Neural network control for earthquake structural vibration reduction using MRD

Abstract

Structural safety of building particularly that are intended for exposure to strong earthquake loads are designed and equipped with high technologies of control to ensure as possible as its protection against this brutal load. One of these technologies used in the protection of structures is the semi-active control using a Magneto Rheological Damper device. But this device need an adequate controller with a robust algorithm of current or tension adjustment to operate which is further discussed in the following of this paper. In this study, a neural network controller is proposed to control the MR damper to eliminate vibrations of 3-story scaled structure exposed to Tōhoku 2011 and Boumerdès 2003 earthquakes. The proposed controller is derived from a linear quadratic controller designed to control an MR damper installed in the first floor of the structure. Equipped with a feedback law the proposed control is coupled to a clipped optimal algorithm to adapt the current tension required to the MR damper adjustment. To evaluate the performance control of the proposed design controller, two numerical simulations of the controlled structure and uncontrolled structure are illustrated and compared.

Keywords

MR damper / semi-active control / earthquake vibration / neural network / linear quadratic control

Introduction

During the last decade, the attraction of using control technologies in structural civil engineering to avoid destruction is due to the increase of available strong ground motions. These motions are the causes of several economics and lives loss estimated by 10000 people die each year. One of the tragedies ground motions in the north of Africa is the Agadir earthquake of February 29, 1960, at 23:40 (local time) with esteemed magnitude of 6.25 on the Richter scale of magnitude. This tragedy caused the death of more than 12000 people and injured 12000 others who accounted for two-thirds of the population, and destroyed 70% of the city [¹].

Over the past few decades, the researchers focused on safety structural system against this brutal dynamic motions. The objective of control techniques is to reduce the vibration of structural system by an external modification of the system’s structural response. Therefore, the structural vibration control attracted and are attracting increasing attention due to their robustness to eliminate this vibration and preserve structure from ground motions destruction. This vibration control system can have many forms, passive, semi-active, active, and hybrid one [²–⁵].

The passive control system is the oldest system of control introduced by Ref. [⁶]. This system consist of one or many passive devices embedded to the structure to reduce dynamical motions, designed to modify the structural characteristics of the system (stiffness or damping). The passive control system is preferred for its simplicity on the one side and its economy on the other side. However, these passive devices have a limited level dynamic motions performance control [⁷]. In spite of these devices limitation, the passive control was a subject of several theoretical and experimental studies. In Ref. [⁸], High Damping Seismic Isolator composed by Lead Rubber Bearing Isolator and High Coefficient friction Slider was proposed to control multistory building exposed to tow seismic components.

The active control with a necessity of more power to operate assured more effectiveness and control performance is preferred to control vibrations of structures affected by strong ground motions [⁹]. An active control technique was introduced to a tuned masse damper based on the first mode natural frequency of the structure and the tuned mass damper. This system was designed to control a tall building against strong wind load [¹⁰]. Battaini et al. [¹¹] suggested an active masse damper with a fuzzy logic controller for vibration control of three-story structure subjected to a ground motion.

Furthermore, semi-active control methods shown great potential for strong ground motions in civil engineering and have received considerable attention in the last few decades. A semi-active device is an adjustable and basically adaptable to a feedback control. These systems are widely used because of their real-time and low energy operating, facility of implanting in real structures and can be used in a variety of devices and technologies, such as electrorheological (ER) dampers and magnetorheological (MR) dampers [¹²,¹³]. The use of MR damper in structural protection against strong ground motions, which provides wide dynamic range, seems to be able to prevent these structures from destruction, due to the mechanical characteristics of the MR fluid. This fluid subjected to the current intensity or magnetic field can change its viscosity in milliseconds. Thus, due to this rheological property, the MR damper covered and combined the advantages of each passive and active dampers. This behavior makes it useful in many structural protection applications. Oliveira et al. [¹⁴] investigated a semi-active control based on Magneto Rheological damper installed on a 2-DOF. The effectiveness of the proposed semi-active control is demonstrated for both experimental and numerical simulations. Yoshida et al. [¹⁵] used MR damper as semi-active device to control a lateral and torsional motions in asymmetric 2-story structure model excited by a shake table reproducing a seismic excitation. The comparison between the constant voltage and the controlled voltage applied to MR damper responses shown the effectiveness of the semi-active control system.

