Neural Network-Based Tracking Control of Uncertain Robotic Systems: Predefined-Time Nonsingular Terminal Sliding Mode Approach

Abstract

This article investigates the predefined time trajectory tracking control of uncertain nonlinear robotic systems. A radial basis function neural network (RBFNN) is used to estimate uncertainties in the robotic system dynamics. To avoid the singularity of terminal sliding-mode control (TSMC), a modified sliding variable is adopted. In order to realize that the tracking errors can converge to a small neighborhood of the origin in
predefined time
, within which the maximum convergence time can be adjusted by explicit parameters in advance, a nonsingular TSMC based on the RBFNN is proposed. Experiments on a ROKAE platform demonstrate the effectiveness and advantage of the proposed control method.

Full text

10510 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
Neural Network-Based Tracking Control of
Uncertain Robotic Systems: Predefined-Time
Nonsingular Terminal Sliding-Mode Approach
Yizhuo Sun , Yabin Gao, Member, IEEE, Yue Zhao, Member, IEEE, Zhuang Liu, Jiahui Wang ,
Jiyuan Kuang, Fei Yan, Member, IEEE, and Jianxing Liu , Senior Member, IEEE
Abstract—This article investigates the predefined time
trajectory tracking control of uncertain nonlinear robotic
systems. A radial basis function neural network (RBFNN)
is used to estimate uncertainties in the robotic system dy-
namics. To avoid the singularity of terminal sliding-mode
control (TSMC), a modified sliding variable is adopted. In
order to realize that the tracking errors can converge to a
small neighborhood of the origin in predefined time, within
which the maximum convergence time can be adjusted by
explicit parameters in advance, a nonsingular TSMC based
on the RBFNN is proposed. Experiments on a ROKAE plat-
form demonstrate the effectiveness and advantage of the
proposed control method.
Index Terms—Neural networks (NN), nonsingular termi-
nal sliding-mode control (NTSMC), predefined time control,
trajectory tracking control, uncertain robotic systems.
I. INTRODUCTION
MANIPULATORS, a typical class of robotic systems,
have been increasingly playing important roles in a
wide range of fields, such as medical rehabilitation, welding,
sorting, and rescue. In these scenarios, the manipulators face
model uncertainties and external disturbances [1], which add
Manuscript received September 20, 2021; revised December 8, 2021,
January 14, 2022, and February 22, 2022; accepted March 11, 2022.
Date of publication March 29, 2022; date of current version May 2, 2022.
This work was supported in part by the National Key R&D Program of
China under Grant 2019YFB1312001, in part by the Natural Science
Foundation of China under Grant 62022030 and Grant 62103118, in
part by the Fundamental Research Funds for the Central Universities
under Grant 2020HIT, in part by the Degree and Postgraduate Education
Reform Project of Harbin Institute of Technology under Grant 21MS003,
in part by the Self-Planned Task of State Key Laboratory of Advanced
Welding and Joining (HIT), and in part by the China Postdoctoral Sci-
ence Foundation under Grant 2021M700037 and Grant 2021T140160.
(Corresponding authors: Yabin Gao; Yue Zhao.)
Yizhuo Sun, Yabin Gao, Yue Zhao, Zhuang Liu, Jiyuan Kuang, and
Jianxing Liu are with the Department of Control Science and En-
gineering, Harbin Institute of Technology, Harbin 150001, China (e-
mail: syz_hit@hit.edu.cn; yabingao@hit.edu.cn; yue.zhao@hit.edu.cn;
liuz@hit.edu.cn; 20b904022@stu.hit.edu.cn; jx.liu@hit.edu.cn).
Jiahui Wang is with the College of Intelligent Systems Science and
Engineering, Harbin Engineering University, Harbin 150001, China (e-
mail: jiahui_wang@hrbeu.edu.cn).
Fei Yan is with the School of Information Science and Technol-
ogy, Southwest Jiaotong University, Chengdu 611756, China (e-mail:
fyan@swjtu.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TIE.2022.3161810.
Digital Object Identifier 10.1109/TIE.2022.3161810
difficulty to stability analysis and control design. To deal with
such an uncertain system, many control strategies have been
proposed, such as fuzzy control [2], neural network (NN)
control [3], active disturbance rejection control (ADRC) [4],
and sliding-mode control (SMC) [5].
As is well known, SMC is widely implemented in uncertain
systems because of its good robustness and fast transient
response [6]. To achieve a finite-time convergence and improve
the convergence performance in sliding modes, terminal SMC
(TSMC) has been researched. In [7], in order to solve the
problem that tracking errors of rigid manipulator converge to
zero in finite time, a multi-input/multi-output (MIMO) terminal
sliding-mode (TSM) controller was proposed. In [8], a propor-
tion integration differentiation (PID) fast TSMC based on fuzzy
adaptive law was proposed to improve the system robustness
to model uncertainty and external disturbances. Although TSM
controllers can achieve finite-time convergence, they would
cause singularity problems. To solve this problem, a lot of efforts
have been devoted to the study of nonsingular terminal sliding-
mode control (NTSMC) [9]. To improve the anti-interference
and fault compensation ability of the robot’s tracking control,
a control method based on adaptive backstepping nonsingular
fast TSMC [10] was proposed. In [11], a switched sliding-mode
controller was proposed, which used different sliding-mode
controllers in different regions to achieve global high-speed
convergence and global nonsingularity. However, both the TSM
and the NTSM have the same problem: Chattering. To deal with
this problem, some scholars have given a high-order sliding-
mode control (HOSMC) scheme [12], [13]. The HOSMC is
an extension of the traditional SMC. It puts the discontinuity
term in the high-order derivative of the sliding-mode surface so
that the chattering of the low-order sliding-mode surface can be
eliminated essentially. Thus, the smooth output of the system
can be finally realized. For example, in [12], a second-order
sliding-mode (SOSM) control strategy was implemented in
neutral-point-clamped power converter to improve the steady-
state characteristics. The SOSM algorithm for manipulators
was proposed in [13] to solve motion control problems.
In some application scenarios, robot tasks need to meet
time constraints, that is, to achieve finite-time tracking. Cor-
respondingly, there are fruitful results on the problem of finite-
time convergence [14], [15]. However, the time of finite-time
convergence is related to the initial values of the system. Thus,
0278-0046 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

