In recent years, designing seismic buildings to withstand a broad range of earthquakes has become desirable to increase earthquake resilience. Many studies focused on developing of various practical structural control strategies to strengthen the seismic buildings. Structural control can be divided into four categories: passive, active, semi-active, and hybrid control systems by installing control devices to the structure. A semi-active control system has recently generated significant interest from researchers due to the combination of passive and active control systems characteristics. Researchers have proposed various schematic concepts for installing control devices. Connecting two buildings through control devices, called coupled buildings control, has also been explored. The structures are connected by fluid viscous-elastic dampers (Zhang and Xu, 1999, 2000; Yang et al., 2003) and friction dampers (Bhaskararao and Jangid, 2006; Ng and Xu, 2006) to improve the performance of the couple buildings. In Japan, a hybrid passive control system connecting a base-isolated building to a non-base isolated building (free wall) was first proposed by Obayashi Corporation and has been deeply investigated (Murase et al., 2013; Taniguchi et al., 2016; Hayashi et al., 2018; Nakamura et al., 2021). It is demonstrated that a hybrid passive control system is effective for long-period and pulse-like ground motions and has high redundancy and resilience against many disturbances.

Active control devices implemented a more advanced strategy to connect multiple high-rise buildings (Seto and Matsumoto, 1999). In numerous studies, semi-active control devices have also been utilized as connecting dampers (Yan and Zhou, 2006; Bharti et al., 2010; Uz and Hadi, 2014; Kim, 2016). In the past, magnetorheological (MR) dampers employing MR fluids to provide control capability have been identified as potential devices for semi-active control of seismic buildings due to their large force capacity, high dynamic range with the low power requirement.

Few investigations have been conducted about the effect of the distribution of dampers between coupled buildings. Bhaskararao and Jangid (2006) analyzed the optimal force of the friction dampers for decreasing the seismic response of coupled buildings system. In addition, they confirmed that it is not necessary to utilize connecting dampers on all floors, but a small number of dampers connected in proper locations can significantly reduce the seismic responses of the coupled buildings system. On the other hand, Bharti et al. (2010) proposed coupled buildings connected with MR dampers. Both studies suggested that providing linkages at all floor levels is unnecessary. Nevertheless, no optimal distribution solution was delivered.

Bigdeli et al. (2012) presented optimization algorithms for locating a limited number of viscous dampers to reduce seismic response of coupled buildings during different earthquakes. Uz and Hadi (2014) also proposed a multi-objective optimization approach to obtain an optimal design method for minimizing the number of MR dampers between the coupled buildings.

The semi-active control algorithms for MR dampers can be classified into two categories. The first category is based on a mathematical model to operate. These methods contain Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG), H₂/LQG strategies, and Lyapunov’s direct approach control methods (Dyke, 1996; Dyke et al., 1996; Chang and Zhou, 2002; Xu and Zhang, 2002; Bharti et al., 2010). The second category is known as non-model-based control method. It relies on the system’s measured responses and the accurate mathematical model is not necessary. This category includes neural network and fuzzy logic control (FLC) method (Yan and Zhou, 2006; Gu and Oyadiji, 2008; Uz and Hadi, 2014; Al-Fahdawi et al., 2019).

Researchers have interested in fuzzy logic control theory over the past years. Because the fuzzy sets and rules require a comprehensive understanding of the system dynamics, it is necessary to pre-determine the system properly.

One of the main disadvantages of fuzzy controllers is their inability of learn. As a result, the use of knowledge and experience of controller database specialists is essential. To overcome this problem, a learning process can be applied and modify the fuzzy logic controller. Several methods based on learning-capable fuzzy controllers have been proposed. The genetic algorithm (GA) inspired by evolutionary theory is one of the effective methods for designing fuzzy controllers to develop an appropriate fuzzy controller. (Yan and Zhou, 2006; Uz and Hadi, 2014). GA can be applied as a multi-objective optimization tool to reduce both the displacement response and acceleration response in structural vibration control applications.

Another method is the combination of neural networks and fuzzy logic in neuro-fuzzy controllers. The neural-fuzzy controller combines neural networks and a fuzzy inference system that compensates for the fuzzy control deficiencies through neural network training and adaptability.

In the previous research, Gu et al. (2017, 2019) have successfully used the neuro network control algorithm to operate the MR elastomer isolator. It can be identified that the semi-active control system in which the MR-based devices in combination with the neuro network system or the neuro-fuzzy logic controller can effectively reduce the responses of the structure.

On the other hand, the adaptive neuro-fuzzy inference system (ANFIS) is one of the most important and widely used types of neuro-fuzzy networks (Jang, 1993). In the previous studies, the ANFIS has been proved as an effective method to modify the fuzzy membership function and to create the fuzzy rules (Gu and Oyadiji, 2008; César and Barros, 2016; Al-Fahdawi et al., 2019).

