Fuzzy logic based mrac for a second order system

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  • 1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME & TECHNOLOGY (IJEET)ISSN 0976 – 6545(Print)ISSN 0976 – 6553(Online)Volume 4, Issue 2, March – April (2013), pp. 13-24 IJEET© IAEME: www.iaeme.com/ijeet.aspJournal Impact Factor (2012): 3.2031 (Calculated by GISI)www.jifactor.com ©IAEME FUZZY LOGIC BASED MRAC FOR A SECOND ORDER SYSTEM 1 2 Rajiv Ranjan , Dr. Pankaj Rai 1 (Assistant Manager (Projects)/Modernization & Monitoring, SAIL, Bokaro Steel Plant, India 2 (Head of Deptt., Deptt. Of Electrical Engineering, BIT Sindri, Dhanbad ,India ABSTRACT In this paper a fuzzy logic approach for model reference adaptive control (MRAC) scheme for MIT rule in presence of first order and second order noise has been discussed. This noise has been applied to the second order system. Simulation is done in MATLAB- Simulink and the results are compared for varying adaptation mechanisms due to variation in adaptation gain with and without noise. The result shows that system is stable. Keywords: Adaptive Control, MRAC (Model Reference Adaptive Controller), Adaptation Gain, MIT rule, Fuzzy Logic, Noise 1. INTRODUCTION Robustness in Model reference adaptive Scheme is established for bounded disturbance and unmodeled dynamics. Adaptive controller without having robustness property may go unstable in the presence of bounded disturbance and unmodeled dynamics. Fuzzy logic controller gives the better response even in the present of bounded disturbance and unmodeled dynamics. Model reference adaptive controller has been designed to control the nonlinear system. MRAC forces the plant to follow the reference model irrespective of plant parameter variations to decrease the error between reference model and plant to zero[2]. Effect of adaption gain on system performance for MRAC using MIT rule for first order system[3] and for second order system[4] has been discussed. Comparison of performance using MIT rule & and Lyapunov rule for second order system for different value of adaptation gain is discussed [1]. Even in the presence of bounded disturbance and unmodeled dynamics system show stability in chosen range of adaptation gain[5]. 13
  • 2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME Now a days, Fuzzy logic control is used for nonlinear, higher order and time delaysystem[6]. Fuzzy logic is widely used as a tool for uncertain, imprecise decision makingproblems [7].Fuzzy logic controller located in error path of MRAC without disturbances hasbeen discussed [8]. In this paper adaptive controller for second order system using MIT rule in thepresence of first order and second order bounded disturbance and unmodeled dynamics hasbeen discussed first and then fuzzy logic controller has been applied. Simulation has beendone for different value of adaptation gain in MATLAB with fuzzy logic control and withoutfuzzy logic control and accordingly performance analysis has been discussed for MIT rulefor second order system in the presence of bounded disturbance and unmodeled dynamics.2. MODEL REFERENCE ADAPTIVE CONTROL Model reference adaptive controller is shown in Fig. 1. The basic principle of thisadaptive controller is to build a reference model that specifies the desired output ofthe controller, and then the adaptation law adjusts the unknown parameters of the plant sothat the tracking error converges to zero [6] Figure. 13. MIT RULE There are different methods for designing such controller. While designing an MRACusing the MIT rule, the reference model, the controller structure and the tuning gains for theadjustment mechanism is selected. MRAC begins by defining the tracking error, e, which isdifference between the plant output and the reference model output:system model e=y(p) −y(m) (1)The cost function or loss function is defined asF (θ) = e2 / 2 (2)The parameter θ is adjusted in such a way that the loss function is minimized. Therefore, it isreasonable to change the parameter in the direction of the negative gradient of F, i.e 14
  • 3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME 1 2 (3) J (θ ) = e (θ ) 2 dθ δJ δe = −γ = −γe (4) dt δθ δθ – Change in γ is proportional to negative gradient of J J (θ ) = e(θ ) dθ δe (5) = −γ sign(e) dt δθ  1, e > 0  where sign(e) =  0, e = 0 − 1, e < 0 From cost function and MIT rule, control law can be designed.4. MATHEMATICAL MODELLING IN PRESENCE OF BOUNDED ANDUNMODELED DYNAMICS Model Reference Adaptive Control Scheme is applied to a second order system usingMIT rule has been discussed [3] [4]. It is a well known fact that an under damped secondorder system is oscillatory in nature. If oscillations are not decaying in a limited time period,they may cause system instability. So, for stable operation, maximum overshoot must be aslow as possible (ideally zero).In this section mathematical modeling of model reference adaptive control (MRAC) schemefor MIT rule in presence of first order and second order noise has been discussedConsidering a Plant: = -a - by + bu (6)Consider the first order disturbance is = - +Where is the output of plant (second order under damped system) and u is the controlleroutput or manipulated variable.Similarly the reference model is described by: = - - y+ r (7)Where is the output of reference model (second order critically damped system) and r isthe reference input (unit step input). 15
