Wavelet PSO-Based LQR Algorithm for Optimal ... - Semantic Scholar

Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-46140

WAVELET PSO-BASED LQR ALGORITHM FOR OPTIMAL STRUCTURAL CONTROL USING ACTIVE TUNED MASS DAMPERS Mahdi Abdollahirad Istanbul Technical University, Department of Civil Engineering, Istanbul, Maslak 34469, Turkey [email protected]

Arcan Yanik Istanbul Technical University, Department of Civil Engineering, Istanbul, Maslak 34469, Turkey [email protected]

Unal Aldemir Istanbul Technical University, Earthquake Engineering and Disaster Management Institute, Istanbul, Maslak 34469, Turkey [email protected]

ABSTRACT

Mitigating the seismic responses of major structures such as tall buildings, wind sensitive bridges, and poten-tially susceptible structures has been studied comprehensively over the years [1519]. The passive tuned mass damper (PTMD) is effective in reducing the vibration of structures caused by earthquakes with limited band frequency [20-21]. TMD system relies on the damping forces introduced through the inertia force of a secondary system attached to the main structure by spring and dashpot to reduce the response of the main structure. Lin et al. (2010) used initially accelerated passive TMD to suppress structural peak responses under near-fault ground motions [22]. This study showed that the PTMD initial velocity is effective in vibration control of structures under pulse-like ground motion. However, due to the limitation of PTMD stroke as well as the applied force, the initial velocity cannot be too large in practical applications. To overcome these shortcomings, the active control can be applied to TMD. Extensive reviews on using the active tuned mass damper (ATMD) can be found in civil engineering literature [23-24]. Chey et al. (2010) proposed a semi-ATMD to mitigate the response of structures. In this study, the stiffness of the resettable device design was combined with rubber bearing stiffness [25]. Moreover, some researchers developed an advanced control law for Semi-Active resettable devices [26]. Also, other researcher proposed a new resettable device control to reduce the response of structures [27-28].

This study presents a new method to find the optimal control forces for active tuned mass damper. The method uses three algorithms: discrete wavelet transform (DWT), particle swarm optimization (PSO), and linear quadratic regulator (LQR). DWT is used to obtain the local energy distribution of the motivation over the frequency bands. PSO is used to determine the gain matrices through the online update of the weighting matrices used in the LQR controller while eliminating the trial and error. The method is tested on a 10-story structure subject to several historical pulse-like near-fault ground motions. The results indicate that the proposed method is more effective at reducing the displacement response of the structure in real time than conventional LQR controllers. INTRODUCTION With the development of construction techniques, it is possible to build large-span bridges, pipelines, dams, and other essential structures in seismically active regions or through active faults. The strong earthquakes, which have occurred in recent years, have resulted in severe damage to the near-fault buildings and bridges and loss of life. Hence, research on the characterization and analytical modeling of near-fault seismic ground motions, as well as the study of their influences on the performance of engineering structures, has become an active area in both seismology and the earthquake engineering field [1-14].

