Design & Implementation Of Fuzzy Parallel Distributed .
IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE)e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 12, Issue 4 Ver. II (Jul. – Aug. 2017), PP 20-28www.iosrjournals.orgDesign & Implementation of Fuzzy Parallel DistributedCompensation Controller for Magnetic Levitation SystemMilad Gholami1, Zohreh Alzahra Sanai Dashti2, Masoud Hajimani31, 2(Department of Electrical Engineering, Qazvin Branch, Islamic Azad University Qazvin, Iran)3(Department of Electrical Engineering, Alborz University Qazvin (Mohammadiyeh), Iran)Corresponding Author: Milad GholamiAbstract: This study applies technique parallel distributed compensation (PDC) for position control of aMagnetic levitation system. PDC method is based on nonlinear Takagi-Sugeno (T-S) fuzzy model. It is shownthat this technique can be successfully used to stabilize any chosen operating point of the system. All derivedresults are validated by experimental and computer simulation of a nonlinear mathematical model of the system.The controllers which introduced have big range for control the system.Keywords: Parallel Distributed Compensation, Magnetic Levitation System, --------------------------------------- ---------Date of Submission: 21-07-2017Date of acceptance: ----------------------------------- ----------I. IntroductionThere are different ways to provide non- contact surfaces on of which is through magnetic suspension,usually known as (Maglev). They utilize it in many critical ways and various engineering systems such asmagnetic high speed passenger trains, low friction ball bearings, suspension wind tunnel models. Based on thesource of suspension, Maglev systems can be classified as attractive or repulsive branches. These systems usuallyfunction as open or transitory loops and are defined through complicated differential equations showing thedifficulties of handling the system. Therefore designing a feedback controller to decide the precise location of themagnetic levitation is an important and complex task [1,2].Controlling magnetic levitation has been a matter ofgreat importance and continual concern of many researchers particularly in the current decade. Furthermore, theyhave utilized the linearization techniques to design control rules for levitation systems [3,4]. In 2004, Z. J. Yang,consider Robust position control of a magnetic levitation system via dynamic surface control technique [5].Reference [6] explain Robust Output Feedback Control of a Class of Nonlinear Systems Using a DisturbanceObserver. Yang et al. introduced adaptive robust output-feedback control method in 2008 [7]. In 2009, Linproposed a robust dynamic sliding-mode controller utilizing adaptive recurrent neural network [8]. RafaelMorales planed Nonlinear Control for Magnetic Levitation Systems Based on Fast Online AlgebraicIdentification of the Input Gain in 2011[9]. References [10-12] depict other kinds of nonlinear controllers basedon nonlinear methods. Other approaches used to control magnetic levitation system are Robust output feedbackstabilization of a field-sensed magnetic suspension system [13], linear controller design[14], as well as Simulationand control design of an uniaxial magnetic levitation system [15].In 2011, Chih-Min Lin designed SoPC-BasedAdaptive PID Control System for Magnetic Levitation System [16]. Mamdani and logical type Nero-Fuzzy basedon sliding mode separately and two types of FLEXNFIS designed for control magnetic levitation systems [17].Also, RBF sliding mode controller has been designed by Aliasghary and his colleagues in 2008 [18]. In thispaper, we consider a magnetic levitation system and propose a parallel distributed compensation (PDC) controllerfor magnetic levitation system. Recently, the Takagi–Sugeno (T–S) fuzzy model based controllers have beenapplied to several applications, such