A feedback control system usually implements active and semiactive control of seismically excited structures. The objective of the control system is described by a performance index, including weighting matrix norms. The choice of weighting matrices is usually based on engineering experience. A new procedure for weighting matrix components choice based on the parametric optimization method is developed in this study. It represents a twostep optimization process. In the first step a discretetime control system is synthesized according to a quadratic performance index. In the second step the weighting coefficients are obtained using the results of the first step. Numerical simulation of a typical structure subjected to earthquakes is carried out in order to demonstrate the effectiveness of the proposed method. It shows that applying the proposed technique provides a choice of the weighting matrices and results in enhanced structural behaviour under different earthquakes. Copyright © 2004 John Wiley Sons, Ltd.
THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS Struct Design Tall Spec Build 13, 5572 (2004) Published online June 2004 in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.233 A NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES IN ACTIVE CONTROLLED STRUCTURES G AGRANOVICH1, Y RIBAKOV2,3* AND B BLOSTOTSKY2 Department of Electric Engineering, Faculty of Engineering, College of Judea and Samaria, Ariel, Israel Department of Civil Engineering, Faculty of Engineering, College of Judea and Samaria, Ariel, Israel Institute for Structural Concrete and Building Materials, University of Leipzig, Germany SUMMARY A feedback control system usually implements active and semi-active control of seismically excited structures The objective of the control system is described by a performance index, including weighting matrix norms The choice of weighting matrices is usually based on engineering experience A new procedure for weighting matrix components choice based on the parametric optimization method is developed in this study It represents a twostep optimization process In the first step a discrete-time control system is synthesized according to a quadratic performance index In the second step the weighting coefficients are obtained using the results of the first step Numerical simulation of a typical structure subjected to earthquakes is carried out in order to demonstrate the effectiveness of the proposed method It shows that applying the proposed technique provides a choice of the weighting matrices and results in enhanced structural behaviour under different earthquakes Copyright â 2004 John Wiley & Sons, Ltd INTRODUCTION Active and semi-active damping of seismically excited structures is usually implemented by a feedback control system (Housner et al., 1997; Spencer et al., 1999) The optimal control forces are generally calculated according to the structural behaviour, which is measured during the earthquake and transferred to a computer These forces are further produced by actuators or dampers installed in the structure Recent feedback control development methods are based on optimal control theories (Antsaklis and Mitchel, 1997; Doyle et al., 1989) These methods require the following mathematical description of the problem First of all mathematical models of the structure and of the excitation should be obtained A performance index for structural behaviour and control rules should then be chosen The performance index is a measure of the control forces and the regulated variables describing the structural behaviour Minimization of this performance index yields an optimal control law According to well-known modern approaches the performance index has a form of various matrix norms, such as L2, H2, and H (Antsaklis and Mitchel, 1997; Doyle et al., 1989; Spencer et al., 1994; Dyke et al., 1995) In most practical optimization problems these indices not directly describe the problem, because they have no direct physical sense Indeed, an integral of squared state vector or control forces vector is very similar to energy But actually arguing about energy minimization has again no physical sense, because generally the performance indexes include a sum of such two squares The sum yields a compromise between the required and the dissipated energy However, a reasonable * Correspondence to: Dr Ing Yuri Ribakov, Universitat Leipzig, Marschnerstrasse 31, 04109 Leipzig, Germany E-mail: ribakov@wifa.uni-leipzig.de Copyright â 2004 John Wiley & Sons, Ltd Received December 2002 Accepted February 2003 56 G AGRANOVICH ET AL question is which energy is more important and how it affects the structural response to earthquakes Moreover, sometimes an apparent improvement of the performance index leads to a worse structural response An additional criterion is proposed in the current study in order to improve the performance index and to design a control system, providing more efficient control and yielding further decrease in structural response to earthquakes Spencer et al (1999) described several direct criteria for structural control of seismically excited buildings However, the feedback optimal control solutions are known for performance indices in the form of matrix norms and for linear structural models only Hence these optimization problems are commonly employed for structural control optimization A classical