# Neural Terminal Sliding-Mode Control for Uncertain Systems with Building Structure Vibration

Authors: Jianhui Wang; Wenli Chen; Zicong Chen; Yunchang Huang; Xing Huang; Wenqiang Wu; Biaotao He; Chunliang Zhang
Journal: Complexity
 (2019)

---

## Abstract

Building structures occasionally suffer from unpredictable earthquakes, which can cause severe damage and can threaten human lives. Thus, effective control methods are needed to protect against structural vibration in buildings, and rapid finite-time convergence is a key performance indicator for vibration control systems. Rapid convergence can be ensured by applying a sliding-mode control method. However, this method would result in chattering issue, which would weaken the feasibility of the physical implementation. To address this problem, a neural terminal sliding-mode control method is proposed. The proposed method is combined with a terminal sliding-mode and a hyperbolic tangent function to ensure that the considered system can be stabilized in finite-time without chattering. Finally, the control effect of the proposed method is compared with that of LQR (linear quadratic regulator) control and switching function control. The simulation results showed that the proposed method can ensure rapid convergence while the chattering issue can be eliminated effectively. And the structural building vibration can be suppressed effectively too.


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## Body

## 1. Introduction

Earthquakes are a natural phenomenon that can cause severe damage to building structures and can threaten human lives.Today, it remains difficult to predict when an earthquake may occur, and seismic waves are uncertain. Both factors pose challenges for building structure controllers during design. Thus, it is worth researching advanced antiseismic technology to protect building structures from vibration.During the past few decades, many modern control methods, including linear quadratic regulator (LQR) control [1–3], sliding-mode control [4–6], and semiactive control [7–10], have proven active and effective measures in the area of structural vibration. These methods can stabilize the system. However, building structure vibrations cannot be effectively suppressed in finite-time, which might cause severe damage to the building structure.To guarantee the rapid convergence of the system, sliding-mode control is applied for structural vibration control. According to the current deviation of the system and various derivative values, the vibration control method can change by jumping in the transient process. Therefore, the system can enter the sliding plane and obtain sliding-mode motion quickly, which ensures the rapid convergence of the system [11–14]. Meanwhile, the application of sliding-mode control is always accompanied by chattering, which may weaken the physical implementation rate.By applying the hyperbolic tangent function, direct switching of the control input can be avoided, and chattering can be eliminated. This approach allows the system to respond to the outside world in a timely manner and facilitates physical implementation. However, most control methods use Lyapunov stability, which belongs to the field of asymptotic stability research; in this field, the stable time goes toward infinity, which is meaningless for the control of structural vibrations [15–19].As research on the combination of Lyapunov stability theory and the theorem of homogeneity has developed, finite-time stable control methods have progressed. Finite-time stable control has been widely applied, for example, in the position control of synchronous permanent magnet motors and in high-precision guidance laws [20–23]. However, the finite-time stable control method is rarely applied to structural vibrations.However, uncertainties that could affect system stability exist in practical systems. The guarantee of finite-time stability when considering a system with uncertainty is an outstanding research area. Neural networks can be used to approximate any nonlinear function with generalization. Thus, a Radial Basis Function (RBF) neural network is adopted to address this problem; see [24–28].The neural terminal sliding-mode control method is proposed in this work to address an uncertain system with structural vibrations. An RBF neural network is combined with a terminal sliding-mode and the hyperbolic tangent function to handle an uncertain system with structural vibrations. The finite-time stabilization method is used to stabilize the building structural system in finite-time. As shown below, the main compensatory mechanism and contributions of the proposed schemes are summarized as follows.Most traditional control methods were researched without considering uncertainty and rapid convergence performance, which may result in serious damage to building structures [3–5]. Rapid convergence can be ensured by applying a sliding-mode control method; however, uncertainty affects rapid convergence, which poses a considerable challenge regarding controller design. Thus, a combination of a RBF neural network and terminal sliding-mode control is proposed to solve the uncertainty issues and ensure rapid convergence.In practice, chattering weakens the feasibility of physical implementation, which means that stable timeliness cannot be guaranteed and substantial structural damage may occur [8–10]; therefore, the feasibility of physical applications is important to determine. In this paper, finite-time control is combined with a hyperbolic tangent function to eliminate chattering and prevent structural building vibrations in finite time.The remainder of this paper is organized as follows. In Section2, models and a problem formulation are proposed. Then, the neural terminal sliding-mode control method is proposed to address the uncertain system with structural vibrations. In Section 3, the control effect of the proposed method is compared with the LQR control method and switching function control. The simulation results showed that the proposed method can effectively eliminate chattering, ensure rapid convergence, and effectively suppress structural building vibration. Finally, conclusions are given in Section 4.