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Copyright © 2015 by ASME

Comprehensive studies have been done to determine the optimal actuator force for the active vibration control systems. The most widespread methods are linear quadratic regulator (LQR), LQG, H2, H∞, sliding mode control, pole assignment, Clippes Optimal Control and Bang-Bang control. Most control methods are based on the optimization technique of maximizing the performance using less control energy under certain constraint and most optimization algorithms used in control design are traditional methods. In this approach, there are difficulties associated in selecting the suitable continuous differentiable cost function [29]. Unlike traditional optimization methods, evolutionary algorithms such as genetic algorithm (GA) find an optimal solution from the complex and possibly discontinuous solution space. In the field of structural control, GAs have been applied to obtain gains for the optimal controller [30], reduced order feedback control [31], optimal damper distribution [32], and design and optimize the different parameters of the ATMD control scheme [33]. Aldemir (2010) introduced a simple integral type quadratic functional as the performance index to suppress the seismic vibrations of buildings [34]. He used the method of calculus of variations to minimize the proposed performance index and obtain the optimal control force. Also, Aldemir et al. (2012) proposed simple methods to obtain the suboptimal passive damping and stiffness parameters from the optimal control gain matrix to control structural response under earthquake excitation [35]. Bitaraf et al. (2012) proposed a directadaptive-control method to control the behavior of an undamaged and a damaged structure using semi-active and active devices [36]. Their method is defined in a way to optimize the response of the controlled structure. Some researchers proposed a new approach for nonlinear system identification and control based on modular neural networks [37]. Their proposed method reduced the computational complexity of neural identification by decomposing the whole system into several subsystems. Particle swarm optimization (PSO) is a global optimization technique that has been developed by [38]. The underlying motivation of PSO algorithm was the social behavior observable in nature, such as flocks of birds and schools of fish to guide swarms of particles towards the most promising regions of the search space. In terms of population initialization with random solutions, PSO is similar to GAs. GA has been widely used as an optimization technique in the engineering field [39-44]. In comparison with GA, PSO does not use crossover and mutation operator. PSO as an optimization tool provides a population based search procedure in which individuals called particles change their position (state) with time. Each particle in the swarm represents a candidate solution to the optimization problem. Particle swarms had not been used in the field of structural engineering until recently, where limited studies have been performed in the areas of structural shape optimization [45-46] as well as topology optimization [47]. Some researchers implemented the PSO algorithm for constrained structural optimization of plane and space truss structures [48-49] whereas [50] tried a heuristic PSO scheme for the optimization of truss structures. Another researcher [51]

, employed PSO algorithm to find the optimum parameters of a TMD subject to stationary base excitations. PSO has been also combined with mathematical methods in various ways. Some researchers presented hybrid PSO-gradient algorithm for global structural optimization [52]. In another studyPSO-based neural network to generate artificial pulse-like near-field ground motions was proposed [53]. Researchers also combined PSO algorithm with radial basis function neural network to predict Parkinson’s disease tremor [54]. Wavelet analysis is commonly used by the engineering community for signal processing to show the frequency content of a signal, for example, in earthquake engineering [55-56], structural engineering [57-59], transportation engineering [6061], dam engineering [62] , medical science [63], and image analysis [64]. The LQR is used widely to determine the appropriate control force by many researchers [65-66]. However, the traditional LQR algorithm does not consider external force vibrations such as earthquakes or wind. To simulate realistic circumstances, the excitation must be known prior to determining the optimal control force to achieve more reliable solutions. The effect of specific earthquakes has been accounted for in a few studies [67-68]. For example some researchers introduced a method based on updating weighting matrices from a database of earthquakes [69]. Nonetheless, in these studies, offline databases were still required. Some other researchers proposed a wavelet-based time-varying adaptive LQR algorithm to updated weighting matrixes online using scalar multiplier or factor according to the energy in the different frequency bands over a time window [70]. However, although in this method the weighting matrixes change when the resonance occurs, the scalar multiplier was chosen offline and it has constant value. Therefore, in this study offline data were still obliged. The design parameters of weighting matrices in LQR control system should be selected by the designer. As these matrices directly affect the optimal control performance many discussions have been done to select these matrices called eigenstructure assignment [71-72]. Obviously, it is not easy to determine the appropriate weighting matrices for an optimal control system and a suitable systematic method is not presented for this goal. In other words, there is no direct relationship between weighting matrices and control system characteristics, and selecting these matrices is done by using trial and error based on designer’s experience. In an LQR problem with the quadratic performance index J, the choice of the parameters in penalty matrices Q and R is crucial for the performance of feedback controlled system under consideration. In this article, we present a new method to find optimal control force of ATMD via PSO-based LQR and wavelet analysis with application to the control of a civil structure. In this study, the use of DWT as a time-frequency tool helps to provide time-varying energy in different frequency bands. Furthermore, the weighting matrices are updated online using PSO algorithm according to the energy in each frequency band over a time window. The application of the proposed

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0 𝐴𝐴 = � −𝑀𝑀−1 𝐾𝐾

approach to a number of pulse-like near-fault ground motions is presented. This method is developed and compared with conventional LQR methods. It demonstrates that proposed method is effective in reducing the response of structures during earthquake motion. Moreover, in comparison with similar methods, it shows that by using wavelet PSO-basedLQR (WPSOB-LQR) not only does the displacement response of structures decrease significantly, but also the total energy demand reduces.

