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

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The optimal mode-scheduling problem 2

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The autonomous problem 3

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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) 6

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Problem classifications Linear vs. nonlinear Timing optimization vs. sequencing optimization Off line vs. on line 7

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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. 8

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The timing optimization problem 9

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The gradient Define the costate equation Variational arguments: 10

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Steepest descent algorithm with Armijo step size 11

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Principle of sufficient descent 13

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The Steepest Descent Algorithm with Armijo Step size 14

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Modification: descent algorithm with Armijo step size 15

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The timing optimization problem 16

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Constrained algorithm: 17

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On-line setting 18

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Asymptotic convergence – meaningless. Instead, approach to stationary points 21

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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 26

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The sequencing optimization problem 29

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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.) 30

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Sensitivity analysis and optimality function Gradient insertion 31

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Optimality functions and steepest descent (Polak) 33

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Sufficient descent 37

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S. Almer, S. Mriethoz, M. Morari, “Optimal Sampled Data Control of PWM Systems Using Piecewise Affine Approximations”, Proc. 49 th CDC, Atlanta, 2010 Example 39

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PWM problem (Almer at al. [2], 2010 CDC) 41

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The scheduling optimization problem 43

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Adding a switching cost H. Kawashima, Y. Wardi, D. Taylor, and M. Egerstedt, 2012 ADHS TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A AA A A A 47 The PWM problem, variable number of cycles

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Model for switching energy 48

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Energy cost: Optimal control problem: 49

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This is in the form 51

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w=0.5 52

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w=0.9 53

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w=0.1 54

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Thank you 56

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