Modelling and Simulation Study on Fractional and Integer Order PI Controller for a SISO Process

Transcription

1 International Journal of Science, Engineering and Technology Reearch (IJSETR), Volume 3, Iue 3, March 2014 Modelling and Simulation Study on Fractional and Integer Order PI Controller for a SISO Proce S.Diya and P.nbumalar btract In recent year it i remarkable to ee increaing number of tudie related to the theory and application of fractional order controller, pecially PI λ and PI λ D μ controller in many area of cience and engineering. Reearch actiitie are focued on deeloping new method and analyi for fractional order controller a an extenion to claical control theory. Thi paper deal with the deignof fractional orderpi λ controller for a SISO (two interacting tank leel) proce. Integer order Proportional integral (IO-PI) controller and fractional order proportional integral (FO-PI) controller ha been deigned for integer order ytem(ios) by approximated M contrained integral gain optimization (- MIGO) and fractional M contrained integral gain optimization (F-MIGO) techniquerule repectiely. Further the performance of cloed loop repone of IOS with fractional and integer order PI controller ha been analyzed baed on integral quared error ()criterion. From the imulation tudy it ha been inferred that the deigned FO-PI controller how better performance than the IO-PI controller in term of et point tracking, robutne, diturbance rejection and tability analyi. Keyword Fractional order controller, integer order controller and two interacting tank leel ytem I. INTRODUCTION The PID controller i by far the mot dominating form of feedback in ue today. Due to it functional implicity and performance robutne, the proportional-integral-deriatie controller ha been widely ued in the proce indutrie. Deign and tuning of PID controller hae been a large reearch area eer ince Ziegler and Nichol preented their method in Specification, tability, deign, application and performance of the PID controller hae been widely treated ince then. On the other handin recent year, fractional order dynamic ytem and controller baed on fractional order calculu hae gained an increaing attention in control community [1]. Thi i mainly due to the fact that many real phyical ytem are well characterized by fractional order differential equation inoling non integer order deriatie. S.Diya, Department of Electronic and Intrumentation Engineering, SRM Unierity, Kattankulathur, Tamil Nadu, India,Mobile No P.nbumalar, Department of Electronic and Intrumentation Engineering B.S.bdur Rahman Unierity, Vandalur, Chennai-48, Tamil Nadu, India, Mobile No The concept of fractional order PI λ D μ controller which ha an integrator of real order λ and differentiator of real order μ i propoed by Podlubny [2]. The fractional order controller tructure i gien by C() =K P + K i α + K d μ wherek P i the proportional contant, K i i the integration contant and K d i the differentiation contant. The PI λ D μ algorithm i repreented by a fractional integro-differential equation of type a follow c t = K P e t + K i D λ e t + K d D μ e(t) where D i the integro-differential operator[3], e(t) i the controller input and c(t) i the controller output. Clearly, depending on the alue of the order λ and μ, the numerou choice for the controller' type can be made. For intance, taking λ=1 and μ=1 yield the claical PID controller. Moreoer, the election of λ=1 and μ=0 lead to the PI controller, λ=0 and μ=1 gie the PD controller, and alo λ=0 and μ=0 reult in P controller. The paper i organized a follow: Section II gie a brief decription of the proce modelling. In ection III, integer and fractional order PI controller i deigned for IOS uing F-MIGO and -MIGO tuning technique. The comparatie imulation reult for the control performance for fractional and integer order PI controller baed on criterionare preented in ection IV. Finally, reult and concluion were dicued in ection V.. Proce decription II. PROCESS MODELING The two interacting tank proce a hown in Fig 1, conit of two identical cylindrical tank with equal croectional area (). Thee two tank are connected by cylindrical pipe of uniform cro ectional area (α). From the reeroir the proce liquid i pumped into the firt tank through a control ale. The input flow to tank 1 i q i. The two tank are interconnected through manual ale. The objectie i tocontrol the leel of liquid in tank 2 by arying the inflow rate (q i ) in the tank 1. ll Right Reered 2014 IJSETR 369

