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TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A. The optimal mode-scheduling problem. The autonomous problem. Application areas. Automotive powertrain control (Wang, Beydoun , Cook , Sun , and Kolmanovsky )

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  1. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

  2. The optimal mode-scheduling problem

  3. The autonomous problem

  4. Application areas • Automotive powertrain control (Wang, Beydoun, Cook, Sun, and Kolmanovsky) • Switching circuits (Almer, Mariethoz, and Morari;DeCarlo et al.; Kawashima et al.) • Telecommunications (Rehbinder and Sanfirdson; Hristu-Varsakelis) • Switching control between subsystems or data sources (Lincoln and Rantzer; Brockett) • Mobile robotics (Egerstedt)

  5. Problem classifications • Linear vs. nonlinear • Timing optimization vs. sequencing optimization • Off line vs. on line

  6. Theoretical developments • Problem definition: Branicky, Borkar, and Mitter • Maximum principle: Piccoli; Shaikh and Caines; Sussmann • Algorithms: Xu and Antsaklis; Shaikh and Caines; Attia, Alamir, and Canudas de Wit; Bengea and DeCarlo; Egerstedt et al.; Caldwell and Murphy; Gonzalez, Vasudevan, Kamgarpour, Sastry, Bajcsy, and Tomlin • Control: Bengea and DeCarlo; Almer, Mariethoz, and Morari; Kawashima et al.

  7. The timing optimization problem

  8. The gradient Define the costate equation Variational arguments:

  9. Steepest descent algorithm with Armijo step size

  10. Principle of sufficient descent

  11. The Steepest Descent Algorithm with Armijo Step size

  12. Modification: descent algorithm with Armijo step size

  13. The timing optimization problem

  14. Constrained algorithm:

  15. On-line setting

  16. Asymptotic convergence – meaningless. Instead, approach to stationary points

  17. Example: A mobile robot tracking a target (goal) while avoiding two obstacles. The robot predict the future movement of the target by linear approximation given its position and velocity

  18. The sequencing optimization problem

  19. Current approaches: • Geometric approaches (Shaikh and Caines) • Relaxation algorithms (Bengea and DeCarlo, Caldwell and Murphy) • Gradient techniques (Xu and Antsaklis, Gonzalez and Tomlin, Attia et al., Egerstedt et al.)

  20. Sensitivity analysis and optimality function Gradient insertion

  21. Optimality functions and steepest descent (Polak)

  22. Sufficient descent

  23. Example S. Almer, S. Mriethoz, M. Morari, “Optimal Sampled Data Control of PWM Systems Using Piecewise Affine Approximations”, Proc. 49th CDC, Atlanta, 2010

  24. PWM problem (Almer at al. [2], 2010 CDC)

  25. The scheduling optimization problem

  26. Adding a switching cost H. Kawashima, Y. Wardi, D. Taylor, and M. Egerstedt, 2012 ADHS The PWM problem, variable number of cycles TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA

  27. Model for switching energy

  28. Energy cost: Optimal control problem:

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