Inter national J our nal of P o wer Electr onics and Dri v e System (IJPEDS) V ol. 11, No. 1, March 2020, pp. 291 301 ISSN: 2088-8694, DOI: 10.11591/ijpeds.v11.i1.pp291-301 r 291 Design of fractional order contr ollers using constrained optimization and r efer ence tracking method Manoj D . P atil 1 , K. V adirajacharya 2 , Swapnil Khubalkar 3 1 Department of Electrical Engineering, Annasaheb Dange Colle ge of Engineering and T echnology , Ashta, India 2 Department of Electrical Engineering, Dr . Babasaheb Ambedkar T echnological Uni v ersity , Lonere, Raig ad, India 3 Department of Electrical Engineering, G. H. Raisoni Colle ge of Engineering, Nagpur , India Article Inf o Article history: Recei v ed Jun 7, 2019 Re vised Jul 8, 2019 Accepted No v 12, 2019 K eyw ords: Constrained optimization Controller Fractional order control T uning ABSTRA CT In recent times, fractional order controllers are g aining more interest. There are se v eral fractional order controllers are a v ailable in literature. Still, tuning of these controllers is one of the main issues which the control community is f acing. In thi s paper , online tuning of fi v e dif ferent fractional order controllers is discussed viz. tilted proportional- inte gral-deri v ati v e (T -PID) controller , fractional order proport ional-inte gral (FO-PI) controller , fractional order proportional-deri v ati v e (FO-PD) controller , fractional order proportional-inte gral-deri v ati v e (FO-PID) controller . A reference tracking method is proposed for tuning of fractional order controllers. First order with dead time (FO WDT) system is used to check feasibility of the control strate gy . This is an open access article under the CC BY -SA license . Corresponding A uthor: Manoj D. P atil, Assistant Professor , ADCET , Ashta, Maharashtra, T el: + 91 9763530440 Email: mdpatileps@gmail.com 1. INTR ODUCTION The proportional-inte gral-deri v ati v e (PID) controllers are broadly used in man y indus trial applications [1]. The simplicity of structure and easily implementable tuning strate gies are behind their success. Man y industrial systems can be pres ented by first order with dead time (FO WDT) models and design of controllers for such systems is important [2]. Zie gler -Nichols tuning, some nature inspired optimization technique lik e genetic algorithm, particle sw arm optimization, ant colon y algorithm ha v e widely used techniques for PID controllers [3, 4]. No w-a-days, generalized form of PID controller is getting v ery popular which is kno wn as fractional order PID (FO-PID) controller [6]. In these controller strate gies, inte gral and dif ferentiators are of fractional order . T o use t he fractional order is a step closure to real life because the natural processes are mostly fractional [7, 8]. Ho we v er , the fractional nature may be v ery small. Clearly , the use of inte ger order in place of fractional order will dif fer significantly to the actual situation. Ho we v er , the inte ger order models are welcomed due to the absence of solutions to the fractional models. Recently , the fractional order models are analyzed and implemented successfully in [9, 10]. It is sho wn that the performance of con v entional PID controllers can be impro v ed using the e xtra de gree of freedom from the use of fractional inte grator and dif ferentiator [12, 13]. Fractional order controllers are being increasingly used for controlling FO WDT systems [14]. Unlik e traditional PID controller , the order of inte gration and/or dif ferentiation in fract ional controllers are real and unkno wn parameters to be tuned. This leads to solving nonlinear equations to design fractional controller . There are dif fere nt optimization methods are tried to find the parameters of these J ournal homepage: http://ijpeds.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
