1. Guaranteed Stable Projection-Based Model Reduction for Indefinite and Unstable Linear Systems Brad Bond Luca Daniel MIT Speaker: Tuck, Fang Gong

2. Overview MotivationMotivation BackgroundBackground Extraction and Stability Stable Model Reduction Stabilizing ProjectionsStabilizing Projections Problem Formulation Stability Constraints Existence of Stabilizing Projections Computation of Stabilizing Projections ExamplesExamples ConclusionConclusion

3. Motivation Passive physical systems, described by Partial Derivatives Equations (PDEs) Field-solver PDEs Discretization Linear system -1,000,000 equations -Potentially unstable & unstructured Reduced order model -Stable Ideally

4. Overview MotivationMotivation BackgroundBackground Extraction and Stability Stable Model Reduction Stabilizing ProjectionsStabilizing Projections Problem Formulation Stability Constraints Existence of Stabilizing Projections Computation of Stabilizing Projections ExamplesExamples ConclusionConclusion

5. Background BUT we have little control over this stepBUT we have little control over this step Linear may be unstable / unstructuredLinear may be unstable / unstructured Field-solver PDEs Discretization Linear system Passive physical systems (describe by PDEs)

6. Stability L(x) : Lyapunov functionL(x) : Lyapunov function Characterize stability of linear systemCharacterize stability of linear system Stability constraints for a linear systemStability constraints for a linear system * M. Vidyagasar. Nonlinear Systems Analysis. Prentice Hall, New York,1978. S. Sastry. Nonlinear Systems: Analysis, Stability, and Control. Springer, New York, 1999. Lyapunov Function: http://en.wikipedia.org/wiki/Lyapunov_function P is a symmetric positive definite matrix

7. Stable Model Reduction Congruence transform (Galerkin Projection), PRIMACongruence transform (Galerkin Projection), PRIMA Construct V for accuracyConstruct V for accuracy However …However …

8. Stable model reduction Stabilizing Projections

9. Intro : Stabilizing Projections

10. Overview MotivationMotivation BackgroundBackground Extraction and Stability Stable Model Reduction Stabilizing ProjectionsStabilizing Projections Problem Formulation Stability Constraints Existence of Stabilizing Projections Computation of Stabilizing Projections ExamplesExamples ConclusionConclusion

11. Problem Formulation I

12. Problem Formulation II

13. Stabilizing Projections: Stability Constraints

14. Difficult to solve: Quadratic in U

15. Stabilizing Projections: Linear Constraints

16. Existence of Solutions

18. Existence of Solutions: Unstable Systems E > 0 A+A T < 0 Space of stabilizing U

19. Stabilizing Projections: Existence Summary

20. Computing Solutions: Efficient Solutions

21. Efficient Optimal Solutions

22. Stabilizing Projections Summary

23. Overview MotivationMotivation BackgroundBackground Extraction and Stability Stable Model Reduction Stabilizing ProjectionsStabilizing Projections Problem Formulation Stability Constraints Existence of Stabilizing Projections Computation of Stabilizing Projections ExamplesExamples ConclusionConclusion

24. Example: Interconnect

25. Example: Inductor

26. Example: Power Grid

27. Example: MEMS Linearization

28. Comparison to other Methods

30. Conclusion