robust adaptive control of MIMO pure feedback nonlinear system via important dynamic surface control technique

1. Received June 30, 2019, accepted July 11, 2019, date of publication July 16, 2019, date of current version August 5, 2019.
Digital Object Identifier 10.1109/ACCESS.2019.2929119
Robust Adaptive Control of MIMO Pure-Feedback
Nonlinear Systems via Improved Dynamic
Surface Control Technique
YANG ZHOU 1, WENHAN DONG1, SHUANGYU DONG 2, YONG CHEN1,
RENWEI ZUO 1, AND ZONGCHENG LIU 1
1Aeronautics Engineering College, Air Force Engineering University, Xi’an 710038, China
2SMZ Telecom Pty., Ltd., Melbourne, VIC 3130, Australia
Corresponding author: Zongcheng Liu (liu434853780@163.com)
This work was supported in part by the National Natural Science Foundation of China under Grant 61304120,
Grant 61473307, and Grant 61603411.
ABSTRACT This paper presents a global dynamic surface control (DSC) method for a class of uncertain
multi-input/multi-output (MIMO) pure-feedback nonlinear systems with non-affine functions possibly being
in-differentiable. It is well known that the traditional DSC method is commonly used for reducing the design
complexity of the backstepping control method; however, the regulation results of the DSC method are semi-
global uniformly ultimately bounded (SGUUB). An improved DSC (IDSC) method is first designed in this
paper so that the results are global uniformly ultimately bounded (GUUB). Comparing with the traditional
DSC method, the parameters of first-order filters in IDSC are time varying rather than constants. The control
design for MIMO pure-feedback nonlinear systems researched is much more complex than the SISO cases,
and the presence of in-differentiable non-affine functions considered in this paper makes the control design
even more difficult. Therefore, we proposed the IDSC method, which can significantly reduce the complexity
of the control design for the MIMO pure-feedback nonlinear systems in cooperation with the backstepping
method, and it is proved that IDSC can guarantee the GUUB of all the signals of the system. Finally,
the simulation results are provided to demonstrate the effectiveness of the designed method.
INDEX TERMS Robust adaptive control, improved dynamic surface control (IDSC), pure-feedback systems,
MIMO nonlinear systems.
I. INTRODUCTION
Over the past decades, adaptive-control schemes were exten-
sively used to cope with the control problems of nonlinear
systems with unknown nonlinearities. Although the existent
approaches can provide an effective methodology to con-
trol those uncertain nonlinear systems, most of the results
are only suitable for the single-input/single-output (SISO)
nonlinear systems [1]–[17]. For MIMO nonlinear systems,
where uncertain interconnection usually exists among vari-
ous inputs and outputs, the control problem becomes much
more complex. In order to cope with the control problem of
MIMO interconnected nonlinear system, a class of adaptive
backstepping design approaches were proposed [18]–[22].
For instance, by effectively combining the backstep-
ping approach and adaptive fuzzy-logic control, Lee [21]
The associate editor coordinating the review of this manuscript and
approving it for publication was Wei Xu.
presented a robust adaptive control method for a class
of MIMO nonlinear systems with couplings among input
channels. In the work of Chen et al. [22], by using the
backstepping design approaches, a novel adaptive neural
control design approach was proposed for a class of non-
linear MIMO time-delay systems in block-triangular form.
In order to improve the tracking performance and robustness
of backstepping control, Sui et al proposed a novel finite
time control method and supplied it to the MIMO nonlinear
systems [23]–[25]. Although the backstepping control design
proposed by the aforementioned literatures had provided
effective process for the control problem of MIMO inter-
connected nonlinear system, there existed limitations in the
existing works as follows: 1) there is few work on the control
of MIMO pure-feedback systems with in-differentiable non-
affine functions for three reasons that, firstly, MIMO sys-
tems control design is much more complicated than SISO
ones, secondly, pure-feedback systems control design is
96672 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ VOLUME 7, 2019

2. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
much more difficult than strict-feedback ones, thirdly, a few
results for systems with in-differentiable non-affine functions
can be found, while unfortunately they cannot be used for
MIMO systems directly. 2) DSC method is always consid-
ered as the best choice for solving the control problems on
MIMO systems, however, the regulation results are only
SGUUB for all the researches related to DSC method since
the DSC is basically a SGUUB method. The details are
discussed as follows.
For nonlinear systems in pure-feedback forms, the con-
trol design become quite difficult because pure-feedback
nonlinear systems contain non-affine functions, which have
no affine appearance of the variables to be used as virtual
and actual control. To overcome the design difficulty of
pure-feedback system, several adaptive backstepping con-
trol approaches were developed for SISO [26]–[28] and
MIMO [29], [30] nonlinear systems. For example, to over-
come the design difficulty from non-affine structure of pure-
feedback system, mean value theorem was exploited to
deduce affine appearance of state variables, and the adap-
tive neural tracking control for a class of non-affine pure-
feedback systems with multiple unknown state time-varying
delays was proposed [26]. The implicit function theorem
was employed to demonstrate the existence of an ideal con-
troller that can achieve control objective, and neural net-
work or fuzzy system is used to construct this unknown ideal
implicit controller [27], [28]. In the work of Tong et al. [29],
the filtered signals were introduced to circumvent algebraic
loop problem existing in the controller design for the non-
linear pure-feedback systems, and an adaptive fuzzy output
feedback control law was proposed for a class of uncertain
MIMO pure-feedback nonlinear systems with immeasurable
states. In the work of Sui et al. [30], an adaptive fuzzy output
feedback tracking control approach was developed for a class
of MIMO stochastic pure-feedback nonlinear systems with
input saturation based on the backstepping recursive design
technique. However, the problem ‘‘explosion of complexity’’
arising due to repeated differentiation of intermediate vari-
ables made the backstepping method difficult to implement
in practice [31]. Thus, dealing with the control problem
of MIMO pure-feedback nonlinear system by backstepping
method will make the controller design much more compli-
cated. Furthermore, it is difficult to apply the above method
to the nonlinear system with non-affine non-differentiable
functions.
To cope with the aforementioned problem ‘‘explosion of
complexity’’, the dynamic surface control (DSC) method
were proposed by Swaroop et al. [32] to improve the back-
stepping control method. Subsequently the DSC method
was applied in the control design of pure -feedback nonlin-
ear systems and the stability analysis of DSC control was
provided [33]–[36]. By combining the DSC method and
mean value theorem, the robust stabilization problem was
discussed for a class of non-affine pure-feedback systems
with unknown time-delay functions and perturbed uncertain-
ties [33]. By using the piecewise functions to model the
non-affine functions to an affine form, Liu et al. [34]–[36]
proposed a novel adaptive DSC control scheme for pure-
feedback nonlinear systems with the non-affine functions
being non-differentiable. Furthermore, to improve the tran-
sient performance of DSC method, predictors were incorpo-
rated into the DSC design, which used the prediction errors
instead of tracking errors to update the adaptive parame-
ters [37]–[39]. However, the above methods are all restricted
to the SISO nonlinear systems. Moreover, though the DSC
method has simplified the complex control design process of
the backstepping method by introducing a first-order filter to
estimate the differential of the virtual control law, the perfor-
mance of the DSC control schemes will be affected inevitably
unfortunately, for the reason that the state variables have to be
limited in a series of compact sets to guarantee the stability
of the filters, that is, the DSC method can only guarantee
boundedness of system signals semi-globally [32]. In order
to overcome the disadvantage of dynamic surface control,
several global control methods were proposed. In [40], non-
linear adaptive filters instead of the first-order low pass
ones were introduced to avoid the repeated differential of
the virtual control signals. In [41], by combining the neural
controller with robust controller, a new switching mechanism
was designed to pull the transient states back into the neural
approximation domain from the outside, thereby realizing the
global boundedness of the closed loop system signals. How-
ever, the methods proposed by [43], [44] can only be applied
to the control problem of SISO strict-feedback system, which
are not applicable to MIMO pure-feedback system. To the
best of our knowledge, few global dynamic surface methods
have been proposed for MIMO pure feedback systems.
Motivated by the above discussion, in this paper, a novel
robust adaptive improved dynamic surface control (IDSC)
approach is proposed for a class of MIMO pure-feedback
nonlinear systems with in-differentiable non-affine func-
tions. In particular, the system investigated in this paper
has a more general control structure with each subsystem
being of completely non-affine pure-feedback form, and cou-
plings are nonlinearly existed in every subsystem equation.
By using the IDSC method, a systematic design procedure
is then developed for the design of a novel robust adaptive
IDSC control.
The main contributions of this paper are summarized as
follows.
1) In order to overcome the disadvantage of traditional
DSC method, a global dynamic surface control method
is first designed in this paper by introducing first order
sliding mode differentiator. Comparing with the tradi-
tional DSC method, a globally uniformly ultimately
bounded result is achieved due to the approximation
performance of first order sliding mode differentia-
tor, and the limitation of compact sets is removed
simultaneously.
2) The controllable condition for MIMO non-affine pure-
feedback nonlinear systems has been given as shown as
Assumption 1 which can guarantee the controllability
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3. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
of systems with only a relaxed condition that the non-
affine functions are continuous.
3) The restrictive conditions that the uncertain non-affine
functions must be derivable and the sign of the gain
function must be known are removed for MIMO non-
affine pure-feedback system, which means that our
approach can be applied more widely.
The remainder of this paper is organized as follows.
Section II gives the problem formulation and preliminaries.
In Section III, the adaptive tracking controller is designed for
the MIMO nonlinear system with the non-affine functions
being in-differentiable. The stability analysis of the closed-
loop system is given in Section IV. The simulation examples
are given to demonstrate the effectiveness of the proposed
method in Section V and followed by Section VI which
concludes this paper.
II. PROBLEM DESCRIPTION AND PRELIMINARIES
Consider the following MIMO nonlinear systems with each
subsystem having the completely non-affine pure-feedback
form [42]:





