Controller Design for Fractional Order Systems

Transcription

1 Controller Deign for Fractional Order Sytem A Thei Submitted in Partial Fulfilment of the Requirement for the Award of the Degree of Mater of Technology in Control & Automation by Ankuh Kumar Department of Electrical Engineering National Intitute of Technology, Rourkela Rourkela , Odiha, INDIA May 2013

2 Controller Deign for Fractional Order Sytem A Thei Submitted in Partial Fulfilment of the Requirement for the Award of the Degree of Mater of Technology in Control & Automation by Ankuh Kumar (Roll 211EE3149) Under the Guidance of Prof. Subhojit Ghoh Department of Electrical Engineering National Intitute of Technology, Rourkela Rourkela , Odiha, INDIA

3 DEPARTMENT OF ELECTRICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA, ODISHA

4 ACKNOWLEDGEMENT Firt and foremot, I am truly indebted to my upervior Prof. Subhojit Ghoh for hi inpiration, excellent guidance and unwavering confidence throughout my tudy, without which thi thei would not be in the preent form. I alo thank him for hi graciou encouragement throughout the work. I expre my gratitude to Prof. Sandip Ghoh, Prof. Suovan Samanta and Prof. Somnath Maity for their helpful and timulating comment. I am alo very much obliged to Prof. A. K. Panda, Head of the Department of Electrical Engineering, NIT Rourkela for providing all poible facilitie toward thi work. I would alo like to thank all my friend, epecially, Rameh, Khuhal, Dineh, Pankaj, Zeehan, Praanna, Ankeh,Soumya, Mahendra and Smruti for extending their technical and peronal upport and making my tay pleaant and enjoyable. Lat but not the leat, I mention my indebtedne to my father and mother for their love and affection and epecially their confidence which made me believe me. Ankuh Kumar Rourkela, May 2013 ii

5 ABSTRACT In recent time, the application of fractional derivative ha become quite apparent in modeling mechanical and electrical propertie of real material. Fractional integral and derivative ha found wide application in the control of dynamical ytem, when the controlled ytem or/and the controller i decribed by a et of fractional order differential equation. In the preent work a fractional order ytem ha been repreented by a higher integer order ytem, which i further approximated by econd order plu time delay (SOPTD) model. The approximation to a SOPTD model i carried out by the minimization of the two norm of the actual and approximated ytem. Further, the effectivene of a fractional order controller in meeting a et of frequency domain pecification i determined baed on the frequency repone of an integer order PID and a fractional order PID (FOPID) controller, deigned for the approximated SOPTD model. The advent of fuzzy logic ha led to greater flexibility in deigning controller for ytem with time varying and nonlinear characteritic by exploiting the ytem obervation in a linguitic manner. In thi regard, a fractional order fuzzy PID controller ha been developed baed on the minimization different optimal control baed integral performance indice. The indice have been minimized uing genetic algorithm. Simulation reult how that the fuzzy fractional order PID controller i able to outperform the claical PID, fuzzy PID and FOPID controller. iii

6 CONTENTS CERTIFICATE ACKNOWLEDGMENT ABSTRACT CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF ACRONYMS i ii iii iv vi viii ix CHAPTER 1: Introduction Introduction of Fractional Order Sytem Literature Review on Fractional order controller Objective Thei Organization. 6 CHAPTER 2: Fractional Order Sytem: Repreentation and Approximation Introduction OF Fractional Order Calculu Definition of Fractional Order Propertie of Fractional Order Differentiation Frequency and Time Domain Analyi of Fractional Order Linear Sytem Frequency Domain Analyi of Linear Fractional Order Sytem Time Domain Analyi of Fractional Order Sytem Integer order Approximation of Fractional Order Sytem Outaloup recurive filter Model Reduction Technique for Fractional Order Sytem Genetic Algorithm 15 iv

7 CHAPTER 3: Fractional Order Controller Deign Introduction to Fractional Order Controller Advantage of Fractional Order controller Fractional order PID Controller tuning Frequency domain Analyi Time domain Analyi Simulation Reult and Dicuion 23 CHAPTER 4: Fuzzy Controller Deign Introduction of Fuzzy controller Fuzzy Fractional order Controller Fuzzy memberhip function and Rule bae Simulation Reult Comparion and Study of Different Controller CHAPTER 5: Concluion and Future work Concluion Future work. 34 REFERENCES 35 v

8 LIST OF TABLES Table 2.1: Approximated reduced order model of (2.12) obtained uing GA. 16 Table 3.1: Deign of a fractional order PID and Integer order PID controller of plant with different performance indice.. 22 Table 3.2: Comparion of cloed loop performance of plant with fractional order PID and integer order PID controller for different performance indice.. 22 Table 4.1: Rule bae Table 4.2: Deign of a Fuzzy FOPID and PID controller of plant with different performance indice. 28 Table 4.3: Comparion of cloed loop performance of plant with Fuzzy FOPID and Fuzzy PID controller for different performance indice.. 29 vi

