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MULTIOBJECTIVE OPTIMAL STRUCTURAL VIBRATION CONTROL USING fuzzy logic control system

MULTIOBJECTIVE OPTIMAL STRUCTURAL VIBRATION CONTROL USING fuzzy logic control system
MULTIOBJECTIVE OPTIMAL STRUCTURAL VIBRATION CONTROL USING fuzzy logic control system

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to be chosen on the basis of an experimental trial-and-error study of the control objective.The initial design of an FLC can be optimized using an optimization procedure(e.g.,GAs). TWO-BRANCH TOURNAMENT GA

The GA is a computational representation of natural selec-tion,making the analogy that in survival of the?ttest an in-dividual more?t in its environment is akin to a more optimal design(Goldberg1989;Holland1992).This analogy includes representing designs as individuals in a population,performing a selection(survival of?ttest),and crossover(mating)of a generation of these designs(from a mating pool)to create children,who in turn become the population in the next gen-eration.An individual design is represented by a chromosome, generally a binary string of1s and0s that represent the design parameter values(value of the design variables)for each in-dividual.

The GA has also been adopted for multiobjective optimi-zation(Fonseca and Fleming1995;Crossley et al.1999).Most of these algorithms are based on a?tness function formulated as either a weighted sum of all objectives or some form of ranking,which assigns a better?tness value to designs based on their dominance.Among its many features,the GA uses a population of points to conduct its search.Because the GA is evaluating an entire population in each generation,this pro-vides an opportunity to generate a set of nondominated designs in one run.The two-branch tournament algorithm is based on this feature of the https://www.sodocs.net/doc/0b17792587.html,e of a selection operator for mul-tiobjective optimization is the major difference of the two-branch tournament approach(Crossley et al.1999)when com-pared to other GA-based multiobjective optimization methods. It generates a representation of Pareto-optimal designs,rather than a single Pareto-optimal design,making an ef?cient use of the GA’s population-based search.

Chromosome Representation

Each design is represented by an n-bit-long chromosome. Here,n is the sum of the length required to represent each design variable,which can be determined by

(r?1)p r

r

2<(U?L)?10?2(1) where U and L=upper and lower bound of the design vari-able,respectively;p r=required decimal precision;and r= required length in bits used to represent the design variable.

Initial Population

The GA starts from a population of chromosomes as a set of initial designs.The initial population is chosen randomly. Population size(i.e.,the total number of chromosomes in the population)has been chosen such that the number of design points are suf?ciently large that it encompasses the global op-timum solution.A larger population results in convergence in fewer generations.

Fitness Function

The GA uses a?tness function value for the selection op-erator;therefore,this function re?ects both the objective and a penalty for constraint violation.The?tness function has been constructed in the manner of a sequential unconstrained min-imization technique(i.e.,an objective with external penalty functions to handle the constraints).Because the GA does not require derivatives,or even continuity of the function,several options are available to describe the?tness function(Fonseca and Fleming,1995).For the present study,the?tness func-

tions,which have to be minimized,have been obtained by combining the objective functions?i and the constraint func-tion g i for n cons number of constraints,given by

n cons

f=??c max[0,g](2)

11i i

?

i=1

n cons

f=??c max[0,g](3)

22i i

?

i=1

where?i,c i,and g i=i th objective function,i th penalty coef-?cient,and i th constraint violation,respectively.In this study, to simplify the selection of the coef?cients,c i,all the con-straints have been formulated in a scaled form and the same value of c i has been used for all the constraints.In the scaled form,constraint functions have been posed as given in(4)to enforce a value greater than the allowed value,or as in(5)to enforce a value less than the allowed value.These functions are negative valued when the constraints are satis?ed and pos-itive valued when violated

actual value

g=1??0(4)

i allowed

actual value

g=?1?0(5)

i allowed

Selection Criteria

The two-branch tournament selection approach is such that designs are competitive on one of the two objectives.The pro-cedure for this selection is shown in Fig.1and enumerated in the following steps:

1.Place the entire population of the current generation in

the pot.

2.Select two individuals randomly from the pot without

replacement.

https://www.sodocs.net/doc/0b17792587.html,pute the?rst?tness,and copy the better performing

individual to the parent pool.

4.Repeat steps2and3until the pot is empty.

5.Re?ll the pot with the population from the current gen-

eration.

