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

Authors: Aydin Azizi
Journal: Complexity
 (2020)

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## Abstract

This 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. The mentioned design has been implemented considering the presence of the Weiner process and existence of uncertainty in the structure and actuator. Then, the obtained general control law has been generalized to control the ATMD system. The results show that the designed controller is effective to reduce the effect of the unwanted impused vibrations on the building.


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## Body

## 1. Introduction

Stochastic factors are one of the inherent aspects of most dynamical systems that occur in different ways such as external force and changes in the inherent parameters of the system. Stochastic Differential Equations (SDE) in financial mathematics and economics have been extensively investigated [1–6]. Also, the effect of stochastic factors in science and mechanical and electronic engineering has been considered [7–20].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 [21–27]. 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 [28–30]. The studied structure is an 11-story building equipped with an ATMD system at the upmost story. ATMD has been used to reduce unwanted vibrations. This 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.The dynamic behavior of the structure in the active and passive mode and considering the uncertainties has been studied. The 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.


## 2. Model Description

The physical and geometrical models of the studied system are presented in this section. The studied model is an 11-story building, the schematic view of which is shown in Figure1. The mass of each story is denoted by mi, where i represents the story number. Moreover, the stiffness and damping coefficient for each story are represented by Ki and Ci, respectively. The 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 xi. 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 11th story is equipped with an ATMD system. The mass of this system, its stiffness, and damping coefficient are denoted by m12, K12, and C12. After installation, the DoF of system increases to 12.Figure 1
Schematic view of the studied building with ATMD.The dynamic equation governing the system behavior is presented in the following equation [31]:(1)MX¨+CX˙+KX=U+DX¨g,where M, K, and C denote mass, stiffness, and damping matrices, respectively. Moreover, X=x1,x2,x3,…,x11,x12T, X˙, and X¨ represent the structural displacement, velocity, and acceleration, respectively. The term U=0,0,…,0,bx,tu,−bx,tuT is the control force vector of the ATMD system responsible for controlling the vibrations of the 11th story. The term Dx¨g denotes the effects of base excitation on the system, where x¨g is the acceleration of base excitations and D=−m1,m2,m3,…,m11,m12T. The nominal matrix values for M, K, 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.(2)M=m1000000000000m2000000000000m3000000000000m4000000000000m5000000000000m6000000000000m7000000000000m8000000000000m9000000000000m10000000000000m11000000000000m12,K=k1+k2−k20000000000−k2k2+k3−k30000000000−k3k3+k4−k40000000000−k4k4+k5−k50000000000−k5k5+k6−k60000000000−k6k6+k7−k70000000000−k7k7+k8−k80000000000−k8k8+k9−k90000000000−k9k9+k10−k100000000000−k10k10+k11−k110000000000−k11k11+k12−k120000000000−k12k12,C=c1+c2−c20000000000−c2c2+c3−c30000000000−c3c3+c4−c40000000000−c4c4+c5−c50000000000−c5c5+c6−c60000000000−c6c6+c7−c70000000000−c7c7+c8−c80000000000−c8c8+c9−c90000000000−c9c9+c10−c100000000000−c10c10+c11−c110000000000−c11c11+c12−c120000000000−c12c12.


