A Case Study on Designing a Sliding Mode Controller to Stabilize the Stochastic Effect of Noise on Mechanical Structures: Residential Buildings Equipped with ATMD

)is study aims to stabilize the unwanted fluctuation of buildings as mechanical structures subjected to earth excitation as the noise. In this study, the ground motion is considered as a Wiener process, in which the governing stochastic differential equations have been presented in the form of Ito equation. To stabilize the vibration of the system, the ATMD system is considered and located on the upmost story of the building. A sliding mode controller has been utilized to control the ATMD system, which is a robust controller in the presence of uncertainty. For this purpose, the design of a sliding mode controller for the general dynamic system with Lipschitz nonlinearity and considering the Ito relations has been accomplished. )e mentioned design has been implemented considering the presence of the Weiner process and existence of uncertainty in the structure and actuator. )en, the obtained general control law has been generalized to control the ATMD system. )e results show that the designed controller is effective to reduce the effect of the unwanted impused vibrations on the building.

Earthquake is an example of a natural phenomenon that influences the dynamics of a structures and building. Dynamic behavior of structures in the presence of earthquakes has been extensively studies in [21][22][23][24][25][26][27]. In the mentioned studies, the dynamic behavior of the structures has been investigated under the influence of a particular earthquake. But in this paper, we aim to examine the dynamic behavior of the structure subjected to White Gaussian Noise (WGN) because the density function of its power spectrum is constant at all frequencies. So, unlike the previous studies, a new formulation should be considered.
For this purpose, Itô formulation is considered to solve governing SDE of the structure [28][29][30]. e studied structure is an 11-story building equipped with an ATMD system at the upmost story. ATMD has been used to reduce unwanted vibrations.
is system has been controlled by means of the sliding mode controller. For this reason, the sliding mode controller has been designed for the general and nonlinear Lipschitz dynamic system in the presence of actuator and system uncertainties and based on the Itô theory. Finally, the designed controller has been generalized to control the ATMD system. e dynamic behavior of the structure in the active and passive mode and considering the uncertainties has been studied. e effect of various controller parameters on the dynamic behavior of the structure has been also investigated. Moreover, the effect of the controlled system and different parameters of the controller on the basin of attraction was also studied. At the end, the controller's robustness to structural and actuator uncertainty has been studied and the obtained results have been presented.

Model Description
e physical and geometrical models of the studied system are presented in this section. e studied model is an 11story building, the schematic view of which is shown in Figure 1. e mass of each story is denoted by m i , where i represents the story number. Moreover, the stiffness and damping coefficient for each story are represented by K i and C i , respectively. e degree of freedom (DoF) of the system is considered along the horizontal direction, since the ground displacement effects are horizontally applied to the structure. As shown in Figure 1, the displacement of each story is denoted by x i . Upon ground excitation, each story experiences a vibration and, given that the first mode of vibration is more likely to occur, the largest displacement occurs in the topmost story. To alleviate this deformation caused by earthquakes, the 11 th story is equipped with an ATMD system. e mass of this system, its stiffness, and damping coefficient are denoted by m 12 , K 12 , and C 12 . After installation, the DoF of system increases to 12. e dynamic equation governing the system behavior is presented in the following equation [31]: , and [C] are also presented in the following. However, note that uncertainty will be also considered for these values later in the control design process.

