Developed comparative analysis of metaheuristic optimization algorithms for optimal active control of structures

Abstract

A developed comparative analysis of metaheuristic optimization algorithms has been used for optimal active control of structures. The linear quadratic regulator (LQR) has ignored the external excitation in solving the Riccati equation with no sufficient optimal results. To enhance the efficiency of LQR and overcome the non-optimality problem, six intelligent optimization methods including BAT, BEE, differential evolution, firefly, harmony search and imperialist competitive algorithm have been discretely added to wavelet-based LQR to seek the attained optimum feedback gains. The proposed approach has not required the solution of Riccati equation enabling the excitation effect in controlling process. Employing this advantage by each of six mentioned algorithms to three-story and eight-story structures under different earthquakes led to define (1) the best solution, (2) convergence rate and (3) computational effort of all methods. The purpose of this research is to study the aforementioned methods besides the superiority of ICA in finding the optimal responses for active control problem. Numerical simulations have confirmed that the proposed controller is enabling to significantly reduce the structural responses using less control energy compared to LQR.

1 Introduction

Applications of soft computing methods in different field of civil engineering have been used in many researches recently [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. The idea of using active control strategy as one of these methods for civil structures has been introduced in early of 70s. The active control system can change the structural dynamic properties by adding an auxiliary input to the structure. Smart structure is the name that refers to this system [15,16,17,18,19].

One of the most important subjects in active control is to find out the optimum control force, so that using the least control energy, the structural responses have been remained under the allowable values [17]. Respectively, successfully implementation of the smart technology has been laid in an effective algorithm choice that adjusted the applied control forces. LQR, LQG, sliding mode control, pole assignment, H₂, and H_∞ are known as the common control algorithms [18], furthermore, LQR has been selected as the most effective, simple application and popular controller in active and semi-active structural control.

The objective of LQR is to minimize a quadratic cost function establishing a reasonable balance between the reduction of responses and required control forces. During the minimization procedure, the external excitation has not participated in solving the Riccati algebraic equation avoiding LQR as truly optimal. An applicable solution to deal with the mentioned difficulties has been passed through the intelligent optimization methods. The metaheuristic capabilities in the global optimal solution achievement for nonlinear and complex problems have provided a lot of flexibility in design controller.

Electrical and mechanical engineering have proposed multi-objective approaches for selecting LQR weights in controlling of double-inverted pendulum system [20]. Wang et al. [21] has applied Artificial Bee Colony (ABC) algorithm in weight selection of LQG for large antenna servo, and to avoid the repeated adjustment of LQR weights, accordingly, ABC has been used to determine Q and R for inverted pendulum system [22].

According to the literature, few studies have been performed in structural control as well as LQR weight selection based on metaheuristic algorithms. GA as an effective optimization technique has been used for LQR weighting matrix selection [23,24,25,26,27,28,29,30,31]. In the field of active and passive control design, GA has been used successfully for response mitigation of coupled buildings [16], finding the optimal size and location of passive devices [32, 33], number and location of active control devices and sensors [17, 34]. Joghataie and Mohebbi [26] have proposed an optimal control algorithm based on nonlinear Newmark integration and distributed genetic algorithm (DGA) used to optimally select weights in the proposed controller. Amini et al. [35] has presented a method to find the optimal control forces for active tuned mass damper. In this research, to eliminate the trial and error, particle swarm optimization (PSO) has been used to determine feedback gains with online updates of LQR weighting matrices. To optimize the number, location and total driving force of required actuators, an improved particle swarm algorithm has been presented in [36]. Amini and Ghaderi [37] have described an improved version of ant colony optimization (ACO) to find the optimal locations of fuzzy logic-controlled MR dampers. Harmony search (HS) has been utilized in the field of structural control as well as in [38]. Hybridization of HS and ACO for optimal locating of dampers has been discussed in [39], and a comprehensive review of the researches on the optimal damper location problem based on optimization algorithms has been presented in [40]. Several attempts have been made to determine the optimal parameters of the tuned mass damper (TMD) via metaheuristic algorithms. Mohebbi and Joghataie [41] have developed a DGA-based control algorithm for minimizing these parameters. Few researchers [35, 42,43,44,45] have used an improved version of PSO for optimizing the required parameters of TMD. Leung and Zhang [46] have also used an improved version of PSO for optimizing the required parameters of TMD. Differential evolution (DE) algorithm has been applied to obtain optimum parameters of passive device by minimizing H₂ and H_∞ norms of the benchmark system equipped with TMD [47]. Bagheri and Amini [48] have proposed the controller employing the pattern search method in minimizing LQR performance index. Amini and Bagheri [49] have presented an approach for the optimal control of structures estimating the control forces conducted by the performance index optimization through the colonial competitive algorithm.

