One challenge in the application of control systems to

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One challenge in the application of control systems to
INTEGRATED DEVICE PLACEMENT AND CONTROL DESIGN IN CIVIL
STRUCTURES USING GENETIC ALGORITHMS
Ping Tan1, Shirley J. Dyke,2 Andy Richardson3, and Makola Abdullah4
ABSTRACT
One challenge in the application of control systems to civil engineering structures is appropriate integration of a control system into a structure to achieve effective performance. Placement
of control devices is strongly linked to the performance of a control system, and the most appropriate device placement scheme is strongly dependent on the performance objectives of the control system. Additionally, for the most effective control system, the placement scheme should be
integrated with the design of the controller rather than sequential. This paper proposes an integrated technique to place devices and design controllers based on the use of genetic algorithms. The
approach is flexible, allowing the designer to base the placement scheme on performance goals
and/or system requirements. Active control devices are used and an H2/LQG controller based on
acceleration feedback is selected for this study based on previous successes with this approach in
civil engineering systems. To illustrate the proposed methodology, two numerical examples are
considered. The first example considers a 40-story shear building, and the second is a full-scale,
irregular, 9-story building. Control is achieved through the placement of one or more active control devices placed on various floors in an active bracing configuration. The improvements in the
effectiveness of the proposed methodology as compared to previously developed techniques are
demonstrated through comparative studies.
Keywords: active control, genetic algorithms, optimal placement, protective systems
1. Postdoctoral Research Assoc., Dept. of Civil Engineering, City College of the City University of New York (research performed while a Postdoctoral Researcher at Washington University in St. Louis) email: [email protected]
2. Edward C. Dicke Professor of Engineering, Dept. of Civil Engr., Washington University in St. Louis, St. Louis, Missouri,
63130, email: [email protected] (Corresponding Author).
3. Structural Engineer, PBS&J, 5300 West Cypress Street, Suite 200, Tampa, Florida 33607, email: [email protected]
4. Associate Professor, Dept. of Civil Engr., Florida A&M University, Tallahassee, FL 32210,
email: [email protected]
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INTRODUCTION
To improve the safety and serviceability of structures excited by dynamic loads, significant effort
has been directed toward the possibility of employing protective systems in civil engineering structures (Housner et al., 1997; Spencer and Sain, 1997; Spencer and Nagarajaiah, 2003). To sustain
the design and improve the performance of these systems, the development of methods for optimal
utilization of these types of devices is an important research topic. One of the challenges of this
technology is in achieving the most effective performance of these systems while considering cost
through appropriate integration of the control system into a structure. The capabilities of the control
system strongly depend on the placement of the control devices.
Researchers have been investigating techniques for device placement since the introduction
of the concept of structural control. Chang and Soong (1980) placed a limited number of active
devices in a structure for modal control by minimizing a performance index. Lindberg and Longman (1984) discussed the appropriate number and placement of devices based on independent
modal space control. Vander Velde and Carnignan (1984) considered structural failure modes to
place the devices, and Ibidapo (1985) used structural mode shapes as optimality criteria for device
placement. Cheng and Pantiledes (1988) computed the controllability index associated with each
story of a building, from which the actuator locations were provided. Most of these methods do
not explicitely relate placement to performance and the methods are often problem specific.
Search algorithms have also been utilized for device placements. Takewaki (1997) used a
gradient-based approach to search for an optimal solution that would minimize a desired system
transfer function. Teng and Liu (1992) and Xu and Teng (2002) developed an incremental algorithm for placement of active/passive control devices. Wu et al. (1997) investigated both iterative
and sequential approaches to placement of passive devices in irregular structures. Zhang et al.
(1992) and Lopez et al. (2002) proposed sequential search methods for determining the optimal
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damper placement. Although these algorithms are computationally efficient, they may results in
local optima being found (Goldberg, 1999). As an alternative, a guided random search method,
the simulated annealing method, was employed to place devices (Chen et al. 1991; Liu et al.
1997). Although this method has an advantage over heuristic-based algorithms in seeking global
optima, it does not always provide efficient solution techniques.
