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Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
Performance analysis of a second order system using mrac
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Performance analysis of a second order system using mrac

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  • 1. INTERNATIONAL JOURNAL OF ELECTRICALISSN 0976 – 6545(Print),International Journal of Electrical Engineering and Technology (IJEET), ENGINEERINGISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME & TECHNOLOGY (IJEET)ISSN 0976 – 6545(Print)ISSN 0976 – 6553(Online)Volume 3, Issue 3, October - December (2012), pp. 110-120 IJEET© IAEME: www.iaeme.com/ijeet.aspJournal Impact Factor (2012): 3.2031 (Calculated by GISI) ©IAEMEwww.jifactor.com PERFORMANCE ANALYSIS OF A SECOND ORDER SYSTEM USING MRAC Rajiv Ranjan1, Dr. Pankaj Rai2 1 (Assistant Manager (Projects)/Modernization & Monitoring, SAIL, Bokaro Steel Plant, India, rajiv_er@yahoo.com) 2 (Head of Deptt., Deptt. Of Electrical Engineering, BIT Sindri, Dhanbad ,India, pr_bit2001@yahoo.com) ABSTRACT In this paper model reference adaptive control (MRAC) scheme for MIT rule and Lyapunov rule has been discussed. These rules have been applied to the second order system. Simulation is done in MATLAB- Simulink for different value of adaptation gain and the results are compared for varying adaptation mechanisms due to variation in adaptation gain. Keywords: adaptive control, MRAC (Model Reference Adaptive Controller ), adaptation gain, MIT rule, Lyapunov rule 1. INTRODUCTION Traditional non-adaptive controllers are good for industrial applications, PID controllers are cheap and easy for implementation [1]. Nonlinear process is difficult to control with fixed parameter controller. Adaptive controller is best tool to improve the control performance of parameter varying system. Adaptive controller is a technique of applying some system identification to obtain a model and hence to design a controller. Parameter of controller is adjusted to obtained desired output [2].Model reference adaptive controller has been developed to control the nonlinear system. MRAC forcing the plant to follow the reference model irrespective of plant parameter variations. i.e decrease the error between reference model and plant to zero[5]. MRAC implemented in feedback loop to improve the performance of the system [3].There are many adaptive control schemes [4] but in this paper mainly MRAC control approach with MIT rule and Lyapunov rule has been discussed. Effect of adaption gain on system performance for MRAC using MIT rule 110
  • 2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEMEfor first order system[11] and for second order system[12] has been discussed. Compression ofperformance using MIT rule & and Lyapunov rule for first order system for different value ofadaptation gain is discussed [13],[14]. In this paper adaptive controller for second order systemusing MIT rule and Lyapunov rule has been discussed first and then simulated for different valueof adaptation gain in MATLAB and accordingly performance analysis is discussed for MIT ruleand Lyapunov rule for second order system .2. MODEL REFERENCE ADAPTIVE CONTROLModel reference adaptive controller is shown in Fig. 1. The basic principle of this adaptivecontroller is to build a reference model that specifies the desired output of the controller, andthen the adaptation law adjusts the unknown parameters of the plant so that the tracking errorconverges to zero [6] Figure 13. MIT RULEThere are different methods for designing such controller. While designing an MRAC using theMIT rule, the designer selects the reference model, the controller structure and the tuning gainsfor the adjustment mechanism. MRAC begins by defining the tracking error, e. This is simply thedifference 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 1 2 (3) J (θ ) = e (θ ) 2 111
  • 3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME dθ δJ δe = −γ = −γe (4) dt δθ δθ – Change in 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 MODELLINGModel Reference Adaptive Control Scheme is applied to a second order system using MIT rule.It is a well known fact that an under damped second order system is oscillatory in nature. Ifoscillations are not decaying in a limited time period, they may cause system instability. So, forstable operation, maximum overshoot must be as low as possible (ideally zero).This can automatically reduce the transient period of the system and improve the systemperformance. A critically damped second order system gives a characteristic without anyoscillations and this characteristic is similar to the first order system. But it is not feasible toachieve such system practically. In this paper a second order under damped system with largesettling time, very high maximum overshoot and with intolerable dynamic error is taken as aplant. The object is to improve the performance of this system by using adaptive control scheme.For this purpose, a critically damped system is taken as the reference model. Let the second ordersystem be described by: Y ( s)Where system = kG ( s ) where K is constant. U ( s)and Y ( s) = koG ( s) U c (s)using plant Gm ( s ) = ko G ( s ) 1 2 dθ δeLet cost function as J (θ ) = e (θ )  → = −γ e (6) 2 dt δθγ usually kept small 112
