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MIMO LQG/LTR Control

MIMO LQG/LTR Control. for the Earthmoving Vehicle Powertrain Simulator (EVPS). group seminar on Oct. 19th, 2000 [presenter] Rong Zhang [advisor] Prof.. Andrew Alleyne [team partner] Eko Prasetiawan [project sponsor] Caterpillar. A LLEYNE R ESEARCH G ROUP. Overview.

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MIMO LQG/LTR Control

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  1. MIMO LQG/LTR Control for the Earthmoving Vehicle Powertrain Simulator (EVPS) group seminar on Oct. 19th, 2000 [presenter] Rong Zhang [advisor] Prof.. Andrew Alleyne [team partner] Eko Prasetiawan [project sponsor] Caterpillar ALLEYNE RESEARCH GROUP

  2. Overview • 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions ALLEYNE RESEARCH GROUP, M&IE/UIUC

  3. 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions 1. Problem statement • Introduction to the Earthmoving Vehicle Powertrain • An analogy between passenger vehicle powertrain and EVP • EVPS schematic and I/O list • Need for coordination • A tracking example ALLEYNE RESEARCH GROUP, M&IE/UIUC

  4. An analogy • Passenger Vehicle • Prime mover:Usually Spark-Ignition type engine (gas) • Torque Converter:Mechanical gearbox • Resistance speed control:Brake • Earthmoving Vehicle • Prime mover:Usually Compression-Ignition type engine (diesel) • Torquepressure converter:Hydraulic pump • Resistance speed control:Flow valve ALLEYNE RESEARCH GROUP, M&IE/UIUC

  5. EVPS schematic... 1 4 3 5 2 3 2 1 ALLEYNE RESEARCH GROUP, M&IE/UIUC

  6. … and I/O list • A MIMO control system • Controlled outputs: load speeds (3) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  7. Need for coordination! • A tracking example Tracking references Node 1: A rising stepNode 2: 0Node 3: 0 Using only one input: Flow Valve 1... nm1(rpm) nm2(rpm) nm3(rpm) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  8. 5 control inputs without coordination... [Q] How to take actions at the right time, right direction and right amount?[A] Coordination needed ! ALLEYNE RESEARCH GROUP, M&IE/UIUC

  9. 2. Introduction to LQG/LTR control • 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions • Pole placement? • “Performance” vs. “Cost” • LQR controller -- “optimal” feedback law • LQR estimator -- “optimal” filter by Kalman • LQG controller design • Optimal controller + Optimal estimator • LQG/LTR controller design • Optimal + Optimal  Robustness ALLEYNE RESEARCH GROUP, M&IE/UIUC

  10. Pole placement? Poles will be here! [Q1] Where should the target poles be placed? Too slow? poor performance! Too fast? expensive controller and surprising power bill! [Q2] Is there an “optimal” controller balancing both Performance and Cost? “Punishment philosophy” ALLEYNE RESEARCH GROUP, M&IE/UIUC

  11. LQR controller In this method, pole locations are not designed directly. Instead, find a good u=-Kx that minimizes: • Q and R are Performance Index or “Punishment Matrices” • Want a quicker state convergence? make Q bigger to punish large states! • Want to keep control efforts within saturation range or at a lower cost? make R bigger to punish overacting inputs! [Solution] Theoretical: ARE equation finds us a good K Practical: Matlab command ‘lqr’ ALLEYNE RESEARCH GROUP, M&IE/UIUC

  12. LQR estimator Not all the states are available, how to construct them from y? An estimator Find a good L(ue=-LCe) that minimizes: [Solution] Theoretical: ARE equation Practical: Matlab command‘lqr’ ‘kalman’ If Qe and Re are determined by process and measurement noise level...A Kalman Filter! ALLEYNE RESEARCH GROUP, M&IE/UIUC

  13. LQG = LQR control + Kalman filter • It’s a Optimal + Optimal design, but is it “optimal” in the sense of robustness? No! LQG/LTR = LQG + Robustness recovery • Using a recovery procedure (r=0 to inf), to make the LQG closed-loop closer to that of the Target Loop: the ideal LQR loop with full-state feedback. [Solution] Theoretical: Loop Transfer Recovery procedure Practical: Matlab command ‘ltru’ ‘ltry’ ALLEYNE RESEARCH GROUP, M&IE/UIUC