SUN et al.: NEURAL NETWORK-BASED TRACKING CONTROL OF UNCERTAIN ROBOTIC SYSTEMS 10511
some scholars have proposed a control strategy called the fixed-
time convergence, whose convergence time is independent of
the initial values [16]. In [17], a fixed-time HOSMC approach
was presented for power systems. In [18], the fixed-time con-
sensus problem of high-order nonlinear multiagent systems was
studied. The robust fixed-time consensus problem under an undi-
rected topology was investigated in [19]. In [20], a continuous
TSM algorithm was applied to the servo motor system to achieve
fixed-time convergence. However, the upper bound of fixed time
convergence is a complex function related to the controller
parameters, which makes it impossible to obtain the desired
convergence time directly when tuning the parameters. To solve
this problem, Sánchez-Torres et al. [21] proposed a concept of
predefined-time stability of dynamical systems, of which the
convergence time could be acquired directly by tuning the pa-
rameter of predefined time. Then, a second-order backstepping
controller that can realize predefined-time stability was proposed
in [22]. Ni and Shi [23] divided the predefined-time stability
into strong predefined-time stability and weak predefined-time
stability. An NN is usually used to learn and approximate any
unknown nonlinear function, which only requires relatively little
system information. It has an extensive applications, such as
power converters [24] and wind turbine systems [25]. In [26],
a Hammerstein-type NN was used to approximate the energy
storage system. In [27], an NN-based supervisor controller was
proposed to achieve a high-performance control of the maglev
train system with uncertain disturbance force and time-delay.
In [28], the NN and extended-state observer were adopted to
improve the control accuracy of the system with uncertain-
ties. Hence, these typical results motivate us to develop some
NN-based control applications for the tracking of the interested
robotic systems with uncertainties. Meanwhile, the use of the
radial basis function NN (RBFNN) integrated with other control
methods is also quite popular. In [29], an model predictive
control (MPC) based on the RBFNN was proposed to solve the
constraints of the actuator and the disturbances in the altitude
subsystem of an aspirated hypersonic vehicle. The problem of
upper limb exoskeleton trajectory tracking under input satura-
tion was solved in [30] by an adaptive controller based on NN.
As introduced in the article, the existing finite-time control
methods [31]–[33] can only realize the finite-time or fixed-time
convergence depending on initial conditions or controller pa-
rameters without explicit time constant, but the problem of the
predefined-time convergence has not been solved yet. Compared
with the existing two kinds of convergence performance, the con-
trol scheme proposed in this article can set the upper bound of the
convergence time in advance, and then realize the improvement
of tracking accuracy by adjusting some controller parameters.
However, the traditional TSMC does not have such advantages.
In this article, we propose a new NTSMC strategy combining
the RBFNN, which can guarantee the practical predefined-time-
tracking performance of the manipulator system with parameter
uncertainty. Compared with the existing works, the main con-
tributions of this article are summarized as follows:
1) A novel RBFNN-based predefined-time NTSMC strategy
is developed for trajectory tracking of uncertain manipu-
lator systems, which can achieve excellent robustness and
tracking performance.
2) New techniques are developed to design the predefined-
time NTSMC law to achieve the practical predefined-
time tracking performance of the manipulator system.
Meanwhile, the maximum convergence time can be
predefined. By using the predefined-time NTSMC law,
the singularity problem of traditional TSMC can be
avoided.
3) Some explicit conditions for determining the controller
parameters are established. The practical predefined-time
convergence of the manipulator system is analyzed by
using the Lyapunov function method.
Besides, the effectiveness of the proposed control strategy is
verified by performance tests on the ROKAE platform under dif-
ferent predefined-time conditions. Compared with the traditional
NTSMC, the experimental results show that the proposed control
algorithm can help to achieve the higher tracking accuracy and
better torque performance.
The rest of this article is organized as follows. In Section II, the
dynamic model of uncertain robotic systems is formulated and
preliminary knowledge is presented. In Section III, a predefined-
time NTSM controller based on the RBFNN is proposed and
the stability analysis is presented. In Section IV, experiments
based on the ROKAE platform are completed. Finally, Section V
concludes this article.
Notation: The superscript “” is the transpose of a
matrix or a vector. For vectors x=[x1,x
2,...,x
r]∈Rr
and k=[k1,k
2,...,k
r]∈Rr,|x|<|k|means |xi|<|ki|
∀i∈[1,...,r].xk=[xk1
1,...,x
kr
r]. In order to simplify the
expression, the following special symbols are used: sig(x)k=
[|x1|k1sign(x1),|x2|k2sign(x2),...,|xr|krsign(xr)].xk=
diag {|x1|k1,|x2|k2,...,|xr|kr}.x=diag{x1,x
2,...,x
r}.
exp(k)denotes an exponential function based on the natural
constant e.
II. SYSTEM DESCRIPTION OF MANIPULATORS
A. Dynamic System of Manipulator
The dynamics of the uncertain manipulators is described by
M(q)¨q+C(q, ˙q)˙q+G(q)+F(˙q)=τ(t)(1)
where q(t)∈Rr,˙q∈Rrand ¨q(t)∈Rrare the position, veloc-
ity, and acceleration vectors, respectively. τ∈Rris the input
torque, M(q)=M0(q)+ΔM(q)∈Rr×risasymmetriciner-
tia matrix, and M0(q)is the nominal section of inertia matrix
and ΔM(q)is the uncertain section. C(q, ˙q)∈Rrrepresents the
centripetal and Coriolis torques matrix; G(q)∈Rris gravity;
F(˙q)∈Rris frictional force. According to the nonsingularity
of matrix M0(q), system (1) can be rewritten as
¨q=M−1
0τ−M−1
0[C(q, ˙q)˙q+G(q)+F+ΔM(q)¨q].(2)
Define x1(t)=q(t),x2(t)= ˙q(t)and Δ=−M−1
0
[C(q, ˙q)˙q+G(q)+F(˙q)+ΔM(q)¨q]. System (2) can be
rewritten as
˙x1=x2
˙x2=M−1
0τ+Δ (3)
where x1=[x11,x
12,...,x
1r]and x2=[x21,x
22,...,x
2r].
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