6.Select two individuals randomly from the pot without

replacement.

https://www.sodocs.net/doc/0b17792587.html,pute the second?tness,and copy the better perform-

ing individual to the parent pool.

8.Repeat steps6and7until the pot is empty.

At the end of the above steps,the parent pool is full.A new population is generated from this population of the parent pool using crossover and mutation operations.

Crossover

The crossover operator is used to produce two offspring from the selected parents.To select the parents for crossover from the new population,a random number between0and1 is generated.If this random number is less than the probability of crossover,then the chromosome is selected for crossover (Michaelwicz1996).The selected chromosomes are randomly paired for crossover.A single crossover site is selected ran-domly.

Mutation

To maintain variability of the population,a mutation oper-ation is also performed on certain individuals.The mutation is performed on a bit-by-bit basis,with a certain probability of mutation.This operation is also performed with the help of

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FIG. 1.

Flowchart of Two-Branch Tournament GA

a random number between 0and 1.If the random number is

less than the probability of mutation,then the bit under con-sideration will be switched (i.e.,0to 1or 1to 0).Selection of Nondominated Designs

This algorithm generates the two best designs,one for each objective,in each generation.Thus,a large number of feasible designs are obtained at the end of the algorithm.All of these obtained feasible designs are not Pareto-optimal designs.Each of the designs in the list generated by GA is compared with all other designs for dominance.If design 1is better than de-sign 2for both the objectives,then design 1dominates over design 2or,in other words,one can say design 2has been dominated by design 1.If a design has been dominated by any other design,the design will be discarded or else it will be retained in the list.After a comparison of all of the designs in this way,all the retained designs are nondominated designs and form a Pareto-optimal set of designs.

BENCHMARK PROBLEM

The example taken for this study is the ?rst generation benchmark problem on structural control given in Spencer et al.(1998).Details of this benchmark problem are available in Spencer et al.(1998),but a brief description has been given here for completeness.It is a three-story,single-bay,model building with an active mass driver at the third ?oor.The eval-uation model is a high-?delity,linear,time-invariant 28-state state-space representation of the input-output model of the physical structure.The model has been presented in continu-ous time as follows:

x ˙(t )=Ax (t )?B u (t )?E x ¨(t )(6)g y (t )=C x (t )?D u (t )?F x ¨(t )?v (t )

(7)m y y y g z (t )=C x (t )?D u (t )?F x ¨(t )

(8)

z z z g In these equations,x is the state vector;x ¨g is the scaler ground

acceleration;u is the scaler control input;y m =[x m ,x ¨a 1,x ¨a 2,

x ¨a 3,x ¨am ,x ¨g ]T

is the vector of measured responses;z =[x 1,x 2,x 3,x m ,x ˙1,x ˙2,x ˙3,x ˙p ,x ¨a 1,x ¨a 2,x ¨a 3,x ¨am ]T is the vector of control-lable responses;v is the vector of measurement noises (sen-sors);and A ,B ,E ,C y ,D y ,F y ,C z ,D z ,and F z are matrices given for the benchmark problem that represent the structural properties and position vectors.Eq.(6)represents the state-space equations of motion,and (7)and (8)are the output and controlled response equations,respectively.EV ALUATION CRITERIA

The evaluation criteria for this example problem has been given in detail in Spencer et al.(1998),and a brief description has been presented here for completeness.Stochastic Evaluation Criteria

The excitation (ground acceleration)x ¨g is a stationary ran-dom process with a power spectral density

2224

S (4?????)

0g g g S (?)=(9)

¨¨x x 222222

g g (???)?4???

g g g where the natural frequency ?g and the damping ratio ?g are

variables lying in the intervals 20rad/s ??g ?120rad/s and 0.3??g ?0.75.The spectral intensity S 0[(10)]has been chosen such that it keeps a constant RMS value of the ground acceleration x ¨g =0.12g ,irrespective of changes in ?g and ?g

0.03?g

2S =

g (10)

02

??(4??1)

g g When this random ground disturbance has been applied to the structure,the performance of the controller has been mea-sured by the following nondimensional criteria:

???d d d 123

J =max

,,(11)1??

????,?g g

x x x 3o 3o 3o

???¨¨¨x x x a 1a 2a 3

J =max

,,(12)2?

?

????,?¨¨g g

x ¨x

x a 3o a 3o a 3o ?x m J =max

(13)3????,?g g

x 3o ?˙x m J =max

(14)4????,?˙x g g

3o ?¨x am J =max

(15)

5??