Recently, several buildings around the world are protected by a semi-active devices. However, the semi-active control system can’t depend only on the device robustness. On the other hand, an effective control system is based on strong and robust law control which make quickly as possible the adjustment of the control force desired in real time. Due to this, most of the researches have focused on concepts of control law techniques development [¹⁶]. These control laws are classed in two classes depending on the needs of mathematical model. The first class which needs mathematical model of the system based on Lyapunov stability is called classical law. The second class which doesn’t need mathematical model is called intelligent law. Artificial neural network is the most used intelligent law especially in the nonlinear systems [¹⁷,¹⁸] based on the input and output parameters [¹⁹,²⁰].

Zizouni et al. [²¹] proposed a linear quadratic regulator (LQR) to control a 3-story scaled structure equipped by MR damper. Liu et al. [²²] proposed a robust linear quadratic regulator (RLQR) which can considered a disturbance in the system keeping the simplicity of the LQR control. A performed linear quadratic Gaussian controller was investigated in flexible beam control against dynamic load by Ref. [²³] in which the authors proposed a conversion of robustness problem into a mathematical optimization of the affected system by a non-Gaussian white noise. Using MR damper device to control vibrations of single and multiple DOF, Neelakantan and Washington presented a sliding mode controller in Ref. [²⁴]. In Ref. [²⁵] the authors discussed a fuzzy Logic Controller to control vibration of 76-story building under wind excitation using an Active Tuned Mass Damper.

Hybrid LQR-PID control was designed for active tuned mass damper to control vibration of 10-story seismic excited structure in which the numerical simulations proved the performance of the hybrid LQR-PID control comparing to the LQR to reduce seismic responses of the excited structure [²⁶]. In Ref. [²⁷] Schurter and Roschke described a new approach of neuro-fuzzy technique for seismic vibration reduction of single and multiple DOF scaled-building using MR damper. The effectiveness of this approach was proved by numerical tests and simulations in which the responses are compared to the passive control of MR damper.

The main objective of this paper is to use a MR damper to reduce vibration of seismically excited 3-story scaled structure. A neuro-controller consisting of a training algorithm based on linear quadratic controller is developed and used to suppress vibration of the tested structure subjected to Tōhoku 2011 and Boumerdès 2003 earthquakes, respectively. The conventional linear controller is converted to an ANN based on LQC where the expression of the control law is replaced by a suitable ANN. The key features of the proposed intelligent control scheme is that a minimal learning variables are employed to approximate an ideal control law based on LQC controller which include external disturbances. These features guarantee a good transient and the control performance of the structural system is improved. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control configuration.

The reminder of this paper is assigned as follow: the model of the scaled structure as well as the corresponding propriety’s matrices are presented in Section 2. The mechanical model of the MR damper predicting the behavior of this device is descripted in Section 3. In Section 4, the training controller algorithm (neural-network) and the trainer controller algorithm (LQC) are designed. Herein Section 5, numerical simulations and compared results of the controlled and uncontrolled structure under seismic excitations are given. At last the effectiveness and performance of the controller are discussed in brief conclusion.

The mathematical model of motion of structures

The model used in this study is a 3-story scaled-structure equipped with an MR damper located between the ground and first floor as shown in Fig. 1. Jansen and Dyke proved that in this location the control force been more effective than the control force of MR damper located in the upper floors [²⁸]. This structure is seismically subjected to {{\displaystyle \ddot{x}}}_g .

In general, the equation of dynamic motion for the structural system presented in Fig. 1 can be written as

(1){{\displaystyle M}}_{\mathrm{s}}\ddot{\mathit{x}}+{{\displaystyle C}}_{\mathrm{s}}\dot{\mathit{x}}+{{\displaystyle K}}_{\mathrm{s}}\mathit{x}={{\displaystyle M}}_{\mathrm{s}}\mathit{\Lambda }{{\displaystyle \ddot{\mathit{x}}}}_{\mathit{g}}+\mathit{\Gamma }{{\displaystyle \mathit{f}}}_{\mathrm{M}\mathrm{R}},

where \ddot{\mathit{x}}, \dot{\mathit{x}}, and x are respectively acceleration, velocity, and displacement vector of the floor’s relative to the ground, M_s, C_s, and K_s are respectively mass, damping, and stiffness matrix of the structure system, f_MR is MR damper generated force, \mathit{\Gamma }={\left[-\mathrm{1}\mathrm{}\mathrm{0}\mathrm{}\mathrm{0}\right]}^{\mathrm{T}} is the MR damper’s vector position, \mathit{\Lambda }={\left[-\mathrm{1}\mathrm{}\mathrm{1}\mathrm{}\mathrm{1}\right]}^{\mathrm{T}} is the vector of earthquake acceleration effect and {{\displaystyle \ddot{\mathit{x}}}}_{\mathrm{g}} is the earthquake acceleration.

