Concrete structure based on fuzzy DC

The traditional centralized control scheme cannot meet the stability and seismic requirements of HRB structures. DC scheme and FC technology are utilized to obtain structural risks in HRBs, thereby improving the stability and safety of HRB structures. Traditional centralized control schemes face significant challenges in maintaining stability due to their reliance on a single controller, communication delays, lack of local adaptation, and difficulties in managing complex, nonlinear systems. These limitations can lead to vulnerabilities, inefficiencies, and instability, particularly in dynamic and large-scale environments. DC approaches often provide a more robust solution by distributing decision-making and enhancing system resilience. The equilibrium motion model of the building is shown in Eq. (1).

(1) $$ \left\{\begin{array}{l}M\overset{}{\ddot{X}}(t)+C\dot{X}(t)+ KX(t)={D}_sF(t)+{B}_sU(t)\\ {}X\left({t}_0\right)={X}_0,\dot{X}\left({t}_0\right)={\dot{X}}_0\end{array}\right. $$

In Eq. (1), K, C, and M all represent the building stiffness, damping matrix, and building mass, respectively; $ \overset{"}{X}(t) $ means the building displacement vector; $ {D}_s $ denotes the external interference matrix; $ U(t) $ expresses the actuator control vector; $ F(t) $ indicates the external excitation vector; $ {B}_s $ refers to the actuator positioning matrix; $ X\left({t}_0\right) $ stands for the initial displacement of the structure; and $ \dot{X}\left({t}_0\right) $ means the initial velocity vector of the structure. In the analysis of concrete structure models, if the vibration damping ratio parameter of the first period of the concrete structure is set to 2%, and the remaining periods are all below 10%. The expression of the vibration damping ratio in the first period is shown in Eq. (2).

(2) $$ {\xi}_i=\mathit{\min}\left\{\frac{\omega_i}{50{\omega}_1},0.1\right\}\hskip1em \left(i=2,\cdots, n\right) $$

At this moment, the damping matrix is expressed as shown in Eq. (3).

(3) $$ C=M\Phi \Lambda \left[2{\xi}_1{\omega}_1,2{\xi}_2{\omega}_2,\cdots, 2{\xi}_n{\omega}_n\right]{\Phi}^{-1} $$

In Eq. (3), $ \Phi $ denotes the formation matrix; $ {\xi}_i $ is the damping ratio of the i period; $ {\omega}_n $ indicates the natural frequency of the i period. The concrete structure equation is shown in Eq. (4).

(4) $$ \left\{\begin{array}{c}\dot{Z}(t)= AZ(t)+ BU(t)+ DF(t)\\ {}Z\left({t}_0\right)={Z}_0={\left[\begin{array}{l}{X}_0\\ {}{\dot{X}}_0\end{array}\right]}_{2n\times l}\end{array}\right. $$

In Eq. (4), A, B, and D are expressed as shown in Eq. (5).

(5) $$ \left\{\begin{array}{c}A={\left[\begin{array}{cc}{0}_{n\times n}& {I}_{n\times n}\\ {}-{M}^{-1}K& -{M}^{-1}C\end{array}\right]}_{2n\times 2n}\\ {}B={\left[\begin{array}{l}{0}_{n\times p}\\ {}{M}^{-1}{B}_s\end{array}\right]}_{2n\times p}n\times \\ {}D={\left[\begin{array}{l}{0}_{n\times q}\\ {}-\left\{1\right\}\end{array}\right]}_{2n\times q}\end{array}\right. $$

In Eq. (5), M ⁻¹K = Φ refers to the formation matrix; $ {0}_{n\times n} $ , $ {0}_{n\times p} $ , and $ {0}_{n\times q} $ express the n × n, n × p, and n × q dimensional zero matrix; B_s stands for the actuator displacement matrix; and $ {I}_{n\times n} $ expresses the identity matrix. Distributed structure control system is different from centralized structure system control. The internal structure of a distributed system is composed of multiple substructure systems, each of which has an independent controller and can perform separate SC based on the concrete structure. The principle is shown in Figure 1. Parallel DC is a control strategy where multiple independent controllers operate simultaneously to manage different subsystems or components within a larger system. Each controller makes decisions based on local information and conditions, allowing for real-time responses and enhanced system performance.