performance index form is an L2 one with an infinite upper horizon: J (u) = y T (t )Qy(t ) + u T (t ) Ru(t )dt (1) where y is a vector of structural displacements, velocities and accelerations, u is a control forces vector, and Q and R are symmetrical non-negative definite weighting matrices describing the balance of the structural behaviour and of the control action (Dyke et al., 1995; Norgaard et al., 2000) In any performance index described by Equation (1) the relative magnitudes of the control forces (components of u) and of the regulated variables (components of y) should also be taken into account The matrices Q and R usually have a diagonal form and give different weights to components of vectors y and u These weights take into account the different physical nature of the components and different requirements to their values Spencer et al (1999), Dyke et al (1995), Dyke and Spencer (1997), Battaini et al (2000) and others investigated the influence of different weighting coefficients on the effectiveness of optimum control algorithms applied to earthquake-excited buildings Generally most of the coefficients are equal to zero For example, in Dyke and Spencer (1997) only the top storey acceleration weight is non-zero, whereas in Battaini et al (2000) only in the two lower storeys are absolute displacements weights non-zero Generally the matrices Q and R are assumed based on practical experience in structural seismic design An algorithm for weighting matrix components choice based on parametrical optimization method is described in this paper DESCRIPTION OF THE PROPOSED METHOD As mentioned above, generally the weighting matrices Q and R selection (Equation 1) is based on engineering experience Technical constraints on variables and controls can also be taken into account Usually this choice is made by a trial and error method For more qualitative choice of the performance index weighting matrices the following parametrical optimization method is proposed The proposed approach is applicable to various weighted performance indices Its application to acceleration LQG control design of seismically excited structural control is considered in this study The LQG approach is an output feedback design method that has been shown to be effective for design of acceleration feedback control strategies for this class of systems (Spencer et al., 1994; Dyke et al., 1995, 1996; Battaini et al., 2000) Let Jopt be a direct criterion for control strategies evaluation, for example one or several of those described by Spencer et al (1999) Thus two criteria are obtained The first one is J with a known feedback control solution, and the second one is Jopt, for which the feedback control solution is unknown A compromise solution is to use the first criterion (J) as a working criterion for the second one (Jopt) The above-mentioned working criteria contain some weighting parameters Let these parameters be defined by W In this case the second criterion will be a function of weighting parameters W of the Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES 57 first one, i.e Jopt(W) For example, for the performance index J described in Equation (1) W collects the matrices Q and R or their components Thus the problem is reduced to a choice of W, at which the optimal control according to criterion J provides a minimum value of the criterion Jopt(W) This approach enables application of well-developed numerical parametric optimization methods for solution of the problem described by Nelder and Mead (1965) and Gill et al (1981) According to the proposed method, the optimal control synthesis problem should be solved at each step The general linear model of the controlled structure according to the proposed optimization method can be described as follows: x (t ) = Ac x (t ) + Bc u(kt ) + Ec xg (t ) (2) where: x(t) is the state space vector of the systems continuous part, which includes the vectors of story displacements and velocities of the structure, and the state vectors of the actuators and the measurement subsystems; u(kt) is the control signal, which is an output signal of a digital controller for sampling times kt (k = 0, 1, 2, ); t is the controllers sampling period; xg(t) is the ground acceleration; and the matrices Ac, Bc and Ec describe the continuous part of the whole system The control system should be realized in a digital form, hence the differential equations (2) are transformed to an equivalent system of finite-difference equations (based on an equivalent transformation technique described in Antsaklis and Mitchel, 1997) as follows: x (kt ) = Ax (kt ) + Bu(kt ) + Exg (kt ) (3) where t t 0 A = e Act , B = e Act Bc dt , E = e Act Ec dt (4) The output vector contains structural displacements, velocities and accelerations: y(kt ) = Cx (kt ) + Du(kt ) + Fxg (kt ) (5) where matrices C, D and F describe the dependence between the output vector and the structures state vector and excitations The measurement vector ym (kt ) = Cmx (kt ) + Dm u(kt ) + Fm xg (kt ) + v(kt ) (6) contains the floor accelerations of the structure Matrices Cm, Dm and Fm describe the parameters of the measurement subsystem According to the LQG design approach (Dyke et al., 1995, 1996; Battaini et al., 2000) the ground acceleration xg(kt) and the measurement noise v(kt) are taken to be a stationary white noise with known intensity An infinite horizon