  • 4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEMELet the controller be described by the law: u = θ1 r − θ 2 y p (8) e = y p − y m = G p Gd u − Gm r (9)  b  k  y p = G p Gd u =  2  (θ 1 r − θ 2 y p )  s + as + b  s + k  bkθ 1 yp = 2 r ( s + as + b)( s + k ) + bkθ 2 bkθ 1 e= 2 r − Gm r ( s + as + b)( s + k ) + bkθ 2 ∂e bk = 2 r ∂θ 1 ( s + as + b)( s + k ) + bkθ 2 ∂e b 2 k 2θ 1 =− 2 r ∂θ 2 [( s + as + b)( s + k ) + bkθ 2 ] 2 bk =− 2 yp ( s + as + b)( s + k ) + bkθ 2 If reference model is close to plant, can approximate: ( s 2 + as + b)(s + k ) + bkθ 2 ≈ s 2 + a m s + bm bk ≈ b (10) ∂e bm = b / bm 2 uc ∂θ 1 s + a m s + bm (11) ∂e bm = −b / b m 2 y plant ∂θ 2 s + a m s + bmController parameter are chosen as θ1 = bm /b and θ 2 = ( b − bm )/bUsing MIT dθ 1 ∂e  bm  (12) = −γ e = −γ  2   s + a s + b u c e dt ∂θ 1  m m  dθ 2 ∂e  bm  = −γ e =γ 2 s +a s+b y plant e  (13) dt ∂θ 2  m m Where γ = γ x b / bm = Adaption gainConsidering a =10, b = 25 and am =10 , bm = 1250 16
  • 5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME5. MIT RULE IN PRESENT OF BOUNDED DISTRUBANCE AND UNMODELEDDYNAMICSConsider the first order disturbance: 1 Gd = s +1Time response for different value of adaption gain for MIT rule in presence of first orderdisturbance is given below: Figure 2 Figure 3 Figure 4 Figure 5 17
  • 6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEMESimulation results with different value of adaptation gain for MIT rule in presence of firstorder bounded and unmodeled dynamics is summarized below: Without any In presence of first order bounded and unmodeled controller dynamics γ =0.1 γ =2 γ =4 γ =5 Maximum 61% 0 27% 30% 35% Overshoot (%) Undershoot 40% 0 15% 18% 21% (%) Settling Time 1.5 26 8 6 4 (second)In the presence of first order disturbance the adaptation gain increases the overshoot andundershoot with decrease in settling time. This overshoot and undershoot are due to the firstorder bounded and unmodeled dynamics. It shows that even in the presence of first orderbounded and unmodeled dynamics, system is stable.Consider the second order disturbance: 25 Gd = 2 s + 30s + 25 Time response for different value of adaption gain for MIT rule in presence of first orderdisturbance is given below: Figure 6 Figure 7 18
  • 7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME Figure 8 Figure 9Simulation results with different value of adaptation gain for MIT rule in presence of secondorder bounded and unmodeled dynamics is summarized below: Without any In presence of second order bounded and unmodeled controller dynamics γ =0.1 γ =2 γ =4 γ =5 Maximum 61% 0 35% 40% 45% Overshoot (%) Undershoot 40% 0 10% 15% 20% (%) Settling Time 1.5 30 12 10 7 (second)In the presence of first order disturbance the adaptation gain increase the overshoot andundershoot with decrease in settling time. This overshoot and undershoot are due to thesecond order bounded and unmodeled dynamics. It shows that even in the presence of firstorder bounded and unmodeled dynamics, system is stable.6. FUZZY LOGIC CONTROLLER The structure of fuzzy logic consists of three functional block as fuzzyficationinterface, fuzzy interface engine and defuzzyfication.. Design of fuzzy logic controllerinvolves the appropriate of a parameter set. Parameter set includes the input and outputs offuzzy logic controller, the number of linguistic terms and the respective membershipfunctions for each linguistic variable, the interface mechanism, the rule and the fuzzifiactionand defuzzyfication method.The fuzzy controller implements a rule base made set of IF-THEN type rule. An example ofIF-THEN rule is following.IF e is negative big(NB) and de is negative big(NB) THEN u is positive big(PB).In this case we have considered five number of triangular membership functions. Rule tableis shown in table 1. 19
  • 8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME Table no: 1 de e NB NS Z PS PB NB PB PB PB PS Z NS PB PB PS Z NS Z PB PS Z NS NB PS PS Z NS NB NB PB Z NS NB NB NB7. FUZZY ADAPTIVE CONTROLLER To achieve better time response fuzzy logic controller is applied with adaptivecontroller. Proposed method improves the rise time, settling time, overshoot and steady stateerror. Fuzzy logic controller is located in error path of MRAC in presence of first order andsecond bounded and unmodeled dynamics to minimize the error. Proposed adaptive MRACis shown in figure 10. ym e Reference Model u Plant Noise Controller r Fuzzy Logic Figure 1010. FUZZY LOGIC CONTROL WITH MIT RULE IN PRESENT OF BOUNDEDDISTRUBANCE AND UNMODELED DYNAMICSConsider the first order disturbance: 1 Gd = s +1Time response for different value of adaption gain for MIT rule with fuzzy logic controller inpresence of first order disturbance is given below: 20