## 3. 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:(3)x˙1=x2,x˙2=fx,t+bx,tu+hx,tv˙,where fx,t, bx,t, and hx,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. The objective of the controller design is for the x1 to track xd. To this end, the dynamic error of e=x1−xd should be defined. However, since xd is considered zero in structural control, the error is defined as e=x1 and the corresponding dynamic error is defined as follows:(4)e˙1=x2−x˙d,e˙2=fx,t+bx,tu+hx,tv˙−x¨d.The Lyapunov function is considered asV=1/2Es2. From a mathematical viewpoint and based on Ito’s theory, equation (4) can be reformulated in the form of a differential equation as follows:(5)de1=x2−x˙ddt,de2=fx,t+bx,tu−x¨ddt+hx,tdv,x¨d=0.Given thatxd=x˙d=x¨d=0, the abovementioned equation is presented as follows for simplification purposes:(6)de1=x2dt,de2=fx,t+bx,tudt+hx,tdv.The abovementioned equation represents an Itô SDE used instead of equation (4) and considering the Wiener process. The terms fx,t+bx,tu and hx,t represent the drift function and diffusion, respectively [28, 29].The sliding surface was considered ass=e2+λe1 in the design of the sliding mode controller, from which s˙=e˙2+λe2 and ds=de2+λe2dt can be derived. In the final form and according to equation (4), the ds equation can be rewritten to obtain equation (5):(7)ds=fx,t+bx,tu+λe2dt+hx,tdv.Assumingy=gx,t and employing Itô ’s differentiation formula for dy, we have [28, 29](8)y=gx,t,dy=∂g∂tdt+∂g∂xdx+12∂2g∂x2dx2.Based on Ito’s formula, differentiatings2 produces ds2=2sds+dsds. Therefore, the dV value emerges as follows:(9)dV=Esfx,t+bx,tu+λe2dt+hx,tdv+12fx,t+bx,tu+λe2dt+hx,tdv2.By including the following relations proposed by [32, 33] in the calculations,(10)dt⋅dt=0,dt⋅dv=0,dv⋅dv=dt.And also considering the properties of the Wiener process [28],(11)Ehx,tdv=0,Ehx,tdv2=h2x,tdt.According to equations (10) and (11), the expected value of dV is determined as follows:(12)dV=Esfx,t+bx,tu+λe2dt+12h2x,tdt.Dividing the abovementioned relation bydt, V˙ is obtained as follows:(13)V˙=Esfx,t+bx,tu+λe2+12Eh2x,t.The stability condition for the sliding mode controller is defined asV˙<0 based on the Lyapunov second method for stability [34]. Assuming the systems involve no uncertainty, the controller stability is only guaranteed by considering fx,t+bx,tu+λe2=−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.The structural and actuator uncertainties have been considered in the controller model in this study. To this end, some of the inequalities associated with thefx,t and bx,t functions along with their nominal values should be taken into account.Assumingf^x,t as the nominal values for fx,t and Fx,t is a positive function expressed as follows:(14)fx,t−f^x,t≤Fx,t.As a result, the following equations hold true:(15)sfx,t−sf^x,t≤sFx,t,sfx,t≤sf^x,t+sFx,t.Ifbx,t