model reduction of dynamical systems - uclouvain · model reduction of dynamical systems p. van...
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Model Reduction of Dynamical Systems
P. Van Dooren, CESAME, Université catholique de Louvain
Francqui lezing 30 April 2010
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Motivation
Predict a storm surge in the North sea (Verlaan-Heemink ’97)
60.000 variables, 15 inputs (buoys and radars)
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Motivation
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What models ?
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What models ?
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Explicit Discrete Linear Time Invariant Systems
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What norm ?
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What norm ?
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In continuous-time :
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Convolution map S from inputs to outputs
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Hankel map H : past inputs to future outputs
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Hankel map H : past inputs to future outputs
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Hankel map factorization
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Gramians derived from the Hankel map
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Approximation via balanced truncation
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Numerical procedure
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Square root approach
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Dense Stein solvers (exact)
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Dense Stein solvers (approximate)
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Interpolation approach (continuous-time)
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Should also approximate Gramians
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Krylov subspaces
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Rational interpolation and moment matching
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Implicit continuous LTI systems
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This includes modal matching
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Tangential interpolation
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Tangential interpolation
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H2 optimal approximations
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How evaluate this norm ?
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Gradients are easier
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Leads to a fixed point iteration
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A PDE example on a FE mesh
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We will assume
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Fixed point iteration often converges …
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but can also be erratic
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Approximation errors
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Multilevel idea
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Experiments
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Experiments
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Convergence is delicate
Basins of attraction of different local minima of low order error function
Order 1 approximationApproximation error vsInitial interpolation point
Order 2 approximationBasin of attraction vsInitial interpolation points
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A mechanical application
Modeling of mechanical structures
Identification/calibration (cheap sensors)
Simulation/validation (prognosis)
Model reduction
Control (earthquakes, large flexible structures)
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Passive / Semi-Active Fluid Dampers
Passive fluid dampers contain bearings and oil absorbing seismic energy. Semi-active dampers work with variable orifice damping.(Picture courtesy Steven Williams)
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More examples of Control Mechanisms
Engineering Structures, Vol. 17, No. 9, Nov. 1995.
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The Future: Fine-Grained Semi-Active Control
Dampers are based on Magneto-Rheological fluids with viscosity that changes in milliseconds, when exposed to a magnetic field.
New sensing and networking technology allows to do fine-grained real- time control of structures subjected to winds, earthquakes or hazards.(Pictures courtesy Lord Corp.)
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This technology starts to be applied…
Dongting Lake Bridge has now MR dampers to control wind-induced vibration (Pictures courtesy of Prof. Y. L. Xu, Hong Kong Poly.)
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Second order system models
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Reduced order model
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Start by simplifying the model …
Simplify bykeeping only concrete substructure
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and then reduce the state dimension …
i.e. reduce the number of equations describing the “state” of the system
26400 2nd order eqs
20 2nd order eqs
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Use of 2nd order models
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Clamped beam example
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Interpolation of large scale systems
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Interconnected systems
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Interconnected systems
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Several examples
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General interconnected systems
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Realize interconnected systems
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Example
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Closed loop Gramians
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Constrained Gramians
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Constrained Krylov spaces
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Time-varying linear systems
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H2 approximation
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Error function is a linear map
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depends on the reduced order model
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Gradients are given by
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Nonlinear systems
Look for a simple energy function
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Example
Chemical vapor deposition reactor
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Conclusions
• Model reduction of linear time invariant systems isquite sophisticated and efficient these days
• Algorithmic aspects are the issue right now
• Time-varying extensions exist (for discrete-time case)
• Nonlinear extensions exist but lack well established results
• There are many successful test cases
• Model reduction is stil quite hot …
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BBC report
of Sept. 2002