Another versatile global optimization approach is to use genetic algorithms (GA). This approach has shown significant promise in its ability to solve problems where the objective function
is not a continuous function of the design variables and the variable space is discrete (Goldberg,
1999). Singh and Moreschi (2002) examined placement of passive energy dissipation devices in a
multistory building for seismic protection. Wongprasert and Symans (2004) employed GA for
identifying the optimal damper distribution to control the nonlinear seismic response of a 20-story
benchmark building. Simpson and Hansen (1996) used GAs to optimize the placement of actuators to actively control interior harmonic sound levels in a cylinder with floor structure. Ponslet
and Eldred (1996) employed a GA approach to design an isolation system. Abdullah et al. (2001)
combined genetic algorithms and a gradient-based optimization technique to design the optimal
position of direct velocity feedback control controllers in buildings. Li et al. (2001) have developed a multi-level genetic algorithm to solve a multi-tasking and multi-model optimization problem simultaneously.
In practice, a placement design procedure should be simple and efficient, and require only an
acceptable amount of computational effort. Control devices should be placed in the structure with
due consideration to the designer’s design objectives, including performance goals such as response reduction, as well as system resources (number of devices, force capacity, actuator dynamics, device stroke/velocity, etc) and cost. Additionally, the placement strategy should allow other
information such as actuator dynamics and disturbance characteristics to be integrated into the design procedure. To appropriately account for all of these issues, the placement strategy should be
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integrated with the controller design. Thus, a combinatorial optimization problem can be considered in which control algorithm parameters and the device locations are determined simultaneously. However, in nearly all of the previous placement studies identified, controller design and
device placement are performed separately. Generally this may not result in the most effective and
efficient control design, although in certain situations the optimal control system, or a control system with similar performance, may be identified.
This paper demonstrates the use of genetic algorithms (GA) for integrated placement of control devices and controller design. This integrated approach is shown to improve upon some previously developed techniques to place devices. It is also demonstrated to be flexible, allowing the
designer to base the placement scheme on a variety of performance goals and system resources.
Additionally, the designer could easily incorporate actuator dynamics and characteristics of the
disturbance, if known, into the design procedure. For this study, the structure is assumed to remain linear, and active control devices are employed. As shown by Yoshida and Dyke (2004), this
assumption is often appropriate even in the case of a structure exhibiting nonlinear behavior. An
H2/LQG controller based on absolute acceleration feedback is selected. To validate the proposed
approach two numerical examples are presented in which the proposed approach is employed to
place active control devices and the system performance is evaluated. In both examples, control is
achieved through the placement of devices in an active bracing configuration. The numerical results also indicate the methodology is simple, effective and efficient.
METHODOLOGY
Integrated Design and Device Placement Procedure
Genetic algorithms is a heuristic random search technique based on the concept of natural selection and natural genetics of a population. Thus, GA is a population-based method for searching
large combinatorial design spaces to identify a globally optimal or near-optimal combination of
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design variables (Goldberg, 1999). Although there is no guarantee that the optimal solution will
always be obtained, the robust behavior and random nature of methods based on GAs make them
attractive for this application.
Genetic algorithms typically require the use of coded strings which represent an underlying set
of parameters. These parameters are coded into a chromosome, which has an associated fitness
value determined by the selected performance function of interest. The corresponding fitness function is used to reflect the degree of ‘goodness’ of each chromosome. In each generation, the fitness
of each chromosome is determined, and a new generation is developed by combining features of
chromosomes with the highest fitness values. GA proceed by reproducing chromosomes to search
for those with increasing fitness. In searching for the best design, GA work with a population, which
is comprised of different chromosomes simulating natural evolution by means of random genetic
changes that produce successively better approximations, ultimately leading to a final design solution.
The three basic operations included in GA are selection, cross-over and mutation. Selection
determines which individuals are chosen for mating and how many offspring each selected individual produces. Crossover and mutation operations create new designs for further exploration in
the search space. The rules for mating, crossover and mutation are defined in a probabilistic manner. As a new population is selected, the performance function is evaluated for each new design to
determine its fitness. Reproduction, crossover, and mutation are subsequently applied to successive populations. The power of GA lies in their robustness as well as the efficiency and simplicity
of the technique. GA are especially suitable for problems in which the fitness function is not a
continuous and differentiable function of the design variables or the variable design space is discrete.
GA are employed here for solving a mixed optimization problem, in which both the positions
of control devices and controller gains are to be optimized. The proposed integrated approach
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uses GA to simultaneously place the devices and design the controller (i.e., select the weighting
matrices). Figure 1 provides a flow chart of the proposed design procedure. Note that the third
column, describing the steps involved in evaluating each string, is the primary contribution of the
proposed method as compared to previously proposed GA techniques for civil structures.
