Model Reduction of Dynamical Systems

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Model Reduction of Dynamical Systems Real-Time
Control

• Ahmed Sameh
• Department of Computer Science
• Purdue University

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Model Reduction

• Collaborative Research (Medium ITR)
• Purdue University
• A. Grama, C. Hoffmann, A. Sameh (CS), Sozen (CE)
• Rice University
• A. Antoulas (ECE), D. Sorensen (CAAM)
• Florida State University
• K. Gallivan (CS)
• Catholic University of Louvain (Belgium)
• P. Van Dooren (ME)

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Outline

• Mathematical modeling and simulation
• Model reduction
• Research goals
• Examples of existing structure control mechanisms
• Future directions
• Structural simulations two case studies of
  structure-fluid interaction.
• Conclusion

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Physical Process
Mathematical Modeling
Simulation
understanding process behavior
prediction modification
feasibility of process control
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Examples of applications in science engineering

• Flex model of the space station.
• Structure response of high-rise buildings to
  earthquakes and wind.
• Simulation and control of MEMS.
• Electronic circuit simulation.
• Climate modeling.

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Model reduction an example replace a
large-scale system of differential equations,
.
x(t) Ax(t) Bu(t) y(t) Cx(t)
x(t) ? RN u(t) ? Rm y(t) ? Rp
by one of substantially lower dimension that
has nearly the same response characteristics



A WTAV ? Rn C CV B WTB
WTV In n
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Research Goals

• Development and implementation of a library of
  parallel algorithms for those sparse matrix
  computations that arise in model reduction
  schemes for large-scale dynamical systems.

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Example Obtain the dominant invariant subspace
of (PQ), where P and Q are given by the
Lyapunov eqns
AP PAT BBT 0 ATQ QA CTC 0
without explicitly obtaining P Q.
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• Development of robust algorithms for open
  problems in
• model reduction of structured dynamical systems.

Example
M, C, K are symmetric
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• 3. Development and validation of control
  algorithms based on reduced models.
• 4. Implementation of real-time control algorithms
  on sensor-actuator microgrids (as distributed
  computational platforms).
• 5. Development of an environment for validation
  of large-scale structural simulations and control.

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Examples of Control Mechanisms
Engineering Structures, Vol. 17, No. 9, Nov. 1995.
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Multistep Pendulum Dampers
The Yokohama Landmark Tower, one of the tallest
buildings in Japan relies on multistep pendulum
dampers (2) to damp dominant vibration mode of
0.185 Hz. Pictured on the right is a model of the
pendulum (Picture credits Steven Williams). .
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Examples Active Mass Damper in the Kyobashi
Seiwa Building
An Active Mass Damper consists of a mass whose
motion (displacement, velocity, acceleration) is
controlled, in this case, by a turn-screw
actuator. Eigenvalue analysis of the structure
shows that the dominant vibration mode is in
transverse direction with a period of 1.13 s. and
second eigenvalue in the torsional direction with
a period of 0.97s. This two-mass active damper
damps these two modes (Picture courtesy Bologna
Fiere).
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Passive Control Base Isolation
Base isolation is a mature technology, commonly
used in bridges. Pictured left is a base isolator
in use on a building at the Kajima Research
Facility. Pictured on the right are base
isolators used in a viaduct in Nagoya. These
structures rely on (passive) base isolation to
control the structure in the event of ground
motion (Picture credits Steven Williams).
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Passive / Semi-Active Fluid Dampers
Pictured left is a passive fluid damper with
bottom casing containing the bearings and oil
used to absorb seismic energy. Pictured right is
a semiactive damper with variable orifice damping
(Picture credits Steven Williams).
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The Future Fine-Grained Semi-Active Control.
A new class of dampers based on
Magnetorheological Fluids (fluids capable of
changing their viscosity characteristics in
milliseconds, when exposed to magnetic fields,
courtesy Lord Corp.), coupled with considerable
advances in sensing and networking technology,
present great potential for fine-grained
real-time control for robust structures. These
control mechanisms enhance resilience of
structures subjected to traditional hazards such
as high winds and earthquakes, in addition to
man-made hazards.
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Emerging Frontiers
The Dongting Lake Bridge is being retrofitted
with MR dampers to control wind-induced vibration
(picture source Prof. Y. L. Xu, Hong Kong Poly.)
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Structural Simulations case study-I (C.
Hoffmann, S. Kilic, M. Sozen)