as Design and Implementation of Fuzzy Control on a Two-Wheel InvertedPendulum [19], T-S Fuzzy Model-Based Adaptive Dynamic Surface Control for Ball and Beam System [20] andT-S Fuzzy Maximum Power Point Tracking Control of Solar Power Generation Systems [21]. The fuzzycontrollers have been widely and successfully utilized for controlling many nonlinear systems. In General, anonlinear system can be transformed into a T–S fuzzy model, and then, using linear matrix inequality (LMI)approaches the parallel distributed compensation (PDC) fuzzy controller design is accomplished [22-24].Morerecently, authors in [25] utilized the neural adaptive controller for controlling magnetic levitation system. Their works isinspired from the method presented in [26]. Parallel distributed compensation have been utilized in [27] for speed control ofdigital servo system. This work is the extended version of authors from their early work in [28]. The rest of thepaper is organized as follows. Section II, contains the mathematical model of the magnetic levitation system.Section III deals with the parallel distributed compensation controller in detail. Section IV discusses thesimulation results of the proposed control schemes. In section V, we design and implement the controller for amagnetic levitation system model of Feedback company series (33-210). Finally, the conclusion is given inSection VI.DOI: 10.9790/1676-1204022028www.iosrjournals.org20 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for MagneticII. Mathematical Model Of The SystemIn this paper the magnetic levitation system is considered. It consists of a ferromagnetic ball suspendedin a voltage-controlled magnetic field. The vertical motion is only considered. Keeping the ball at a prescribedreference level is the objective. The schematic diagram of the system is shown in Figure 1. The dynamic model ofthe system can be written as [3]:dp vdt i dv mg c c dt p d ( L( p )i )Ri edt 2mL( p ) L1 2CP WherePthe ball positionvthe ball's velocityRthe coil’s resistanceithe current through the electromagnetethe applied voltagemthe mass of the levitated objectWe chose the states and control input as: x1 p, x2 v,magnetic levitation system are as follows:gcdenotes the gravitycthe magnetic force constantLthe coil’s inductance that is a nonlinearfunction of ball’s position (p)L1a parameter of systemx3 i and u e. Thus, the state space equations ofx1 x2x2 gc x3 c x3 m x1 2 R2C x2 x3 ux3 () LL x12Ly x1The parameters of the magnetic levitation system are as follows [3]. The coil’s resistance R 28.7 , theinductance L1 0.65H, the gravitational constant gc 9.81milliseconds-2, the magnetic force constant C 1.24 102and the mass of the ball m 11.87gand x1d 0.01mthe desired value ofx1.Figure1. Magnetic levitation systemIII. Parallel Distributed CompensationInitially, a given nonlinear plant is represented by the Takagi-Sugeno fuzzy model. Expressing the jointdynamics of each fuzzy inference (rule) by a linear system model is the main characteristics of the T-S fuzzymodel. Particularly, description of the Takagi-Sugeno fuzzy systems is done by fuzzy IF-THEN rules, whichlinear input-output relations of a system is locally represented by. The fuzzy system is of the following form [29,30]:Rule iIF q1(t) is Mi1, and qi(t) is Mir THENx(t ) Ai x(t ) Bi u (t )(4)FOR i 1,2, ,rDOI: 10.9790/1676-1204022028www.iosrjournals.org21 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for MagneticWhere asqT (t ) [q1 (t ), q2 (t ),., qn (t )] uT (t ) [u1 (t ), u2 (t ),., un (t )]Where x(t), u(t)and q(t)are the state vector, the input vector and the assumptive variables’ vector, respectively.r is the number of model rules, Mir is the fuzzy set and x(t ) Ai x(t ) Biu (t ) is the output for every i rule where Aiand Biare the state vector matrix, respectively. A R n n , B R1 n Parallel Distributed