292 r ISSN: 2088-8694 controllers [15]. The con v entional tuning rules are not applicable here. A genetic algorit hm is proposed for tuning in [16]. P article sw arm optimization is proposed in [17]. In [18], art ificial bee colon y algorithm is proposed. Some other literature for tuning is reported in [19, 20]. Sti ll, the area of tuning the fractional order controllers is opened. In [21], iterati v e feedback tuning algorithm is emplo yed to tune a type of FO-PID controller . The paper [22] deals with an optimal design of a ne w type v ariable coef ficient FO-PID controller by using heuristic optimization algorithms. Although man y studies ha v e tried to correct the performance of the system’ s transient and steady state responses. It is ob vious that handling the step response and reference tracing together will bring out a better control response. Ho we v er , there are no studies using dif ferent controller parameters of the system with uncertain plant structure. The major contrib ution of the paper is to fill this g ap by presenting a no v el approach. In present paper , a reference tracking optimization method is proposed for tuning the fractional controllers. T uning of dif ferent fractional controllers is sho wn using this method. In the proposed method, e v en though the controller has changed the optimization algorithm remains same. Only , the controller structure has to be changed. If the plant gets changed, still the refe rence tracking optimization method w orks. Hence, the proposed method can pro vide the solution for tuning of all other fractional controllers. The or g anization of the paper is as: some preliminary information is gi v en in Section II. Section III introduces the dif ferent controller structures. In Section IV , the tuning methodology is proposed. In Section V , simulation results are gi v en. The paper has been concluded in Section VI. 2. PRELIMIN ARIES Fractional calculus enables the use of real po wers of the inte gration or dif ferentiation operator . The basics of fractional calculus are described in [23, 24]. Riemann-Liouville (RL) defined the fractional operator as, a D t f ( t ) = 1 ( m )   d dt ! m Z t a f ( ) ( t ) m +1 d (1) where, ( m 1) < m , is a real number , m is an inte ger , and ( : ) is the Euler’ s g amma function. The Laplace transform of (2) gi v es, L f a D t f ( t ) g = s F ( s ) (2) s has to be approximated in inte ger order for realization. There are se v eral approximation techniques are a v ailable [26]. 2.1. Method of A ppr oximation An y transfer function can be approximated by interlacing real poles and zeros [27]. The inte ger order approximation can be obtained within the desired band of frequenc y by using Oustaloup et al. method [24]. Here, poles and zeros are obtained as follo ws, D N ( s ) = ! u ! h N k = N 1 + s=! 0 k 1 + s=! k (3) where, ! k = ! b ! h ! b ( k + N +1 = 2+ = 2) (2 N +1) and ! k 0 = ! b ! h ! b k + N +1 = 2 = 2 (2 N +1) . The number of poles and zeros are a v ailable as 2N+1, the approximation order is considered as N. 3. DESIGN OF FRA CTION AL ORDER CONTR OLLERS V arious controller strate gies, which are designed based on the fractional calculus, are discussed here. 3.1. Contr oller 1. T -PID In [23], a dif ferent structure of fractional order controller is gi v en, where the proportional term of PID controller is changed with a tilted term with a transfer function s 1 =m . The ne w transfer function of T -PID controller closely approximate an optimal transfer function, hence pro vides impro v ement in controller . The T -PID controller allo ws smaller ef fects of plant parameter v ariation and better disturbance rejection ratio