As mentioned earlier, those studies applied metaheuristics involved in the field of seeking the optimal number and location of sensors and actuators, setting the parameters of passive TMD systems and selecting the LQR weighting matrices for active control systems [27,28,29, 31, 50,51,52,53,54,55]. However, few studies have been conducted in active control problem focusing on the classical methods’ shortcoming such as sub-optimality problem and the effect of external excitation in control forces calculation. On the other hand, some of metaheuristics have not been explored in control problem yet.

In this research, wavelet transform is used to decompose the external excitation into different frequency bands, thus the elements of feedback gain matrix are searched by six optimization techniques in each frequency range. Achieving the best solution, convergence rate and computational effort of these algorithms are compared through two numerical examples:

• imperialist competitive algorithm (ICA) as a socio-politically motivated algorithm,

• differential evolution (DE) as an evolutionary method,

• BAT, BEE and firefly (FA) as natural-inspired algorithms, and

• harmony search (HS) as music-inspired metaheuristic.

According to the knowledge of the authors, this comparison has not discussed the structural active control problem yet; therefore, the formulation has not required the solution of Riccati algebraic equation and could regard the external excitation effect.

The current study has focused on the state space equation of motion and formulation of LQR control problem, followed by outlining the proposed approach, respectively, first wavelet transform is briefly introduced and the metaheuristics’ usage in wavelet-based LQR controller has been detailed. Also, the model structures and used earthquakes are introduced; furthermore, an overview on components and operators of metaheuristics has been demonstrated to present and compare the simulation results.

2 System equations and linear quadratic regulator

A review of performed studies in structural control has shown the efficiency and applicability of LQR making it the most popular control techniques. For a building equipped with an active tendon control subjected to a ground excitation, the state space description of system motion is expressed as follows:

$$\dot{X}\left( t \right) = AX\left( t \right) + BU\left( t \right) + E\ddot{X}_{g} \left( t \right),$$

(1)

$X\left( t \right) = \left[ {q\left( t \right),\dot{q}\left( t \right)} \right]^{T}$ is called the state vector, where $q(t)$ and $\dot{q}\left( t \right)$ denote the n-dimensional vector of the relative displacements and velocities. A (plant matrix), B (control matrix) and E (excitation matrix) are given by the following equations:

$$A = \left[ {\begin{array}{*{20}c} O & I \\ { - M^{ - 1} K_{s} } & { - M^{ - 1} C_{d} } \\ \end{array} } \right]_{2n \times 2n} ,$$

(2)

$$B = \left[ {\begin{array}{*{20}c} O \\ {M^{ - 1} H} \\ \end{array} } \right]_{2n \times n} ,$$

(3)

$$E = \left[ {\begin{array}{*{20}c} O \\ { - M^{ - 1} \varGamma } \\ \end{array} } \right]_{2n \times n} ,$$

(4)

where M, C_d and K_s are $n \times n$ matrices of mass, damping and stiffness of the structure, respectively, $U(t)$ is $m \times 1$ control force vector applied by m actuators and $\ddot{X}_{g} \left( t \right)$ representing the time history of ground acceleration. $\varGamma$ is $n \times 1$ influencing vector of the external forces and $H$ is defined as $n \times m$ location matrix of actuators.