10512 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
B. Predefined-Time Stability
Consider a system described as follows:
˙x(t)=g(x(t)),g(0) = 0,t
0=0,x(t0)=x0(4)
where x∈Rris state vector; x0is the initial state; and g:Rr→
Rris a continuous nonlinear function. The following definitions
and lemma are provided for analysis.
Definition 1 ([23], [34]): System (4) is predefined-time con-
vergent if there exists a constant Tf>0and a function T(x(0))
satisfying T(x(0)) ≤Tf∀x(0) ∈Rr,then the system states
reach the origin at T(x(0)), i.e., limt→T(x(0)) x(t)=0.
Definition 2: For all initial values x(0) of system (4), if there
exists a vector ι=[ι1,...,ι
r],ιi>0, and a constant Tf>0
satisfying as |x(t)|≤ι, t ≥Tf,then the dynamic system in (4)
is called practical predefined-time convergent.
Lemma 1 ([21]): The system
˙
ψ=−1
Ts
Ξp(ψ)
with Ξp(ψ)= 1
pexp(|ψ|p)|ψ|1−psign(ψ)for 0<p<1, and
Ts>0is predefined-time stable with settling-time Ts. That is,
ψ(t)=0for t>T
sindependent of the initial value x0.
Remark 1: In order to distinguish the difference between the
fixed-time convergence and the predefined-time convergence,
let us take an example of a method of the fixed-time conver-
gence presented in [16]. As presented in [16], the dynamic
system in (4) is fixed-time convergent if there is a positive
definite continuous function V(x)in the domain Rsatisfying
˙
V(t)≤−k1Vα(x)−k2Vβ(x),x∈R\{0},where k1,k2,α,
and βare positive constants and satisfy α<1and β>1.The
state of the system can converge to the equilibrium point in fixed
time, which is independent of initial values. The convergence
time Tis called the settling-time and can be expressed as
T≤1
k1(1 −α)+1
k2(1 −β).
Obviously, it can be obtained that the controller parameters
will affect the upper bound of the fixed-time convergence time
although this upper bound is independent of initial values. In the
predefined-time convergence, a time constant is introduced, and
the upper bound of the convergence time is only related to this
constant. Then, this will greatly facilitate the users to design the
upper bound of the convergence time. In addition, considering
that uncertainties such as disturbances in actual systems, we
introduce the definition of the practical predefined-time conver-
gence in Definition 2, which is extended from Definition 1. It
also provides a theoretical basis for engineering applications.
C. RBFNN Approximation
The RBFNN has the ability to approximate unknown smooth
functions. It plays a role in nonlinear compensation and param-
eter identification. In this article, the uncertainty part Δof the
manipulator system is approximated through the RBFNN
Δ(z)=θ∗Φ(z)+(5)
where z=[e
1,e

2]are the input variables, e1and e2are de-
signed later, θ∗=[θ∗
1,...,θ
∗
r]is the ideal RBFNN weight
vector, is the approximation error and is bounded, i.e.,
||≤max for max >0,Φ(z)=[Φ
1(z),...,Φr(z)];Φi(z)=
[Φi1(z),...,Φil(z)]is the RBF vector and lis the NN node
number. Φij(z)is often selected as
Φij (z)=exp −(zi−μij )(zi−μij )
b2
ij (6)
where zi=[e1i,e
2i],μi=[μi1,...,μ
il]is the central param-
eter of each neuron in the hidden layer and bi=[bi1,...,b
il]
(i=1,...,r)is the width of the RBF.
III. CONTROLLER DESIGN AND STABILITY ANALYSIS
A. NTSM Controller Design
For the system in (3), an NTSMC strategy with NN is designed
for the purpose of trajectory-tracking control. Define the system-
tracking error as
e1=x1−xd
e2=x2−˙xd(7)
where e1=[e11,e
12,...,e
1r]and e2=[e21,e
22,...,e
2r].
The derivative of the error system could be written as
˙e1=e2
˙e2=M−1
0τ+θ∗Φ(z)+−¨xd.(8)
A nonsingular terminal sliding variable is constructed as
s=⎧
⎪
⎨
⎪
⎩
e2+1
nTcexp(e1n)(c1ι−ne1+c2
×ι−1−nsig(e1)2),if ˆs=0and |e1|≤ι
ˆs, else
(9)
where n=[n1,n
2,...,n
r],0<n
i<1
2,c1=1+n,c2=
−n,ˆs=e2+1
nTcexp(e1n)sig(e1)(1−n). The predefined
settling time is defined as Tcand ιis the convergence accuracy.
The RBFNN-based practical predefined-time NTSMC is pro-
posed as
τ=⎧
⎪
⎨
⎪
⎩
M0(G1−G2−G3),if ˆs=0
M0(G1−G4−G3),if ˆs=0 and |e1|>ι
M0(G1−G5−G3),if ˆs=0 and |e1|≤ι
(10)
where
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
G1−ˆ
θΦ−maxsign(s)+¨xd
G21
nT2
cexp(2e1)nsig(e1)1−n
−1−n
n2T2
cexp(2e1)nsig(e1)1−2n
G31
mTds1−2mexp(s)2msign(s)−λsign(s)
G41−n
nTcexp(e1n)e1−n˙e1+1
Tcexp(e1n)˙e1
G5sig(e1)n−1e2
Tcexp(e1n)(c1ι−ne1
+c2ι−1−nsig(e1)2)+ 1
nTcexp(e1n)
×(c1ι−ne2+2c2ι−1−ne2e1)
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