In the stat space the system will be written as [²⁹]

(2)\begin{array}{c}\hfill \dot{\mathit{z}}=A\mathit{z}+B\mathit{f}+E{{\displaystyle \ddot{\mathit{x}}}}_g,\hfill \\ \hfill \mathit{y}=C\mathit{z}+D\mathit{f}+G{{\displaystyle \ddot{\mathit{x}}}}_g,\hfill \end{array}

where z, f, and y are respectively variable state vector, input force, and outputs measured vector, A, B, C, D, E, and G are stat space representation matrices depended on the output vectors presented as follow

(3)\mathit{A}=\left[\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill -{{\displaystyle M}}_{\mathrm{s}}^{-\mathrm{1}}{{\displaystyle K}}_{\mathrm{s}}\hfill \end{array}\begin{array}{c}\hfill I\hfill \\ \hfill -{{\displaystyle M}}_{\mathrm{s}}^{-\mathrm{1}}{{\displaystyle C}}_{\mathrm{s}}\hfill \end{array}\right],

(4)\mathit{B}=\left[\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill {{\displaystyle M}}_{\mathrm{s}}^{-\mathrm{1}}\mathrm{\Gamma }\hfill \end{array}\right],

(5)\mathit{C}=\left[\begin{array}{c}\hfill \mathrm{1}\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill -{{\displaystyle M}}_{\mathrm{s}}^{-\mathrm{1}}{{\displaystyle K}}_{\mathrm{s}}\hfill \end{array}\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill \mathrm{1}\hfill \\ \hfill -{{\displaystyle M}}_{\mathrm{s}}^{-\mathrm{1}}{{\displaystyle C}}_{\mathrm{s}}\hfill \end{array}\right],

(7)\mathit{E}=-\left[\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill \mathrm{\Lambda }\hfill \end{array}\right],

(8)\mathit{G}=\left[\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\right].

Nonlinear model of MR damper

Numerous studies have been performed on the application of MR damper in controlling structures subjected to seismic load. Therefore, the dynamic behavior of this device have attracted considerable attention of many searchers in both analytical and experimental. Over the last several years, MR damper became the most attractive device in the structural vibration control due to their advantages. Requiring less than 50 W of power, this device can operate only with a battery, inexpensive by the fact that the force is adjusted depending on the magnetic field variation and responds in milliseconds [³⁰].

The first model described the MR damper nonlinear behavior is the Bingham model [³⁰]. This model was extended by the Bingham extended model [³¹]. Contrariwise, the hysteretic Bouc-Wen model proposed by Bouc and generalized by Wen described better this behavior based on a mathematical differential equation [³²]. This model was also extended basing on an experimental validation by Ref. [³³]. The following equations described the nonlinear behavior of MR damper (presented in Fig. 2)

(9){{\displaystyle c}}_{\mathrm{1}}\ddot{y}={{\displaystyle c}}_{\mathrm{0}}\left(\dot{x}-\dot{y}\right)+{{\displaystyle k}}_{\mathrm{0}}\left(x-y\right)+\alpha z,

(10)\dot{z}=-\gamma \left|\dot{x}-\dot{y}\right|z{\left|z\right|}^{n-\mathrm{1}}-\beta \left(\dot{x}-\dot{y}\right){\left|z\right|}^n+A\left(\dot{x}-\dot{y}\right),

(11)\dot{y}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle c}}_{\mathrm{0}}+{{\displaystyle c}}_{\mathrm{1}}}}\left[\alpha z+{{\displaystyle c}}_{\mathrm{0}}\dot{x}+{{\displaystyle k}}_{\mathrm{0}}\left(x-y\right)\right].

