# Robust Adaptive Neural Backstepping Control for a Class of Nonlinear Systems with Dynamic Uncertainties

Authors: Hongyan Yang; Huanqing Wang; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper is concerned with adaptive neural control of nonlinear strict-feedback systems with nonlinear uncertainties, unmodeled dynamics, and dynamic disturbances. To overcome the difficulty from the unmodeled dynamics, a dynamic signal is introduced. Radical basis function (RBF) neural networks are employed to model the packaged unknown nonlinearities, and then an adaptive neural control approach is developed by using backstepping technique. The proposed controller guarantees semiglobal boundedness of all the signals in the closed-loop systems. A simulation example is given to show the effectiveness of the presented control scheme.


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## Body

## 1. Introduction

In the past decades, much attention has been paid on the control design of complex nonlinear systems [1–11]. Many remarkable control approaches in this area have been developed, including adaptive backstepping technique [1–3], fault tolerant control [12–17], and and fuzzy control [18–29]. In particular, adaptive backstepping approach has played an important role in the control of strict-feedback nonlinear systems. Generally, adaptive backstepping provides a systematic control approach to solve the tracking or regulation control problems of uncertain nonlinear systems, in which the classic adaptive control is applied to deal with the unknown parameter and backstepping technique is used to construct controller. The main feature of adaptive backstepping control is that it can handle the control problems of nonlinear systems without the requirement of matching condition. Adaptive backstepping technique was provided in [1] to obtain global stability and asymptotic tracking performance for parametric strict-feedback systems with overparameterization, and the overparameterization was overcome by applying the tuning functions in [2]. Then, a backstepping-based design was extensively utilized to control different types of nonlinear systems [30–35]. All the above control methods, however, assume that the nonlinear functions of the control systems are either known or bounded by known functions multiplying uncertain parameters. This restriction makes the aforementioned methods inapplicable to the control of the systems with unknown continuous nonlinear functions.On the other hand, approximation-based adaptive neural (or fuzzy) backstepping control has received increasing attention in recent years. In general, approximation-based adaptive backstepping technique is an effective control approach for handling the control problem of highly uncertain complex nonlinear strict-feedback systems, in which neural networks or fuzzy systems are utilized to model the unknown nonlinear functions. So far, there exist some elegant results; see, for example, [36–54] and the references therein. By applying adaptive neural control together with backstepping, in [36–43], many control approaches are developed for single-input and single-output (SISO) nonlinear systems or multi-input and multioutput (MIMO) nonlinear systems. Alternatively, several fuzzy adaptive control strategies [19, 44–55] were developed to deal with the control problem of uncertain nonlinear systems with strict-feedback form. However, the above adaptive neural or fuzzy backstepping control approaches required the controlled strict-feedback nonlinear systems to be free of the unmodeled dynamics and dynamic disturbances. As stated in [56, 57], the unmodeled dynamics and dynamic disturbances often appear in practical systems [58, 59] due to the measurement noise, modeling errors, external disturbances, modeling simplifications, or changes with time variations, and they are also the resources of the instability of the considered systems. Therefore, some researchers have concentrated on the problem of control design for nonlinear systems with unmodeled dynamics and dynamic disturbances. In [56, 57], the problem of adaptive backstepping control was investigated for a class of nonlinear systems with dynamics uncertainties, in which the nonlinear functions were assumed to be linear combinations of the known functions with unknown parameters. Furthermore, by using the approximation properties of fuzzy logic systems, Tong et al. [58, 59] developed several fuzzy adaptive control approaches for nonlinear systems in strict-feedback form, where the number of adaptation laws depends on the number of fuzzy base functions. The more fuzzy rules are applied to improve approximation accuracy, the more adaptive parameters will be needed, and, in this way, the online learning time may be very large.Inspired by previous works, this paper focuses on the problem of adaptive neural control for nonlinear strict-feedback systems with unmodeled dynamics and dynamic disturbances. During the controller design, a dynamic signal is introduced to handle the unmodeled dynamics and RBF neural networks are used to approximate the unknown nonlinearities, and then an adaptive neural control scheme is systematically derived via backstepping. The proposed controller guarantees that all the signals in the closed-loop systems are semiglobally uniformly ultimately bounded (SGUUB) in the sense of mean square. Compared with the control approaches [58, 59], the main contributions of this paper are summarized as follows: (
1
) the strict limitation to the dynamic disturbances is relaxed, which can refer to Remark 3; (
2
) by estimating the norm of the weight vector of neural networks basis functions, the number of adaptive parameters is not more than the order of the considered nonlinear system. As a result, the burdensome computation is significantly alleviated, which makes our control design more suitable for the practical applications.The remainder of the paper is organized as follows. Section2 begins with the problem formulation and some preliminaries. A backstepping-based adaptive control scheme is design in Section 3. In Section 4, a numerical example is given. Finally, the conclusion of this paper is shown in Section 5.


## 2. Problem Formulation and Preliminaries

In this paper, we consider a class of nonlinear strict-feedback systems described by(1)
z
˙
=
q
(
z
,
x
)
,
x
˙
i
=
g
i
(
x
¯
i
)
x
i
+
1
+
f
i
(
x
¯
i
)
+
Δ
i
(
x
,
z
,
t
)
,
i
=
1
,
…
,
n
-
1
,
x
˙
n
=
g
n
(
x
¯
n
)
u
+
f
n
(
x
¯
n
)
+
Δ
n
(
x
,
z
,
t
)
,
where x
=
[
x
1
,
x
2
,
…
,
x
n
]
T
∈
R
n, u
∈
R, and y
∈
R are the system state, control input, and system output, respectively, x
¯
i
=
[
x
1
,
x
2
,
…
,
x
i
]
T
∈
R
i, f
i
(
·
), and g
i
(
·
) are unknown smooth functions, and Δ
i (i
=
1,2
,
…
,
n) are nonlinear dynamic disturbances. The z-dynamics in (1) denotes the unmodeled dynamics.Remark 1.
It is worth noting that many practical systems such as the electromechanical system [59] transformable into (1) have been investigated extensively during the last decades from both theoretical and practical viewpoints; see, for example, [56–59].In order to facilitate the control design later, the below assumptions are imposed on the system (1).Assumption 2.
For the dynamic disturbancesΔ
i (i
=
1,2
,
…
,
n) in (1), there exist unknown nonnegative smooth functions ϕ
i
1
(
·
) and ϕ
i
2
(
·
), such that
(2)
|
Δ
i
(
x
,
z
,
t
)
|
≤
ϕ
i
1
(
|
(
x
1
,
…
,
x
i
)
|
)
+
ϕ
i
2
(
|
z
|
)
.Remark 3.