is defined as(16)0<b0<bx,t<bM,where b0 and bM are positive values representing the upper and lower bounds of the bx,t function, the following equations are necessary for the controller design:(17)sFx,t≤sbx,tb0Fx,t,(18)sλe2=sλbb0e2+sλe21−bb0,sλe2≤sλbb0e2+sλe2b1b0−1b,attention to⟶sλe2b1b0−1b≤sλe2b1b0−1bM,So,sλe2≤sλbb0e2+sλe2b1b0−1bM,(19)sf^x,t=sbb0f^x,t+sf^x,t1−bb0,sf^x,t≤sbb0f^x,t+sf^x,tb1b0−1b,attention to⟶sf^x,tb1b0−1b≤sf^x,tb1b0−1bM,So,sf^x,t≤sbb0f^x,t+sf^x,tb1b0−1bM.The following inequality holds true considering equations (15), (17), and (19):(20)sfx,t≤sbb0f^x,t+sf^x,tb1b0−1bM+sbx,tb0Fx,t.Equation (21) also holds true considering equations (18) and (20):(21)V˙≤sλbb0e2+sλe2b1b0−1bM+sbb0f^x,t+sf^x,tb1b0−1bM+sbb0Fx,t+sbx,tu+12h2x,t,V˙≤sbb0λe2+f^x,t+sbx,tu+sb1b0−1bMλe2+f^x,t+sbb0Fx,t+12h2x,t.By definingu as follows, the relation V˙≤−θEs2+1/2Eh2x,t holds true:(22)u=−1b0λe2+f^x,t+θ×s−signsη+ψ,where η represents a positive value and ψx,t=1/b0−1/bMλe2+f^x,t+1/b0Fx,t. In fact, u=u1+u2, where u1 and u2, which are obtained as follows, cause the nominal and uncertainty terms to be negative definite, respectively:(23)u=u1+u2,λ1b0e2+1b0f^x,t+u1=−θb0s,u1=−1b0λe2+f^x,t+θ×s,sbx,tu2+sbx,t1b0−1bMλe2+f^x,t+1b0Fx,t=−ηbx,ts,su2+s1b0−1bMλe2+f^x,t+1b0Fx,t=−ηs,ψx,t=1b0−1bMλe2+f^x,t+1b0Fx,t,su2=−ηs−sψ,su2=−sη+ψ,u2=−signsη+ψ.Theory.
assumeΔ as the following set:(24)Δ=s∈REs2≤H2θ,where θ is a positive value. In fact, V˙≤−θEs2+1/2Eh2x,t represents the attraction set for the trajectory st. Assuming at t=0, the trajectory st lies outside the attraction set, i.e., Es2t=0>H2/2θ, we obtain(25)V˙≤−θEs2t=0+12Eh2x,t<0,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 ofs for the moment t=0, it will continue to be in this area.
The designed sliding mode controller is then applied to the system, as shown in Figure1, based on the abovementioned equations. The equation governing the dynamic behavior of the 11th story is expressed as follows:(26)x˙1=x2,x˙2=−1M11K11+K12x11+C11+C12x˙11−K11x10−K12x12−C11x˙10−C12x˙12−x¨g+bx,tM11u.
Thefx,t for this equation is defined as follows:(27)fx,t=−1M^11+ΔM11K^11+ΔK11+K^12+ΔK12x11+C^11+ΔC11+C^12+ΔC12x˙11−K^11+ΔK11x10−K^12+ΔK12x12−C^11+ΔC11x˙10−C^12+ΔC12x˙12,where M^11, K^11, K^12, C^11, and C^12 denote nominal values and ΔM11, ΔK11, ΔK12, ΔC11, and ΔC12 represent the maximum value for the respective uncertainty term.
In this case,Fx,t is defined as follows:(28)Fx,t=1M^11−ΔM^11ΔK^11+ΔK^12x11+ΔC^11+ΔC^12x˙11+ΔK^11x10+ΔK^12x12+ΔC^11x˙10+ΔC^12x˙12.
Moreover,bx,t is assumed as bx,t=1+αsint/M21 in this case, meaning that the system’s actuator includes the uncertainty 1+αsint. Under these conditions, b0 and bM are expressed as follows:(29)b0=1−αM^21+ΔM^21,bM=1+αM^21−ΔM^21.
Finally, the term associated with the control forceu is conveniently determined using equation (22).