SUN et al.: NEURAL NETWORK-BASED TRACKING CONTROL OF UNCERTAIN ROBOTIC SYSTEMS 10513
Fig. 1. An illustration of the convergence time Tdand Tcfor a second-
order system as an example.
where m=[m1,m
2,...,m
r],0<m
i<1
2,λis a diagonal
matrix, of which the element satisfies λi≥˜
θ
iΦi, and Tdis the
predefined settling time.
Remark 2: Regarding the controller parameters Tcand Td,
they portray two stages of the convergence time of the tracking
error variables under the designed predefined-time SMC. As
shown in Fig. 1 for a second-order system with e1∈Rrand
e2∈Rr, the convergence time includes two segments in terms
of the two stages. Tddenotes the maximum time of the first
segment of the tracking error variables moving from the initial
point to the sliding surface. Tcdenotes the maximum time of
the second segment of tracking error variables moving from the
point reached (point A) to a small neighborhood near the origin
O. Tdand Tcare set in advance by the user.
B. Stability Analysis
This subsection provides the theoretical results for the reach-
ability analysis of the desired sliding mode and the convergence
analysis of the tracking errors in the sliding mode.
Theorem 1: For system (8), the proposed controller (10) can
drive the system trajectory onto the sliding surface s=0in (9)
within a predefined time Td, where the parameter Tdis user-
predefined.
Proof: The time-derivative of sin (9) is
˙s=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙e2+sig(e1)n−1e2
Tcexp (e1n)(c1ι−ne1
+c2ι−1−nsig(e1)2+1
nTcexp (e1n)
×c1ι−ne2+2c2ι−1−ne2e1
if ˆs=0and |e1|≤ι
˙e2+1−n
nTcexp (e1n)e1−n˙e1+1
Tcexp (e1n)˙e1,else.
(11)
Consider the candidate Lyapunov function as
V1=1
2ss+1
2
r

i=1
tr(˜
θi
P−1
i˜
θi)(12)
where ˜
θi=θ∗
i−ˆ
θiis the weighted estimation error, Piis a
symmetric positive definite matrix, and ˆ
θiis the estimate of the
optimal weight θ∗
i, and the adaptive law is constructed as
˙
ˆ
θi=PiΦisi(13)
where Φiis the RBF vector, which can be obtained by (6). siis
the sliding variable, which is obtained by (9).
The proof consists of the following three parts.
Part 1: In this case, ˆs=0and |e1|>ι. By substituting (11)
into the first derivative of V1and expanding ˙
V1,wehave
˙
V1=s
Y1−1−n
nTc
exp(e1n)e1−n˙e1+1
Tc
exp(e1n)˙e1
+Y2+1−n
nTc
exp(e1m)e1−m˙e1
+1
Tc
exp(e1n)˙e1+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi)
≤−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi)−sλsign(s)−
r

i=1
|si|
×(max −)−
r

i=1
siˆ
θi
Φi+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi)
(14)
where
Y1−ˆ
θΦ−maxsign(s)+¨xd−λsign(s)
Y2−1
mTds1−2mexp(s)2msign(s)+θ∗Φ(Z)+.
Part 2: In this case, ˆs=0,|e1|>ι, and e2=˙e1=
−1
nTcexp(e1n)sig(e1)(1−n). Substituting (11) into the first
derivative of V1in (12) yields
˙
V1=sY1−1−n
n2T2
c
exp(2e1n)sig(e1)1−2n
+1
nT 2
c
exp(2e1n)sig(e1)1−n+Y2
+1−n
n2T2
c
×exp(2e1n)sig(e1)1−2n+1
nT2
c
×exp(2e1n)sig(e1)1−n+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi)
≤−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi)
−sλsign(s)−
r

i=1
|si|
×(max −)−
r

i=1
siˆ
θi
Φi+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi).(15)
Part 3: In this case, ˆs=0and |e1|<ι. Substituting (11) into
the first derivative of V1in (12), we can obtain that
˙
V1=s˙e2+sig(e1)m−1e2
Tc
exp(e1n)
×c1ι−ne1+c2ι−1−nsig(e1)2)
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

10514 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
+1
nTc
×exp(e1n)(c1ι−ne2
+2c2ι−1−ne2e1+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi)
≤−
r

i=1
1
miTd
|si|2−2miexp |si|2mi−sλsign(s)
−
r

i=1
|si|×(max −)−
r

i=1
siˆ
θi
Φi
+
r

i=1
tr(˜
θi
P−1
i
˙
˜
θi).(16)
Summarizing the results in (14)–(16), since siˆ
θi
Φi=
tr(ˆ
θi
Φisi), by substituting (13) and λinto (14), (15), and (16),
respectively, ˙
V1can be calculated as
˙
V1≤−
r

i=1
|si|(max −)
−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi)≤0.(17)
According to (17), it can be obtained that sand ˜
θare bounded.
In the following, let us prove that the trajectories of the
tracking error variable ecan be enforced to the sliding-mode
surface within the predefined time Td. Consider the following
Lyapunov function:
V2=1
2ss(18)
where V2=r
i=1 V2i=1
2r
i=1 sisi. Similar to the analysis
of V1,V2will also be proved in three parts.
Part 1: In this case, ˆs=0and |e1|>ι. By substituting (11)
into the first derivative of V2in (18), we have
˙
V2=sY1−1−n
nTc
×exp(e1n)e1−n˙e1
+1
Tc
exp(e1n)˙e1+Y2−¨xd+1−n
nTc
×exp(e1m)e1−m˙e1+1
Tc
exp(e1n)˙e1
≤−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi).(19)
Part 2: In this case, ˆs=0and |e1|>ι. By substituting (11)
into the first derivative of V2in (18), we can obtain that
˙
V2=
sY1−1−n
nTc
exp(e1n)e1−n˙e1+1
Tc
exp(e1n)˙e1
+Y2+1−n
nTc
exp(e1n)e1−n˙e1+1
Tc
exp(e1n)˙e1
≤−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi).(20)
Part 3: In this case, ˆs=0and |e1|<ι. Substituting (11) into
the first derivative of V2in (18) yields
˙
V2=s˙e2+sig(e1)m−1e2
Tc
exp(e1n)c1ι−ne1
+c2ι−1−nsig(e1)2+1
nTc
exp (e1n)
×c1ι−ne2+2c2ι−1−ne2e1
≤−
r