Int J Po w Elec & Dri Syst, V ol. 11, No. 1, March 2020 : 291 – 301 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 r 293 as compared to con v entional PID controller . The T -PID controller is obtained as, u ( t ) = K P D e ( t ) + K I Z e ( t ) + K D D e ( t ) (4) Applying Laplace transform, u ( s ) = K P s + K I =s + K D s (5) where, K P ; K D ; K I are proportional, deri v ati v e, and inte gral constant, u ( s ) is control signal, D is deri v ati v e operator . The structure of T -PID controller is sho wn in Figure 1. It sho ws t ilted component with proportional component which is k e y f actor in this controller structure. Figure 1. Structure of T -PID controller 3.2. Contr oller 2. FO-PID FO-PID controller used to minimize the error of the system. The error e ( t ) is scaled by g ain K P in the proportional control. The steady state error is minimized by inte gral control. The transients in the system are tak en care by deri v ati v e control. In FO-PID, in addition to K P ; K D ; and K I , tw o more parameters - inte grator order , - order of dif ferentiator are added to gi v e more de gree of freedom due to which impro v ement can be seen in the controller’ s performance. By adding up all these terms, FO-PID controller is achie v ed. The FO-PID controller is gi v en by , u ( t ) = K P e ( t ) + K I D e ( t ) + K D D e ( t ) (6) Applying Laplace transform, u ( s ) = K P + K I s + K D s ; ( ; > 0) (7) When = 1, = 1, PID controller is obtained. When = 1, = 0, it becomes PI controller . At = 0, = 1, the PD controllers is obtained. The generalization of fractional controllers is sho wn in Figure 2. It sho ws the dif ferent structures a v ailable in the PID domain. FO-PID can be an ywhere in the plane. The structure of FO-PID controller is depicted in Figure 3. Figure 2. FO-PID generalization Figure 3. FO-PID controller Design of fr actional or der contr oller s using ... (Manoj D. P atil) Evaluation Warning : The document was created with Spire.PDF for Python.
294 r ISSN: 2088-8694 3.3. Contr oller 3. FO-PI When the = 0 and 2 > > 0 , then the FO-PI controller is obtained as, u ( s ) = K P + K I s (8) The FO-PI controller is depicted in Figure 4. The dif ference is that deri v ati v e part is not present, here. If steady state is k e y issue the FO-PI controller is the best choice. 3.4. Contr oller 4. FO-PD When the = 0 and 2 > > 0 , then the FO-PD controller is obtained as, u ( s ) = K P + K D s (9) Here, inte gral part is not in the controller structure. If the plant is ha ving transients then this controller i s best choice. The FO-PD controller is depicted in Figure 5. Figure 4. FO-PI controller Figure 5. FO-PD controller 4. PR OPOSED TUNING OF THE CONTR OLLERS Se v eral techniques are a v ailable for optimizing the parameters of controllers. Here, const rained optimization method with reference tracking is proposed t o optimize the controller parameters. Constrained minimization is the problem of finding a feasible solution that is optimum to a problem subject to constraints. The aim of constrained optimization is to con v ert the problem into an smaller subproblems which can then be solv ed by iterati v e methods. The characteristic of these methods is to translate the constrained problem to unconstrained problem by using an objecti v e function for constraints. In such a w ay , the s olution is found out for constrained problem using a sequence of unconstrained optimizations. The strate gy of the simulated model is sho wn in Figure 6. The constraints are k ept on the plant response, including maximum o v ershoot, settling time, ri se time as sho wn in Figure 7. Also, the reference tracking signal is pro vided as a goal to be achie v ed as sho wn in Figure 8. The model is simulated and the