The subsystems of the DC system will effectively control based on the data conditions of the structural layer and form a complete closed-loop SC system. The unpredictability of HRB control characteristics arises from a combination of dynamic environmental conditions, complex structural interactions, human activities, and limitations in control systems. Understanding and mitigating these factors are crucial for enhancing the reliability and safety of HRBs. The choice between precise mathematical and analytical solutions depends on the problem at hand. For complex systems where analytical solutions are not feasible, precise mathematical methods are often better. Conversely, when an analytical solution exists, it can provide valuable insights and exact results. In many cases, a combination of both methods can be beneficial—using analytical solutions to gain insights and validate numerical results from precise mathematical approaches. Therefore, FC technology is introduced to determine SC parameters. The fuzzy domain set is defined as U, and its expression is shown in Eq. (6).

(6) $$ U=\left\{{u}_1,{u}_2,\cdots {u}_n\right\} $$

In Eq. (6), $ {u}_1,{u}_2,\cdots {u}_n $ indicate the domain set. In FC, the selection of membership function will affect the overall effectiveness of FC, and the membership function suitable for FC will be ultimately obtained through empirical, subjective, and objective data support. The membership function is determined by fuzzy statistics, and the variable set of cloud top boundary belonging to the set U is defined as A*. The element u_i of the subtest subset in n belongs to A*, which is expressed as $ {n}_{\dot{A}} $ . A stable $ {n}_{\dot{A}}/n $ ratio can be obtained through multiple experiments, and then u_i belongs to the fuzzy set A membership, which is expressed in Eq. (7).

(7) $$ {\mu}_A\left({u}_i\right)=\underset{n\to \infty }{\lim}\frac{n_{\dot{A}}}{n} $$

In Eq. (7), A denotes the borderless and $ {A}^{\cdotp } $ consistent fuzzy set; n means the amount of tests; u_i represents the amount of $ {u}_i\in {A}^{\cdotp } $ in the n test. The FC system is shown in Figure 3.

Figure 3.

Schematic diagram of the fuzzy control system structure.

In Figure 3, the FC part mainly uses the maximum membership function method to determine accurate control values. This method is convenient and simple and is suitable for the field of concrete structure control. By processing the externally received analog signal through FC, it is converted into digital information in real time and transmitted to the sensor to achieve a dynamic control effect. The data rates of sensors can range from very low (1 Hz) for simple measurements to several kHz for complex applications such as LIDAR and cameras. The choice of sensor and its data rate often depends on the specific application requirements, including the need for real-time processing and the type of information being captured. In specific FC, due to the fact that different inference results in fuzzy reasoning correspond to different degrees of membership, the maximum one of these degrees of membership is used as the output, as shown in Eq. (8).

(8) $$ {u}_0=\max {\mu}_A\left(\mathrm{u}\right)\hskip1em u\in U $$

In the process of fuzzy reasoning, there are multiple outputs corresponding to the max membership degree, and the mean of all corresponding values is selected as the final output value, as expressed in Eq. (9).

(9) $$ {u}_0=\frac{1}{J}\sum \limits_{j=1}^J{u}_j $$

Among them, u_j and J are expressed as shown in Eq. (10).