Al-Fahdawi et al. (2019) investigated the effectiveness of the ANFIS controller which is operated as a single-input single-output feedback system under three schemes. It can be identified that the MR dampers operated by the ANFIS controller can effectively reduce the responses of two connecting buildings. César and Barros (2016) demonstrated the semi-active control system in which the ANFIS controller is implemented to optimize the fuzzy inference rule as two-input single-output feedback system. It reveals that the ANFIS with two-input single-output feedback system can alleviate the responses of the seismic building.

In this study, the ANFIS controller implemented as two-input single-output feedback system is used to establish parameters for fuzzy membership functions and creates fuzzy rules. Moreover, the four types of the feedback in the semi-active control system are chosen to compare the numerical results under the ground motions. In order to assess the effectiveness of the proposed semi-active control system, three types of strategies: passive-off, passive-on and semi-active control algorithms (four schemes of ANFIS and LQR) are investigated under the different ground motions. The numerical results of the three control strategies are examined and in comparison with the uncontrolled (unconnected) strategy.

Murase et al. (2013) proposed a hybrid passive control system with a base-isolated building connecting to a non-base isolated building by oil dampers. In this study, MR dampers are utilized as connecting dampers instead of oil dampers. The system is modeled using the MDOF mass-spring-dashpot model as depicted in Figure 1.

The base-isolated layer consists of a natural rubber isolator and oil dampers. The floor mass of the main structure, the free wall, and the base isolation floor is 170 t, 100 t, and 510 t, respectively. The story height of each building is 3.5 m across all stories.

The fundamental natural period of main structure is 6.73 s, while those of the superstructure of the base-isolated building and the free wall are 3.01 and 2.13 s. The stiffness distribution of each building is trapezoidal (the top-to-bottom story stiffness ratio is 1/3). The damping ratio of the base-isolated layer for a rigid super-structure is 0.15, and the structural damping ratio of the super-structure is set to 0.02 (stiffness-proportional damping), and the damping ratio of the free wall is 0.03 (stiffness-proportional damping).

Spencer et al. (1997) proposed “a phenomenological model to characterize the behavior of an MR damper prototype”. Figure 2 illustrates a simple mechanical idealization of an MR damper based on a Bouc-Wen hysteresis model, which was developed and shown to accurately predict the behavior of an MR damper over a broad range of inputs. It is governed by the following simultaneous equations. F d = c 1 y ˙ + k 1 x − x 0 (1) z ˙ = − γ x ˙ − y ˙ z z n − 1 − β x ˙ − y ˙ z n + A x ˙ − y ˙ (2) y ˙ = α z + c 0 x ˙ + k 0 x − y / c 0 + c 1 (3) where z is the evolution variable, which is the scale factor of Bouc-Wen hysteresis, α, γ, β, n, and A are the correlation coefficients of the hysteresis parameter that control the shape of the hysteresis loop.

y is internal displacement, c ₁ is viscous damping coefficient at low velocity, c ₀ is viscous damping observed at large velocities, k ₁ is stiffness of the accumulator, and k ₀ is spring stiffness coefficient at high velocity. Furthermore, x is the relative displacement of the spring, x ₀ is the initial relative displacement, and k ₁ , k ₀ , γ, β, n, and A are constant.

The parameters dependent on the applied voltage are expressed as follows: c 0 = c 0 a + c 0 b u c 1 = c 1 a + c 1 b u α = α a + α b u (4)

The dynamics involved in reaching rheological equilibrium in the MR damper are accounted for through the first-order filter. u ˙ = − η u − v (5a)

v is the voltage applied to the current generator.

In this study, damper parameters are chosen to have a capacity of 600 kN at a maximum voltage of Vmax = 3 Volts. The mechanical properties of the MR damper are listed in Table 1 (Bharti et al., 2010).

The governing equations of the motion are presented in matrix form below. M x ¨ + C x ˙ + K x = P d F d − M r x ¨ 0 (5b) where M , C , and K are the mass, damping, and stiffness matrices, respectively. The expression for the displacement vector with respect to the ground is: x = x 1 , x 2 , x 3 , … , x 67 (6)

x ¨ and x ˙ are acceleration and velocity vectors, respectively.