To begin, an initial population of feasible designs (Old-Gen) is randomly generated. In the
context of placement of control devices, a feasible design is a specific distribution of devices, and
a chromosome represents a possible arrangement of control devices. The initial population is selected according to their fitness for production of offspring, using both cross-over and mutation
operations. These offspring are inserted into the population, replacing the parents and producing a
new generation. In evaluating each chromosome of a population, a search is conducted using a
specified objective function to determine the control gains. Each control design is evaluated by
calculating its performance criterion, and the selection operations are subsequently implemented.
The strings which yields the best fitness, designated ‘Best-Gen’ are kept, and the procedure is repeated. With each successive generation, the ‘Best-Gen’ population is selectively updated. The
selection, breeding and updating process continues until the specified number of generation, i is
exceeded. The string that yields the best fitness value is then identified.
A limitation of GA is that there is no guarantee that the optimal solution is always obtained.
However, in practice, and particularly in the problem considered herein, efficiency of an algorithm yielding a near-optimal solution often supersedes the need for the optimal solution. For example, if a designer can obtain a placement yielding near-optimal performance (within a few
percent of the fitness of the optimal solution) in two hours of CPU time, it would not be worthwhile to spend two weeks of searching all possible combinations to find the optimal solution.
Thus, GA provide an attractive tool for control system design.
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Control Design
Consider an actively controlled structure excited by a seismic input. Assuming that the controller
provides adequate control action to keep the structure itself in the linear range, the equation of
motion can be written as
Mx·· + Cx· + Kx = – MΓ x·· g + Λu
(1)
where M , C and K are the mass, damping and stiffness matrices of the system, respectively, x is
the vector of relative displacements, Γ describes the influence of the ground excitation, x·· g is the
ground acceleration vector, u is the vector of control forces, and its coefficient matrix, Λ , is the
matrix determined due to the location of control devices. Defining the state vector
T T T
x s = [ x x· ] , this equation can be written in state space form as
x· s = Ax s + Bu + Ex·· g
(2)
z
z
z
z = C x s + D u + F x·· g
(3)
y
y
y
y = C x s + D u + F x·· g
(4)
where A and B are the system matrices, z is the regulated output vector, which is obtained from
z
z
z
the mapping matrices, C , D and F . Similarly, y is the measurement vector, which is obtained
y
y
y
from the mapping matrices, C , D and F .
For control design, the input excitation x·· g may be modeled as a filtered, stationary white
noise. The disturbance shaping filter can be written in state space form, and the states of the system model can be augmented accordingly (Dyke et al., 1996a,b). If the number of states in the
structural model is high, controllers can also be designed using a reduced-order system model.
An H2/LQG controller is selected to reduce the responses of the structure based on its performance in previous applications (Yi et al., 2001; Dyke et al., 1995, 1996a–c, 1998). In this ap-
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proach, an infinite horizon performance index is selected to weight the vector of regulated outputs
given above by z
τ
T
1
J = lim --- E ∫ ( z Q
0
τ→∞ τ
T
z + u R u ) dτ
(5)
where R is an identity matrix, and Q = qI is a weighting matrix with equal weighting placed on
each of the regulated outputs. Thus we can design controllers by adjusting the weighting factor q
and the selected regulated outputs. Further, the ground excitaion and measurement noise are assumed to be independent, and the measurement noises are identically distributed. For the designs
herein, the ratio of the power spectral densities is taken to be γ = S wj wj ⁄ S vi vi = 25 , where w j is
the jth term of the disturbance vector and v i is the ith term in the measurement noise vector.
The control and estimation problems can be performed separately, according to the separation
principle (Stengel 1986; Skelton 1988), yielding a controller of the form
u = – Kx̂ s
(6)
where x̂ s is the Kalman filter estimate of the state vector, and K is the full state feedback gain
matrix. The Kalman Filter optimal estimator is given by
y
y
x̂ s = Ax̂ s + Bu + L ( y – C x̂ s – D u )
(7)
where L is the observer gain matrix of the stationary Kalman filter. Calculations to determine K
and L were performed using the MATLAB routine lqry.m and lqew.m within the control toolbox.