9 LIST OF FIGURES Figure 1.1: Infinite line tranmiion ytem... 2 Figure 2.1: Frequency repone of Fractional Order ytem 4 Figure 2.2: Step repone of Fractional Order model.. 5 Figure 2.3: Frequency repone FO model and higher integer order G(). 8 Figure 2.4: Step repone FO model and higher integer order G()... 9 Figure 2.5: Step repone of higher order model and reduce SOPTD 17 Figure 2.6: Frequency repone of higher order model and reduce SOPTD.. 17 Figure 3.1: General form of a fractional order PID controller 18 Figure 3.2: Step Repone of FOPID and PID Controller uing GA, while conidering IAE a objective function 23 Figure 3.3: Step Repone of FOPID and PID Controller uing GA, while conidering ISE a objective function 23 Figure 3.4: Step Repone of FOPID and PID Controller uing GA, while conidering ITAE a objective function. 24 Figure 3.5: Step Repone of FOPID and PID Controller uing GA, while conidering ITSE a objective function 24 Figure 4.1: Structure of the Fuzzy Fractional Order PID controller. 26 Figure 4.2: Memberhip function for error, fractional rate of error and FLC output.. 27 Figure 4.3: Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA, while conidering IAE a objective function 29 Figure 4.4: Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering ISE a objective function. 30 Figure 4.5: Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA, while conidering ITAE a objective function. 30 vii

10 Figure 4.6: Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering ITSE a objective function. 31 Figure 4.7: Step Repone of Controller Deign uing GA, while conidering IAE a objective function. 31 Figure 4.8: Step Repone of Controller Deign uing GA, while conidering ISE a objective function Figure 4.9: Step Repone of Controller Deign uing GA, while conidering ITAE a objective function.. 32 Figure 4.: Step Repone of Controller Deign uing GA, while conidering ITSE a objective function. 33 viii

11 LIST OF ACRONYMS Lit of Acronym FO FOPTD SOPTD PID FOPID IAE ISE ITAE ITSE FLC MF : Fractional Order : Firt Order Plu Time Delay : Second Order Plu Time Delay : Proportional-Integral-Derivative : Fractional Order Proportional-Integral-Derivative : Integral Abolute Error : Integral Square Error : Integral Time Abolute Error : Integral Time Square Error : Fuzzy Logic Controller : Memberhip Function ix

12 Chapter 1 Introduction INTRODUCTION 1.1 Introduction of Fractional Order Sytem Fractional calculu provide an excellent intrument for the decription of memory and hereditary propertie of variou material and procee. Thi i the primary advantage of fractional derivative in comparion to claical integer order model, where uch dynamic not taken into account. The advantage of fractional derivative become more appealing in the modeling of mechanical, electrical and electro-mechanical propertie of real material, a well a in the decription of rheological propertie of rock, and in many other field. Recent time have wide application of field fractional integral and derivative alo in the theory of control of dynamical ytem, where the controlled ytem or/and the controller i decribed by a et of fractional differential equation. The mathematical modeling and imulation of ytem and procee, baed on the decription of their propertie in term of fractional derivative, naturally lead to differential equation of fractional order the neceity to olve uch equation to obtain the repone for a particular input. Thought in exitence for more than 300 year, the idea of fractional derivative and integral ha remained quite a trange topic, very hard to explain, due to abence of a pecific tool for the olution of fractional order differential equation. For thi reaon, thi mathematical tool could be judged far from reality. But many phyical phenomena have intrinic fractional order decription and o fractional order calculu i neceary to replicate their input-output characteritic. Fractional order calculu allow u to decribe and model a real object more accurately than the claical integer method. Detail of pat and preent progre in the analyi of dynamic ytem modeled by FODE can be found in [5 6]. PID (proportional integral derivative) controller, which have been dominating indutrial controller, have been modified uing the notion of a Fractional Order integrator and differentiator. It ha been hown that the incorporation of two degree of freedom from the ue of a Fractional Order integrator and differentiator provide a greater degree of flexibility and hence make it poible to further improve the performance of traditional PID controller. National Intitute of Technology, Rourkela Page 1

13 Chapter 1 Introduction Fractional calculu have found wide in different unrelated topic uch a: tranmiion line theory in Infinite line tranmiion ytem, chemical analyi of aqueou olution, deign of heat-flux meter, vico elaticity, dielectric polarization, electromagnetic wave, rheology of oil, growth of intergranular groove on metal urface, quantum mechanical calculation, and diemination of atmopheric pollutant. One of the prime application of fractional calculu in electrical engineering ha been on the modeling an infinite length tranmiion line (Figure 1). Conidering an infinite length tranmiion line with erie and hunt impedance repreented by Z a and Z b repectively, where Infinite line tranmiion ytem: za za za Figure 1.1: Infinite Line Tranmiion Sytem Equivalent impedance Z = z a z b When z a = R and 1 z b = C R Z = 1/ 2 (Fractional order ytem) C 1.2 Literature Review on Fractional order controller Thi ection provide a brief urvey of the tate of the art technique in fractional order controller deign. Podlubny given a more flexible tructure by extended in traditional notion of PID controller [11] with the controller gain define the fractional differ-integral a deign variable. And given everal intelligent technique for efficient tuning of uch fractional order controller. To deign controller dominant pole placement baed optimization National Intitute of Technology, Rourkela Page 2