(10) $$ \left\{\begin{array}{l}{u}_j=\underset{u\in U}{\max}\left({\mu}_A(u)\right)\\ {}J=\left|\left\{u\right\}\right|\end{array}\right. $$

FC with magnetorheological fluid damper

In the field of concrete structure control, traditional semiactive control strategies cannot meet the requirements of fast and short response changes during earthquakes, and earthquake adjustment parameters can be calculated in a relatively short time. The fuzzy DC technology is applied to the magnetorheological fluid damper (MRFD), which is widely used in the field of concrete structures. It has the advantages of low energy consumption, fast response, and can be combined with microcomputer control technology. The motion state diagram of the MRFD with two parallel plates is shown in Figure 4.

Figure 4.

Schematic diagram of the motion state of magnetorheological fluid damper with two parallel plates.

According to the constitutive equation of magnetic current in the stable shear field, this state can be described, as shown in Eq. (11).

(11) $$ \tau ={\tau}_y\mathit{\operatorname{sgn}}\left(\dot{\gamma}\right)+\eta \dot{y} $$

In Eq. (11), $ {\tau}_y $ denotes the yield of the magnetorheological fluid; $ \tau $ means the total shear stress of the magnetorheological fluid; and $ \dot{y} $ refers to the parameters of the magnetized fluid. When the particles do not meet magnetic saturation, the shear yield under the theory of magnetorheological fluid is expressed in Eq. (12).

(12) $$ {\tau}_y=\sqrt{6}{\phi \mu}_0{M}_s^{1/2}{H}_0^{2/3} $$

In Eq. (12), M_s stands for the magnetization intensity of the saturated particle state; $ \phi $ means the particle content of the magnetorheological fluid; $ {\mu}_0 $ refers to the true permeability; and H ₀ represents the applied magnetic field intensity. In the entire study of MRFD, their structure is mainly composed of a cylinder body, a piston, a magnetic fluid variant, and an electromagnetic coil. Taking the common shear valve-type MRFD as the object of study, it can be considered a valve-type damper and shear damping force, as expressed in Eq. (13).

(13) $$ F=\left[\frac{3\pi \eta L\left({D}^2-{d}^2\right)}{4{Dh}^3}+\frac{L\pi D\eta}{h}\right]v+\left[\frac{3\pi L\left({D}^2-{d}^2\right)}{h}+ L\pi D\right]{\tau}_y\mathit{\operatorname{sign}}(v) $$

In Eq. (13), $ \eta $ stands for the apparent density of the magnetorheological fluid; $ \mathit{\operatorname{sign}}(v) $ indicates the piston motion control function; h means the particle gap of the magnetorheological fluid; D means the piston diameter; d refers to the internal diameter of the piston; and L stands for the effective length of the piston. According to Eq. (12) and Eq. (13), the shear-type MRFD model can be simplified, as shown in Eq. (14).

(14) $$ {F}_{sv}=\frac{3\eta L{\left[\pi \left({D}^2-{d}^2\right)\right]}^2}{4{Dh}^3}v+\frac{3 L\pi \left({D}^2-{d}^2\right)}{4h}{\tau}_y\mathit{\operatorname{sgn}}(v) $$

According to Eq. (14), the model is composed of two terms. The first term mainly considers fluid power viscosity, which is a fixed value. In the second item, the yield strength of magnetorheological fluid is mainly considered, which is related to variable Coulomb damping. Therefore, the adjustable multiple of MRFD is shown in Eq. (15).

(15) $$ {\beta}_v=\frac{bh^2{\tau}_y}{4\eta Q}=\frac{Dh^2{\tau}_y}{\eta \left({D}^2-{d}^2\right)v} $$

In Eq. (15), v denotes the piston speed and b expresses the Coulomb damping characteristic parameter. By adjusting the effective length and damping force of the piston appropriately, the SC adaptability of the damper can be changed. However, in practical control, the relationship between damping force and adjustable multiple is contradictory, and it is necessary to fully consider the actual needs of the concrete structure to meet the seismic control requirements of the concrete structure. At the same time, it is necessary to control strategies, and the selection of control strategies will directly affect the control effect of MRFD on concrete structures. In the research, fuzzy DC strategy is mainly used and compared with traditional LQR control strategy. LQR can obtain the optimal control law with linear feedback of the state, which makes it easy to form closed-loop optimal control. LQR optimal control can achieve good performance indicators of the original system at a low cost.