Determination of Control Gain
To integrate the controller design with device placement, for each string the controller is designed by adjusting the coefficient of the regulated response weighting matrix, q, while considering a specific objective. The objective function on which the design is based can be stated in terms
of the control resources (i.e., forces) needed or the system performance (i.e., response reduction).
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These two approaches for determining the control gain are designated as approach 1 and 2, respectively. The corresponding objective functions used to determine the optimal control gain are
defined by
J g1 = max ( σ fk ) ≤ [ f l ]
for
k = 1, 2, …, r
(8)
where σ f k is the RMS force of the kth device and [ f l ] is the RMS device force limit, and
maxσ
z ij
i, j
J g2 = -------------------- ≤ [ zl ]
maxσ uzij
for
i = 1, 2, …, n , j = 1, 2…, m
(9)
i, j
where σ zij is the controlled value of the specified regulated RMS response of the jth frame at ith
floor, and σ uz ij is the corresponding uncontrolled response. Here [ z l ] is the chosen level of response reduction. The response used in Eq. (9) may be absolute acceleration, relative displacement or interstory drift, depending on the control objective. RMS responses are determined by
solving the Lyapunov equation (Soong and Grigoriu, 1993) and the excitation is assumed to be a
stationary filtered white noise.
For purposes of illustration, assume that the second objective function is used. The objective
function declines monotonously with increasing weighting coefficient until a specified level, or
threshold, in performance is achieved. Once this is achieved, an objective function value (i.e.,
sum of all RMS forces) is assigned to that string, and we continue on to the next string. Each
string requires a different control force to achieve a set performance level. The string using the
smallest control force, represents the best control device placement scheme. Alternatively, if the
first objective function had been selected, the threshold would be determined by evaluating the
maximum control force required for each design, and the selected objective function value would
correspond to the selected response quantity. Both approaches are adopted and demonstrated in
the numerical examples presented herein. If the threshold value is set too aggressively, it may not
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be possible to reach that threshold, typically because the controller is unstable. Thus experience,
or prior knowledge of the system, is helpful.
Performance Function
The fitness of a particular arrangement of control devices has an associated performance
function. Performance functions are used by the GA to evaluate the reduction in certain response
quantities of design interest or control resources used, and make appropriate comparisons of the
resulting designs. Note that the performance function selected should not correspond to the same
responses used in the objective function for control gain determination. Using the same response
for both would lead to circular logic in design of the controller. In this paper, the following performance criteria are considered. The first criterion is a displacement based criterion, and is given as
maxσ
z ij
i, j
J 1 = -------------------maxσ
uz ij
i, j
for
i = 1, 2, …, n , j = 1, 2…, m
(10)
in which σ zij is the controlled RMS relative displacement of the jth frame at ith floor, and σ uz ij is
the corresponding uncontrolled relative displacement. The second criterion is an acceleration
based criterion, given as
maxσ ··
z ij
i, j
J 2 = -------------------maxσ ··
for i = 1, 2, …, n , j = 1, 2…, m
(11)
uz ij
where σ ·· is the controlled RMS accelerations of the jth frame at ith floor, and σ
z ij
··
uz ij
is the corre-
sponding uncontrolled absolute acceleration. The third criterion based on the interstory drifts is
given as
maxσ
d ij
i, j
J 3 = -------------------maxσ
ud
i, j
ij
for i = 1, 2, …, n , j = 1, 2…, m
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(12)
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where σ dij is the controlled RMS interstory drift of the jth frame at ith floor, and σ ud ij is the corresponding uncontrolled interstory drift. An additional criterion based on the control force is
J 4 = sum ( σ u k )
for k = 1, 2, …, r
(13)
where σ uk is the kth RMS control force.
A majority of the time in optimization techniques based on GAs is typically spent on the
evaluation of the fitness function for each individual. The fitness of a particular arrangement of
control devices is defined as a function of the associated performance function values. RMS responses and control forces are determined using the Lyapunov equation (Soong and Grigoriu,
1993). The GA codes using in this analysis are implemented in MATLAB (Pohlheim, 1999).
NUMERICAL EXAMPLES
In the following sections, two numerical examples considering full-scale buildings are presented to demonstrate the proposed approach and to assess the achievable performance in each.
When available, comparisons of the proposed technique to that obtained by other researchers are
presented. Although placement of sensors is generally an important step in the design of a controller, it is not a focus of this research effort. Thus, in each design considered herein, the absolute acceleration of several floors of the example buildings are used for feedback.