14 Chapter 1 Introduction problem have been attempted uing Differential Evolution in Maiti et al. [16], Biwa et al. [17] and Invaive Weed Optimization with Stochatic Selection (IWOSS) in Kundu et al. [18]. Maiti et al. [19] alo tuned a FOPID controller for table minimum phae ytem by minimizing an integral performance index i.e. ITAE criteria with Particle Swarm Optimization (PSO). A imilar approach ha been adopted for optimization of a weighted um of Integral of Abolute Error (IAE) and ISCO to find out the controller parameter with GA by Cao, Liang & Cao [20] and with PSO by Cao & Cao [21]. Cai, Pan & Du [22] tuned a controller by minimizing the ITAE criteria uing multi-parent cro over evolutionary algorithm. Luo & Li [23] tuned a imilar ITAE baed controller with Bacterial Foraging oriented by Particle Swarm Optimization (BF-PSO). Meng & Xue [24] deigned a controller uing a multi-objective GA which minimize the infinity-norm of the enitivity (load diturbance uppreion), and complementary enitivity function (high frequency meaurement noie rejection), rie time and percentage of maximum overhoot and additionally meet the pecified gain cro-over frequency, phae margin and io-damping property rather than minimizing thee a a ingle objective with a controller tuning methodology uing Self-Organizing Migrating Algorithm (SOMA), which i an extenion of that propoed by Monje et al. [26] uing contrained Nelder- Mead Simplex algorithm. Zhao et al. [27] tuned a controller for inter-area ocillation in power ytem by minimizing a weighted um of the weighted ummation like Zamani et al. [29]. Kadiyala, Jatoth & Pothalaiah [28] deigned PSO baed optimization problem for minimizing a weighted um of, teady-tate error to deign a controller for aerofin control ytem. A PSO baed imilar approach can be found in Sadati, Zamani & Mohajerin [32] for SISO and MIMO ytem. Sadati, Ghaffarkhah & Otadabba [30] deigned a Neural Network baed FOPID controller by minimizing the Mean Square Error (MSE) of the cloed loop ytem while weight of the Neural Network and fractional order are determined in the learning phae and the controller gain are adapted with change in the error. Ou, Song & Chang [31] deigned a FOPID controller for Firt Order Plu Time Delay (FOPTD) ytem uing Radial Bai Function (RBF) neural network where the controller gain and differential-integral order can be determined from the time contant and delay of the proce after the neural network i trained with a large et of FOPID parameter and ytem parameter. Weighted um of everal time-domain and frequencydomain criteria baed optimization approach ha been ued to tune a FOPID controller with PSO National Intitute of Technology, Rourkela Page 3

15 Chapter 1 Introduction for an automatic voltage regulator by Ghartemani et al. [32] and Zamani et al. [25], [32]. The approach in [33] alo propoe an -optimal FOPID controller by putting the infinity norm of the weighted enitivity and complementary enitivity function a an inequality contraint to the objective function that in [25]. Lee & Chang [34-35] ued Improved Electromagnetim with Genetic Algorithm (IEMGA) to minimize the Integral of Squared Error (ISE) while earching for optimal parameter. Pan et al. [36] ued evolutionary algorithm for time domain tuning of controller to cope with the network induced packet drop and tochatic delay in NCS application. Recent advent of few non-pid type intelligent fractional order controller have been hown to be more effective over the exiting technologie. Efe [37] ued fractional order integration while deigning an Adaptive Neuro-Fuzzy Inference Sytem (ANFIS) baed liding mode control. Delavari et al. [38] propoed a fuzzy fractional liding mode controller and tuned it parameter with GA. Barboa et al. [39] incorporated fuzzy reaoning in fractional order PD controller. Arena et al. [40-41] introduced a new Cellular Neural Network (CNN) with FO cell and tudied exitence of chao in it. Valerio & Sa da Cota [42] tudied fuzzy logic baed approximation of variable complex and real order derivative with and without memory. In the preent tudy, the tuning of a new fuzzy FOPID controller ha been attempted with GA and the cloed loop performance are compared with an optimal controller. The inputoutput MF(Memberhip function) and differ-integral of the FO fuzzy PID controller are tuned while minimizing weighted um of variou error indice and control ignal imilar to that in Cao, Liang & Cao [21] and Cao & Cao [22] with a imple ISE criteria. While [37-38] focue on fractional order fuzzy liding mode controller. The preent work i concerned with the fuzzy analogue of the conventional PID controller, which i widely ued in the proce control indutry. In Barboa et al. [39], the fractional fuzzy PD controller i invetigated in term of digital implementation and robutne. However the tuning methodology i complex and might not alway enure optimal time domain performance. The performance improvement i even more for complicated and ill-behaved ytem which have been enforced to obey a et of deired control objective with GA in the preent formulation. National Intitute of Technology, Rourkela Page 4

16 Chapter 1 Introduction 1.3 Objective Baed on the literature urvey and the cope outlined in the previou ub-ection, the following objective are framed for the preent work: Repreentation fractional order linear ytem by an integer order ytem. Repreentation of a fractional order ytem by an approximated lower order (integer) ytem with time delay. Deign of fractional order controller baed on frequency domain pecification. Deign of fuzzy fractional order controller baed on the minimization of time domain baed integral performance indice. 1.4 Thei Organization The thei conit of five chapter organized a follow: Chapter 1 give an introduction about fractional order ytem with pecific tre on fractional order controller. Chapter 2 provide a decription on the time and frequency domain repreentation of fractional order ytem along with the repreentation of fractional order by it integer order equitant. Chapter 3 provide the deign and the neceity of fractional order controller. In chapter 4, deign of fuzzy fractional order and fuzzy integer order controller are carried out and correponding repone are compared. Chapter 5 provide concluion and future work. National Intitute of Technology, Rourkela Page 5