Due to the actuator capacity curbs and economic efficiency, the response mitigation under allowable values with less control energy is desirable. The linear quadratic regulator is a minimized quadratic performance index to make a good trade-off between limiting the structural response and reducing the control force consumption. This quadratic performance index “J” is formulated as follows:

$$J = \int\limits_{{t_{0} }}^{{t_{f} }} {[X(t)^{T} QX(t) + U(t)^{T} RU(t)]{\text{d}}t} ,$$

(5)

where Q and R are referred to state and control force weighting matrices and tf is the duration of earthquake. Essentially, Q is non-negative semi-definite and R is the positive definite matrices. The solution has been provided by the feedback control law and the control force calculated by the following equation based on the system’s states:

$$\left\{ {U(t)} \right\}_{n \times 1} = - [K]_{n \times 2n} \left\{ {X(t)} \right\}_{2n \times 1} ,$$

(6)

where K is the feedback gain matrix (Eq. 7) and the symmetric matrix P is the solution of algebraic Riccati equation (ARE):

$$K = - R^{ - 1} B^{T} P,$$

(7)

$$A^{T} P + PA - PBR^{ - 1} B^{T} P + Q = - \dot{P}.$$

(8)

3 Wavelet-based optimal active control formulation

The wavelet transform (WT) as one of the most effective time–frequency analysis tools can be used to detect the local frequency content of earthquakes. WT is a transformation decomposing the signal into a superposition of the elementary basic function $\psi_{s,\tau } (t)$ called mother wavelet. The wavelets are generated from scaling and translation of this mother wavelet. The continuous wavelet transform (CWT) of the signal x(t) is expressed as

$$wt(s,\tau ) = \frac{1}{\sqrt s }\int_{ - \infty }^{ + \infty } {x(t)\psi^{*} } \left( {\frac{t - \tau }{s}} \right){\text{d}}t,$$

(9)

where $\psi^{*}$ is the complex conjugation of $\psi$ and wt (s,$\tau$) is called the wavelet coefficient. In Eq. (9), s is the scale factor representing the frequency content of signal and translation factor $\tau$ discriminating the location of wavelets during the time. The discrete version of wavelet transform (DWT) has been used by discretizing the scale and translation parameter. Using dyadic values for $\tau$ and s (Eq. 10), the corresponding $wt_{j,k} (t)$ is as Eq. (11). j and k are positive integers.

$$s = 2^{ - j} ,\,\,\,\tau = 2^{ - j} k,$$

(10)

$$wt_{j,k} (t) = 2^{j/2} \psi (2^{j} t - k).$$

(11)

If the earthquake signal $\ddot{x}_{g} \left( t \right)$ is decomposed to I level using DWT:

$$\ddot{x}_{g} \left( t \right) = A_{I} \left( t \right) + \mathop \sum \limits_{i \le I} D_{i} \left( t \right),$$

(12)

D_i is the ith detail and A_I is the Ith level of approximate signal. Adopting this approach, LQR performance index has been computed as follows:

$$X_{i} \left( t \right) = \left[ {q_{i} \left( t \right),\dot{q}_{i} \left( t \right)} \right] ,$$

(13)

$$X_{I} \left( t \right) = \left[ {q_{I} \left( t \right),\dot{q}_{I} \left( t \right)} \right],$$

(14)

$$\begin{aligned} J & = \int_{0}^{{t_{f} }} {[X_{I} (t)^{T} Q\,} X_{I} (t) + U_{I} (t)^{T} RU_{I} (t)]{\text{d}}t \\ & \quad + \sum\limits_{i \le I} {\int_{0}^{{t_{f} }} {[X_{i} (t)^{T} Q\,} X_{i} (t) + U_{i} (t)^{T} RU_{i} (t)]{\text{d}}t,} \\ \end{aligned}$$

(15)

$$X = X_{I} (t) + \sum\limits_{i \le I} {X_{i} (t)} ,$$

(16)

$$U = U_{I} (t) + \sum\limits_{i \le I} {U_{i} (t)} .$$

(17)

In this study, to enhance the controller performance, DWT has been applied to decompose the excitation into different frequency bands. For decomposition process, signal is decomposed using Daubechies wavelet of order 10 (db 10) mother wavelet in five levels.

Defined wavelet-based performance index (Eq. 15) has been selected as objective function and gain matrix elements considered as design variables of optimization process. Then metaheuristic algorithm has been employed to search the optimum gain matrix for each frequency band to calculate the control force in each domain (Eq. 18), thus total control force is also calculated with Eq. (19). $K_{i}$ is the gain matrix at level i.

$$\begin{aligned} U_{I} (t) = - K_{I} X_{I} (t), \hfill \\ U_{i} (t) = - K_{i} X_{i} (t), \hfill \\ \end{aligned}$$

(18)

$$U(t) = - K_{I} X_{I} (t) - \sum\limits_{i \le I} {K_{i} X_{i} (t)} .$$

(19)

Adopting this approach, the earthquake excitation has been participated in solving the Riccati equation and calculating the control forces. The proposed controller has been employed by different optimization algorithms to increase the accuracy toward the optimum controller gains comparing different aspects of metaheuristics in dealing with the optimal active control problem.