response is check ed with initial paramet ers. The simulation produces an un-optimized step response and the initial data for tuning. The plots are updated so that the design requirements can satisfied. Int J Po w Elec & Dri Syst, V ol. 11, No. 1, March 2020 : 291 – 301 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 r 295 Figure 6. Proposed scheme of optimization process of fractional controllers Figure 7. Step response specifications Figure 8. Reference tracking specifications The optimization algorithm are di vided as lar ge and medium scale. The lar ge scale algori thm requires linear algebra that does not store full matrices. This can be done by storing sparse matrices internally . A lar ge-scale algorithm can be used for a small problem. The medium-scale algorithm creates full matrices internally and requires dense linear algebra. If a problem is significantly lar ge, then full matrices require a lar ge amount of memory and a long time to e x ecute. Whereas, a medium-scale algorithm pro vides e xtra functionality such as additional constraint types which tends to pro vide better performance. The guidelines for constrained optimization with reference tracking is as follo ws: Step I: Start the process with the use of the ‘ inter ior point ’ algorithm which i s a lar ge-scale algorithm. ‘ inter ior point ’ not only handles sparse lar ge problems b ut also a small dense problems. The algorithm satisfies constraints at all iterations, and can reco v er from ‘ I nf = N aN ’ results. After that go for a dif ferent algorithms. The other algorithm may f ail, as some algorithms may use more time and memory , while some algorithms may not accept an initial point [28, 29]. If the problem did not solv e then mo v e to w ards step II. Step II: In this step, use sequential programming ‘ sq p ’ algorithm should be used. A ‘ sq p ’ algorithm is one of the nonlinear programming methods. ‘ sq p ’ can satisfy bounds at all iterati ons. The algorithm can reco v er easily from ‘ I nf = N aN ’. Here, a quadratic programming subproblem is solv ed at e v ery major iteration. This method allo ws constrained optimization [30]. An o v ervie w of ‘ sq p ’ is found in [31]. If the results are not as per the requirement then go for Step III. Step III: The ‘ activ e set ’ algorithm can tak e lar ge steps that gi v es speed to the algorithm. The algorithm is ef fecti v e for problems with nonsmooth constraints. In this method, acti v e constraints are included in canceling operation. These three steps will gi v e the required results. 5. RESUL TS AND DISCUSSION Systems wi th first order plus dead time models ha v e been classified in three cate gories based on relati v e dead time which is function of time T and dead time as gi v en belo w [32]: 1. Lag dominated systems - < 0 : 1 ; where = + T 2. Lag delay balanced systems - 0 : 1 < < 0 : 6 3. Delay dominated systems - > 0 : 6 Here, the results are sho wn for lag dominated systems only . Whereas, the methodology is tested ag ainst lag delay balanced and delay dominated system also. Design of fr actional or der contr oller s using ... (Manoj D. P atil) Evaluation Warning : The document was created with Spire.PDF for Python.
296 r ISSN: 2088-8694 5.1. Case 1. with PID The step response specification is sho wn in Figure 9 and the reference tracking specifications achie v ed with PID controller is sho wn in Figure 10. Optimization progress with PID controller is sho wn in T able 1. The obtained PID parameters are K P = 1 : 659 ; K I = 0 : 315 ; K D = 0 : 173 . Figure 9. Step response specifications with PID controller Ti me  (s eco nd s) Am pl itu de 0 10 20 30 40 50 60 70 80 90 100 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 Figure 10. Reference tracking specifications with