## 1. Introduction

Since the nature of an earthquake is its unpredictable characteristics, the traditional passive design approaches do not guarantee the minimum damages with economic designs [

1,

2,

3]. Consequently, buildings and bridges are continuously getting smarter, using intelligent control devices, as well as state-of-the-art structural health monitoring technologies [

4,

5,

6]. In the past decade, the vibration control of adjacent buildings under seismic and wind loads has gained significant attention. The most frequently proposed solution for controlling the adjacent building is to couple them with actuators or dampers, which involves the installation of the control devices, such as semi-active dampers between two buildings to improve the performances of both structures. New studies have also revealed the promising potentials of a special type of dampers, triangular-plate added damping and stiffness (TADAS), for seismic vibration control applications [

7,

8], which can be used for pounding hazard mitigation, as well.

A significant number of studies in the literature have used semi-active control algorithms, based on LQR [

9], LQG [

10] and FLC [

11], to control magnetorheological (MR) dampers [

12], and of particular interest are the optimal controls using metaheuristic and neural network algorithms [

13,

14,

15]. Despite the advances, coupling adjacent buildings is not always the best solution, particularly when the dimensions, as well as the other dynamic properties, are not similar [

16]. Besides, the second building resists the vibration induced in the first building, which may cause the collapse of both, in the case of miscalculations due to local failures under strong ground motions, particularly if one of the adjacent buildings is controlled using based-isolations [

17].

Most of the recent studies have used simplified, one-dimensional shear frames or symmetric 3D buildings [

18,

19], neglecting the geometry and irregularity of structures, as well as bidirectional seismic loading. Earthquake loads can be applied in two directions that may cause simultaneous translational and torsional vibrations in buildings with considerable irregularity [

20,

21]. Such coupled vibrations can cause nonlinear behavior and severe damage in the external frame members, particularly, the corner columns and the bracing system [

22,

23]. Furthermore, for those structures built on a soft soil medium, different dynamic characteristics result in different responses due to the soil–structure interaction (SSI) [

24,

25,

26]. Considering the SSI, Farshidifar and Soheili [

27] studied the seismic response of actively controlled single and multi-story buildings; however, similar studies need to be carried out to consider the asymmetric properties of structures, as well as the pounding hazard of adjacent high-rise buildings. Therefore, despite the advantages of the proposed methods for regular buildings under the unidirectional seismic loads, such solutions are not always the best solution when two irregular buildings with different dynamic characteristics need to be controlled for possible pounding hazards.

As an alternative to a wired sensors network for smart structures, wireless sensor networks (WSNs) have attracted several researchers [

28]. With lower cost, an array of WSNs can effectively be used for monitoring the structural deterioration, as well as controlling vibration during an earthquake, by using artificial neural networks [

29,

30]. The data that is acquired using WSNs is essentially the same as the traditional counterpart that has been challenged recently due to deployment difficulties, as well as reliability issues during an extreme event. Nowadays it is feasible to mimic the swarm behaviors in nature and share important response data with the adjacent buildings using a wireless sensors network and via cloud-based computing [

31]. Keeping this in mind, the idea of swarm-based parallel control (SPC) of such buildings is proposed in this paper. With this approach, the response data from each wireless sensor is transferred to the adjacent buildings to update the responses without physical structural links.

Figure 1 illustrates such information flow that is inspired by nature. Using the similar concept for adjacent buildings, the control force can be determined for each building with respect to the other surrounding structures, meaning that using the proposed algorithm, the optimal control forces are first determined based on the response of each building individually, and then, it is modified based on the responses of the other adjacent structures. In this study, the fuzzy inference system (FIS) is used to interpret the data using nonlinear mappings. The nature-inspired information flow among the swarms can be seen in

Figure 1, while the information flow between each adjacent building is illustrated in

Figure 2.

The goal of this study is to introduce and validate the performance of the proposed algorithm for adjacent buildings, using semi-active MR dampers that are wirelessly controlled by utilizing cloud-based computation. In addition, irregularity and soil–structure interactions are considered in this study. Two pairs of MR dampers are installed on each floor in two directions to suppress coupled translational–torsional motions, which are controlled by implementing the concept of the Internet-of-Things (IoT) and wireless sensor networks (WSNs). Robustness of the proposed method is evaluated and verified using three example cases: a single multi-story building considering the SSI, two adjacent buildings coupled using MR dampers as well as active actuators, and five adjacent buildings controlled using the proposed SPC. The LQR-based active control system is considered to provide benchmark results for comparison purposes only, and the mechanism and advantages of the proposed method over the active tendon controller are not discussed in this study.

