1. Optimal Distributed State Estimation and Control, in the Presence of Communication Costs Nuno C. Martins nmartins@umd.edu AFOSR, MURI Kickoff Meeting, Washington D.C., September 29, 2009 Department of Electrical and Computer Engineering Institute for Systems Research University of Maryland, College Park

2. Setup is a network whose nodes might comprise of: Linear dynamic systems Sensors with transmission capabilities Receivers including state estimator A Simple Configuration: Introduction

3. Setup is a network whose nodes might comprise of: Linear dynamic systems Sensors with transmission capabilities Receivers including state estimator A Simple Configuration: Applications: -Tracking of stealthy aerial vehicles via (costly) highly encrypted channels. Introduction

4. Setup is a network whose nodes might comprise of: Linear dynamic systems Sensors with transmission capabilities Receivers including state estimator A Simple Configuration: Applications: -Tracking of stealthy aerial vehicles via (costly) highly encrypted channels. -Distributed learning and control over power limited networks. NSF CPS: Medium 1.5M Ant-Like Microrobots - Fast, Small, and Under Control PI: Martins, Co PIs: Abshire, Smella, Bergbreiter Introduction

5. Setup is a network whose nodes might comprise of: Linear dynamic systems Sensors with transmission capabilities Receivers including state estimator A Simple Configuration: Applications: -Tracking of stealthy aerial vehicles via (costly) highly encrypted channels. -Distributed learning and control over power limited networks. - Optimal information sharing in organizations. Introduction

6. Setup is a network whose nodes might comprise of: Linear dynamic systems Sensors with transmission capabilities Receivers including state estimator A Simple Configuration: Ultimately, we want to tackle general instances of the multi-agent case.

7. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Optimal solution: time Erasure Transmit A New Method for Certifying Optimality

8. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Optimal solution: time Erasure Transmit Numerical method to compute Optimal thresholds A New Method for Certifying Optimality

9. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Optimal solution (a modified Kalman F.): Erasure? yes no Execute K.F. A New Method for Certifying Optimality

10. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Past work: A New Method for Certifying Optimality

11. Frigyes Riesz Issai Schur Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Past work: Key to our proof is the use of majorization theory. A New Method for Certifying Optimality

12. … Tandem Topology Recent Extensions

13. … Tandem Topology Optimal Modified K.F. Threshold policyMemoryless forward Recent Extensions

14. … Tandem Topology Optimal Modified K.F. Threshold policyMemoryless forward Control with communication costs (Lipsa, Martins, Allerton’09) Recent Extensions

15. Multiple-stage Gaussian test channel Problems with Non-Classical Information Structure

16. Multiple-stage Gaussian test channel Lipsa and Martins, CDC’08 Problems with Non-Classical Information Structure

17. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Extensions: … Future directions: -More General Topologies, Including Loops Summary and Future Work

18. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Extensions: … Future directions: -More General Topologies, Including Loops -Optimal Distributed Function Agreement with Communication Costs and Partial Information Summary and Future Work

19. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Extensions: … Future directions: -More General Topologies, Including Loops -Optimal Distributed Function Agreement with Communication Costs and Partial Information -Game convergence and performance analysis Summary and Future Work

20. Major results: Nonlinear, non-convex. Optimality was a long standing open problem. Solution is provided in: G. M. Lipsa, N. C. Martins, “Certifying the Optimality of a Distributed State Estimation Scheme via Majorization Theory”, submitted to TAC, 2009 Extensions: … Future directions: -More General Topologies, Including Loops -Optimal Distributed Function Agreement with Communication Costs and Partial Information -Include Adversarial Action (Game Theoretic Approach) Summary and Future Work Thank you