change of overhoot i very mall. Figure 7 how the tep repone of GC3 control ytem for gain change. It i obviou that the ytem i not robut to the change of gain gain.5...gain -.-.gian t/ Fig6.Unit tep repone of G C control ytem for deferent gain gain gain gain t/ Fig7.Unit tep repone of G C3 control ytem for deferent gain 26. The author Publihed by Atlanti Pre 296

5 Application in Buck Converter of the Setting Method In order to verify the control effect of obtained fractional PI λ D μ controller to drive the BUCK converter driving contant power load, the imulation ha been carried on by MATLAB oftware. The principle diagram of BUCK converter driving contant power lord i hown in figure 8. Parameter of main circuit are R = 5 Ω, L = 5mH, C = 5μF and vin = 2V. The output voltage v and power P are V and W repectively. Fig8. Principle diagram of BUCK converter Uing the above method and genetic algorithm to optimize the parameter of fractional order controller, the obtained PI λ D μ controller i 277. Gcf ()... (9).8 The traditional PID controller obtained by Z - N etting method i [] Gc ( ) () The curve of tep repone of the fractional order and the traditional integer order control are hown in figure 9. Obviouly, fractional PI λ D μ controller i ignificantly better than the integer order PID controller in repone peed, adjut time and overhoot, etc. 4 tep repone tranient repone of load change 2 8 Z-N method fractional order PID control Z-N method Vo 6 4 fractional order PID control Vo t Fig9. Step repone t x -3 Fig. Repone curve while load change Figure i the repone curve of fractional order PI λ D μ and conventional PID control trategy for load reitance jump from 5Ω to 2.5 Ω. It i hown that fractional PI λ D μ controller ha tronger robutne when the ytem load i changed. Concluion The fractional order controller ha le enitivity and more robutne for the change of parameter of the controlled ytem. The tatic and dynamic characteritic of the control ytem would be dramatically improved. The Bode ideal tranfer function i applied to parameter etting of fractional order PID controller in thi article. The example prove the method can not only obtain atifactory dynamic and tatic characteritic, alo can improve the robutne of ytem to gain change. Acknowledgement Thi work wa financially upported by the Key Scientific Reearch Project of College and Univeritie in Department of Education of Henan Province (4A22). 26. The author Publihed by Atlanti Pre 297

6 Reference []Barboa R, Duarte F and Tenreiro Machado J. A Fractional Calculu Perpective of Mechanical Sytem Modeling. In Control 22 5th Portuguee Conference on Automatic Control, September 22, [2]Xub, Tuyq, Liu L B. Journal of Electronic Meaurement and Intrument, 28, 22(5): [3]Podlubny I. IEEE Tran Automatic Control, 999, 44 (): [4] Youfi N, Melchior P,et al. Nonlinear Dynamic,24,76(): [5]Lurie J. Three-Parameter Tunable Tilt-Integral-Derivative (TID) Controller. US Patent US53767, 994. [6]Luo Y, Zhang T, Lee B, et al. IEEE Tranaction on Control Sytem Technology, 24, 22(): [7]Barboa,R..S.,Tenreiro Machado, J.A.,Ferreira,I.M. Nonlinear Dyn. 38(/4), 35-32(24) [8]Vinagre,B.M, Monje, C.A., Calderon, A.J., et al. The Fractional Integrator a a Reference Function. In Firt IFAC Workhop on Fractional Differentiation and it Application (pp.5-55) Bordeaux, France:ENSEIRB(24). [9]Dingyu Xue,Chunna Zhao, Yangquan Chen. Fractional Order PID Control of a DC-Motor with Elatic Shaft : A Cae Study. Proc of the 26 American Control Conf. Minneapoli, 26 : []Dingyu Xue. Computer Aided Deign of Control Sytem. Verion 2. Beijing: Tinghua Univerity Pre, 26. ( In Chinee) []Shitao Ren, Liliang Liu, Yunxia Shen. Electrical Switch,2,6:9-2.( In Chinee) 26. The author Publihed by Atlanti Pre 298