  14. Optimal + Optimal  Robustness LQR with Full-state feedback LQG with measurements feedback Singular Value Bode Plot  Singular Value Bode Plot  r = 0 (no recovery) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  15. Loop transfer recovery... Closer to the target loop Singular Value Bode Plot  Singular Value Bode Plot  r = 1 (small recovery) r = 105 (large recovery) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  16. 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions 3. EVPS LQG/LTR design • Plant Model (14 states) to Design Plant Model (17 states) • To insure 0 tracking errors to step inputs, the PM is augmented by 3 free integrators. • LQG design • Good “Punishment Matrices” are found and tested • LQG/LTR design • Robustness or the ideal LQR is recovered ALLEYNE RESEARCH GROUP, M&IE/UIUC

  17. EVPS System Plant ModelA14x14 Design Plant ModelA17x17 Three 1/s’ added to insure 0 tracking error ALLEYNE RESEARCH GROUP, M&IE/UIUC

  18. LQG Controller LQG/LTR Controller ALLEYNE RESEARCH GROUP, M&IE/UIUC

  19. 4. EVPS LQG/LTR performance • 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions • Simultaneous tracking • Different nodes track different speed references • The total flow demand changes • Disturbance rejection • One of the 3 nodes is subject to a pressure disturbance • The TOTAL flow demand does not change • The distribution of pressures among the 3 nodes is changed ALLEYNE RESEARCH GROUP, M&IE/UIUC

  20. Simultaneous speed-tracking • nm1tracking +/- 100rpm reference • nm2being regulated • nm3tracking - 60rpm reference nm1(rpm) nm2(rpm) nm3(rpm) Opposite direction Same direction ALLEYNE RESEARCH GROUP, M&IE/UIUC

  21. Pressures of simu-tracking • pd1increased to push through more flow • pd2unchanged to maintain the same flow • pd3decreases to push through less flow pd1(MPa) pd2(MPa) pd3(MPa) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  22. Control inputs of simu-tracking • Throttle when total flow demand • Pump when total flow demand • Flow 1 when speed reference 1 • Flow 2 compensates for pressure resulted from total flow • Flow 3 when speed reference 3 ALLEYNE RESEARCH GROUP, M&IE/UIUC

  23. Pressure disturbance at node 1 Pressure step as disturbance is applied at node 1 only pd1(MPa) pd2(MPa) Neighbor node pressure doesn’t change significantly pd3(MPa) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  24. Speeds of disturb. rejection • nm1decreases when disturbance pressure squeezes out some flow; then regulated by the controller • nm2increases by pressure disturbance squeezes in some flow from neighbor node; then regulated by the controller • nm3 increases by pressure disturbance squeezes in some flow from neighbor node; then regulated by the controller nm1(rpm) nm2(rpm) nm3(rpm) ALLEYNE RESEARCH GROUP, M&IE/UIUC

  25. Control inputs of disturb. rejection total flow demand not changed! • Throttle compensates for small total pressure • Pump doesn’t need to change much • Flow 1 to fight disturbance pressure • Flow 2 compensates for upstream pressure caused by load 1 • Flow 3 compensates for upstream pressure caused by load 1 ALLEYNE RESEARCH GROUP, M&IE/UIUC

  26. 1. Problem statement • 2. Introduction to LQG/LTR control • 3. EVPS LQG/LTR design • 4. EVPS LQG/LTR performance • 5. Conclusions 5. Conclusions • An LQG/LTR MIMO controller is successfully designed and implemented • The system: 14 states, 9 measurements, 5 inputs • The design plant model with free integrators: 17 states • The LQG/LTR controller: 17 states, 9 inputs, 5 outputs • It has satisfying tracking and disturbance rejecting performance • It’s robustness and working range are subject to further validation • Model reduction technique will be used to simplify the controller ALLEYNE RESEARCH GROUP, M&IE/UIUC

  27. References • M. Athans, "A tutorial on the LQG/LTR method," presented at American Control Conference, Seattle, WA, 1986. A quick start. • B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods. Eaglewood Cliffs, New Jersey: Prentice-Hall, 1990. A textbook. • J. C. Doyle and G. Stein, "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Trans. Automat. Contr., vol. AC-26, pp. 4-16, 1982. A classic. • A. Saberi, B. M. Chen, and P. Sannuti, Loop Transfer Recovery: Analysis and Design. London: Springer-Verlag, 1993. A monograph. • Matlab manual online “Robust Control Toolbox” at:http://www.mathworks.com/access/helpdesk/help/pdf_doc/robust/robust.pdf A useful tool. ALLEYNE RESEARCH GROUP, M&IE/UIUC

  28. Control 2 1 Implement Hydr. Pump Engine 5 4 3 Drive Steering An earthmoving vehicle powertrain ALLEYNE RESEARCH GROUP, M&IE/UIUC

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