This assumption is similar to the one in [58, 59] in which ϕ
i
1
(
·
) and ϕ
i
2
(
·
) are known. Assumption 2, however, does not require them to be known. Therefore, Assumption 2 relaxes the restriction in the existing results.Assumption 4 (see [59]).
The unmodeled dynamics in (1) is exponentially input-to-state practically stable (exp-ISpS); that is, for the system z
˙
=
q
(
z
,
x
), there exists an exp-ISpS Lyapunov function V
(
z
) such that
(3)
α
1
(
|
z
|
)
≤
V
(
z
)
≤
α
2
(
|
z
|
)
,
∂
V
(
z
)
∂
z
q
(
z
,
x
)
≤
-
c
0
V
(
z
)
+
μ
(
|
x
|
)
+
d
0
,
where α
1, α
2, and μ are of class K
∞-functions and c
0 and d
0 are known positive constants.Assumption 5 (see [50]).
For1
≤
i
≤
n, the signs of g
i
(
x
¯
i
) are known, and there exists unknown positive constant b such that
(4)
0
<
b
≤
|
g
i
(
x
¯
i
)
|
<
∞
,
∀
x
¯
i
∈
R
i
.Remark 6.
Equation (4) implies that g
i
(
x
¯
i
) are either strictly positive or negative. Without loss of generality, it is supposed that 0
<
b
≤
g
i
(
x
¯
i
). In addition, since b is not required in the designed controller, its true value is not required to be known.Lemma 7 (see [59]).
IfV is an exp-ISpS Lyapunov function for a control system, that is, (3) hold, then, for any constants c
¯ in (
0
,
c
0
), any initial condition x
0
=
x
0
(
0
), and any function μ
¯
(
x
1
)
≥
μ
(
|
x
1
|
), there exists a finite time T
0
=
T
0
(
c
¯
,
r
0
,
z
0
), a nonnegative function D
(
t
) defined for all t
≥
0, and a signal described by
(5)
r
˙
=
-
c
¯
r
+
μ
¯
(
x
1
(
t
)
)
+
d
0
,
r
(
0
)
=
r
0
,
such that D
(
t
)
=
0 for all t
≥
T
0,
(6)
V
(
z
(
t
)
)
≤
r
(
t
)
+
D
(
t
)
.
For all t
≥
0, the solutions are defined. Without loss of generality, this paper takes μ
¯
(
·
) as μ
¯
(
s
)
=
s
2
μ
0
(
s
2
), where μ
¯
(
·
) is a nonnegative smooth function. Therefore, the dynamical r defined by (5) becomes
(7)
r
˙
=
-
c
¯
r
+
x
1
2
μ
0
(
|
x
1
2
|
)
+
d
0
,
r
(
0
)
=
r
0
,
where μ
0 is a nonnegative smooth function.Throughout this paper, RBF neural networks are applied to model the unknown continuous nonlinear functions. In [60], it has been indicated that, with enough node number l, the RBF neural networks ϕ
*
T
ξ
(
X
) can model the continuous function f
(
X
) within a compact set Ω
X
⊂
R
q to arbitrary accuracy ɛ
>
0 as
(8)
f
(
X
)
=
ϕ
*
T
ξ
(
X
)
+
δ
(
X
)
,
∀
X
∈
Ω
X
∈
R
q
,
in which ϕ
* denotes the ideal weight vector and is specified as
(9)
ϕ
*
:
=
arg
min
⁡
ϕ
∈
R
¯
N
{
sup
⁡
X
∈
Ω
X
|
f
(
X
)
-
ϕ
T
ξ
(
X
)
|
}
δ
(
X
) depicts the approximation error satisfying |
δ
(
X
)
|
≤
ɛ, ϕ
*
=
[
ϕ
1
,
ϕ
2
,
…
,
ϕ
l
]
T
∈
R
l is the weight vector, and ξ
(
X
)
=
[
ξ
1
(
X
)
,
ξ
2
(
X
)
,
…
,
ξ
l
(
X
)
]
T is the basis function vector with l being the number of the neural networks nodes and l
>
1. The basis function ξ
i
(
X
) is taken as the Gaussian function in the below form:
(10)
ξ
i
(
X
)
=
exp
⁡
[
-
(
X
-
μ
i
)
T
(
X
-
μ
i
)
η
i
2
]
,
(
i
=
1,2
,
…
,
l
)
,
where μ
i
=
[
μ
i
1
,
μ
i
2
,
…
,
μ
i
q
]
T and η
i are the center of the receptive field and the width of the Gaussian function, respectively.


## 3. Adaptive Neural Control Design

In this section, the adaptive backstepping control design for system (1) is proposed. As usual, in the backstepping approach, the following coordinate transformation is made:
(11)
z
i
=
x
i
-
α
i
-
1
,
i
=
1,2
,
…
,
n
,
where α
0
=
0, α
i is the virtual control signal and will be constructed at Stepi, and the actual controller u will be designed at Stepn. Now, we begin the controller design procedure.Step 1.
Consider the following subsystem:(12)
z
˙
=
q
(
z
,
x
)
,
x
˙
1
=
g
1
(
x
1
)
x
2
+
f
1
(
x
1
)
+
Δ
1
(
x
,
z
,
t
)
.
Based on z
1
=
x
1, then choose Lyapunov functions as
(13)
V
1
=
1
2
z
1
2
+
1
λ
0
r
+
b
2
γ
1
θ
~
1
2
,
where μ
¯
(
x
1
)
=
x
1
2
μ
0
(
x
1
2
), λ
0 and γ
1 are positive design parameters, and θ
~
1
=
θ
1
-
θ
^
1 is the parameter error with θ
^
1 being the estimation of θ
1 which is defined later.By taking (12) into account, we have
(14)
V
˙
1
=
z
1
(
g
1
(
x
1
)
x
2
+
f
1
(
x
1
)
+
Δ
1
)
+
1
λ
0
r
˙
-
b
γ
1
θ
~
1
θ
^
˙
1
≤
z
1
(
g
1
(
x
1
)
x
2
+
f
1
(
x
1
)
)
+
|
z
1
|
|
Δ
1
|
+
1
λ
0
r
˙
-
b
γ
1
θ
~
1
θ
^
˙
1
.
By Assumption 2, it follows that
(15)
V
˙
1
≤
z
1
(
g
1
(
x
1
)
x
2
+
f
1
(
x
1
)
)
+
|
z
1
|
ϕ
11
(
|
x
1
|
)
+
|
z
1
|
ϕ
12
(
|
z
|
)
+
1
λ
0
(
x
1
2
μ
0
(
x
1
2
)
+
d
0
)
-
c
¯
λ
0
r
-
b
γ
1
θ
~
1
θ
^
˙
1
.