PID controller 5.2. Case 2. with T -PID Step response specification is sho wn in Figure 11 whereas reference tracking specifications achie v ed with T -P ID controller is depicted in Figure 16. Optimization progress with T -PID control ler is sho wn in T able 2. The obtained T -PID parameters are K P = 0 : 2 ; K I = 0 : 312 ; K D = 1 : 144 ; = 0 : 8 . T able 1. Optimization Progress with PID Controller Iteration F- count Reference T racking Specifications (minimum) Step Response Specification (Upper) ( < =0) Step Response Specification (Lo wer) ( < =0) 0 8 217.2816 -0.0396 -0.3833 1 32 51.6097 -0.0146 -0.4510 2 49 50.4433 -0.0142 -0.4093 3 57 14.5348 -0.0101 -0.2185 4 65 6.4976 -0.0093 0.0017 5 75 5.0369 -0.0089 0.0028 6 84 3.8441 -0.0090 0.0021 7 92 3.8441 -0.0090 0.0021 Figure 11. Step response specifications with T -PID controller Ti me  (s eco nd s) Am pl itu de 0 10 20 30 40 50 60 70 80 90 100 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 Figure 12. Reference tracking specifications with T -PID controller Int J Po w Elec & Dri Syst, V ol. 11, No. 1, March 2020 : 291 – 301 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 r 297 T able 2. Optimization Progress with T -PID Controller Iteration F- count Reference T racking Specifications (minimum) Step Response Specification (Upper) ( < =0) Step Response Specification (Lo wer) ( < =0) 0 8 325.7320 -0.0357 -0.0521 1 32 288.3060 -0.0394 -0.0493 2 46 248.7523 -0.0347 -0.0344 3 54 296.1709 -0.0135 -0.0015 4 62 177.7636 -0.0119 0.0018 5 76 124.0190 -0.0107 0.0062 6 94 123.9967 -0.0104 0.0076 7 104 103.6499 -0.0080 0.0105 8 112 71.6489 -0.0086 0.0103 9 121 66.5799 -0.0123 -0.0012 10 129 35.0891 -0.0113 0.0030 11 137 18.5853 -0.0076 0.0105 12 145 5.1998 -0.0104 0.0069 13 153 4.0939 -0.0103 0.0075 14 161 3.8244 -0.0104 0.0071 15 169 3.6999 -0.0103 0.0074 16 180 3.5236 -0.0101 0.0089 17 199 3.5099 -0.0100 0.0091 18 211 3.5039 -0.0100 0.0092 19 223 3.5039 -0.0100 0.0092 5.3. Case 3. with FO-PI Step response specification is sho wn in Figure 13 whereas reference tracking specifications achie v ed with FO-PI controller is depicted in Figure 14. Optimization progress with FO-PI controller is sho wn in T able 3. The obtained FO-PI parameters are K P = 1 : 73 ; K I = 3 : 49 ; = 0 : 55 . Figure 13. Step response specifications with FO-PI controller Ti me  (s eco nd s) Am pl itu de 0 10 20 30 40 50 60 70 80 90 100 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 Figure 14. Reference tracking specifications with FO-PI controller T able 3. Optimization Progress with FO-PI Controller Iteration F- count Reference T racking Specifications (minimum) Step Response Specification (Upper) ( < =0) Step Response Specification (Lo wer) ( < =0) 0 8 3.0496e+04 -0.4158 -0.4947 1 16 6.9294e+03 -0.1180 -0.2182 2 24 2.9033e+03 -0.0369 -0.1182 3 32 1.9057e+03 -0.0044 -0.0610 4 40 1.6598e+03 0.0014 -0.0253 5 53 1.1243e+03 -0.0248 -0.0173 6 65 1.1886e+03 -0.0220 -0.0135 7 76 1.4792e+03 0.0023 -0.0098 8 111 1.5002e+03 0.0020 -0.0096 9 145 1.5214e+03 0.0019 -0.0096 10 161 1.6484e+03 0.0013 -0.0090 11 183 1.6540e+03 0.0012 -0.0089 12 224 1.6540e+03 0.0012 -0.0089 Design of fr actional or der contr oller s using ... (Manoj D. P atil) Evaluation Warning : The document was created with Spire.PDF for Python.