performance index (Equation 1) takes in this case the following form: J = lim E ẩ y T (kt )Q1T Q1 y(kt ) + u T (kt ) R1T R1u(kt ) Tặ ẻ kt Ê T (7) The square root form of the index weight matrices Q = Q1TQ1 and R = R1TR1 is chosen to avoid the following two problems in parametrical optimization application The first problem is positive definiteness of index weight matrices constraint, which requires application of much more complicated Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 58 G AGRANOVICH ET AL parametrical optimization methods with constraints The second one is the big difference in weight coefficient values, impairing a convergence property of the parametrical optimization process The separation principle allows the control and estimation problems to be considered separately, yielding a discrete-time dynamic controller (Stengel, 1986) Hence the optimal control law is obtained as follows: -1 u(kt ) = - Kx (kt ), K = ( R1T R1 + BT SB) BT SA (8) with the same gain matrix as the deterministic LQ2 - control, where S is the solution of the Riccati algebraic equation given by -1 A T SA - S - A T SB( R1T R1 + B T SB) B T SA + Q1T Q1 = (9) A Kalman steady-state estimation x (kt) of the system state vector is obtained from the filter equation: x (kt + t ) = Ax (kt ) + Bu(kt ) + L[ ym (kt ) - Dm u(kt ) - Cm x (kt )] (10) where x (kt) is the optimal estimate of the systems state vector x(kt) The filter gain matrix L is determined in the following way: L = ( PCmT + EQm FmT )( Rm + PCmT ) -1 -1 APAT - P + EQm E T + ( PCmT + EQm FmT )( Rm + PCmT ) (Cm P + Fm Qm E T ) (11) (12) where Qm and Rm are the intensity matrices of the ground acceleration xg(kt) and the measurement noise v(kt) white-noise approximations, respectively When the performance index (Equation 7) represents a working criterion, Equations (8)(14) yield an optimal feedback control LQG optimization problem solution of this index, minimized with linear dynamic constraints (Equations 3, and 6) Following Spencer et al (1999), each proposed control strategy is evaluated for four historical earthquake records: (i) El Centro (California, 1940), (ii) Hachinohe (Hachinohe City, 1968), (iii) Northridge (California, 1994), (iv) Kobe (Hyogo-ken Nanbu, 1995) The appropriate responses have being used to calculate the evaluation criteria The evaluation criteria (Spencer et al., 1999) are divided into four categories: building responses, building damage, control devices, and control strategy requirements The first three categories have both peak- and norm-based criteria Small values of the evaluation criteria are generally more desirable Depending on the purpose and priorities of designing one of the criteria proposed in Spencer et al (1999) or their combination, a direct optimization criterion Jopt can be chosen As a representative example an optimization problem with a desirable minimum peak inter-storey drift ratio over the time history of each earthquake is considered: El Centro ẽ ễễ di (t ) Hachinohe ễễ J1 = max èmax t ,i hi Northridge ễ ễ ễể Kobe ễ (13) under constraints on a peak forces value generated by all the control devices over the time history of each earthquake: Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES El Centro ẽ ễễ Hachinohe ễễ J11 = max èmax ui Ê Umax t ,i Northridge ễ ễ ễể Kobe ễ 59 (14) where Ê t Ê tmax is the time-history earthquake range, Ê i Ê Nfloors is the building floors range, di(t) is the inter-storey drift of the above-ground level over the time history of each earthquake, and hi is the height of the associated storey The direct criterion was chosen in the following form: J opt (W ) = J1 + r( J11 ) (15) where r(J11) is a penalty function, which possesses zero value if inequality (14) is valid and reaches a high positive value otherwise The direct criteria (Equations 13, 14 and 15) calculation for the linear structure model (Equations 3, and 6) with feedback control (Equations and 10) consists of the following main steps: (i) Weighted matrices Q1 and R1 value assignment Note that all or some of those matrices elements are parameters of the direct optimization criterion Jopt(W) (ii) Feedback control (Equations and 10) parameters K and L calculation using Equations (8), (9), (11) and (12) (iii) Controlled structure simulations over the time history of each earthquake and criteria calculation using Equations (13), (14) and (15) An integral optimization algorithm consists of three main blocks (see Figure 1) Note that the proposed procedure is very similar to a neural net training (Norgaard et al., 2000) In a similar way the proposed algorithm is based on real excitation data, which is obtained from the historical earthquake records However, in this case instead of comparison with desired output an optimal control is realized It is obvious that for a particular earthquake this control will be optimal if in the criterion (Equations 13 and 14) only this particular earthquake record is treated Optimization of the Structural parameters and initial values of W0 assignment Stepwise procedure: Wn + = Wn + s ( Jopt ( Wn )) and minimum Wopt = arg Min( Jopt ( W )) search Optimized system simulation and analysis Figure Parametrical optimization algorithm Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 60 G AGRANOVICH ET AL criterion (13) yields a control, providing the best structural response to the worst-case earthquake