0 𝐵𝐵 = � −1 � 𝑀𝑀 𝐿𝐿

0 𝐸𝐸 = � −1 � 𝑀𝑀 𝐻𝐻

(1)

𝑡𝑡

𝐽𝐽 = ∫0 𝑓𝑓�𝑥𝑥 𝑇𝑇 (𝑡𝑡)𝑄𝑄𝑄𝑄(𝑡𝑡) + 𝑢𝑢𝑇𝑇 𝑅𝑅𝑅𝑅(𝑡𝑡)�𝑑𝑑𝑑𝑑

{𝑥𝑥} = �

𝑞𝑞(𝑡𝑡) � 𝑞𝑞̇ (𝑡𝑡)

(2)

THE PROPOSED METHOD Excitations, such as earthquakes, are extremely nonstationary in amplitude and frequency. Therefore, the probability of existence of the natural frequency of the system, in the frequency domain of ground motion, is high, which causes resonance in the system. The weighting matrices are determined offline and are not updated when the structure is subjected to external excitations. Therefore, to account for the effect of resonance, the classical LQR has to be modified by adapting weighting matrices. The control effort will be more costly if the components of R are large relative to those of Q, for the whole period of external force. Therefore, the response in these undesirable frequency bands must be mitigated without increasing the gain over all the frequency bands, by changing the control energy weighting matrices in these bands. In other words, the weighting matrices should change if the local frequency content of earthquake is close to the natural frequency of the system. In this study, the real-time DWT controller is updated at regular time steps from the initial time (t0) until the current time (tc) to achieve the local energy distribution of the motivation over frequency bands. The time interval under consideration (t0, tc) is subdivided into time-window bands. The time of ith window is

(3)

(4)

{x} is the state vector of dimension 2(m + n) × 1, and

{u} = −{G}{x}

(9)

The matrices Q and R are called the response and control energy weighting matrices, respectively. The weighting matrices Q and R are selected in classical LQR at the first step. The control force is given by Eqn. (5) where the gain matrix is obtained from the discrete algebraic Riccati equation.

Displacement vector is defined as q(t) = (m+n)×1 and qi is the displacement of ith floor relative to ground (i = 1, 2, ..., N), qd is the displacement of damper relative to ground, control force vector, u(t), is of the order l×1, and fe(t) is the external dynamic force vector of dimension r × 1, L and H are (n+m) × l and (n+m)×r location matrices, which define locations of the control forces and the external excitations, respectively. A statespace representation of Equation (1) can be written as:

where

(8)

The classical LQR algorithm can determine the optimal active control forces for the linear system with the aim of minimizing the cost function. The cost function is a quadratic function of the control effort and the state. The cost function is defined as

where M, C, and K are the mass, damping, and stiffness matrices of the structure, respectively, of the order (n+m)×(n+m). If “q” in Eqn. (1) is taken as the relative displacement with respect to the ground, then mass matrix M is considered to be diagonal. Damping matrix C takes a form similar to K.

{x˙} = [A]{x+ [B]{u} + [E] f e

(7)

matrix, control location, and external excitation location matrices, respectively. The matrices “0” and “I” in Eqns. (6) to (8) denote the zero and identity matrices of size (m+n)×( m+n), respectively.