17 Chapter 2 Fractional Order Sytem: Repreentation and Approximation FRACTIONAL ORDER SYSTEM: Repreentation and Approximation 2.1 Introduction OF Fractional Order Calculu Fractional order calculu i an area where the mathematician deal with derivative and integral from non-integer order. There are different definition of Fractional Order differentiation and integration. Some of the definition extend directly from integer-order calculu. The welletablihed definition include the Grünwald-Letnikov definition, the Cauchy integral formula, the Caputo definition and the Riemann-Liouville definition [1-4]. The definition will be ummarized firt, and then their propertie will be given Definition of Fractional Order Calculu Definition 2.1 (Cauchy Fractional Order integration formula). Thi definition i a general extenion of the integer-order Cauchy formula (2.1) where C i the mooth curve encircling the ingle-valued function f(t). Definition 2.2 (Grünwald Letnikov definition). The definition i defined a [6] lim 1 (2.2) where 1 repreent the coefficient of the polynomial 1. The coefficient can alo be obtained recurively from 1, 1 j=1,2. (2.3) National Intitute of Technology, Rourkela Page 6

18 Chapter 2 Fractional Order Sytem: Repreentation and Approximation Definition 8.3 (Riemann Liouville Fractional Order differentiation). The Fractional Order integration i defined a [7] (2.4) where 0 < α < 1and a i the initial time intance, often aumed to be zero, i.e., a = 0. The differentiation i then denoted a. The Riemann Liouville definition i the mot widely ued definition in fractional order calculu. The ubcript on both ide of D repreent, repectively, the lower and upper bound in the integration. Definition 8.4 (Caputo definition of Fractional Order differentiation). Caputo definition i given by [2] (2.5) where α=m γ,m i a integer, and 0 < γ 1. Similarly Caputo Fractional Order integration i defined a, γ < 0 (2.6) Propertie of Fractional Order Differentiation The Fractional Order differentiation ha the following propertie [8]: 1. The Fractional Order differentiation with repect to t of an analytic function i alo analytical. 2. The Fractional Order differentiation i exactly the ame with integer-order one, when α=n i an integer. Alo. (2.7) 3. The Fractional Order differentiation i linear; i.e., for any contant a,b one ha []= t t (2.8) National Intitute of Technology, Rourkela Page 7

19 Chapter 2 Fractional Order Sytem: Repreentation and Approximation 4. Fractional Order differentiation operator atify the commutative-law, and alo atify [ ]= [ ]= (2.9) 5. The Laplace tranform of Fractional Order differentiation i defined a L L (2.) In particular, if the derivative of the function f (t) are all equal to 0 at t = 0, one ha L L. 2.2 Frequency and Time Domain Analyi of Fractional Order Linear Sytem The Fractional Order ytem i the direct extenion of claical integer-order ytem. The Fractional Order ytem i etablihed upon the Fractional Order differential equation, and the Fractional Order tranfer function of a ingle variable ytem can be defined a (2.11) where, are real number and the order, of the numerator and the denominator can alo be real number. The analyi of the Fractional Order Laplace tranformation and their invere i very complicated. The cloed-form olution to the problem are not poible in general. Fractional Order ytem tranfer function give a (2.12) National Intitute of Technology, Rourkela Page 8

20 Chapter 2 Fractional Order Sytem: Repreentation and Approximation Frequency Domain Analyi of Linear Fractional Order Sytem It can be een that, when jω i ued to ubtitute for the variable in the Fractional Order tranfer function model (2.12), the frequency domain repone can be eaily evaluated. Thu, Fractional Order Bode diagram, Nyquit plot, and Nichol chart can be eaily evaluated. In figure 2.1 frequency repone of Fractional Order ytem repreented in (2.12) i given. 50 Bode Diagram 0 Magnitude (db) Phae (deg) Frequency (rad/ec) Figure 2.1 : Frequency repone of Fractional Order ytem Time Domain Analyi of Fractional Order Sytem The evaluation of the time domain repone of a Fractional Order ytem i more complicated. Let u conider a pecial form of a Fractional Order differential equation [8] (2.13) where u(t) can be repreented by a certain function and it Fractional Order derivative. Aume alo that the output function y(t) ha zero initial condition. The Laplace tranform can be ued to find the tranfer function National Intitute of Technology, Rourkela Page 9

21 Chapter 2 Fractional Order Sytem: Repreentation and Approximation... (2.14) Conider the Grünwald Letnikov definition in (2.2). The dicrete form of it can be rewritten a (2.15) The numerical olution to the Fractional Order differential equation (2.12) i given a (2.16) For the general form of the Fractional Order tranfer function in (2.12), the right-hand ide can equivalently be evaluated firt by uing numerical method. The final olution can be obtained from (2.16).Uing equation (2.16) tep repone of fraction-order ytem i obtained a in figure Amplitude Time [ec] Figure 2.2 : Step repone of Fractional Order model National Intitute of Technology, Rourkela Page

22 Chapter 2 Fractional Order Sytem: Repreentation and Approximation 2.3 Integer order Approximation of Fractional Order Sytem A Fractional order linear time invariant (LTI) ytem i mathematically equivalent to an infinite dimenional LTI filter. Thu a fractional order ytem can be approximated uing higher order polynomial having integer order differ-integral operator. The realization of FO differintegrator in integer order can be carried in two way [8]. Continuou time realization Dicrete time realization Here we are ued continuou time realization to realize our Fractional Order ytem into integer order ytem by uing Outaloup recurive filter [9] Outaloup recurive filter Outaloup recurive filter give a very good fitting to the Fractional Order element within a choen frequency band. Let u aume that the expected fitting range i (, ). The filter can be written a (2.17) where the pole, zero, and gain of the filter can be evaluated from a ; ; (2.18) where γ i the order of the differentiation, 2N 1 i the order of the filter, and the frequency fitting range i given by (, ). The filter can be deigned uch that it may fit very well within the frequency range of the fractional order differentiator. Integer order tranfer function of fractional term 0.603, 0.501, 0.242, 0.789, of Fractional Order ytem are repectively given a National Intitute of Technology, Rourkela Page 11