i=1
1
miTd
|si|2−2miexp(|si|2mi).(21)
Observing the results in (19)–(21), let us define ψi=2V2i,
then we have ˙
ψi≤− 2
miTdψi1−miexp(ψimi).For the ith joint,
suppose that at time t1i,ψi(t1i)=0, and V2i(t1i)=0. Accord-
ing to Lemma 1, the time t1iof the system trajectory reaching
the sliding manifold si=0satisfies
t1i≤Td(1 −lim
ψi(0)→∞ exp(−ψi(0)mi)) Td∀i∈[1,...,r].
(22)
Hence, V2(t1)=0with t1=max
it1i. This ends the proof. 
Remark 3: In the proof of Theorem 1, we have proven that
sand ˜
θare bounded. Since the uncertainty of the manipulator
system is assumed to be bounded, a parameter λgreater than the
difference between the ideal value of the NN and the estimated
value can be found. Thus, the trajectory of the system in (7) can
reach the sliding surface within the predefined time.
Theorem 2: Consider the sliding manifold s(e1,e
2)=0in
(9). Within the predefined time Tc, the tracking errors e1and
e2can converge to small neighborhoods Ω1={e1||e1|≤ι}
and Ω2={e2||e2|≤(1
nTc)exp(ιn)ι1−n}of the origin, re-
spectively, where ιand nare defined in (9).
Proof: In the sliding mode, we have s≡0. Hence, on the one
hand, recalling the sliding variable sin (9), let us first consider
ˆs=0. In this case, ˆs=s=0and thus we have
e2=−1
nTc
exp(e1n)sig(e1)(1−n).(23)
Consider the Lyapunov function candidate: V31 =1
2e2
1,where
V31 =r
i=1 V31i=r
i=1 1
2e2
1i. The time-derivative of V3is
˙
V31 =e
1˙e1=−
n

i=1
1
niTc
exp(2V31i
ni
2)(2V31i)
2−ni
2.(24)
Define ςi=2V31i. Then, (24) can be written as
˙ςi=−2
niTcexp(ςi
ni
2)ς
2−ni
2.For the ith joint, suppose that
at time t2i>0,V32i(t2i)=0and ςi(t2i)=0. The convergence
time of tracking errors e1iand e2iis calculated as
t2i≤Tc(1 −lim
ψi(0)→∞ exp(−ςi(0)
ni
2)) = Tc∀i∈[1,...,r].
(25)
Hence, we have V32(t2)=0, where t2=max
it2i. This means
that when the system trajectories (e1,e
2) can converge to the
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

SUN et al.: NEURAL NETWORK-BASED TRACKING CONTROL OF UNCERTAIN ROBOTIC SYSTEMS 10515
origin within the predefined time Tcaccording to Lemma 1, in
the sliding mode s=ˆs=0.
On the other hand, when s=0but ˆs=0, we know that e1and
e2are within the small neighborhoods Ω1and Ω2near the origin,
respectively, according to (9). Furthermore, let us consider the
following Lyapunov function:
V32 =1
2e2
1.(26)
The time derivative of the function V32 is
˙
V32 =e
1˙e1=−1
nTc
exp ((|e1|)n)
×(1 + n)ι−ne2
1−nι−1−ne3
1.(27)
Let ˙
V32 =0and ˙
V32 =r
i=1 ˙
V32i=r
i=1 1
2e
1i˙e1i, it can
be obtained that the sign of ˙
V32iis the opposite of the
sign of (1 + ni)ι−nie2
1i−niι−1−nie3
1i.LetG(e1i)=(1+
ni)ιni
ie2
1i−niι−1−ni|e1i|3. It is easy to get that the function
G(e1i)is monotonically increasing in the interval [0,2(1+ni)ιi
3ni)
and monotonically decreasing in the interval (2(1+ni)ιi
3ni,ι
i].
Once e1i=0,G(e1i)=0, and |e1i|=ιi, it goes that
G(e1i)=ι2−mi
i>0. Hence, we can obtain that ˙
V32i≤0(the
“=” is taken if and only if G(e1i)=0). Therefore, we have
˙
V32 ≤0∀|e1|≤ι(28)
which means that e1and e2will converge to the origin asymp-
totically.
In all, from the results in (25) and (28), we know that when
the sliding mode s=0is reached, ˆs=0is then achieved within
the predefined time Tc. That is to say, the tracking errors e1
and e2will reach a small neighbor of the origin within the
predefined time Tc. Afterwards, the tracking errors e1and e2will
asymptotically converge to the origin. Therefore, the practical
predefined-time convergent of the tracking errors e1and e2
can be ensured, as introduced in Definition 2. This ends the
proof. 
Remark 4: When the predefined time Tcand Tdare given,
the final steady tracking errors can be reduced by adjusting mi
and ni. In this article, different values (in a unit frame) of mi
and nifor each link are allowed in the designed control law, so
as to achieve more flexible adjustment.
Remark 5: Since the manipulator dynamics is a strongly
coupled system, adjusting the parameters n=[n1,...,n
r]and
m=[m1,...,m
r]corresponding to a single joint will also
affect the control accuracy of other joints. Therefore, during
the experiment, the selection of the parameters is an important
issue. According to the designed controller, the parameters ni
and mishould satisfy 0<n
i<1
2,0<m
i<1
2. In the process
of parameter tuning, the values of niand micorresponding
to each joint are first approached to 0.5 as much as possible,
because the larger the niand mi, the smaller the corresponding
initial torque. Then, one can appropriately reduce the values of
niand miof the corresponding joints according to the obtained
tracking accuracy. In the actual process, the method we used is
to adjust nifirst, and then adjust mi.
Fig. 2. Experimental platform ROKAE.
IV. EXPERIMENTAL RESULTS
A. Experimental Setup
Experiments are carried out on the ROKAE robot platform,
as shown in Fig. 2, to verify the effectiveness of the proposed
NTSM controller based on the NN. The ROKAE robot is a redun-
dant flexible joint manipulator with seven degrees of freedom.
In the Linux environment, users can read information or send
instructions to robot through the network. The ROAKE robot
information can be read and transmitted in real time by invoking
the corresponding Application Programming Interface provided
by ROKAE. Real-time data is transmitted through UDP with
a maximum communication frequency of 1 kHz. Three groups
of experiments are conducted under the same initial position
and desired trajectory conditions: In the controller with NN, the
tracking performance of the system with the predefined time
of 6 and 8 s are verified, and in the controller without NN,
the performance control experiment with the predefined-time
of 8 s is set. Table I shows the parameters of the controllers
corresponding to the three experiments and the initial states of
the ROKAE robot.
In order to highlight the superiority of the proposed controller,
some comparative experiments were provided here. The com-
pared controller [35]–[37] below is set to Group 4 as
τ=−M0fNN +σ
γsig2−γ(e2)+K11
γ
σe2γ−1s
+K12tanh s
ρ2−¨xd
where the parameters are selected as σ=3,γ=103
79 ,K11 =
diag{4,5,10,20},K12 =diag{6,8,10,20}, and ρ=0.1.The
parameters of fNN are the same as those of Group 1.
B. Experimental Analysis
Figs. 3–5, respectively, show the trajectory tracking, con-
vergence error, and torque curves corresponding to the NTSM
controller with the NN and the predefined time of 8 s. Figs. 6–
8, respectively, show the trajectory tracking, convergence error,
and torque curves corresponding to the NTSM controller with
the NN and the predefined time of 6 s. Figs. 9–11, respectively,
show the trajectory tracking, convergence error, and torque
curves corresponding to the NTSM controller without the NN
and the predefined time of 8 s. Figs. 12–14, respectively, show
the trajectory tracking, convergence error, and torque curves
corresponding to the traditional NTSM controller (Group 4).
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

