MULTIOBJECTIVE OPTIMAL STRUCTURAL VIBRATION CONTROL USING fuzzy logic control system

1330/JOURNAL OF STRUCTURAL ENGINEERING/NOVEMBER2001to 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 GAThe GA is a computational representation of natural selec-tion,making the analogy that in survival of thefittest an in-dividual morefit 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 offittest),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 afitness function formulated as either a weighted sum of all objectives or some form of ranking,which assigns a betterfitness 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 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 efficient use of the GA’s population-based search.Chromosome RepresentationEach 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(rϪ1)p rr2<(UϪL)ϫ10Յ2(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 PopulationThe 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 sufficiently large that it encompasses the global op-timum solution.A larger population results in convergence in fewer generations.Fitness FunctionThe GA uses afitness function value for the selection op-erator;therefore,this function reflects both the objective and a penalty for constraint violation.Thefitness 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 thefitness function(Fonseca and Fleming,1995).For the present study,thefitness 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 byn consf=␾ϩc max[0,g](2)11i i͸i=1n consf=␾ϩc max[0,g](3)22i i͸i=1where␾i,c i,and g i=i th objective function,i th penalty coef-ficient,and i th constraint violation,respectively.In this study, to simplify the selection of the coefficients,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 satisfied and pos-itive valued when violatedactual valueg=1ϪՅ0(4)i allowedactual valueg=Ϫ1Յ0(5)i allowedSelection CriteriaThe 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 inthe pot.2.Select two individuals randomly from the pot withoutreplacement.pute thefirstfitness,and copy the better performingindividual to the parent pool.4.Repeat steps2and3until the pot is empty.5.Refill the pot with the population from the current gen-eration.6.Select two individuals randomly from the pot withoutreplacement.pute the secondfitness,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.CrossoverThe 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.MutationTo 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 ofJOURNAL OF STRUCTURAL ENGINEERING/NOVEMBER2001/13311332/JOURNAL OF STRUCTURAL ENGINEERING /NOVEMBER2001FIG. 1.Flowchart of Two-Branch Tournament GAa random number between 0and 1.If the random number isless than the probability of mutation,then the bit under con-sideration will be switched (i.e.,0to 1or 1to 0).Selection of Nondominated DesignsThis 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 PROBLEMThe example taken for this study is the first 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 floor.The eval-uation model is a high-fidelity,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 )ϩB u (t )ϩE x ¨(t )(6)g y (t )=C x (t )ϩD u (t )ϩF x ¨(t )ϩv (t )(7)m y y y g z (t )=C x (t )ϩD u (t )ϩF x ¨(t )(8)z z z g In these equations,x is the state vector;x ¨g is the scaler groundacceleration;u is the scaler control input;y m =[x m ,x ¨a 1,x ¨a 2,x ¨a 3,x ¨am ,x ¨g ]Tis 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 CRITERIAThe 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 CriteriaThe excitation (ground acceleration)x ¨g is a stationary ran-dom process with a power spectral density2224S (4␨␻␻ϩ␻)0g g g S (␻)=(9)¨¨x x 222222g g (␻Ϫ␻)ϩ4␨␻␻g g g where the natural frequency ␻g and the damping ratio ␨g arevariables lying in the intervals 20rad/s Յ␻g Յ120rad/s and 0.3Յ␨g Յ0.75.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 ␨g0.03␨g2S =g (10)02␲␻(4␨ϩ1)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 123J =max,,(11)1ͭͮ␴␴␴␻,␨g gx x x 3o 3o 3o␴␴␴¨¨¨x x x a 1a 2a 3J =max,,(12)2ͭͮ␴␴␣␻,␨¨¨g gx ¨xx a 3o a 3o a 3o ␴x m J =max(13)3ͭͮ␴␻,␨g gx 3o ␴˙x m J =max(14)4ͭͮ␴␻,␨˙x g g3o ␴¨x am J =max(15)5ͭͮ␴␻,␨¨x g ga 3oJOURNAL OF STRUCTURAL ENGINEERING /NOVEMBER 2001/1333FIG. 2.Membership Function Used for Input and Output Variables ofFLCwhere ␴represents the RMS value of the variable shown as a subscript.The interstory drifts d i are the relative lateral dis-placements between floors (d 1=x 1,d 2=x 2Ϫx 1,d 3=x 3Ϫx 2);x ˙i is the lateral velocity of the i th floor relative to the ground;x ¨ai is the absolute acceleration of the i th floor;x m and x ˙m are the displacement and velocity of the actuator piston relative to the third floor,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-floor displacement and velocity relative to the ground and the absolute acceleration of the third floor,respectively,over all the allowed values of the ␻g and ␨g ,for the