Then, we will deal with the third and fourth terms in (15), respectively. By using 0
≤
|
x
|
-
x
tanh
(
x
/
ϵ
)
≤
0.2785
ϵ
=
ϵ
′, for ∀
ϵ
>
0, we have
(16)
|
z
1
|
ϕ
11
(
|
x
1
|
)
=
|
z
1
ϕ
11
(
|
x
1
|
)
|
-
z
1
ϕ
11
(
|
x
1
|
)
tanh
⁡
(
z
1
ϕ
11
(
|
x
1
|
)
ϵ
11
)
+
z
1
ϕ
11
(
|
x
1
|
)
tanh
⁡
(
z
1
ϕ
11
(
|
x
1
|
)
ϵ
11
)
≤
ϵ
11
′
+
z
1
ϕ
11
(
|
x
1
|
)
tanh
⁡
(
z
1
ϕ
11
(
|
x
1
|
)
ϵ
11
)
≤
z
1
ϕ
^
11
(
x
1
)
+
ϵ
11
′
,
where ϵ
11
′
=
0.2785
ϵ
11 and ϕ
^
11
(
x
1
)
=
ϕ
11
(
|
x
1
|
)
tanh
(
z
1
ϕ
11
(
|
x
1
|
)
/
ϵ
11
) is a smooth function.By using the same derivations as [58], one has
(17)
|
z
1
|
ϕ
12
(
|
z
|
)
≤
|
z
1
|
ϕ
¯
12
(
r
)
+
1
4
z
1
2
+
d
1
(
t
)
≤
z
1
ϕ
¯
12
(
r
)
tanh
⁡
(
z
1
ϕ
¯
12
(
r
)
ϵ
12
)
+
ϵ
12
′
+
1
4
z
1
2
+
d
1
(
t
)
≤
z
1
ϕ
^
12
(
x
1
,
r
)
+
ϵ
12
′
+
1
4
z
1
2
+
d
1
(
t
)
,
where ϵ
12
′
=
0.2785
ϵ
12
,
ϕ
¯
12
(
r
)
=
ϕ
12
∘
α
1
-
1
(
2
r
), d
1
(
t
)
=
(
ϕ
12
∘
α
1
-
1
(
2
D
(
t
)
)
)
2, and ϕ
^
12
(
x
1
,
r
)
=
ϕ
¯
12
(
r
)
tanh
(
z
1
ϕ
¯
12
(
r
)
/
ϵ
12
).Subsequently, substituting (16) and (17) into (15) gives
(18)
V
˙
1
≤
z
1
(
g
1
(
x
1
)
z
2
+
g
1
(
x
1
)
α
1
+
f
^
1
(
Z
1
)
)
-
z
1
2
2
+
d
0
λ
0
-
c
¯
λ
0
r
-
b
γ
1
θ
~
1
θ
^
˙
1
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
)
,
where the function f
^
1
(
Z
1
) is defined as
(19)
f
^
1
(
Z
1
)
=
f
1
(
x
1
)
+
ϕ
^
11
(
x
1
)
+
ϕ
^
12
(
r
)
+
3
4
z
1
+
1
λ
0
(
x
1
μ
0
(
x
1
2
)
)
,
where Z
1
=
[
x
1
,
r
]
T
∈
Ω
Z
1
⊂
R
2.Since the smooth functionf
^
1
(
Z
1
) is unknown, it cannot be implemented in practice. By employing RBF neural network in ϕ
1
T
ξ
(
Z
1
) to approximate f
^
1
(
Z
1
), we have
(20)
f
^
1
(
Z
1
)
=
ϕ
1
T
ξ
1
(
Z
1
)
+
δ
1
(
Z
1
)
,
|
δ
1
(
Z
1
)
|
≤
ϵ
13
,
where δ
1
(
Z
1
) denotes an approximation error and ϵ
13
>
0 is a given positive constant.Next, the following result can be obtained by substituting (20) into (18):
(21)
V
˙
1
≤
g
1
(
x
1
)
z
1
z
2
+
g
1
(
x
1
)
z
1
α
1
+
z
1
ϕ
1
T
ξ
1
(
Z
1
)
+
z
1
δ
1
(
Z
1
)
-
z
1
2
2
+
d
0
λ
0
-
c
¯
λ
0
r
-
b
γ
1
θ
~
1
θ
^
˙
1
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
)
.
By using
(22)
z
1
ϕ
1
T
ξ
1
(
Z
1
)
≤
b
θ
1
2
η
1
2
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
z
1
2
+
η
1
2
2
,
z
1
δ
1
(
Z
1
)
≤
z
1
2
2
+
ϵ
13
2
2
,
one has
(23)
V
˙
1
≤
g
1
(
x
1
)
z
1
z
2
+
g
1
(
x
1
)
z
1
α
1
+
b
θ
1
2
η
1
2
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
z
1
2
+
η
1
2
2
+
ϵ
13
2
2
+
d
0
λ
0
-
c
¯
λ
0
r
-
b
γ
1
θ
~
1
θ
^
˙
1
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
)
with θ
1
=
b
-
1
∥
ϕ
1
∥
2 being an unknown parameter.Construct the virtual control signalα
1 as
(24)
α
1
=
-
k
1
z
1
-
θ
^
1
2
η
1
2
z
1
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
,
where k
1 and η
1 are positive design constants.By taking Assumption5 into account, one has
(25)
g
1
(
x
1
)
z
1
α
1
≤
-
k
1
b
z
1
2
-
θ
^
1
2
η
1
2
b
z
1
2
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
.
Further, by substituting (25) into (23), we obtain
(26)
V
˙
1
≤
-
b
k
1
z
1
2
-
c
¯
λ
0
r
+
b
θ
~
1
γ
1
(
γ
1
2
η
1
2
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
z
1
2
-
θ
^
˙
1
)
+
g
1
(
x
1
)
z
1
z
2
+
η
1
2
2
+
ϵ
13
2
2
+
d
0
λ
0
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
)
.
Next, we choose the adaptive law in the following form:
(27)
θ
^
˙
1
=
γ
1
2
η
1
2
ξ
1
T
(
Z
1
)
ξ
1
(
Z
1
)
z
1
2
-
σ
1
θ
^
1
,
where η
1 and σ
1 are design parameters.By using (27), we can rewrite (26) as
(28)
V
˙
1
≤
-
b
k
1
z
1
2
-
c
¯
λ
0
r
+
g
1
(
x
1
)
z
1
z
2
+
b
σ
1
θ
~
1
θ
^
1
γ
1
+
η
1
2
2
+
ϵ
13
2
2
+
d
0
λ
0
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
)
.