  • 4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEMETuning γ is crucial to adaptation rate and stability.Considering a Plant: ‫ݕ‬௣ = -a‫ݕ‬ሶ௣ - by + bu ሷ (7)Where ‫ݕ‬௣ 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: ‫ݕ‬௠ = -ܽ௠ ‫ݕ‬ሶ௠ - ܾ௠ y + ܾ௠ r ሷ (8)Where ‫ݕ‬௠ is the output of reference model (second order critically damped system) and r is thereference input (unit step input).Let the controller be described by the law: u = θ1r − θ 2 y p (9) e = y p − ym = G p u − Gm r (10)  b  y p = G pu =  2 (θ 1 r − θ 2 y p )  s + as + b  bθ 1 yp = 2 r s + as + b + bθ 2 bθ1 e= 2 uc − Gm r s + as + b + bθ 2 ∂e b = 2 r ∂θ1 s + as + b + bθ 2 ∂e b 2θ1 =− r ∂θ 2 ( s 2 + as + b + bθ 2 ) 2 b =− 2 yp s + as + b + bθ 2 If reference model is close to plant, can approximate: s 2 + as + b + bθ 2 ≈ s 2 + a m s + bm ∂e bm (11) = b / bm 2 uc ∂θ 1 s + a m s + bm ∂e bm = −b / b m 2 y plant (12) ∂θ 2 s + a m s + bm 113
  • 5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEMEController parameter are chosen as θ1 = bm /b and θ 2 = ( b − bm )/bUsing MIT dθ 1 ∂e  bm  (13) = −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 (14) dt ∂θ 2  m m Where γ = γ x b / bm = Adaption gainConsidering a =10, b = 25 and am =10 , bm = 12505. SIMULATION RESULTS FOR MIT RULETo analyze the behavior of the adaptive control the following model has designed in Matlab/SimulinkTime response for different value of adaption gain for MIT rule is given below: Figure 2 Figure 3 Figure 4 114
  • 6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME Figure 5 Figure 6 Figure 7 Figure 8The time response characteristics for the plant and the reference model are studied. It is observed that thecharacteristic of the plant is oscillatory with overshoot and undershoot whereas the characteristic of thereference model having no oscillation. Dynamic error between these reduced to zero by using modelreference adaptive control technique. 115
  • 7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEMEResults with different value of adaptation gain for MIT rule is summarized below: Without any With MRAC controller γ =0.1 γ =2 γ =3 γ =5Maximum 61% 0 0 0 0Overshoot (%)Undershoot (%) 40% 0 0 0 0Settling Time 1.5 22 2.4 2.25 2.1(second)Without controller the performance of the system is very poor and also having high value of undershootand overshoot(fig. 2). MIT rule reduces the overshoot and undershoot to zero and also improves thesystem performance by changing the adaptation gain. System performance is good and stable (fig. 4, fig.5 & fig. 6) in chosen range (0.1< γ >5) . Beyond the chosen range of adaption gain (0.1< γ >5)performance of system is very poor and become unstable (fig.7 & fig. 8). So for only suitable valueof adaptation gain MIT rule can be used to track the plant output closer to the reference output[12].6. LYAPUNOV RULEIn order to derive an update law using Lyapunov theory, the following Lyapunov function isdefined [15],[16]. ଵ ଵ ଵV = ߛ݁ ଶ + (ܾߠଵ − ܾ௠ )ଶ + ଶ௕ (ܾߠଶ + ܽ − ܾܽ௠ )ଶ (15) ଶ ଶ௕The time derivative of V can be found as ሶ ሶܸሶ = ߛ݁݁ሶ + ߠଵ (ܾߠଵ − ܾ௠ ) + ߠଶ (ܾߠଶ + ܽ − ܾܽ௠ ) (16)And its negative definiteness would guarantee that the tracking error converge to zero along thesystem trajectories.For first order system:The process dynamics ‫ݕ‬ሶ௣ + ܽ‫ݕ‬௣ = bu (17)Reference dynamics ‫ݕ‬ሶ௠ + ܽ௠ ‫ݕ‬௠ = ܾ௠ r (18)Inserting the dynamic equations of the plant and reference model in above equation ሶ ሶܸሶ = ߛ݁൫‫ݕ‬ሶ௣ − ‫ݕ‬ሶ௠ ൯+ ߠଵ (ܾߠଵ − ܾ௠ ) + ߠଶ (ܾߠଶ + ܽ − ܾܽ௠ ) ሶ = -ߛܽ௠ ݁ ଶ + (ߛ݁‫ߠ + ݎ‬ଵ )(ܾߠଵ − ܾ௠ ) + (ߠଶ − ߛ݁‫ݕ‬௣ )(ܾߠଶ + ܽ − ܾܽ௠ ) (19) 116
  • 8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEMEEq.(19) give following condition for negative definiteness and thus the update laws: (20) (21) ௗఏమ = ߛ݁‫ݕ‬௣ ௗ௧For MIT rule , above equation can be derived as : ௗఏభ ೘ ௔ = −ߛ݁ (௦ା ௔ ‫ݎ‬ ௗ௧ ೘) ௗఏమ ௔೘ = ߛ݁ (௦ା ௔ ‫ݕ‬௣ ௗ௧ ೘)From the above equation it is observed that MIT rule and Lyapunov theory is similar onlydifference is that MIT rule comprises an additional filter operation with reference model.Lyapunov updates laws for second order system can we written as:ௗ ௔೘ ߠௗ௧ ଵ = − ߛ݁ (௦మ ା௔ ‫ݎ‬ (22) ೘ ௦ା ௕೘ )ௗ ௔೘ ߠ = − ߛ݁ (௦మ ‫ݕ‬ (23)ௗ௧ ଶ ା௔೘ ௦ା ௕೘ ) ௣7. SIMULATION RESULTS FOR LYAPUNOV THEORYTo analyze the behavior of the adaptive control the following model was designed in Matlab/SimulinkTime response for different value of adaption gain for Lyapunov rule is as: 117