10516 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
TABLE I
PARAMETERS USED IN THE THREE GROUPS
Fig. 3. Position tracking trajectories of Group 1. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 4. Position-tracking errors of Group 1. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 5. Control inputs of Group 1. (a) Joint 1. (b) Joint 2. (c) Joint 3.
(d) Joint 4.
Fig. 6. Position-tracking trajectories of Group 2. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

SUN et al.: NEURAL NETWORK-BASED TRACKING CONTROL OF UNCERTAIN ROBOTIC SYSTEMS 10517
Fig. 7. Position-tracking errors of Group 2. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 8. Control inputs of Group 2. (a) Joint 1. (b) Joint 2. (c) Joint 3.
(d) Joint 4.
Fig. 9. Position-tracking trajectories of Group 3. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 10. Position-tracking errors of Group 3. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 11. Control inputs of Group 3. (a) Joint 1. (b) Joint 2. (c) Joint 3.
(d) Joint 4.
Fig. 12. Position-tracking trajectories of Group 4. (a) Joint 1.
(b) Joint 2. (c) Joint 3. (d) Joint 4.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

10518 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
Fig. 13. Position-tracking errors of Group 4. (a) Joint 1. (b) Joint 2.
(c) Joint 3. (d) Joint 4.
Fig. 14. Control inputs of Group 4. (a) Joint 1. (b) Joint 2. (c) Joint 3.
(d) Joint 4.
Fig. 15. Velocity-tracking errors. (a) Group 1. (b) Group 2.
Fig. 15 shows the curves of velocity-tracking errors of each joint
under the control strategies of Groups 1 and 2.
As shown in Fig. 3, each joint can track the desired trajectory
within the predefined time. The tracking accuracy of each joint
is [1 ×10−3,1.5×10−3,5.5×10−3,4×10−3]rad, and the
specific tracking error curves are displayed by Fig. 4. Compared
with Fig. 10,Fig. 4 shows that the proposed NTSM control
strategy with the NN has higher tracking accuracy when the
controller parameters are the same. Meanwhile, the practical
convergence time of each joint is about [5.2, 5.7, 5.5, 6.1] s,
TABLE II
COMPARISONS OF THE TRACKING ERRORS
when the preset upper bound of convergence time is 8 s. In the
case that the predefined time is 6 s, the practical convergence
time of each joint is about [4.54, 3.64, 3.45, 4.32] s, as shown in
Fig. 7. According to Figs. 4 and 7, it is shown that the proposed
control algorithm can achieve different predefined-time con-
vergence under the same initial position and desired trajectory,
which cannot be achieved by the compared control strategy.
By comparing the actual convergence time of Groups 1 and 2
with the predefined convergence time, the proposed control strat-
egy is less conservative in terms of convergence time. Therefore,
it can be seen that under different predefined-time conditions, the
robot can achieve tracking tasks within the corresponding time
limits. It can be seen from Figs. 4 and 13 that the control strategy
proposed in this article can help to achieve the higher steady-state
accuracy than the compared control method in Group 4. The
specific control performance is summarized in Table II.From
Figs. 5 and 14, it can be seen that the control torque produced
by our proposed algorithm is smoother, the chattering is smaller,
and the required maximum torque is smaller. The main reason
for the poorer performance of Group 4 is that the parameters,
such as σand γ, of the sliding surface and controller are scalar
rather than vector. In order to show the effect of the RBFNN on
the estimation of system uncertainty, we set up a comparative
experiment in Group 3. The controller parameters of Groups 1
and 3 are the same. The only difference is that Group 3 does not
use the NN. According to Figs. 9 and 10, it can be concluded that
the performance of the controller without the NN compensation
is significantly worse than that of the controller with the NN.
V. C ONCLUSION
A predefined-time controller based on NN was proposed in
this article to solve the trajectory-tracking problem of uncertain
manipulator systems. The RBFNN was used to estimate the un-
known dynamic function. To accomplish predefined-time track-
ing performance, a novel NTSM control strategy was designed,
where the settling time could be directly tuned as the controller
parameters. The tracking errors can ultimately converge to small
neighborhoods of zero in a predefined time. The effectiveness
of the proposed algorithm is verified by experiments on the
ROKAE platform.
REFERENCES
[1] N. Nikdel, M. Badamchizadeh, V. Azimirad, and M. A. Nazari,
“Fractional-order adaptive backstepping control of robotic manipulators
in the presence of model uncertainties and external disturbances,” IEEE
Trans. Ind. Electron., vol. 63, no. 10, pp. 6249–6256, Oct. 2016.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