conditions In contrast to Equation (13) the following modified criterion is used: J1m El Centro ẽ ễễ di (t ) Hachinohe ễễ = èmax t ,i hi Northridge ễ ễ ễể Kobe ễ (16) It provides the best average result , but not the best structural response to each specific earthquake In this case the direct criterion (15) with the above-described penalty function takes the form J opt (W ) = J1m + r( J11 ) 3.1 (17) NUMERICAL EXAMPLES Description of the structure and preliminary analysis In order to demonstrate affectivity and to verify the proposed optimization procedure, MATLAB-based optimum searches and simulations were carried out A typical six-storey steel office building (DAmore and Astanen-Asl, 1995) designed with UBC-73 (see Figure 2) was chosen for the analysis The structural system consists of a premier welded MR steel frame (Figure 2) Steel ASTM A36 was used for all shapes of columns and grids The stiffness coefficients and floor masses of the building are shown in Table The natural frequencies of the chosen structure are 1ã083, 2ã92, 4ã799, 9ã596, 7ã93 and 6ã478 Hz An initial damping ratio of 2% was assumed for the first vibration mode of the uncontrolled structure columns beams W24x68 W14x95 400 cm W24x68 W14x95 400 cm W24x68 W14x136 400 cm W24x68 W14x136 400 cm W24x102 W14x184 400 cm W24x116 W14x184 520 cm bays ì 610 cm Figure A six-storey structure used for numerical simulation Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES 61 Table Structural parameters of the six-storey building Floor number Floor mass (105 kg) Stiffness coefficient (105 kg/m) 1ã75 1ã75 1ã75 1ã75 1ã75 1ã75 3ã434 0ã865 3ã009 2ã596 2ã183 1ã092 Table Peak inter-storey drifts of the uncontrolled structure (cm) Earthquake record Storey El Centro Hachinohe Northridge Kobe 4ã78 2ã11 3ã17 2ã78 2ã95 1ã61 3ã62 1ã41 1ã87 1ã46 1ã72 1ã00 7ã49 2ã83 3ã84 3ã27 4ã32 2ã66 15ã4 6ã59 9ã3 7ã75 7ã95 4ã34 Table Peak storey accelerations of the uncontrolled structure (m/s2) Earthquake record Storey El Centro Hachinohe Northridge Kobe 5ã2 5ã8 6ã4 7ã7 8ã9 10ã4 3ã2 4ã2 5ã3 5ã2 6ã2 8ã1 9ã6 13ã0 14ã2 13ã8 15ã0 18ã0 13ã8 17ã9 20ã2 20ã5 25ã4 30ã6 Peak inter-storey drifts and story accelerations of the uncontrolled structure under the selected earthquakes are given in Tables and These and following numerical results were obtained using SIMULINK software (MathWorks, 1990) simulation of the structure To this end a version of the Benchmark simulation program for seismically excited buildings (Spencer et al., 1999) modified by the authors was used The above-mentioned four earthquake records were considered with single magnitude level (Spencer et al., 1999), which was equal to Following Spencer et al (1999) and Battaini et al (2000) it was assumed that the noised accelerations of all storeys are available and the control actuators are located at each storey of the structure The dynamics of the measuring instruments and of the control actuators was neglected Similar to Spencer et al (1999), the control force bound Umax in Equation (14) was assumed to be equal to 10,000 N Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 62 3.2 G AGRANOVICH ET AL One- and two-parametric optimization According to the proposed optimization procedure (Figure 1) the vector of optimized parameters W was chosen The weight matrices Q1 and R1 of the working performance index J (Equation 7) have been taken in the following diagonal form: R1 = I6Ơ6 , Q1 = diag{qd I6Ơ6 , qv I6Ơ6 , qa I6Ơ6 } (18) According to Equation (18) the weights of every control force, inter-storey drifts and storey absolute velocities have been assumed to be equal to 1, qd and qv, respectively It should be mentioned that multiplying the criterion (Equation 7) by a constant does not affect the solution Thus, only relative values of the weight parameters are relevant For this reason the control weights in Equation (18) are assumed to be equal to one Note that in the optimization procedures described, for example, in Spencer et al (1999), Dyke et al (1995) and Battaini et al (2000) it is assumed that qd = qv = 0, but R1 and Q1 are diagonal matrices with prescribed numerical values Hence only one optimization parameter qa is used (one-parameter optimization procedure) Some results of the one-parameter qa optimization procedure are presented in Tables 4(ad) In these and the following tables Table 4(a) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric qa optimization under the El Centro earthquake Optimization over El Centro (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 1ã9 0ã7 1ã0 0ã8 0ã8 0ã4 4ã1 4ã7 4ã7 4ã6 4ã3 4ã1 1ã9 0ã7 1ã0 0ã7 0ã8 0ã4 4ã1 4ã7 4ã7 4ã6 4ã2 4ã0 1ã9 0ã7 1ã0 0ã8 0ã8 0ã4 4ã1 4ã7 4ã7 4ã6 4ã2 4ã1 P = 606 Control energy P = 605 P = 605 Table 4(b) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric qa optimization under the Hachinohe earthquake Optimization over Hachnohe (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 1ã3 0ã5 0ã7 0ã5 0ã5 0ã2 1ã7 2ã2 2ã5 2ã9 2ã9 3ã0 1ã3 0ã6 0ã7 0ã5 0ã5 0ã2 1ã7 2ã2 2ã5 2ã9 3ã0 3ã0 1ã3 0ã5 0ã7 0ã5 0ã5 0ã2 1ã6 2ã2 2ã6 2ã9 3ã0 3ã1 P = 352 Copyright â 2004 John Wiley & Sons, Ltd P = 352 P = 353 Struct Design Tall Spec Build 13, 5572 (2004) 63 NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES Table 4(c) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric qa optimization under the Nothridge earthquake Optimization over Nothridge (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 6ã0 2ã4 3ã1 2ã4 2ã3 1ã2 88ã2 10ã0 11ã0 11ã0 12ã0 13ã0 6ã1 2ã1 2ã8 2ã1 2ã0 1ã0 8ã8 10ã0 10ã0 10ã0 10ã0 10ã0 6ã0 2ã1 2ã8 2ã1 2ã0 1ã0 8ã7 10ã0 10ã0 10ã0 10ã0 11ã0 P = 1090 Control energy P = 1470 P = 1450 Table 4(d) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric qa optimization under the Kobe earthquake Optimization over Kobe (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 7ã1 2ã5 3ã3 2ã5 2ã5 1ã2 6ã5 7ã2 7ã0 8ã2 8ã7 9ã1 7ã4 2ã9 4ã0 3ã1 3ã2 1ã6 6ã2 7ã1 7ã8 9ã5 11ã0 11ã0 7ã4 3ã0 4ã1 3ã2 3ã2 1ã6 6ã1 7ã1 7ã9 9ã7 11ã0 11ã0 P = 3800 P = 3770 P= 1Ê i Ê6 tf xiÂ(t ) fi (t ) dt P = 3780 (19) is the total energy required for the control of the structure, where fi(t) is the control force developed by the ith control device and xiÂ(t) is the velocity in the ith control device during the earthquake First an ideal optimization has been performed A real earthquake record was used as an input signal After the parameters of the performance index have been obtained, the same earthquake record has been applied in order to validate the efficiency of the obtained parameters The results of this optimization are shown in Tables 4(ad) (columns and 3) It is obvious that such optimization is unavailable for application, because it requires prior knowledge of the future earthquake The ideal optimization has been performed for the following two reasons First, it enables comparison of the subsequent results of a real optimization with an ideal structural behaviour Secondly, it is possible to show that the values of the optimized parameters essentially depend on the earthquakes record and not only on the earthquakes peak ground acceleration (PGA) Columns and 5, and and 7, in Tables 4(ad) present results of one-parametric optimization over the four chosen earthquakes according to criteria (15) and (17) The weighting coefficient of storey accelerations qa (Equation 18) was selected as an optimized parameter Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 64 G AGRANOVICH ET AL The optimization for each of these two criteria includes about 3040 steps Analysis of the optimization process for the criterion (15) shows that for the order of 1020 steps the process tends to reduce maximal inter-storey drift under the Kobe earthquake having the highest PGA After that the peak inter-storey drift values for the Northridge and Kobe earthquakes are similar The subsequent optimization steps tend to provide a compromise between minimum values for these two earthquakes The one-parameter optimization is not ideal Nevertheless for both criteria (15) and (17) and for each of the four considered earthquakes it yields a close structural response compared to the ideal control (Tables 4ad) However, for the Northridge earthquake it requires higher control energy compared to the ideal control Tables 5(ad) present the results of a two-parametric optimization The weighting coefficients of inter-storey drifts qd and storey accelerations qa (Equation 18) were selected as optimized parameters The process of step optimization for each of two criteria (15 and 17) contains about 5060 steps and yields the following results: qd = 7.17 Ơ 107 and qa = 101 Applying two-parametric optimization yields a decrease in the inter-storey drifts, compared to the one-parametric one; however, it results in an essential increase of floor accelerations It should be noted that the addition of a third optimized parameter qv does not yield any significant improvement compared to the two-parametric optimization results Table 5(a) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric qd, qa optimization under the El Centro earthquake Optimization over El Centro (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 0ã7 0ã3 0ã4 0ã3 0ã3 0ã1 3ã2 3ã2 4ã3 5ã3 6ã1 6ã6 0ã7 0ã3 0ã4 0ã3 0ã2 0ã1 4ã5 5ã4 7ã5 9ã2 11ã0 11ã0 0ã7 0ã3 0ã4 0ã3 0ã2 0ã1 3ã3 3ã3 4ã6 5ã6 6ã6 7ã1 P = 489 Control energy P = 643 P = 511 Table 5(b) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric qd, qa optimization under the Hachinohe earthquake Optimization over Hachnohe (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 0ã6 0ã3 0ã3 0ã2 0ã2 0ã1 1ã5 2ã2 3ã1 3ã9 4ã5 4ã8 0ã6 0ã3 0ã3 0ã2 0ã2 0ã1 1ã1 1ã4 2ã0 2ã4 2ã8 3ã0 0ã6 0ã3 0ã3 0ã2 0ã2 0ã1 1ã0 1ã2 1ã6 2ã0 2ã3 2ã5 P = 735 Copyright â 2004 John Wiley & Sons, Ltd P = 298 P = 269 Struct Design Tall Spec Build 13, 5572 (2004) 65 NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES Table 5(c) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric qd, qa optimization under the Nothridge earthquake Optimization over Nothridge (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 2ã8 1ã1 1ã4 1ã0 0ã9 0ã5 8ã0 10 14 16 19 20 2ã8 1ã1 1ã4 1ã0 0ã9 0ã5 8ã1 10 14 16 19 20 2ã8 1ã1 1ã4 1ã0 0ã9 0ã5 6ã3 6ã9 9ã2 11 13 14 P = 1640 Control energy P = 1640 P = 1500 Table 5(d) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric qd, qa optimization under the Kobe earthquake Optimization over Kobe (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 2ã0 0ã8 0ã9 0ã7 0ã7 0ã3 4ã2 5ã6 7ã8 9ã3 11 12 2ã0 0ã8 0ã9 0ã7 0ã7 0ã3 3ã8 5ã1 6ã7 8ã1 9ã2 9ã8 2ã0 0ã8 1ã0 0ã7 0ã7 0ã3 3ã6 4ã6 6ã3 7ã5 8ã7 9ã3 P = 2790 P = 2750 P = 2630 Roof displacement and roof acceleration time histories in the uncontrolled structure and in the structure with one- and two-parametric optimization (criterion 17) under the El Centro earthquake are shown in Figures and