The force generalized by the system is

(12){{\displaystyle f}}_{\mathrm{M}\mathrm{R}}={{\displaystyle c}}_{\mathrm{0}}\left(\dot{x}-\dot{y}\right)+{{\displaystyle k}}_{\mathrm{0}}\left(x-y\right)+{{\displaystyle k}}_{\mathrm{1}}\left(x-{{\displaystyle x}}_{\mathrm{0}}\right)+\alpha z,

where x and \dot{x}, are respectively displacement and velocity of the damper, f_MR and z are respectively the generated force and the hysteretic component, k₀ and k₁ are the accumulator stiffness respectively at low and high velocity, c₀ and c₁ are the viscous damping respectively at low and high velocity, g, b, n, and A are parameters given the shape and scale of the hysteresis loop.

(13)\alpha ={{\displaystyle \alpha }}_{\mathrm{a}}+{{\displaystyle \alpha }}_{\mathrm{b}}u,

(14)c_1={c_1}_{\mathrm{a}}+{c_1}_{\mathrm{b}}u,

(15)c_0=c_{0\mathrm{a}}+c_{0\mathrm{b}}u,

(16)\dot{u}=-\eta (u-v),

where u is a phenomenological variable enveloping the system, v is the command voltage applied to the control circuit and h is a time response factor.

Control algorithm and training rule

The trainer controller

The linear quadratic controller is one of the most commonly classical control algorithm used in seismic control vibration. Among the optimal feedback control algorithm in linear system it is the most popular controller because of its simplicity of integration in the system and its facility of implementation. Despite the fact that this controller didn’t maintain desired performance in parameter perturbations or external disturbances presence, which is the practical case [³⁴,³⁵]. The optimal control forces are obtained by quadratic cost function minimization noted J(u) in which the gain control matrix is obtained. Moreover, the optimal feedback control law can be determined by solving the matrix Riccati equation differential or algebraic [³⁶].

(17)J\left(u\right)={\displaystyle {\int }_{{{\displaystyle t}}_{\mathrm{0}}}^{{{\displaystyle t}}_{\mathrm{1}}}\left[x{\left(t\right)}^{\mathrm{T}}·Q\left(t\right)·x\left(t\right)+u{\left(t\right)}^{\mathrm{T}}·R\left(t\right)·u\left(t\right)\right]}\mathrm{d}t,

where R and Q are the design parameters performance of x(t), u(t) which are respectively the state vector and the output vector and the desired control force is defined as

(18){{\displaystyle f}}_{\mathrm{d}}=-Kx,

where K is the state feedback gain and f_d is the output of the command which the desired control force.

(19)K=R^{-\mathrm{1}}\left(N^{\mathrm{T}}+B^{\mathrm{T}}P\right),

where P is a semi-positive defined matrices and it is the solution of the Riccati differential equation given by

(20)Q-{PBB}^{\mathrm{T}}P+PA+A^{\mathrm{T}}P=\mathrm{0.}

According to Refs. [²¹] and [³⁷], A, B, Q, R are known real coefficient matrices.

Figure 3 shows the basic block diagram of the classical LQC controller for structure vibration control.

The training controller

The artificial neural network is an intelligent system composed by an interconnected layers. In each layer there are a processing units called neurons which can operate as network [³⁸]. The main objective of this controller is to construct a strict feedback law which can tracks a desired trajectory in real time. Nevertheless, by the potential of learning it can be used to approximate any continuous function with required accuracy. The proposed neural network is trained to imitate the linear quadratic trainer controller in which the inputs are the state variables x₁,{{\displaystyle \dot{x}}}_{\mathrm{1}} and its output is the required control force (f_d). Therefore, one of the widely used neural network algorithm is a four layers algorithm with two hidden layer for neurons interconnected and multiplied by a weight, input layer and the last layer for output performant [³⁹].

The proposed intelligent LQC controllers, as shown in Fig. 4, consists of an input layer and an output layer, and between them two hidden layers are used. In the input layer there are two nodes, nevertheless in each hidden layer we proposed three neural nodes. Finally in the output layer, only one output node has been chosen since there is only a single output signal in the LQC controller. All the nodes of a defined layer are connected by weights to all the nodes in the next layer. The selected structure has been designed by means of trial and error after numerous simulation tests. During simulation tests, some more simple structures of ANN with single hidden layer were investigated. Unfortunately, none of them gave as good performance tracking as structure with two hidden layers shown in Fig. 4 which is recommended for difficult cases.

The output neuron has the linear activation function, whereas the neurons in hidden layers have the sigmoid shapes activation function. The backpropagation algorithm is used to perform the ANN where the output signal is compared with the desired output signal (the required force f_d), and the error between them is processed back through the network to update the weights and biases of the neural network until the output error becomes smaller than a minimum threshold defined by a corresponding performance function (see Fig. 5).

