Seismic vibration control using a magneto-rheological damper is a technique that interests several researchers around the world. This technique offers suitable structural building protection, ensuring human safety against earthquake excitation damages. The robustness of these devices depended in many cases on the designed law of control. Over the years, research focused on the development and modelling of various controllers to enhance the structural vibration elimination of buildings. The emphasis of this paper is on the evaluation of semi-active control robustness to reduce the displacements of a three-storey tested structure. The semi-active control device is a magneto-rheological fluid damper installed on the ground floor of the earthquake's excited structure and is controlled by an adaptive non-linear controller coupled to a clipped optimal algorithm to drive the current. The proposed controller is a sliding-mode controller reinforced by an adaptive technique to perform the control gain choice and overcome the chattering problem. The present law of adaptation is a switching conditional law between two laws offering the required gain depending on the system state. The numerical simulation results prove the effectiveness of the proposed semi-active control strategy in attenuating the displacements of the tested structure.

Earthquake excitations cause building structure damage, city destruction, and human deaths. They directly affect buildings or their structural components through the brutal motion of the ground-propagating vibration. During the last century, earthquakes caused deaths and homelessness of millions of people and destroyed several cities. Algerian towns were affected by strong earthquakes during this period: Algiers in 1916, El Asnam in 1980, Constantine in 1985, Tipasa in 1989, Mascara in 1994, Algiers in 1996, and Boumerdès in 2003

The passive control system is based principally on isolation or energy dissipation devices designed to develop an opposite control force to the motion of the structure induced by earthquake excitation. The greatest advantages of these devices are that they are simple, economic, and do not need energy to operate, but they have limited robustness against strong dynamical loads, which limits their utilization. The passive control technologies have been widely employed for a large number of buildings in the world. In

Recently, major attention has been paid to active structural control research.

The hybrid control system can also be considered one of the structural control systems which are a combination of the aforementioned systems. Generally, the hybrid system is composed of an active system combined with a passive system. Thereby, the hybrid system offers the advantages of both coupled systems. Remarkable searches have been made over the last decade. An eight-storey structure is controlled using a hybrid control composed of viscous liquid dampers as a passive control and an active tuned mass damper. The active tuned mass damper is installed on the top floor, and on each one of the other floors a viscous liquid damper is placed. To prove the effectiveness of the proposed hybrid control system in eliminating structural vibrations, this structure was subjected to the Kobe, El Centro, Hachinohe, and Northridge earthquakes

However, in order to tune the control force of the active or semi-active control device in real time, the design of a robust controller is considered an essential step. In the existing literature, several control techniques and algorithms are quoted. Many papers have investigated the use of non-linear controllers in vibration control. In

In the civil engineering structural semi-active control, the principal key of reliability and efficiency is the suitable choice of the adopted controller. However, the sliding-mode controller is one of the most used controllers in this field because of its several promises, especially in a complex presence. Otherwise, in the presence of a switching part, this controller exposed an obvious stability phenomenon called chattering caused by an infinite oscillation frequency. Furthermore, the adaptation is the widely adopted solution to suppress the chattering and offer more stability to the sliding-mode controller. Ideally, the desired law of adaptation must offer perfect adjustment of the control gain depending on the system state using low computational energy

In this paper, an adaptive non-linear controller is proposed to reduce undesirable vibrations of a scaled structure caused by seismic excitation. The controller is designed to control a semi-active magneto-rheological damper coupled to a clipped algorithm current driver. The sliding-mode controller is reinforced by a switching adaptive law to ensure stability and to suppress the chattering phenomenon. The suggested law offers adaptation of the boundary layer thickness in the sliding surface of the controller. Therefore, the switching jump length is adapted to maintain the desired stability of the system, although the proposed adaptation law switches between two laws which calculated the gain of control depending on the system position compared to the sliding surface. Under the 1940 El Centro earthquake record, the effectiveness of the proposed semi-active control strategy is evaluated. In order to prove the robustness of the control, the numerical simulation results of the controlled and uncontrolled structures are compared and discussed. The rest of the paper is organized as follows: Sect. 2 is reserved for the system presentation in which the magneto-rheological damper and the test scaled structure modelling are illustrated, where the mathematical model of the controlled system is explained. Next, in Sect. 3 the controller and the adaptive law are formulated, ensuring the Lyapunov stability criterion. The numerical example of the 1940 El Centro earthquake is presented and the comparison of the results of the two cited cases is shown in Sect. 4. Finally, this paper is concluded by the drawn conclusion in the last section.

The magneto-rheological device is considered one of the interesting semi-active control devices. The properties offered by the magneto-rheological fluid attracted several numerical and experimental studies and investigations. The main investigations focused on the mathematical modellization of the high non-linear behaviour of the controllable fluid. Nonetheless, in the presence of the current field, this fluid can change its physical properties in milliseconds only. In addition, the device offered simplicity, reliability, and suitable stability in temperature variation

Cross section of the MR damper body.

Therefore, from the invention of the magneto-rheological fluid, several studies presented mathematical modelling for the non-linear behaviour of this fluid

Generalized Bouc–Wen model of the magneto-rheological damper.