uncon-trolled case.Deterministic Evaluation CriteriaThe 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)6ͫͭͮͬx x x earthquake recordst3o 3o 3o͉x ¨(t )͉͉x ¨(t )͉͉x ¨(t )͉a 1a 2a 3J =maxmax,,(17)7ͫͭͮͬx ¨x ¨x ¨earthquake recordsta 3o a 3o a 3o ͉x (t )͉m J =max max(18)8ͫͭͮͬx earthquake recordst3o ͉x ˙(t )͉m J =max max(19)9ͫͭͮͬx ˙earthquake recordst3o͉x ¨(t )͉am J =maxmax(20)10ͫͭͮͬx ¨earthquake recordsta 3owhere x 3o and x ˙3o =largest peak uncontrolled values of thedisplacement and velocity relative to the ground,respectively;and x ¨a 3o =largest peak uncontrolled absolute acceleration for the corresponding earthquake record.FLC DESIGNThe FLC has been designed using five 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 finer 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 defined 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 defines the slope of the membership function;and c is the position of the center of the membership function1␮=(21)x 2bx Ϫc 1ϩͯͯaThe membership functions for the input and output variables have been shown in Fig.2.Two inputs,namely,the third-floor absolute acceleration and the third-floor pseudovelocity,have been used in this study.The choice of a velocity and an ac-celeration component for feedback can be explained in thecontext of the state of the system in the fundamental mode ofvibration.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 refined using the information about velocity and acceleration in the case of the fundamental mode of vibration.The inference rules were further fine-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-1334/JOURNAL OF STRUCTURAL ENGINEERING /NOVEMBER2001FIG. 3.Inference Rules for FLC Used in This Study TABLE 1.Specifications 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 7Aggregation Maximum Fuzzy inference Mamdani type (MATLAB 1999)DefuzzificationCenter of gravity (COG)zation.The adopted inference rules in this study are shown inFig.3.Specifications of the FLC used in this study are given in Table 1.OPTIMIZATION OF FLCIn 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 ProblemThe 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 fitness function f 1and f 2as given by (2)and (3).The parameters specified 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 floor relative to the ground,␴˙x 3o the RMS absolute acceleration of the third floor,peak dis-␴¨x 3o placement x 3o and peak velocity x ˙3o of the third floor relative to the ground,and the peak absolute acceleration x ¨3o of the third floor,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 Ϫ0.5,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 fine-tuned after the first 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)specified 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 first five 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 five 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:␴Յ␴(22)x x m mmax␴Յ␴(23)¨¨x x am ammax␴Յ␴(24)u u maxmax ͉u (t )͉Յu (25)maxtmax ͉x (t )͉Յx (26)m m maxtmax ͉x ¨(t )͉Յx ¨(27)am am maxtwhere 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 floor,respectively;and =maximum limit of the ␴x ¨¨x am ammax maxRMS 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 coefficient 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 OptimizationThe 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-ified 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 Ϯ3V ,and sensor noise of 0.01-V RMS (i.e.,0.3%of the span of the ADC).Two inputs,namely,the third-floor absolute acceleration and the third-floor 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-JOURNAL OF STRUCTURAL ENGINEERING /NOVEMBER 2001/1335FIG.4.SIMULINK Model for Building with FLCTABLE 2.Initial Parameters and Constraint LimitsParameters and constraintsValue(a)Worst-Case RMS uncontrolled response of third floor(␨g =0.3,␻g =37.3rad/s)␴x 3o 1.31cm␴˙x 3o 47.9cm/s ␴¨x 3o1.79g(b)Peak uncontrolled response of third floor for 1940El Centro earthquakex 3o 3.37cmx ˙3o 131.0cm/s x ¨3o5.05g(c)Peak uncontrolled response of third floor for 1968Hachinohe earthquakex 3o 1.66cmx ˙3o 58.3cm/s x ¨3o2.58g (d )Actuator constraints␴u max 1.0V ␴x m max3.0cm ␴¨x m a max 2.0g u max 3.0V x m max 9.0cm x ¨am max6.0gFIG. 5.Pareto-Optimal Performance Index J 6versus J 7erations 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 RESULTSThe 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 twoobjectives 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 and1336/JOURNAL OF STRUCTURAL ENGINEERING /NOVEMBER2001FIG. 6.Variation of Performance Index J 1versus J 2for Pareto-OptimalSolutionsFIG.7.Variation of Control Signal for Worst-Case Stability Test ofFLCBienkiewicz 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.CONCLUSIONSContributions 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 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