Analysis of Fractional order PID controller for Ceramic Infrared Heater

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1 06 IJEDR Volume 4, Issue ISSN: Analysis of Fractional order PID controller for Ceramic Infrared Heater Vineet Shekher, V.S. Guta, Sumit Saroha EEE, Deartment SRM, University, NCR Camus, Ghaziabad, India Abstract - In this aer, on the basis of comutational scheme, a controller is designed to satisfy the robustness roerty with resect to gain variation and desired hase margin criteria for PID controller. In study, numerical comutation of tuning formulae and the relationshi between design secification and design arameter are both discussed by taking an examle of ceramic infrared heating system. In the design secification, the controller arameters and the lant conditions consider and a fair comarison with an otimal design integer order PID (IOPID controller has been done via simulation to show the controllers dynamic erformance, stability and robustness. IndexTerms - PID, IOPID, FOC, FOPTD, FOPID, Ceramic Infrared Heater I. INTRODUCTION In, the recent year the alications of fractional calculus have been attracting more and more researchers in the field of engineering and science [4, 7]. The orders of fractional calculus are real number. Today, many researchers has been focused on fractional was roosed [4] in as a generalization of PID controller, where exanding of the derivative and integrals to fractional orders which are adjusted to the frequency resonse of the control system directly and continuously. This aer resents a mathematical comutational tuning scheme of FOC for certain temerature system used in industry. The main contribution of this aer includes. Fractional order PID controller with and as unity is roosed (IOPID and mathematical comutation has been done on first order lus time delay system.. Mathematical comutational has been done for fractional order PID controller and imlementated on first order lus time delay system 3. According to the systematic design and simulation, a fair comarison of control erformance has been done with IOPID controller. 4. From the simulation results, it is found that FOPID controller outerforms than the IOPID controller. order controllers and obtained some useful results. The fractional order PID controller ( PI D II. PROBLEM FORMULATION The energy resources in the country are not evenly distributed with snow fed hydro resources concentrated in the north, monsoon deendant hydro in the south and coal reserves in central India. As a result, long transmission lines are constructed from the generating stations located close to the energy sources to the load centers and there is long haulage of ower. The Indian ower grids are also characterized by a well meshed network. Power flow between two areas may not only be direct but there may also be loo flows. A number of flow-gates, which are corridors comrising of a grou of trunk lines, have been identified by the system oerators for monitoring the ower flows. Design Secification of Control lant and controllers a Control Plant: Because of small delay time in large number of temerature system, a tyical first order lus time delay lant has been discussed as k G( s e Ts Ls It is an aroximately model of a large number of industrial temerature lant. For the ceramic infrared heating system, transfer function with the value of gain (k variation of 3.96 to 4., time constant of 40 sec and lag time of 7 sec has been taken by using S-shaed aroximation. The tyical FOPTD lant transfer function for ceramic infrared heater taken as [ ] G ( s e 40s 7s ( ( b Controllers: The fractional order roortional integral derivative controller (FOPID has the following form IJEDR60007 International Journal of Engineering Develoment and Research ( 43

2 06 IJEDR Volume 4, Issue ISSN: Where, (0, and (0, Ti C( s Tds s Clearly, this is a secific form of the most common differentiator of order. By considering the value of and PI D (3 controller which includes an integrator of order, the controller form become the IOPID in the following exression as i CIOPID ( s T Td s (4 s Where, i and d reresents roortional, integral and derivative gain resectively [4][6]. and a Design Consideration: By considering, the tuning method resented by Monje, Vinagre and their colleague used in the aer []. Monje and Vinagre et. al, consider the five design criteria algorithm for design secification. These design criteria obtained by getting the value of required hase margin and gain margin as m By getting the hase margin (, critical frequency m c oint on the Nyquist curve of lant at which arg[( G( ] 80 o c g and critical frequency ( as follow. a. Phase Margin and Gain Crossover Frequency m G( j c c, five design criteria of Monje Vinagre et. al methods are given The two frequency domain secifications are used to measure the robustness i.e gain margin and hase margin. The hase margin is related to the daming of the system, thus the following equation should be satisfied C( j G( j 0 and Where cg is the gain crossover frequency. b. Robustness due to variation in the gain of Plant The hase is forced to be flat at following constraint cg [ Arg( C( j G( j ] (5 cg cg cg cg m and the hase lot is almost constant within the interval around d( Arg( C( jcg G( jcg d As, er the hase lot around the secified frequency cg cg 0 (6 cg to satisfy the is locally flat, which imlies that the system will be more robust to variation of gain and ste resonse is almost constant within the interval with constant overshoots. c. High Frequency Noise Rejection The following condition must be satisfied to the robustness due to high frequency noise C( j G( j T( j A C( j G( j (7 Where A is the desired value of the noise attenuation for frequency is d. Good outut disturbance rejection rad/sec. The following constraint must be satisfied to ensure a good outut disturbance rejection. t S( j B C( j G( j (8 IJEDR60007 International Journal of Engineering Develoment and Research ( 44