ẋj,ij = fj,ij (x̄1,(ij−ηj1), . . . , x̄m,(ij−ηjm), xj,ij+1) + dj,ij (t),
ẋj,ρj = fj,ρj (X, ūj) + dj,ρj (t)
yj = xj,1, ij = 1, 2, . . . , ρj − 1, j = 1, 2, . . . , m
(1)
where xj,ij ∈ R denotes the ij th state of the j th subsystem;
uj ∈ R is the input of the j th subsystem; yj ∈ R is the output
of the j th subsystem; fj,ij is the unknown nonlinear functions;
ρj is the order of the j th subsystem; dj,ij (t) ∈ R is the external
disturbance; X = [x̄1,ρ1 , x̄2,ρ2 , . . . , x̄m,ρm ]T ∈ R
Pm
k=1 ρk is
the vector of all state variables in the system; and x̄j,ij =
[xj,1, xj,2, . . . , xj,ij ]T ∈ Rij , ūj = [u1, u2, . . . , uj]T ∈ Rj,
ηjl = ρj − ρl, l = 1, 2, . . . , m. For convenience, we denote
4j,ij = [x̄1,(ij−ηj1), . . . , x̄m,(ij−ηjm)]T , ij = 1, 2, . . . , ρj −1 and
4j,ρj = [XT , ūT
j−1]T .
There exist three cases to be considered for the order
differences ηjl [42], [43]: 1) when j = l, that is ηjl = 0, then
the state vector x̄j,(ij−ηjl) = x̄j,ij exists in (1); 2) when j 6= l
and ij − ηjl ≤ 0, then the state vector x̄l,(ij−ηjl) does not exist,
and does not appear in(1); 3) when j 6= l and ij − ηjl > 0,
then state vector x̄l,(ij−ηjl) exists in(1).
Assumption 1: Define Fj,ij (4j,ij , xj,ij+1) = fj,ij (4j,ij ,
xj,ij+1) − fj,ij (4j,ij , 0), ij = 1, 2, . . . , ρj. Denote xj,ρj+1 = uj,
j = 1, 2, . . . , m. We assume that functions fj,ij (4j,ij , 0) and
Fj,ij (4j,ij , xj,ij+1) satisfy:
fj,ij (4j,ij , 0) ≤ θj,ij ϕj,ij (4j,ij ) (2)
and











Fj,ij
(4j,ij )xj,ij+1 + Cj,ij
(4j,ij ) ≤ Fj,ij (4j,ij , xj,ij+1)
≤ F̄j,ij (4j,ij )xj,ij+1 + C̄j,ij (4j,ij ), xj,ij+1 ≥ 0
F0
j,ij
(4j,ij )xj,ij+1 + C0
j,ij
(4j,ij ) ≤ Fj,ij (4j,ij , xj,ij+1)
≤ F̄0
j,ij
(4j,ij )xj,ij+1 + C̄0
j,ij
(4j,ij ), xj,ij+1 < 0
(3)
where θj,ij are unknown constants, and ϕj,ij (4j,ij ) are some
known positive continuous functions, Fj,ij
(4j,ij ), F̄j,ij (4j,ij ),
F0
j,ij
(4j,ij ) and F̄0
j,ij
(4j,ij ) are unknown positive continu-
ous functions, while Cj,ij
(4j,ij ), C̄j,ij (4j,ij ), C0
j,ij
(4j,ij ) and
C̄0
j,ij
(4j,ij ) are unknown continuous functions. Assume that
there exist unknown positive constants Fj,ijm, Fj,ijM , F̄j,ijm,
F̄j,ijM , F0
j,ijm, F0
j,ijM , F̄0
j,ijm, F̄0
j,ijM , Cj,ijM , C̄j,ijM , C0
j,ijM , C̄0
j,ijM
which satisfy
0 ≤ Fj,ijm ≤ Fj,ij
(4j,ij ) ≤ Fj,ijM (4)
0 ≤ F̄j,ijm ≤ F̄j,ij (4j,ij ) ≤ F̄j,ijM (5)
0 ≤ F0
j,ijm ≤ F0
j,ij
(4j,ij ) ≤ F0
j,ijM (6)
0 ≤ F̄0
j,ijm ≤ F̄0
j,ij
(4j,ij ) ≤ F̄0
j,ijM (7)
0 ≤ |Cj,ij
(4j,ij )| ≤ Cj,ijM (8)
0 ≤ |C̄j,ij (4j,ij )| ≤ C̄j,ijM (9)
0 ≤ |C0
j,ij
(4j,ij )| ≤ C0
j,ijM (10)
0 ≤ |C̄0
j,ij
(4j,ij )| ≤ C̄0
j,ijM (11)
Remark 1: It should be pointing out that ϕj,ij (4j,ij ) can
be called as ‘‘core function’’ for it contains the deep-rooted
information from the uncertain nonlinearity fj,ij (4j,ij , 0).
Much more details on the rationality of assumption for
fj,ij (4j,ij , 0) can be seen in [44], which also gives some illus-
trations on the selection of function fj,ij (4j,ij , 0), and it is also
worth to mentioning that ϕj,ij (4j,ij ) can be chosen as neural
core functions, such as Gaussian functions, if fj,ij (4j,ij , 0)
can be approximated by neural networks [44]. This means
the completely unknown nonlinearity which can tackled by
neural networks can be also solved by choosing the core
function ϕj,ij (4j,ij ) as neural core functions.
It is noticeable that MIMO system [42], where each subsys-
tem is assumed to satisfy gj,ij
< |∂fj,ij (x̄j,ij , xj,ij+1 )
∂xj,ij+1 | ≤
ḡj,ij , (gj,ij
, ḡj,ij ∈ R), requires that fj,ij (x̄j,ij , xj,ij+1 ) must be
differentiable for the sake of using the mean value theo-
rem. In practice, non-smooth nonlinearities exist in a wide
range of real control systems [45]–[47], which leads to in-
differentiable for fj,ij (x̄j,ij , xj,ij+1 ). Thus, the methods pro-
posed in [42] would come across troubles when applied to
those systems. Contrastively, (3) is more general in the sense
that the derivative of fj,ij (x̄j,ij , xj,ij+1 ) is not involved in the
assumption.
From(3), it can be found that there exist unknown functions
ϑj,ij (4j,ij , xj,ij+1) and ϑ0
j,ij
(4j,ij , xj,ij+1) taking values in the
closed interval [0, 1] and satisfying





