## 2. Analytical Equations of Motion Considering Soil–structure Interaction

Figure 3 shows a multi-story shear frame building considering the soil–structure interaction simulation. In three-dimensional space, a building can be idealized as a (3 × n Story + 5)-DOF system. This is because each story has three degrees of freedom, plus five degrees of freedom for the base of the structure, swaying motion in the

x- and

y-directions, rocking about the

x and

y-axes, and twisting about the

z-axis. As shown in

Figure 3, pairs of dashpots and springs are defined to consider the SSI effects in the simulations. In this figure,

${x}_{b}$ denotes the base displacement,

${\varphi}_{y}$ represents the rocking motions about the y-axis, and the relative displacements of stories are shown as

${x}_{i}$. Note that the settlement in the

z-direction is not considered, and the twisting motion is not visible from the side view.

The control devices placement is shown in

Figure 4 for the MR devices, and due to the eccentricity with respect to the center of mass,

${d}_{i}$, each device generate a moment to resists the torsional motion. Irregularity in plan and elevation is described by the mass eccentricity with respect to the center of rigidity,

${e}_{x}$ and

${e}_{y}$.

The governing equations of motion for a controllable n-story building can be expressed as follows in general,

where

$\left[M\right]$,

$\left[C\right]$ and

$\left[K\right]$ are the dynamic properties of the building, and

$\left\{x\left(t\right)\right\}$,

$\left\{\dot{x}\left(t\right)\right\}$ and

$\left\{\ddot{x}\left(t\right)\right\}$ are the relative (to the ground motion) displacement, velocity and acceleration vectors, respectively.

$\left[\gamma \right]$ is the coefficient matrix of the input controlling forces, and

$\left\{\delta \right\}$ is the ground motion coefficient vector. The governing equation of motion can be re-written in state-space form as explained in [

32]:

where

It should be noted that the above-mentioned equations are in the continuous-time format; however, for the simulation with Matlab software, they are converted to the discrete-time format. A step-by-step procedure for computing the response of the system in the discrete-time domain is presented in the reference book by Cheng et al. [

33]. The ground motions coefficients are given in

$\left\{\delta \right\}$ for two directions, and the coefficient of the input controlling forces on each degree of freedom, including rotation,

$\left\{u\left(t\right)\right\}$, can be described using the

$\left[\gamma \right]$ matrix as [

32,

34]:

where

${\left[\gamma \right]}_{i}$ is

Then, the mass and stiffness matrices considering the SSI effects can be assembled as:

where

where

$\left[0\right]$ is a zero matrix, and

${\left[m\right]}_{i}^{*}$ is the mass matrix of the

$i\mathrm{th}$ story [

32].

In the above-mentioned equations,

${m}_{i}$ is the mass of the

$i\mathrm{th}$ story, the mass moment inertia about each axis in three-dimension is defined as follows for a floor slab with dimensions of

$a\times b$, and coordinates of

$\left({x}_{m,j},{y}_{m,j}\right)$. The center of the mass coordinate is given as

$({\overline{X}}_{m},{\overline{Y}}_{m})$.

The stiffness matrix of each floor is not diagonal like the mass matrix [

32].

where

Similarly, the center of stiffness is located at (

${\overline{X}}_{k},{\overline{Y}}_{k})$, and that of the

$j\mathrm{th}$ lateral resisting member with the coordinate

$\left({x}_{k,j},{y}_{k,j}\right)$ has the stiffness of

${k}_{x,j}$, and

${k}_{y,j}$ in two directions. The

$j\mathrm{th}$ column is assumed to have

${k}_{x,j}={k}_{y,j}=12E{I}_{j}/{h}^{3}$. Thus, the stiffness matrix is assembled as follows [

32,

34]:

The damping matrix is constructed in the same way as for the stiffness matrix, but the superstructure damping is determined using Rayleigh’s method [

35] without SSI. The SSI parameters are determined using the equations given in [

25,

32,

34], which are summarized in

Table 1. In this study, two cases with and without SSI effects are investigated. The soft soil is assumed to have the Poisson’s ratio of 0.33, the density of 2400 kg/m

^{2}, shear-wave velocity and modulus of 500 m/s and 6 × 108 N/m

^{2}, respectively. Details of the SSI models, as well as the methodology, are presented by Nazarimofrad and Zahrai [

34].

## 3. Swarm-Based Parallel Control (SPC)

In this study, three different control alternatives for adjacent buildings are presented and discussed. One most commonly used method is to control each building independently without any information flow (Case-I in this paper). Another approach is to couple two adjacent buildings side-by-side by actuators or additional dampers, such as hydraulic or MR dampers (Case-II in this paper). Both methods are discussed, and the results are compared with the proposed swarm-based parallel control (SPC), which is Case-III in this paper. The SPC algorithm is the FLC for each building, with an additional updating agent that is another FLC to consider the response of the other buildings simultaneously.