Noting
(29)
b
σ
1
θ
~
1
θ
^
1
γ
1
=
-
b
σ
1
θ
~
1
2
γ
1
+
b
σ
1
θ
~
1
θ
1
γ
1
≤
-
b
σ
1
θ
~
1
2
2
γ
1
+
b
σ
1
θ
1
2
2
γ
1
,
then the following inequality holds:
(30)
V
˙
1
≤
-
b
k
1
z
1
2
-
c
¯
λ
0
r
-
b
σ
1
θ
~
1
2
2
γ
1
+
d
0
λ
0
+
D
1
+
g
1
(
x
1
)
z
1
z
2
,
where D
1
=
η
1
2
/
2
+
b
σ
1
θ
1
2
/
2
γ
1
+
ϵ
13
2
/
2
+
ϵ
11
′
+
ϵ
12
′
+
d
1
(
t
).Step 2.
Based onz
2
=
x
2
-
α
1, then the time derivative of z
2 is given by
(31)
z
˙
2
=
x
˙
2
-
α
˙
1
=
g
2
(
x
¯
2
)
x
3
+
f
2
(
x
¯
2
)
+
Δ
2
-
∂
α
1
∂
x
1
(
g
1
(
x
1
)
x
2
+
f
1
(
x
1
)
+
Δ
1
)
-
∂
α
1
∂
θ
^
1
θ
^
˙
1
-
∂
α
1
∂
r
r
˙
=
g
2
(
x
¯
2
)
x
3
+
(
f
2
(
x
¯
2
)
-
∂
α
1
∂
x
1
f
1
(
x
1
)
)
+
Δ
¯
2
-
∂
α
1
∂
x
1
g
1
(
x
1
)
x
2
-
∂
α
1
∂
θ
^
1
θ
^
˙
1
-
∂
α
1
∂
r
r
˙
,
where Δ
¯
2
=
Δ
2
-
(
∂
α
1
/
∂
x
1
)
Δ
1.Construct the Lyapunov function(32)
V
2
=
V
1
+
1
2
z
2
2
+
b
2
γ
2
θ
~
2
2
.The derivative ofV
2 is
(33)
V
˙
2
≤
-
b
k
1
z
1
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
z
2
[
∂
α
1
∂
θ
^
1
g
2
(
x
¯
2
)
x
3
+
g
1
(
x
1
)
z
1
+
f
2
(
x
¯
2
)
-
∂
α
1
∂
x
1
f
1
(
x
1
)
-
∂
α
1
∂
x
1
g
1
(
x
1
)
x
2
-
∂
α
1
∂
θ
^
1
θ
^
˙
1
-
∂
α
1
∂
r
r
˙
]
+
|
z
2
Δ
¯
2
|
+
D
1
+
d
0
λ
0
-
b
γ
2
θ
~
2
θ
^
˙
2
.
By Assumption 2, we have
(34)
|
z
2
Δ
¯
2
|
=
|
z
2
|
|
Δ
2
-
∂
α
1
∂
x
1
Δ
1
|
≤
|
z
2
|
(
|
Δ
2
|
+
|
∂
α
1
∂
x
1
|
|
Δ
1
|
)
≤
|
z
2
|
(
ϕ
21
+
|
∂
α
1
∂
x
1
|
ϕ
11
)
+
|
z
2
|
(
ϕ
22
(
|
z
|
)
+
|
∂
α
1
∂
x
1
|
ϕ
12
(
|
z
|
)
)
.
Similar to the estimation methods in (35), the following results can be obtained:
(35)
|
z
2
|
(
ϕ
21
+
|
∂
α
1
∂
x
1
|
ϕ
11
)
≤
z
2
ϕ
^
21
(
x
¯
2
,
θ
^
1
,
r
)
+
ϵ
21
′
,
|
z
2
|
(
ϕ
22
(
|
z
|
)
+
|
∂
α
1
∂
x
1
|
ϕ
12
(
|
z
|
)
)
≤
|
z
2
|
(
ϕ
22
∘
α
1
-
1
(
2
r
)
+
|
∂
α
1
∂
x
1
|
ϕ
12
∘
α
1
-
1
(
2
r
)
)
+
|
z
2
|
ϕ
22
∘
α
1
-
1
(
2
D
(
t
)
)
+
|
z
2
|
|
∂
α
1
∂
x
1
|
ϕ
12
∘
α
1
-
1
(
2
D
(
t
)
)
≤
|
z
2
|
ϕ
¯
22
(
x
1
,
θ
^
1
,
r
)
+
z
2
2
4
(
1
+
(
∂
α
1
∂
x
1
)
2
)
+
d
2
(
t
)
≤
z
2
ϕ
^
22
(
x
1
,
θ
^
1
,
r
)
+
ϵ
22
′
+
z
2
2
4
(
1
+
(
∂
α
1
∂
x
1
)
2
)
+
d
2
(
t
)
,
where ϕ
^
21
(
x
¯
2
,
θ
^
1
,
r
)
=
(
ϕ
21
+
|
∂
α
1
/
∂
x
1
|
ϕ
11
)
tanh
(
z
2
(
ϕ
21
+
|
∂
α
1
/
∂
x
1
|
ϕ
11
)
/
ϵ
21
), ϵ
21
′
=
0.2785
ϵ
21, ϕ
¯
22
(
x
1
,
θ
^
1
,
r
)
=
ϕ
22
∘
α
1
-
1
(
2
r
)
+
|
∂
α
1
/
∂
x
1
|
ϕ
12
∘
α
1
-
1
(
2
r
), ϕ
^
22
(
x
1
,
θ
^
1
,
r
)
=
ϕ
¯
22
(
x
1
,
θ
^
1
,
r
)
tanh
(
z
2
ϕ
¯
22
(
x
1
,
θ
^
1
,
r
)
/
ϵ
22
), and d
2
(
t
)
=
∑
j
=
1
2
‍
(
ϕ
j
2
∘
α
1
-
1
(
2
D
(
t
)
)
)
2.Substituting (35) into (33) gives
(36)
V
˙
2
≤
-
b
k
1
z
1
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
z
2
[
(
1
+
(
∂
α
1
∂
x
1
)
2
)
g
2
(
x
¯
2
)
x
3
+
g
1
(
x
1
)
z
1
+
f
2
(
x
¯
2
)
-
∂
α
1
∂
x
1
f
1
(
x
1
)
-
∂
α
1
∂
x
1
g
1
(
x
1
)
x
2
-
∂
α
1
∂
θ
^
1
θ
^
˙
1
-
∂
α
1
∂
r
r
˙
+
ϕ
^
21
(
x
¯
2
,
θ
^
1
,
r
)
+
ϕ
^
22
(
x
1
,
θ
^
1
,
r
)
+
z
2
4
(
1
+
(
∂
α
1
∂
x
1
)
2
)
]
+
ϵ
21
′
+
ϵ
22
′
+
d
2
(
t
)
+
D
1
+
d
0
λ
0
-
b
γ
2
θ
~
2
θ
^
˙
2
≤
-
b
k
1
z
1
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
z
2
[
g
2
(
x
¯
2
)
z
3
+
g
2
(
x
¯
2
)
α
2
+
f
^
2
(
Z
2
)
]
-
z
2
2
2
+
ϵ
21
′
+
ϵ
22
′
+
d
2
(
t
)
+
D
