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A Market-Based Control Solution for Semi-Active Structural Control

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Lynch, Law
S
ource: Computing in Civil and Building Engineering: Proceedings of the Eight International Conference,
Stanford, CA, USA, August 14-16, 2000.









A Market-Based Control Solution for Semi-Active Structural Control

Jerome Peter Lynch
1
and Kincho H. Law
2



Abstract

A decentralized control algorithm for structural control system design has
been developed. This novel approach is termed market-based control. As a potential
alternative to the classical Linear Quadratic Regulator (LQR), this method
decentralizes the decision process of the control system and allows for independent
and autonomous control formulation to occur directly upon the control device. This
paper outlines the theoretical formulation of both approaches and presents their
performance on a semi-active controlled structural model.


Introduction

Early research efforts in the field of structural control concentrated heavily
upon active structural control systems. These systems limit structural deflections by
employing actuators to apply forces directly to the structure. While remarkable
progress was made, it was determined that the approach had many technological and
economic limitations when used in protecting structures during earthquakes. To
overcome these limitations, a semi-active approach to structural control was
formulated. In this elegant approach, actuators are no longer used to apply forces to
a structure directly. Rather, the forces needed for control are generated indirectly by
devices that change the overall damping and stiffness properties of the structure.
With small energy consumption characteristics, semi-active devices are an especially
attractive solution for limiting earthquake deflections.

Numerous semi-active devices have been proposed and constructed. In
particular, researchers at Kajima Corporation, Japan, have successfully designed and
tested a variable damping device known as the Semi-active Hydraulic Damper

1
Ph.D. Student, Department of Civil and Environmental Engineering, Stanford University,
Stanford, CA 94305. Email: jplynch@stanford.edu
2
Professor, Department of Civil and Environmental Engineering, Stanford University,
Stanford, CA 94305. Email: law@cive.stanford.edu
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Lynch, Law
(SHD) [Kurata et al., 1999]. The SHD device is a variable damper whose damping
coefficient can be changed by changing the orifice opening between the two
hydraulic chambers of the damper. Joint research efforts between the University of
Notre Dame and Washington University have produced a variable damping device
known as the magnetorheological damper [Spencer et al., 1998]. The damping
coefficient of the magnetorheological damper changes when a magnetic field around
the piston chamber causes a viscosity change of the dampers fluid. The SHD device
can provide a maximum damping force of 1000 kN using only 70W of power while
the magnetorheological damper can provide a 200 kN damping force using 20 to
50W of power.

In practice, a variable damping device is strategically placed at the apex of a
lateral resisting frames V-brace. With dampers located at various floors throughout
the structure, they are centrally controlled by a controller located within the
structure. Kajima has recently installed SHD devices in a five story structure in
Shizuoka, Japan with an SHD device at the first four floors [Kurata et al., 1999].
Accelerometers located upon each floor feedback information of the structures
dynamics to the controller which uses a traditional Linear Quadratic Regulation
(LQR) algorithm to generate commands for the control devices.

It is certain that semi-active devices will continue to evolve into more
compact, cheaper and efficient control devices. With the arrival of small and
inexpensive control devices, structures will be able to deploy hundreds of these
devices for vibration control during earthquakes. The means of centrally controlling
each semi-active device will no longer be an economically efficient solution for
systems with an abundance of devices. As an alternative, a decentralized control
method needs to be formulated for the control of structures that employ a large
number of control devices. In such an approach, a central controller will no longer
be needed to regulate the structure during an excitation since each semi-active device
will have on-board computational means for formulating a semi-optimized control
solution. By decentralizing the control solution, the control algorithm will become
model independent. This will lead to robust control of the structure if device or
structural failure was to occur.


The Classical LQR Technique of Structural Control

Structures are often idealized as one-dimensional lumped mass shear models
in which each floor of the structure represents one degree of freedom. Under an
excitation, the structures dynamics are defined by the response of the system at each
degree of freedom. In control vernacular, such a system is often referred to as a
multi-input multi-output system (MIMO). Transforming the dynamics of the system
from the continuous time domain to the complex plane via Laplace transforms, the
dynamic response of the system can be represented by the roots of the characteristic
equation of the system. These roots, often termed the poles of the system, will
generally fall in the left half side of the complex plane and correspond to the various
modes of response of the structure. Figure 1 shows a general one degree of freedom
structure represented in the complex plane.
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Figure 1 Physical System Converted to the Complex Plane


To improve the response of the structure to dynamic disturbances, a
controller is employed which causes the movement of the systems roots to more
desirable locations. In MIMO systems, pushing roots to a desired location can be an
arduous procedure when considering the control effort necessary to drive the roots to
those locations. To handle this complexity, optimal control techniques have been
developed, such as LQR, which are a systematic guide of pole placement that allows
for weighing the control response against control effort [Franklin et al., 1990].


Optimal control techniques are a powerful tool for determining optimal
control performance through the minimization of a cost function of the system. The
form of the cost function is flexible and can contain terms that represent different
weighted costs of system parameters. The cost function for LQR optimal control is

The state vector used in the cost function is from a one dimensional ordinary
differential equation representation of the system. The state vector X(t) contains both
the displacement vector of the structure, x(t), and the velocity vector v(t). The
control vector of the system is represented by U(t). The matrices Q and R represent
weighting upon the energy associated with the structural response and the control
input respectively.

To ensure that a minimum of the cost function, J, can be found, the weighting
matrices Q and R must be positive definite so that the n-dimensional shape of the
cost functions surface is upward convex and a global minimum point of the surface
exists. Proof of a minimum point can be given by considering the Taylor-series
expansion of the cost function about the minimum point [Stengel, 1994]. The
response and control terms of the cost function are quadratic to ensure that the cost
functions surface has a defined slope at all points. With no cusp points, a minimum
in the overall cost function is guaranteed given the positive definite condition on Q
and R. In a one dimensional example, consider if the cost is proportional to the
system displacement. The cost function of the displacement will have no slope
defined at the zero point since a cusp is present. Furthermore, a quadratic cost
function has the additional benefit of placing harsher penalties on large
displacement