The simulation shows that using one-parametric qa optimization yields a decrease of up to 70% and 60% in roof displacements and accelerations respectively, compared to the uncontrolled structure Applying two-parametric qd, qa optimization yields a further essential decrease in roof displacements compared to one-parametric optimization; however, the accelerations are almost twice as high as in the uncontrolled structure 3.3 Six- and twelve-parametric optimization Supposition regarding the equality of weight matrices diagonal elements (Equation 18) used in the previous numerical example is restrictive Relaxation of this restriction may lead to further improvement in structural behaviour Let us assume the weight matrices Q1 and R1 of the working performance index J in Equation (7) have the following form: R1 = I6Ơ6 , Q1 = diag{Qd , Qv , Qa } Copyright â 2004 John Wiley & Sons, Ltd (20) Struct Design Tall Spec Build 13, 5572 (2004) 66 G AGRANOVICH ET AL Figure Roof displacement time history under the El Centro earthquake (optimization according to criterion 17) Figure Roof acceleration time history under the El Centro earthquake (optimization according to criterion 17) where Qd, Qv, Qa are diagonal Ơ matrices Then different weights of every inter-storey drift Qd, velocity Qv and acceleration Qa are allowed It is obvious that each of the above-mentioned three weight matrices include six optimized parameters Thus, in the examined structure the number of parameters varies from to 18 The simulation results of six-parametric optimization for acceleration weights Qa are presented in Tables 6(ad) Similar to Spencer et al (1999), Dyke et al (1995, 1996), the matrices Qd and Qv have been assumed to be zero The optimal parameter values for criterion (15) are Qa = diag{00007, 00171, 381, 242, 557, 1570} and for criterion (17) the optimal parameters are Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 67 NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES Table 6(a) Peak inter-storey drifts and storey accelerations of the controlled structure with six-parametric Qa optimization under the El Centro earthquake Optimization over El Centro (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 1ã5 0ã6 0ã8 0ã8 0ã7 0ã3 3ã1 5ã2 5ã0 4ã3 4ã0 4ã1 1ã7 0ã6 0ã9 0ã6 0ã7 0ã7 3ã3 5ã4 5ã0 5ã2 4ã5 4ã0 1ã8 0ã6 0ã9 0ã8 1ã3 0ã4 3ã2 5ã4 4ã8 4ã4 3ã7 3ã7 P = 683 Control energy P = 632 P = 640 Table 6(b) Peak inter-storey drifts and storey accelerations of the controlled structure with six-parametric Qa optimization under the Hachinohe earthquake Optimization over Hachinohe (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 0ã9 0ã4 0ã5 0ã6 0ã3 0ã4 1ã2 2ã1 2ã0 2ã2 2ã4 3ã1 1ã1 0ã5 0ã6 0ã4 0ã5 0ã4 1ã2 2ã1 2ã5 2ã8 2ã7 2ã5 1ã1 0ã4 0ã6 0ã5 0ã7 0ã3 1ã1 2ã0 2ã2 2ã3 2ã2 2ã2 P = 381 Control energy P = 346 P = 427 Table 6(c) Peak inter-storey drifts and storey accelerations of the controlled structure with six-parametric Qa optimization under the Northridge earthquake Optimization over Northridge (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 5ã5 2ã1 2ã8 2ã1 2ã1 2ã2 11 11 12 11 9ã4 5ã4 2ã0 2ã8 1ã9 2ã3 1ã9 6ã8 11 11 12 10 8ã7 5ã4 1ã9 2ã8 2ã4 3ã3 1ã2 6ã8 11 10 9ã4 8ã2 8ã3 P = 1480 Copyright â 2004 John Wiley & Sons, Ltd P = 1620 P = 2270 Struct Design Tall Spec Build 13, 5572 (2004) 68 G AGRANOVICH ET AL Table 6(d) Peak inter-storey drifts and storey accelerations of the controlled structure with six-parametric Qa optimization under the Kobe earthquake Optimization over Kobe (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 3ã6 1ã3 2ã4 3ã0 3ã0 2ã2 3ã8 7ã5 6ã7 6ã4 6ã7 7ã0 6ã0 2ã4 3ã3 2ã2 2ã8 2ã5 3ã8 7ã8 7ã7 8ã7 8ã6 7ã7 4ã4 1ã7 2ã2 2ã2 3ã3 1ã1 3ã9 8ã2 7ã7 7ã0 6ã5 6ã6 P = 3860 Control energy P = 3920 P = 3810 Qa = diag{198, 00212, 298, 1090, 3480, 3370} The step optimization process for each of these two criteria includes about 450550 steps The results show the advantage of criterion (17) Peak responses obtained applying this criterion are closer to the ideal optimization (columns and and columns and in Tables 6ad, respectively) However, the improvement is not significant, and for all the selected earthquakes, except the Kobe one, applying criterion (17) requires higher control energy compared to criterion (15) Applying the sixparametric optimization reduces the higher floor accelerations, compared to the two-parametric one; however, the inter-storey drifts are higher (Tables 5a6d) Finally a twelve-parametric optimization has been carried out Different weighting coefficients of inter-storey drift Qd and acceleration Qa were selected as optimized parameters The process of step optimization for each of two criteria (15 and 17) contains about 15001700 steps and yields the following results: Qd = diag{125 Ơ 10 , 536 Ơ 10 , 59 Ơ 10 , 18 Ơ 10 , Ơ 10 , Ơ 10 } Qa = diag{003, 016, 014, 08, 920, 46 Ơ 10 } for criterion (15) and Qd = diag{467 Ơ 10 , 421 Ơ 10 , 196 Ơ 10 , 502 Ơ 10 , 73 Ơ 10 , 986 Ơ 10 } Qa = diag{004, 0195, 056, 119 , 458, 357 Ơ 10 } for criterion (17) The peak values of the inter-storey drifts and floor accelerations in the structure with twelveparametric optimization are shown in Tables 7(ad) Note that applying the twelve-parametric optimization results in low inter-storey drifts as in case of the two-parametric one and in relatively low floor accelerations It is important that for all of the selected earthquakes the required