International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 9 ROBUST MODEL REFERENCE ADAPTIVE CONTROL FOR A SECOND ORDER SYSTEM Rajiv Ranjan 1 , Dr. Pankaj Rai 2 1 (Assistant Manager (Projects)/Modernization & Monitoring, SAIL, Bokaro Steel Plant, India,
[email protected]) 2 (Head of Deptt., Deptt. Of Electrical Engineering, BIT Sindri, Dhanbad ,India,
[email protected]) 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]. INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY (IJEET) ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), pp.09-18 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2012): 3.2031 (Calculated by GISI) www.jifactor.com IJEET © I A E M E International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 10 In this paper adaptive controller for second order system using MIT rule in the presence of first order and second order bounded disturbance and unmodeled dynamics has been discussed first and then simulated for different value of adaptation gain in MATLAB and accordingly performance analysis is discussed for MIT rule for 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 this adaptive controller is to build a reference model that specifies the desired output of the controller, and then the adaptation law adjusts the unknown parameters of the plant so that the tracking error converges to zero [6] Figure 1 3. MIT RULE There are different methods for designing such controller. While designing an MRAC using the MIT rule, the designer selects the reference model, the controller structure and the tuning 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) = e 2 / 2 (2) The parameter θ is adjusted in such a way that the loss function is minimized. Therefore, it is reasonable to change the parameter in the direction of the negative gradient of F, i.e (3) ) ( 2 1 ) ( 2 θ θ e J = International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 11 δθ δ γ δθ δ γ θ e e J dt d − = − = (4) – Change in γ is proportional to negative gradient of J (5) From cost function and MIT rule, control law can be designed. 4. MATHEMATICAL MODELLING IN PRESENCE OF BOUNDED AND UNMODELED DYNAMICS Model Reference Adaptive Control Scheme is applied to a second order system using MIT rule has been discussed [3] [4]. It is a well known fact that an under damped second order 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 as low as possible (ideally zero). In this section mathematical modeling of model reference adaptive control (MRAC) scheme for MIT rule in presence of first order and second order noise has been discussed Considering 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 controller output 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 is the reference input (unit step input). Let the controller be described by the law: ¦ ¹ ¦ ´ ¦ < − = > = − = = 0 , 1 0 , 0 0 , 1 ) ( where ) ( ) ( ) ( e e e e sign e sign e dt d e J δθ δ γ θ θ θ International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 12 (8) (9) Where d G = Noise or disturbances. If reference model is close to plant, can approximate: (10) (11) Controller parameter are chosen as 1 θ = m b /b and 2 θ = ( m b b − )/b Using MIT (12) (13) Where γ = ' γ x m b b / = Adaption gain Considering a =10, b = 25 and m a =10 , m b = 1250 ( ) r bk k s b as s bk y y r k s k b as s b u G G y r G u G G y y e y r u p p d p p m d p m p p 2 2 1 2 1 2 2 1 ) )( ( θ θ θ θ θ θ + + + + = − | ¹ | \ | + | ¹ | \ | + + = = − = − = − = p m y bk k s b as s bk r bk k s b as s k b e r bk k s b as s bk e r G r bk k s b as s bk e 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 ) )( ( ] ) )( ( [ ) )( ( ) )( ( θ θ θ θ θ θ θ θ + + + + − = + + + + − = ∂ ∂ + + + + = ∂ ∂ − + + + + = plant m m m m c m m m m m m y b s a s b b b e u b s a s b b b e b bk b s a s bk k s b as s + + − = ∂ ∂ + + = ∂ ∂ ≈ + + ≈ + + + + 2 2 2 1 2 2 2 / / ) )( ( θ θ θ e y b s a s b e e dt d e u b s a s b e e dt d plant m m m c m m m | | ¹ | \ | + + = ∂ ∂ − = | | ¹ | \ | + + − = ∂ ∂ − = 2 2 2 2 1 1 γ θ γ θ γ θ γ θ International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 13 5. SIMULATION RESULTS FOR MIT RULE WITHOUT BOUNDED DISTURBANCE AND UNMODELED DYNAMICS To analyze the behavior of the adaptive control the designed model has been simulated in Matlab-Simulink. The simulated