## 2. Modeling and Analysis of Building Structures

### 2.1. Modeling Building Structures under Earthquake Conditions

A building structure optimized for interlaminar shear is used for modeling. Thus, the n-layer building structure can be simplified as a building structure with n degrees of freedom. Under earthquake conditions, the dynamic equation can be described as follows:(1)
M
D
¨
+
C
D
˙
+
K
D
=
F
x
¨
g
+
Q
Uwhere D
=
[
d
1
,
d
2
,
d
3
,
.
.
.
,
d
n
]
T is the displacement vector of the structure, d
i
(
i
=
1,2
,
.
.
.
,
n
) is the displacement of theith floor (relative to the ground) of the building structure, K is the stiffness matrix, M is the mass matrix, F
=
-
M
I is the transformation matrix of the ground seismic acceleration, where I
=
1
1
⋯
1
T is the unit column of n
×
1, C is the damping matrix, x
¨
g is the ground seismic acceleration, Q is a matrix denoting the location of the actuators, and U is the control input.Defining a state-space vector(2)
Y
=
y
1
,
y
2
Twhere y
1
=
[
y
11
,
y
12
,
…
,
y
1
n
]
T
=
D and y
2
=
[
y
21
,
y
22
,
.
.
.
,
y
2
n
]
T
=
D
˙. Equation (1) is rewritten as follows:(3)
Y
˙
=
A
r
Y
+
W
r
x
¨
g
+
B
r
Uwhere A
r
=
0
I
n
×
n
-
M
-
1
K
-
M
-
1
C
,
W
r
=
0
M
-
1
F
,
B
r
=
0
M
-
1
Q.According to the rank criterion, the controllability of the building structure under earthquake conditions can be certified. Thus, structural vibration can be restrained by designing a control variable setting as follows:(4)
U
=
Q
-
1
C
D
˙
+
K
D
+
M
Vwhere V
=
[
u
1
,
u
2
,
.
.
.
,
u
n
]
T is a variable derived from (2) and (3) as follows:(5)
Y
˙
=
0
I
n
×
n
0
0
Y
+
0
I
n
×
n
V
-
M
-
1
F
x
¨
gSettingf
=
-
x
¨
g, the system is composed of a mutually independent subsystem, which is described as follows:(6)
y
˙
11
=
y
21
y
˙
21
=
u
1
-
f
y
˙
12
=
y
22
y
˙
22
=
u
2
-
f
⋯
y
˙
1
n
=
y
2
n
y
˙
2
n
=
u
n
-
fThe seismic wave is uncertain. Thus, further analysis is difficult. Neural finite-time stable control is used to handle this problem.


### 2.2. Approximation System Research

Analyzing the system above, seismic wavex
¨
g is assumed to be known. The RBF neural network is chosen to approximate x
¨
g because it can effectively approximate the unknown variable [29]. The network algorithm is designed as follows:(7)
h
j
=
e
x
p
-
x
-
c
j
2
2
b
j
2
(8)
f
=
W
∗
T
h
x
+
εwhere x is an input of the network, j is thejth joint of the hidden-layer of the network, h
=
h
1
,
⋯
,
h
j
,
⋯
h
N
T
,
N
≥
0, and W
∗ is the desirable permission of the network, ε is the approximate error of the network, and ε
≤
ε
N. According to the above analysis, the model can be described as follows: (9)
f
y
=
W
∗
T
h
y
f
^
y
=
W
^
T
h
yBecause functionf is unknown, to offset the effect of function f on the system, the RBF neural network is used to approximate the function, which allows (4) to be rewritten as follows:(10)
U
=
Q
-
1
C
D
˙
+
K
D
˙
+
M
V
+
F
f
^