According to the results of this study, the effectiveness of each algorithm in vibration reduction through various seismic excitations has been evaluated by numerical simulations. This formula is able to be easily performed with any metaheuristic algorithm providing the great flexibility in the active controller design (Fig. 1).

Fig. 1

The flowchart of proposed optimal controller

4 Numerical study

To investigate the performance of proposed controller, three- and eight-story shear frames have been equipped with active tendon system selected as numerical examples. The schematic model of mentioned structures and dynamical properties of these structures are summarized in Table 1 and Fig. 2. The placed actuators in all stories have applied the control force.

Table 1 Dynamical properties of studied structures

Fig. 2

Aforementioned structures equipped with active tendons: a 3-dof; b 8-dof structures

The displacement response of structure’s upper story (Fig. 2) was calculated by earthquake excitation (Fig. 3) for uncontrolled and controlled structures, and required control forces were obtained. Then the performance of each controller has been evaluated in terms of reducing the displacement response and the required control force compared to uncontrolled mode. Finally, the advantages of different optimization algorithms are compared in optimized controller design from a variety of aspects. Comparing the performance of the controller in addition to the structure response, nine indicators of performance in Table 2 have been used for more accurate comparison.

Fig. 3

The time histories of Kobe, Parkfield and Landers earthquakes

Table 2 Formulas of benchmark indices [56]

The time history of the earthquake records is introduced as the input of the system and the velocity and displacement responses of the structure is received as the output of the program. Based on the output values and using control rule, Eq. 19 of the time history of the control forces applied to the structure was calculated.

To demonstrate the effectiveness and potential application of controller for different earthquakes, aforementioned structures are analyzed under three well-known historical earthquakes: Kobe (PGA = 0.345 g), Landers (PGA = 0.245 g) and Parkfield (PGA = 0.357 g) ground motions. The details of the earthquakes are shown in Fig. 3.

In addition to wavelet-based LQR performance index used as objective function of optimization, nine benchmark indices obtained from the results are calculated (Table 2) to evaluate the controller performance by reduction of different design parameters such as interstory drift, story displacement, acceleration, base shear and needed control force compared to the uncontrolled cases.

$d_{i} (t)$, $x_{i} (t)$, $\ddot{x}\left( t \right)$, $h_{i}$ and $m_{i}$ are the interstory drift, displacement, acceleration, height and mass of the ith story, respectively. $f_{l} (t)$ is the control force produced by lth actuator and W is the structure weight. c and uc superscripts are related to controlled and uncontrolled cases.

In the following, the basic steps of aforementioned intelligent optimization methods are expressed as pseudo-code (a brief review on the operators and parameters of metaheuristic algorithm). Then the controller based on these algorithms is applied to the prior structures under various earthquakes and the performance of the optimal smart systems is compared to the uncontrolled one, LQR controller and one another.

4.1 The ICA-based optimal controller

Imperialist competitive algorithm inspired by imperialist competition has been developed by Atashpaz-Gargari and Lucas [57]. The pseudo-code of ICA is shown in Fig. 4.

Fig. 4

The pseudo-code of ICA

ICA has required specific parameters to effectual performance affecting the quality of results and algorithm capability. In this study, ICA parameters are set as in Table 3

Table 3 The values of ICA parameters

The maximum number of iterations has been selected based on the required accuracy of optimization; accordingly, in this study, 50 iterations have been selected for ICA. The process has started to solve vibration control problems with 30 (for 3-dof structure) and 50 (for 8-dof structure) population size. In the following, ICA’s performance in confronting with active control problem has been studied by numerical simulations.

4.1.1 Vibration control of structures with ICA-based optimal controller

In example 1, the responses of 3-dof building are controlled. The needed control force and controlled displacement of top story due to Kobe earthquake are presented with the corresponding uncontrolled ones in Fig. 5. The comparative results are drawn in Table 4.