When n-degree-of-freedom (N-DOF) systems with “m” ATMDs are subjected to external excitation and control forces, they govern equations of motion and can be written as

q = [q1q2q3 ...qn qd]

(6)

2(m+n)×2(m+n), 2(m+n)×l, and 2(m+n)×r are the system

THE EQUATIONS OF MOTION OF ATMD AND QUADRATIC COST FUNCTION OF CLASSICAL LQR

M 𝑞𝑞 ∙∙(t) + Cq˙(t) + Kq(t) = L.u(t) + H. f e(t)

𝐼𝐼 � −𝑀𝑀−1 𝐶𝐶

(5)

{G} is the gain matrix;

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for real-time control of structures, because it can detect the time of earthquake frequency change efficiently. The exclusive range of frequency of Dj is denoted as follows:

(ti-1, ti) in which the signal can be decomposed into timefrequency bands by wavelet. Using DWT with a multiresolution analysis (MRA) algorithm, the exact decomposition of signals over time-window bands is obtained in real time. The local energy content at different frequency bands over the considered time window are given by the MRA. Clearly, a frequency contains maximum energy is domain frequency of that window. When the domain frequency of each window approaches the natural frequency of the system, resonance occurs in the structure, which causes a subsequent high displacement response. The capability of the wavelet for carrying out timefrequency analysis has been exploited in this article to resolve frequency locally in time. Wf (a, b), CWT of a signal f(t), is defined a

(10)

𝑊𝑊𝑓𝑓 (𝑎𝑎, 𝑏𝑏) =

frequency range of level

𝑗𝑗 = [𝑓𝑓1 𝑗𝑗, 𝑓𝑓2 𝑗𝑗] where f1j and f2j are expressed as follows: f1 j = (2−j−1)/t, f2 j = (2−j)/t

Here 𝛹𝛹 ∗ is the complex conjugate of a base function (“𝛹𝛹 ”). “a” and “b” are scale and translation parameters, respectively. The scaling parameter, “a,” represents the frequency content of the wavelet. The translation parameter, “b,” represents the location of wavelet in time. Thus, in comparison to the Fourier transform, the basic function of the wavelet transform retains the time zone, as well as the frequency zone. The main idea of DWT is the same as that of CWT. In DWT, the scale parameter “a” and the translation parameter “b” are discretized by using the dyadic scale, that is:

𝑡𝑡

𝐹𝐹

𝐽𝐽𝑖𝑖 = ∫𝑡𝑡 𝑖𝑖−1�𝑥𝑥 𝑇𝑇 (𝑡𝑡)𝑄𝑄𝑖𝑖 𝑥𝑥(𝑡𝑡) + 𝑢𝑢𝑇𝑇 𝑅𝑅𝑖𝑖 𝑢𝑢(𝑡𝑡)�𝑑𝑑𝑑𝑑

(16)

{u} = −[G]i{x}

(17)

𝑖𝑖

ti and

(11)

the control equation is defined as:

where [G]i, the gain matrix of the ith window, is employed to obtain the desired control of the response. The gain matrix [G]i, in the steady-state case, is obtained using the Riccatti algebraic equation, similar to the traditional LQR problem, using updated weighting matrices for [Q]i and [R]i. The 2(n + m) × 2(n + m) state weight matrix Q is considered to be diagonal with the following structure:

where z is a set of positive integers. In this study, the following relationship is used to compute the pseudo-frequencies corresponding to that scale (MATLAB, 2008): 𝑐𝑐 𝐹𝐹𝑎𝑎 = 𝛼𝛼.∆

(12)

[𝑄𝑄] = �

where “α” is a scale, “∆” is the sampling period, Fc is the frequency maximizing the Fourier amplitude of the wavelet modulus (center frequency in Hz), and Fa is the pseudofrequency corresponding to the scale “a,” in Hz. In the case of DWT, the wavelet plays the role of dyadic filter. The DWT analyzes the signal by implementing a wavelet filter of a particular frequency band to propagate along a time axis. The signal can be decomposed to wavelet details and wavelet approximations at various levels, as follows:

𝑄𝑄11 0

0 � 𝑄𝑄22

(18)