1
+
d
0
λ
0
-
b
γ
2
θ
~
2
θ
^
˙
2
,
where z
3
=
x
3
-
α
2 and the function f
^
2
(
Z
2
) is specified as
(37)
f
^
2
(
Z
2
)
=
g
1
(
x
1
)
z
1
+
f
2
(
x
¯
2
)
-
∂
α
1
∂
x
1
f
1
(
x
1
)
+
ϕ
^
21
(
x
1
,
θ
^
1
,
r
)
+
ϕ
^
22
(
x
¯
2
,
θ
^
1
,
r
)
+
z
2
4
[
1
+
(
∂
α
1
∂
x
1
)
2
]
+
z
2
2
-
∂
α
1
∂
x
1
g
1
(
x
1
)
x
2
-
∂
α
1
∂
θ
^
1
θ
^
˙
1
-
∂
α
1
∂
r
r
˙
with Z
2
=
[
x
¯
2
,
θ
^
1
,
r
]
T
∈
Ω
Z
2
∈
R
4 and Ω
Z
2 being some known compact set in R
4.To compensate for the unknown nonlinear functionf
^
2
(
Z
2
), a neural network ϕ
2
T
ξ
2
(
Z
2
) is utilized to model f
^
2
(
Z
2
) such that, for any given positive constant ϵ
23,
(38)
f
^
2
(
Z
2
)
=
ϕ
2
T
ξ
2
(
Z
2
)
+
δ
2
(
Z
2
)
,
|
δ
2
(
Z
2
)
|
≤
ϵ
23
,
where ϵ
23 denotes approximation error.Then, substituting (38) into (36), one has
(39)
V
˙
2
≤
-
b
k
1
z
1
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
g
2
(
x
¯
2
)
z
2
z
3
+
g
2
(
x
¯
2
)
z
2
α
2
+
z
2
ϕ
2
T
ξ
2
(
Z
2
)
+
z
2
δ
2
(
Z
2
)
-
z
2
2
2
+
ϵ
21
′
+
ϵ
22
′
+
d
2
(
t
)
+
D
1
+
d
0
λ
0
-
b
γ
2
θ
~
2
θ
^
˙
2
.
By using
(40)
z
2
ϕ
2
T
ξ
2
(
Z
2
)
≤
b
θ
2
2
η
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
z
2
2
+
η
2
2
2
,
z
2
δ
2
(
Z
2
)
≤
z
2
2
2
+
ϵ
23
2
2
,
it can be easily verified that
(41)
V
˙
2
≤
-
b
k
1
z
1
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
g
2
(
x
¯
2
)
z
2
α
2
+
b
θ
2
2
η
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
z
2
2
+
g
2
(
x
¯
2
)
z
2
z
3
+
η
2
2
2
+
ϵ
21
′
+
ϵ
22
′
+
ϵ
23
2
2
+
d
2
(
t
)
+
D
1
+
d
0
λ
0
-
b
γ
2
θ
~
2
θ
^
˙
2
with θ
2
=
b
-
1
∥
ϕ
2
∥
2 being an unknown constant.Furthermore, the virtual controlα
2 is constructed as
(42)
α
2
=
-
k
2
z
2
-
θ
^
2
2
η
2
2
z
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
,
where k
2
>
0 and η
2 are the design constants.Then, the following result can be easily obtained:(43)
g
2
(
x
¯
2
)
z
2
α
2
≤
-
k
2
b
z
2
2
-
θ
^
2
2
η
2
2
b
z
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
.By applying (43), (41) can be rewritten as
(44)
V
˙
2
≤
-
∑
i
=
1
2
‍
k
i
b
z
i
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
b
θ
~
2
γ
2
(
γ
2
2
η
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
z
2
2
-
θ
^
˙
2
)
+
g
2
(
x
¯
2
)
z
2
z
3
+
η
2
2
2
+
ϵ
21
′
+
ϵ
22
′
+
ϵ
23
2
2
+
d
2
(
t
)
+
d
0
λ
0
+
D
1
.
Define the adaptive law as
(45)
θ
^
˙
2
=
γ
2
2
η
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
z
2
2
-
σ
2
θ
^
2
,
where η
2, γ
2, and σ
2 are design parameters.Combining (44) with (45) produces
(46)
V
˙
2
≤
-
∑
i
=
1
2
‍
k
i
b
z
i
2
-
b
σ
1
θ
~
1
2
2
γ
1
-
c
¯
λ
0
r
+
b
σ
2
θ
~
2
θ
^
2
γ
2
+
g
2
(
x
¯
2
)
z
2
z
3
+
η
2
2
2
+
ϵ
21
′
+
ϵ
22
′
+
ϵ
23
2
2
+
d
2
(
t
)
+
D
1
+
d
0
λ
0
≤
-
∑
i
=
1
2
‍
(
k
i
b
z
i
2
+
b
σ
i
θ
~
i
2
2
γ
i
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
i
=
1
2
‍
D
i
+
g
2
(
x
¯
2
)
z
2
z
3
,
where D
i
=
η
i
2
/
2
+
b
σ
i
θ
i
2
/
2
γ
i
+
ϵ
i
3
2
/
2
+
ϵ
i
1
′
+
ϵ
i
2
′
+
d
i
(
t
), i
=
1,2, and the result b
σ
2
θ
~
2
θ
^
2
/
γ
2
≤
-
b
σ
2
θ
~
2
2
/
2
γ
2
+
b
σ
2
θ
2
2
/
2
γ
2 has been used in the above equation.Step i (3
≤
i
≤
n
-
1).