2 International Journal of Science, Engineering and Technology Reearch (IJSETR), Volume 3, Iue 3, March 2014 Whereq o12 and q o2 are the inflow and outflow of tank 2, which i gien by q 012 = β 12 a 12 2g( 1 t 2 t ) (6) q 02 = β 2 a 2 2g 2 (t) (7) Subtituting (6) and (7) in equation (5) gie d 2 (t) dt = β 12 a 12 2g( 1 t 2 (t)) β 2a 2 2g 2 t (8) Linearized integer order model at different operating point are lited in table 1. TBLE: 1 INTEGER ORDER MODEL Fig.1 Two interacting tank leel ytem Where, h 1, h 2 = height of fluid in tank 1 and tank 2 repectiely (cm). = cro ectional area of tank 1 and tank 2 (cm). u 1 = input to tank 1 (). q i = flow rate of fluid into tank 1(cm 2 /). q o12 = flow rate of fluid between tank 1 and tank 2 (cm 2 /). k 1 = pump flow rate into tank 1(cm 3 /). β 12 = ale ratio between tank 1 and tank 2. β 2 = ale ratio at the outlet of tank 2. a 12 = cro ectional area of jointed pipe between tank 1and tank 2 (cm 2 ). a 2 = cro ectional area of pipe at the outlet of tank 2 (cm 2 ). g = graity (cm/ 2 ). Region Operating point Height (h 1 ) cm Height (h 2 ) cm Integer order ytem P i () ( ) e 3.96 ( ) e 8.66 ( ) e 13.4 (2.1-3) e 18.5 (3-4.5) e 28.9 B. Mathematical modelling of two interacting tank leel proce The non-linear equation decribing the open loop dynamic of the TITL proce i deried uing the ma balance equation and Bernoulli principle a follow, ccording to ma balance equation, rate of accumulation= inflow rate- outflow rate. For tank 1 d 1 dt = q i q o12 (1) Where q i and q o12 are the inflow and outflow of tank 1, which i gien by q i = k 1 u 1 (2) q 012 = β 12 a 12 2g( 1 t 2 (t)) (3) Subtituting (2) and (3) in equation (1) gie d 1 (t) = k 1u 1 dt β 12a 12 2g( 1 t 2 t ) Similarly for tank 2 (4) d 2 dt = q o12 q o2 (5) III. CONTROLLER DESIGN. Fractional order PI λ controller (FOC) deign Let P() and C() be the plant and controller tranfer function repectiely. The tranfer function of the fractional order PI λ controller i C f = K P + K i (9) λ The fractional order PIcontroller in time domain i repreented by a: c t = K P e t + K i D t α (10) whered α t i the fractional integro-differential operator. The general calculu operator (including fractional order and integer order) i defined a: d / dt, R( ) 0 adt 1, R( ) 0 t ( d ), R( ) 0 a wherea and t are repectiely the lower and upper bound, and α i the order of deriatie or integral, which can be non-integer, or een complex number ll Right Reered 2014 IJSETR 370

3 International Journal of Science, Engineering and Technology Reearch (IJSETR), Volume 3, Iue 3, March 2014 The tuning rule for FO-PI controller baed on Fractional M contrained integral gain optimization method (F- MIGO) for generic FOPDT model [4], [5] i gien by K P = 1 k p τ T i = T α = τ ifτ if 0.4 τ < if 0.1 τ < ifτ < 0.1 (11) (12) (13) IV. SIMULTION RESULT ND DISCUSSION. Set point tracking: Fig 2 how the ero performance of IOS with integer and fractional order PI controller. From the figure it i obered that the oerhoot of IOS with FO-PI controller i ignificantly reduced when compared with IO-PI controller for IOS. Table 3 how the performance analyi of cloed loop repone of integer and fractional order PI controller baed on criterion. τ = L L + T (14) wherek p and T are the teady tate gain and time contant and L i the delay. B. Integer order controller(ioc) deign The tranfer function of integer order PI controller i repreented by C f = K P + K i (15) The tuning rule for integer order PI controller baed on pproximated M contrained integral gain optimization method (-MIGO) for generic FOPDT model [6] i gien by K = T k P k p L K i = K 0.33L + 6.8LT 10L + T 1 17 (16) The controller parameter of fractional and integer order PI controller for integer order ytem (IOS) of are hown in Table 2. TBLE: 2 CONTROLLER PRMETERS Fig 2: Set point tracking performance of fractional/integer order controller for IOS TBLE 3: PERFORMNCE NLYSIS FOR SET POINT TRCKING Region FOC with IOS IOC with IOS ( ) ( ) ( ) (2.1-3) Region Integer order PI controller C i () Fractional order PI controller C f () (3-4.5) B. Robutne Tet ( ) ( ) ( ) (2.1-3) The uncertaintie in the plant parameter like change in the gain or the time contant are conidered for checking the robutne of both control ytem. By apply ±5% change in gain the unit tep repone of each control ytem for the parametric uncertaintie are hown in Fig. 3 and 4. From thee figure, it i hown that the robutne of the FOC for IOS i better than one of the IOC for IOS. The performance indice of robutne tet for two control ytem are hown in Table 4 and 5, repectiely. (3-4.5) ll Right Reered 2014 IJSETR 371