??,?¨x g g

a 3o

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FIG. 2.Membership Function Used for Input and Output Variables of

FLC

where ?represents the RMS value of the variable shown as a subscript.The interstory drifts d i are the relative lateral dis-placements between ?oors (d 1=x 1,d 2=x 2?x 1,d 3=x 3?

x 2);x ˙i is the lateral velocity of the i th ?oor relative to the ground;x ¨ai is the absolute acceleration of the i th ?oor;x m and x ˙m are the displacement and velocity of the actuator piston relative to the third ?oor,respectively,and x ¨am is the absolute acceleration of the actuator piston.The normalization con-stants and are the worst-case RMS values of the ?,?,?˙¨x x x 3o 3o a 3o third-?oor displacement and velocity relative to the ground and the absolute acceleration of the third ?oor,respectively,over all the allowed values of the ?g and ?g ,for the uncon-trolled case.

Deterministic Evaluation Criteria

The ground acceleration is one of the two historical earth-quake records:NS component of the 1940El Centro and 1968Hachinohe earthquakes.The performance of the controller un-der the applied seismic excitations has been measured by the following nondimensionalized criteria:

?d (t )??d (t )??d (t )?

123J =max max

,,(16)6????x x x earthquake records

t

3o 3o 3o

?x ¨(t )??x ¨(t )??x ¨(t )?

a 1a 2a 3J =

max

max

,,(17)7????x ¨x ¨x ¨earthquake records

t

a 3o a 3o a 3o ?x (t )?m J =max max

(18)8????x earthquake records

t

3o ?x ˙(t )?m J =max max

(19)9????x ˙earthquake records

t

3o

?x ¨(t )?am J =

max

max

(20)

10????

x ¨earthquake records

t

a 3o

where x 3o and x ˙3o =largest peak uncontrolled values of the

displacement and velocity relative to the ground,respectively;and x ¨a 3o =largest peak uncontrolled absolute acceleration for the corresponding earthquake record.FLC DESIGN

The FLC has been designed using ?ve membership func-tions for each of the input variable (acceleration and velocity)and seven membership functions for the output variable (con-trol input u )for ?ner input-output mapping.The input subsets are NL =negative large,NE =negative,ZE =zero,PO =positive,and PL =positive large.The output subsets are NL =negative large,NE =negative,NS =negative small,ZE =zero,PS =positive small,PO =positive,and PL =positive large.A generalized bell-shaped membership function has been used because it can approximate almost all other types of membership functions based on its parameter [(21)].The shape of the generalized bell shape membership function has been de?ned by the parameters a ,b ,and c (MATLAB 1999).Here,a is the half-width of the membership function at 0.5mem-bership grade;b de?nes the slope of the membership function;and c is the position of the center of the membership function

1

?=

(21)

x 2b

x ?c 1?

??

a

The membership functions for the input and output variables have been shown in Fig.2.Two inputs,namely,the third-?oor absolute acceleration and the third-?oor pseudovelocity,have been used in this study.The choice of a velocity and an ac-celeration component for feedback can be explained in the

context of the state of the system in the fundamental mode of

vibration.These feedback components help in generating the initial inference rule base (e.g.,if velocity is zero and accel-eration is high,the structure is at its extreme position and control action is not needed because it is going to return to its neutral position due to the restoring force).On the other hand,if the acceleration is zero and the velocity is high,then the structure is in its neutral position and control action should be applied so that it remains close to its neutral position in order to reduce the maximum displacement.At the intermediate states (i.e.,between the extreme and neutral position),if ve-locity and acceleration are of the same sign,the structure is returning to its neutral position due to its restoring force,and,if the acceleration and velocity are of opposite sign,then the structure is moving toward its extreme position and accord-ingly the control action should be applied.Thus,both a ve-locity feedback and an acceleration feedback are necessary for an improved decision on control action.Initially,the inference rules have been selected on the basis of the available data for the controlled structural response and control command for the linear quadratic Gaussian controller presented by Spencer et al.(1998).These inference rules have then been re?ned using the information about velocity and acceleration in the case of the fundamental mode of vibration.The inference rules were further ?ne-tuned during the optimization of the controller,observing the contribution from various rules and the position of various membership functions after one run of the optimi-

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FIG. 3.Inference Rules for FLC Used in This Study TABLE 1.