The MRFD action position matrix is defined as B_s, and MRFD dampers are arranged between two adjacent floors. The control drive is labeled as $ {u}_j(t) $ , the forward force applied to the j layer is marked as $ {u}_j(t) $ , and the reaction force applied to the $ j-1 $ layer is denoted as $ -{u}_j(t) $ . The control model is expressed in Eq. (16).

(16) $$ {B}_sU(t)=-{C}_d\dot{X}+{B}_s{U}_{sy} $$

In Eq. (16), B_s means the actuator positioning matrix; C_d indicates the viscous damping coefficient; $ {U}_{sy} $ refers to the damping control force vector; $ U(t) $ stands for the SC force; and $ \dot{X} $ expresses the relative velocity of the damper movement. The entire fuzzy DC principle is shown in Figure 5.

Figure 5.

Schematic diagram of fuzzy decentralized control process.

Multifloor fuzzy DC with LQR

Comparing DC and centralized control helps in understanding their respective strengths and weaknesses, enabling the selection of the most appropriate control strategy based on specific system requirements, complexity, and operational goals. At present, HRBs mainly rely on DC, and each controller is independent of each other to achieve adaptive control of complex environments in HRBs. In the DC of HRBs, the more controllers are divided into different floors, the better the control effect, but the lower the economic efficiency is. Therefore, it is necessary to fully consider the actual situation of the floors. DC is the division of floors into multiple floor control units, which are controlled independently of each other. Due to the higher the floor, the higher the uncertainty factor of the floor, and the more floor structure control parameters are required, an additional FC controller is designed in each floor structure control. In the research, the actuator adopts a direct drive form and has good performance in current HRBSC. The positioning moment of the actuator is shown in Eq. (17).

(17) $$ {B}_s={\left[\begin{array}{ccc}1& 0& 0\dots 0\\ {}0& 1& 0\dots 0\\ {}\vdots & \vdots & \vdots \ddots \vdots \\ {}0& 0& 0\dots 1\end{array}\right]}_{n\times n} $$

In the control of HRB structures, it is usually necessary to consider the effects of building mass, vibration acceleration, and other factors on concrete structure control. In this regard, each domain is divided into seven fuzzy subsets, including seismic acceleration, interlayer displacement, and so on, corresponding to one to seven top-level discrete controls, namely negative big, negative medium, negative small, zero, positive small, positive medium, and positive big. NB, NM, NS, ZE, PS, PB, and PB represent seven levels. The control coefficient is opposite to the domain name, and the maximum control force in concrete structure control can be controlled through a proportional factor. Based on the experience of membership selection introduced in the following section, the interlayer displacement, seismic acceleration, and control force are plotted as membership function curves. The acceleration membership function curve is shown in Figure 6.

Figure 6.

Acceleration membership relationship.

The function of FC is that when the concrete structure is faced with seismic action and the acceleration is PB corresponding to the interstory displacement (ISD), the floor is in the most unfavorable seismic state, and the building seismic hierarchy of hazard controls is the largest at this moment. When the acceleration level under seismic excitation is NB, and the ISD level of the building is PB, the ISD is opposite to the seismic acceleration, the corresponding building hierarchy of hazard controls is the smallest, and the concrete structure control is the safest. The corresponding hierarchy of FC is shown in Table 1.

Table 1.

Fuzzy control rules table

The maximum control force of the floor is set to 5 m/s ² × 1.1 × 10⁶ kg = 5.5 × 10⁶ N. In fuzzy DC, concrete structure control is mainly divided into multiple sub-SC units, and different structural units can adopt different control strategies. Figure 7 shows the three strategies used in fuzzy DC.

Figure 7.

Schematic diagram of decentralized control strategy.