Fig. 5

The results of ICA controller for 3-dof’s top story under Kobe earthquake: a displacement; b control force

Table 4 The comparison between ICA and LQR for 3-dof under Kobe earthquake

On the whole, the results have shown that ICA controller can find more optimal objective function than LQR. The maximum displacement of 3-dof roof under Kobe earthquake has been reduced 11% with only 91% of maximum control force needed for LQR. Maximum drift, acceleration and velocity have also been reduced by ICA controller. According to Table 5, the maximum displacement of 8-dof roof under Parkfield earthquake is approximately the same as LQR with only using of 50% of maximum control force (Fig. 6).

Table 5 Comparison between ICA and LQR for 8-dof under Parkfield earthquake

Fig. 6

Results of ICA controller for 8-dof’s top story under Parkfield earthquake: a displacement; b control force

4.2 The DE-based optimal controller

Differential evolution algorithm has been proposed by Storn and Price [58]. The main concepts of DE are shown in the pseudo-code given in Figure 7

Fig. 7

The pseudo-code of DE

The parameters shown in Table 6 are considered for DE, in this paper (Fig. 8).

Table 6 The values of DE parameters

Fig. 8

Results of DE controller for 3-dof top story under Landers earthquake: a displacement; b control force

4.2.1 Vibration control of structures with DE-based optimal controller

The results of structural displacement and control force for 8-dof top story due to Parkfield earthquake calculated by DE controller are shown in Fig. 9.

Fig. 9

Results of DE controller for 8-dof top story under Parkfield earthquake: a displacement; b control force

The results have indicated that DE controller has reduced both the displacement and control forces simultaneously for 3-dof and 8-dof (Tables 7, 8).

Table 7 Comparison between DE and LQR for 3-dof under Landers earthquake

Table 8 Comparison between DE and LQR for 8-dof under Parkfield earthquake

4.3 BAT-based optimal controller

The standard BAT algorithm has been introduced by Xin-She Yang [51] based on the echolocation behavior of microbats to find and hunt for prey. The procedure of BAT is shown through the stages in pseudo-code (Fig. 10).

Fig. 10

The pseudo-code of BAT algorithm

In this paper, to solve the optimal active control problem, the parameters of Table 9 have been selected for BAT algorithm; therefore, the related subsection has presented the results of BAT-based controller to studied structures.

Table 9 The values of BAT parameters

Table 10 Comparison between BAT and LQR for 3-dof under Parkfield earthquake

4.3.1 Vibration control of structures with BAT-based optimal controller

The 8-dof’s displacement and control force in the cases of uncontrolled, LQR and BAT controller are presented in Fig. 12 and Table 11.

Table 11 Comparison between BAT and LQR for 8-dof under Parkfield earthquake

According to Figs. 11 and 12, and Tables 10 and 11, this controller has performed better than conventional LQR in reducing the displacements and control force for 3-dof and 8-dof structures.

Fig. 11

The results of BAT controller for 3-dof top story under Parkfield earthquake: a displacement; b control force

Fig. 12

The results of BAT controller for 8-dof top story under Parkfield earthquake: a displacement; b control force

4.4 The BEE-based optimal controller

In this study, BEE algorithm [59] (as fourth metaheuristic technique) has been used as an optimization algorithm inspired by the natural foraging behavior of honey bees to find the food resources. Pseudo-code of this algorithm is drawn in Fig. 13.

Fig. 13

The pseudo-code of BEE algorithm

The algorithm required the number of parameters as in Table 12.

Table 12 The values of BEE parameters

The time history of displacement response and control force for 3-dof shear frame under Landers earthquake is shown in Fig. 14. Also, Fig. 15 shows the time history of response and needed control force for 8-dof excited by Parkfield earthquake.

Fig. 14

Results of BEE controller for 3-dof top story under Landers earthquake: a displacement; b control force

Fig. 15

Results of BEE controller for 8-dof top story under Parkfield earthquake: a displacement; b control force

4.4.1 Vibration control of structures with BEE-based optimal controller

BEE has outstandingly performed LQR in terms of controller effort and structural responses (Tables 13, 14).