23 Chapter 2 Fractional Order Sytem: Repreentation and Approximation National Intitute of Technology, Rourkela Page 12 (2.19) (2.20) (2.21) (2.22) (2.23) Equivalent total integer order tranfer function G() of Fractional Order ytem (2.12) uing the Outaloup Recurive Filter i given a ) ( = g ) ( = g ) ( = g ) ( = g ) ( = g

24 Chapter 2 Fractional Order Sytem: Repreentation and Approximation G( ) = (2.24) The bode diagram of Fractional Order ytem and integer order ytem i depicted in figure 2.3. It i oberved that bode diagram Fractional Order ytem and integer order ytem (higher order) converge a. And the correponding tep repone of Fractional Order derivative ytem and integer order approximated ytem i hown in figure Bode Diagram M a g n itu d e ( d B ) P h a e ( d e g ) Ga() G() Figure 2.3: Frequency repone FO model and higher integer order G() National Intitute of Technology, Rourkela Page 13

25 Chapter 2 Fractional Order Sytem: Repreentation and Approximation Step Repone Ga() G() 2.5 Amplitude Time (ec) Figure 2.4: Step repone FO model and higher integer order G() 2.4 Model Reduction Technique for Fractional Order Sytem A oberved in the previou ection if the integer-order approximation i ued to fit the Fractional Order tranfer function model with the ue of the Outaloup recurive filter, the order of the reulting integer order ytem could be extremely high. Thu, a low-order approximation to the original problem can be found uing the optimal model reduction method. Recall that the expected reduced-order model given by [12].,... (2.25)... An objective function for minimizing the -norm of the deviation between the tranfer function of higher order and approximated ytem can be defined min ^, Where ϕ i the et of parameter to be optimized (2.26) = [,.,,, ] (2.27) National Intitute of Technology, Rourkela Page 14

26 Chapter 2 Fractional Order Sytem: Repreentation and Approximation For an eay evaluation of the criterion J, the delayed term in the reduced-order model can be further approximated by a rational function / () uing the Pade approximation technique. Thu, the revied criterion can then be defined by min ^ The -norm computation can be evaluated recurively uing an optimization algorithm []. (2.28) 2.5 Genetic Algorithm Thi ection provide a brief decription about genetic algorithm (GA) [14-15] and it application in the minimization of (2.28). Genetic algorithm i a tochatic optimization proce inpired by natural evolution. During the initialization phae, a random population of olution vector with uniform ditribution i created over the whole olution domain. The population i encoded a a double vector. Fitne evaluation: Since the purpoe of uing genetic algorithm i to determine a reduce order model with minimizing objective function (2.23) from the earch pace. Reproduction: Individual tring are copied baed on the fitne and ent to the mating pool. The reproduction operation i implemented uing roulette wheel arrangement. Croover: During croover operation, two tring elected at random from the mating pool undergo croover with a certain probability at a randomly elected croover point to generate two new tring. Mutation: Depending on whether a randomly generated number i larger than a predefined mutation probability or not, each bit in the tring obtained after croover i altered (changing 0 to 1 and 1 to 0). In each generation, the fittet member fitne function value i compared with that of the previou fittet one. If a very inignificant improvement i een for ome ucceive generation then the algorithm i topped, otherwie all the operation decribed above are carried out till a model i obtained with a deired objective function. National Intitute of Technology, Rourkela Page 15

27 Chapter 2 Fractional Order Sytem: Repreentation and Approximation Table 2.1 : Approximated reduced order model of (2.12) obtained uing GA. Type of ytem FO Sytem FOPTD Sytem SO Sytem SOPTD Sytem Third Order Sytem Fourth Order Sytem Reduced- order Model,,ɩ,, ,, ,,,, ,, ,, Objective Function value In the preent work obtain a lower order integer equivalent of fractional order ytem by, Genetic Algorithm ued to minimizing the objective function J different type model obtained integer order approximation are conidered. The correponding approximated model are given in Table 2.1. It i oberved that objective function value i minimum for SOPTD model, a expected tep repone (figure 2.5) and bode plot (figure 2.6) of SOPTD model cloe to higher order integer order ytem (2.19) with minimum error. Now reduce order integer order model of higher order model i rewritten a.... (2.30) National Intitute of Technology, Rourkela Page 16

28 Chapter 2 Fractional Order Sytem: Repreentation and Approximation Step Repone Higher Integer order ytem Reduce SOPTD ytem Am plitude Time (ec) Figure 2.5: Step repone of higher order ytem and reduce SOPTD ytem 50 0 Bode Diagram P ha e (de g) M agnitude (db) x Higher Integer order ytem Reduce SOPTD ytem Frequency (rad/ec) Figure 2.6: Frequency repone of higher order ytem and reduce SOPTD ytem National Intitute of Technology, Rourkela Page 17