Mathematical Modeling
Without loss of generality, the controller is initially designed for a general system, after which it is applied to the studied structural control. Consider the mathematical model presented in the following equation: where f(x, t), b(x, t), and h(x, t) represent continuous functions satisfying the Lipschitz condition. In this case, u, v, and _ v � dv/dt indicate the control force, standard Wiener process, and white Gaussian noise, respectively. e objective of the controller design is for the x 1 to track x d . To this end, the dynamic error of e � x 1 − x d should be defined. However, since x d is considered zero in structural control, the error is defined as e � x 1 and the corresponding dynamic error is defined as follows: e Lyapunov function is considered as V � 1/2E(s 2 ). From a mathematical viewpoint and based on Ito's theory, equation (4) can be reformulated in the form of a differential equation as follows: Given that the abovementioned equation is presented as follows for simplification purposes: e abovementioned equation represents an Itô SDE used instead of equation (4) and considering the Wiener process. e terms f(x, t) + b(x, t)u and h(x, t) represent the drift function and diffusion, respectively [28,29].
e sliding surface was considered as s � e 2 + λe 1 in the design of the sliding mode controller, from which _ s � _ e 2 + λe 2 and ds � de 2 + λe 2 dt can be derived. In the final form and according to equation (4), the ds equation can be rewritten to obtain equation (5): Assuming y � g(x, t) and employing Itô 's differentiation formula for dy, we have [28,29] Complexity y � g(x, t), Based on Ito's formula, differentiating s 2 produces ds 2 � 2sds + dsds. erefore, the dV value emerges as follows: By including the following relations proposed by [32,33] in the calculations, And also considering the properties of the Wiener process [28], According to equations (10) and (11), the expected value of dV is determined as follows: Dividing the abovementioned relation by dt, _ V is obtained as follows: e stability condition for the sliding mode controller is defined as _ V < 0 based on the Lyapunov second method for stability [34]. Assuming the systems involve no uncertainty, the controller stability is only guaranteed by considering f(x, t) + b(x, t)u + λe 2 � − θs, limiting the region of attraction associated with the sliding surface. However, given the presence of uncertainty in most of dynamical systems, the system uncertainties are included in the controller design equations in the following part. e structural and actuator uncertainties have been considered in the controller model in this study. To this end, some of the inequalities associated with the f(x, t) and b(x, t) functions along with their nominal values should be taken into account.
Assuming f(x, t) as the nominal values for f(x, t) and F(x, t) is a positive function expressed as follows: As a result, the following equations hold true: where b 0 and b M are positive values representing the upper and lower bounds of the b(x, t) function, the following equations are necessary for the controller design: e following inequality holds true considering equations (15), (17), and (19): Equation (21) also holds true considering equations (18) and (20): 4 Complexity By defining u as follows, the relation _ (22) where η represents a positive value and ψ(x, t) where u 1 and u 2 , which are obtained as follows, cause the nominal and uncertainty terms to be negative definite, respectively: eory. assume Δ as the following set: where θ is a positive value. In fact, represents the attraction set for the trajectory s(t). Assuming at t � 0, the trajectory s(t) lies outside the attraction set, i.e., E[s 2 (t � 0)] > H 2 /2θ, we obtain which indicates a declining rate for V as it tends to enter the Δ set and ultimately remains in this region. If it is located in the domain of s for the moment t � 0, it will continue to be in this area. e designed sliding mode controller is then applied to the system, as shown in Figure 1, based on the abovementioned equations. e equation governing the dynamic behavior of the 11 th story is expressed as follows: e f(x, t) for this equation is defined as follows: where M 11 , K 11 , K 12 , C 11 , and C 12 denote nominal values and ΔM 11 , ΔK 11 , ΔK 12 , ΔC 11 , and ΔC 12 represent the maximum value for the respective uncertainty term. In this case, F(x, t) is defined as follows: Moreover, b(x, t) is assumed as b(x, t) � (1+ α sin t)/M 21 in this case, meaning that the system's actuator includes the uncertainty (1 + α sin t). Under these conditions, b 0 and b M are expressed as follows: Finally, the term associated with the control force u is conveniently determined using equation (22).

Results and Discussion
In this section, the obtained results will be presented and the control of horizontal displacement of the 11-story building will be discussed. Given that the first vibrational mode of the building is easily excited, the largest displacement in this mode is experienced by the topmost story. erefore, vibration analysis is conducted on the 11 th story. e physical and geometrical specifications of the studied building are given in Table 1. In the first scenario, the uncertainty was assumed as and α � 0. e resulting horizontal displacement for story 11 is demonstrated in Figure 2 for both active and passive cases. As shown in this figure, the vibrational amplitude of the controlled system is considerably smaller than that of the uncontrolled system. e variations of the control force u with respect to time are demonstrated in Figure 3. As shown, the chattering phenomenon is apparent within some time intervals. Chattering is mainly caused by the term sign(s) in the control force relation. In fact, the discontinuity and undifferentiability of this function at point s � 0 is responsible for the chattering phenomenon. e chattering around the zero point is highly harmful, since, in addition to the force magnitude, its sign also changes. However, at nonzero points, the chattering only causes a decrease or increase in the force magnitude. In our case, as shown, chattering occurs at nonzero points.
To resolve the chattering problem and satisfy the continuity and Lipschitz condition, the term tanh(s)/ε was used as an approximation of the sign(s) function for the function Table 1: e physical and geometrical specifications of the studied building.