The diagonal matrix component Q11 and Q22 contain the weights associated with the relative displacements and velocities, respectively. In this study, the weighting matrix for the response states is assumed to have a constant value at all times and the control energy weighting matrices are updated for every time window by a scalar multiplier and can be defined as

[R]i = δi[I] 𝑓𝑓(𝑡𝑡) = 𝐴𝐴𝐽𝐽 + ∑𝑗𝑗≤𝐽𝐽 𝐷𝐷𝑗𝑗

(15)

where t is the time step of f(t). To mitigate the displacement responses of the structure, a high control force is needed. To decrease the responses of the structure, it is suitable to find optimal values of the control energy weighting matrix [R]. To obtain the optimum response of the system without consuming a lot of energy, the weighting matrices [R] are calculated via PSO algorithm on the resonant band of frequency and the TMD switch off if the displacement of structure is less than the allowable displacement. The advantage of this local optimal solution is that it has the ability to change the value of the matrix [R] on a special frequency, in contrast to the classic LQR which is a global optimal solution. To achieve this, the cost function integral is developed with weighting matrices of each of the windows, [Q]i and [R]i. The cost function is given by

∞ 1 𝑡𝑡−𝑏𝑏 ∫ 𝑓𝑓(𝑡𝑡)𝛹𝛹 ∗ � 𝑎𝑎 � 𝑑𝑑𝑑𝑑 �𝑡𝑡|𝑎𝑎| −∞

a = 2j,b = k.2j j,kεz

(14)

(19)

(13) where δi is a scalar parameter used to scale the weighting matrix and is obtained via PSO algorithm based on the time-

where Dj denotes the wavelet detail and AJ stands for the wavelet approximation, respectively. DWT can be very useful

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where X p and Fp are the controlled responses and control forces of structure calculated using WPSOB-LQR, and Xc and Fc are the controlled responses and control forces of structure calculated through classic LQR. The control energy weighting matrices are reduced when the structure has a significant high value of displacement response. This reduction of weighting matrices sets off the lesser displacement without penalty. Therefore, the positive aspect of proposed method is that the gain matrices are calculated adaptively by using the time-varying weighting matrices depending on online response characteristics instead of a priori (offline) choice of the weights as in the classical case.

frequency analysis of a response state. Hence, the scalar parameter of gain matrix can be written as:

δi ≠ 1

δi = 1

if the frequency of excitation is close to the natural frequency of system, Otherwise.

The value of the δi has been proposed as less than one when the resonance happens. This makes it possible to change the weighting matrices for different frequency bands. PSO algorithm starts with a random population (swarm) of individuals (particles) in the search space and works on the social behavior in the swarm. The position and the velocity of the ith particle in the d-dimensional search space can be represented as Xi =[xi,1, xi,2, ..., xi,d] and Vi = [vi,1, vi,2, ..., vi,d], respectively. Each particle has its own best position (pbest) Pi = [pi,1, pi,2, ..., pi,d] corresponding to the personal best objective value obtained so far at time t. The global best particle (gbest) is denoted by Pg, which represents the best particle found so far at time t in the entire swarm. The new velocity of each particle is calculated as follows:

vi,j(t + 1) = wvi,j(t) + c1r1(pi,j − xi,j(t)) +c2r2(pg,j − xi,j(t))

(20)

where j = 1, 2,..., d; c1 and c2 are acceleration coefficients; w is the inertia factor; and r1 and r2 are two independent random numbers uniformly distributed in the range of [0, 1]. Thus, the position of each particle is updated in each generation according to the following:

xi,j(t + 1) = xi,j(t) + vi,j(t + 1)

(21)