control energy for the twelve-parametric optimization is the lowest, compared to other cases Roof displacement and acceleration time histories of the structure under the El Centro earthquake for the uncontrolled structure and for the cases of two-, six- and twelve-parametric optimization are Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 69 NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES Table 7(a) Peak inter-storey drifts and storey accelerations of the controlled structure with twelve-parametric Qd, Qa optimization under the El Centro earthquake Optimization over El Centro (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 0ã7 0ã3 0ã4 0ã3 0ã2 0ã1 4ã9 6ã2 7ã6 9ã2 11ã0 12ã0 0ã7 0ã3 0ã4 0ã3 0ã2 0ã1 5ã7 6ã8 5ã7 4ã9 5ã2 5ã4 0ã7 0ã4 0ã4 0ã3 0ã2 0ã1 13ã0 7ã4 5ã5 4ã9 5ã2 5ã5 P = 647 Control energy P = 462 P = 511 Table 7(b) Peak inter-storey drifts and storey accelerations of the controlled structure with twelve-parametric Qd, Qa optimization under the Hachinohe earthquake Optimization over Hachinohe (ideal) Storey Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 0ã6 0ã4 0ã3 0ã2 0ã2 0ã1 4ã5 2ã6 2ã6 2ã2 2ã0 2ã0 0ã6 0ã3 0ã4 0ã2 0ã2 0ã1 1ã1 1ã9 2ã0 2ã0 1ã9 1ã9 0ã6 0ã4 0ã3 0ã2 0ã2 0ã1 3ã4 2ã4 2ã3 2ã1 1ã9 1ã9 P = 278 Control energy P = 246 P = 254 Table 7(c) Peak inter-storey drifts and storey accelerations of the controlled structure with twelve-parametric Qd, Qa optimization under the Northridge earthquake Optimization over Northridge (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 2ã8 1ã9 1ã3 1ã0 0ã9 0ã5 14ã0 8ã7 9ã1 10ã0 11ã0 11ã0 2ã8 1ã3 1ã6 1ã0 0ã9 0ã5 8ã9 11ã0 9ã7 11ã0 11ã0 11ã0 2ã8 1ã6 1ã3 1ã0 0ã9 0ã5 19ã0 10ã0 9ã5 10ã0 11ã0 11ã0 P = 1500 Copyright â 2004 John Wiley & Sons, Ltd P = 1390 P = 1420 Struct Design Tall Spec Build 13, 5572 (2004) 70 G AGRANOVICH ET AL Table 7(d) Peak inter-storey drifts and storey accelerations of the controlled structure with twelve-parametric Qd, Qa optimization under the Kobe earthquake Optimization over Kobe (ideal) Storey Control energy Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) Drift (cm) Acc (m/s2) 2ã0 1ã5 0ã9 0ã7 0ã6 0ã3 7ã0 5ã4 6ã7 7ã2 8ã2 9ã2 2ã0 0ã9 1ã1 0ã7 0ã6 0ã3 4ã1 6ã1 7ã0 7ã0 7ã8 8ã1 2ã0 1ã1 0ã9 0ã7 0ã6 0ã3 11ã0 6ã4 6ã8 7ã2 7ã6 8ã1 P = 2830 P = 2490 P = 2550 Figure Roof displacement time history under the El Centro earthquake (optimization according to criterion 17) shown in Figures and It demonstrates that applying twelve-parametric optimization yields the most effective reduction in structural response Similar results were obtained for the three other selected earthquakes CONCLUSIONS A new procedure for control design of seismically excited structures was developed and verified The procedure represents a two-step optimization process At the first step a discrete-time control system is synthesized according to a quadratic performance index At the second step the weighting coefficients for the performance index used in the first one is carried out It means that the second criterion is a working criterion for the first one Numerical simulations of a typical six-storey steel office building were carried out in order to demonstrate the effectiveness of the proposed optimization procedure Optimum search and simulaCopyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES 71 Figure Roof floor acceleration time history under the El Centro earthquake (optimization according to criterion 17) tions were carried out by means of MATLAB and SIMULINK-based programs The numerical simulation showed high efficiency of the proposed method Its main advantage is providing a choice of the index weighting coefficients in complicated control problems of multistorey structures, when the trial and error method and intuition are ineffective Applying the proposed algorithm is an efficient way to further improve the structural response to earthquakes Further investigation of the proposed algorithm, including laboratory tests, is required in order to make it useful for practical applications ACKNOWLEDGEMENTS The Centre of Scientific Absorption of the Ministry of Absorption, State of Israel, supported the research The financial support of the Humboldt Foundation, Germany, is greatly appreciated REFERENCES Antsaklis PJ, Mitchel AM 1997 Linear Systems McGraw-Hill: New York Battaini M, Yang G, Spencer BF Jr 2000 Bench-scale experiment for structural control Journal of Engineering Mechanics, ASCE 126(2): 140148 DAmore E, Astanen-Asl A 1995 Seismic behavior of six-story instrumented building under 1987 and 1994 Northridge earthquakes Report No UCB/CE: Steel 95/03 Department of Civil Engineering, University of California: Berkeley, CA Doyle JC, Glover K, Khargonekar P, Francis B 1989 State-space solutions to standard H2 and H control problems IEEE Transactions on Automatic Control 34: 831847 Dyke SJ, Spencer BF Jr 1997 A comparison of semi-active control strategies for the MR damper In Proceedings of the IASTED International Conference, Intelligent Information Systems, Bahamas, 810 December 1997 Dyke SJ, Spencer BF Jr, Quast P, Sain MK, Kaspari DC Jr, Soong TT 1995 Acceleration feedback control of MDOF structures Journal of Engineering Mechanics, ASCE 122(9): 897971 Dyke SJ, Spencer BF Jr, Quast P, Kaspari DC Jr, Sain MK 1996 Implementation of active mass driver using acceleration feedback control Microcomputers in Civil Engineering, Special Issue on Active