The mathematical generalized Bouc–Wen model is described by the following equations:

The parameter values of the augmented Bouc–Wen model are shown in Table

Parameter values of the augmented Bouc–Wen model.

To show the typical hysteresis behaviour of the MR damper, a sinusoidal displacement is applied to the device with a frequency of 2.5 Hz and 1.5 cm amplitude. By contrast, a different constant voltage level is associated with this sinusoidal displacement. The chosen voltage levels are 0, 0.5, 0.75, and 1 V, respectively.

The generated force of the MR damper function of time is illustrated in Fig.

Generated force of the simulated Bouc–Wen model under sinusoidal excitation.

Force–displacement hysteresis loop of the simulated Bouc–Wen model under sinusoidal excitation.

Force–velocity hysteresis loop of the simulated Bouc–Wen model under sinusoidal excitation.

The proposed semi-active control is evaluated using a three-storey scaled structure subjected to a ground motion excitation

Tested scaled structure model.

The dynamic equation of motion of the system in Fig.

Since its appearance in the 1950s the sliding-mode controller has been considered one of the most interesting non-linear controllers. The robustness of this controller, especially in the presence of the uncertainties or the system condition variations, makes it one of the most used controllers. However, several investigations and studies are interested in structural vibration control using a sliding-mode controller, whereas the high-frequency oscillation of the switch controller part caused the chattering problem which can affect the system stability and precision

Therefore, various solutions were investigated to suppress the undesirable chattering effect and to perform the sliding-mode controller. Furthermore, several tested techniques proved their efficiency as the boundary layer technique using the saturation function

Consider the non-linear system

The sliding-mode controller (SMC) is composed of two components as follows:

In this stage, the switching part of the sliding-mode controller in Eq. (

The second Lyapunov candidate function of Eq. (

Therefore, the schematic block of the proposed semi-active adaptive control is illustrated in Fig.

The schematic block of the semi-active adaptive control.

When

To prove the robustness of the proposed semi-active control strategy using the magneto-rheological damper controlled by an adaptive sliding-mode controller, a numerical test is proposed. Based on the Matlab/Simulink software, the results were carried out and plotted. The tested scaled structure of Fig.

The NS (north–south) time-scaled component of the 1940 El Centro earthquake.

The numerical displacement results of the compared two cases of controlled and uncontrolled structures under the 1940 El Centro excitation are clearly shown for the three floors in Fig.

Displacement responses of the first, second, and third floors to the 1940 El Centro earthquake.

Structure floor acceleration responses to the 1940 El Centro earthquake.

Nevertheless, the time adjustment of the adaptive gain of Eqs. (

Time response of adaptive control gain under the 1940 El Centro earthquake.

Zoomed time response voltages applied to the MR damper during the 1940 El Centro earthquake.

The effectiveness of the proposed adaptive semi-active control strategy is proven using the displacement floor reduction ratio given by

The peak displacement reduction ratio of the SMC and ASMC during the 1940 El Centro earthquake.

Values of performance indices of the SMC and ASMC during the 1940 El Centro earthquake.

In addition, two groups of evaluation indices are calculated to confirm the robustness of the semi-active adaptive sliding-mode controller in suppressing structural vibrations. The first group of the index (

The displacement floor reduction ratio results are given in Table

Using an earthquake record as input excitation to investigate the performance of the adaptive control approach, the responses of the tested structure are examined. The displacement responses of the structure are reduced reasonably compared to the responses of the uncontrolled structure under the selected excitation, as observed from the comparison of the plotted responses of the uncontrolled structure with those of the controlled structure, using the magneto-rheological damper with an adaptive sliding-mode controller in which the semi-active adaptive strategy provides adequate vibration reduction. In addition, through system computation, the stability of the system is observed and the chatter effect is completely reduced. The switching law offers more stability to the controlled system than the previously tested laws

A semi-active adaptive control has been proposed for the vibration response reduction of seismically excited structures. The proposed non-linear sliding-mode controller is designed to calculate the desired control force. In fact, the classical sliding-mode controller provides chatters in numerical simulations. The adaptive proposed law is stability reinforcement to the classical sliding-mode controller. Through numerical simulation investigation of a tested scaled structure, it is noted that the proposed adaptive sliding-mode control approach performs very well with the stability of the system output. The compared numerical simulation results of the displacement responses of the tested structure of the controlled and uncontrolled cases have shown clearly the robustness of the proposed strategy. The effectiveness of the adaptive sliding control using a semi-active magneto-rheological damper to suppress structural vibrations is verified by nine performance indices under the 1940 El Centro earthquake excitation. In addition, the comparison of the sliding-mode controller and the adaptive sliding-mode controller results using the peak displacement reduction ratio and the performance indices proves the reliability of the adaptive approach.

The code in this research is available upon request by contact with the corresponding author.

The paper was prepared with the contributions of all authors. KZ, AS and IKB conducted simulation model establishment. KZ and LF are responsible for theoretical analysis and paper writing. LF and MD verified the model mathematical and numerical stability. MD and IKB supervised and reviewed the manuscript.

The contact author has declared that none of the authors has any competing interests.

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This paper was edited by Daniel Condurache and reviewed by three anonymous referees.