### 2.3. Terminal Sliding-Mode Controller Design

u
i is theith controller, and y
1
i
,
y
2
i denotes the displacement of theith floor (relative to the ground) of the building structure. Thus, the structural building model with uncertain earthquakes can be described as follows: (11)
y
˙
1
i
=
y
2
i
y
˙
2
i
=
-
f
~
+
u
iRemark 1.
f is the actual seismic wave, and f
^ is the estimated value approximated by RBF; therefore, f
~ is the error of the actual value and the estimated value. Thus, f
~
=
f
-
f
^.
The sliding-mode function is designed as follows:(12)
s
=
P
E
-
Zwhere e
=
y
1
i
-
y
1
d
i
,
e
˙
=
y
2
i
-
y
2
d
i, P
=
p
,
1, E
=
x
-
x
d
=
e
e
˙
T, and Z
=
z
t
z
˙
t
T. Thus, we can obtain(13)
s
=
p
e
-
p
z
t
+
e
˙
-
z
˙
twhere p must satisfy the Hurwitz condition (p
>
0).Remark 2.
ForY of the system to track desired state y
1
d
i
,
y
2
d
i within appointed time T, [30] proposes a method of constructing terminal function Z
t. The method is designed as follows.
To achieve global robustness,E
0
=
Z
0, i.e., e
0
=
z
0
,
e
˙
0
=
z
˙
0. To achieve the convergence of appointed time T, z
t
=
0
,
z
˙
t
=
0
,
z
¨
t
=
0. Therefore, the terminal function can be described as follows: (14)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
+
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
-
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
0
≤
t
≤
T
0
,
t
≥
Twhere b
i
j
i
,
j
=
0,1
,
2 is a coefficient that can be obtained by solving the equation.
The derivative of (13) can be obtained as follows:(15)
s
˙
=
p
e
˙
-
p
z
˙
t
+
e
¨
-
z
¨
t
Ife
¨
=
y
˙
2
i
-
y
˙
2
d
=
-
f
~
+
u
-
y
˙
2
d, then(16)
s
˙
=
p
e
˙
-
p
z
˙
t
-
f
~
+
u
-
y
˙
2
d
-
z
¨
tRemark 3.
The hyperbolic tangent function is as follows:(17)
tanh
⁡
x
ε
=
e
x
/
ε
-
e
-
x
/
ε
e
x
/
ε
+
e
-
x
/
εwhere ε
>
0 and the speed of the inflection point of the hyperbolic tangent smooth function depends on the value of ε.
The controller is designed as follows:(18)
u
=
-
p
e
˙
+
p
z
˙
t
+
y
˙
2
d
+
z
¨
t
-
η
tanh
⁡
s
εwhere η is a positive design parameter.
Derived from (16) and (18):(19)
s
˙
=
-
f
~
-
η
tanh
⁡
s
ε
A Lyapunov function is defined as follows:(20)
V
=
1
2
s
2
+
1
2
λ
W
~
T
W
~
Then, the derivative of (20) can be obtained as follows:(21)
V
˙
=
s
s
˙
-
1
λ
W
~
T
W
^
˙