Compensation theory (PDC) is a compensation for every fuzzy model rule. Theresulting overall fuzzy controller, which is nonlinear in general, is a fuzzy blending of each individual linearcontroller shares the same fuzzy sets with the fuzzy system (4). PDC controller is often shown as below: WhereFiare the feedback gains.The fuzzy control rules have a linear controller (state feedback laws in this case) in theconsequent parts. We can use other controllers, for example, output feedback controllers and dynamic outputfeedback controllers, insteadof the state feedback. Thus, fuzzy controller is as follow:Rule iIF q1(t) is Mi1, and qi(t) is Mir THENu (t ) Fi x(t )(7)FOR i 1,2, ,ru (t ) ri 0 wi (t ) Fi x(t ) ri 0wi (t ) pwi (t ) M ij (q j (t ))j 1 The termMij (qj (t)) is the grade of membership (qj (t)) in Mij.Where the feedback gains with state feedback laws(Ackerman) will be attained as below:Fi qmiT ( Ai ),i 1, 2,., n. Where qmi is as follow:qmi [0, 0,., 0,1] ciT ci [ Bi , Ai Bi ,., Ai m 1 Bi ]i 1, 2,., n. The equilibrium of a fuzzy control system (4) is globally asymptotically stable if there exists a common positivedefinite matrix p the following two conditions are satisfied:AiT PAi P 0i 1, 2,., r. A. PDC Controller for Magnetic Levitation SystemIn section we used Takagi-Sugeno fuzzy model based on PDC for Magnetic levitation system. It shouldbe noted that the ball equilibrium points is between 1 to 2 centimeters. The state space (3) can be model bySugeno rule as follow:Rule iIF X1(t) is Mi. THENx(t ) Ai x(t ) Bi u (t )(13)FOR i 1,2, ,rMatrixes Ai and Bi computed by linearization equilibrium points as follow: 0 cx 2Ai 3ei 2 mx3ei 0 0 Bi 0 1 2C L1 x1ei DOI: 10.9790/1676-120402202810cx2 3ei2Lx1ei cx3ei 22mx1ei R 2C L1 x1ei , i 1, 2,., 5. 0 www.iosrjournals.org22 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for MagneticWhere equilibrium points as follow:x1e1 0.010, x2 e1 0, x3e1 0.3064x1e 2 0.012, x2 e 2 1.e 10 , x3e 2 0.3064x1e 3 0.014, x2e 2 1.e 8 , x3e 2 0.3677x1e 4 0.016, x2 e 2 1.e 6 , x3e 2 0.4290x1e 5 0.018, x2e 2 1.e 4 , x3e 2 0.4903 Considering the fact that equilibrium point of X1 is between 1 and 2 centimeters, the membership function of thesystem was determined as such a distance. Figure 2 shows the membership 0.0050.010.0150.020.025x 1 [cm]Figure 2. Membership functionsIn PDC method for each rule model a proper control rule is considered where is as follow:Rule iIF x1(t) is Mi. THENu (t ) Fi x(t )(16)FOR i 1, 2, , 5.The feedback gains with state feedback laws will be attained as below:F1 [-1.30e 04F2 [-1.46e 04F3 [-1.60e 04F4 [-1.75e 04F5 [-1.89e 04-2.75e 02-3.24e 02-3.73e 02-4.22e 02-4.74e 02-0.98e 02-0.99e 02-1.00e 02-1.01e 02-1.01e 02]]]]](17)Considering relation (12) matrix P is a definite positive matrix which is shown as follow:55.19 97.85 2.5p 2.5 0.0771 2.632 55.19 2.632 544.50 IV. Simulation ResultsIn this section, the results of simulation are shown. The Figure 3 shows the position of ball. Figures 4and 5 show the states system and Figure. 6 shows the signal control (the applied voltage) for system. In Figures 3,4, 5 and 6 we compare PDC controller result with PID controller and Mamdani Neuro-Fuzzy controller.Figure 3. Position of Ball for x1d 0.01mFigure 4. Simulation result x2 for x2d 0 rad/sDOI: 10.9790/1676-1204022028www.iosrjournals.org23 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for MagneticFigure 5. Simulation result x3 for x3d 0.3064 mAFigure 6. Simulation result signal controlResults show that PDC controller delivers the ball to the desired position much more quickly than MamdaniNeuro-Fuzzy and PID controllers and has shorter settling time than other controllers. In table I we compare PDCcontroller result with other controllers.TABLE I. COMPARISON RESULTS CONTROLLERSControllerMaximum signal control (volt) Settling time (Sec)PDC[purpose]130.11PID[18]-8000.9SLIDING