According to the architecture of the neural network depicted in Fig. 4, the output of node in the first hidden layer is

(21){{\displaystyle a}}_i\left(k\right)=f\left({\displaystyle {\sum }_{j=\mathrm{1}}^{\mathrm{3}}{{\displaystyle w}}_{ij}\left(k\right){{\displaystyle e}}_j\left(k\right)+{{\displaystyle a}}_i\left(k-\mathrm{1}\right)}\right),

where f\left(·\right) is the activation function, which is chosen as a logarithmic sigmoid function: f\left(x\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\mathrm{e}}^{-x}}}.

The output of the second hidden layer is calculated as

(22){{\displaystyle b}}_i\left(k\right)=g\left({\displaystyle {\sum }_{j=\mathrm{1}}^{\mathrm{3}}{{\displaystyle w}}_{ij}\left(k\right){{\displaystyle a}}_j\left(k\right)+{{\displaystyle b}}_i\left(k-\mathrm{1}\right)}\right),

where f\left(·\right) is the activation function, which is chosen as tangent sigmoid function: g\left(x\right)={\displaystyle \frac{\mathrm{1}-{\mathrm{e}}^{-\mathrm{2}x}}{\mathrm{1}+{\mathrm{e}}^{-\mathrm{2}x}}}.

And the output of the controller is

(23)u\left(k\right)={\displaystyle {\sum }_{i=\mathrm{1}}^{\mathrm{3}}{{\displaystyle w}}_i}\left(k\right){{\displaystyle b}}_k\left(k\right),

where a_i and b_i are respectively the input and output nodes of the hidden layer, k is sampling times, w_ij is the node weight and u is the measured output of the controller.

To evaluate the robustness of the proposed neural network learning controller closed-loop error between the measured output and expected value output derived from the linear controller is chosen:

(24)e\left(t\right)={{\displaystyle u}}_{\mathrm{m}}\left(t\right)-{{\displaystyle u}}_{\mathrm{e}}\left(t\right),

where e(t) is the learning strategy error, u_m(t) and u_e(t) are respectively the measured output and the expected output. Finally, the control block diagram for the proposed controller is inserted in the control configuration (see Fig. 6).

As indicated previously, the desired force control can’t by directly generated by MR damper. Contrariwise, a clipped optimal algorithm is designed to convert a control force into a control voltage signal. The clipped optimal control is an on-off switch algorithm control based on a Heaviside step function adjustment, in which if the MR damper control force is adjusted the desired control force is obtained by the on-off voltage in real time as Refs. [¹², ²¹, ⁴⁰].

(25)v={{\displaystyle v}}_{\max }H\left(\left({{\displaystyle f}}_{\mathrm{d}}-{{\displaystyle f}}_{\mathrm{M}\mathrm{R}}\right){{\displaystyle f}}_{\mathrm{M}\mathrm{R}}\right)

where v_max is the maximum applied voltage, H(·) is a Heaviside step function, f_MR is a MR damper force, and f_d is a desired control force signal calculated by the proposed controller.

Numerical example

The performance of the proposed control algorithm is evaluated in this study by one example through numerical simulation. The model is a 3-story building previously presented with an integrated MR damper installed between the first floor and the ground [²¹, ⁴¹]. The tested scaled structure is subjected to two time-scaled seismic excitations, the first one is the Tōhoku 2011 presented in Fig. 7 and the second one is the Boumerdès 2003 presented in Fig. 8.

The parameters of the scaled structure and the MR damper are defined as:

\left[{{\displaystyle M}}_{\mathrm{s}}\right]=\left[\begin{array}{c}\hfill \mathrm{98.3}\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill \mathrm{98.3}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill \mathrm{98.3}\hfill \end{array}\right],\mathrm{}\left(\mathrm{k}\mathrm{g}\right)

\left[{{\displaystyle C}}_{\mathrm{s}}\right]=\left[\begin{array}{c}\hfill \mathrm{175}\hfill \\ \hfill -\mathrm{50}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\begin{array}{c}\hfill -\mathrm{50}\hfill \\ \hfill \mathrm{100}\hfill \\ \hfill -\mathrm{50}\hfill \end{array}\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill -\mathrm{50}\hfill \\ \hfill \mathrm{50}\hfill \end{array}\right],\mathrm{}\left(\mathrm{N}·\mathrm{s}/\mathrm{m}\right)

