Model Reduction of Dynamical Systems

& Real-Time Control

Ahmed Sameh

Department of Computer Science

Purdue University

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)

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

Physical Process

MathematicalModeling

Simulation

understandingprocess behavior

prediction & modification

feasibility ofprocess control

Examples of applications in science & engineering

1. Flex model of the space station.

2. Structure response of high-rise buildings to

earthquakes and wind.

3. Simulation and control of MEMS.

4. Electronic circuit simulation.

5. Climate modeling.

Model reduction – an example: replace a large-scale system of differential equations,

by one of substantially lower dimension that has nearly the same response characteristics:

x(t) RN ; u(t) Rm ; y(t) Rp

x(t) = Ax(t) + Bu(t)

y(t) = Cx(t)

.

~A = WTAV Rn ; C = CV ; B = WTB WTV = In ; n < < N

~~

)()(

)()()(

tx~C~

ty

tuB~

tx~A~

tx~

Research Goals:

1. 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.

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.

Example:

x

xE,y

uB

x

x

M

K

x

x

M

MC

)0(

00

0

0

M, C, K are symmetric

2. Development of robust algorithms for open problems inmodel reduction of structured dynamical systems.

)( E)t(y

)t(Bu)t(f x(t))tω,(K)t(x)tω,(C)t(x)tω,(M

tx

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.

Examples of Control Mechanisms

Building Control Mechanism Damping Fr., Effective Damper Mass.

CN Tower, Toronto (533m).

Passive Tuned Mass Damper

John Hancock Bldg, Boston (244m).

Two Passive Tuned Dampers

0.14 Hz, 2 x 300t, 4% damping ratio

Sydney Tower (305m) Passive Tuned Pendulum 0.1, 0.5Hz, 220t

Rokko Island P&G, Kobe (117m)

Passive Tuned Pendulum 0.33 – 0.62Hz, 270t

Yokohama Landmark Tower (296m)

Active Tuned Mass Dampers (2)

0.185Hz, 340t

Shinjuku Park Tower, Tokyo (227m)

Active Tuned Mass Dampers (3)

330t

TYG Building, Atsugi (159m)

Tuned Liquid Dampers (720)

0.53 Hz, 18.2t

Engineering Structures, Vol. 17, No. 9, Nov. 1995.

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).

.

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).

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).

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).

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.

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.)

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. 

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

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.

Acquired Model

Check Against Public Data

Resulting Mesh (Partial View)

Column Model

Dual Wing Impact with Wing Skin and Fuel

IBM Regatta Power4 platform with 8 processors

Model size: 1.2M elements

Run time: 20 hours

Detail study of wing impacting 4 rows of columns

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

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.

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.

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.

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.

Structural Simulations: case study-II (comparison with experiments)

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

Experimental Verification

Impact Experiment

Impact Experiment

Smooth Particle Hydrodynamics(SPH)

Smooth Particle Hydrodynamics(SPH)

Smooth Particle Hydrodynamics(SPH)

Simulation Web Site

http://www.cs.purdue.edu/homes/cmh/simulation

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)

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).

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.