Fj,ij (4j,ij , xj,ij+1)
= (1−ϑj,ij (4j,ij , xj,ij+1))(Fj,ij
(4j,ij )xj,ij+1+Cj,ij
(4j,ij ))
+ϑj,ij (4j,ij , xj,ij+1)(F̄j,ij (4j,ij )xj,ij+1+C̄j,ij (4j,ij )), xj,ij+1 ≥0
Fj,ij (4j,ij , xj,ij+1)
= (1−ϑ0
j,ij
(4j,ij , xj,ij+1))(F0
j,ij
(4j,ij )xj,ij+1 + C0
j,ij
(4j,ij ))
+ϑ0
j,ij
(4j,ij , xj,ij+1)(F̄0
j,ij
(4j,ij )xj,ij+1 + C̄0
j,ij
(4j,ij )), xj,ij+1 0
(12)
96674 VOLUME 7, 2019

4. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
To make the control system design succinct, define func-
tions Gj,ij (4j,ij , xj,ij+1) and Hj,ij (4j,ij , xj,ij+1) as follows
Gj,ij (4j,ij , xj,ij+1)
=









(1 − ϑj,ij (4j,ij , xj,ij+1))Fj,ij
(4j,ij )
+ϑj,ij (4j,ij , xj,ij+1)F̄j,ij (4j,ij ), xj,ij+1 ≥ 0
(1 − ϑ0
j,ij
(4j,ij , xj,ij+1))F0
j,ij
(4j,ij )+
ϑ0
j,ij
(4j,ij , xj,ij+1)F̄0
j,ij
(4j,ij ), xj,ij+1 0
(13)
Hj,ij (4j,ij , xj,ij+1)
=











(1 − ϑj,ij (4j,ij , xj,ij+1))Cj,ij
(4j,ij )
+ϑj,ij (4j,ij , xj,ij+1)C̄j,ij (4j,ij ), xj,ij+1 ≥ 0
(1 − ϑ0
j,ij
(4j,ij , xj,ij+1))C0
j,ij
(4j,ij )
+ϑ0
j,ij
(4j,ij , xj,ij+1)C̄0
j,ij
(4j,ij ) , xj,ij+1 0
(14)
Using (13), (14), we can model the non-affine terms
Fj,ij (4j,ij , xj,ij+1) as follows
Fj,ij (4j,ij , xj,ij+1)=Gj,ij (4j,ij , xj,ij+1)xj,ij+1+Hj,ij (4j,ij , xj,ij+1)
(15)
From (15), the systems (1) can be represented as follows:



ẋj,ij = fj,ij (4j,ij , 0) + Gj,ij (4j,ij , xj,ij+1)xj,ij+1
+Hj,ij (4j,ij , xj,ij+1) + dj,ij (t),
yj = xj,1, ij = 1, 2, . . . , ρj, j = 1, 2, . . . , m
(16)
According to (4)-(11), we can further have
0 ≤ Gjm ≤ Gj,ij (4j,ij , xj,ij+1) ≤ GjM
0 ≤ |Hj,ij (4j,ij , xj,ij+1)| ≤ C∗
j,ij
(17)
where
Gjm = min{Fj,ijm, F̄j,ijm, F0
j,ijm, F̄0
j,ijm}
GjM = max{Fj,ijM , F̄j,ijM , F0
j,ijM , F̄0
j,ijM }
C∗
j,ij
= max{|Cj,ijM + C̄j,ijM |, |C0
j,ijM + C̄0
j,ijM |}.
Remark 2: Obviously, the derivative of fj,ij (4j,ij , xj,ij+1)
is not involved in the aforementioned modeling process,
so fj,ij (4j,ij , xj,ij+1) need not to be differentiable and the
assumption that the sign of gain function must be known is
removed.
Assumption 2: The desired trajectory yd = [yd1,
yd2, . . . , ydm]T ∈ Rm are sufficiently smooth functions of t
and yd , ẏd and ÿd are bounded.
Assumption 3: For 1 ≤ ij ≤ ρj, 1 ≤ j ≤ m, there exist
unknown positive constants d∗
j,ij
such that

6. dj,ij (t)

8. ≤ d∗
j,ij
.
Lemma 1 [34]: Consider the dynamic system of the form
χ̇(t) = −aχ(t) + pw(t), where a and p are positive constants
and w(t) is a positive function. Then, for any given bounded
initial condition χ(t0) ≥ 0, we have χ(t) ≥ 0, ∀t ≥ 0.
Lemma 2 [34]: Hyperbolic tangent function tanh(·) will
be used in this paper, and it is well known that tanh(·) is
continuous and differentiable, and it fulfills



0 ≤ |q| − q tanh
q
υ
≤ 0.2785υ
0 ≤ q tanh
q
υ
(18)
for any q ∈ R and ∀υ 0.
Lemma 3 [48]: The first order sliding mode differentiator
is designed as
ρ̇0 = ζ0 = −τ0 |ρ0 − f (t)|
1
2 sign(ρ0 − f (t)) + ρ1
ρ̇1 = −τ1sign(ρ1 − ζ0) (19)
where ρ0, ρ1 and ζ0 are the state variables of the differentiator,
τ0 and τ1 are the designed parameters of the first order sliding
mode differentiator, and f (t) is a known function. Then,
ζ0 can approximate the differential term ˙
f (t) to any arbitrary
accuracy if the initial deviations ρ0 − f (t0) and ζ0 − ˙
f (t0) are
bounded.
Lemma 4: For any x ∈ R, the following inequality holds

13. |x|
1
2 sign(x) −
x tanh
x
µ
1
2
tanh
x
µ

18. ≤ γ (20)
where µ is a designed parameter and γ is any unknown
positive constants.
Proof: See the Appendix.
III. ADAPTIVE TRACKING CONTROLLER DESIGN
In this section, adaptive tracking control for MIMO sys-
tem (1) is presented based on the backstepping approach. The
recursive design procedure for each subsystem contains ρj
steps. At each recursive ij, the virtual stabilizing control αj,ij
is designed to make the system toward equilibrium position.
Finally, the actual control law uj is designed in step ρj.
To avoid repeatedly differentiating αj,ij , which leads to the
so-called ‘‘explosion of complexity’’, in the sequel, the
DSC technique [32] is employed.
To start, consider the following change of coordinates:
(
ej,1 = xj,1 − ydj
ej,ij = xj,ij − αj,ijf
(21)
where ej,ij is the tracking error of every subsystem and αj,ijf
is the output of the following first-order filters:
τj,ij+1α̇j,ij+1f + αj,ij+1f = αj,ij (22)
with αj,ij as the input and αj,ij+1f (0) = αj,ij (0).
By defining the output error of the filter as yj,ij+1 =
αj,ij+1f − αj,ij , it yields α̇j,ij+1f = −(yj,ij+1/τj,ij+1) and
the boundedness of yj,ij+1 will be proved in the following
part.
Lemma 5: Let 1
τj,ij+1
= 2ˆ
α̇2
j,ij
+ 2εj,ij1, where εj,ij1 is a
positive design constant and ˆ
α̇j,ij is the estimate of α̇j,ij which
defined later. Then we can have:
yj,ij+1| ≤ y∗
j,ij+1 (23)
where y∗
j,ij+1 is any positive constant.
Proof: Consider the following quadratic Lyapunov func-
tion candidate:
Vyj,ij+1 =
1
2
y2
j,ij+1 (24)
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19. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
The time derivative of Vyj,ij+1 is
V̇yj,ij+1 = yj,ij+1ẏj,ij+1
= yj,ij+1(α̇j,ij+1f − α̇j,ij )
= −
y2
j,ij+1
τj,ij+1
− yj,ij+1α̇j,ij (25)
To avoid the tedious analytic computation, the following
first order sliding mode differentiator according to Lemma 3
and 4 is adopted to estimate the differential term α̇j,ij :