and Hybrid Structural Control 11: 305323 Gill PE, Murray W, Wright MH 1981 Practical Optimization Academic Press: London Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) 72 G AGRANOVICH ET AL Housner G, Bergman LA, Caughey TK, Chassiakov AG, Claus RO, Masri SF, Skelton RE, Soong TT, Spenser BF, Yao JTP 1997 Structural control: past, present, and future Journal of Engineering Mechanics, ASCE 123(9): 897971 MathWorks 1990 MATLAB Users Guide MathWorks: Natick, MA Nelder JA, Mead R 1965 A simplex method for function minimization Computer Journal 7: 308313 Norgaard M, Ravn O, Poulsen NK, Hansen LK 2000 Neural Networks for Modelling and Control of Dynamic Systems Springer: Berlin Soong TT 1990 Active Structural Control: Theory and Practice Wiley: New York Spencer BF Jr, Suhardjo J, Sain MK 1994 Frequency domain optimal control for aseismic protection Journal of Engineering Mechanics, ASCE 120(1): 135159 Spencer BF Jr, Christenson RE, Dyke SJ 1999 Next generation benchmark control problem for seismically excited buildings Proceedings of the Second World Conference on Structural Control, Vol Wiley: Chichester; 13511360 Stengel RF 1986 Stochastic Optimal Control: Theory and Application Wiley: New York Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 5572 (2004) [...]... (optimization according to criterion 17) tions were carried out by means of MATLAB and SIMULINK-based programs The numerical simulation showed high efficiency of the proposed method Its main advantage is providing a choice of the index weighting coefficients in complicated control problems of multistorey structures, when the ‘trial and error’ method and intuition are ineffective Applying the proposed algorithm... structure Applying two-parametric qd, qa optimization yields a further essential decrease in roof displacements compared to one-parametric optimization; however, the accelerations are almost twice as high as in the uncontrolled structure 3.3 Six- and twelve-parametric optimization Supposition regarding the equality of weight matrices diagonal elements (Equation 18) used in the previous numerical example... Systems, Bahamas, 8–10 December 1997 Dyke SJ, Spencer BF Jr, Quast P, Sain MK, Kaspari DC Jr, Soong TT 1995 Acceleration feedback control of MDOF structures Journal of Engineering Mechanics, ASCE 122(9): 897–971 Dyke SJ, Spencer BF Jr, Quast P, Kaspari DC Jr, Sain MK 1996 Implementation of active mass driver using acceleration feedback control Microcomputers in Civil Engineering, Special Issue on Active and... Department of Civil Engineering, University of California: Berkeley, CA Doyle JC, Glover K, Khargonekar P, Francis B 1989 State-space solutions to standard H2 and H• control problems IEEE Transactions on Automatic Control 34: 831–847 Dyke SJ, Spencer BF Jr 1997 A comparison of semi -active control strategies for the MR damper In Proceedings of the IASTED International Conference, Intelligent Information... parameters Thus, in the examined structure the number of parameters varies from 6 to 18 The simulation results of six-parametric optimization for acceleration weights Qa are presented in Tables 6 (a d) Similar to Spencer et al (1999), Dyke et al (1995, 1996), the matrices Qd and Qv have been assumed to be zero The optimal parameter values for criterion (15) are Qa = diag{0◊0007, 0◊0171, 381, 242, 557 ,... 2630 Roof displacement and roof acceleration time histories in the uncontrolled structure and in the structure with one- and two-parametric optimization (criterion 17) under the El Centro earthquake are shown in Figures 3 and 4 The simulation shows that using one-parametric qa optimization yields a decrease of up to 70% and 60% in roof displacements and accelerations respectively, compared to the uncontrolled... is an efficient way to further improve the structural response to earthquakes Further investigation of the proposed algorithm, including laboratory tests, is required in order to make it useful for practical applications ACKNOWLEDGEMENTS The Centre of Scientific Absorption of the Ministry of Absorption, State of Israel, supported the research The financial support of the Humboldt Foundation, Germany,... Tall Spec Build 13, 55 72 (2004) 69 NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES Table 7 (a) Peak inter-storey drifts and storey accelerations of the controlled structure with twelve-parametric Qd, Qa optimization under the El Centro earthquake Optimization over El Centro (‘ideal’) Storey 1 2 3 4 5 6 Optimization according to (15) Optimization according to (17) Drift (cm) Acc (m/s2) Drift (cm) Acc... ¥ 10 8 } Qa = diag{0◊04, 0◊195, 0◊56, 119 ◊ , 4◊58, 3◊57 ¥ 10 3 } for criterion (17) The peak values of the inter-storey drifts and floor accelerations in the structure with twelveparametric optimization are shown in Tables 7 (a d) Note that applying the twelve-parametric optimization results in low inter-storey drifts as in case of the two-parametric one and in relatively low floor accelerations It... earthquake (optimization according to criterion 17) Figure 4 Roof acceleration time history under the El Centro earthquake (optimization according to criterion 17) where Qd, Qv, Qa are diagonal 6 ¥ 6 matrices Then different weights of every inter-storey drift Qd, velocity Qv and acceleration Qa are allowed It is obvious that each of the above-mentioned three weight matrices include six optimized parameters