  • 9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME Figure 11 Figure 12 Figure 13 Figure 14Simulation results with different value of adaptation gain for MIT rule in presence of firstorder bounded and unmodeled dynamics is summarized below: With Fuzzy Logic γ =0.1 γ =2 γ =4 γ =5 Maximum 0 15% 20% 26% Overshoot (%) Undershoot 0 3% 6% 8% (%) Settling Time 15 7 5 3 (second) 21
  • 10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEMEWith fuzzy logic controller, it is observed that overshoot, undershoot and settling timeconsiderably has been improved even in presence of first order bounded and unmodeleddynamics, showing stability of the system.Consider the second order disturbance: 25 Gd = 2 s + 30s + 25Time response for different value of adaption gain for MIT rule with fuzzy logic controller inpresence of second order disturbance is given below: Figure 15 Figure 16 Figure 17 Figure 18 22
  • 11. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEMESimulation results with different value of adaptation gain for MIT rule in presence of secondorder bounded and unmodeled dynamics is summarized below: With Fuzzy Logic γ =0.1 γ =2 γ =4 γ =5 Maximum 0 16% 20% 24% Overshoot (%) Undershoot 0 3% 4% 6% (%) Settling Time 17 8 6 4 (second)With fuzzy logic controller, it is observed that overshoot, undershoot and settling timeconsiderably has been improved even in presence of second order bounded and unmodeleddynamics, showing stability of the system.8. CONCLUSION The use of Fuzzy logic controller gives that better result as compared to conventionalModel Reference adaptive Controller. Comparison of result between conventional MRACand with fuzzy logic controller shows considerable improvement in overshoot, undershootand settling time. Time response is studied in the presence of first order & second order bounded andunmodeled dynamics using MIT rule with varying the adaptation gain using fuzzy controller.It is observed that, disturbance added in the conventional MRAC has some oscillations at thepeak of signal is considered as a random noise. This noise has been reduced considerably bythe use of fuzzy logic controller. It can be concluded that fuzzy logic controller shows betterresponse and can be considered good enough for second order system in presence of firstorder & second order bounded and unmodeled dynamics.REFERENCES[1] R.Ranjan and Dr. Pankaj rai “Performance Analysis of a Second Order System usingMRAC” International Journal of Electrical Engineering & Technology, Volume 3, Issue 3,October - December (2012), pp. 110-120. Published by IAEME[2] Slontine and Li, “Applied Nonlinear Control”, p 312-328, ©1991 by Prentice HallInternational Inc[3] P.Swarnkar, S.Jain and R. Nema “Effect of adaptation gain on system performancefor model reference adaptive control scheme using MIT rule” World Academy of science,engineering and technology, vol.70, pp 621-626, Oct’2010[4] P.Swarnkar, S.Jain and R. Nema “Effect of adaptation gain in model referenceadaptive controlled second order system” ETSR-Engineering, Technology and appliedscience research,, vol.1, no,-3 pp 70-75, 2011[5] R.Ranjan and Dr. Pankaj rai “ROBUST MODEL REFERENCE ADAPTIVECONTROL FOR A SECOND ORDER SYSTEM.’ International Journal of ElectricalEngineering & Technology, Volume 4, Issue 1, (Jan-Feb 2013), Page no. 9-18. Published byIAEME. 23
  • 12. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME[6] S.R.Vaishnav, Z.J.Khan, “Design and Performance of PID and Fuzzy logic controllerwith smaller rule base set for highr order system” World Congress on engineering andCoputer Science, 2007[7] M.Muruganandam, M. Madheshwaran “Modeling and simulation of modified Fuzzylogic controller for various type of DC motor drives” International conference for control,automation, communication and energy conservation, 2009 [8] S.Z.Moussavi, M. Alasvandi & Sh, Javadi “Speed Control of Permanent Magnet DCmotor by using combination of adaptive controller and fuzzy controller” International Journalof computer application , Volume 25, No-20 August 2012. [9] Jasmin Velagic, Amar Galijasevic," Design of fuzzy logic control of permanentmagnet DC motor under real constraints and disturbances", IEEE International Symposiumon Intelligent Control, 2009. [10] A. Suresh kumar, M. Subba Rao, Y.S.Kishore Babu, "Model reference linear adaptivecontrol of DC motor Using Fuzzy Controller", IEEE Region 10 Conference, 2008.[11] VenkataRamesh.Edara, B.Amarendra Reddy, Srikanth Monangi and M.Vimala,“Analytical Structures For Fuzzy Pid Controllers and Applications” International Journal ofElectrical Engineering & Technology (IJEET), Volume 1, Issue 1, 2010, pp. 1 - 17, Publishedby IAEME.[12] M.Gowrisankar and Dr. A. Nirmalkumar, Srikanth Monangi and M.Vimala,“Implementation & Simulation of Fuzzy Logic Controllers for the Speed Control OfInduction Motor and Performance Evaluation of Certain Membership Functions”International Journal of Electrical Engineering & Technology (IJEET), Volume 2, Issue 1,2011, pp. 25 - 35, Published by IAEME. 24