1. Raffaello D’Andrea Cornell University Design and Control of Interconnected Systems

4. Examples Power generation and distribution Vehicle platoons Satellite formation flight Paper processing Adaptive optics MEMS data storage Optical switching “Smart” structures and so on... Common thread: Distributed sensing and actuation capabilities Highly structured interconnection topology

5. General Problem Class PLANT CONTROLLER Stability, performance, robustness Requirements: GiGi vivi didi uiui yiyi zizi wiwi GiGi uiui yiyi ~ wiwi ~ vivi ~

6. Basic building block, one spatial dimension Simplest case: Homogeneous Systems

7. PERIODIC CONFIGURATION

8. BOUNDARY CONDITIONS

9. INFINITE EXTENT SYSTEM

10. 2D, 2D BOUNDARY CONDITIONS

11. 2D, 1D BOUNDARY CONDITIONS

12. 2D, NO BOUNDARY CONDITIONS

13. Results for linear and piece-wise linear systems Theorem: If the following semidefinite program has a solution: where N and the are fixed, and only a function of the basic building block, then D’Andrea ’98, D’Andrea & Dullerud ‘03 all interconnected systems are well-posed, stable, and

14. Basic building block: control design Design controller that has the same structure as the plant:

15. PERIODIC CONFIGURATION

16. 2D, 2D BOUNDARY CONDITIONS

17. Properties of design Controller has the same structure as the plant Finite dimensional, convex optimization problem Optimization problem size is independent of the number of units

18. Arbitrary interconnections, heterogeneous components

20. Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution: Langbort, Chandra, & D’Andrea ’03 Chandra, Langbort, & D’Andrea ‘03 if the subsystems are not interconnected:

21. Theorem: the interconnected system is well-posed, stable, and if the following coupled semidefinite programs have a solution: Langbort, Chandra, & D’Andrea ’03 Chandra, Langbort, & D’Andrea ‘03 if the subsystems are not interconnected: When working with linearized dynamics, results generalize to control system design

22. Summary Semidefinite programming a powerful tool for control design and analysis of interconnected systems Generalization of powerful results for single systems: linear, piece-wise linear, nonlinear Leads to distributed semidefinite programs, whose structure is captured by interconnection topology