3 06 IJEDR Volume 4, Issue ISSN: Where B is the desirable value of sensitivity function for which the frequency is s rad/sec. III. DESIGN ANALYSIS a. Design of IOPID controller By considering the FOPDT system for ceramic infrared heater, whose oen loo transfer function P( s C( s G ( s The frequency resonse for ceramic IR heater system as Where =3.96 to 4., T=40 sec, L=7 sec The gain and hase of the lant are as follow k G ( s e Tj G ( j k Lj ( T Arg G j T L [ ( ] tan ( Controller Design: As er the FOPID controller, the value of integrator order ( resectively then IOPID controller obtain as[0] i CIOPID ( s T Td s s and differentiator order ( In this study, a method has been roosed to obtain the roortional gain constant ( and the constant of derivative gain( d on the Nyquist curve of lant (.Let the m be the required hase margin and G ( s G s at which arg[( G( ] 80 o and define gain margin as Then, in order to make the hase margin of the system equal to must satisfied. g m c k G ( j m c c are taken as unity, the constant of integral gain ( i c be the frequency of the critical oint and C( j G ( j, the following equation c c jm C( jc e kc cosm jkc sinm G ( j According to IOPID controller transfer function(4, we can get the frequency resonse as The gain and hase of controller are as follow, The oen loo frequency resonse given as c i C( j j d (9 j C( j i d d i Arg[ C( j] tan P( j C( j G ( j The gain and hase of the oen loo frequency resonse as follows P( j ( T i d (0 ( ( IJEDR60007 International Journal of Engineering Develoment and Research ( 45

4 06 IJEDR Volume 4, Issue ISSN: [ ( ] tan d Arg P j i tan ( T L (3 According to the design secification (i and (ii, the robustness to gain variation in the lant, we can establish an equation about Where, i and d as k B A A B i T c LB c A k B c A A B d T LB A k B c A A tan tan ( ct L c m B T c From, the Nyquist curve, we set the gain margin and hase margin as follow by taking Nyquist and bode lot of system as 0.08 rad/sec, c m =60 0 By solving the equation (4, (5 and (6, we get =.85, i =0.0855, d =9.074 The IOPID controller obtained as CIOPID, i and d directly (4 (5 ( s (7 s Figure. Block diagram of feedback control system with IOPID Controller Figure. Nyquist lot of system G (s IJEDR60007 International Journal of Engineering Develoment and Research ( 46

5 06 IJEDR Volume 4, Issue ISSN: Figure 3. Bode lot of system G (s Figure 4. Ste resonse of the system for C IOPID for hase margin (60 0 and ω c=0.08 rad/sec b. Design of PI D Controller This section reresents the develoment of a tuning method of PI D arameter uncertainty structure. All arameters of the PI D Five unknown arameters of the PI D controller for first order lus time delay system with gain controller are calculated to satisfy the erformance of the lant. controller are estimated solving five non-linear equations that satisfy five design criteria []. Bode lot of FOPTD systems with gain arameter uncertainty structure are successfully combined with five design criteria controller. The hase and amlitude of the lant in frequency domain taken as, to obtain the PI D Arg[ G ( j] tan ( T L k G ( j ( T FOPID Controller design From fractional order PID controller transfer function (3, we can get its frequency resonse as follows, i C( j ( j ( j C( j ( j ( j i d ( j cos( jsin( k cos( jsin( i cos( jsin( d k cos( cos( j i sin( sin( d i d a k i cos( d cos( (8 b i sin( d sin( (9 d IJEDR60007 International Journal of Engineering Develoment and Research ( 47

