Design of different reference model based model reference adaptive controller for inversed model non-minimum phase system

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1 MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS ISSN: (Print) (Online) Vol. 4 No. June 07 pp DOI: 0.880/ep.0400 Licensed under CC BY-NC 4.0 A publication of IIETA Design of different reference odel based odel reference adaptive controller for inversed odel non-iniu phase syste Mita Pal * Gauta Sarkar Ranjit Kuar Barai Taal Roy Electrical Engineering Departent Jadavpur University Kolkata India Dept. of Electrical Engineering MCKV Institute of Engineering Liluah Howrah India Eail: itapal9@gail.co ABSTRACT This paper deonstrates the unique property of Model Reference Adaptive Controlled (MRAC) syste. Reference Model is used in the MRAC control structure plays a ajor role for the transient characteristics of the output response of controlled plant. An inversion based Non-iniu phase (NMP) syste always gives unbounded response MRAC schee not only stabilize this inverse NMP syste it also force the output trajectory to follow the path of the reference odel. Lyapunov stability theory based control logic has been applied to obtain the required control paraeters with adjustable gains. Keywords: Lyapunov Stability Theory Model Reference Adaptive Control Non-Miniu Phase Syste Reference Model.. INTRODUCTION In control syste engineering any approaches have been taken to track the desired trajectory. The purpose of the tracking control syste is to force the output of the controlled plant to follow the desired trajectory as close as possible. Very recent developents of tracking control for noniniu phase syste have been studied. Robust sliding ode learning control schee has been developed for chattering free convergence []. Tracking error can be iniized in NMP syste by using filtered basis function has been discussed in []. An additive-state-decoposition-based tracking control ethod has been used to obtain the output feedback tracking of uncertain non-iniu phase syste is represented in [3]. Aong these tracking control systes discussed in the papers provide no scope for designer to shape the output trajectories of the controlled plant as their choice. But Model Reference Adaptive control schee has this option to select the naics of reference odel as per control designer s requireent. The output response of the plant always follows the output trajectory of the reference odel in this control fraework. Moreover MRAC schee has been successfully applied in various fields in control syste engineering. Lyapunov based Model Reference Adaptive schee has been for Aerial vehicle D.C.otor has been proposed in [4] [5]. To suppress the vibration of piezoelectric sart structures [6] a direct MRAC is suggested. Model reference adaptive control for an ionic polyer etal coposite in underwater applications is experientally ipleented in Labview fraework [7]. A echatronics elevator syste driven by peranent agnet synchronous otor is developed and it has been experientally controlled by iniu energy odel reference schee [8]. A decentralized Model Reference Adaptive control schee for Twin rotor MIMO syste has been investigated [9] in siulation environent. A case stu like slow level process has been successfully perfored by MRAC schee [0]. But in the biological environent the applications are praiseworthy Model reference control for soft tissue operation [] or glucose regulation in blood [] have been efficiently siulated. It has been also checked that the perforance of MRAC is far better than Fuzzy Logic Controller and PI controller [3]. Though the observed research works highlighted the effectiveness of MRAC schee but they haven t shown any coparison between reference odel output and the controlled output of the plant. That interesting part of Model Reference Adaptive Control Schee will be focused in this presentation. The objective of this deonstration is to check whether the output trajectory of the MRAC controlled inversion based NMP plant truly follow the path of the Reference Model output response or not. Under daped over daped critically daped and un-daped plants have been chosen as reference odel for this control structure and step signal has been taken as final desired stea state value of output response. As the adaptive schee introduced non-linearity in the syste naics [4] Lyapunov stability theory has been applied to get the control paraeters of the controller. Variable gains of the control paraeters have been adjusted to get the appropriate control paraeters. 75