The optimal control effort is obtained for each window with updated weighting matrix of the control effort [R] via the algebraic Riccatti equation. The updated weighting matrix and optimal control gains are obtained for each window, independent of the neighboring windows. In other words, the solution to the modified optimal control problem need not consider the transition conditions between two windows. In the proposed method, the total duration of the external excitation is subdivided into a number of time windows. For each of these windows, the cost function is minimized subject to the constraint given by Equation (1), by updating the weight matrices in real time. Figs. 1 and 2 show the flowchart of the PSO algorithm and proposed method for determining control energy weighting matrices and optimal control force. The fitness or objective function for the PSO algorithm for each ground motion is as follows: 𝑋𝑋 (𝑖𝑖)

𝐹𝐹 (𝑖𝑖)

𝐽𝐽 = � 𝑋𝑋𝑝𝑝(𝑖𝑖) + 𝐹𝐹𝑝𝑝(𝑖𝑖) � /2 𝑐𝑐

𝑐𝑐

Figure 1. FLOWCHART OF THE PROPOSED METHOD

(22)

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The parameter δ used for scaling the weighting matrices is determined via PSO algorithm when the central frequency of each window band is close to the natural frequency of the MDOF system, and for others frequencies is assumed to be 1. Hence, the weighting matrix component [R], equals to δ[I] for resonance frequency bands and for the rest of the frequency bands it is kept as [I]. In as much as the diagonal matrices Q11 and Q22 contain the weights associated with the relative displacements and the relative velocities, respectively, these matrices are chosen as

Q11 = diag(1,1,1,1,1,1,1,1,1,1,0.001) Q22 = diag(1,1,1,1,1,1,1,1,1,1,0.001)

[23]

The state-weighting matrix Q is chosen and is similar to classic LQR for all frequency bands. Daubechies wavelet of order 4 (db4), is used as a mother wavelet to decompose the time history of acceleration for different window bands, to determine the frequency distribution of each band. The Daubechies wavelets have reasonably good localization in time and frequency to capture the effects of local frequency content in a time signal, and allow for fast decomposition by using MRA. The signals recorded in real time are decomposed for each interval window, which is considered as 1 second for updating. The updating interval has been decided upon based on the practical consideration in implementation of the updating as well as (i) expected range of local frequency content which is dependent on the excitations (in this case earthquakes) and (ii) the possible range of the structural natural period. The gain matrices are updated for each window by solving the Riccatti equation. Therefore, the control forces and controlled responses are calculated. The MATLAB software is utilized to calculate all computation. For numerical example, the response of structure of the typical MDOF structure with only one ATMD on the top floor is compared with the corresponding uncontrolled ones under Imperial Valley-06 (El Centro Array-06, 1979), Imperial Valley06 (Agrarias, 1979), and Loma Prieta (Oakland, 1989) ground motions in Figs. 3–5.

Figure 2. FLOWCHART OF THE PSO ALGORITHM.

NUMERICAL EXAMPLES In this section, to investigate the potential application of the proposed PSO-based modified LQR control, the results of dynamic analysis of the typical MDOF structure with only one ATMD on the top floor are discussed. To illustrate the potential application of the proposed method an example of a 10-storey shear-frame building structure which has been excited by a number of nearfault pulse-like earthquake ground motions has been considered. The structure represents a typical shear building with 10 stories, which has the lumped masses at each storey. The masses and the stiffness at each storey are supposed to be 10 ton and 2 × 106 KN/m, respectively. The modal damping ratio is uniformly assumed to be 2% for each mode. The TMD frequency is assumed to be close to the building’s first modal frequency; that is, 2.11 rad/s. The mass and damping ratio of TMD is supposed to be 3% of the total mass of the system and 7%, respectively. The ATMD switches on when the displacement of the top floor is more than the allowable displacement of structure, which is assumed to be one thousandth of the height of the structure.

Figure 3. RESULTS FOR THE 1981 (El Centro-06) (CONTROLLED & UNCONTROLLED DISPLACEMENT.)

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Figure 4. RESULTS FOR THE (CONTROLLED & UNCONTROLLED DISPLACEMENT.)