### 2.4. Stability Analysis

Derived from (21):(22)
V
˙
=
s
-
f
~
-
η
tanh
⁡
s
ε
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
s
f
~
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
s
W
~
h
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
W
~
s
h
+
W
^
˙
λLemma 4 (see [31]).
For any givenx, ε
>
0 and there is an inequality: (23)
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
≥
0
The explanation for Lemma4 is as follows.
According to the definition of the hyperbolic tangent function, we have the following:(24)
x
tanh
⁡
x
ε
=
x
e
x
/
ε
-
e
-
x
/
ε
e
x
/
ε
+
e
-
x
/
ε
=
1
e
2
x
/
ε
-
1
x
e
2
x
/
ε
-
1
Thus, it can be said that(25)
e
2
x
/
ε
-
1
≥
0
,
x
≥
0
e
2
x
/
ε
-
1
<
0
,
x
<
0
Hence,(26)
x
e
2
x
/
ε
-
1
≥
0
Then,(27)
x
tanh
⁡
x
ε
=
1
e
2
x
/
ε
+
1
x
e
2
x
/
ε
-
1
≥
0
Thus, the consequence can be achieved such that(28)
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
≥
0which completes the proof.Settings
h
=
W
^
˙
/
λ, so W
^
˙
=
-
λ
s
h:(29)
V
˙
=
-
s
η
tanh
⁡
s
ε
≤
0From the above discussion, system stability can be proven.From the expression of functionZ
t, when t
=
0, z
0
=
e
0 and z
˙
0
=
e
˙
0 (i.e., s
0
=
P
E
0
-
Z
0
=
0). Hence, the initial system state is already on the sliding surface, which eliminates the arrival phase of the sliding film and ensures the global robustness and stability of the closed-loop system.Remark 5.
Because the system has global robustness, i.e.,E
t
=
Z
t, design function Z
T
=
0 can be used to ensure that E
T
=
0 so that the tracking error converges to zero within finite time T.
To achieve convergence for a specified timeT, it is necessary to ensure that when t
=
T, z
T
=
0, z
˙
T
=
0 and z
¨
T
=
0, where z
¨
T
=
0 meets the requirements of (14). Therefore, when 0
≤
t
≤
T, function z
t and its derivative can be written as follows:(30)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
-
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
-
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
The derivative of (30) is as follows:(31)
z
˙
t
=
e
˙
0
+
e
¨
0
t
+
3
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
2
+
4
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
3
+
5
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
4
The derivative of (31) is as follows:(32)
z
¨
t
=
e
¨
0
+
6
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
+
12
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
2
+
20
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
3
Whent
=
T, we can obtain z
T
=
0. Thus,(33)
z
T
=
e
0
+
e
˙
0
T
+
1
2
e
¨
0
T
2
+
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
+
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
+
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
=
1
+
b
00
+
b
10
+
b
20
e
0
+
T
1
+
b
01
+
b
11
+
b
21
e
˙
0
+
T
2
1
2
+
b
02
+
b
12
+
b
22
e
¨
0
=
0
The necessary conditions for the establishment ofz
T
=
0 are as follows: (34)
1
+
b
00
+
b
10
+
b
20
=
0
1
+
b
01
+
b
11
+
b
21
=
0
0.5
+
b
02
+
b
12
+
b
22
=
0
Similarly, whent
=
T, the necessary condition for z
˙
T
=
0
,
z
¨
T
=
0 is as follows:(35)
3
b
00
+
4
b
10
+
5
b
20
=
0
1
+
3
b
01
+
4
b
11
+
5
b
21
=
0
1
+
3
b
02
+
4
b
12
+
5
b
22
=
0
(36)
6
b
00
+
12
b
10
+
20
b
20
=
0
6
b
01
+
12
b
11
+
20
b
21
=
0
1
+
6
b
02
+
12
b
12
+
20
b
22
=
0
Three ternary equations can be sorted by the above equations:(37)
b
00
+
b
10
+
b
20
=
-
1
3
b
00
+
4
b
10
+
5
b
20
=
0
6
b
00
+
12
b
10
+
20
b
20
=
0
(38)
b
01
+
b
11
+
b
21
=
-
1
3
b
01
+
4
b
11
+
5
b
21
=
-
1
6
b
01
+
12
b
11
+
20
b
21
=
0
(39)
b
02
+
b
12
+
b
22
=
-
0.5
3
b
02
+
4
b
12
+
5
b
22
=
-
1
6
b
02
+
12
b
12
+
20
b
22
=
-
1
According toB
x
=
G, to express the ternary equations, the above three equations can be written as the following three forms:(40)
B
1
x
1
=
G
1
,
B
1
=
1
1
1
3
4
5
6
12
20
,
G
1
=
-
1
0
0
(41)
B
2
x
2
=
G
2
,
B
2
=
1
1
1
3
4
5
6
12
20
,
G
2
=
-
1
-
1
0
(42)
B
3
x
3
=
G
3
,
B
3
=
1
1
1
3
4
5
6
12
20
,
G
3
=
-
0.5
-
1
-
1
The solution to the above equations is as follows:(43)
b
00
=
-
10
b
10
=
15
b
20
=
-
6
b
01
=
-
6
b
11
=
8
b
21
=
-
3
b
02
=
-
1.5
b
12
=
1.5
b
22
=
-
0.5
Moreover, through further solutions:(44)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
10
T
3
e
0
+
6
T
2
e
˙
0
+
3
2
T
e
¨
0
t
3
+
15
T
4
e
0
+
8
T
3
e
˙
0
+
3
2
T
2
e
¨
0
t
4
-
6
T
5
e
0
+
3
T
4
e
˙
0
+
1
2
T
3
e
¨
0
t
5
0
≤
t
≤
T
0
,
t
>
Twhere L
∞ is defined as a finite function set.Remark 6.
From (30), we know V
˙
≤
0, which means that Lyapunov function V
t must be nonincreasing with respect to the time variable. Because the initial value V
0 is finite, it follows that V
t
∈
L
∞. Thus, V
t is bounded, which allows us to obtain the following:(45)
l
i
m
t
→
∞
V
t
=
0
Derived from (21), it is determined that s and W
~ are bounded.
Becauses
=
P
E
-
Z and the closed-loop system enters the sliding stage at the initial moment, we can determine that s
0
=
0, i.e., E
t
≡
Z
t. Because s is bounded and P is a constant, Z
t can converge within finite time T, and E
t can converge within finite time T. Therefore, according to the preset terminal function Z
t, the output error can be guaranteed to converge to zero within a finite time to ensure the system can be stabilized within a finite time.