## 4. 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. Therefore, vibration analysis is conducted on the 11th story. The physical and geometrical specifications of the studied building are given in Table 1. In the first scenario, the uncertainty was assumed as ΔMi=0.01Mi, ΔKi=0.01Ki, ΔCi=0.01Ci, and α=0. The 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.Table 1
The 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
C
12
438
×
10
3
Ns
/
mFigure 2
The resulting horizontal displacement for 11th story controlled and uncontrolled cases.The variations of the control forceu 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 signs in the control force relation. In fact, the discontinuity and undifferentiability of this function at point s=0 is responsible for the chattering phenomenon. The 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.Figure 3
Variation of control parameteru for Figure 2.To resolve the chattering problem and satisfy the continuity and Lipschitz condition, the termtanhs/ε was used as an approximation of the signs function for the function bx,tu, where the continuity and differentiability conditions are satisfied at point s=0. The diagram of horizontal displacements for different ε values is demonstrated in 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.Figure 4
(a) Controlled system behavior for different magnitudes ofε. (b) Detailed view of Figure 4(a).
(a)
(b)The variations of forceu 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 signs and tanhs/0.001 functions are consistent, which is due to the accurate approximation of the signs function by tanhs/0.001.Figure 5
(a) The variations of forceu corresponding to Figure 4 for different ε. (b) Detailed view of Figure 5(a).
(a)
(b)Moreover, the phase diagram for the 11th story is also demonstrated in Figure 6 for different cases. As shown, by increasing ε, the region of attraction is also extended. This region was almost similar for both signs and tanhs/0.001 functions.Figure 6
Phase portrait of horizontal displacement of 11th story considering functions signs and tanhs/ε. (a) signs. (b) ε=0.1. (c). ε=0.01. (d) ε=0.001.
(a)
(b)
(c)
(d)The effect ofhx,t on the dynamic behavior of the structure and the region of attraction is discussed in this part. The vibrational amplitude of the uncontrolled system for different hx,t values is shown in Figure 7. As demonstrated, the displacement amplitude is linearly increased by increasing hx,t.Figure 7
Vibrational behavior of 11th story in the passive mode and for different magnitudes of hx,t.Also, the attraction domain for the mentioned figure is presented in Figure8. As is clear from this figure, the domain of attraction set is extended with increasing hx,t.Figure 8
Phase portrait of horizontal displacement of 11th story for passive case and different values of hx,t. (a) hx,t=1. (b) hx,t=2. (c) hx,t=3.
(a)
(b)
(c)The horizontal displacement for the controlled case is demonstrated in Figure9 with respect to different hx,t values. As shown, by increasing hx,t, the vibrational amplitude is also increased, but its value is substantially smaller compared to the uncontrolled case. The 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 hx,t increases.Figure 9
The horizontal displacement of 11th story for the controlled case and for different values of hx,t.Figure 10
The force variations corresponding to Figure9.Moreover, the region of attraction corresponding to Figure9 is demonstrated in Figure 11. As shown, similar to the uncontrolled case, the region of attraction increases by increasing hx,t.Figure 11
Phase portrait of 11th story for the controlled case and different hx,t. (a) hx,t=1. (b) hx,t=2. (c) hx,t=3.
(a)
(b)
(c)The effects ofη and λ on the behavior of a controlled system is discussed in this section. The 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 horizontal displacement of the system to increase substantially.Figure 12
The variations in horizontal displacement for different values ofη and λ=1. (a) The variations in horizontal displacement for η=1, η=10, and η=100 and λ=1. (b) Detailed view of Figure 12(a). (c) The variations in horizontal displacement for η=1, η=1000, and η=100000 and λ=1. (d) Detailed view of Figure 12(c).
(a)
(b)
(c)
(d)Additionally, the region of attraction for anη value of 100,000 is shown in Figure 13. As shown, the region of attraction is considerably expanded at this η value.Figure 13
Phase portrait and basin of the attraction set for the active case withη=100000 and λ=1.The displacement amplitude of the system for differentλ values and a fixed η value of 1 is demonstrated in Figure 14. As shown, the displacement amplitude is substantially decreased by increasing lambda. The 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.Figure 14
The horizontal displacement amplitude of the active system for differentλ values and η=1.Figure 15
The phase portrait of 11th story for the active system considering η=1 and λ=5.This section discusses the controller robustness to system uncertainty. In the first scenario, only the system uncertainties were considered and the actuator uncertainties were neglected. The variations of horizontal displacement for differentΔ 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.Figure 16
The variations of horizontal displacement of 11th story for the active system and different values of Δ.This is due to the fact thatFx,t is also increased by increasing Δ. However, as expressed in equation (22), presence of Fx,t in this equation virtually increases the term ψx,t+η, and as shown in Figure 12, increasing η increases the controller robustness. The 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.Figure 17
The region of the attraction set for Figure16 and considering different values of Δ=5% and Δ=10%. (a) The region of the attraction set for Δ=5%. (b) The region of the attraction set for Δ=10%.
(a)
(b)The results for the case with actuator uncertainty andΔ=10% were extracted and presented in Figure 18. In this case, bx,t=1+αsint. The results for different α values are shown in this figure, which indicates the significant controller robustness in presence of actuator uncertainty.Figure 18
The horizontal displacement of 11th story for the active system considering uncertainty with Δ=1% and different values of α.The region of attraction in this case is depicted in Figure19. As shown, the extent of region of attraction is increased by increasing α.Figure 19
Related phase portrait to Figure18. (a) α=0.05. (b) α=0.1. (c) α=0.2.
(a)
(b)
(c)The results in presence of actuator and system uncertainties are shown in Figure20. 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. The region of attraction for this case is demonstrated in Figure 21. As shown, the extent of region of attraction is increased by increasing Δ.Figure 20
The horizontal displacement of 11th story for the active system considering uncertainty with α=0.2 and different values of Δ. (a) Δ=5%. (b) Δ=10%.Figure 21
Related phase portrait of Figure20.
(a)
(b)


## 5. Conclusion

This study has examined the dynamic behavior of an 11-story building equipped with an ATMD system. The stimulation force has been applied on the structure in the form of a Wiener process and an earthquake. The 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.The designed controller has been further developed to control the ATMD system. The dynamic behaviors of the structure in the active and passive modes have been simulated. The 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. The results show that reduction ofε and increase of η reduce the structural vibration amplitude.



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