ρ̇j,ij0 = ζj,ij0
= −j,ij0|ρj,ij0 − αj,ij |
1
2 sign(ρj,ij0 − αj,ij ) + ρj,ij1
ρ̇j,ij1 = −j,ij1sign(ρj,ij1 − ζj,ij0)
(26)
where ρj,ij0, ρj,ij1 and ζj,ij0 are the states of the system(26),
and j,ij0, j,ij1 are positive design constants.
By virtue of the approximation property of the first order
sliding mode differentiator, we have
|ζj,ij0 − α̇j,ij | ≤ εj,ij01 (27)
where εj,ij1 is any positive constant.
Define
ζ̂j,ij0 = −j,ij0((ρj,ij0 − αj,ij ) tanh(
ρj,ij0 − αj,ij
µj,ij
))
1
2
× tanh(
ρj,ij0 − αj,ij
µj,ij
) + ρj,ij1 (28)
where ζ̂j,ij0 is the estimate of the auxiliary variable ζj,ij0.
According to Lemma 4, we can know that

22. ζj,ij0 − ζ̂j,ij0

25. ≤ εj,ij02 (29)
Then we can further have

28. ζ̂j,ij0 − α̇j,ij

31. ≤

34. ζ̂j,ij0 − ζj,ij0 + ζj,ij0 − α̇j,ij

37. ≤ εj,ij01 + εj,ij02 = εj,ij0 (30)
hence ζ̂j,ij0 can be regarded as the estimate of α̇j,ij . Denote
ˆ
α̇j,ij = ζ̂j,ij0 and 1
2τj,ij+1
= ˆ
α̇2
j,ij
+ εj,ij1, then we arrive at
−
y2
j,ij+1
2τj,ij+1
− yj,ij+1α̇j,ij
= −y2
j,ij+1(ˆ
α̇2
j,ij
+ εj,ij1) − yj,ij+1α̇j,ij
≤
1
4
− εj,ij1y2
j,ij+1 − |ˆ
α̇j,ij ||yj,ij+1| − yj,ij+1α̇j,ij
≤
1
4
− εj,ij1y2
j,ij+1 + |εj,ij0||yj,ij+1| (31)
Then we can further rewrite (25) as
V̇yj,ij+1 ≤ −
y2
j,ij+1
2τj,ij+1
+
1
4
− εj,ij1y2
j,ij+1 + |εj,ij0||yj,ij+1|
≤ −(
1
2τj,ij+1
+ εj,ij1 −
1
2
)y2
j,ij+1 +
ε2
j,ij0
2
+
1
4
≤ −Cj,ij+1Vyj,ij+1 + C0
j,ij+1 (32)
where Cj,ij+1 = ( 1
τj,ij+1
+ 2εj,ij1 − 1) and C0
j,ij+1 =
ε2
j,ij0
2 + 1
4 . Therefore, by appropriately online-tuning the design
parameters j,ij0 and j,ij1, the output error of filter (22)
can be regulated to an arbitrary small range. Thus, we have
|yj,ij+1| ≤ y∗
j,ij+1.
The proof is completed.
Remark 3: By virtue of the approximation property of the
first order sliding mode differentiator (26), an auxiliary vari-
able ζj,ij0 is designed to estimate α̇j,ij .However, it is noticeable
that the tracking differentiator based on Lemma 3 is discon-
tinuous owing to the sign functions that are employed, which
can affect the closed loop performance severely. To overcome
this problem, ζ̂j,ij0 is then designed according to Lemma 4 by
employing hyperbolic tangent function to ensure the feasi-
bility in backstepping process. Therefore the stability and
boundedness of the output error yj,ij+1 of filter (22) have been
proved according to (32), that is, yj,ij+1 can be render to arbi-
trary small by appropriately tuning the design parameter j,ij0
and j,ij1. In the relevant literatures, a series of compact sets
are generally defined to analyze the boundedness of the out-
put error of filter (please refer to [46]–[50] for details), which
can only guarantee the semi-global boundedness of yj,ij+1.
However, the novel scheme of Lemma 5 has removed the
restriction for the initial conditions of system variables, which
can guarantee the global boundedness of yj,ij+1. Moreover,
the stability analysis process will be simplified greatly for the
reason that the stability of yj,ij+1 have been proved at each
recursive step.
Noting xj,ij+1 = ej,ij+1 + αj,ij+1f and yj,ij+1 = αj,ij+1f −
αj,ij , we have:
xj,ij+1 = ej,ij+1 + αj,ij + yj,ij+1 (33)
Construct the virtual control laws αj,ij (ij = 1, . . . , ρj − 1)
and the actual control law uj as follows:
αj,ij = −kj,ij ej,ij − δ̂j,ij tanh(
ej,ij
ςj,ij
) − ωj,ij α̇j,ijf
× tanh(
ej,ij α̇j,ijf
ςj,ij
) −
θ̂j,ij ej,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) (34)
uj = −kj,ρj ej,ρj − δ̂j,ρj tanh(
ej,ρj
ςj,ρj
) − ωj,ρj α̇j,ρjf
× tanh(
ej,ρj α̇j,ρjf
ςj,ρj
) −
θ̂j,ρj ej,ρj
2a2
j,ρj
ϕ2
j,ρj
(4j,ρj ) (35)
where kj,ij 0, ςj,ij 0, aj,ij 0, θ̂j,ij is the estimate of the
unknown constant θ∗
j,ij
, and θ∗
j,ij
= G−1
jm θ2
j,ij
.
The adaptive laws are chosen as:
˙
δ̂j,ij = γj,ij ej,ij tanh(
ej,ij
ςj,ij
) − σj,ij γj,ij δ̂j,ij (36)
˙
θ̂j,ij =
βj,ij e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) − σj,ij βj,ij θ̂j,ij (37)
96676 VOLUME 7, 2019

38. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
where, σj,ij 0, γj,ij 0, βj,ij 0 and ωj,ij ≥ G−1
jm are
the design parameters. According to Lemma 1, for any given
bounded initial condition δ̂j,ij (0) ≥ 0, θ̂j,ij (0) ≥ 0 we have
δ̂j,ij (t) ≥ 0, θ̂j,ij (t) ≥ 0 for ∀t ≥ 0.
Step ij (1 ≤ ij ≤ ρj − 1): Denote αj,1f = ydj. Then, noting
ej,ij = xj,ij − αj,ijf , Fj,ij (4j,ij , xj,ij+1) = fj,ij (4j,ij , xj,ij+1) −
fj,ij (4j,ij , 0) and(15), we have.
ėj,ij = ẋj,ij − α̇j,ijf
= fj,ij (4j,ij , xj,ij+1) + dj,ij (t) − α̇j,ijf
= fj,ij (4j,ij , 0) + Gj,ij (4j,ij , xj,ij+1)xj,ij+1
+ Hj,ij (4j,ij , xj,ij+1) + dj,ij (t) − α̇j,ijf (38)
Consider the stabilization of subsystem (38) and the fol-
lowing quadratic Lyapunov function candidate
Vej,ij =
1
2
e2
j,ij
(39)
The time derivative of Vej,ij along (38) is
V̇ej,ij = ej,ij ėj,ij
= ej,ij (fj,ij (4j,ij , 0) + Gj,ij (4j,ij , xj,ij+1)xj,ij+1
+ Hj,ij (4j,ij , xj,ij+1) + dj,ij (t) − α̇j,ijf ) (40)
Utilizing (17) and Assumption 3, we can rewrite (40) as:
V̇ej,ij ≤ ej,ij fj,ij (4j,ij , 0) + ej,ij Gj,ij (4j,ij , xj,ij+1)xj,ij+1
+ |ej,ij |C∗
j,ij
+ |ej,ij |d∗
j,ij
− ej,ij α̇j,ijf
≤
θ2
j,ij
e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) +
a2
j,ij
2
+ |ej,ij |Gjmδ∗
j,ij
+ ej,ij Gj,ij (4j,ij , xj,ij+1)xj,ij+1 − ej,ij α̇j,ijf (41)
where aj,ij is any positive constant, δ∗
j,ij
= G−1
jm (C∗
j,ij
+ d∗
j,ij
)
with δ̂j,ij being its estimate.
Substituting (33) and (34) into (41) yields
V̇ej,ij ≤ Gj,ij (4j,ij , xj,ij+1)(ej,ij+1 + yj,ij+1)ej,ij − kj,ij Gjme2
j,ij
− Gjmωj,ij ej,ij α̇j,ijf tanh(
ej,ij α̇j,ijf
ςj,ij
) − ej,ij α̇j,ijf
− Gjmδ̂j,ij ej,ij tanh(
ej,ij
ςj,ij
) + |ej,ij |Gjmδ∗
j,ij
+
a2
j,ij
2
−
Gjmθ̂j,ij e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) +
θ2
j,ij
e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) (42)
Noting that ωj,ij ≥ G−1
jm , we get
V̇ej,ij ≤ Gj,ij (4j,ij , xj,ij+1)(ej,ij+1 + yj,ij+1)ej,ij +
a2
j,ij
2
+ |ej,ij α̇j,ijf | − ej,ij α̇j,ijf tanh(
ej,ij α̇j,ijf
ςj,ij
)
+ |ej,ij |Gjmδ∗
j,ij
− Gjmδ̂j,ij ej,ij tanh(
ej,ij
ςj,ij
)
−
Gjmθ̃j,ij e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij ) − kj,ij Gjme2
j,ij
(43)
where θ̃j,ij = θ∗
j,ij
− θ̂j,ij .
Consider the following Lyapunov function candidate
Vj,ij = Vej,ij +
Gjmδ̃2
j,ij
2γj,ij
+
Gjmθ̃2
j,ij
2βj,ij
(44)
where δ̃j,ij = δ∗
j,ij
− δ̂j,ij .
From (43), the time derivative of Vj,ij is
V̇j,ij ≤ V̇ej,ij −
Gjmδ̃j,ij
˙
δ̂j,ij
γj,ij
−
Gjmθ̃j,ij
˙
θ̂j,ij
βj,ij
≤ Gj,ij (4j,ij , xj,ij+1)(ej,ij+1 + yj,ij+1)ej,ij − kj,ij Gjme2
j,ij
+ Gjmδ∗
j,ij
[|ej,ij | − ej,ij tanh(
ej,ij
ςj,ij
)] +
a2
j,ij
2
+ [|ej,ij α̇j,ijf | − ej,ij α̇j,ijf tanh(
ej,ij α̇j,ijf
ςj,ij
)]
−
Gjmδ̃j,ij
γj,ij
[˙
δ̂j,ij − γj,ij ej,ij × tanh(
ej,ij
ςj,ij
)]
−
Gjmθ̃j,ij
βj,ij
[ ˙
θ̂j,ij −
βj,ij e2
j,ij
2a2
j,ij
ϕ2
j,ij
(4j,ij )] (45)
According to Lemma 2 and (36), (37), it follows from (45)
that
V̇j,ij ≤ Gj,ij (4j,ij , xj,ij+1)(ej,ij+1 + yj,ij+1)ej,ij − kj,ij Gjme2
j,ij
+ 0.2785ςj,ij + 0.2785Gjmδ∗
j,ij
ςj,ij +
a2
j,ij
2
+ Gjmσj,ij δ̃j,ij δ̂j,ij + Gjmσj,ij θ̃j,ij θ̂j,ij (46)
Step ρj: Noting that ej,ρj = xj,ρj − αj,ρjf , the dynamics of
ej,ρj -subsystem can be written as
ėj,ρj = fj,ρj (X, ūj) + dj,ρj (t) − α̇j,ρjf (47)
Similarly, choosing the quadratic function Vej,ρj as Vej,ρj =
e2
j,ρj
/2 and noting 4j,ρj = [XT , ūT
j−1]T , we have
V̇ej,ρj = ej,ρj [fj,ρj (4j,ρj , 0) + Gj,ρj (4j,ρj , uj)uj
+ Hj,ρj (4j,ρj , uj) + dj,ρj (t) − α̇j,ρjf ]
≤ Gj,ρj (4j,ρj , uj)ujej,ρj + |ej,ρj |Gjmδ∗
j,ρj
− ej,ρj α̇j,ρjf +
θ2
j,ρj
e2
j,ρj
2a2
j,ρj
ϕ2
j,ρj
(4j,ρj ) +
a2
j,ρj
2
(48)
where δ∗
j,ρj
= G−1
j,ρjm(C∗
j,ρj
+d∗
j,ρj
) with δ̂j,ρj being its estimate.
Substituting actual control law (35) into (48) yields
V̇ej,ρj ≤ −kj,ρj Gjme2
j,ρj
−
Gjmθ̃j,ρj e2
j,ρj
2a2
j,ρj
ϕ2
j,ρj
(4j,ρj ) +
a2
j,ρj
2
− Gjmδ̂j,ρj ej,ρj tanh(
ej,ρj
ςj,ρj
) + |ej,ρj |Gjmδ∗
j,ρj
− ej,ρj α̇j,ρjf tanh(
ej,ρj α̇j,ρjf
ςj,ρj
) + |ej,ρj α̇j,ρjf | (49)
VOLUME 7, 2019 96677

39. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
Consider the following Lyapunov function candidate:
Vj,ρj = Vej,ρj +
Gjmδ̃2
j,ρj
2γj,ρj
+
Gjmθ̃2
j,ρj
2βj,ρj
(50)
where δ̃j,ρj = δ∗
j,ρj
− δ̂j,ρj , θ̃j,ρj = θ∗
j,ρj
− θ̂j,ρj .
From (49), the time derivative of Vj,ρj is
V̇j,ρj ≤ V̇ej,ρj −
Gjmδ̃j,ρj
˙
δ̂j,ρj
γj,ρj
−
Gjmθ̃j,ρj
˙
θ̂j,ρj
βj,ρj
≤ −kj,ρj Gjme2
j,ρj
+
a2
j,ρj
2
+ Gjmδ∗
j,ρj
[|ej,ρj | − ej,ρj tanh(
ej,ρj
ςj,ρj
)]
−
Gjmδ̃j,ρj
γj,ρj
[˙
δ̂j,ρj − γj,ρj ej,ρj tanh(
ej,ρj
ςj,ρj
)]
+ [|ej,ρj α̇j,ρjf | − ej,ρj α̇j,ρjf tanh(
ej,ρj α̇j,ρjf
ςj,ρj
)]
−
Gjmθ̃j,ρj
βj,ρj
[ ˙
θ̂j,ρj −
βj,ρj e2
j,ρj
2a2
j,ρj
ϕ2
j,ρj
(4j,ρj )] (51)
According to Lemma 2 and (36), it follows from (51) that
V̇j,ρj ≤ −kj,ρj Gjme2
j,ρj
+ 0.2785Gjmδ∗
j,ρj
ςj,ρj + 0.2785ςj,ρj
+ Gjmσj,ρj δ̃j,ρj δ̂j,ρj + Gjmσj,ρj θ̃j,ρj θ̂j,ρj +
a2
j,ρj
2
(52)
The design process of the adaptive tracking controller has
been completed.
IV. STABILITY ANALYSIS
In this section, the main results of this paper are stated,
the stability analysis of the closed-loop system is given,
and the global boundedness of all the signals will be
proved.
Theorem 1: Consider the non-affine pure-feedback nonlin-
ear system (1) under Assumptions 1-3. The virtual control
laws are determined as (34), and the adaptation laws are given
by (36) and (37). The actual adaptive controller is constructed
by (35) with the adaptation laws given by (36) and (37). Given
initial conditions δ̂j,ij ≥ 0, θ̂j,ij ≥ 0 there exist kj,ij , aj,ij , ςj,ij ,
σj,ij , γj,ij , βj,ij , ωj,ij j,ij0, j,ij1 and εj,ij1 which can make that:
1) all of the signals in the closed-loop system are global
bounded;
2) the tracking error ē1 = [e1,1, e2,1, . . . , em,1]T can be
regulated to an arbitrary small neighborhood of the
origin.
Proof: 1) Consider the Lyapunov function as follows:
V =
m
X
j=1
ρj
X
ij=1
Vj,ij (53)
According to (46) and(52), the time derivative of V is:
V̇ ≤
m
X
j=1
ρj
X
ij=1
[ − kj,ij Gjme2
j,ij
+ Gjmσj,ij δ̃j,ij δ̂j,ij + 0.2785ςj,ij
+ Gjmσj,ij θ̃j,ij θ̂j,ij + 0.2785Gjmδ∗
j,ij
ςj,ij +
a2
j,ij
2
]
+
m
X
j=1
ρj−1
X
ij=1
[Gj,ij (4j,ij , xj,ij+1)(ej,ij+1+yj,ij+1)ej,ij ] (54)
It is noteworthy that the unknown coupling term
Gj,ij (4j,ij , xj,ij+1) in (54) contains state variables of every
subsystem. Instead of approximating the coupling term by
using the RBFNNs, we utilize (17) and Young’s inequality
to remove it from the inequality (54). Thus, both the circular
control construction problem and coupling problem are over-
come and the online computation load is lightened greatly.
Due to the virtue of (17) and (23), (54) can be rewritten as
V̇ ≤
m
X
j=1
ρj
X
ij=1
[ − kj,ij Gjme2
j,ij
+ Gjmσj,ij δ̃j,ij δ̂j,ij + 0.2785ςj,ij
+ Gjmσj,ij θ̃j,ij θ̂j,ij + 0.2785Gjmδ∗
j,ij
ςj,ij +
a2
j,ij
2
]
+
m
X
j=1
ρj−1
X
ij=1
[GjM (|ej,ij+1| + y∗
j,ij+1)|ej,ij |] (55)
Invoking the following inequalities:
δ̃j,ij δ̂j,ij = δ̃j,ij (δ∗
j,ij
− δ̃j,ij ) ≤
δ∗2
j,ij
2
−
δ̃2
j,ij
2
θ̃j,ij θ̂j,ij = θ̃j,ij (θ∗
j,ij
− θ̃j,ij ) ≤
θ∗2
j,ij
2
−
θ̃2
j,ij
2
(56)
GjM

41. ej,ij+1

45. ej,ij

47. ≤
GjM e2
j,ij
2
+
GjM e2
j,ij+1
2
GjM y∗
j,ij+1

49. ej,ij

51. ≤
G2
jM y∗2
j,ij+1cj,ij0
2
+
e2
j,ij
2cj,ij0
(57)
where cj,ij0 is arbitrary positive constant, we have
V̇ ≤
m
X
j=1
ρj
X
ij=1
Gjm[−kj,ij e2
j,ij
−
1
2
σj,ij δ̃2
j,ij
−
1
2
βj,ij θ̃2
j,ij
]
+
m
X
j=1
ρj−1
X
ij=1
[
GjM
2
(e2
j,ij
+ e2
j,ij+1) +
e2
j,ij
2cj,ij0
] + ξ0 (58)
where
ξ0 =
m
X
j=1
ρj
X
ij=1
(
1
2
Gjmσj,ij δ∗2
j,ij
+
1
2
Gjmβj,ij θ∗2
j,ij
+
a2
j,ij
2
)
+ 0.2785Gjmδ∗
j,ij
ςj,ij + 0.27857ςj,ij
+
m
X
j=1
ρj−1
X
ij=1
G2
jM y∗2
j,ij+1cj,ij0
2
(59)
96678 VOLUME 7, 2019

52. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
Setting kj,ij = G−1
jm (GjM + 1/(2cj,ij0) + cj,ij1), with cj,ij1
being arbitrary positive constant, we can rewrite (58) as
V̇ ≤ −
m
X
j=1
ρj
X
ij=1
[−cj,ij1e2
j,ij
−
1
2
Gjmσj,ij δ̃2
j,ij
−
1
2
Gjmβj,ij θ̃2
j,ij
] + ξ0
≤ −ξ1V + ξ0 (60)
where ξ1 =
m
P
j=1
minij=1,...,ρj {2cj,ij1, σj,ij γj,ij , σj,ij βj,ij }.
Integrating (60) over [0, t], we have
V(t) ≤ (V(0) − ξ2)e−ξ1t
+ ξ2
≤ V(0) + ξ2 (61)
where ξ2 = ξ0/ξ1 are positive constant.
It is noticeable that ξ2 = ξ0/ξ1 can be made arbitrarily
small by reducing ςj,ij , cj,ij0, and meanwhile increasing cj,ij1,
σj,ij γj,ij and βj,ij . From (61) we can know that V, ej,ij δ̃j,ij
and θ̃j,ij are bounded and the closed-loop system is stable.
δ̂j,ij = δ∗
j,ij
− δ̃j,ij , θ̂j,ij = θ∗
j,ij
− θ̃j,ij are bounded because
of the boundedness of δ∗
j,ij
, δ̃j,ij , θ∗
j,ij
and θ̃j,ij . Since ej,1 =
xj,1 −yd1 and yd1 being bounded, xj,1 is bounded. Since αj,1 is
a function of bounded signals ej,1, δ̂j,1, θ̂j,1 yd1 and ẏd1, so αj,1
is also bounded. From xj,2 = ej,2 +αj,1 +yj,2, it can be known
xj,2 is bounded. Similarly, αj,ij−1 and xj,ij , i = 3, . . . , ρj, are
bounded. Therefore, all the signals of the closed-loop system
are bounded.
2) Since Vej,ij = e2
j,ij
/2 and according to(53), we have
m
X
j=1
ρj
X
ij=1
e2
j,ij
2
≤ V (62)
Using the first inequality in(61), the following inequality
holds:
lim
t→∞
kē1k ≤
√
2V ≤
p
2ξ2 (63)
Note that ξ2 depends on the design parameters kj,ij , ςj,ij ,
σj,ij , γj,ij , βj,ij , ωj,ij , j,ij0, j,ij1 and εj,ij1. Therefore, by appro-
priately online-tuning the design parameters, the tracking
error kē1k can be regulated to an arbitrary small neighbor-
hood of the origin.
The proof is completed.
According to Theorem 1, it can be easily found that the
method proposed in this paper can regulate the signals of (1)
to an arbitrarily small neighborhood of the origin. However,
the finite-time control problem is not considered in this paper,
for the reason that most of the existing finite-time control
methods are semi-globally stable [23]–[25]. If a globally
stable finite-time control method can be developed, we can
not only guarantee the global results, but also achieve faster
convergence speed and smaller steady-state tracking error.
This will be the focus of our future work.
V. SIMULATION RESULTS
In this section, two simulation examples are provided to illus-
trate the effectiveness and merits of the proposed adaptive
control approach.
Example 1: Numerical example.
Consider the following MIMO non-affine pure-feedback
nonlinear system:









ẋ1,1 = f1,1(41,1, x1,2)
ẋ1,2 = f1,2(X, u1) + d1(t)
ẋ2,1 = f2,1(X, ū2) + d2(t)
yj = xj,1, j = 1, 2
(64)
where dj(t) = 0.1 cos(0.01t) cos(xj,1), j = 1, 2, and
f1,1(41,1, x1,2), f1,2(X, u1), f2,1(X, ū2) are described as
follows:













f1,1(41,1, x1,2) = x1,1 + x1,2 +
x3
1,2
3
+ 0.2sign(x1,2)
f1,2(X, u1) = x1,1x1,2 + x2,1 + ϕ1(u1) +
ϕ1(u1)3
7
f2,1(X, ū2) = x1,1x1,2 + x2,1 + ϕ1(u1) + ϕ2(u2) +
ϕ2(u2)3
7
(65)
where
ϕ1(u1) =











(u1 − 0.5) +
(u1 − 0.5)3
7
, u1 ≥ 0.5
0, −1u1 0.5
(u1 + 1) +
(u1 + 1)3
7
, u1 ≤ −1
(66)
ϕ2(u2) =





0.5, u2 ≥ 0.5
u2, −1 u2 0.5
−1, u2 ≤ −1
(67)
Clearly, system (64) consists of two subsystems (ρ1 = 2;
ρ2 = 1). Since 1 − ρ12 = 0, the state vector x̄2,(1−ρ12) does
not appear in (64). It can be seen that the non-affine functions
of the above system is in-differentiable with respect to x1,2,
u1 and u2, since non-smooth nonlinearity is present.
Take yd1 = 0.75 sin(t) + 0.25 sin(0.5t) and yd2 =
0.75 cos(2t) + 0.25 cos(t) as our reference trajectories with
the initial value yd1(0) = 0, yd2(0) = 1. The control objective
is to make the outputs y1 and y2 track the desired trajectories
yd1 and yd2.
According to Theorem 1, the adaptive controllers is
designed as (34), (35) and the adaptation laws are provided
by (36), (37), with k1,1 = k1,2 = k2,1 = 8, ς1,1 = ς1,2 =
ς2,1 = 0.75, ω1,1 = ω1,2 = ω2,1 = 1, γ1,1 = γ2,1 = γ1,2 =
0.5, β1,1 = β2,1 = β1,2 = 0.5, σ1,1 = σ2,1 = σ1,2 = 0.06,
1,10 = 40, 1,11 = 0.2ε1,11 = 0.1, µ1,1 = 10 and
a1,1 = a2,1 = a1,2 = 0.5. The initial conditions are chosen
as, [ x1,1(0) x1,2(0) x2,1(0) ]T = [ 0 0 1 ]T , and δ̂1,1(0) =
δ̂1,2(0) = δ̂2,1(0) = 0, θ̂1,1(0) = θ̂1,2(0) = θ̂2,1(0) = 0.
In order to highlight the superiority of our method,
we selected the method of Zhao and Lin [51] for comparison.
For details of Zhao’s design method, please refer to the
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53. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
FIGURE 1. System outputs y1 and desired trajectory yd1 of example 1.
FIGURE 2. System outputs y2 and desired trajectory yd2 of example 1.
literature [51]. The simulation results are shown in Figs. 1-9.
It can be seen from Figs. 1-4 that better tracking performance
than [51] is obtained. Figs. 5-8 show the boundedness of
δ̂1,1, δ̂1,2, δ̂2,1, θ̂1,1, θ̂1,2, θ̂2,1, and u1, u2. It can be observed
from these results that excellent control performance has been
achieved even though the non-affine function of system (64)
is in-differentiable.
To highlight the global ability of our method, we also
selected a particularly large initial value (x1(0) = 6) for the
system. As can be seen from Fig. 9, although the initial value
of the system differs greatly from the desired initial value,
our method can still achieve the stability and well tracking
performance of the control system with an acceptable error
range. However, when the initial value of the system exceeds
the compact set of [51], the system under the control of [51]
is unable to converge and the simulation results cannot be
obtained, for the reason that the method proposed by [51] can
only guarantee the semi-global uniformly ultimately bound-
edness of the systems.
Example 2: Physical example.
In this section, to illustrate the practicability of the pro-
posed controller, tracking problem of a robotic manipulator
with two degrees of freedom (DOF) is simulated in this
subsection. To propose a dynamic model for the robotic
manipulator in Fig. 10, following equations are written
FIGURE 3. Tracking errors e1,1 of example 1.
FIGURE 4. Tracking errors e2,1 of example 1.
FIGURE 5. Control inputs u1 of example 1.
base on [52]:
q̈1
q̈2
=
M11 M12
M21 M22
−1
v1(u1)
v2(u2)
−
−hq̇2 −h (q̇1 + q̇2)
hq̇1 0
q̇1
q̇2
(68)
where
M11 = a1 + 2a3 cos (q2) + 2a4 sin (q2)
M12 = M21 = a2 + a3 cos (q2) + a4 sin (q2)
M22 = a2
h = a3 sin (q2) − a4 cos (q2)
96680 VOLUME 7, 2019

54. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
FIGURE 6. Control inputs u2 of example 1.
FIGURE 7. Adaptive parameters δ̂ of example 1.
FIGURE 8. Adaptive parameters θ̂ of example 1.
and
a1 = I1 + m1l2
c1 + Ie + mel2
ce + mel2
1
a2 = Ie + mel2
ce
a3 = mel1lce cos δe
a4 = mel1lce sin δe
The dead-zone model is selected as:
v(u) =











(u − 0.5) +
(u − 0.5)3
7
, u ≥ 0.5
0, −1 u1 0.5
(u + 1) +
(u + 1)3
7
, u ≤ −1
(69)
FIGURE 9. System output y1 and desired trajectory yd1 of example 1.
FIGURE 10. Robotic Manipulator with two DOF.
In the simulation, the following parameter values are used:
m1 = 1, me = 2
l1 = 1, lc1 = 0.5, lce = 0.6
I1 = 0.12, Ie = 0.25, δe =
π
6
Let y = [q1, q2]T , u = [u1, u2]T , x =
q1, q̇1, q2, q̇2
T
.
Then (68) can be written as the following state-space form:





ẋ11 = x12
ẋ12 = f12(x, u1, u2) + d1
y1 = x11





ẋ3 = x4
ẋ4 = f22(x, u1, u2) + d2
y2 = x21
(70)
The control objective is to force the system output q1 and q2
to track the desired trajectories y1d = sin(t) and y2d = sin(t),
respectively.
According to Theorem 1, the adaptive controllers are
designed as (34), (35) and the adaptation laws are provided
by (36), (37), with k1,1 = k1,2 = k2,1 = k2,2 = 8, ς1,1 =
ς1,2 = ς2,1 = ς2,2 = 0.25, ω1,1 = ω1,2 = ω2,1 = ω2,2 = 1,
γ1,1 = γ1,2 = γ2,1 = γ2,2 = 0.5, β1,1 = β1,2 =
β2,1 = β2,2 = 0.06, σ1,1 = σ1,2 = σ2,1 = σ2,2 = 0.06,
VOLUME 7, 2019 96681

55. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
FIGURE 11. System output and desired trajectory of example 2.
FIGURE 12. System output and desired trajectory of example 2.
FIGURE 13. Tracking errors of example 2.
1,10 = 2,10 = 10, 1,11 = 2,11 = 0.2, ε1,11 = ε2,11 = 0.1,
a1,1 = a1,2 = a2,1 = a2,2 = 5 and µ1,1 = µ2,1 = 10.
The initial conditions are chosen as δ̂1,1(0) = δ̂1,2(0) =
δ̂2,1(0) = δ̂2,2(0) = 0, θ̂1,1(0) = θ̂1,2(0) = θ̂2,1(0) =
θ̂2,2(0) = 0, [x1,1 (0) x1,2 (0) x2,1(0) x2,2(0)]T = [1 0 0.5 0]T .
The simulation results are shown in Figs. 11-17. It can be
seen from Fig. 11-13 that fairly good tracking performance
is obtained. Figs. 14-16 show the boundedness of δ̂1,1, δ̂1,2,
δ̂2,1, δ̂2,2, θ̂1,1, θ̂1,2, θ̂2,1, θ̂2,2,and u1, u2. It can be observed
from these results that excellent control performance has
been achieved even though the non-affine function of sys-
tems (70) are in-differentiable. To show the advantage of the
FIGURE 14. Control inputs of example 2.
FIGURE 15. Adaptive parameters of example 2.
FIGURE 16. Adaptive parameters of example 2.
improved DSC, the estimate performance of first-order fil-
ter and the sliding mode differentiator should be provided
in Fig. 17. It can be seen that the estimate error is greatly
reduced by using the sliding mode differentiator.
The design parameters have various influences on the per-
formance of the proposed scheme. In particular, the large
positive design constant j,ij0 and the small positive design
constant j,ij1 are influence factors of the first order sliding
mode differentiators that could make the approximation per-
formance better. Besides, the purpose of setting the adapta-
tion gain σj,ij is to adjust the convergence rate of adaptive
parameters δ̂j,ij and θ̂j,ij , and larger σj,ij can lead to a faster
96682 VOLUME 7, 2019

56. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
FIGURE 17. Estimate errors y1,2 of the sliding mode differentiator and
first-order filter.
convergence rate. In Lemma 2 and 4, The smaller ςj,ij and
µj,ij are, the closer the hyperbolic tangent function to the sign
function is. In addition, the design parameter ωj,ij ≥ G−1
jm
does not affect the size of tracking error ej,ij , and we can tune
its value from trial simulations since the positive constant
G−1
jm is unknown.
VI. CONCLUSION
In this paper, a novel adaptive tracking controller has been
presented for a more general class of MIMO non-affine
pure-feedback nonlinear systems. By modeling the non-
affine nonlinear functions appropriately, the assumption that
the non-affine functions must be differentiable is removed,
and only a continuous condition is required. The IDSC tech-
niques have been proposed in this paper which can signif-
icantly reduce the complexity of control design for MIMO
pure-feedback nonlinear systems in cooperation with back-
stepping method, and it is proved that IDSC can guarantee
the GUUB of all the signals of system. Robust compensators
are employed to circumvent the influences of approximation
errors and disturbances. Finally, according to the simulation
results, the signals in the closed-loop system are guaranteed
to be GUUB, and the system outputs are proven to converge
to a small neighborhood of the desired trajectory. As a conse-
quence, the feasibility and effectiveness of our approach are
proved.
APPENDIX
A. PROOF OF LEMMA 4
To obtain the conclusion, two cases are discussed as follows:
Case 1: For any x ∈ R, |x|
1
2 +
x tanh
x
µ
1
2
≥ 1

61. |x|
1
2 −
x tanh
x
µ
1
2

66. ≤

71. |x|
1
2 −
x tanh
x
µ
1
2

76. ·

81. |x|
1
2 +
x tanh
x
µ
1
2

86. =

90. |x| − x tanh
x
µ

94. (71)
According to Lemma 1, one has

99. |x|
1
2 −
x tanh
x
µ
1
2

104. ≤ |x| − x tanh
x
µ
≤ 0.2785µ
(72)
Consider the property of the sign function, we know that

109. |x|
1
2 sign(x) −
x tanh
x
µ
1
2
sign(x)

114. =

119. |x|
1
2 −
x tanh
x
µ
1
2

124. ≤ 0.2785µ (73)

129. x tanh
x
µ
1
2
sign(x) −
x tanh
x
µ
1
2
sign
x
µ

134. = 0 (74)
Noting that µ is a positive constant, it holds that

139. x tanh
x
µ
1
2
sign
x
µ
−
x tanh
x
µ
1
2
tanh
x
µ

144. ≤
x tanh
x
µ
1
2

148. sign
x
µ
− tanh
x
µ

152. (75)
When x 0,

156. sign
x
µ
− tanh
x
µ

160. =

164. 1 −
e(x/µ) − e−(x/µ)
e(x/µ) + e−(x/µ)

168. =
2e−(x/µ)
e(x/µ) + e−(x/µ)
≤ e−(x/µ)
(76)
When x 0,

172. sign
x
µ
− tanh
x
µ

176. =

180. −1 −
e(x/µ) − e−(x/µ)
e(x/µ) + e−(x/µ)

184. =
2e(x/µ)
e(x/µ) + e−(x/µ)
≤ e(x/µ)
(77)
In view that the difference between the sign function and
the hyperbolic tangent function shows exponential growth,
one reaches

189. x tanh
x
µ
1
2
sign
x
µ
−
x tanh
x
µ
1
2
tanh
x
µ

194. ≤
x tanh
x
µ
1
2
e−|x/µ|
≤ γ ∗
(78)
where γ ∗ is a positive constant.
Therefore, we can obtain that

199. |x|
1
2 sign(x) −
x tanh
x
µ
1
2
tanh
x
µ

204. ≤

209. |x|
1
2 sign(x) −
x tanh
x
µ
1
2
sign(x)

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215. Y. Zhou et al.: Robust Adaptive Control of MIMO Pure-Feedback Nonlinear Systems via IDSC Technique
+

220. x tanh
x
µ
1
2
sign(x)
−
x tanh
x
µ
1
2
sign
x
µ

225. +

230. x tanh
x
µ
1
2
sign
x
µ
−
x tanh
x
µ
1
2
tanh
x
µ

235. ≤ 0.2785µ + γ ∗
= γ (79)
Case 2: For any x ∈ R, |x|
1
2 +
x tanh
x
µ
1
2
1
It is easy to know that |x| 1, and then we further have

240. |x|
1
2 sign(x) −
x tanh
x
µ
1
2
tanh
x
µ

245. ≤ γ (80)
This completes the proof.
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YANG ZHOU received the B.Sc. degree in elec-
trical engineering and automation from Air Force
Engineering University, Xi’an, China, in 2017,
where he is currently pursuing the M.S. degree
in control theory and engineering. His research
interests include fight control, adaptive control,
and neural networks.
WENHAN DONG received the B.Sc. degree in
electrical engineering and automation and the
M.Sc. and Ph.D. degrees in control theory and
engineering from Air Force Engineering Univer-
sity, Xian, China, in 2000, 2003, and 2006, respec-
tively, where he is currently a Professor with the
College of Aeronautics Engineering. His research
interests include adaptive control and flight
simulation.
SHUANGYU DONG received the B.Eng. degree
in electrical engineering and automation from
Xi’an Jiao Tong University, Xi’an, China, in 2015,
and the M.Eng. degree in electrical engineering
from the University of Melbourne, Melbourne,
Australia, in 2017. She is currently an Engineer
with SMZ Telecom Pty., Ltd., Melbourne. Her
research interests include deep learning and adap-
tive control.
YONG CHEN received the B.Sc. degree in electri-
cal engineering and automation, the M.Sc. degree
in navigation, guidance and control, and the Ph.D.
degree in control science and engineering from
Air Force Engineering University, Xi’ an, China,
in 2006, 2009, and 2012, respectively, where he
is currently with the College of Aeronautics and
Astronautics Engineering. His research interests
include flight control, control allocation, and adap-
tive neural control.
RENWEI ZUO received the B.Sc. degree in detec-
tion guidance and control from the Nanjing Uni-
versity of Aeronautics and Astronautics, Nanjing,
China, in 2016, and the M.Sc. degree in control
science and engineering from Air Force Engineer-
ing University, Xi’an, China, in 2018, where he is
currently pursuing the Ph.D. degree with the Aero-
nautics Engineering College. His research interests
include flight control, adaptive control, and neural
networks.
ZONGCHENG LIU received the B.Sc. degree
in electrical engineering and automation and the
M.Sc. and Ph.D. degrees in control theory and
engineering from Air Force Engineering Uni-
versity, Xi’an, China, in 2009, 2011, and 2015,
respectively, where he is currently a Lecturer with
the Aeronautics Engineering College. His research
interests include flight control, intelligent and
autonomous control, and neural networks.
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