29 Chapter 3 Fractional Order Controller Deign FRACTIONAL ORDER CONTROLLER DESIGN 3.1 Introduction to Fractional Order Controller Controlling indutrial plant require atifaction of wide range of pecification. So, wide range of technique are needed. Motly for indutrial application, integer order controller are ued for controlling purpoe. Now day fractional order (FOPID) controller i ued for indutrial application to improve the ytem control performance. The mot common form of a fractional order PID controller i the controller [11]. FOPID controller provide extra degree of freedom for not only the need of deign controller gain (,, ) but alo deign order of integral and derivative. The order of integral and derivative are not necearily integer, but any real number. A hown in Fig. 3.1, The FOPID controller generalize the conventional integer order PID controller and expand it from point to plane. Thi expanion could provide much more flexibility in PID control deign. following form [17] The tranfer function of uch a controller ha the (3.1) Figure 3.1 : General form of a fractional order PID controller It i Clear, by electing λ = 1 and µ = 1, a claical PID controller can be recovered. Uing λ = 1, µ = 0, and λ = 0, µ = 1, repectively correpond to the conventional PI & PD controller. All thee claical type of PID controller are pecial cae of the PI D controller. National Intitute of Technology, Rourkela Page 18

30 Chapter 3 Fractional Order Controller Deign 3.2 Advantage of Fractional Order controller A compared to an integer order controller, a fractional order i uppoed to offer the following advantage [11] If the parameter of a controlled ytem change, a fractional order controller i le enitive than a claical PID controller. FOC have two extra variable to tune. Thi provide extra degree of freedom to the dynamic propertie of fractional order ytem. 3.3 Fractional order PID Controller tuning Baed on the derived SOPTD model (2.30), the deign of PID controller with integer order and Fractional Order dynamic carried out. In the preent work Fractional Order PID controller are tuned baed on 1) Frequency domain pecification 2) Time domain baed optimial control tuning Frequency domain Analyi Monje-Vinagre propoed an optimization method fractional controller tuning for tuning of FOPID controller [13]. In thi method, a propoed tuning rule i baed on a pecified deirable behavior of the controlled ytem related to pecified value of the following objective. 1. No teady-tate error 2. Specified gain croover frequency 0 (3.2) 3. Specified phae margin repreented a (3.3) 4. Robutne againt variation of gain of the plant, o around the gain cro over frequency National Intitute of Technology, Rourkela Page 19

31 Chapter 3 Fractional Order Controller Deign phae of the open loop tranfer function mut be contant. 0 (3.4) 5. For rejecting high-frequency noie, at high frequencie the cloed loop tranfer function mut have mall magnitude. Thu it i required that at ome pecified frequency it magnitude be le than ome pecified gain. at frequency w w rad/ (3.5) 6. The enitivity function mut have a mall magnitude at low frequencie. To reject output diturbance and tack reference it hould atify the following at frequency w w rad/ (3.6) The cloed-loop ytem i required to meet the above define ix pecification by properly tuning five parameter of FOPID. All five nonlinear equation ( ) need to be olved imultaneouly, to find out FOPID unknown five parameter (,,,,μ). In thi ection, an FOPID controller i deigned for the approximated reduce integer order SOPTD model, obtained in previou chapter i.e..... (3.7) The fallowing deigning pecification are conidered for deigning FOPID and integer order PID controller. Phae Margin =80 degree Gain croover frequency=0.3 rad/ Robutne to variation in the gain of the plant mut be fulfilled A=-20 db at w w =rad/ B=-20 db at w w =.01 rad/ National Intitute of Technology, Rourkela Page 20

32 Chapter 3 Fractional Order Controller Deign By olving five deign criteria ( ) Fractional order PID controller i obtained a (3.8) Putting λ=1 and µ=1 and olving upper five equation ( ) correponding integer order PID controller i obtained a Time domain Analyi (3.9) For deigning controller baed on time domain, controller aim at minimization of different integral performance indice namely 1) Integral quare error ISE e tdt 2) Integral abolute error IAE et dt 3) Integral time-quare error ITSE e tdt 4) Integral time-abolute error ITAE et dt Starting from random initialized parameter, GA progreively minimize different integral performance indice iteratively while finding optimal et of parameter for the FOPID and PID controller. The algorithm terminate if the value of the objective function doe not change appreciably over ome ucceive iteration. For model (3.7) the parameter of FOPID and PID are calculated for different performance indice uing GA and the reult are depicted in Table 3.1. Step repone pecification for different FOPID and PID are given in Table 3.2. National Intitute of Technology, Rourkela Page 21

33 Chapter 3 Fractional Order Controller Deign Table 3.1: Deign of a fractional order PID and Integer order PID controller of plant with different performance indice. Type of controller Performance Index Minima of performance indice λ µ FOPID IAE PID IAE FOPID ISE PID ISE FOPID ITAE PID ITAE FOPID ITSE PID ITSE Table 3.2: Comparion of cloed loop performance of plant with fractional order PID and integer order PID controller for different performance indice. Type of Performance Rie time Peak time Peak overhoot Setting time controller Index (ec) (ec) (%) (ec) FOPID IAE PID IAE FOPID ISE PID ISE FOPID ITAE PID ITAE FOPID ITSE PID ITSE National Intitute of Technology, Rourkela Page 22

34 Chapter 3 Fractional Order Controller Deign 3.4 Simulation Reult and Dicuion PID FOPID 1 Amplitude Time(ec) Figure 3.2 :Step Repone of FOPID and PID Controller uing GA,while conidering IAE a objective function FOPID PID Figure 3.3 :Step Repone of FOPID and PID Controller uing GA,while conidering ISE a objective function. National Intitute of Technology, Rourkela Page 23

35 Chapter 3 Fractional Order Controller Deign PID FOPID 1 Amplitude Time(ec) Figure 3.4 :Step Repone of FOPID and PID Controller uing GA,while conidering ITAE a objective function PID FOPID Amplitude Time(ec) Figure 3.5 :Step Repone of FOPID and PID Controller uing GA,while conidering ITSE a objective function. National Intitute of Technology, Rourkela Page 24

