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

The optimal mode-scheduling problem 2

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

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

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

The timing optimization problem 9

The gradient Define the costate equation Variational arguments: 10

Steepest descent algorithm with Armijo step size 11

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Principle of sufficient descent 13

The Steepest Descent Algorithm with Armijo Step size 14

Modification: descent algorithm with Armijo step size 15

The timing optimization problem 16

Constrained algorithm: 17

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

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

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

Model for switching energy 48

Energy cost: Optimal control problem: 49

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This is in the form 51

w=0.5 52

w=0.9 53

w=0.1 54

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Thank you 56