2 The paper is organized as follows; Section discusses brief theoretical back ground of Model Reference Adaptive Control. Matheatical background of control theory is represented by section 3. Section 4 and 5 present nuerical exaple and shows the siulated result respectively siulation results are discussed in section 6 and finally conclusions are drawn in section 7. where Y is reference odel output y is plant output r is reference input signal Let Control input: ur (3). MODEL REFERENCE ADAPTIVE CONTROL Model Reference Adaptive Control is a particular class of adaptive control syste. In this class of control syste a plant odel is selected called reference odel which is used to describe the desired characteristics of the plant to be controlled. 3. MRAC EMPLOYING LYAPUNOV STABILITY THEORY FOR SECOND ORDER PLANT Figure. Model reference adaptive control syste Two different approaches are applied in Model Reference Adaptive Control: Indirect Model Reference Adaptive Control and Direct Model Reference Adaptive Control. In indirect Model Reference adaptive Control unknown plant paraeter is estiated using a plant odel before the control input of the controlled plant. In direct approach appropriate control logic is applied to get the adjustable control paraeter autoatically and the output error becoes iniized. In these two forulations of MRAC Indirect approach needs identification odel as well as Reference Model whereas direct odel uses the reference odel only [5]. Direct Model Reference adaptive control consists of reference odel the plant the controller and the adjustent echanis. Error signal is produced by coparing the reference odel output and the output of the controlled plant. Adjustent Mechanis block calculate the appropriate control paraeters by using MIT rule[6] or Liapunov s stability theory[7] and with the help of these paraeters Controller block produce control input of the plant. Reference Model of the control schee is selected according to the desired perforance index so that the plant output follow the desired output trajectory. Tracking error is siply difference between the plant output and the reference odel outputs refer to figure. Considering nd order Reference Model: d y a br () And a nd order Plant Model: d y a bu () Figure. Block diagra Lyapunov based direct odel reference adaptive control where and are the control paraeters Tracking error: e y y (4) By subtracting the equation () fro equation () we get d e d y d y a bu ( a ) br (5) Replacing u in equation(5) fro equation (3)we get a b( r ) a br (6) d e By adding and deducting a with equation (6) we get a a b r b a a b r (7) d e a ( ) ( b a a ) ( b b ) r (8) d e d e de a ( b a a ) ( b b ) r (9) Assuing the initial values of the control paraeter and and integrating equation (9) with respect to t we get de ae ( b a a) y ( b b ) r (0) 76

3 Fro the error naics it is observed that the tracking error will go to zero if b a a and b b. The paraeter adjustent rule thus achieve the goal. Let the Lyapunov function for the error naics v e e b a a b b where b 0. b b () ( b a a) ( b b) V ( e ) e () b b This function is zero when e is zero and the controller paraeters are equal to the correct values. For a valid Lyapunov function tie derivative of lyapunov function ust be negative. The derivative is given by de d d e ( b a a) ( b b) (3) Substituting the value of equation (3) we get de fro equation (0) into 4. NUMERICAL EXAMPLE FOR SIMULATION This section presents nuerical exaple to deonstrate Model Reference Adaptive controlled inversed Noniniu phase syste [7].Reference Models used in this control stucture are of different daping nature. nd order non-iniu phase syste whose representstion is X AX Bu Y CX Du where A D. B 0 Reference odels are listed below. 4 0 C and Transfer function Nature of daping 5 S S 5 Under daped 5 S 0S 5 Critically daped 5 S 0S 5 Over daped 5 S 5 Undaped e[ ae ( b a a ) y ( b b ) r] d d ( b a a) ( b b) ae ( b a a ) ye ( b b ) re d d ( b a a) ( b b) (4) (5) 5. SIMULATION RESULT Inversion based NMP syste has been used as unstable syste to be controlled. Four reference odels with different daping nature have been attached with the proposed control structure and unit step signal has been applied for coand input. The following figures deonstrate output response of the plant before and after the application of MRAC. d ( ae b a a )( ye ) d ( b b )( re ) (6) If the paraeters are updated as follows d re (7) d ye (8) Figure 3.Unit step response of unstable inverse NMP syste without control re s (9) ye s (0). and are the control paraeters with adjustable gain Figure4. Unit step response of MRAC controlled inverse NMP syste using un-daped reference ode 77