6 06 IJEDR Volume 4, Issue ISSN: According to secification (a, the hase value b Arg[ C( j] tan a ( P j c According to secification (a we get the magnitude of C( j a b Arg[ P( j ] Arg[ C( j ] Arg[ G( j ] c c c m b a tan tan ( ct Lc m ( P j c as P( j G( j C( j c c c (0 k a b ( t 0 ( According to secification (b we get As d Arg( P( jc 0 d a. bu b. au T L 0 ba a T c Where au i.. c cos( d.. c cos( (3 bu i.. c sin( d.. c sin( (4 As er the secification (c we get the high frequency noise rejection as k at bt T ( jc 0 k. at j( T k. bt Where at i.. t cos( d.. t cos( (6 bt i.. t sin( d.. t sin( (7 As er the secification (d we get Good disturbance rejection as ( (5 ( T S( jc 0 (. as (. bs T (8 Where as i.. s cos( d.. s cos( (9 bs i.. s sin( d.. s sin( (30 Steady state gain k does not have any effect on hase lot of the lant. In order to design robust PI D controller should be satisfied with transfer function of FOPID, namely must be taken at the oint x. The constraint of hase margin and gain margin should be satisfied at oint y shows the minimum hase margin. c IJEDR60007 International Journal of Engineering Develoment and Research ( 48

7 06 IJEDR Volume 4, Issue ISSN: Figure 5. Bode lot of a FOPTD lant By using Equation (0,,, 5, 8 five unknown arameter,,, and i d can be solved by using FMINCON otimization toolbox of Mat Lab. Equation ( is considered as a main equation and other equations are taken as non-linear constraints for otimization. Value of the all five unknown arameters are calculated to obtain the PI D controller to control the ceramic IR heater as =0.6073, i=6.94,d=0.045, =0.785, = and transfer function of fractional order PID controller given as C( s FOPID s s Figure 6. Block diagram of feedback control system with FOPID controller Ninteger is a toolbox of Mat Lab intended to hel develoing fractional ( or non-integer order controllers for single inut- single outut lant and to access their erformance.the ste resonse of the lant with FOPID controller obtained by using nintblock of Mat Lab develoed by Valerio, D. IJEDR60007 International Journal of Engineering Develoment and Research ( 49

8 06 IJEDR Volume 4, Issue ISSN: Figure 7.Ste resonse of the system for C FOPID for hase margin (60 0 and ω c=0.08 rad/sec The ste resonse of the system shows that the system is more effective and robust to gain change and overshoot of the ste resonses is almost constant. Bode lot, Magnitude lots of T(s and S(s of the system obtained in Mat Lab. It shows that hase of the system are almost flat and almost constant within an interval around c with secified constraints. Figure 8. Magnitude of T(s for C FOPID (sg(s From the figure of bode lot, T(s and S(s, one can conclude that the controller satisfies the robust erformance of the system. IV. RESULT Figure (9: Magnitude of S(s for C FOPID (sg(s Ste resonse secification and erformance indices of C IOPID and C FOPID Controller Ste Resonse Secification IOPID Controller FOPID Controller Rise Time Peak Time 6 4 IJEDR60007 International Journal of Engineering Develoment and Research ( 50