Furthermore, r is influence vector with all elements equal to unity, x ¨ 0 is ground acceleration, P d   is a matrix for the position of the dampers, and F d is the damping force vector of MR dampers obtained from Eq. 1. The complete mass, damping, and stiffness matrices of the structural system are: M = M m O 1 O 2 M w (7) C = C m O 1 O 2 C w (8) K = K m O 1 O 2 K w (9) where O _{1 (41×26)} and O _{2 (26×41)} are null matrices, M _m and M _w are the mass matrices of the main structure, and the free wall which can be described as follows. M m = m b , m m 1 , m m 2 , m O O ⋱ m 39 , m m 40 , m (10) M w = m 1 , w m 2 , w m 3 , w O O ⋱ m 25 , w m 26 , w (11)

K _m and K _w are stiffness matrices of the main structure and the free wall, respectively. K m = k b , m + k 1 , m − k 1 , m − k 1 , m k 1 , m + k 2 , m − k 2 , m − k 2 , m k 2 , m + k 3 , m − k 3 , m   ⋱ − k 39 , m k 39 , m + k 40 , m − k 40 , m − k 40 , m k 40 , m (12) K w = k 1 , w + k 2 , w − k 2 , w − k 2 , w k 2 , w + k 3 , w − k 3 , w − k 3 , w k 3 , w + k 4 , w − k 4 , w   ⋱ − k 25 , w k 25 , w + k 26 , w − k 26 , w − k 26 , w k 26 , w (13)

The governing equation of motion is: y ˙ t = A y t + B s F d + E x ¨ 0 (14) where y is the state variable, A is the system matrix composed of structural mass, stiffness and damping matrices, B _d is the distribution matrix of the damper, and E is the matrix of excitation force. The matrices are expressed as follows: y ˙ = x ˙ x ¨ (15) y = x x ˙ (16) A = O I − M − 1 K − M − 1 C (17) B s = O M − 1 P d (18) E = O − r (19) where I and O are the identity and null matrices, respectively.

In this study, the effect of the bending is neglected and the coupled buildings control system does not overturn during the ground motions.

Different sets of evaluation criteria are utilized to evaluate the buildings’ performance in the structural control benchmark problem (Ohtori et al., 2004). The evaluation criterion minimizes the non-dimensionalized peak response due to the earthquake records, in which the smaller values of the evaluation criteria indicate superior performance. The set of evaluation criteria applied in this study to compare the performance of the structural control is a different method based on the building peak responses, as shown in the following equations: J 1 = max i max t x i t max i max t x i , u n c t r l t (20) J 2 = max i max t x ¨ i t max i max t x ¨ i , u n c t r l t (21) with the range i = [1 40], i = [1 26] (only above the ground level) for the main structure and free wall, respectively.

Where x t , x ¨ t are the displacement and absolute acceleration of the ith floor in each building under the control strategies, respectively, and x i ,   u n c t r l t and x ¨ i ,   u n c t r l t are the displacement and absolute acceleration of the ith floor in each building under the uncontrolled strategy.

Fukumoto and Takewaki, (2017) proposed five cases of the location of the connecting dampers in the adjacent coupled buildings. In this study, modal analysis determines the locations of connecting dampers. The model analyses of the main structure and free wall are displayed in Figure 3. The dampers are installed on the 5th, 7th, 9th, 11th, 13th, 18th, 21st, 24th, and 26th floors.

In this study, the Adaptive Neuro-Fuzzy Inference system (ANFIS) is utilized to improve the learning ability of the fuzzy logic controller. It makes the system less dependent on expert knowledge. ANFIS is composed of five layers and the schematic of ANFIS architecture is shown in Figure 4. It uses the Takagi and Sugeno’s fuzzy if-then rules as follows:

Rule 1: If x is A ₁ , y is B ₁ , then f 1 = p 1 x + q 1 y + r 1

Rule 2: If x is A ₂ , y is B ₂ , then f 2 = p 2 x + q 2 y + r 2

Rule N: If x is A _n , y is B _n , then f n = p n x + q n y + r n

where x and y are model inputs and A 1 … A n and B 1 … B n are fuzzy sets. f 1 … f n are the fuzzy outputs, and p i , q i , and r i are node parameters.

The node functions in the same layer are of the same function family as explained below: ( O i j is the output of the i node in the j layer).

Layer 1: Every node i in this layer is a square node with a node function O i 1 = u A i x ,   i = 1 ,   2 (22) O i 1 = u B i − 2 y ,   i = 3 ,   4 (23) where u A i and u B i − 2 are weights obtained from the fuzzy membership function.

Layer 2: Every node i in this layer is labeled π in which w _i are calculated by multiplying the membership values as follows. O i 2 = w i = u A i x × u B i y ,   i = 1 ,   2 (24)

Layer 3: The nodes in this layer is circle nodes labeled N. The normalized firing strength is calculated in this layer. It is the ratio of the firing strength of the ith rule to the total sum of all firing strengths: O i 3 = w ¯ i = w i w 1 + w 2 ,   i = 1 ,   2 (25)

Layer 4: Every node i in this layer is a square node with a node function O i 4 = w ¯ i f i = w ¯ i p i x + q i y + r i ,   i = 1 ,   2 (26) where w ¯ i is the normalized firing strength from the previous layer (3rd layer) and p i ,   q i , r i are the parameters in the node.