  • 9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME Figure. 9 Figure. 10 Figure. 11 Figure. 12 Figure. 13 Figure. 14In case of MIT rule, if adaptation gain is increased to chosen range (0.1< γ >5) the performanceof the system become very poor and system also become unstable (fig.7 & fig. 8). In this sectionLyapunov rule is used for designing of system in chosen range as well as beyond the chosenrange and then performance of the system is analyzed for different value of adaptation gain. 118
  • 10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME Without any With Lyapunov controller γ =0.1 γ =2 γ =3 γ =5Maximum 61% 0 0 0 40%Overshoot (%)Undershoot 40% 0 0 0 30%(%)Settling Time 1.5 18 1.9 1.6 1.4(second)In chosen range of adaption gain (0.1< γ >5) performance of system by using Lyapunov rule isbetter as compared to the MIT rule response (fig. 10, fig.11 & fig. 12).Beyond the chosen range performance of the system has also been analyzed for γ =7 and γ =10and it is observed that system having little oscillation but system is stable (fig. 13 & fig. 14).System is even also stable for the higher value of adaption gain γ =50.8. CONCLUSIONAdaptive controllers are very effective where parameters are varying. The controller parametersare adjusted to give desired result. This paper describes the MRAC by using MIT rule andLyapunov rule for second order system.Time response is studied for second order system using MIT rule and Lyapunov rule withvarying the adaptation gain. It is observed that in case of MIT rule, if adaptation gain increasesthe time response of the system also is improved for the chosen range of adaption gain andfurther system is unstable in the upper range. In Lyapnove rule, system is stable beyond thechosen range of adaptation gain. So with suitable value of adaptation gain in MIT rule andLyapunov rule plant output can be made close to reference model. It can be concluded thatperformance using Lyapunov rule is better than the MIT rule.REFERENCES[1] K.J.Astrom and T. Hagglund ”PID controllers, theory , design and tuning .InstrumentSociety of America, North California.[2] Marc Bosdan and Shankar Sastry “Adaptive control stability , convergence andRobustness” PH, New Jersy.[3] C.Quck”Direct model reference adaptive PI controller using the gradations approach”Procedding of IEEE’93 ,pp.447-450[4] G.Tao “Adaptive Control Design and Analysis” John Wiely and sons, New York,2003,[5] Slontine and Li, “Applied Nonlinear Control”, p 312-328, ©1991 by Prentice HallInternational Inc.[6] Karl J. astrom , Bjorn Wittenmark “Adative Control” Pearson Education Asia,2001.[7] Jan Erik Stellet, Influence of Adaptation Gain and Reference Model Parameters onSystem Performance for Model Reference Adaptive Control, World Academy of Science,Engineering and Technology 60 2011[8] R.Prakash, R.Anita, ROBUST MODEL REFERENCE ADAPTIVE PI CONTROL,Journal of Theoretical and Applied Information Technology 119
  • 11. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME[9] Rathikarani Duraisamy and Sivakumar Dakshinamurthy, An adaptive optimisationscheme for controlling air flow process with satisfactory transient performance, Maejo Int. J. Sci.Technol. 2010, 4(02), 221-234[10] Prasanta Sarkar1 Sandip Das2, APPLICATIONS OF TIME MOMENTS INESTIMATION AND CONTROL, XXXII NATIONAL SYSTEMS CONFERENCE, NSC 2008,December 17-19, 2008[11] P.Swarnkar, S.Jain and R. Nema “Effect of adaptation gain on system performance formodel reference adaptive control scheme using MIT rule” World Academy of science,engineering and technology, vol.70, pp 621-626, Oct’2010[12] P.Swarnkar, S.Jain and R. Nema “Effect of adaptation gain in model reference adaptivecontrolled second order system” ETSR-Engineering, Technology and applied science research,,vol.1, no,-3 pp 70-75, 2011[13] P.Swarnkar, S.Jain and R. Nema “Comparative analysis of mit rule and lyapunov in formodel reference adaptive control scheme” Innovative system design and engineering, vol.2 pp.154-162, Apr 2011[14] J.E.Stellet “Influence of Adaption gain and reference model parameters on systemperformance for for model reference adaptive control” World Academy of science, engineeringand technology, , pp 1761-1773, 2011.[15] J.J. E. Sloatine, Applied nonlinear Control. Prentice – Hall, Englewood Cliffs, 1991.[16] G.Dumont “ Eece 574 – adaptive control – lecturer notes model reference adaptivecontrol” Mar. 2011 120

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