1. INTERNATIONAL Electrical EngineeringELECTRICAL ENGINEERING International Journal of JOURNAL OF and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME & TECHNOLOGY (IJEET)ISSN 0976 – 6545(Print)ISSN 0976 – 6553(Online)Volume 4, Issue 1, January- February (2013), pp.09-18 IJEET© IAEME: www.iaeme.com/ijeet.aspJournal Impact Factor (2012): 3.2031 (Calculated by GISI)www.jifactor.com ©IAEME ROBUST MODEL REFERENCE ADAPTIVE CONTROL FOR A SECOND ORDER SYSTEM 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 has been proposed in presence of first order and second order noise. The noise has been applied to the second order system. Simulation is done in MATLAB-Simulink and the results are compared for varying adaptation mechanisms for different value of adaption gain in presence of noise and without noise, which show that system is stable. Keywords: adaptive control, MRAC (Model Reference Adaptive Controller), adaptation gain, MIT rule, Noise, Disturbances. 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. Model reference adaptive controller has been developed to control the nonlinear system. MRAC forces the plant to follow the reference model irrespective of plant parameter variations. i.e 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 & Lyapunov rule for second order system for different value of adaptation gain is discussed [1]. 9
2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEMEIn this paper adaptive controller for second order system using MIT rule in the presence offirst order and second order bounded disturbance and unmodeled dynamics has beendiscussed first and then simulated for different value of adaptation gain in MATLAB andaccordingly performance analysis is discussed for MIT rule for second order system in thepresence 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 designer selects the reference model, the controller structure and thetuning gains for the adjustment mechanism. MRAC begins by defining the tracking error, e.This is simply the difference 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 as F (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 10
3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 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).Let the controller be described by the law: 11
4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 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 2Where Gd = Noise or disturbances. 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 θ 1 = 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 12
5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME5. SIMULATION RESULTS FOR MIT RULE WITHOUT BOUNDEDDISTURBANCE AND UNMODELED DYNAMICSTo analyze the behavior of the adaptive control the designed model has been simulated inMatlab-Simulink.The simulated result for different value of adaptation gain for MIT rule is given below: Figure 2 Figure 3 Figure- 4 13
6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME Figure 5 Figure 6The time response characteristics for the plant and the reference model are studied. It isobserved that the characteristic of the plant is oscillatory with overshoot and undershootwhereas the characteristic of the reference model having no oscillation. Dynamic errorbetween these reduced to zero by using model reference adaptive control technique. Results with different value of adaptation gain for MIT rule is summarized below: Without any With MRAC controller γ =0.1 γ =2 γ =4 γ =5 Maximum 61% 0 0 0 0 Overshoot (%) Undershoot 40% 0 0 0 0 (%) Settling 1.5 22 2.4 2.25 2.1 Time (second)Without controller the performance of the system is very poor and also having high value ofundershoot and overshoot(fig. 2). MIT rule reduces the overshoot and undershoot to zero andalso improves the system performance by changing the adaptation gain. System performanceis good and stable (fig. 3, fig. 4, fig. 5 & fig. 6) in chosen range (0.1< γ <5). 14
7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME6. 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 7 Figure 8 Figure 9 Figure 10 15
8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (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 =5Maximum 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, if the adaptation gain increases the overshoot andundershoot increases, but the settling time decreases. This overshoot and undershoot are dueto the first order bounded and unmodeled dynamics. It shows that even in the presence of firstorder bounded 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 11 Figure 12 16
9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME Figure 13 Figure 14Simulation 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 second order disturbance, if the adaptation gain increases the overshootand undershoot increases, but the settling time decreases. This overshoot and undershoot aredue to the second order bounded and unmodeled dynamics. It shows that even in the presenceof second order bounded and unmodeled dynamics, system is stable.7. CONCLUSION Adaptive controllers are very effective where parameters are varying. The controllerparameters are adjusted to give desired result. This paper describes the MRAC by using MITrule in the presence of first order & second order bounded and unmodeled dynamics.Time response is studied in the presence of first order & second order bounded andunmodeled dynamics using MIT rule with varying adaptation gain. It has been observedthat, disturbance added in the conventional MRAC has some oscillations at the peak ofsignal, hence these disturbances can be considered as a random noise. These oscillationsreduce with the increase in adaptation time. These overshoots and undershoots are due to thepresence of bounded and unmodeled dynamics or noise. It can be concluded that even in thepresence of bounded and unmodeled dynamics, system performance is stable. 17
10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEMEREFERENCES[1] Rajiv Ranjan and Dr. Pankaj rai “Performance Analysis of a Second Order Systemusing MRAC”, International Journal of Electrical Engineering & Technology(IJEET),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.Prakash, R.Anita, ROBUST MODEL REFERENCE ADAPTIVE PI CONTROL,Journal of Theoretical and Applied Information Technology[6] Kreisselmier, G and Narendra k.s “Stable Model Reference Adaptive Control in thepresence of bounded disturbances” IEEE Trans. Automat. Contr, AC-29, pp,202-211, 1984[7] Petros.A.Ioannou & Jing Sun, “Robust Adaptive Control”, Prentice Hal, secondedition, 1996[8] Samson, C.: “Stability analysis of adaptively controlled system subject to boundeddisturbances”, Automatica 19, pp. 81-86, 1983.[9] Ioannou, P. A., and G. Tao, ‘Dominant richness and improvement of performance ofrobust adaptive control’, Automatica, 25, 287-291 (1989).[10] Narendra.K.S and A.M. Annaswamy, “Stable Adaptive systems” Prentice-Hall,Second Edition, 1989[11] Kreisselmier, G., and Anderson, B.D.O. “Robust model reference adaptive control.”IEEE Transactions on Automatic Control 31:127-133, 1986.[12] Minakshi DebBarma, Sumita Deb, Champa Nandi and Sumita Deb, “MaximumPhotovoltaic Power Tracking Using Perturb & Observe Algorithm In Matlab/SimulinkEnvironment” International Journal of Electrical Engineering & Technology (IJEET),Volume 1, Issue 1, 2010, pp. 71 - 84, Published by IAEME[13] Dr. V.Balaji and E.Maheswari, “Model Predictive Control Techniques For Cstr UsingMatlab” International Journal of Electrical Engineering & Technology (IJEET), Volume 3,Issue 3, 2012, pp. 121 - 129, Published by IAEME [14] Dr. Manish Doshi, “Analysis Of Intelligent System Design By Neuro AdaptiveControl” International Journal Of Advanced Research In Engineering & Technology(IJARET), Volume2, Issue1, 2011, pp. 1 - 11, Published by IAEME 18
Be the first to comment