Various control objectives are considered for the designs discussed herein. Some controllers
are designed to minimize the relative displacements and interstory drifts of each floor by equally
weighting the corresponding regulated responses of each floor. In the design of other controllers,
the control objective is to minimize the absolute accelerations of each floor by equally weighting
the absolute accelerations. Values for these performance functions, J 1 – J 4 , are also obtained to
evaluate the resulting designs for each example. Control designs which are open-loop unstable are
eliminated due to the fact that they cannot be implemented. Additionally, although it would be relatively simple to incorporate actuator dynamics into the placement scheme as discussed in Fig. 1,
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and would yield improved results in real-world applications (Dyke et al., 1995), we neglect actuator dynamics to focus on validation of the proposed approach.
Example 1: 40-story Shear Building
This example employs a 40-story, shear-building model, selected from Abdullah et al. (2001).
The structural properties of the each floor of the structure are: mass m i = 1290 ton; stiffness
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k i = 10 kN/m; and damping constant c i = 100 kN/(m/s). Controllers are designed using a
reduced-order model considering only the first eight modes, but are evaluated using the full system
model. A Kanai-Tajimi disturbance shaping filter (Soong and Grigoriu, 1993) is used for the design
and the parameters were selected as ω g = 17 rad/sec and ς g = 0.3 (Ramallo et al., 2002).
In this example, 10 actuators are to be placed. Each actuator has a maximum RMS control
force of 4000 kN. In situations in which this limit is achieved, several smaller devices could be
placed in parallel to achieve this force. Each generation contains 25 strings, and 200 generations
are considered.
Table 1 provides a comparison of the results of three search algorithms designed to minimize
the relative displacement, and provides the associated system performance and resources required. Cases A and B correspond to the results obtained using two approaches described in Abdullah et al. (2001). Case C corresponds to the results of the proposed methodology. The variation
of the performance and resulting positions of devices with generation of evolution are shown in
Fig.2. The results indicate that, for a given level of RMS control force, significant improvements
in the control performance can be achieved when devices are placed using an integrated procedure. Note that the total control forces required for each design are quite close. In Cases A and B
some devices are not using their full capacity, while a small number of control devices achieve
most of the control action. However, a more uniform distribution of RMS control forces can be
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achieved with the proposed procedure. Thus, the proposed methodology allows for the capacities
of devices to be fully utilized in real-world applications, resulting in a more cost-effective design.
Example 2: 9 Story Irregular Building
For a regular, symmetric building as discussed in Example 1, the control performance is primarily
dependent on the floors where actuator are placed, and the placement within the floors is not critical
to the performance. However, in the case of a building with an irregular plan (in which the lateral
and torsional motions are coupled), the number of potential control device locations is considerably
larger, making the problem more complex. Not only the floor, but also the placement within the
floor, are essential variables to consider for achieving effective performance in an irregular building.
The challenges and opportunities regarding the control of irregular structures are discussed in more
detail by Yoshida (2003), Yoshida et al., (2003) and Yoshida and Dyke (2005).
Example 2 considers the full scale, 9 story irregular building used by Yoshida (2003) and
Yoshida and Dyke (2005) for semiactive control studies (Obayashi, 2002). The main structural
system of this building is steel reinforced concrete (SRC) and the plan view is shown in Fig. 3. On
each floor there are five bays in the x-direction and four bays in the y-direction. An important feature of this building is that the distribution of shear walls makes this structure irregular, demonstrating coupling the lateral and torsional motions.
A two-dimensional, linear, lumped-parameter model of this building is employed to capture
and focus on the x-translation and the torsional responses of the building (Yoshida, 2003; Yoshida
and Dyke, 2005). The first two calculated natural frequencies are 0.83 Hz (translation in x-direction), and 1.29 Hz (torsion). The damping is assumed to be 2% for all modes. The parameters
required to generate a lumped parameter model of this structure (including mass, moment of inertia, lateral and torsional stiffnesses) are provided in Table 2 for each floor along with the resulting
frequencies of the model. Control devices can be installed within each frame on each floor, for a
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total of 45 viable device locations. For feedback to the controller, the absolute acceleration of
each floor is measured at Frame A and Frame E for a total of 18 acceleration responses.
To provide a basis for comparison, the performance of the control designs resulting from implementation of the proposed approach are compared to four possible placement schemes of 9 devices. The designs and results are shown in Table 3 for both drift- and acceleration-based designs.