## 2.1. Modeling Building Structures under Earthquake Conditions

A building structure optimized for interlaminar shear is used for modeling. Thus, the n-layer building structure can be simplified as a building structure with n degrees of freedom. Under earthquake conditions, the dynamic equation can be described as follows:(1)
M
D
¨
+
C
D
˙
+
K
D
=
F
x
¨
g
+
Q
Uwhere D
=
[
d
1
,
d
2
,
d
3
,
.
.
.
,
d
n
]
T is the displacement vector of the structure, d
i
(
i
=
1,2
,
.
.
.
,
n
) is the displacement of theith floor (relative to the ground) of the building structure, K is the stiffness matrix, M is the mass matrix, F
=
-
M
I is the transformation matrix of the ground seismic acceleration, where I
=
1
1
⋯
1
T is the unit column of n
×
1, C is the damping matrix, x
¨
g is the ground seismic acceleration, Q is a matrix denoting the location of the actuators, and U is the control input.Defining a state-space vector(2)
Y
=
y
1
,
y
2
Twhere y
1
=
[
y
11
,
y
12
,
…
,
y
1
n
]
T
=
D and y
2
=
[
y
21
,
y
22
,
.
.
.
,
y
2
n
]
T
=
D
˙. Equation (1) is rewritten as follows:(3)
Y
˙
=
A
r
Y
+
W
r
x
¨
g
+
B
r
Uwhere A
r
=
0
I
n
×
n
-
M
-
1
K
-
M
-
1
C
,
W
r
=
0
M
-
1
F
,
B
r
=
0
M
-
1
Q.According to the rank criterion, the controllability of the building structure under earthquake conditions can be certified. Thus, structural vibration can be restrained by designing a control variable setting as follows:(4)
U
=
Q
-
1
C
D
˙
+
K
D
+
M
Vwhere V
=
[
u
1
,
u
2
,
.
.
.
,
u
n
]
T is a variable derived from (2) and (3) as follows:(5)
Y
˙
=
0
I
n
×
n
0
0
Y
+
0
I
n
×
n
V
-
M
-
1
F
x
¨
gSettingf
=
-
x
¨
g, the system is composed of a mutually independent subsystem, which is described as follows:(6)
y
˙
11
=
y
21
y
˙
21
=
u
1
-
f
y
˙
12
=
y
22
y
˙
22
=
u
2
-
f
⋯
y
˙
1
n
=
y
2
n
y
˙
2
n
=
u
n
-
fThe seismic wave is uncertain. Thus, further analysis is difficult. Neural finite-time stable control is used to handle this problem.


## 2.2. Approximation System Research

Analyzing the system above, seismic wavex
¨
g is assumed to be known. The RBF neural network is chosen to approximate x
¨
g because it can effectively approximate the unknown variable [29]. The network algorithm is designed as follows:(7)
h
j
=
e
x
p
-
x
-
c
j
2
2
b
j
2
(8)
f
=
W
∗
T
h
x
+
εwhere x is an input of the network, j is thejth joint of the hidden-layer of the network, h
=
h
1
,
⋯
,
h
j
,
⋯
h
N
T
,
N
≥
0, and W
∗ is the desirable permission of the network, ε is the approximate error of the network, and ε
≤
ε
N. According to the above analysis, the model can be described as follows: (9)
f
y
=
W
∗
T
h
y
f
^
y
=
W
^
T
h
yBecause functionf is unknown, to offset the effect of function f on the system, the RBF neural network is used to approximate the function, which allows (4) to be rewritten as follows:(10)
U
=
Q
-
1
C
D
˙
+
K
D
˙
+
M
V
+
F
f
^


## 2.3. Terminal Sliding-Mode Controller Design

u
i is theith controller, and y
1
i
,
y
2
i denotes the displacement of theith floor (relative to the ground) of the building structure. Thus, the structural building model with uncertain earthquakes can be described as follows: (11)
y
˙
1
i
=
y
2
i
y
˙
2
i
=
-
f
~
+
u
iRemark 1.
f is the actual seismic wave, and f
^ is the estimated value approximated by RBF; therefore, f
~ is the error of the actual value and the estimated value. Thus, f
~
=
f
-
f
^.
The sliding-mode function is designed as follows:(12)
s
=
P
E
-
Zwhere e
=
y
1
i
-
y
1
d
i
,
e
˙
=
y
2
i
-
y
2
d
i, P
=
p
,
1, E
=
x
-
x
d
=
e
e
˙
T, and Z
=
z
t
z
˙
t
T. Thus, we can obtain(13)
s
=
p
e
-
p
z
t
+
e
˙
-
z
˙
twhere p must satisfy the Hurwitz condition (p
>
0).Remark 2.
ForY of the system to track desired state y
1
d
i
,
y
2
d
i within appointed time T, [30] proposes a method of constructing terminal function Z
t. The method is designed as follows.
To achieve global robustness,E
0
=
Z
0, i.e., e
0
=
z
0
,
e
˙
0
=
z
˙
0. To achieve the convergence of appointed time T, z
t
=
0
,
z
˙
t
=
0
,
z
¨
t
=
0. Therefore, the terminal function can be described as follows: (14)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
+
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
-
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
0
≤
t
≤
T
0
,
t
≥
Twhere b
i
j
i
,
j
=
0,1
,
2 is a coefficient that can be obtained by solving the equation.
The derivative of (13) can be obtained as follows:(15)
s
˙
=
p
e
˙
-
p
z
˙
t
+
e
¨
-
z
¨
t
Ife
¨
=
y
˙
2
i
-
y
˙
2
d
=
-
f
~
+
u
-
y
˙
2
d, then(16)
s
˙
=
p
e
˙
-
p
z
˙
t
-
f
~
+
u
-
y
˙
2
d
-
z
¨
tRemark 3.
The hyperbolic tangent function is as follows:(17)
tanh
⁡
x
ε
=
e
x
/
ε
-
e
-
x
/
ε
e
x
/
ε
+
e
-
x
/
εwhere ε
>
0 and the speed of the inflection point of the hyperbolic tangent smooth function depends on the value of ε.
The controller is designed as follows:(18)
u
=
-
p
e
˙
+
p
z
˙
t
+
y
˙
2
d
+
z
¨
t
-
η
tanh
⁡
s
εwhere η is a positive design parameter.
Derived from (16) and (18):(19)
s
˙
=
-
f
~
-
η
tanh
⁡
s
ε
A Lyapunov function is defined as follows:(20)
V
=
1
2
s
2
+
1
2
λ
W
~
T
W
~
Then, the derivative of (20) can be obtained as follows:(21)
V
˙
=
s
s
˙
-
1
λ
W
~
T
W
^
˙