298 r ISSN: 2088-8694 5.4. Case 4. with FO-PD Step response specification is sho wn in Figure 15 whereas reference tracking specifications achie v ed with FO-PD controller is sho wn in Figure 16. Optimization progress with FO-PD controller is sho wn in T able 4. The obtained FO-PD parameters are K P = 107 : 8 ; K D = 1 : 968 ; = 0 : 63 . Figure 15. Step response specifications with FO-PD controller Ti me  (s eco nd s) Am pl itu de 0 10 20 30 40 50 60 70 80 90 100 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 Figure 16. Reference tracking specifications with FO-PD controller T able 4. Optimization Progress with FO-PD Controller Iteration F- count Reference T racking Specifications (minimum) Step Response Specification (Upper) ( < =0) Step Response Specification (Lo wer) ( < =0) 0 8 4.6123e+04 -0.5694 -0.6106 1 37 4.2841e+04 -0.5977 -0.5954 2 90 4.2834e+04 -0.5977 -0.5951 3 98 1.1651e+04 -0.1515 -0.2874 4 106 4.9829e+03 0.0738 -0.1398 5 114 2.3560e+03 0.0450 -0.0682 6 122 2.0831e+03 0.0241 -0.0318 7 130 2.1208e+03 0.0119 -0.0138 8 138 2.2145e+03 0.0049 -0.0051 9 146 2.3234e+03 0.0014 -0.0013 10 154 2.3443e+03 2.4580e-04 -1.3125e-04 11 162 2.3429e+03 2.4999e-05 -3.0894e-06 12 163 2.3429e+03 2.4999e-05 -3.0894e-06 5.5. Case 5. with FO-PID Step response specification is sho wn in Figure 17 whereas reference tracking specifications achie v ed with FO-PID controller is sho wn in Figure 18 . Optimization progress with FO-PID controller is sho wn in T able 5. The obtained FO-PID parameters are K P = 107 : 8 ; K I = 0 : 05 ; K D = 73 : 9 ; = 0 : 51 ; = 0 : 62 . Figure 17. Step response specifications with FO-PID controller Time  (second s) Ampli tude 0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 18. Reference tracking specifications with FO-PID controller Int J Po w Elec & Dri Syst, V ol. 11, No. 1, March 2020 : 291 – 301 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 r 299 T able 5. Optimization Progress with FO-PID Controller Iteration F- count Reference T racking Specifications (minimum) Step Response Specification (Upper) ( < =0) Step Response Specification (Lo wer) ( < =0) 0 8 3.3061e+04 -0.4414 -0.5273 1 16 8.4492e+03 -0.0965 -0.2408 2 24 3.6500e+03 0.0549 -0.1193 3 32 2.1838e+03 -0.0070 -0.0522 4 40 2.0712e+03 -0.0034 -0.0240 5 48 2.1917e+03 -0.0013 -0.0099 6 56 2.3034e+03 -6.4314e-04 -0.0033 7 64 2.3730e+03 -4.5363e-05 -6.6447e-04 8 72 2.3881e+03 1.2499e-06 -4.1323e-05 9 81 2.3873e+03 1.0911e-06 -1.2722e-05 10 90 2.3855e+03 1.2882e-06 -3.8773e-06 11 91 2.3855e+03 -3.8773e-06 1.2882e-06 Hence, all the a v ailable controllers can be tuned with the help of method discussed in this paper . Only the controller structure need to be changed in the simulation and according to constrained and reference tracking method e v ery type of fractional order controller can be optimized. All the fi v e controllers are compared with each other and FO-PID controller is found to be superior in all the fi v e cases. Since, the FO-PID controller is the combination of fractional inte grator and dif ferentiator . Fractional dif ferentiator tak es care of transient response while fractional inte grator looks after the steady state response. 6. CONCLUSION In this paper , dif ferent structures of FO-PID controllers are designed and applied to FO WDT proces s. The tuning methods of fractional controllers in v olv es comple x equations. So, it is dif ficult to find a solution to the problem. Because of that, the tuning method is simplified so that the controller can be tuned v ery easily and ef fecti v ely . So that, the constrained optimization algorithm with reference tracking specifications is found to pro vide solution to tune these controller strate gies. The proposed tuning algorithm can be applied to dif ferent control strate gies and dif ferent plant structures. REFERENCES [1] S. Y in, X. Li, H. Gao, and O. Kaynak, “Data-based techniques focused on modern industry: An o v ervie w , ” IEEE T r ans. Ind. Electr on., v ol. 62, no.1, pp.657-667, Jan 2015. [2] C. Anil and R. P adma Sree, “T uning of PID controllers for inte grating systems using direct synthes is method, ” ISA T r ans., v ol. 57, pp. 211-219, 2015. A v ailable: 10.1016/j.isatra.2015.03.002. [3] A. O‘Dwyer , Handbook of PI and