In the entire concrete structure control, three SC strategies are adopted. In SC strategy 1, the model is divided into three subcontrol units: first to seventh floors, eighth to 14th floors, and 15th to 20th floors. In SC strategy 2, according to the characteristics of the floor structure, it is divided into 5 subcontrol units: first to fourth floors, fifth to eighth floors, nineth to 12th floors, 13th to 16th floors, and 17th to 20th floors. In SC strategy 3, it belongs to completely DC, where actuators are set in each layer and an FC controller is set in the single-layer unit structure to achieve SC of the entire building. At the same time, in the control of multifloor concrete structures, considering that the FC controller has a relatively average effect in concrete structure control, if the actuator acts on the interlayer, the FC controller relies on empirical data and cannot meet the control requirements. For this issue, the application of layer MRFD in multifloor structure control will be investigated to achieve effective control.

Suppose that we have a stabilizable full-order model in Eq. (5) to minimize the performance index

$$ \mathrm{J}={\int}_0^{\infty}\left[{X}^T(t){Q}_cX(t)+{U}^T(t){R}_cU(t)\right]\mathrm{d}t $$

where $ {Q}_C={Q}_C^T={H}^TH\ge 0 $ (positive semidefinite) and $ {R}_C={R}_C^T>0 $ (positive definite). This performance index, commonly called quadratic performance index, says that we wish to find an optimal control law $ {U}^{\ast }(t) $ such that the integral-squared-error of the deviations of the state trajectories from the nominal is kept small without using a great deal of control energy. The weighting matrices $ {Q}_C $ and $ {R}_C $ are chosen by the designer to dictate the relative importance of the states and control energy. Without going into the optimization theory required to prove what the solution is, we simply claim that a unique optimal control law that minimizes J exists and is given by (Lu et al., Reference Lu2003).

$$ {U}^{\ast }(t)={R}_c^{-1}{B}^T{P}_cX(t)={K}_cX(t) $$

where $ {P}_C $ is a constant, symmetric, positive semidefinite matrix which is the solution to the algebraic Riccati equation.

$$ {A}^T{P}_C+{P}_CA-{P}_C{BR}_C^{-1}{B}^T{P}_C+{Q}_C=0. $$

The closed-loop regulator is asymptotically stable and the minimum value of J is $ {X}^T(0){P}_CX(0) $ .

FC based on MRFD

To verify the FC effect of MRFDs, a concrete structure benchmark model platform was selected for relevant experiments. The benchmark model platform for concrete structure was selected for relevant experiments. The experiment followed the American construction industry standard of a 20-story seismic resistant steel structure building as the experimental model. Assuming that the horizontal stiffness of the floor slab is infinitely large, the horizontal and lateral stiffness between floors were preserved. In the experiment, only the vertical impact of earthquake was considered, and the vertical freedom of the frame was not studied. At the same time, to better analyze the seismic effect of concrete structures, without considering the influence of rotational degrees of freedom, a shear model was adopted for experiments. Meanwhile, when selecting seismic signals, FEMA guidelines should be followed. Table 2 shows the basic data of the benchmark building model.

Table 2.

Basic data of benchmark architectural model

In the experiment, two types of seismic excitation waves were selected to simulate the SC effect, one being El Centro wave and the other being Kobe wave, with accelerations of 3.42 and 8.18 m/s², respectively. And it was compared with the traditional LQR structure control technology to verify the actual application effect of the research technology. Figure 9 shows the curves of two types of waves.

Figure 9.

Acceleration curves of two types of seismic waves.

Figure 9 (a) shows the acceleration time history results of EI seismic wave. From the curve results, the acceleration time history remained high within 5 seconds before the earthquake and gradually decreased thereafter. Figure 9 (b) shows the acceleration time history results of Kobe seismic wave. According to the curve results, the acceleration time history value was low in the first 5 s, increased in 7 to 13 s, then gradually decreased, and the seismic fluctuation gradually decreased. Figure 10 shows the relationship between SC force, interlayer displacement, and seismic acceleration.