According toz
i
=
x
i
-
α
i
-
1, the dynamics of z
i is
(47)
z
˙
i
=
g
i
(
x
¯
i
)
x
i
+
1
+
f
i
(
x
¯
i
)
+
Δ
i
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
(
g
j
(
x
¯
j
)
x
j
+
1
+
f
j
(
x
¯
j
)
+
Δ
j
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
θ
^
j
θ
^
˙
j
-
∂
α
i
-
1
∂
r
r
˙
=
g
i
(
x
¯
i
)
x
i
+
1
+
f
i
(
x
¯
i
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
(
g
j
(
x
¯
j
)
x
j
+
1
+
f
j
(
x
¯
j
)
)
+
Δ
¯
i
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
θ
^
j
θ
^
˙
j
-
∂
α
i
-
1
∂
r
r
˙
,
where Δ
¯
i
=
Δ
i
-
∑
j
=
1
i
-
1
‍
(
∂
α
i
-
1
/
∂
x
j
)
Δ
j.Consider the Lyapunov functionV
i as
(48)
V
i
=
V
i
-
1
+
1
2
z
i
2
+
b
2
γ
i
θ
~
i
2
.
By using the derivations similar to those used in the former steps, we can obtain
(49)
V
˙
i
≤
-
∑
j
=
1
i
-
1
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
z
i
(
g
i
(
x
¯
i
)
x
i
+
1
+
f
i
(
x
¯
i
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
f
j
(
x
¯
j
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
g
j
(
x
¯
j
)
x
j
+
1
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
θ
^
j
θ
^
˙
j
-
∂
α
i
-
1
∂
r
r
˙
+
g
i
-
1
(
x
¯
i
-
1
)
z
i
-
1
∑
j
=
1
i
-
1
‍
)
+
|
z
i
Δ
¯
i
|
+
d
0
λ
0
+
∑
j
=
1
i
-
1
‍
D
j
-
b
γ
i
θ
~
i
θ
^
˙
i
.
Similar to (34), we have
(50)
|
z
i
Δ
¯
i
|
≤
|
z
i
|
(
|
Δ
i
|
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
∂
x
j
|
|
Δ
j
|
)
≤
|
z
i
|
(
ϕ
i
1
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
∂
x
j
|
ϕ
j
1
)
+
|
z
i
|
(
ϕ
i
2
(
|
z
|
)
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
∂
x
j
|
ϕ
j
2
(
|
z
|
)
)
.
Furthermore, the following inequalities can be easily verified by repeating the same arguments as (35):
(51)
|
z
i
|
(
ϕ
i
1
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
∂
x
j
|
ϕ
j
1
)
≤
z
i
ϕ
^
i
1
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
+
ϵ
i
1
′
,
(52)
|
z
i
|
(
ϕ
i
2
(
|
z
|
)
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
∂
x
j
|
ϕ
j
2
(
|
z
|
)
)
≤
z
i
ϕ
^
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
+
z
i
2
4
[
1
+
∑
j
=
1
i
-
1
‍
(
∂
α
i
-
1
∂
x
j
)
2
]
+
ϵ
i
2
′
+
d
i
(
t
)
,
where ϕ
^
i
1
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
=
(
ϕ
i
1
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
/
∂
x
j
|
ϕ
j
1
)
tanh
(
z
i
(
ϕ
i
1
+
∑
j
=
1
i
-
1
‍
|
∂
α
i
-
1
/
∂
x
j
|
ϕ
j
1
)
/
ϵ
i
1
), ϵ
i
1
′
=
0.2785
ϵ
i
1, ϕ
^
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
=
ϕ
¯
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
tanh
(
z
i
ϕ
¯
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
/
ϵ
i
2
), ϕ
¯
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
=
ϕ
i
2
∘
α
1
-
1
(
2
r
)
+
∑
j
=
1
i
-
1
‍
|
∂
i
-
1
/
∂
x
j
|
ϕ
j
2
∘
α
1
-
1
(
2
r
)
,
ϵ
i
2
′
=
0.2785
ϵ
i
2, and d
i
(
t
)
=
∑
j
=
1
i
‍
(
ϕ
j
2
∘
α
1
-
1
(
2
D
(
t
)
)
)
2, noting that d
i
(
t
)
≥
0 for all t
≥
0.Substituting (51) and (52) into (49) results in
(53)
V
˙
i
≤
-
∑
j
=
1
i
-
1
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
z
i
(
g
i
(
x
¯
i
)
z
i
+
1
+
g
i
(
x
¯
i
)
α
i
+
f
^
i
(
Z
i
)
)
-
z
i
2
2
+
ϵ
i
1
′
+
ϵ
i
2
′
+
d
i
(
t
)
+
d
0
λ
0
+
∑
j
=
1
i
-
1
‍
D
j
-
b
γ
i
θ
~
i
θ
^
˙
i
,
where the function f
^
i
(
Z
i
) is defined by
(54)
f
^
i
(
Z
i
)
=
f
i
(
x
¯
i
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
f
j
(
x
¯
j
)
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
x
j
g
j
(
x
¯
j
)
x
j
+
1
-
∑
j
=
1
i
-
1
‍
∂
α
i
-
1
∂
θ
^
j
θ
^
˙
j
-
∂
α
i
-
1
∂
r
r
˙
+
g
i
-
1
(
x
¯
i
-
1
)
z
i
-
1
+
ϕ
^
i
1
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
+
ϕ
^
i
2
(
x
¯
i
,
θ
^
¯
i
-
1
,
r
)
+
z
i
4
[
1
+
∑
j
=
1
i
-
1
‍
(
∂
α
i
-
1
∂
x
j
)
2
]
+
z
i
2
,
where Z
i
=
[
x
¯
i
,
θ
^
¯
i
-
1
,
r
]
T
∈
Ω
Z
i
∈
R
2
i, x
¯
i
=
[
x
1
,
…
,
x
i
]
T, and Ω
Z
i is some known compact set in R
2
i.Currently, a neural networkϕ
i
T
ξ
i
(
Z
i
) is utilized to model f
^
i
(
Z
i
) such that, for a given ϵ
i
3
>
0, f
^
i
(
Z
i
) can be expressed as
(55)
f
^
i
(
Z
i
)
=
ϕ
i
T
ξ
i
(
Z
i
)
+
δ
i
(
Z
i
)
,
|
δ
i
(
Z
i
)
|
≤
ϵ
i
3
.