36 Chapter 3 Fractional Order Controller Deign Figure 3.2 how tep repone of PID and FOPID with IAE performance index reult how that FOPID give fater repone than PID but ettling time i more. Figure 3.3 how tep repone of PID and FOPID with ISE performance index. FOPID reache teady tate fater with low peak overhoot. Figure 3.4 how tep repone of PID and FOPID with ITAE performance index, teady tate i reached at earlier for FOPID. Figure 3.5 tep repone of PID and FOPID with ITSE performance index FOPID give better performance. In next chapter Fuzzy FOPID and Fuzzy PID are deigned for ame performance indice and reult are compared. National Intitute of Technology, Rourkela Page 25

37 Chapter 4 Fuzzy Controller Deign FUZZY CONTROLLER DESIGN 4.1 Introduction of Fuzzy controller In the preent tudy, the extra of freedom provided by fractional rate of error in the deign of conventional FLC baed PID controller [43]. It i logical that the fractional rate of error introduce ome extra degree of flexibility in the input variable of FLC and can be tuned alo like the input-output caling factor a the FLC gain and hape of the memberhip function (MF) to get enhanced cloed loop performance. The preent tudy tet the effectivene of the propoed fuzzy FOPID controller at producing better performance compared to claical PID, fuzzy PID and even FOPID controller. 4.2 Fuzzy Fractional order Controller The tructure of the fuzzy PID conidered here i a combination of fuzzy PI and fuzzy PD controller (Figure 4.1). In integer order fuzzy PID controller, the input are the error and the derivative of error and the FLC output i multiplied by caling factor a and it integral multiplied with b and then ummed to give the total controller output [44]. But in the preent cae the integer order rate of the error at the input to the FLC i replaced by it fractional order counterpart (µ). Alo the order of the integral i replaced by a fractional order (λ) at the output of the FLC repreent a fractional order ummation (integration) of the FLC output. Figure 4.1: Structure of the Fuzzy Fractional Order PID controller National Intitute of Technology, Rourkela Page 26

38 Chapter 4 Fuzzy Controller Deign 4.3 Fuzzy memberhip function and Rule bae The propoed FLC baed FOPID controller ue a two dimenional linear rule bae (Table:4.1) for the error, and fractional rate of change of error and the FLC output with tandard triangular memberhip function and Mamdani type inferencing [43]. The triangular memberhip function i choen over the other type like Gauian, trapezoidal, bell-haped, π-haped etc. a it i eaier to implement in practical hardware. In Fig.4.2, the fuzzy linguitic variable NB, NS, Z, PS, and PB with range [-1 1] repreent negative big, negative mall, zero, poitive mall and poitive big repectively. The FLC output i determined by uing centre of gravity method by defuzzification. NB NS Z PS PB Figure 4.2: Memberhip function for error, fractional rate of error and FLC output Table 4.1: Rule bae E DE NB NS Z PS PB NB NB NS NS NB NS Z Z NB NS Z PS PB PS Z PS PB PB PS National Intitute of Technology, Rourkela Page 27

39 Chapter 4 Fuzzy Controller Deign Genetic Algorithm ha been ued here to find the optimum et of value for the controller parameter. The variable that contitute the earch pace for the fractional fuzzy PID controller are {a,,,,λ,µ}. The interval of the earch pace for thee variable are {a,,, } [0,] and {λ,µ} [0,2]. For tuning the GA, a large number of election and croover trategie were teted through pilot run to determine the mot uitable one. Single point croover i ued in conjunction with elitit trategy baed Roulette wheel election. The elitit trategy, which i able to preerve uperior tring, i incorporated in the preent work by replacing the wort tring of a particular generation with the bet tring of the previou generation. The number of chromoome in the initial population and the maximum number of generation i et at 30 and 150 in each run. The croover and mutation probability wa fixed at 0.8 and 0.1, repectively. Table 4.2: Deign of a Fuzzy FOPID and PID controller of plant with different performance Indice. Type of controller Performance Index Minima of performance indice Controller parameter λ µ Fuzzy FOPID Fuzzy PID Fuzzy FOPID Fuzzy PID Fuzzy FOPID Fuzzy PID Fuzzy FOPID Fuzzy PID IAE IAE ISE ISE ITAE ITAE ITSE ITSE National Intitute of Technology, Rourkela Page 28

40 Chapter 4 Fuzzy Controller Deign Table 4.3: Comparion of cloed loop performance of plant with Fuzzy FOPID and Fuzzy PID controller for different performance indice. Type of controller Performance Index Rie time (ec) Peak time (ec) Peak overhoot (%) Setting time (ec) Fuzzy IAE FOPID Fuzzy PID IAE Fuzzy ISE FOPID Fuzzy PID ISE Fuzzy ITAE FOPID Fuzzy PID ITAE Fuzzy ITSE FOPID Fuzzy PID ITSE Simulation Reult Fuzzy PID Fuzzy FOPID 1 Amplitude Time(ec) Figure 4.3 :Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering IAE a objective function. National Intitute of Technology, Rourkela Page 29

41 Chapter 4 Fuzzy Controller Deign Fuzzy PID Fuzzy FOPID Figure 4.4 :Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering ISE a objective function Fuzzy FOPID Fuzzy PID 1 Amplitude Time(ec) Figure 4.5 :Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering ITAE a objective function. National Intitute of Technology, Rourkela Page 30

