1. Multiobjective control: safety and efficiency Hamsa Balakrishnan, David Culler, Edward A. Lee, S. Shankar Sastry, Claire Tomlin (PI) University of California at Berkeley December 17 2009

2. Multiobjective control Target Set Maneuver sequencing is accomplished by stringing together capture sets, starting from the target set and working backwards Avoid sets can be combined with capture sets to guarantee safety Unsafe Set

3. Example: Collision Avoidance Pilots instructed to attempt to collide vehicles [Gabe Hoffmann]

4. Back-Flip: Results

5. Decentralized optimization based on dual decomposition Consider the following simple example: aircraft, each with dynamics Private (local) cost function: Global cost function:

6. Centralized Optimization

7. Dual Decomposition: Overview One primal problem leads to many dual problems The dual decomposition method –solves one instance of these many dual problems –breaks the primal problem into a set of small problems –provides a lower bound on the global optimal solution If the problem is convex, dual decomposition returns the global optimal solution

8. Primal Problem Dual Problem Decoupled Primal Problem Dual Decomposition: Overview 1 2 3 1 2 3 1’ 2’ 2’’ 3’ 1 3 1’ 2’ 2’’ 3’ 2

9. Decentralized results after 5 iterations