Properties
Values M 1 , M 2 , . . . , M 11 255 × 10 3 kg M 12 77 × 10 3 kg K 1 , K 2 , . . . , K 11 25 × 10 6 N/m K 12 205 × 10 3 N/m C 1 , C 2 , . . . , C 11 216 × 10 3 Ns/m    Figure 4. As shown, the displacement amplitude is increased by increasing ε. However, this increase is negligible compared to the displacement of the uncontrolled system. e variations of force u corresponding to Figure 4 for different ε values are shown in Figure 5. As indicated, the amplitude of force u at ε � 0.1 is smaller compared to other ε values. Moreover, the chattering phenomenon is also fully resolved in this case, while it is still observed at other cases, for example, ε � 0.001. As shown in Figures 4 and 5, the vibrational amplitudes for both sign(s) and tanh(s/0.001) functions are consistent, which is due to the accurate approximation of the sign(s) function by tanh(s/0.001).
Moreover, the phase diagram for the 11 th story is also demonstrated in Figure 6 for different cases. As shown, by increasing ε, the region of attraction is also extended. is region was almost similar for both sign(s) and tanh(s/0.001) functions. e effect of h(x, t) on the dynamic behavior of the structure and the region of attraction is discussed in this part. e vibrational amplitude of the uncontrolled system for different h(x, t) values is shown in Figure 7. As    and different h(x, t).
demonstrated, the displacement amplitude is linearly increased by increasing h(x, t). Also, the attraction domain for the mentioned figure is presented in Figure 8. As is clear from this figure, the domain of attraction set is extended with increasing h(x, t).
e horizontal displacement for the controlled case is demonstrated in Figure 9 with respect to different h(x, t) values. As shown, by increasing h(x, t), the vibrational amplitude is also increased, but its value is substantially smaller compared to the uncontrolled case. e force variations corresponding to Figure 9 is demonstrated in the diagram of Figure 10. As shown, the amplitude of force variations also experiences an increase as h(x, t) increases.
Moreover, the region of attraction corresponding to Figure 9 is demonstrated in Figure 11. As shown, similar to the uncontrolled case, the region of attraction increases by increasing h(x, t).
e effects of η and λ on the behavior of a controlled system is discussed in this section. e variations in horizontal displacement of the system are demonstrated in Figure 12 for different η values. As shown, for η values lower than 1000, increasing the η values is not significantly effective, but increasing this parameter to 100,000 causes the 12 Complexity horizontal displacement of the system to increase substantially. Additionally, the region of attraction for an η value of 100,000 is shown in Figure 13. As shown, the region of attraction is considerably expanded at this η value. e displacement amplitude of the system for different λ values and a fixed η value of 1 is demonstrated in Figure 14. As shown, the displacement amplitude is substantially decreased by increasing lambda. e region of attraction for a λ value of 5 is shown in Figure 15. Consider this figure suggests that the region of attraction becomes more limited for a lambda value of 5.
is section discusses the controller robustness to system uncertainty. In the first scenario, only the system uncertainties were considered and the actuator uncertainties were neglected. e variations of horizontal displacement for different Δ values are demonstrated in Figure 16. As shown, the controller is highly robust to uncertainties and even within some time intervals, and the horizontal displacement of the system is decreased by increasing Δ.
is is due to the fact that F(x, t) is also increased by increasing Δ. However, as expressed in equation (22), presence of F(x, t) in this equation virtually increases the term ψ(x, t) + η, and as shown in Figure 12, increasing η increases the controller robustness.
e regions of attraction for Δ � 5% and Δ � 10% are plotted in Figure 17. As shown, despite its larger extent for Δ � 10% compared to Δ � 5%, the region of attraction has become more compact.  14 Complexity e results for the case with actuator uncertainty and Δ � 10% were extracted and presented in Figure 18. In this case, b(x, t) � 1 + α sin t. e results for different α values are shown in this figure, which indicates the significant controller robustness in presence of actuator uncertainty. e region of attraction in this case is depicted in Figure 19. As shown, the extent of region of attraction is increased by increasing α.
e results in presence of actuator and system uncertainties are shown in Figure 20. In this case, α is set to be 0.2. As shown, the controller exhibits a high robustness in presence of actuator and system uncertainties. e region of attraction for this case is demonstrated in Figure 21. As shown, the extent of region of attraction is increased by increasing Δ.

Conclusion
is study has examined the dynamic behavior of an 11story building equipped with an ATMD system. e stimulation force has been applied on the structure in the form of a Wiener process and an earthquake. e sliding mode scheme has been used to control the ATMD system. So, considering Lipschitz nonlinearity and based on the Ito formulation, a sliding mode controller in the presence of the uncertainty for the general dynamical system with the second-order governing stochastic differential equation has been designed. e designed controller has been further developed to control the ATMD system. e dynamic behaviors of the structure in the active and passive modes have been simulated. e presented results demonstrate the high ability of the sliding mode controller to reduce unwanted vibrations of buildings under stimulation of the ground. Also, the results show that the designed controller has a good robustness in the presence of structural and actuator uncertainties. In addition, the effects of the controller parameters on system behavior have been studied. e results show that reduction of ε and increase of η reduce the structural vibration amplitude.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.