Because it is not reasonable to compare the results using GA to all combinations of device placements, Cases D1–D4 and A1–A4 are several placement schemes selected for comparative purposes based on practical experience, in which cases the devices are placed along the edges of the
lower storys. Cases D5 and A5 correspond to the proposed scheme with the device RMS force
limit set to 150 kN. Note that placement schemes D5 and A5 perform significantly better than the
other cases in Table 3. These results indicate that the proposed approach yields a superior control
system compared to the other arrangements for this irregular building. Thus, appropriate placement of the devices in the building is important for the design of a high performance, cost-effective solution, especially in the case of an irregular structure.
Implementation Studies
As discussed previously, the proposed integrated approach can be implemented for a variety
of design goals. For instance, the designer may have a specific performance goal to achieve, or a
specific force/cost of the system. Next a few sample design scenarios are considered to demonstrate the variety of approaches that can be employed. The irregular structure studied in Example
2 is used for these implementation studies.
First we consider the results of the proposed technique when implemented using specific
numbers of control devices. Table 4 shows the resulting placements for various device numbers
for both drift-based and acceleration-based designs. Observe that as the number of devices increases, the results do not always include the locations in the previous case with fewer devices.
For example, when one device is placed using acceleration weighting, the optimal location is po-
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sition 26, but the most effective placement scheme using three devices does not include position
26. Although there is no guarantee that we have achieved the global minimum in the evaluation
criteria, this implies that a sequential design approach does not generally yield the most effective
placement because the addition of each device changes the system dynamics.
Next we examine the effect of varying the force limit. The results for placement of 9 devices
using various force limits are provided in Table 5. The placement of devices were determined for
several chosen levels of RMS control forces. At a low level of force, devices tend to be placed at
the same position. However, when the RMS force limit is increased, the devices are spread
throughout the structure. Clearly a larger control force than the chosen force level is required at
the position in which more than one actuator is clustered, and a single device cannot provide the
required force if the limit is too low. Moreover, most of the actuators are concentrated at frame A
(i.e., locations 1, 6, 11, 16, 21, 26, 31, 36, or 41) when the force limit is low. With an increase in
allowable control force, the proposed approach placed devices at other frames to more effectively
reduce both torsional and translational motions.
Another case is investigated to identify the placement scheme to achieve specific, quantitative performance objective. Table 6 shows the performance and selection of nine device locations
for various levels of response reduction. Type A controllers focus on minimizing interstory drifts
and type B controllers are based on minimizing absolute accelerations. This study provides an example to study the relationship between system cost (i.e., number of devices required) and performance.
To demonstrate the performance of the proposed methodology for real earthquakes, the El
Centro and Northridge earthquakes are used. The Northridge record is used to determine the number of devices because this earthquake requires larger control forces. Using the total control force
required at each control device location to realize the performance, the number of control devices
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which have a capacity of 1,000kN are determined, as shown in Table 7. Note that the number of
devices roughly doubles for each increase in performance level.
Structural responses for the two earthquakes are provided in Table 8 for all six controllers
along with the corresponding percent of the uncontrolled value in parenthesis. The results indicate
that the type A controllers can reduce drifts more effectively than the type B controllers, while the
type B controllers can reduce accelerations more efficiently than the type A controllers. Controller 3A and 3B achieve the best performance at the expense of higher control forces. However,
note that all controllers can reduce both drift and acceleration responses substantially for the two
earthquakes.
CONCLUSIONS
An integrated technique for control device placement and controller design has been proposed and demonstrated. The approach employs genetic algorithms to identify an optimal or nearoptimal solution to the placement problem. In addition, the approach was shown to be flexible, allowing the designer to base the placement scheme on performance goals and/or control system resources (i.e., system costs). This offers the benefit of yielding a more practical solution in which
the standard sized (i.e., off-the-shelf) control devices can be implemented with known force capacities and there is a higher degree of uniformity in the forces required by the various devices.
Active control devices in conjunction with H2/LQG controllers were considered and acceleration
feedback was used in all control designs. The effectiveness of the proposed methodology has been
demonstrated through numerical studies and, when possible, the results were compared to those
of other researchers and placement schemes based on practical experience. The results of these
numerical studies also show that the most effective distribution of actuators is highly dependent
on the desired level of performance and the device force limit. Furthermore, the methodology allows for the designer to consider the trade-off between economy and performance.
