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P1111223206

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P1111223206

  1. 1. By Dr.Warsame H.Ali Roy G. Perry College of Engineering, Prairie View A&M University, Prairie View, TX A CASCADED PLUS STATE FEEDBACK OPTIMAL CONTROLLER FOR A THREE PHASE INDUCTION MOTOR BASED ON VECTOR CONTROL ALGORITHM 07/22/15ICGST 2012 Presented July 17
  2. 2. Outline Problem Statement Overview of Literature - induction motor - Linearization - Optimal Control  Proposed solution Method used Simulation results Analysis Conclusion & Recommendation Q/A 07/22/15ICGST 2012 Presented July 17
  3. 3. Problem Statement • To design an optimal 3-phase Induction motor controller based on vector control algorithm that rejects disturbances, eliminates steady state errors and improves dynamic performance. Design consists of proposed - state feedback gain which provides optimal control - plant cascaded with integrator to mitigate errors. 07/22/15ICGST 2012 Presented July 17
  4. 4. Overview of Literature Induction motors • Construction - Stator - Rotor Advantages of induction motors  Simple design - Series of three windings in the exterior stator Low cost - due to simple design Reliability - due to simple design - no brushes to replace like a dc motor 07/22/15ICGST 2012 Presented July 17
  5. 5. Induction motor characteristics 1. The induced torque is zero at synchronous speed 2. The curve is nearly linear between no-load and full load 3. The rotor resistance is much greater than the reactance, so the rotor current, torque increase linearly with the slip. 07/22/15ICGST 2012 Presented July 17
  6. 6. Motor Drives Induction motors can only rotate at a synchronous speed (Ns) speed that is related to the frequency by; Ns= 120 x f/p where p = # of poles in the stator f = supply frequency in hertz • To operate induction motors at speeds rather than the main supply speed an electric drive is needed  Electric drives are an interconnected combination of equipment that provides a means of adjusting the operating speed of a mechanical load. 07/22/15ICGST 2012 Presented July 17
  7. 7. Electric drives consists of two main components - controller - inverter Controllers are devices that use algorithms to control the performance of electric motors in a predetermined way Inverters supply the motor with power as prescribed by the controller Control algorithms are divided into two major categories; - Scalar Control - Vector Control 07/22/15ICGST 2012 Presented July 17
  8. 8. Optimal Control In optimal control, one tries to find a controller that provides the best possible performs with regards to a certain criteria with respect to some given measure of performance.  An example is a controller that uses the least amount of control-signal energy to take the output to zero. In this case the measure of performance (optimality criterion) would be the control-signal energy. Linear Quadratic Regulator The optimal control method chosen for the controller is a Linear Quadratic Regulator. 07/22/15ICGST 2012 Presented July 17
  9. 9. In a state feedback version of the LQR, all the states can be measured and are available for control. The cost is described by a quadratic function that is a function of state and control variables LQR guarantees the stability of the closed loop system as long as the system is controllable and observable. The matrices A, B, C and D of a linear quadratic regulator are constants based on the state space model, matrices Q and R are weighting matrices which are also constants ẋ = Ax +Bu y = Cx +Du . 07/22/15ICGST 2012 Presented July 17
  10. 10. Linearization Jacobian Linearization of nonlinear systems is mostly used for nonlinear system control because of the well established and design analysis tools for linear systems. The linearization method approximates the solution of the non-linear induction motor by decomposing the non linear model into a linear component and a non-linear component The Jacobian linearization matrix does not fully represent the dynamic system accurately because after linearization there is an affine non linear term that is disregarded in controller computations.  The non linear component is considered as a disturbance to the linear component and as a result disregarded in controller computations. 07/22/15ICGST 2012 Presented July 17
  11. 11. Proposed solution  A current fed induction motor mathematical model is developed and its performance compared to an existing voltage model 07/22/15ICGST 2012 Presented July 17
  12. 12. The induction motor nonlinear differential equations are linearized using Jacobian linearization and transformed into state space model. s p e e d time 07/22/15ICGST 2012 Presented July 17
  13. 13. The plant in state space model a) Generic cascaded controller scheme b) Controller design a b 07/22/15ICGST 2012 Presented July 17
  14. 14. Pre-defined Controller from ẋ = Ax +Bu y = Cx 1 = u, A1 = 0, B1 = 1, C1 = [1;1;1], Eẋ C = 1 Controller and plant cascaded 07/22/15ICGST 2012 Presented July 17
  15. 15. Augmented System State space model for the augmented system is ẋe (t) = Ae + Beu1(t) + Eer(t) ye(t) = y1(t) = Cexe(t) Where 07/22/15ICGST 2012 Presented July 17