SUN et al.: NEURAL NETWORK-BASED TRACKING CONTROL OF UNCERTAIN ROBOTIC SYSTEMS 10519
[2] Y. Gao, J. Liu, Z. Wang, and L. Wu, “Interval type-2 FNN-based quan-
tized tracking control for hypersonic flight vehicles with prescribed
performance,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 51, no. 3,
pp. 1981–1993, Mar. 2021.
[3] W. He, G. Shuzhi Sam, Y. Li, C. Effie, and S. N. Yee, “Neural network
control of a rehabilitation robot by state and output feedback,” J. Intell.
Robotic Syst., vol. 80, no. 1, pp. 15–31, 2015.
[4] W. Xue, R. Madonski, K. Lakomy, Z. Gao, and Y. Huang, “Add-on
module of active disturbance rejection for set-point tracking of motion
control systems,” IEEE Trans. Ind. Appl., vol. 53, no. 4, pp. 4028–4040,
Jul./Aug. 2017.
[5] S. Jia and J. Shan, “Finite-time trajectory tracking control of space ma-
nipulator under actuator saturation,” IEEE Trans. Ind. Electron., vol. 67,
no. 3, pp. 2086–2096, Mar. 2020.
[6] J. Liu, L. Wu, C. Wu, W. Luo, and L. G. Franquelo, “Event-triggering
dissipative control of switched stochastic systems via sliding mode,”
Automatica, vol. 103, pp. 261–273, 2019.
[7] Z. Man, A. Paplinski, and H. Wu, “A robust MIMO terminal sliding
mode control scheme for rigid robotic manipulators,” IEEE Trans. Autom.
Control, vol. 39, no. 12, pp. 2464–2469, Dec. 1994.
[8] G. Zhong, C. Wang, and W. Dou, “Fuzzy adaptive PID fast terminal sliding
mode controller for a redundant manipulator,” Mech.Syst. Signal Process.,
vol. 159, no. 6, 2021, Art. no. 107577.
[9] Y. Feng, X. Yu, and Z. Man, “Non-singular terminal sliding mode control
of rigid manipulators,” Automatica, vol. 38, no. 12, pp. 2159–2167, 2002.
[10] M. Van, M. Mavrovouniotis, and S. S. Ge, “An adaptive backstepping
nonsingular fast terminal sliding mode control for robust fault tolerant
control of robot manipulators,” IEEE Trans. Syst., Man, Cybern. Syst.,
vol. 49, no. 7, pp. 1448–1458, Jul. 2019.
[11] F. Zhang, “High-speed nonsingular terminal switched sliding mode control
of robot manipulators,” IEEE/CAA J. Automatica Sinica, vol. 4, no. 4,
pp. 775–781, 2017.
[12] X. Shen et al., “High-performance second-order sliding mode control for
NPC converters,” IEEE Trans. Ind. Inform., vol. 16, no. 8, pp. 5345–5356,
Aug. 2020.
[13] A. Ferrara and G. P. Incremona, “Design of an integral suboptimal
second-order sliding mode controller for the robust motion control of
robot manipulators,” IEEE Trans. Control Syst. Technol., vol. 23, no. 6,
pp. 2316–2325, Nov. 2015.
[14] M. Galicki, “Finite-time trajectory tracking control in a task space of
robotic manipulators,” Automatica, vol. 67, pp. 165–170, 2016.
[15] J.Pliego-Jiménez, M. A. Arteaga-Pérez, and M. López-Rodríguez, “Finite-
time control for rigid robots with bounded input torques,” Control Eng.
Pract., vol. 102, 2020, Art. no. 104556.
[16] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of
linear control systems,” IEEE Trans. Autom. Control, vol. 57, no. 8,
pp. 2106–2110, Aug. 2012.
[17] J. Ni, L. Liu, C. Liu, X. Hu, and T. Shen, “Fixed-time dynamic surface
high-order sliding mode control for chaotic oscillation in power system,”
Nonlinear Dyn., vol. 86, pp. 401–420, 2016.
[18] X. You, C. Hua, K. Li, and X. Jia, “Fixed-time leader-following consensus
for high-order time-varying nonlinear multiagent systems,” IEEE Trans.
Autom. Control, vol. 65, no. 12, pp. 5510–5516, Dec. 2020.
[19] H. Hong, W. Yu, G. Wen, and X. Yu, “Distributed robust fixed-time
consensus for nonlinear and disturbed multiagent systems,” IEEE Trans.
Syst., Man, Cybern. Syst., vol. 47, no. 7, pp. 1464–1473, Jul. 2017.
[20] H. Hou, X. Yu, L. Xu, K. Rsetam, and Z. Cao, “Finite-time continuous
terminal sliding mode control of servo motor systems,” IEEE Trans. Ind.
Electron., vol. 67, no. 7, pp. 5647–5656, Jul. 2020.
[21] J. D. Sánchez-Torres, E. N. Sanchez, and A. G. Loukianov, “Predefined-
time stability of dynamical systems with sliding modes,” in Proc. Amer.
Control Conf., 2015, pp. 5842–5846.
[22] E. Jiménez-Rodríguez, J. D. Sánchez-Torres, and A. G. Loukianov,
“Predefined-time backstepping control for tracking a class of mechanical
systems,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 1680–1685, 2017.
[23] J. Ni and P. Shi, “Global predefined time and accuracy adaptive neural net-
work control for uncertain strict-feedback systems with output constraint
and dead zone,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 51, no. 12,
pp. 7903–7918, Dec. 2021.
[24] Y. Yin et al., “Observer-based adaptive sliding mode control of NPC con-
verters: An RBF neural network approach,” IEEE Trans. Power Electron.,
vol. 34, no. 4, pp. 3831–3841, Apr. 2019.
[25] H. Li, D. Zhang, and S. Y. Foo, “A stochastic digital implementation of
a neural network controller for small wind turbine systems,” IEEE Trans.
Power Electron., vol. 21, no. 5, pp. 1502–1507, Sep. 2006.
[26] D. Xu, J. Liu, X. Yan, and W. Yan, “A novel adaptive neural network con-
strained control for a multi-area interconnected power system with hybrid
energy storage,”IEEE Trans. Ind. Electron., vol. 65, no. 8, pp. 6625–6634,
Aug. 2018.
[27] Y. Sun, J. Xu, G. Lin, W. Ji, and L. Wang, “RBF neural network-based
supervisor control for maglev vehicles on an elastic track with networktime
delay,” IEEE Trans. Ind. Inform., vol. 18, no. 1, pp. 509–519, Jan. 2022.
[28] J. Y. Lau, W. Liang, and K. K. Tan, “Motion control for piezoelectric-
actuator-based surgical device using neural network and extendedstate ob-
server,” IEEE Trans. Ind. Electron., vol. 67, no. 1, pp. 402–412, Jan. 2020.
[29] H. An, Z. Guo, G. Wang, and C. Wang, “Neural adaptive control of air-
breathing hypersonic vehicles robust to actuator dynamics,” ISA Trans.,
vol. 116, pp. 17–29, 2021.
[30] W. He, Z. Li, Y. Dong, and T. Zhao, “Design and adaptive control for
an upper limb robotic exoskeleton in presence of input saturation,” IEEE
Trans. Neural Netw. Learn. Syst., vol. 30, no. 1, pp. 97–108, Jan. 2019.
[31] M. L. Corradini and A. Cristofaro, “Nonsingular terminal sliding-mode
control of nonlinear planar systems with global fixed-time stability guar-
antees,” Automatica, vol. 95, pp. 561–565, 2018.
[32] L. Yin, Z. Deng, B. Huo, and Y. Xia, “Finite-time synchronization for
chaotic gyros systems with terminal sliding mode control,” IEEE Trans.
Syst., Man, Cybern. Syst., vol. 49, no. 6, pp. 1131–1140, Jun. 2019.
[33] T. Madani, B. Daachi, and K. Djouani, “Modular-controller-design-based
fast terminal sliding mode for articulated exoskeleton systems,” IEEE
Trans. Control Syst. Technol., vol. 25, no. 3, pp. 1133–1140, May 2017.
[34] E. Jiménez-Rodríguez, J. D. Sánchez-Torres, D. Gómez-Gutiérrez, and A.
G. Loukianov, “Predefined-time stabilization of high order systems,” in
Proc. Amer. Control Conf., 2017, pp. 5836–5841.
[35] M. Tran and H. J. Kang, “A novel adaptive finite-time tracking control for
robotic manipulators using nonsingular terminal sliding mode and RBF
neural networks,” Int. J. Precis. Eng. Manuf., vol. 17, no. 7, pp. 863–870,
2016.
[36] L. Sun and Y. Liu, “Extended state observer augmented finite-time trajec-
tory tracking control of uncertain mechanical systems,” Mech. Syst. Signal
Process., vol. 139, 2020, Art. no. 106374.
[37] S. Zhang, P. Yang, L. Kong, W. Chen, Q. Fu, and K. Peng, “Neu-
ral networks-based fault tolerant control of a robot via fast terminal
sliding mode,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 51, no. 7,
pp. 4091–4101, Jul. 2021.
Yizhuo Sun received the B.M. degree in au-
tomation from the Hefei University of Technol-
ogy, Xuancheng, China, in 2016, and the M.E.
degree in control theory and control engineering
from the University of Electronic Science and
Technology of China, Chengdu, China, in 2019.
He is currently working toward the Ph.D. degree
in control theory and control engineering from
Harbin Institute of Technology, Harbin, China.
His research interests include sliding-mode
control and robotics control.
Yab i n Gao (Member, IEEE) received the B.M.
degree in information management and infor-
mation system and the M.E. degree in software
engineering from Bohai University, Jinzhou,
China, in 2012 and 2015, respectively, and the
Ph.D. degree in control science and engineering
from the Harbin Institute of Technology, Harbin,
China, in 2020.
He is currently a Lecturer with the Department
of Control Science and Engineering, Harbin In-
stitute of Technology.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.