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ACKNOWLEDGEMENT
This research is partially supported by National Science Foundation Grant CMS 97–33272
(Grant Monitor, Dr. Shih-Chi Liu). Additionally the authors would like to thank Dr. Osamu Yoshida of Obayashi Corporation for the 9-story full scale building model used herein.
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Start
Start
Form Model
(with Actuator
Dynamics or
Disturbance Char.)
Place
Sensors
Reduce
Model
Apply
GA
Evaluate
Final
Design
Old-Gen
Old-Gen
Crossover
Crossover
Mutate
Mutate
New-Gen
New-Gen
Evaluate
Evaluate
Each
Each String
String
Selection
Selection
Find
Find Q
Q
LQG Design
Compute
perforance criterion
Mutate
Mutate
Best-Gen
Best-Gen
Update
Update
P<i
End
End
Figure 1. Flow Chart Describing the
Design Procedure.
22
Tan et al.
Story
J1
Generation
Generation
(a)
(b)
Figure 2. Example Evolutionary Curves for the
Proposed Methodology (a)Performance
and (b) Position of Devices.
23
Tan et al.
D
17,050
X
CM
13,200
C
B
A
1
2
3
5
4
25,600
E
6,000 6,600 6,000 7,000
34,400
6,400 6,400 6,400 7,600
Y
7,600
6
Figure 3. 9-Story Irregular Building (Plan View).
24
Tan et al.
Table 1: Performance Comparison for
Placement Schemes.
J4
(1,000kN)
q
0.4763 0.5992
28.65
109.05
123489
12 14 26 36
0.4213 0.5621
28.42
109.2
11111
22223
0.3535 0.5638
28.63
109.4
Case
Device
Locations
A
3 4 7 10 11 15
19 24 30 36
B
C
J1
25
J2
Tan et al.
Table 2: Structural Parameters (x-Direction).
Total
Moment of
Stiffness
Inertia
Floor
Weight
i
Wi
Ii
(kN)
(kgcm )
9
15556
2.50E+12
9206
2.65E+10
8
10198
1.64E+12
11516
3.07E+10
k xi
2
Torsional
Stiffness
k θi
(kN/cm) (kNcm/rad)
7
10118
1.63E+12
11408
3.25E+10
6
10205
1.64E+12
11731
3.32E+10
5
10295
1.66E+12
12257
3.48E+10
4
10294
1.66E+12
13182
3.83E+10
3
10382
1.67E+12
15172
4.35E+10
2
10470
1.68E+12
16301
4.63E+10
1
10983
1.77E+12
18309
5.34E+10
26
Tan et al.
Table 3: Comparison of Placement Schemes.
Interstory Drift Based Designs
Case
Device Locations
J3
q
D1
112233678
0.3962
1012.6
D2
1 1 5 5 6 6 10 10 11
0. 3968
1012.3
D3
1 1 6 6 11 11 16 16 21
0.2917
1012.8
D4
26 35 35 36 36 41 41 42 42
0.5601
1011.9
1 6 6 11 16 20 22 26 33
0.2463
1012.8
D5
*
Acceleration Based Designs
Case
Device Locations
J2
q
A1
112233678
0.4598
107.7
A2
1 1 5 5 6 6 10 10 11
0.3969
107.8
A3
1 1 6 6 11 11 16 16 21
0.3966
108.2
A4
26 35 35 36 36 41 41 42 42
0.4822
107.4
A5*
5 10 14 16 20 21 22 26 31
0.2523
108.2
*. Designed using proposed approach.
27
Tan et al.
Table 4: Placement and Performance for Various
Numbers of Devices.
Interstory Drift Based Designs
No. of
Devices
Device Locations
J3
q
1
16
0.7628
1011.5
3
11 16 21
0.4839
1012
6
1 6 11 16 21 26
0.3224
1012.5
9
1 6 6 11 16 20 22 26 33
0.2484
1012.8
1 2 6 8 11 14 17 19 22 25 26 31 0.1936
1013.1
12
Acceleration Based Designs
No. of
Devices
Device Locations
J2
q
1
26
0.7665
106.9
3
16 21 31
0.4923
107.4
6
10 11 16 25 26 26
0.3322
107.8
9
5 10 14 16 20 21 22 26 31
0.2523
108.2
1 5 5 8 8 15 17 20 21 21 28 31 0.2284
108.4
12
28
Tan et al.