MODE[12]250.5FEL[18]250.3RBF[18]1000.25MAMDANI NEURO FUZZY[17]2500.2ANFIS[17]550.28Basic flexible OR-type NFIS[17]540.25Parallel OR-type FLEXNFIS[17]550.6Basic flexible AND-type NFIS[17] 580.32Parallel AND-type FLEXNFIS[17] 650.3Table I show that PDC controller has shorter settling time than other controllers and it has less Maximum signalcontrol competed with other controllers. PDC controller as compared with sliding mode, FEL, RBF, MamdaniNeuro fuzzy, ANFIS and flexible fuzzy controller, has got much easier planning and mathematical calculation.V. Design And ImplementationFigure7. Magnetic levitation system (testbed)In this section, we introduced a magnetic levitation system model of Feedback company series (33-210)and we have design and implemented a PDC controller for it. Here, we inspired from the presented approaches in[31, 32, and 33] in our experimental evaluation of PDC controller for magnetic levitation system.Figure 7 showsthe system. The dynamic model of the system can be written as follow:DOI: 10.9790/1676-1204022028www.iosrjournals.org24 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for Magnetice R.i L ii x L x t 0 0 x2 tdx vdtdvim mg c( ) 2dtx The where x denotes the ball position,v is the ball's velocity, R is the coil’s resistance, i is the currentthrough the electromagnet, e is the applied voltage, m is the mass of thelevitated object, gc denotes the gravityand c is the magnetic force constant. L Is the coil’s inductance that is a nonlinear function of ball’s position ( x )and L0 is a parameter of system xe the desired value of x . In table II shows Parameter values.TABLE II. PARAMETER VALUESParametervaluesL10.65HR28.7 c1.477 10 4gc9.81m sec 2m0.021kgxe[ 1 2]cmObtaining and fitting frequency response data to estimate a plant model is a common practice in control design.Closed-loop frequency response data for the plant were obtained and plotted in the bode diagram shown inFigure.8.Figure 8. Frequency response data for the plantFrom the diagram, the experimental model of the maglev system was approximated as:GP exp ( s) 1.6ss( 1)( 1)30.530.5 It should be noted that the ball equilibrium points is between -1 to -2 centimeters. The dynamic model of thesystem linearized around 5 equilibrium points. Therefore there would be 5 subsystems. Each subsystem wasapproximated to a second degree system.x 1cme1x 1.2cme2x 1.4cme3x 1.6cme4x 1.8cme5 The state space equations (19) can be model by Sugeno rule as follow:Rule iIF X (t) is Mi. THEN (22)x(t ) Ai x(t ) Bi u (t )FOR i 1,2, ,rMatricesAi and Bi computed by linearization equilibrium points as follow:DOI: 10.9790/1676-1204022028www.iosrjournals.org25 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for Magnetic1 0 0 B1 48.8 1 0 1 0 0 B2 48.8 0 B3 48.8 1 0 0 B4 48.8 1 0A5 15.20 0 B5 48.8 0A1 30.5 0A2 26.1 0A3 19.6 0A4 17.6Considering the fact that equilibrium point of X is between -1 and -2 centimeters, the membership function of thesystem was determined as such a distance. Figure 9 shows the membership functions.The control rule is asfollow:Rule iIF X (t) is Mi. THENu (t ) Fi x(t )(23)FOR i 1, 2, , 5-1-0.50x 1 [cm]Figure 9. Membership functions.The feedback gains with state feedback laws will be attained as below:F1 [2.6405 0.073904]F2 [2.5530 0.073904]F3 [2.4858 0.073904]F4 [2.4187 0.073904]F5 [2.3515 0.073904] The figure 9 shows simulation of PDC controller. Figure 10. Simulation of PDC controllerThe Figures 10 and 11show results of experimental and Figures 12 and 13 show results computer simulation.Figures 10 and 12 show the position of ball. Figures 10 and 12 show signal control (the applied voltage) forsystem.DOI: 10.9790/1676-1204022028www.iosrjournals.org26 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for Magnetic-0.6-0.8x 1[cm]-1-1.2-1.4-1.6-1.8-202000400060008000 10000 12000 14000 16000time[ms]18000Figure11. Result