• Simulate the effects of crashing an air frame
  loaded with fuel (simulating a Boeing 757) into a
  reinforced concrete frame similar to the one
  supporting the Pentagon building using LS-Dyna
• Model the columns to reproduce the behavior of
  spirally reinforced columns including the
  difference in material response of the concrete
  within and outside the spiral reinforcement.
• Exclude effects of fuel explosion and subsequent
  fire damage

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Aircraft Meshing

• Needed structural elements
• Ribs, stringers,
• Floor,
• Tank enclosure.
• Shell and beam elements.
• Fluid modeled by (partial) filling of elements in
  a (moving) Eulerian grid of air.

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Acquired Model
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Check Against Public Data
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Resulting Mesh (Partial View)
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Column Model
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Dual Wing Impact with Wing Skin and Fuel

• IBM Regatta Power4 platform with 8 processors
• Model size 1.2M elements
• Run time 20 hours

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• Detail study of wing impacting 4 rows of columns

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Full Impact with Fuselage, Wing Structure, and
Fuel

• Fuselage model includes the floor system and
  stringer beams
• Wing structure includes spars, fuel compartments,
  and fuel

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Full Impact with Fuselage, Wing Structure, and
Fuel .

• Coarse model 300K elements, 0.20 sec. real time,
  IBM Regatta Power4 platform with 8 processors, 24
  hours run time.
• Detailed model 1.2M elements, 0.25 sec. real
  time, IBM Regatta Power4 platform with 8
  processors, 68 hours run time.

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Results from Simulations (1)

• The simulation demonstrates that the number of
  columns destroyed in the facade of the building
  does not have to correspond to the full wing
  span.
• The tips of the wings, having less mass, are cut
  by the columns rather than the wing cutting the
  columns.

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Results from Simulation (2)

• The simulation suggests that the reinforced
  concrete column core will cut into the fuselage
  until the fuel tanks reach it, at which time the
  column is destroyed.

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Results from Simulation (3)

• The simulation shows the deceleration of the
  plane after impact as witnessed by the buckling
  of the fuselage near the tail structure.

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Structural Simulations case study-II
(comparison with experiments)

• Investigate the fluid (water)-reinforced concrete
  interaction at high speed impact.

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Experimental Verification
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Impact Experiment
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Impact Experiment
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Smooth Particle Hydrodynamics(SPH)
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Smooth Particle Hydrodynamics(SPH)
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Smooth Particle Hydrodynamics(SPH)
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Simulation Web Site
http//www.cs.purdue.edu/homes/cmh/simulation
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Conclusions

• Work has been initiated on several fronts
• Acquiring actual high-rise structural models
  (Sozen)
• Developing novel model reduction techniques and
  application on the above acquired full models
  (Antoulas, Gallivan, Sorensen, Van Dooren)
• Development of sparse matrix parallel algorithms
  needed for model reduction (Grama, Sameh)

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Conclusions.

• Development of simulation environment using both
  the full and reduced models (Hoffmann, Sameh,
  Sozen).
• Development of control algorithms for full and
  reduced model (Gallivan, Van Dooren)
• Implementation of the real-time control
  algorithms on the sensor-actuator microgrid
  (Grama, Sameh).
• Dense sensor-actuator instrumentation of model
  structures, and validation by scale experiments
  (Grama, Sameh, Sozen).

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Conclusions.

• Year-1 Targets
• Develop simulation methodology and demonstrate
  its use as a validation tool.
• Demonstrate viability of model reduction for
  real-time control.
• Instrument test structures and develop
  infrastructure for data gathering and
  assimilation.