4 7. CONCLUSIONS Figure 5. Unit step response of MRAC controlled inverse NMP syste using under daped reference odel This paper deonstrates that the plant which has been used as reference odel of MRAC has a significant role in tracking control syste. It provides uch flexibility to the designer as they can get the option to odify the plant output as per their requireents. Here reference odels with different daping ratio have been chosen to verify the role of the reference odel in MRAC structure. An inversion based Non-iniu phase syste has been selected for the unstable plant and step input signal has been taken as set point. Lyapunov stability theory has been applied to obtain the control paraeters of the Model Reference Adaptive Controller. It has been shown that output trajectories of the controlled plant always follow the path provided by the reference odel of this MRAC schee. REFERENCES Figure 6. Unit step response of MRAC controlled inverse NMP syste using critically daped reference odel Figure7. Unit step response of MRAC controlled inverse NMP syste using over-daped reference odel 6. SIMULATION RESULT ANALYSIS Uncontrolled Inversion based NMP syste has shown unbounded output response in figure 3. The siulation stu is perfored in MATLAB environent with four different naics of reference odels like undaped under daped over daped and critically daped characteristics. It has been shown that all the four output responses of controlled plant in figure4 figure5 figure6 and figure7 follow the reference odel trajectories which have been introduced in the MRAC structure. Specially in figure 5 we have checked that the transient characteristics of both the odels alost atched with each other. Stea state errors also found iniu. Stea State Error for three different reference odels: Nature of daping Under daped Critically Over daped daped Stea State Error In case of undaped reference odel though the reference input is step signal output response of the plant follow the oscillatory trajectory of the reference odel. [] Duan M. Raani S.K. Okwudire C.E. (05). Tracking control of non-iniu phase syste using filtered basis functions: a nurbs-based approach Proceedings of the ASME 05 Dynaic Systes and Control Conference Ohio USA pp. -0. [] Do T.M. Man Z. Jin J. Zhang C. Zheng J. Wang H. (05). Sliding ode learning control of noniniu phase non-linear syste International Journal of Robust and Nonlinear Control Vol. 6 No. pp DOI: 0.00/rnc.3406 [3] Wei Z.B. Ren J.R. Quan Q. (06). Further results on additive-state-decoposition-based output feedback tracking control for a class of uncertain non-iniu phase nonlinear systes 06 8th Chinese Control and Decision Conference (CCDC) Yinchuan China pp [4] Orsag M. Korpela C. Bogdan S. Oh P. (03). Lyapunov based odel reference adaptive control for aerial anipulation International Conference on Aircraft Systes Atlanta GA USA pp. -8. [5] Raesh G. Nayak H. (04). Liapunov s stability theory-based odel reference adaptive control for dc otor International Journal of Research in Science and Technology Vol. 4 No. 3 pp [6] Trajkov T.N. Koppe H. Gabbert U. (007). Direct odel reference adaptive control (MRAC) design and siulation for the vibration suppression of piezoelectric sart structures Counications in Nonlinear Science and Nuerical Siulation pp DOI: 0.06/ j.cnsns [7] Brufau-Penella J. Tsiakakis K. Laopoulos T. Puig- Vidal M. (008). Model reference adaptive control for an ionic polyer etal coposite in underwater applications IOP Publishing LtdSart Materials and Structures Vol. 7 No. 4 pp. -9. [8] Chen K.Y. (07). Model reference adaptive iniu-energy control for a echatronic elevator syste Optial Control Application and Methods Vol. 38 No. pp DOI: 0.00/oca.39 [9] Chelihi A. Cheachea M. (04). Model reference adaptive control for twin rotor ultiple-input and ultiple-output syste via inial controller synthesis Journal of Syste and Control Engineering Vol. 8 No. 6 pp

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