Further, similar to (40), we can obtain
(56)
z
i
ϕ
i
T
ξ
i
(
Z
i
)
≤
b
θ
i
2
η
i
2
ξ
i
T
(
Z
i
)
ξ
i
(
Z
i
)
z
i
2
+
η
i
2
2
,
z
i
δ
i
(
Z
i
)
≤
z
i
2
2
+
ϵ
i
3
2
2
,
where θ
i
=
b
-
1
∥
ϕ
i
∥
2 is an unknown constant.Now, construct the virtual control signalα
i as
(57)
α
i
=
-
k
i
z
i
-
θ
^
i
2
η
i
2
z
i
ξ
i
T
(
Z
i
)
ξ
i
(
Z
i
)
,
with k
i
>
0 and η
i being design constants.Then, by substituting (55)–(57) into (53), choosing the adaptive law
(58)
θ
^
˙
i
=
γ
i
2
η
i
2
ξ
i
T
(
Z
i
)
ξ
i
(
Z
i
)
z
i
2
-
σ
i
θ
^
i
with η
i
,
γ
i, and σ
i being the design parameters, and then following the same line as the procedures from (43) to (46), we have
(59)
V
˙
i
≤
-
∑
j
=
1
i
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
j
=
1
i
‍
D
j
+
g
i
(
x
¯
i
)
z
i
z
i
+
1
,
where D
j
=
η
j
2
/
2
+
b
σ
j
θ
j
2
/
2
γ
j
+
ϵ
j
3
2
/
2
+
ϵ
j
1
′
+
ϵ
j
2
′
+
d
j
(
t
), j
=
1,2
,
…
,
i.Step n.
In this step, the actual controlleru is designed. According to z
n
=
x
n
-
α
n
-
1, then we have
(60)
z
˙
n
=
g
n
(
x
¯
n
)
u
+
f
n
(
x
¯
n
)
-
∑
j
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
j
f
j
(
x
¯
j
)
-
∑
j
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
j
g
j
(
x
¯
j
)
x
j
+
1
+
Δ
¯
n
-
∑
j
=
1
n
-
1
‍
∂
α
n
-
1
∂
θ
^
j
θ
^
˙
j
-
∂
α
n
-
1
∂
r
r
˙
,
where Δ
¯
n
=
Δ
n
-
∑
i
=
1
n
-
1
‍
(
∂
α
n
-
1
/
∂
x
i
)
Δ
i. Similarly, choose the following Lyapunov function as
(61)
V
n
=
V
n
-
1
+
1
2
z
n
2
+
b
2
γ
n
θ
~
n
2
.
From (53) and (54), we have
(62)
V
˙
n
≤
-
∑
j
=
1
n
-
1
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
j
=
1
n
-
1
‍
D
j
+
g
n
-
1
(
x
¯
n
-
1
)
z
n
-
1
z
n
+
z
n
(
g
n
(
x
¯
n
)
u
+
f
n
(
x
¯
n
)
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
i
f
i
(
x
¯
i
)
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
i
g
i
(
x
¯
i
)
x
i
+
1
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
θ
^
i
θ
^
˙
i
-
∂
α
n
-
1
∂
r
r
˙
∑
i
=
1
n
-
1
‍
)
-
b
γ
n
θ
~
n
θ
^
˙
n
+
|
z
n
Δ
¯
n
|
.
Using the same estimation methods as (42)–(44), we have
(63)
|
z
n
Δ
¯
n
|
≤
z
n
ϕ
^
n
1
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
)
+
ϵ
n
1
′
+
z
n
ϕ
^
n
2
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
)
+
z
n
2
4
[
1
+
∑
j
=
1
n
-
1
‍
(
∂
α
n
-
1
∂
x
j
)
2
]
+
ϵ
n
2
′
+
d
n
(
t
)
,
where ϕ
^
n
1
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
), ϵ
n
1
′, ϕ
^
n
2
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
), ϵ
n
2
′, and d
n
(
t
) are defined in (51) or (52) with i
=
n. By substituting (63) into (62), one has
(64)
V
˙
n
≤
-
∑
j
=
1
n
-
1
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
j
=
1
n
-
1
‍
D
j
+
ϵ
n
1
′
+
ϵ
n
2
′
+
d
n
(
t
)
+
z
n
(
g
n
(
x
¯
n
)
u
+
f
^
n
(
Z
n
)
)
-
z
n
2
2
-
b
γ
n
θ
~
n
θ
^
˙
n
,
where the function f
^
n
(
Z
n
) is defined by
(65)
f
^
n
(
Z
n
)
=
g
n
-
1
(
x
¯
n
-
1
)
z
n
-
1
+
f
n
(
x
¯
n
)
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
i
f
i
(
x
¯
i
)
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
x
i
g
i
(
x
¯
i
)
x
i
+
1
-
∑
i
=
1
n
-
1
‍
∂
α
n
-
1
∂
θ
^
i
θ
^
˙
i
-
∂
α
n
-
1
∂
r
r
˙
+
ϕ
^
n
1
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
)
+
ϕ
^
n
2
(
x
¯
n
,
θ
^
¯
n
-
1
,
r
)
+
z
n
4
[
1
+
∑
j
=
1
n
-
1
‍
(
∂
α
n
-
1
∂
x
j
)
2
]
+
z
n
2
with Z
n
=
[
x
¯
n
,
θ
^
¯
n
-
1
,
r
]
T
∈
Ω
Z
n
∈
R
2
n and Ω
Z
n being some known compact set in R
2
n. Similarly, neural network ϕ
n
T
ξ
n
(
Z
n
) is employed to model f
^
n
(
Z
n
) such that f
^
n
(
Z
n
)
=
ϕ
n
T
ξ
n
(
Z
n
)
+
δ
n
(
Z
n
), |
δ
n
(
Z
n
)
|
≤
ϵ
n
3. Then, following the same line as used in (40), we have
(66)
z
n
ϕ
n
T
ξ
n
(
Z
n
)
≤
b
θ
n
2
η
n
2
ξ
n
T
(
Z
n
)
ξ
n
(
Z
n
)
z
n
2
+
η
n
2
2
,
z
n
δ
n
(
Z
n
)
≤
z
n
2
2
+
ϵ
n
3
2
2
,
where θ
n
=
b
-
1
∥
ϕ
n
∥
2 denotes an unknown constant and η
n is a design constant.
Subsequently, by combining (64) together with (66), the inequality below holds:
(67)
V
˙
n
≤
-
∑
j
=
1
n
-
1
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
j
=
1
n
-
1
‍
D
j
+
ϵ
n
1
′
+
ϵ
n
2
′
+
d
n
(
t
)
+
z
n
g
n
(
x
¯
n
)
u
+
b
θ
n
2
η
n
2
ξ
n
T
(
Z
n
)
ξ
n
(
Z
n
)
z
n
2
+
η
n
2
2
+
ϵ
n
3
2
2
-
b
γ
n
θ
~
n
θ
^
˙
n
.
At the present stage, construct the real controller u and adaptive law θ
^
˙
n in the following forms:
(68)
u
=
-
k
n
z
n
-
θ
^
n
2
η
n
2
z
n
ξ
n
T
(
Z
n
)
ξ
n
(
Z
n
)
,
(69)
θ
^
˙
n
=
γ
n
2
η
n
2
ξ
n
T
(
Z
n
)
ξ
i
(
Z
n
)
z
n
2
-
σ
n
θ
^
n
,
where k
n, η
n, γ
n, and σ
n are design constants.