For Case-I, the behavior and control of a single building (in the east,

Figure 4) are studied considering the soil–structure interaction (SSI) effects, and the performance of the different control algorithms are compared when the behavior of the adjacent structure is neglected. The response reduction ratio is considered as the performance index, and since the active control has a different control mechanism, the ideal condition is considered for this study. The linear quadratic regulator (LQR) algorithm is used for calculating the optimal control forces that can be applied to the structure through pairs of active tendons that can be installed at the same locations as for the MR dampers. Therefore, the advantages of FLC and SPC are not compared to the active control system. The fuzzy logic control (FLC) is used to efficiently control vibrations using semi-active MR dampers, which is illustrated in

Figure 5. More details, and the background of LQR and FLC algorithms are available from refs. [

32,

36,

37]. The weight matrices for the LQR method are selected as

$R={10}^{-6}\times {I}_{\mathrm{size}\left(\left[\gamma \right]\right)}$, and

$Q={10}^{6}\times {I}_{2n}$, where

$I$ is an identity matrix.

Figure 4 shows five adjacent buildings with different dynamic properties. Instead of coupling buildings (Case-II), response data is shared among the controllers. Thus, an additional control fuzzy logic control layer is considered to avoid pounding, for which the input variables are the relative displacements and the directions. Using this strategy, the optimal required clearance distance can be decreased, which is critical in urban areas, where the cost of space and material is essential. Using this method, the adjacent buildings are not connected physically, and the response data are shared wirelessly. Thus, buildings are independently controlled, and then their relative performances are evaluated using a second fuzzy logic control algorithm. In this paper, the first fuzzy logic control, FIS1, determines the control forces for individual buildings independently, and the second fuzzy logic control, FIS2, determines the control forces based on the relative motions including the torsion in each irregular building. Finally, the optimal MR dampers input voltages are selected by comparing the outputs of FIS1 and FIS2.

The two surface plots of the outputs are shown in

Figure 6 for FIS1 and FIS2 using the Gaussian membership functions for the variables. In

Figure 7, the flowchart of the proposed swarm-based control (SPC) is described in summary. The flowchart includes the following steps:

- Step 1.
Estimating the dynamic properties of the soil as well as the building (e.g., using data-driven-based machine learning models [

38]).

- Step 2.
Assembling the state-space model and equation of motions according to Equations (1)–(12).

- Step 3.
Calculating the response of each of the adjacent buildings, $\{x\left(t\right)$}, by solving the state-space equations in Step 3.

- Step 4.
Normalizing the state vector (i.e., measures responses, $\left\{x\left(t\right)\right\}$ and calculating the control force using the fuzzy logic control FIS1.

- Step 5.
Updating the calculated control force in both directions, using the second fuzzy logic-based updating rules, FIS2, based on the relative displacements of two adjacent buildings. In this step, the outputs of FIS1 and FIS2 are compared, and if the FIS2 output is higher than the FIS1 output, the control force is adjusted based on the FIS2 results.

- Step 6.
Apply the calculated control force, $\left\{u\left(t\right)\right\}$, and continuing until the end of the last time-step.

## 6. Summary and Conclusions

The majority of the studies in seismic vibration control of three-dimensional tall buildings have not considered the irregularity and soil–structure interaction effects on the vibration response. In this paper, a novel bio-inspired seismic vibration control algorithm, called swarm-based parallel control (SPC), is proposed to use the advantages of fuzzy logic-based rules to consider the response of the individual adjacent building in determining the control forces for the group of buildings during extreme events, such as earthquake or wind loads. The main merits of the proposed SPC system can be summarized as:

Despite the other control approaches, such as using coupling devices, the need for the structural links to couple the two adjacent buildings, as well as the complexity, can be eliminated.

Each building can sense and consider the responses of the adjacent building in determining the optimal control force of semi-active devices; therefore, in the case of damage in a building, the other adjacent buildings update the control forces, accordingly.

The proposed SPC can be modified for individual buildings according to the deployed control strategy and devices.

To address the above-mentioned advantages of the proposed controller, numerical simulations are carried out under the seismic loads of seven historic earthquakes. Vibration responses of an irregular 15-story building are studied considering the soil–structure interaction (SSI) effect, as well as the effectiveness of different control methods (Case-I). A sensitivity analysis is also considered to study the performance of each control approach, considering uncertainties in estimating the soil and structural properties. In addition, the idea of coupling two adjacent regular-irregular buildings is discussed using an ideal active control, regardless of the control device type, and the limitations are highlighted (Case-II). It should be noted that for the active control system, the results are obtained based on the LQR algorithm to have a benchmark result for comparison, and the ideal conditions are assumed for either method. Finally, five adjacent buildings with different dynamic properties and no structural connections, considering the SSI effects, are modeled to verify the performance of the proposed SPC algorithm (Case-II).

The results prove that the proposed method has the potential to be paid attention in future developments as part of the bigger concept of the Internet-of-Things (IoT) and smart structures.