Figure 10.

Relationship between structural control force, interlayer displacement, and seismic acceleration.

In Figure 10, the relationship between interlayer displacement, seismic acceleration, and control force was obtained through simulation experiments. When the acceleration was 5 m/s² and the interlayer displacement was 5 cm, both the acceleration and interlayer displacement corresponded to the NB level, which was the most unfavorable state for SC and needed to exceed 15 n/m². Seismic acceleration was negatively correlated with ISD, and as the displacement and seismic acceleration increased, the risk of concrete structures gradually increased. The interlayer displacement curves under different control strategies are shown in Figure 11.

Figure 11.

Time history results of interlayer displacement of the top layer under different control strategies under the action of Centro seismic waves.

Figure 11 (a) shows the displacement results between the top floors under LQR control. From the data results, in uncontrolled concrete structures, as the seismic excitation time increased, the interlayer displacement gradually decreased and tended to stabilize. At the second of seismic excitation, the maximum displacement value achieved by uncontrolled structural interlayer displacement was 9 cm. Using traditional LQR SC, the interlayer displacement was 5 cm at the second. Compared with uncontrolled control, LQR SC could effectively suppress the interlayer displacement caused by seismic excitation. Figure 11 (b) shows the displacement results between the top floors under FC. At the second, fifth, and 14th seconds of seismic excitation, there was significant ISD in the uncontrolled structure, with displacement amounts of 9, 7, and 6 cm, respectively. In FC, the displacement amounts were 4, 2, and 2 cm, respectively. By comparing two SC technologies, FC has improved interlayer displacement control by 21.36% compared to LQR. The acceleration curves under different control strategies are shown in Figure 12.

Figure 12.

Acceleration time history results of top floor buildings under different control strategies under the action of Centro seismic waves.

Under seismic excitation, it was also necessary to consider the impact on building acceleration. Figure 12 (a) and 12 (b) shows the acceleration time history curves of the top floor of the building under LQR control and FC, respectively. In Figure 12 (a), during the first 5 seconds of seismic excitation, the building model was subjected to the strongest seismic force. In the uncontrolled structure, the maximum acceleration value of -14 m/s² occurred in the third second. In the LQR SC, the vibration of the concrete structure was suppressed, and in the third second, the building acceleration was -5 m/s². In Figure 11 (b), FC strategy 1 was selected for the experiment. The floor acceleration under it was higher than that under LQR control, but the forces between different floors were effectively suppressed, which had a lower impact on seismic effects compared to LQR control. It studied the ISD of the top 20 concrete structure, as shown in Figure 13.

Figure 13.

The time history results of ISD of the lower concrete structure.

Figure 13 (a) shows the acceleration time history curve results of the top floor of the building under LQR control. From the data results, the max interlayer displacement value of -12 cm occurred in the uncontrolled lower floor of the building at the 9th second of seismic excitation. Under LQR control, the maximum interlayer displacement value was -7 cm at the 9th second, and 3 cm, −6 cm, and 3 cm at the 10th, 11th, and 12th seconds, respectively. Compared with no control, the interlayer displacement of the lower layer of the building was suppressed, and the interlayer displacement decreased by 46.65% compared to no control. Figure 13 (b) shows the acceleration time history curve results of the lower floor of the building under FC. From the data results, the max interlayer displacement value without control was -7 cm at 9 s, and at this moment, while the value under FC was −6. At 10, 11, and 12 s, the interlayer displacement values were 3, −5, and 3 cm, respectively. Compared with LQR control, FC had better suppression of interlayer displacement, with a reduction of 51.32% compared to no control. FC had a better effect in the vibration control of HRBs. Simultaneously the control forces between each layer were compared, as shown in Figure 14.

Figure 14.

Control force results between layers.