(AGRASIAS)

Figure 5. RESULTS FOR THE 1989 Loma Prieta (Oakland) (CONTROLLED & UNCONTROLLED DISPLACEMENT.) Figure 6. RESULTS FOR THE 1981 IMPERIAL VALLEY (EL CENTRO-06) Also, to illustrate the potential application of the proposed method, the response of structure, control force and the energy corresponding to the controller for the proposed method are compared with conventional LQR. In this study, the energy of control forces is defined as: 𝑡𝑡

𝐸𝐸𝑐𝑐 = ∫0 𝐹𝐹𝑐𝑐𝑇𝑇 𝑢𝑢̇ 𝑑𝑑𝑑𝑑

(23)

where Fc is the control force of top floor of structure and 𝑢𝑢̇ is an n-dimensional velocity vector. The time history of displacement response, by using different control methods and the time history of required control force for the MDOF under mentioned ground motions are shown in Figs. 6–8. Figs. 6–8 show that the displacement and the total control energy demand, are significantly reduced using our proposed method. In Figs. 6-8 (a), (b), (c) and d represent; controlled displacement , control force , energy demand and main frequency content of ground motion respectively. The control forces for the two systems are not very different for most of the times over the duration except from the peak control. The peak control force increases slightly just on the resonant band of frequency compared to the classic LQR, therefore, the response of structures decreases without higher penalty.

Figure 7.RESULTS FOR THE 1979 IMPERIAL VALLEY AGRASIAS

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Table 1. COMPARISON OF EFFECTIVENESS

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Figure 10. COMPARISON OF THE DISPLACEMENT RESPONSE OF THE STRUCTURES BETWEEN THREE METHODS (1979, Imperial Valley-06 (Agrarias)). CONCLUSION

A new control method using PSO and DWT has been proposed. DWT has been used as a powerful time-frequency tool to provide time-varying energy in different frequency bands. Also, the optimal active control force has been derived by using PSO algorithm based on the minimization of the gain matrices in LQR controller. This method determines the time-varying gain matrices by updating the weighting matrices online, through PSO algorithm, in each frequency band. Therefore, this method does not need prior information about external excitation, thus eliminating the need for an offline database. Also, to obtain the optimum response of structure, the TMD is switched off when the displacement of the structure is less than the allowable displacement. The efficiency of the proposed modified LQR controller is evaluated in terms of the reduction of the response when the 10-storey shear building, with one ATMD on the top floor, is subjected to several Next Generation Attenuation (NGA) projects near fault earthquakes, and results are compared with conventional LQR. The proposed modified LQR controller performs significantly better than the conventional LQR controller in reducing the displacement response of the structure. Moreover, although the peak control force for the proposed method is slightly greater than that of the classical LQR, the total energy demand is reduced. The results show that this method is practicable and worthwhile for vibration control of structures. Moreover, as compared to the method which was presented in [70], a comparison of the displacement and total energy demand illustrates the superiority of the proposed method.

Figure 8. RESULTS FOR THE 1979 Imperial Valley AGRASIAS In addition, to help understand the comparison of responses well, the main frequency content of the earthquake ground motions which were determined by wavelet analysis are shown in Figs. 6d–8d. They show that when the main to the main frequency of structure (1 Hz), frequency of ground motions is close the proposed method can mitigate the high response of structure. Moreover, 29 more earthquake ground motions have been used for numerical study; the obtained results from the presented method and the conventional LQR controller method are given in Table 1. The obtained results indicated that the proposed method is a strong and viable method to the problem of active control in the structures. Finally, to clarify the efficiency and accuracy of the proposed method, the results have been also compared with the outcomes of previous methodology proposed by the authors [70]. Fig. 9 shows that the displacement has decreased more by using the proposed method when compared to the method in [70]. In addition, Fig. 10 shows that in comparison with conventional LQR and proposed method, by using the method in [70] the total energy demands has increased.

Figure 9. COMPARISON OF THE DISPLACEMENT RESPONSE OF THE STRUCTURES BETWEEN THREE METHODS (1979, Imperial Valley-06 (Agrarias)).

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