Speci?cations of FLC Used in This Study Parameters Variables/operators Number fuzzy subsets (membership functions)

Input variables Acceleration x ¨a 35Pseudovelocity x ˙a 35Output variable Control signal u 7

Aggregation Maximum Fuzzy inference Mamdani type (MATLAB 1999)

Defuzzi?cation

Center of gravity (COG)

zation.The adopted inference rules in this study are shown in

Fig.3.Speci?cations of the FLC used in this study are given in Table 1.

OPTIMIZATION OF FLC

In the present study,a multiobjective optimal FLC design approach results in a set of Pareto-optimal designs.Details of the optimization are discussed in the following subsections.Formulation of Optimization Problem

The objectives in the optimization problem are to minimize the maximum of the nondimensionalized peak interstory drift J 6[(16)]and the maximum of the nondimensionalized peak acceleration J 7[(17)]due to the given earthquake excitation;J 6and J 7have been used as the two objective functions ?1and ?2,respectively,to formulate the ?tness function f 1and f 2as given by (2)and (3).

The parameters speci?ed prior to the solution of the opti-mization problem are the properties of the structure and the earthquake excitation signals,which remain constant.In ad-dition,the uncontrolled response of the structure has also been computed and provided as a constant parameter for the opti-mization problem.The uncontrolled response parameters pro-vided were the worst-case stationary RMS displacement ?x 3o and the velocity of the third ?oor relative to the ground,?˙x 3o the RMS absolute acceleration of the third ?oor,peak dis-?¨x 3o placement x 3o and peak velocity x ˙3o of the third ?oor relative to the ground,and the peak absolute acceleration x ¨3o of the third ?oor,corresponding to the respective earthquake.

Design variables of the FLC are the parameters of the input and output membership functions (i.e.,a ,b ,and c )for each of the generalized bell-shaped membership function described earlier.Due to symmetry,parameters of only half of the input and output membership functions have been considered as de-sign variables,and the other half have been computed using the symmetry about the membership function ZE (e.g.,if the membership function parameters a ,b ,c for membership func-tion NE are 0.5,2.0,and ?0.5,then these parameters for membership function PO will be 0.5,2.0,and 0.5).Parameter c for all the ZE input and output membership functions has been taken as 0.0.The inference rules have not been consid-ered as design variables,but these rules have been ?ne-tuned after the ?rst run of the optimization on the basis of the con-tributions from various rules and the position of the various membership functions.

Spencer et al.(1998)speci?ed the design constraints for the maximum capacity of the actuator for the active mass driver (e.g.,maximum limit on displacement,acceleration,and con-trol command for the actuator).These constraints can be di-vided into two groups:(1)for the ?rst ?ve evaluation criteria (i.e.,J 1to J 5,which are the maximum limits on the RMS displacement,acceleration,and control command);and (2)for the last ?ve evaluation criteria (i.e.,J 6to J 10,which are the maximum limits on peak displacement,acceleration,and con-trol command for the actuator).These constraints can be for-mulated as given by the following equations:

???(22)x x m m

max

???(23)¨¨x x am am

max

???(24)u u max

max ?u (t )??u (25)max

t

max ?x (t )??x (26)m m max

t

max ?x ¨(t )??x ¨(27)

am am max

t

where and u max =maximum limit of the RMS control

?u max input and the maximum control input,respectively;and ?x m max

=maximum limit of the RMS displacement and the max-x m max imum displacement of the actuator piston relative to the third ?oor,respectively;and =maximum limit of the ?x ¨¨x am am

max max

RMS absolute acceleration and the maximum absolute accel-eration of the actuator,respectively.

These constraints have been formulated in a scaled form as given in (4)and (5),so that the same value of the penalty coef?cient c i could be used for all the constraints.It is clear from the (22)to (27)that the constraints are nonlinear and discontinuous,which cannot be handled easily by a traditional optimization algorithm.Such constraints can be handled using GA,because GA does not require the computation of deriva-tives or the continuity of the objective function and con-straints.

Solution Procedure for Optimization

The FLC has been implemented in the SIMULINK model (SIMULINK 1998)of the benchmark problem (Spencer et al.1998)using Fuzzy Logic Toolbox (MATLAB 1999)(Fig.4).The parameter used for the SIMULINK model have been spec-i?ed for the benchmark problem (Spencer et al.1998)which are an integration time step of 0.0005s,sampling time of 0.001s,ADC and DAC resolution of 12bit with a saturation at ?3V ,and sensor noise of 0.01-V RMS (i.e.,0.3%of the span of the ADC).