## 2.4. Stability Analysis

Derived from (21):(22)
V
˙
=
s
-
f
~
-
η
tanh
⁡
s
ε
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
s
f
~
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
s
W
~
h
-
1
λ
W
~
T
W
^
˙
=
-
s
η
tanh
⁡
s
ε
-
W
~
s
h
+
W
^
˙
λLemma 4 (see [31]).
For any givenx, ε
>
0 and there is an inequality: (23)
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
≥
0
The explanation for Lemma4 is as follows.
According to the definition of the hyperbolic tangent function, we have the following:(24)
x
tanh
⁡
x
ε
=
x
e
x
/
ε
-
e
-
x
/
ε
e
x
/
ε
+
e
-
x
/
ε
=
1
e
2
x
/
ε
-
1
x
e
2
x
/
ε
-
1
Thus, it can be said that(25)
e
2
x
/
ε
-
1
≥
0
,
x
≥
0
e
2
x
/
ε
-
1
<
0
,
x
<
0
Hence,(26)
x
e
2
x
/
ε
-
1
≥
0
Then,(27)
x
tanh
⁡
x
ε
=
1
e
2
x
/
ε
+
1
x
e
2
x
/
ε
-
1
≥
0
Thus, the consequence can be achieved such that(28)
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
=
x
tanh
⁡
x
ε
≥
0which completes the proof.Settings
h
=
W
^
˙
/
λ, so W
^
˙
=
-
λ
s
h:(29)
V
˙
=
-
s
η
tanh
⁡
s
ε
≤
0From the above discussion, system stability can be proven.From the expression of functionZ
t, when t
=
0, z
0
=
e
0 and z
˙
0
=
e
˙
0 (i.e., s
0
=
P
E
0
-
Z
0
=
0). Hence, the initial system state is already on the sliding surface, which eliminates the arrival phase of the sliding film and ensures the global robustness and stability of the closed-loop system.Remark 5.
Because the system has global robustness, i.e.,E
t
=
Z
t, design function Z
T
=
0 can be used to ensure that E
T
=
0 so that the tracking error converges to zero within finite time T.
To achieve convergence for a specified timeT, it is necessary to ensure that when t
=
T, z
T
=
0, z
˙
T
=
0 and z
¨
T
=
0, where z
¨
T
=
0 meets the requirements of (14). Therefore, when 0
≤
t
≤
T, function z
t and its derivative can be written as follows:(30)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
-
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
-
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
The derivative of (30) is as follows:(31)
z
˙
t
=
e
˙
0
+
e
¨
0
t
+
3
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
2
+
4
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
3
+
5
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
4
The derivative of (31) is as follows:(32)
z
¨
t
=
e
¨
0
+
6
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
+
12
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
2
+
20
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
3
Whent
=
T, we can obtain z
T
=
0. Thus,(33)
z
T
=
e
0
+
e
˙
0
T
+
1
2
e
¨
0
T
2
+
b
00
T
3
e
0
+
b
01
T
2
e
˙
0
