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Transformation of a Mismatched Nonlinear Dynamic Systems into      Strict Feedback Form            by Johanna L. Mathieu a...
Johanna L. Mathieu2012, PostDoc at EEH – Power Systems Laboratory, ETH Zurich2006 – 2012, MS/PhD Student at the University...
4/18Outline1.   Objective2.   Dynamic System Description & Controllability          A bicycle example3.   Control Using F...
5/18 Objective1. Transform a mismatched nonlinear system into a strict   feedback form (also with a mismatched)2. Design t...
6/18 Dynamic System Description              steering angle                   heading angleMIMO SystemTwo inputs:  u1 forw...
7/18ControllabilitySee [Daizhan, C., Xiaoming, H, and Tielong, S., Analysisand Design of Nonlinear Control Systems, 2010]
8/18  Control using Feedback Linearization Dynamic Extension:   See [Sastry’s Nonlinear   Systems, 1999]#Relative degree =...
9/18Control using Feedback Linearization
10/18Transformation into Strict Feedback Form Goal:         Extended state equation  Strict feedback form         (availa...
11/18Transformation into Strict Feedback Form
12/18Dynamic Surface Control (DSC)
13/18Dynamic Surface Control (DSC)
14/18Simulation and Results                     x1 error                                   x1 position error              ...
15/18       Simulation and Results                                  Control saturated:                                    ...
16/18Simulation and Results       Sliding surfaces for x1   Sliding surfaces for x2
17/18Concluding Remarks A new method of defining states was presented for transform  a nonlinear mismatched system to the...
Thank you        Please comments and suggests!CONTROL OF ROBOT AND VIBRATION LABORATORY
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Seminar2012 d

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Seminar2012 d

  1. 1. Transformation of a Mismatched Nonlinear Dynamic Systems into Strict Feedback Form by Johanna L. Mathieu and J. Karl Hedrick Department of Mechanical Engineering, University of California, Berkeley, USA Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME Vol. 133, July 2011, Q2CONTROL OF ROBOT AND VIBRATION LABORATORY Speaker: Ittidej Moonmangmee 3rd years of PhD student Lecturer at STOU December 1, 2012
  2. 2. Johanna L. Mathieu2012, PostDoc at EEH – Power Systems Laboratory, ETH Zurich2006 – 2012, MS/PhD Student at the University of California,Berkeley, USA2006 – 2012, Affiliate at the Lawrence Berkeley NationalLaboratory, Berkeley, California, USA2008, Visiting researcher at the Bangladesh University ofEngineering and Technology Department of Civil Engineering,Dhaka, Bangladesh2005, Research Assistant at the MIT Sea Grant CollegeProgram, Cambridge, Massachusetts, USA2004 – 2005, U.S. Peace Corps Volunteer, Tanzania2000 – 2004, BS Student at the Massachusetts Institute ofTechnology, Cambridge, Massachusetts, USA J. Karl Hedrick (born 1944) is an American control theorist and a Professor in the Department of Mechanical Engineering at the University of California, Berkeley. He has made seminal contributions in nonlinear control and estimation. Prior to joining the faculty at the University of California, Berkeley he was a professor at the Massachusetts Institute of Technology from 1974 to 1988. Hedrick received a bachelors degree in Engineering Mechanics from the University of Michigan (1966) and a M.S. and Ph.D from Stanford University (1970, 1971). Hedrick is the head of the Vehicle Dynamics Laboratory at UC Berkeley. In 2006, he was awarded the Rufus Oldenburger Medal from the American Society of Mechanical Engineers.
  3. 3. 4/18Outline1. Objective2. Dynamic System Description & Controllability  A bicycle example3. Control Using Feedback Linearization4. Dynamic Surface Control (DSC)  Transformation into Strict Feedback Form  Sliding Surface & Control law5. Simulation & Results6. Conclusions
  4. 4. 5/18 Objective1. Transform a mismatched nonlinear system into a strict feedback form (also with a mismatched)2. Design two controllers via (i) Feedback Linearization method (ii) Dynamic Surface Control method to the bicycle tracks a desired trajectory steering angular velocity of the handle bars desired trajectory forward velocity of the bicycle3. Simulate and compare two controllers performance
  5. 5. 6/18 Dynamic System Description steering angle heading angleMIMO SystemTwo inputs: u1 forward velocity of the bicycle u2 angular velocity of the handle barsTwo outputs:
  6. 6. 7/18ControllabilitySee [Daizhan, C., Xiaoming, H, and Tielong, S., Analysisand Design of Nonlinear Control Systems, 2010]
  7. 7. 8/18 Control using Feedback Linearization Dynamic Extension: See [Sastry’s Nonlinear Systems, 1999]#Relative degree = #State = 6 So, it has no zero dynamics  Minimum-phase
  8. 8. 9/18Control using Feedback Linearization
  9. 9. 10/18Transformation into Strict Feedback Form Goal: Extended state equation  Strict feedback form (available for Dynamic Surface Control (DSC) design) Design a controller by Dynamic Surface Control (DSC)
  10. 10. 11/18Transformation into Strict Feedback Form
  11. 11. 12/18Dynamic Surface Control (DSC)
  12. 12. 13/18Dynamic Surface Control (DSC)
  13. 13. 14/18Simulation and Results x1 error x1 position error x2 error x2 position error t MATLAB ode45 Disturbances: Uncertainty bounds: w1 = 0.10 + 0.02r1(t) δ1, δ2, δ4 = 0.2, w2 = 0.15 + 0.02r2(t) δ3 = 0.25 w3 = 0.20 + 0.02r3(t) and δ5, δ6, δ7, δ8 are w4 = 0.10 + 0.02r4(t) change with the where r i(t)  1) (0, function of the state.
  14. 14. 15/18 Simulation and Results Control saturated: -10 to +10u2 (rad/s) Controller gains: k = [10, 10, 1, 1, 10, 10] Filter Time Constant: t τ = [0.05, 0.05, 0.05, 0.05]u4 (rad/s) From the dynamic extension: tu1 (rad/s) t
  15. 15. 16/18Simulation and Results Sliding surfaces for x1 Sliding surfaces for x2
  16. 16. 17/18Concluding Remarks A new method of defining states was presented for transform a nonlinear mismatched system to the strict feedback form Two controller techniques were designed  Feedback linearization (FL) with dynamic extension  Dynamic surface control (DSC) In the disturbance-free case, both FL & DSC performed tracking a desired trajectory In the present of disturbances, the DSC was better to reject it than the FL Tracking performance of the DSC can be designed by using the 1st order filter However, more control effort required for DSC
  17. 17. Thank you Please comments and suggests!CONTROL OF ROBOT AND VIBRATION LABORATORY

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