Figure 14 (a) shows the control effect of each layer under EI seismic wave. From the Figure, there was a certain difference in the control force between FC and LQR control in each floor of HRBs. From the zeroth floor to the fifth floor, the two types of SC had basically the same control force in each floor. At the seventh and eighth floors, due to different SC methods, FC required significantly lower floor control force compared to LQR control. At the seventh floor, the LOR control floor control force was 4.7 × 106 N, while the FC floor control force required at the seventh floor was 2.4 × 106 N. The FC floor control effect was improved by 32.65% compared to LQR control floor control. Figure 13 (b) shows the control effect of each layer under the Kobe seismic wave. According to the data in the figure, under the effect of the Kobe seismic wave, the floor control forces of LOR control and FC were basically the same. However, on the third floor, the floor control forces required by LQR control and FC were 10.0 × 106 N and 11.6 × 106 N, respectively. The possible reason was that the FC adopted sectional control to ensure the stability of the floor earthquake resistance. Compared with LOR control, the floor control effect of FC was increased by 11.56%.

Results of proposed multifloor fuzzy decentralized control

The same seismic excitation was selected as in Section “Concrete structure based on fuzzy DC” for testing, and a 6-story shear-type building model was utilized as the experimental object. Each floor had a height of 3.4 m and a stiffness value of 12000 KN/m. The peak control force of concrete structures under seismic excitation is shown in Table 3.

Table 3.

Peak control force of concrete structures under seismic excitation

Table 3 shows the peak control force results of concrete structures under seismic excitation. In the control of multilayer structures, MRFD and FC were installed in each floor to ensure the control effect of lateral displacement of the floors. From Table 3, there were significant differences in the SC effects among the three types of SCs in multifloor SC. In FC, MRFDs were installed on each floor to reduce lateral forces. In the comparison of lateral displacement, the total lateral displacement without control was 92.65 cm, LQR control was 47.03 cm, and FC was 49.95 cm. Overall, FC dynamically adjusted lateral effects and had the best stability among the three. At the same time, in the comparison of acceleration peaks, the total acceleration peak of FC was 36.65 m/s², and the LQR control was 34.55 m/s². Due to the placement of MRFDs in different floors of FC, the building acceleration was effectively controlled. In the comparison of control forces, the total control force of the six layers under LQR control was 741 KN, while the FC was 718 KN. FC had better control effects. At the same time, in actual building SC, it was also necessary to consider the impact of output variables on SC, including open-loop control with seismic acceleration as input, closed-loop control with floor acceleration and displacement as input, and closed-loop open control with seismic acceleration and displacement as input.

At the same time, under open-loop control, the optimal building displacement control effect was achieved on the 6th floor, with a displacement of 7.9 m/s² on the top floor. From this, reasonable control parameters needed to be selected in practical control to achieve the best control effect. Figure 15 shows the convergence of the desired response under control force. Finally, a 12-story reinforced concrete structure was selected to test the application effect of the proposed technology in practical scenarios. The structural parameters of the building are shown in Table 4.

Table 4.

Structural parameters of actual building scenarios

Figure 15.

the system response indicated by the red dashed line.

The actual application effect of the proposed technology was evaluated by introducing three indicators: maximum acceleration damping rate J1, top displacement damping rate J2, and MRFD damping maximum output J3, as shown in Table 5.

Table 5.

Comparison of control effects in three actual control scenarios

From the data in Table 5, the proposed FC strategy greatly reduced the acceleration and displacement response of the structure under wind load, and was superior to the IQR control strategy, especially in terms of acceleration damping rate J1 control, which was 0.03546, while the LQR was 0.03934. At the same time, FC had a better overall effect in controlling the top displacement damping rate J2 and MRFD damping maximum output J3. In addition, in comprehensive comparison, the proposed FC technology did not damage the concrete structure, while the other two methods both had certain structural damage. The proposed technology had better application effects and met the SC requirements of actual earthquake scenarios in buildings.