\left[{{\displaystyle K}}_{\mathrm{s}}\right]={\mathrm{10}}^{\mathrm{5}}\times \left[\begin{array}{c}\hfill \mathrm{12}\hfill \\ \hfill -\mathrm{6.84}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\begin{array}{c}\hfill -\mathrm{6.84}\hfill \\ \hfill \mathrm{13.7}\hfill \\ \hfill -\mathrm{6.84}\hfill \end{array}\begin{array}{c}\hfill \mathrm{0}\hfill \\ \hfill -\mathrm{6.84}\hfill \\ \hfill \mathrm{6.84}\hfill \end{array}\right],\mathrm{}\left(\mathrm{N}/\mathrm{m}\right)

{{\displaystyle c}}_{\mathrm{0}\mathrm{a}}=\mathrm{21}\mathrm{}\mathrm{N}·\mathrm{s}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}{{\displaystyle c}}_{\mathrm{0}\mathrm{b}}=\mathrm{3.5}\mathrm{}\mathrm{N}·\mathrm{s}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}A=\mathrm{301}\mathrm{;}

{{\displaystyle k}}_{\mathrm{0}}=\mathrm{46.9}\mathrm{}\mathrm{N}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}{{\displaystyle k}}_{\mathrm{1}}=\mathrm{5}\mathrm{}\mathrm{N}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}{{\displaystyle c}}_{\mathrm{1}\mathrm{a}}=\mathrm{283}\mathrm{}\mathrm{N}·\mathrm{s}/\mathrm{c}\mathrm{m}\mathrm{;}

{{\displaystyle c}}_{\mathrm{1}\mathrm{b}}=\mathrm{2.95}\mathrm{}\mathrm{N}·\mathrm{s}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}{{\displaystyle \alpha }}_{\mathrm{a}}=\mathrm{140}\mathrm{}\mathrm{N}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}n=\mathrm{2}\mathrm{;}

{{\displaystyle \alpha }}_{\mathrm{b}}=\mathrm{695}\mathrm{}\mathrm{N}/\mathrm{c}\mathrm{m}\mathrm{;}\mathrm{}\gamma =\mathrm{363}\mathrm{}{\mathrm{c}\mathrm{m}}^{-\mathrm{2}}\mathrm{;}\mathrm{}\beta =\mathrm{363}\mathrm{}{\mathrm{c}\mathrm{m}}^{-\mathrm{2}}\mathrm{;}

\eta =\mathrm{190}\mathrm{}{\mathrm{s}}^{-\mathrm{1}}\mathrm{;}\mathrm{}{{\displaystyle x}}_{\mathrm{0}}=\mathrm{14.3}\mathrm{}\mathrm{c}\mathrm{m}.

Numerical simulations are implemented using MATLAB/Simulink in which the proposed neural network control based on learning of linear quadratic control is evaluated. The comparison of the responses of the controlled and uncontrolled proved the ability of the proposed learning rule coupled with a semi-active control design to eliminate structural earthquake vibrations. The numerical simulations of the first, second, and third floors of the controlled and uncontrolled structure are depicted in Figs. 9–11. Figs. 12 and 17 compared the force responses of the MR damper and shown the performance of the neural network learning under the Tōhoku 2011 and the Boumerdès 2003 earthquakes, respectively. However, the displacement responses of the first, second, and third floors of the compared controlled and uncontrolled structure are presented in Figs. 14–16. Although Figs. 13 and 18 illustrate the error between the forces derived from the NN and linear quadratic controller for both seismic excitations.

Conclusions

The effectiveness of the proposed neural network control in seismic vibration reduction of 3-story scaled structure has been investigated thoroughly in this article. The effectiveness and efficiency control of the neural network controller is proved by the satisfactorily learning accuracy shown in Figs. 13 and 18. The results presented in Table 1 evinced the robustness of the neuro-control strategies combined with the MR damper to reduce structural vibrations caused by seismic excitations. The compared simulation results of the controlled structure and the uncontrolled structure under both of 2011 Tōhoku and 2003 Boumerdès seismic excitations presented in Figs. 9, 10, 11, 14, 15, and 16 shown clearly the performance of the proposed neural network controller in seismic vibrations reduction.

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