10520 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 69, NO. 10, OCTOBER 2022
Yue Zhao (Member, IEEE) received the B.S. de-
gree in electronic information engineering and
M.S. degree in information and communication
engineering from the Harbin Institute of Tech-
nology, Harbin, China, in 2008 and 2010, re-
spectively, and the Ph.D. degree in image and
system from INSA-Lyon, Lyon, France, in 2014.
She is currently an Associate Professor with
the School of Astronautics, Harbin Institute of
Technology. Her research interests include the
reinforced learning algorithm and robotic arm
control.
Zhuang Liu received the B.S. degree in electri-
cal engineering and automation, from the China
University of Mining and Technology, Xuzhou,
China, in 2017, and the M.S. degree in elec-
trical engineering, from the Harbin Institute of
Technology, Harbin, China, in 2019, where he
is currently working toward the Ph.D. degree
in control theory and control engineering from
Harbin Institute of Technology, Harbin, China.
Jiahui Wang (Student, IEEE) received the B.E.
degree in computer science and technology
from the College of Information Technology, Uni-
versity of Science and Technology Liaoning,
Anshan, China, in 2014, and the M.E. degree
in software engineering from Bohai University,
Jinzhou, China, in 2017. She is currently work-
ing toward the Ph.D. degree in control science
and engineering with Harbin Engineering Uni-
versity, Harbin, China.
Jiyuan Kuang received the B.M. degree in con-
trol science and engineering and the M.E. de-
gree in power electronics and power drives from
Shandong University, Jinan, China, in 2016 and
2019, respectively. He is currently working to-
ward the Ph.D. degree in control theory and con-
trol engineering from Harbin Institute of Technol-
ogy, Harbin, China.
His research interests include nonlinear con-
trol and fuel cell power systems.
Fei Yan (Member, IEEE) received the B.S.
and M.S. degrees in mechanical engineering
and automation from Northwestern Polytechni-
cal University, Xi’an, China, in 2004 and 2007,
respectively, and the Ph.D. degree in computer
engineering from the Technical University of
Belfort-Montbeliard, Belfort, France, in 2012.
He is currently an Associate Professor with
the School of Information Science and Technol-
ogy, Southwest Jiaotong University, Chengdu,
China. His research interests include nonlinear
control and artificial intelligence.
Jianxing Liu (Senior Member, IEEE) received
the B.S. degree in mechanical engineering and
the M.E. degree in control science and engi-
neering from the Harbin Institute of Technol-
ogy, Harbin, China, in 2004 and 2010, respec-
tively, and the Ph.D. degree in automation from
the Technical University of Belfort-Montbeliard,
Belfort, France, in 2014.
He is currently a Professor with the De-
partment of Control Science and Engineering,
Harbin Institute of Technology. His research in-
terests include nonlinear control and industrial electronic systems.
Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 21,2022 at 09:31:30 UTC from IEEE Xplore. Restrictions apply.