Layer 5: The single node in this layer is a circle node labeled ∑, which computes the overall output as the summation of all incoming signals. O i 5 = ∑ i w ¯ i f i = ∑ i w i f i ∑ i w i ,   i = 1 ,   2 (27)

In this study, the training data of ANFIS is produced using Linear Quadratic Regulator (LQR) assuming full-state feedback. The LQR control scheme is established on the principle of minimize the cost function J as the following equation: J = ∫ 0 t z T Q z + F d T R F d d t (28) Where Q ∈ R n × n and R ∈ R n × n are the weight factors of z = x   x ˙ T and F d t , which are the state vector and the control damping force, respectively.

The optimal damping force can be written as below: F d = − R − 1 B T P z = − K z (29) where P is the solution of the Riccati matrix differential equation as below: P A + A T P − P B R − 1 B T P + Q = 0 (30)

In this study, the parameters of the LQR are chosen as suggested by (Chang and Zhou, 2002). Q = K 0 0 M (31) R = 0.8 × P d T K − 1 P d (32) where K and M are the stiffness and mass matrices of the coupled system.

An external disturbance defines the range and type of actual excitations, which the fuzzy controller is employed to handle. The training data should be representative of different situations under different excitations during the operation of the controller. As a result, the disturbance used by the controller is an artificial wave (Building Center of Japan, BCJ-L2).

In this study, there are four schemes of feedback. The comparative performance of four schemes of coupled buildings control is DV, VA, DD, and AA, as shown in the following.

DV is the displacement and velocity of the top floor in the main structure.

The primary fuzzy logic controller develops in the current study with two inputs, and the damping force represents the fuzzy outcome. The membership functions of the Adaptive Neuro-Fuzzy Inference system are illustrated in Figure 5. In this study, seven Triangular membership functions are selected. These straight-line membership functions have the advantage of simplicity. In addition, triangular shapes are simple to implement and fast for computation. A trial and error method is used to determine the optimal membership functions for the model. The fuzzy sets for the input variables are denoted as follows: NL = negative large, NM = negative medium, and NS = negative small, ZR = zero, PL = positive large, PM = positive medium, PS = positive small.

Dyke (1996) proposed a clipped-optimal control strategy, also known as clipped voltage law (CVL), for controlling an MR damper using acceleration feedback. The force generated in the MR damper cannot be directly commanded. Only the control voltage v applied to the current driver can be directly controlled. The algorithm for selecting the command signal for the MR damper can be stated as the following equations.

Figures 6, 7 depict the resultant fuzzy surface for the MR dampers on each floor under DV and DD strategies. v = V max H f d − f f (34) v = 0 ,   f o r   f d < f V max ,   f o r   f d ≥ f ∩ f d × f ≥ 0 (35) where f d is the desired optimal damping force, v is the command voltage, V max is the maximum voltage, and H (•) is the Heaviside step function. Figure 8 represents the clipped-optimal control algorithm graphically.

The voltage applied to MR damper should remain constant, when the MR damper provides the desired optimal force (i.e., f = f d ). The voltage applied to the current driver is increased to its maximum level to increase the damper force to match the desired control force, if the magnitude of the force produced by the damper is less than that of the desired optimal force and the two forces have the same sign. Otherwise, the voltage applied to the damper is set to zero.

In this study, a semi-active control system comprises two controllers (a system and a damper controller), as shown in Figure 9. The system controller calculates the desired damping force, and the damper controller serves on the voltage to be applied to the current driver to track the desired damping force.

In this study, the Simulink blocks in MATLAB are utilized to build the Modified Bouc-Wen model of the MR dampers and the semi-active control system. Based on the fourth-order Runge-Kutta method, a MATLAB program is developed to solve the equations of motion for the couple buildings control system subjected to the excitations of the ground motions and the numerical results are presented in this section.

The performance of a semi-active control system for coupled building models with MR dampers has been investigated. Comparing seismic responses under uncontrolled and control strategies reveals that both passive and semi-active control systems can effectively reduce seismic responses. Nevertheless, the performance of the main structure is not identical during all earthquakes.

Based on the results of this study, following conclusion can be drawn:

1) The coupled buildings with base-isolation building and MR dampers, which are regarded as connecting dampers, are effective for seismic response reduction.

In spite of the fact that this study has successfully demonstrated that the proposed control algorithm has the significant reduction for the displacement control of the system, it has the certain limitation in terms of the acceleration control.

Consequently, it would be of interest to develop a more effective method in which ANFIS is integrated with other method that can efficiently adjust the parameters of the membership functions of ANFIS to alleviate both displacement and acceleration of the coupled buildings control system in the future.