  16. 16. With the initial time set to zero and the terminal time is set to infinity the problem is a infinite horizon problem and the feedback law in this case is given by the performance subject to the dynamics ẋe = Aex +Beu It has been shown in classical control theory that LQR optimal control has a linear state feedback u1(t) = – Kexe(t) where K(t) is a properly dimensioned matrix given as Ke(t)= R-1 * Be T * S 07/22/15ICGST 2012 Presented July 17
  17. 17. In which the matrix S > 0 is the solution for the Riccati equation 0= -SAe - Ae T S + SBeR-1 Be T S – Q  While there are two solutions to the algebraic Ricatti equation, the positive definite or positive semi- definite is used to compute the feedback gain.  The resulting closed- loop system becomes ẋe (t) = (Ae +BeKe) xe(t) + Eer(t) which is asymptotically stable due to the property of LQR design. 07/22/15ICGST 2012 Presented July 17
  18. 18. Regular Controller Cascaded Controller 07/22/15ICGST 2012 Presented July 17
  19. 19. 07/22/15ICGST 2012 Presented July 17
  20. 20. Simulation Results and Analysis Time to constant reference (Regular controller ) s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  21. 21. Cascaded controller time taken to reach reference speed s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  22. 22. Simulation Results and Analysis Time in seconds s p e e d Regular controller settling time 07/22/15ICGST 2012 Presented July 17
  23. 23. speed s p e e d Time in sec Cascaded controller settling time 07/22/15ICGST 2012 Presented July 17
  24. 24. Regular controller steady state error s p e e d Time in sec 07/22/15ICGST 2012 Presented July 17
  25. 25. Cascaded controller steady state error s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  26. 26. Disturbance introduced at time 2sec s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  27. 27. Regular controller torque after disturbance t o r q u e Time in seconds 07/22/15ICGST 2012 Presented July 17
  28. 28. Cascaded controller response to a disturbance at time 2 sec t o r q u e Time in seconds 07/22/15ICGST 2012 Presented July 17
  29. 29. Regular controller after disturbance at time 2sec s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  30. 30. Cascaded controller after disturbance at time 2 seconds s p e e d Time in seconds 07/22/15ICGST 2012 Presented July 17
  31. 31. Analysis Dynamic performance - regular controller takes time to settle down - cascaded controller has no settling time • Steady state performance - regular controller has steady state error. - cascaded controller eliminates steady state error • Response to disturbance - regular controller has an increased steady state error after the disturbance. - cascaded controller rejects the effects of disturbance. 07/22/15ICGST 2012 Presented July 17
  32. 32. conclusion  The cascaded plus state feedback controller gives better dynamic performance, steady state performance and responds better to a disturbance introduced to the plant by rejecting the effects of the disturbance. Future Work Implementation of the proposed control scheme. 07/22/15ICGST 2012 Presented July 17
  33. 33. References 1. Jasem M. Tamimi and Hussein M. Jaddu., “Nonlinear optimal controller for a three phase induction motor using quasilinearization,” proceedings of International Symposium on Communications, Control and Signal Processing, March 2006 2. Padmaraja Yedamale., “Speed Control of 3-Phase Induction Motor Using PIC18 Microcontrollers,” Microchip Technology Inc, 2002 3. Dal Y Ohm., “Dynamic Model of Induction Motors for Vector Control,” Drivetech, Inc,. Blacksburg, Virginia. 4. Jaroslav Lepka and Petr Steki., “3- Phase AC Induction Motor Vector Control Using a 56F80x, 56F8100 or 56F8300 Device. Freescale Semiconductor, Inc., 2004 5. P. C. Krause and C.H Thomas., “Simulation of Symmetrical Induction Machinery”. November 1965 6. T. Benmiloud and A Omari., “New Robust Approach of Direct Field Oriented Control of Induction Motor,” World Academy of Science and Engineering Technology 2010 07/22/15ICGST 2012 Presented July 17
  34. 34. 7. N. Mohan, Electric Drives: An Integrative Approach, MNPERE, Minneapolis, Minnesota, 2001. 8. Kevin Warwick, An introduction to control systems, River Edge, New Jersey: World Scientific Publishing Co., 1996 9. Donald E Kirk, Optimal Control theory; An introduction. Mineola, New York: Dover Publications Inc., 1998 10. B.K., Bose, Power Electronics and AC Drives, Prentice-Hall, Englewood Cliffs, New Jersey, 1986. 11. F.L. Lewis, V.L. Syrmos, Optimal Control, Wiley-Interscience, New York, 1995. 12.Dr. Ing and O.I. Okoro., “Matlab Simulation of Induction Machines with Saturable Leakage and Magnetizing Inductances” The Pacific Journal of Science and Technology April 2003. 13.Burak Ozpineci and Leone M. Tolbert., “ Simulink Implementation of Induction machine Model- A modular Approach”.IEEE 14.N. Mohan, Advanced Electric Drives: Analysis, Control and Modeling using Simulink, MNPERE, Minneapolis, Minnesota, 2001. 07/22/15ICGST 2012 Presented July 17
  35. 35. Q/A 07/22/15ICGST 2012 Presented July 17

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