of experimental position of Ball for xe 41.61.824x 10Figure 12. Result of experimental signal control-0.9-1-1.1-1.2x [ms]80001000012000Figure 13. Result of computer simulation position of Ball for xe 1cm2U[v]1.510.5010002000300040005000 6000time[ms]70008000900010000Figure 14. Result of computer simulation signal controlResults show that PDC controller delivers the ball to the desired position. The experimental and simulating resultsare similar with regards to the settling time and the control signal quantity.VI. ConclusionsIn this paper, the intelligent Fuzzy controller for the Magnetic Levitation system has been designedimplemented. This approach has been developed based on Takagi Sugeno Fuzzy model and Fuzzy paralleldistributed system. Designing this technique is to change a nonlinear system into a series of linear subsystems.Each linear subsystem will be controlled independently and separate sub controllers will be included in a Fuzzyfunction basis. The Fuzzy function basis under the control of the linear controllers applies to the system acontrolling signal relevant to the current system condition based on linear modeling.In this paper, suggested method maintains system stability for each equilibrium point. Results show thata PDC controller has shorter settling time than other controllers and it has less Maximum signal control competedwith other controllers. PDC controller as compared with other controllers has got much easier planning andmathematical calculation.References[1][2]L. Dong-kyu, L. Jun-Seong, H. Jae-Hung, and Y. Kawamura, "System modeling of a small flight vehicle using magnetic suspensionand balance system," in World Automation Congress (WAC), 2010, 2010, pp. 1-6.L. Xiaomei, L. Yaohua, L. Congwei, G. Qiongxuan, and Y. Zhenggang, "Study on status of parallel and multiple inverters fortraction system of Maglev," in Electrical Machines and Systems (ICEMS), 2010 International Conference on, 2010, pp. 210-214.DOI: 10.9790/1676-1204022028www.iosrjournals.org27 Page
Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for Magnetic[3][4][5][6][7][8][9][10]D. L. Trumper, S. M. Olson, and P. K. Subrahmanyan, "Linearizing control of magnetic suspension systems," Control SystemsTechnology, IEEE Transactions on, vol. 5, pp. 427-438, 1997.T. Wen and J.-c. Fang, "The exact feedback linerization control for the 2-DOF flywheel suspended by the passive and active hybridmagnetic bearings," in Electronics, Communications and Control (ICECC), 2011 International Conference on, 2011, pp. 2922-2926.Y. Zi-Jiang, K. Miyazaki, S. Kanae, and K. Wada, "Robust position control of a magnetic levitation system via dynamic surfacecontrol technique," Industrial Electronics, IEEE Transactions on, vol. 51, pp. 26-34, 2004.Y. Zi-Jiang, S. Hara, S. Kanae, and K. Wada, "Robust Output Feedback Control of a Class of Nonlinear Systems Using aDisturbance Observer," Control Systems Technology, IEEE Transactions on, vol. 19, pp. 256-268, 2011.Y. Zi-Jiang, K. Kunitoshi, S. Kanae, and K. Wada, "Adaptive Robust Output-Feedback Control of a Magnetic Levitation System byK-Filter Approach," Industrial Electronics, IEEE Transactions on, vol. 55, pp. 390-399, 2008.L. Faa-Jeng, C. Syuan-Yi, and S. Kuo-Kai, "Robust Dynamic Sliding-Mode Control Using Adaptive RENN for MagneticLevitation System," Neural Networks, IEEE Transactions on, vol. 20, pp. 938-951, 2009.R. Morales, V.
dynamics of each fuzzy inference (rule) by a linear system model is the main characteristics of the T-S fuzzy model. Particularly, description of the Takagi-Sugeno fuzzy systems is done by fuzzy IF-THEN rules, which linear input-output relations of a system is locally represented by. The fuzzy system is of the following form [29, 30]: Rule i IF q 1
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
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