Table 5: Placement and Performance for
Various Force Limits.
Interstory Drift Based Designs
J g1
(100kN)
Device Locations
J3
q
1
6 6 11 11 17 17 21 21 27
0.315
1012.4
1.5
1 6 6 11 16 20 22 26 33
0.246
1012.8
3
1 4 6 10 12 17 23 26 31
0.150
1013.7
5
1 5 8 12 18 23 29 31 38
0.119
1014.15
Acceleration Based Designs
J g1
Device Locations
J2
q
1
10 11 11 20 21 21 26 30 31
0.332
107.6
1.5
5 10 14 16 20 21 22 26 31
0.252
108.2
3
3 4 5 9 12 17 20 23 28
0.189
109
5
3 4 4 5 9 11 17 24 29
0.154
109.55
(100kN)
29
Tan et al.
Table 6: Placement and Performance for
Various Response Reductions.
Interstory Drift Based Designs
J g2
Controller
Device Locations
J4
(kN)
q
70
1A
15 16 16 16 16 17 21 21 21
202
1010.8
50
2A
11 16 21 21 26 26 26 26 26
427
1011.5
30
3A
1 6 6 6 11 16 21 21 26
881
1012.5
(%)
Acceleration Based Designs
J g2
Controller
Device Locations
J2
q
70
1B
11 11 26 26 31 31 36 41 41
200
106.2
50
2B
6 16 16 21 21 26 26 36 36
430
106.9
30
3B
3 10 11 1121 21 26 26 30
101
107.9
(%)
30
Tan et al.
Table 7: Position of Control Devices.
Position
Controller 1A Number
15
16
17
21
–
–
1
6
2
6
–
–
Position
Controller 2A Number
11
16
21
26
–
–
4
4
9
20
–
–
Position
1
6
11
16
21
26
Controller 3A Number
10
25
12
11
17
10
Position
Controller 1B Number
11
26
31
36
41
–
4
4
5
3
4
–
Position
6
16
21
26
36
–
Controller 2B Number
5
8
8
10
10
–
Position
3
10
11
21
26
30
Controller 3B Number
10
11
20
20
23
15
31
Tan et al.
Table 8: Responses Due to Historical Earthquakes.
Configuration
Acceleration (m/s2)
Interstory Drift (m)
Maximum
RMS
Maximum
RMS
Total Control
Force (1,000kN)
Maximum
RMS
El Centro Earthquake, N-S Component, Imperial Valley on May 18, 1940
Uncontrolled
0.0350
0.0112
10.42
2.58
–
–
Controller 3A
0.0154(0.44)
0.0028(0.25)
8.203(0.79)
1.278(0.50)
30.6
5.52
Controller 2A
0.0247(0.71)
0.0051(0.45)
7.567(0.73)
1.300(0.50)
12.5
2.76
Controller 1A
0.0307(0.88)
0.0074(0.76)
9.533(0.92)
1.854(0.72)
4.93
1.33
Controller 3B
0.0208(0.60)
0.0028(0.37)
5.532(0.53)
0.797(0.31)
24.0
6.18
Controller 2B
0.0242(0.70)
0.0057(0.50)
7.034(0.68)
1.278(0.50)
11.6
2.69
Controller 1B
0.0317(0.91)
0.0086(0.77)
8.154(0.78)
1.887(0.73)
5.25
1.22
Uncontrolled
0.0140
0.0946
28.1379
3.6263
–
–
Controller 3A
0.0555(0.59)
0.0056(0.40)
17.879(0.64)
1.781(0.49)
74.1
9.33
Controller 2A
0.0825(0.87)
0.0089(0.63)
22.030(0.78)
2.202(0.61)
34.5
4.48
Controller 1A
0.0905(0.96)
0.0103(0.80)
27.369(0.97)
2.826(0.85)
12.53
1.77
Controller 3B
0.0834(0.88)
0.0086(0.61)
13.893(0.49)
1.390(0.38)
69.2
10.84
Controller 2B
0.0795(0.84)
0.0097(0.70)
19.010(0.68)
2.084(0.57)
31.0
4.33
Controller 1B
0.0867(0.92)
0.0120(0.86)
23.518(0.84)
2.698(0.74)
14.71
1.86
Northridge Earthquake, N-S Component, Sylmar on January 17, 1994
32
Tan et al.

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