+
b
02
T
e
¨
0
t
3
+
b
10
T
4
e
0
+
b
11
T
3
e
˙
0
+
b
12
T
2
e
¨
0
t
4
+
b
20
T
5
e
0
+
b
21
T
4
e
˙
0
+
b
22
T
3
e
¨
0
t
5
=
1
+
b
00
+
b
10
+
b
20
e
0
+
T
1
+
b
01
+
b
11
+
b
21
e
˙
0
+
T
2
1
2
+
b
02
+
b
12
+
b
22
e
¨
0
=
0
The necessary conditions for the establishment ofz
T
=
0 are as follows: (34)
1
+
b
00
+
b
10
+
b
20
=
0
1
+
b
01
+
b
11
+
b
21
=
0
0.5
+
b
02
+
b
12
+
b
22
=
0
Similarly, whent
=
T, the necessary condition for z
˙
T
=
0
,
z
¨
T
=
0 is as follows:(35)
3
b
00
+
4
b
10
+
5
b
20
=
0
1
+
3
b
01
+
4
b
11
+
5
b
21
=
0
1
+
3
b
02
+
4
b
12
+
5
b
22
=
0
(36)
6
b
00
+
12
b
10
+
20
b
20
=
0
6
b
01
+
12
b
11
+
20
b
21
=
0
1
+
6
b
02
+
12
b
12
+
20
b
22
=
0
Three ternary equations can be sorted by the above equations:(37)
b
00
+
b
10
+
b
20
=
-
1
3
b
00
+
4
b
10
+
5
b
20
=
0
6
b
00
+
12
b
10
+
20
b
20
=
0
(38)
b
01
+
b
11
+
b
21
=
-
1
3
b
01
+
4
b
11
+
5
b
21
=
-
1
6
b
01
+
12
b
11
+
20
b
21
=
0
(39)
b
02
+
b
12
+
b
22
=
-
0.5
3
b
02
+
4
b
12
+
5
b
22
=
-
1
6
b
02
+
12
b
12
+
20
b
22
=
-
1
According toB
x
=
G, to express the ternary equations, the above three equations can be written as the following three forms:(40)
B
1
x
1
=
G
1
,
B
1
=
1
1
1
3
4
5
6
12
20
,
G
1
=
-
1
0
0
(41)
B
2
x
2
=
G
2
,
B
2
=
1
1
1
3
4
5
6
12
20
,
G
2
=
-
1
-
1
0
(42)
B
3
x
3
=
G
3
,
B
3
=
1
1
1
3
4
5
6
12
20
,
G
3
=
-
0.5
-
1
-
1
The solution to the above equations is as follows:(43)
b
00
=
-
10
b
10
=
15
b
20
=
-
6
b
01
=
-
6
b
11
=
8
b
21
=
-
3
b
02
=
-
1.5
b
12
=
1.5
b
22
=
-
0.5
Moreover, through further solutions:(44)
z
t
=
e
0
+
e
˙
0
t
+
1
2
e
¨
0
t
2
-
10
T
3
e
0
+
6
T
2
e
˙
0
+
3
2
T
e
¨
0
t
3
+
15
T
4
e
0
+
8
T
3
e
˙
0
+
3
2
T
2
e
¨
0
t
4
-
6
T
5
e
0
+
3
T
4
e
˙
0
+
1
2
T
3
e
¨
0
t
5
0
≤
t
≤
T
0
,
t
>
Twhere L
∞ is defined as a finite function set.Remark 6.
From (30), we know V
˙
≤
0, which means that Lyapunov function V
t must be nonincreasing with respect to the time variable. Because the initial value V
0 is finite, it follows that V
t
∈
L
∞. Thus, V
t is bounded, which allows us to obtain the following:(45)
l
i
m
t
→
∞
V
t
=
0
Derived from (21), it is determined that s and W
~ are bounded.
Becauses
=
P
E
-
Z and the closed-loop system enters the sliding stage at the initial moment, we can determine that s
0
=
0, i.e., E
t
≡
Z
t. Because s is bounded and P is a constant, Z
t can converge within finite time T, and E
t can converge within finite time T. Therefore, according to the preset terminal function Z
t, the output error can be guaranteed to converge to zero within a finite time to ensure the system can be stabilized within a finite time.