PID Contr oller T uning Rules. London: Imperial Colle ge Press, 2009. [4] D. V al ´ erio and J. da Costa, “T uning of fractional PID controllers with Zie gler–Nichols-t ype rules, ” Signal Pr ocess., v ol. 86, no. 10, pp. 2771-2784, 2006. A v ailable: 10.1016/j.sigpro.2006.02.020. [5] T . KA CZOREK, F r actional linear systems and electrical cir cuits. Springer International Pu, 2016. [6] M. Shamseldin, M. Sa llam, A. Bassiun y and A. Ghan y , “ A no v el self-tuning fractional order PID control based on optimal model reference adapti v e system, ” Int. J . P ower Electr on. Drive Syst., v ol. 10, no. 1, p. 230, 2019. [7] S. Das, Functional F r actional Calculus. Berlin: Springer Berlin, 2014. [8] K. Sasikala and R. K umar , “Fractional PID controlled cascaded flyback switched mode po wer supply with enhanced time domain response, ” Int. J . P ower Electr on. Drive Syst., v ol. 10, no. 2, p. 909, 2019. [9] P . Prasad, M. P adma Lalitha and B. Sarv esh, “Fractional order PID controlled cascaded re-boost se v en le v e l in v erter fed induction motor system with enhanced res ponse, ” Int. J . P ower Electr on. Drive Syst., v ol. 9, no. 4, p. 1784, 2018. A v ailable: 10.11591/ijpeds.v9.i4.pp1784-1791. [10] S. Khubalkar , A. Junghare, M. A w are And S. Das, “Modeling and control of a permanent-magnet brushless DC motor dri v e using a fractional order proportional inte gral deri v ati v e controller , ” T urkish J . Electrical Engineering & Computer Sci., v ol. 25, pp. 4223-4241, 2017. [11] R. V erma, N. P ande y and R. P ande y , “Electronically tunable fractional order filter , ” Ar abian J . Sci. Eng ., v ol. 42, no. 8, pp. 3409-3422, 2017. A v ailable: 10.1007/s13369-017-2500-8. [12] S. Khubalkar , A. Chopade, A. Junghare, M. A w are and S. Das, “Design and realization of s tand-alone digital fractional order PID controller for b uck con v erter fed DC motor , ” Cir c. Syst. Signal Pr ocess., v ol. 35, no. 6, pp. 2189-2211, 2016. A v ailable: 10.1007/s00034-016-0262-2. Design of fr actional or der contr oller s using ... (Manoj D. P atil) Evaluation Warning : The document was created with Spire.PDF for Python.
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Orban, “ An interior algorithm for nonlinear optimization that combines line search and trust re gion steps, ” Mathematical Pr o gr amming , v ol. 107, no. 3, pp. 391-408, 2005. [30] M. J. D. Po well, “ A f as t algorithm for nonlinearly constrained optimization calculations, ” Lectur e Notes Math. Nu- merical Ana., pp. 144-157, 1978. [31] R. Fletc her , Pr actical methods of optimization. Chichester: John W ile y & Sons, 2008. [32] Y . Chen, T . Bhaskaran, and Xue Dingyu, “Practical T uning Rule De v elopme nt for Fractional Order Proportional and Inte gral Controllers, ” J . Comp. Nonlinear Dynamics, v ol. 3, no. 2, p. 021403, 2008. BIOGRAPHIES OF A UTHORS Mr . Manoj D . P atil w as born in Sangli , Maharashtra, India, in 1987. He has recei v ed his B.E. De gree in Electrical Engineering from Shi v aji Uni v ersity K olhapur , Maharashtra, India in 2009, and the M.E. De gree in Electrical Po wer Systems from Go v ernment Colle ge of E ngineering Aurang abad (which is af filiated to Dr . Babasaheb Ambedkar Marathw ada Uni v ersity Aura ng a bad), Maharashtra, India in 2011. He is w orking as Assistant Professor at Anna saheb Dange Colle ge of Engineering & T echnology , Ashta, Sangli, Maharashtra since July 2011. He is c urrently w orking to w ard the Ph.D. de gree with the Di vision of Electrical Engineering at Dr . Babasaheb Ambedkar T echnological Uni v ersity (B A TU), Lonere, Raig ad, Maharashtra, India. Int J Po w Elec & Dri Syst, V ol. 11, No. 1, March 2020 : 291 – 301 Evaluation Warning : The document was created with Spire.PDF for Python.