9 06 IJEDR Volume 4, Issue ISSN: Peak Overshoot (% Settling Time (5% IAE ISE V. CONCLUSION In this aer, two methods for tuning of PI D unity ower for the integrator and derivative function of PI D controller have been roosed. The first method is based on the idea of using. By solving the equation obtained by taking consideration of the constraint, we get the value of the different three arameters otimized to achieve better ste resonse. The roosed robust tuning method for a PI D controller to control first order lus time delay with arameter uncertainty structure designed. The five design constraints for the benefit of the Monje-Vinagre et. al. method were used to derive five non- of hase extremum of bode enveloes of the lant are linear equations. Value of unknown arameters,,, and i d used to satisfy robust erformance of the system. The simulation results (rise time, eak time, settling time, eak overshoot and erformance indices(iae,ise show that the roosed method of PI D controller has better resonse than IOPID controller for ceramic IR heater. REFERENCES []. Adonis, M and han, MTE. 00. Infrared heating rofile controller. Proceedings of the 3rd International Conference on Control Theory and Alications, Dec., []. Ackland,., 998. Selecting the right infrared temerature sensor. Temerature, InTech, June. [3]. Evalds, P., 00. Power Controls Provide Reliable Industrial Electric Heating. Industrial Heating The International Journal of Thermal Technology, Vol. LXIX, No.0. [4]. Podlubny, L. Dorcak, and I. ostial, On fractional derivatives, fractional-order dynamic system and PIλDμ -controllers, Proc. of the 36th IEEE CDC, San Diego, 999. [5].. J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning, Research Triangle Park, Instrument Society of America, 995. [6]. Podlubny, Fractional-order Systems and Fractional-order Controllers, The Academy of Sciences Institute of Exerimental Physics, UEF-03-94, osice, Slovak Reublic, 994. [7]. Podlubny, Fractional-order systems and PIλDμ -controllers, IEEE Trans. Automatic Control, vol. 44,. 08-4, 999. [8]. C.A. Monje, B.M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse and J.Sabatier, Proosals for Fractional PIλDμ Tuning, The First IFAC Symosium on Fractional Differentiation and its Alications 004, Bordeaux, France, July 9-0, 004. [9]. D.Xue and C.Zhao, Fractional order PID controller design for fractional order systems [J]. Control Theory and Alications, Vol.4,No.5: , 007.(in Chinese [0]. S. E. Hamamci, An algorithm for stabilization of fractional-order time delay systems using fractional- order PID controllers, IEEE Trans. Automat. Control, Vol. 5, , 007. []. N.Tan, O.F.Ozguvenand, M.M. Ozyetkin, Robust stability analysis of fractional order interval olynomials, ISA Transactions, vol. 48,.: 66-7, 009. []. C. A. Monje, Design methods of fractional order controllers for industrial alications. Ph.D. thesis, University of Extremadura, Sain, 006. [3]. D. Valerio, Ninteger v..3. Fractional control toolbox for matlab, 005. [4]. Valério, D. (00. Non-integer order robust control: an alication. In: Student Forum, Porto, 5-8. [5]. Valério, D. and Sá da Costa, J. (00. Time domain imlementations of non-integer order controllers. In: Controlo, Aveiro, [6]. Valério, D. and Sá da Costa, J. (003a. Otimisation of non-integer order control arameters for a robotic arm. In: International Conference on Advanced Robotics, Coimbra. [7]. YangQuan Chen and Ivo Petráš and DingyuXue, Fractional order control a Tutorial, Proceedings of the 009 American Control Conference, St. Louis, MO, USA, 009. [8].. Bettou, A. Charef, F. Mesquine, A new design method for fractional PI λ Dμ controller, IJ-STA, vol.,.44-49, 008 [9]. C. Yeroglu, N. Tan, Develoment of a Toolbox for Frequency Resonse Analysis of Fractional Order Control Systems, 9th Euroean Conference on Circuit Theory and Design, Antalya, August 009. [0]. Tan, N., O zgu ven, O.F., O zyetkin, M.M.: Robust stability analysis of fractional order interval olynomials, ISA Trans., 009, 48, []. Hwang, C., Cheng, Y.C.: A numerical algorithm for stability testing of fractional delay systems, Automatica, 006, 4, []. Xue, D., Chen, Y.Q.: A comarative introduction of four fractional order controllers. Proc. Fourth World Congress, Intelligent Control and Auto, 00, vol. 4, [3]. Yeroglu, C., Onat, C., Tan, N.: A new tuning method for PI λ Dμ controller. ELECO 009 Sixth Int. Conf. Electrical and Electronics Engineering, Bursa, Turkey, 009 IJEDR60007 International Journal of Engineering Develoment and Research ( 5

10 06 IJEDR Volume 4, Issue ISSN: [4]. Tan, N., Yeroglu, C.: Note on fractional-order roortional integral differential controller design IET Control Theory Al., 0, Vol. 5, Issue. 7, [5]. Yeroglu, C., Tan, N., O zyetkin, M.M.: Frequency resonse comutation of fractional order interval transfer functions,int. J. Control Autom. Syst., 00, 8, (5, IJEDR60007 International Journal of Engineering Develoment and Research ( 5