## 3. Control Method Analysis

In this section, three control methods were simulated: LQR control, switching function control, and neural terminal sliding-mode finite-time stable control. Based on a three-layer building structure, the system is subjected to an El earthquake wave. The maximum earthquake acceleration isa
m
a
x
=
3.417
m
/
s
2. The parameters of the proposed control are set asp
=
1, T
=
1, and ε
=
0.02. The mass matrix, damping matrix, stiffness matrix, and the position of the examples are set as follows:(46)
M
=
1
0
4
×
3.456
0
0
0
3.456
0
0
0
3.456
k
g
C
=
1
0
5
×
1.745
-
0.512
-
0.111
-
0.512
1.634
-
0.623
-
0.111
-
0.623
1.122
N
s
/
m
K
=
1
0
6
×
2.4
-
1.2
0
-
1.2
2.4
-
1.2
0
-
1.2
1.2
N
/
m
Q
=
1
0
0
0
1
0
0
0
1Example 1.
No control, LQR control, and proposed control for the three-layer building structure are simulated. The contrast simulation curves of displacement, velocity, acceleration response, and control force are shown in Figures1–4.
According to Table1, the LQR control compared with no control, the maximum displacement of the first, second, and third floors decreased by 59.9%, 67.6%, and 71.9%, respectively. And the maximum velocity decreased by 34.3%, 46%, and 46.4%, respectively. Meanwhile, the proposed control method compared with no control, the maximum displacement of the first, second, and third floors decreased by 97.1%, 98.4%, and 98.7%, respectively. And the maximum velocity decreased by 92.6%, 94.8%, and 94.6%, and maximum acceleration decreased by 64.7%, 56.5%, and 54.5%, respectively.Table 1
Maximum displacement, velocity, and acceleration of in Example1.
Control StrategyParameter
Proposed Control
LQRControl
NoControl
Displacement(mm)
First floor
2
27.7
69.2
Second floor
2
41.1
127
Third floor
2
44.6
159
Velocity(m/s)
First floor
0.023
0.206
0.314
Second floor
0.023
0.239
0.443
Third floor
0.023
0.231
0.431
Acceleration (m/s2)
First floor
1.7019
4.92
4.82
Second floor
1.7019
4.00
3.91
Third floor
1.7019
3.71
3.74Figure 1
Displacement response of Example1.Figure 2
Velocity response of Example1.Figure 3
Acceleration response of Example1.Figure 4
Controlling force of Example1.Example 2.
In Example2, two control methods, i.e., switch control and the proposed control, are simulated for the three-layer building structure. Contrast simulation curves of displacement, velocity, acceleration response, and control force are shown in Figures 5–8, respectively.Figure 5
Displacement response of Example2.Figure 6
Velocity response of Example2.Figure 7
Acceleration response of Example2.Figure 8
Controlling force of Example2.According to Table2, the switch control compared with no control, the maximum displacement of the first, second, and third floors decreased by 78%, 88%, and 90.4%, respectively. The maximum velocity decreased by 84.3%, 88.9%, and 88.6%, respectively, and maximum acceleration decreased by 77.9%, 72.7%, and 71.5%, respectively. Meanwhile, the proposed method compared with no control, the maximum displacement of the first, second, and third floors decreased by 97.1%, 98.4%, and 98.7%, respectively. The maximum velocity decreased by 26.1%, 47.6%, and 46.1%, respectively. And maximum acceleration decreased by 64.7%, 56.5%, and 54.5%, respectively.Table 2
Maximum displacement, velocity, and acceleration of Example2.
Control StrategyParameter
Proposed Control
Switch Control
NoControl
Displacement(mm)
First floor
2
15.2
69.2
Second floor
2
15.2
127
Third floor
2
15.2
159
Velocity(m/s)
First floor
0.0232
0.049
0.314
Second floor
0.0232
0.049
0.443
Third floor
0.0232
0.049
0.431
Acceleration (m/s2)
First floor
1.7019
1.065
4.82
Second floor
1.7019
1.065
3.91
Third floor
1.7019
1.065
3.74Meanwhile, Figure8 clearly shows that strong chattering exists when using the switching function control force, which cannot be removed with these methods.From Figure9, we could know that El Center Acceleration can be approximated by the adopted RBF neural networks effectively. Then, according to the two examples above, both switching function control and the proposed control methods effectively restrain structural vibration. Then, considering practical applications, strong chattering can be effectively resolved by the proposed control method. And the displacements of three-layer buildings can be bounded within a small range. Furthermore, displacement, velocity, and acceleration could tend toward a small range of vibration and stabilize. Therefore, a neural terminal sliding-mode finite-time stable control algorithm can better protect the building structure from earthquake damage and effectively reduce the buffering problem in sliding-mode control.Figure 9
El Center Acceleration and its estimate.


## 4. Conclusion

In this paper, neural terminal sliding-mode control for uncertain systems subject to structural building vibration is proposed. Uncertainties are typically challenging when striving to ensure rapid convergence for controller design, which has been solved by combining an RBF neural network with terminal sliding-mode control. The traditional sliding-mode control inevitably causes chattering problems, which result in considerable challenges regarding physical implementation. Furthermore, it is important to ensure sliding-mode control without chattering issues. Thus, a hyperbolic tangent function is used to design the controller, allowing the stability of the system to be demonstrated. Finally, the control effect of the neural terminal sliding-mode finite-time stable control method is compared with the LQR control and switching function control methods. The feasibility and effectiveness of the proposed methods are verified. However, the input constraints of actual system require further research.



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*Source: 1507051-2019-04-10.xml*