Table 13 Comparison between BEE and LQR for 3-dof under Landers earthquake

Table 14 Comparison between BEE and LQR for 8-dof under Parkfield earthquake

4.5 FA-based optimal controller

The firefly algorithm (FA) has been developed by Xin-She Yang [60] based on the idealization of flashing characteristics of fireflies. The firefly’s behavior is modeled to develop FA by the following process (Fig. 16).

Fig. 16

The pseudo-code of firefly algorithm

In this study, the values shown in Table 15 are considered for FA parameters.

Table 15 The values of FA parameters

4.5.1 Vibration control of structures with FA-based optimal controller

The simulation results of 3-dof and 8-dof structures subject to the Parkfield earthquake calculated by FA controller have been sketched in Figs. 17 and 18.

Fig. 17

Results of FA controller for 3-dof top story under Parkfield earthquake: a displacement; b control force

Fig. 18

The results of FA controller for 8-dof top story under Parkfield earthquake: a displacement; b control force

The proposed controller has reduced the maximum displacement and control force for 3-dof and 8-dof structures (Tables 16, 17).

Table 16 Comparison between the FA and LQR for 3-dof under Parkfield earthquake

Table 17 Comparison between FA and LQR for 8-dof under Parkfield earthquake

4.6 The harmony search-based optimal controller

Geem et al. [61] has pioneered harmony search (HS) by seeking harmony in music making. The following pseudo-code has shown how the harmony in music has inspired finding an optimality in an optimization problem (Fig. 19). HS parameter setting is shown in Table 18.

Fig. 19

The pseudo-code of HS

Table 18 The values of HS parameters

Table 19 Comparison between HS and LQR for 3-dof under Kobe earthquake

4.6.1 Vibration control of structures with HS-based optimal controller

HS is also utilized to solve the numerical examples. The simulation results are depicted in Figs. 20 and 21.

Fig. 20

HS controller results for 3-dof top story under Kobe earthquake: a displacement; b control force

Fig. 21

HS controller results for 8-dof top story under Parkfield earthquake: a displacement; b control force

According to the results, HS controller has reduced the third floor displacement more than 13% compared to LQR. Simultaneously, the applied control force is reduced to 89% (Table 19).

The maximum displacement of the eighth story is reduced from 35 mm in the uncontrolled case to 20 mm in LQR and HS, also the maximum of needed control force for HS controller is 216 kN less than LQR (Fig. 21 and Table 20). The results have represented the power of metaheuristics and superiority of these controllers over the classical LQR in controlling the vibration of 3-dof and 8-dof structures subjected to different earthquake excitations. The next section has contained the results comparing different aspects of studied metaheuristics in dealing with active control problem.

Table 20 The comparison between HS and LQR for 8-dof under Parkfield earthquake

4.7 Comparison of six studied metaheuristics in active control of structures

In this section, the results of six previously introduced optimal controllers are compared. To gain accurate comparison, fixed parameters are used for all algorithms including population size and iteration number. The iteration number for all methods is set as 50. The initial population sizes of all algorithms for 3- and 8-dof structures are set to 30 and 50. The summarized result of all algorithms for 3-dof structure against Kobe earthquake is drawn in Table 21.

Table 21 Comparison between the metaheuristics and LQR for 3-dof under Kobe earthquake

The effectiveness of these controllers in reducing the response of the three-story building due to Landers and Parkfield earthquake is also shown in Tables 22, 23.

Table 22 Comparison between the metaheuristics and LQR for 3-dof under Landers earthquake

Table 23 Comparison between the metaheuristics and LQR for 3-dof under Parkfield earthquake

According to Table 21, the response reduction ratio (ratio of the controlled to uncontrolled response) at top floor of 3-dof shear frame against Kobe earthquake is about 78%, 77%, 77%, 78%, 76% and 78% for ICA, DE, BAT, BEE, FA and HS controllers, respectively. This decrement is about 11%, 9%, 9%, 10%, 5% and 13% compared to LQR, aligned with other similar earthquakes. Comparison between the metaheuristics and LQR for 8-dof structure under Parkfield earthquake is drawn in Table 24.

Table 24 Comparison between the metaheuristics and LQR for 8-dof under Parkfield earthquake

The response reduction ratio at top story of 8-dof structure against Parkfield earthquake is about 46%, 45%, 47%, 46%, 43% and 46% for ICA, DE, BAT, BEE, FA and HS controllers. Almost the same behavior is observed for other earthquakes as well. Consequently, the metaheuristics performance is more optimal than LQR controller; therefore, the maximum response of structure and needed control force are reduced compared to LQR.

