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

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Presentation transcript:

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

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

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

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

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)

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, Lyapunov Function: P is a symmetric positive definite matrix

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

Stable model reduction Stabilizing Projections

Intro : Stabilizing Projections

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

Problem Formulation I

Problem Formulation II

Stabilizing Projections: Stability Constraints

Difficult to solve: Quadratic in U

Stabilizing Projections: Linear Constraints

Existence of Solutions

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

Stabilizing Projections: Existence Summary

Computing Solutions: Efficient Solutions

Efficient Optimal Solutions

Stabilizing Projections Summary

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

Example: Interconnect

Example: Inductor

Example: Power Grid

Example: MEMS Linearization

Comparison to other Methods

Conclusion