4 International Journal of Science, Engineering and Technology Reearch (IJSETR), Volume 3, Iue 3, March 2014 C. Diturbance rejection hown Fig 5, FO-PI controller for IOS ha a good diturbance rejection and attain the et point at the fater rate when compared to IO-PI controller for IOS. The performance analyi of diturbance rejection for integer and fractional order PI controller i hown in Table6. Fig. 3 Robutne tet for fractional PI controller with IOS Fig. 5 Diturbance rejection performance of fractional/integer order controller for IOS Fig. 4 Robutne tet for fractional PI controller with IOS TBLE 4: ROBUSTNESS TEST FOR FO- PI CONTROLLER WITH IOS Region +5% change 0% change in gain -5% change ( ) ( ) ( ) (2.1-3) (3-4.5) TBLE 6: PERFORMNCE NLYSIS FOR DISTURBNCE REJECTION Region FOC with IOS IOC with IOS ( ) ( ) ( ) (2.1-3) (3-4.5) D. Stability analyi It i well known from the general tability theory that a linear time inariant (LTI) ytem i table if all root of characteritic equation are negatie or hae negatie real part. It meanthat they are located on the left half of the complex plane. TBLE 5: ROBUSTNESS TEST FOR IO- PI CONTROLLER WITH IOS Region +5% change 0% change in gain -5% change ( ) ( ) ( ) (2.1-3) (3-4.5) Fig.6 Stability region of LTI FOS with order <q 1 In the fractional-order LTI cae, where the tability of a fractional order control ytem i analyzed ll Right Reered 2014 IJSETR 372

5 International Journal of Science, Engineering and Technology Reearch (IJSETR), Volume 3, Iue 3, March 2014 withmatignon'tability theorem [7],[8], the left half of the plane i different from the integer one. hown in Fig 6, the ertical axi of the complex plane i changed with an angle depending on the fractional order. Intereting notion i that a table fractional ytem may hae root in the right half of complex plane.the pole poition plot of the FOC for IOS obtained uing the MTLB i hown in Fig 7. Thi figure how that all pole of the FOS are in the fractional left half plane and thu the FOS i table. Similarly, the pole poition plot of IOC for IOS i hown in Fig 10. REFERENCES [1] Y. Q. Chen. Ubiquitou fractional order control?.in Proc. FD 2006 Fractional Deriatie and ppl., Porto, Jul , 2006 [2] I. Podlubny, "Fractional-order ytem and PID"-controller," IEEE Tran. utom. Control, ol. 44, no.i, pp , [3] Yang Quan Chen, Io Petra and DingyuXue, Fractional Order Control - Tutorial, merican Control Conference Hyatt Regency Rierfront, St. Loui, MO, US June 10-12, [4] VarhaBhambhani and Yang Quan Chen, Experimental Study of Fractional Order Proportional Integral (FOPI) Controller for Water Leel Control, Proceeding of the 47th IEEE Conference on Deciion and Control, Cancun, Mexico, Dec. 9-11, [5] TriptiBhakaran, YangQuan Chen and Dingy u Xue Practical tuning of fractional order proportional and integral controller, Proceeding of the SME 2007 International Deign Engineering Technical Conference & Computer and Information in Engineering Conference IDETC/CIE 2007, September 4-7, 2007, La Vega, Neada, US. [6] TriptiBhakaran, Practical tuning of fractional order proportional and integral controller,m.sc thei [7] D. Matignon, Generalized Fractional Differential and Difference Equation: Stability Propertie and Modelling Iue Proeerding of Math. Theory of Network and Sytem Sympoium, Padoa, Italy, [8] PrabooNagarajendraNarayanawamy, PalaniappanKanthabhabha, Fractional order PI control trategy for liquid leel ytem, Second World Congre on Nature and Biologically Inpired Computing, Dec , 2010 in Kitakyuhu, Fukuoka, Japan. Fig.9.Pole of FOC for IOS Fig. 10Pole of IOC for IOS V. CONCLUSION ND FUTURE WORK In thi paper, the fractional and integer PI controller i deigned for a cla of linear FOPDT model.from the imulation and performance analyi it i inferred that the deigned fractional order PI controller for integer order ytem how better reult, when compared to the integer order PI controller for integer order ytem in term of et point tracking, robutne, diturbance rejection and tability. Further,the deign and implementation of fractional and integer order controllerfor MIMO proce with will be carried out in future. ll Right Reered 2014 IJSETR 373