Then, repeating the similar procedures as (43)–(46), we can obtain
(70)
V
˙
n
≤
-
∑
j
=
1
n
‍
(
k
j
b
z
j
2
+
b
σ
j
θ
~
j
2
2
γ
j
)
-
c
¯
λ
0
r
+
d
0
λ
0
+
∑
j
=
1
n
‍
D
j
,
where D
j
=
η
j
2
/
2
+
b
σ
j
θ
j
2
/
2
γ
j
+
ϵ
j
3
2
/
2
+
ϵ
j
1
′
+
ϵ
j
2
′
+
d
j
(
t
), j
=
1,2
,
…
,
n.Now, the main result of this research is summarized as follows.Theorem 8.
Consider the system (1) consisting of Assumptions 2–5, the control input (68), and the adaptive laws (58) and (69). Assume that the packaged unknown functions f
^
i
(
Z
i
) (i
=
1,2
,
…
,
n) could be modeled by neural networks ϕ
i
T
ξ
i
(
Z
i
) with the bounded approximation errors. Then, for bounded initial values with θ
^
i
(
0
)
≥
0, all the signals in the closed-loop system are semiglobally boundedness in mean square.Proof.
To give the stability analysis for the closed-loop system, consider the Lyapunov function in the formV
=
V
n, and define
(71)
a
0
=
min
⁡
{
2
k
i
b
,
c
¯
,
σ
i
,
i
=
1,2
,
…
,
n
}
,
b
0
=
d
0
λ
0
+
∑
j
=
1
n
‍
D
j
.
Furthermore, we can rewrite (70) as
(72)
V
˙
≤
-
a
0
V
+
b
0
.
Next, from (72), the following inequality can be easily verified:
(73)
V
(
t
)
≤
(
V
(
0
)
-
b
0
a
0
)
e
-
a
0
t
+
b
0
a
0
,
∀
t
>
0
,
which means that
(74)
V
(
t
)
≤
V
(
0
)
+
b
0
a
0
,
∀
t
>
0
.
Therefore, based on the definition of V in (61), z
j, θ
~
j (j
=
1,2
,
…
,
n), and r are bounded. Because the signal r is bounded, the trajectory z
(
t
) is bounded. Since θ
i are constants, θ
^
i are bounded. Consequently, α
i are also bounded because z
i and θ
^
i are bounded variables. Hence, we conclude that the signals x
i are bounded.


## 4. Simulation Example

A simulation example is presented to show the effectiveness of the proposed control scheme. Consider the second-order nonlinear system as(75)
z
˙
=
-
z
+
x
1
2
+
0.5
,
x
˙
1
=
x
2
+
x
1
2
e
-
0.5
x
1
+
z
x
1
sin
(
x
1
)
,
x
˙
2
=
u
+
x
1
x
2
2
+
x
1
x
2
z
,
where f
1
(
x
1
)
=
x
1
2
e
-
0.5
x
1, f
2
(
x
1
,
x
2
)
=
x
1
x
2
2, Δ
1
=
z
x
1
sin
(
x
1
), and Δ
2
=
x
1
x
2
z. It can be easily verified that Assumption 2 is satisfied. In order to check Assumption 4 holds for z-subsystem in (75), consider V
z
(
z
)
=
z
2, and then
(76)
V
˙
z
(
z
)
=
2
z
(
-
z
+
x
1
2
+
0.5
)
≤
-
2
z
2
+
1
4
ɛ
(
2
z
)
2
+
ɛ
x
1
4
+
ɛ
4
+
z
2
ɛ
.
By choosing ɛ
=
2.5, we have
(77)
V
˙
z
(
z
)
≤
-
1.2
z
2
+
2.5
x
1
4
+
0.625
.
By Defining α
1
(
|
z
|
)
=
0.5
z
2, α
2
(
|
z
|
)
=
2
z
2, c
0
=
1.2, d
0
=
0.625, and μ
(
|
x
1
|
)
=
2.5
x
1
4, Assumption 4 is satisfied. Take c
¯
=
1
∈
(
0
,
c
0
) and define the dynamic signal as follows:
(78)
r
˙
=
-
r
+
2.5
x
1
4
+
0.625
.
By using Theorem 8, the virtual control signal, the real control input, and the adaptive laws are constructed as follows:
(79)
α
1
=
-
k
1
z
1
-
θ
^
1
2
η
1
2
ξ
1
T
(
Z
1
)
ξ
2
(
Z
1
)
z
1
,
u
=
-
k
2
z
2
-
θ
^
2
2
η
2
2
ξ
2
T
(
Z
2
)
ξ
2
(
Z
2
)
z
2
,
θ
^
˙
i
=
γ
i
2
η
i
2
ξ
i
T
(
Z
i
)
ξ
i
(
Z
i
)
z
i
2
-
σ
i
θ
^
i
,
i
=
1,2
.
In the simulation, the design constants are chosen as k
1
=
k
2
=
3, η
1
=
η
2
=
1, γ
1
=
γ
2
=
2, and σ
1
=
σ
2
=
1. The simulation is carried out with the initial conditions [
x
1
(
0
)
,
x
2
(
0
)
,
θ
^
1
(
0
)
,
θ
^
2
(
0
)
]
T
=
[
0.5,0.3,0
,
0
].The simulation results are shown in Figures1–3. Figure 1 shows the trajectories of states x
1 and x
2. Figure 2 displays the trajectory of control input u. Figure 3 shows the trajectories of adaptive parameters θ
^
1 and θ
^
2. From Figures 1–3, we can see that the proposed control approach can guarantee the boundedness of the variables x
1, x
2, u, θ
^
1, and θ
^
2.Figure 1
State variablesx
1 and x
2.Figure 2
The actual control inputu.Figure 3
The adaptive parametersθ
^
1 and θ
^
2.


## 5. Conclusion

In this research, a backstepping-based adaptive neural control scheme has been developed for strict-feedback nonlinear systems with unmodeled dynamics and dynamic disturbances. The proposed adaptive neural controller guarantees that all the signals of the resulting closed-loop system remain semiglobally uniformly ultimately bounded in the sense of mean square. Simulation results have been provided to illustrate the effectiveness of the proposed control scheme. It should be pointed out that the work in this paper does not consider the problem of input nonlinearity and time-delay. Then, they may occur in practical engineering. So, how to control a nonlinear system with input nonlinearity and time-delay is our future research direction.



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*Source: 658671-2014-07-02.xml*