result for different value of adaptation gain for MIT rule is given below: Figure 2 Figure 3 Figure- 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 14 Figure 5 Figure 6 The time response characteristics for the plant and the reference model are studied. It is observed that the characteristic of the plant is oscillatory with overshoot and undershoot whereas the characteristic of the reference model having no oscillation. Dynamic error between 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 controller With MRAC γ =0.1 γ =2 γ =4 γ =5 Maximum Overshoot (%) 61% 0 0 0 0 Undershoot (%) 40% 0 0 0 0 Settling Time (second) 1.5 22 2.4 2.25 2.1 Without controller the performance of the system is very poor and also having high value of undershoot and overshoot(fig. 2). MIT rule reduces the overshoot and undershoot to zero and also improves the system performance by changing the adaptation gain. System performance is good and stable (fig. 3, fig. 4, fig. 5 & fig. 6) in chosen range (0.1<γ <5). International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 15 6. MIT RULE IN PRESENT OF BOUNDED DISTRUBANCE AND UNMODELED DYNAMICS Consider the first order disturbance: d G = 1 1 + s Time response for different value of adaption gain for MIT rule in presence of first order disturbance is given below: Figure 7 Figure 8 Figure 9 Figure 10 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 16 Simulation results with different value of adaptation gain for MIT rule in presence of first order bounded and unmodeled dynamics is summarized below: Without any controller In presence of first order bounded and unmodeled dynamics γ =0.1 γ =2 γ =4 γ =5 Maximum Overshoot (%) 61% 0 27% 30% 35% Undershoot (%) 40% 0 15% 18% 21% Settling Time (second) 1.5 26 8 6 4 In the presence of first order disturbance, if the adaptation gain increases the overshoot and undershoot increases, but the settling time decreases. This overshoot and undershoot are due to the first order bounded and unmodeled dynamics. It shows that even in the presence of first order bounded and unmodeled dynamics, system is stable. Consider the second order disturbance: d G = 25 30 25 2 + + s s Time response for different value of adaption gain for MIT rule in presence of first order disturbance is given below: Figure 11 Figure 12 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 17 Figure 13 Figure 14 Simulation results with different value of adaptation gain for MIT rule in presence of second order bounded and unmodeled dynamics is summarized below: Without any controller In presence of second order bounded and unmodeled dynamics γ =0.1 γ =2 γ =4 γ =5 Maximum Overshoot (%) 61% 0 35% 40% 45% Undershoot (%) 40% 0 10% 15% 20% Settling Time (second) 1.5 30 12 10 7 In the presence of second order disturbance, if the adaptation gain increases the overshoot and undershoot increases, but the settling time decreases. This overshoot and undershoot are due to the second order bounded and unmodeled dynamics. It shows that even in the presence of second order bounded and unmodeled dynamics, system is stable. 7. CONCLUSION Adaptive controllers are very effective where parameters are varying. The controller parameters are adjusted to give desired result. This paper describes the MRAC by using MIT rule 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 and unmodeled dynamics using MIT rule with varying adaptation gain. It has been observed that, disturbance added in the conventional MRAC has some oscillations at the peak of signal, hence these disturbances can be considered as a random noise. These oscillations reduce with the increase in adaptation time. These overshoots and undershoots are due to the presence of bounded and unmodeled dynamics or noise. It can be concluded that even in the presence of bounded and unmodeled dynamics, system performance is stable. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME 18 REFERENCES [1] Rajiv Ranjan and Dr. Pankaj rai “Performance Analysis of a Second Order System using 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 Hall International Inc [3] P.Swarnkar, S.Jain and R. Nema “Effect of adaptation gain on system performance for 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. 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