Also, the comparison of benchmark indices has provided the appropriate information about the advantages of designed controllers. The values of these indices for 3-dof under Parkfield earthquake are shown in Table 25. Based on the calculations, all controllers have performed better than LQR; however, ICA controller performance is optimal than others.

Table 25 The results of benchmark indices for 3-dof under Parkfield earthquake

Regarding other comparative criterion, the root mean square (RMS) of responses is used, accordingly, the results of RMS of displacement for 3-dof under Parkfield earthquake are shown in Fig. 22.

Fig. 22

RMS of 3-dof top story displacement under Parkfield earthquake: a ICA; b DE; c BAT; d BEE; e FA; f HS controllers

The numerical analysis of the above RMS curves has shown total RMS of displacement response (total area in RMS curve) for ICA, DE, BAT, BEE, FA and HS as 23%, 25%, 23%, 23%, 25% and 25% of total RMS of displacement for LQR controller. This percentage is about 83%, 91%, 86%, 85%, 93% and 93% of total RMS of uncontrolled displacement showing ICA’s performance more optimal than other metaheuristics. RMS of needed control force for 8-dof shear frame under Parkfield is shown in Fig. 23.

Fig. 23

RMS of control force for 8-dof under Parkfield earthquake: a ICA; b DE; c BAT; d BEE; e FA; f HS controllers

According to Fig. 23, the total RMS of control force (the covered area of RMS curve) calculated by ICA method is 52% of control force needed for LQR. This percentage for DE, BAT, BEE, FA and HS is 55%, 75%, 70%, 61% and 55%; subsequently, ICA’s performance is also better than other techniques. The convergence rate has significantly determined the comparison of metaheuristics observed and compared from Fig. 24 for 8-dof under Parkfield earthquake.

Fig. 24

Convergence rate of all algorithms for 8-dof under Parkfield earthquakes: a ICA; b DE; c BAT; d BEE; e FA; f HS controllers

Similar behavior has been repeated for 3-dof convergence under various earthquakes showing the out performs of ICA than other methods in terms of convergence rate, respectively, DE and BAT are located in other next ranks.

5 Conclusions

The current study has focused on an optimal configuration of LQR controller applied on 3-dof and 8-dof structures subjected to several historical earthquakes. Based on the results, the metaheuristic optimizer has made control system meet optimal states and control energy, simultaneously. Due to the multiplicity of metaheuristics, the optimization objective function has been performed by six methods and the results have been compared to one another.

• Computational time is an important characteristic of optimization algorithm. Regarding the whole processing time (from the initial to the final configuration), the comparisons have revealed a faster run in HS (50 iterations in about 1 min), also, ICA and DE have required slightly more time than HS; however, BAT and FA are weaker and BEE is the slowest algorithm (4–5 times slower than ICA).

• Considering Fig. 24, FA and BAT have good convergence tendency compared to other algorithms (convergence occurred in about 10 iterations) as a favorable choice for computational burden reduction, meanwhile, the convergence occurred for ICA, DE and BEE is about 40 iterations. Subsequently, regarding the convergence rate, the mentioned metaheuristics have been ranked as FA (first rank) and BAT, ICA, DE and BEE (the next).

• Considering total cost value, ICA and DE have produced better results compared to all the mentioned algorithms. BAT, HS, BEE and FA have second, third, fourth, fifth and sixth rank. Additionally, all the mentioned algorithms have produced better results than LQR. Consequently, each metaheuristic algorithm has advantages and disadvantages in solving active control problem; furthermore, considering the effectiveness of six intelligent controllers, ICA, DE, and HS have performed favorably for the very optimization problem in seeking of the best solution, convergence rate and computational effort. On the contrary, BAT, FA, and BEE have been weakly performed for the problems and leveled the next. The superiority of ICA over other methods in finding the optimal responses for active control problem has been shown as well.

To sum up, the results for active control of structures have shown the potential of wavelet and metaheuristic algorithms in vibration control for building structures depending on the user to select the appropriate algorithm or on the problem types and its requirements such as the quality of solution, convergence rate, computational effort, and consuming time.