Two inputs,namely,the third-?oor absolute acceleration and the third-?oor pseudovelocity have been used for the FLC.The details of the FLC are as discussed in the preceding section.The two-branch tournament GA,discussed in the third section,has been used for the optimization of the controller.Param-eters used for GA have been taken on the basis of guidelines given in Grefenstette (1996)and Williams and Crossley (1998).In this study,the population size has been taken as 100members (designs).An upper limit on the number of gen-

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FIG.

4.

SIMULINK Model for Building with FLC

TABLE 2.

Initial Parameters and Constraint Limits

Parameters and constraints

Value

(a)Worst-Case RMS uncontrolled response of third ?oor

(?g =0.3,?g =37.3rad/s)

?x 3o 1.31cm

?˙x 3o 47.9cm/s ?¨x 3o

1.79g

(b)Peak uncontrolled response of third ?oor for 1940

El Centro earthquake

x 3o 3.37cm

x ˙3o 131.0cm/s x ¨3o

5.05g

(c)Peak uncontrolled response of third ?oor for 1968

Hachinohe earthquake

x 3o 1.66cm

x ˙3o 58.3cm/s x ¨3o

2.58g (d )Actuator constraints

?u max 1.0V ?x m max

3.0cm ?¨x m a max 2.0g u max 3.0V x m max 9.0cm x ¨am max

6.0g

FIG. 5.Pareto-Optimal Performance Index J 6versus J 7

erations has been taken as 500.The probability of crossover and the probability of mutation used in this study are 0.6and 0.003,respectively.A single crossover site has been adopted.Numerical values of various initial parameters and constraints have been given in Table 2.NUMERICAL RESULTS

The optimization of the FLC has resulted in a set of non-dominated (Pareto-optimal)solutions.Variation of the perfor-mance index J 6with J 7,which have been used as the two

objectives of the multiobjective optimal design of the FLC,has been shown in Fig.5.The variation of the RMS perfor-mance index (i.e.,J 1versus J 2,corresponding to these Pareto-optimal designs)has been shown in Fig.6.From Figs.5and 6it is clear that the performance of the FLC in terms of the performance indices J 1,J 2,J 6,and J 7has been found to be better than that obtained using other controllers reported in the literature (Battaini et al.1998;Breneman and Smith 1998;Kim and Ghaboussi 1999;Kose et al.1998;Lu and Skelton 1998;May and Beck 1998;Spencer et al.1998;Young and

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FIG. 6.Variation of Performance Index J 1versus J 2for Pareto-Optimal

Solutions

FIG.7.Variation of Control Signal for Worst-Case Stability Test of

FLC

Bienkiewicz 1998)for the benchmark problem.The modal space sliding-mode control proposed by Adhikari et al.(1998)is an exception.The results show that the performance in this controller in terms of the acceleration response (J 7and J 2)can be improved at the cost of much higher interstory drift re-sponse (J 6and J 1)and vice versa.

The objective of this study was to develop a strategy to get a set of Pareto-optimal designs for the FLC.Performance in-dex J 6and J 7have been chosen as the two objectives to achieve using the full capacity of the actuator.Constraints on the actuator capacity [given by (22)–(27)]have automatically put a bound on the other performance indices (J 3,J 4,J 5,J 8,J 9,and J 10).The performance indices J 3,J 4,J 5,J 8,J 9,and J 10relate to the RMS and peak energy requirement and actuator capac-ity.While these indices may be higher than those obtained in the other studies on the benchmark problem,reported in the literature,the current approach results in improved utilization of the available resources (actuator capacity).

The stability of the FLC has been examined for each of the Pareto-optimal designs using the extreme (worst)initial con-ditions (Casciati 1997;Battaini et al.1998)and has been found to be stable.Results for the design for which the system has taken the longest time (20s)to reach the neutral position is shown in Fig.7(shown only up to 10s for clarity).For all other cases the settling time for this stability test was found to be less.CONCLUSIONS

Contributions emerging from this paper are as follows:?An approach for a multiobjective optimal design of a FLC has been demonstrated.The advantage of this approach is that it gives a set of Pareto-optimal designs,for making an appropriate selection.

