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Raffaello D’Andrea Cornell University

Design and Control of Interconnected Systems. Raffaello D’Andrea Cornell University. Examples. Power generation and distribution Vehicle platoons Satellite formation flight Paper processing Adaptive optics MEMS data storage Optical switching “Smart” structures

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Raffaello D’Andrea Cornell University

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

  2. 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

  3. ~ ~ vi wi General Problem Class CONTROLLER PLANT yi ui wi vi Gi Gi ~ di zi yi ui Stability, performance, robustness Requirements:

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

  5. PERIODIC CONFIGURATION

  6. BOUNDARY CONDITIONS

  7. INFINITE EXTENT SYSTEM

  8. 2D, 2D BOUNDARY CONDITIONS

  9. 2D, 1D BOUNDARY CONDITIONS

  10. 2D, NO BOUNDARY CONDITIONS

  11. 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 all interconnected systems are well-posed, stable, and D’Andrea ’98, D’Andrea & Dullerud ‘03

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

  13. PERIODIC CONFIGURATION

  14. 2D, 2D BOUNDARY CONDITIONS

  15. 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

  16. Arbitrary interconnections, heterogeneous components

  17. Arbitrary interconnections, heterogeneous components

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

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

  20. Summary • Semidefinite programming a powerful tool for controldesign and analysis of interconnected systems • Generalization of powerful results for single systems:linear, piece-wise linear, nonlinear • Leads to distributed semidefinite programs, whosestructure is captured by interconnection topology

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