42 Chapter 4 Fuzzy Controller Deign 1.5 A m p lit u d e Fuzzy PID Fuzzy FOPID Time(ec) Figure 4.6 :Step Repone of Fuzzy FOPID and Fuzzy PID Controller uing GA,while conidering ITSE a objective function. 4.5 Comparion and Study of Different Controller The parameter of fractional order fuzzy logic controller are optimally tuned with GA to handle the model (2.30). Time domain performance of controller (PID, FOPID, fuzzy PID and fuzzy FOPID) with variou integral error indice are compared in thi ectioned Amplitude Fuzzy FOPID Fuzzy PID PID FOPID Time(ec) Figure 4.7: Step Repone of Controller Deign uing GA, while conidering IAE a objective function. National Intitute of Technology, Rourkela Page 31

43 Chapter 4 Fuzzy Controller Deign Fuzzy PID Fuzzy FOPID FOPID PID Figure 4.8: Step Repone of Controller Deign uing GA, while conidering ISE a objective function Amplitude Fuzzy FOPID Fuzzy PID PID FOPID Time(ec) Figure 4.9: Step Repone of Controller Deign uing GA, while conidering ITAE a objective function. National Intitute of Technology, Rourkela Page 32

44 Chapter 4 Fuzzy Controller Deign Amplitude Fuzzy PID PID FOPID Time(ec) Figure 4.: Step Repone of Controller Deign uing GA, while conidering ITSE a objective function. For IAE performance index fuzzy FOPID give fater repone a compared to other. Fuzzy FOPID and fuzzy PID both have teady tate error for ISE performance index. In ITAE performance index fuzzy PID give fater repone with low overhoot and in cae of ITSE all controller performance almot ame. National Intitute of Technology, Rourkela Page 33

45 Chapter 5 Concluion and Future work CONCLUSIONS AND FUTURE WORK 5.1 Concluion In thi thei work, a fractional order ytem i repreented by a higher integer order ytem, which i further approximated by econd order plu time delay (SOPTD) model. The repreented SOPTD model of fractional order ytem i verified both frequency and time domain give the approximate repreentation of fractional-order ytem. Further, the optimal time domain tuning of fractional order PID and claical PID controller baed on Genetic algorithm. Genetic algorithm i ued to minimizing variou integral performance indice. It i oberved that the controller performance depend on the type of proce to be controlled and alo on the choice of integral performance indice. Conidering the uncurtaining in ytem dynamic, a fuzzy fractional order PID and fuzzy PID i deigned. Simulation reult how that the fuzzy fractional order PID controller i able to outperform the claical PID, fuzzy PID and FOPID controller. 5.2 Future work Future work in thi direction will be aimed at improving the performance of fuzzy FOPID controller. It i expected that a fuzzy FOPID controller performance can be achieved by the proper tuning of fuzzy memberhip function and the rule bae. In thi regard, work i planned on the application of evolutionary optimization technique for the optimal configuration of fuzzy fractional order PID controller. National Intitute of Technology, Rourkela Page 34

46 Reference Reference [1] K.B. Oldham, J. Spanier. The Fractional Calculu. NewYork: Academic Pre, [2] I. Podlubny Fractional Differential Equation. San Diego: Academic Pre, [3] Y.Q. Chen, K.L. Moore, Dicretization cheme for fractional differentiator and integrator, IEEE Tran. Circuit Sytem 1: Fundamental Theory Appl. 49 (3) (2002) [4] K.S. Miller, B. Ro An Introduction to the Fractional Calculu and Fractional Differential Equation, NewYork: Wiley, [5] S.G. Samko, A.A.Kilba, O. I. Marichev. Fractional Integral and Derivative and Some of Their Application, Mink: Nauka i Technika, [6] D. Xue, Y.Q. Chen MATLAB Solution to Advanced Applied Mathematical Problem. Beijing: Tinghua Univerity Pre, [7] R. Hilfer Application of Fractional Calculu in Phyic, Singapore: World Scientific, [8] I. Petra, I. Podlubny. P. O Leary. Analogue Realization of Fractional Order Controller Fakulta BERG,TU Koice,2002. [9] A. Outaloup, F. Levron, B. Mathiew, F. Nanot Frequency band complex noninteger differentiator: Characterization and ynthei, IEEE Tranaction on Circuit and Sytem I: Fundamental Theory and Application, 2000, 47(1): [] K. J. Atrom. Introduction to Stochatic Control Theory. London: Academic Pre, 1970 [11] I. Podlubny Fractional-order ytem and controller, IEEE Tranaction on Automatic Control, 1999, 44(1): [12] D. Xue, D.P. Atherton A uboptimal reduction algorithm for linear ytem with a time delay, International Journal of Control, 1994, 60(2): [13] C.A. Monje, B.M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanue and J. Sabatier, Propoal for Fractional PID Tuning, The Firt IFAC Sympoium on Fractional Differentiation and it Application 2004, Bordeaux, France, July 19-20, [14] S.Ghoh, S.Maka Genetic algorithm baed NARX model identification for evaluation of inulin enitivity, Applied Soft Computing 11 (2011) [15] S.Ghoh, Srihari Gude A genetic algorithm tuned optimal controller for glucoe regulation in type 1 diabetic ubject, International Journal for Numerical Method in Biomedical Engineering Int. J. Numer. Meth. Biomed. Engng. (2012). [16] Deepyaman Maiti, Mithun Chakraborty, Ayan Acharya and Amit Konar, Deign of a fractional-order elf-tuning regulator uing optimization algorithm, 11th International Conference on Computer and Information Technology, ICCIT 2008, pp , December 2008, Khulna, Bangladeh. National Intitute of Technology, Rourkela Page 35