?Performance of the FLC in terms of the structural re-sponse (acceleration and interstory drift)has been found to be better than that obtained using other controllers re-ported in the literature for the benchmark problem.

?The major advantage of this controller is its simplicity and the use of a limited number of measured structural responses.

?As the controller design is based on fuzzy logic,it incor-porates an inherent robustness and uncertainty handling capability.

?To incorporate nonlinearities in the model of the structure,the controller need not to be modi?ed because the FLC can also handle nonlinearities.

?As the optimization method is based on GA,the objective function and constraints have to be incorporated in a sin-gle ?tness function.This gives ?exibility in assigning weights to the objective and constraints in a suitable way.REFERENCES

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Young,P.M.,and Bienkiewicz,B.(1998).‘‘Robust controller design for the active mass driver benchmark problem.’’Earthquake Engrg.and Struct.Dyn.,27(11),1149–1164.

NOTATION

The following symbols are used in this paper:

A,B,E,C y,D y,

F y,C z,D z,F z

=state-space matrices for evaluation model;

a=half-width of generalized bell-shaped mem-

bership function at0.5membership grade;

b=slope of generalized bell-shaped membership

function;

c=position of center of generalized bell-shaped

membership function on region of discourse;

c i=i th penalty coef?cient;

d i=interstory drift of i th?oor;

f i=i th?tness function;

g i=i th constraint;

J i=i th evaluation criteria;

k=discrete time step index;

L=lower bound of design variable;

m1,m2=two randomly selected member(designs)of

GA population;

n=length of chromosome in bits;

n cons=number of constraints;

p r=required decimal precision for design varia-

ble;

r=required length in bits to represent design

variable;

S0=spectral intensity to model ground accelera-

tion;

S¨¨x x

g g

=magnitude of constant two-sided spectral

density for white noise used to model ground

excitation;

T=sampling time;

t=time;

U=upper bound of design variable;

u=scalar control input;

u k=scalar control input at time t=kT;

u max=maximum limit of control input;

v=measured noise vector for evaluation model;

x=state vector for evaluation model;

x˙ai=pseudoabsolute velocity of i th?oor;

x¨ai=absolute acceleration of i th?oor;

x˙am=pseudoabsolute velocity of actuator piston;

x¨am=absolute acceleration of actuator piston;

x¨am

max

=maximum limit of absolute acceleration of

actuator piston;

x˙g=pseudoabsolute velocity of ground;

x¨g=absolute acceleration of ground;

x i=displacement of i th?oor relative to ground;

x˙i=velocity of i th?oor relative to ground;

x m=displacement of actuator piston relative to

third?oor;

x˙m=velocity of actuator piston relative to third

?oor;

x m

max

=maximum limit of displacement of actuator

piston relative to third?oor;

x3o=peak third?oor uncontrolled displacement re-

sponse relative to ground for each respective

historical earthquake;

x˙3o=peak third?oor uncontrolled velocity re-

sponse relative to ground for each respective

historical earthquake;

x¨3o=peak third?oor uncontrolled absolute accel-

eration response for each respective historical

earthquake;

y=vector of directly measured response;

z=vector of controlled response;

?g=damping ratio of stationary random process;

?x=membership grade in fuzzy subset corre-

sponding to value x;

?d

i

=RMS interstory drift of i th?oor;

?u=RMS control signal;

?u

max

=maximum limit of RMS control signal;

?¨x

ai

=RMS absolute acceleration of i th?oor;

?¨x

am

=RMS absolute acceleration of actuator piston;

?¨x

am

max

=maximum limit of RMS absolute acceleration

of actuator piston;

?¨x

g

=RMS ground acceleration;

?u

max

=RMS displacement of actuator piston relative

to third?oor;

?˙x

m

=RMS velocity of actuator piston relative to

third?oor;

?x

m

max

=maximum limit of RMS displacement of ac-

tuator piston relative to third?oor;

?x

3o

=worst-case stationary RMS uncontrolled dis-

placement of third?oor relative to ground;

?˙x

3o

=worst-case stationary RMS uncontrolled ve-

locity of third?oor relative to ground;

?¨x

3o

=worst-case stationary RMS uncontrolled ab-

solute acceleration of third?oor;

?i=i th objective function;and

?g=natural frequency of stationary random pro-

cess.

JOURNAL OF STRUCTURAL ENGINEERING/NOVEMBER2001/1337

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