Adaptive neural dynamic surface slidingmode control for uncertain nonlinear systems with unknown input saturation Qiang Chen1, Linlin Shi1, Yurong Nan1, and Xuemei Ren2 Abstract In this
Trang 1Adaptive neural dynamic surface sliding
mode control for uncertain nonlinear
systems with unknown input saturation
Qiang Chen1, Linlin Shi1, Yurong Nan1, and Xuemei Ren2
Abstract
In this article, an adaptive neural dynamic surface sliding mode control scheme is proposed for uncertain nonlinear systems with unknown input saturation The non-smooth input saturation nonlinearity is firstly approximated by a smooth non-affine function, which can be further transformed into an affine form according to the mean value theorem Then, one simple sigmoid neural network is employed to approximate the uncertain nonlinearity including the input saturation, and the approximation error is estimated using an adaptive learning law Virtual controls are designed in each step by combing the dynamic surface control and integral sliding mode technique, and thus the problem of complexity explosion inherent in the conventional backstepping method is avoided With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and comparative simulations are given to illustrate the effectiveness and superior performance
Keywords
Dynamic surface control, integral sliding mode control, neural network, input saturation, nonlinear system
Date received: 8 October 2015; accepted: 21 May 2016
Topic: Robot Manipulation and Control
Topic Editor: Andrey V Savkin
Associate Editor: Jayantha Katupitiya
Introduction
In many practical dynamic systems, lots of nonlinear and
uncertain characteristics are encountered, such as saturation,
hysteresis, dead zone, and so on.1–3Input saturation is well
known as one of the most common smooth input
non-linearities The magnitude of control signal is always limited
due to physical constraints or safety consideration of actual
actuators If the physical input saturation is ignored in the
con-trol process, unfortunately, the designed concon-troller may severely
degrade the system performance or even lead to its instability
So far, much attention has been paid to controllers
design for nonlinear systems with input saturation.4–8 In
the study by Gao and Selmic,4 Chen et al.5 and Chen
et al.,6significant results have been obtained for controlling
saturated nonlinear systems with the bounds of input
saturation being known or estimated in prior Recently,
some research work has been investigated without the prior knowledge of saturation parameter bounds Wen et al.7uses
a smooth non-affine function of the control input signal to approximate the non-smooth saturation function, and a Nussbaum function is introduced to compensate for the nonlinear term arising from the input saturation Due to good approximation abilities of nonlinear functions, neural networks (NNs) are employed in controllers design to
1 College of Information Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang, China
2 School of Automation, Beijing Institute of Technology, Beijing, China Corresponding author:
Yurong Nan, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, Zhejiang, China.
Email: nyr@zjut.edu.cn
International Journal of Advanced
Robotic Systems September-October 2016: 1–14
ª The Author(s) 2016 DOI: 10.1177/1729881416657750
arx.sagepub.com
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
Trang 2approximate the saturated nonlinear systems.8,9However,
in most aforementioned works, multiple NNs are used for
nonlinearity approximation in each step, which may lead to
increasing complexities of the controller design
Sliding mode control (SMC) is regarded as one of the
robust control techniques against matched uncertainties
and bounded disturbances In the study by Zhu et al.,10two
adaptive SMC laws are designed to force the state variables
of the closed-loop system to achieve the attitude
stabiliza-tion The backstepping method relaxes the matching
con-dition at the expense of a high-gain feedback required for
robustness, making it prone to chattering.11In the study by
Taheri et al.,12 backstepping technique is combined with
the SMC to relax the matching condition in SMC design
However, a possible issue in conventional backstepping
method is the problem of complexity explosion caused by
the differentiation operation of virtual controls in each step
To remedy this issue, dynamic surface control (DSC) has
been investigated by introducing a first-order filter in each
recursive design step In the study by Xu et al.,13Wang14
and Li et al.,15a NN-based dynamic surface technique has
been proposed for nonlinear pure-feedback and
strict-feedback systems, respectively However, the effect of
input saturation is not considered in the aforementioned
works Thus, it is a challenge work to develop an effective
robust control scheme for uncertain nonlinear systems with
unknown input saturation
Motivated by the aforementioned discussion, this article
develops a new neural dynamic surface SMC scheme for a
class of uncertain nonlinear systems with unknown input
saturation The main contributions are summarized as
follows
1 We transform the nonlinear pure-feedback system
into the canonical form using the first-order Taylor
expansion and coordinate transformation Besides,
to deal with the smooth input saturation
non-linearity, a smooth non-affine function is used to
approximate the input saturation function
2 Integral sliding mode surface is combined with
DSC to design the controller, in which only one
simple NN is employed for approximating
uncer-tain nonlinearities, and thus the complexity of
con-troller design has been reduced With the proposed
control scheme, no prior knowledge is required on
the bound of input saturation, and the explosion of
complexity in backstepping method is avoided
The rest of this article is organized as follows Problem
formulation and preliminaries are provided in the section
‘‘Problem formulation and preliminaries.’’ Controller
design and stability analysis are given in sections
‘‘Con-troller design’’ and ‘‘Stability analysis,’’ respectively The
section ‘‘Simulations’’ provides comparative simulation
results to validate the proposed scheme Some conclusions
are given in the section ‘‘Conclusion.’’
Problem formulation and preliminaries System description
Consider a class of nonlinear system in the following pure-feedback form
_xi¼ fiðxi;xiþ1Þ ; 1 i n 1 _xn¼ fnðxn;vðuÞÞ
y¼ x1
8
>
where xi¼ ½x1; ;xiT 2 Ri is the vector of states of the ith differential equations, and xn¼ ½x1; ;xnT 2 Rn; fi,
i¼ 1; ; n 1, are unknown smooth functions of
ðx1; xiþ1Þ satisfying fið0; ; 0Þ ¼ 0; y 2 R is the output; vðuÞ 2 R is the control input subject to saturation nonlinearity described as
vðuÞ ¼ satðuÞ ¼ vmax ðuÞ; sgnjuj vmax
u; ; juj vmax
(2) where vmaxis a positive but unknown parameter
Assumption 1 The state variables xi of equation (1) are measurable, and the nonlinear functions fi, i¼ 1; ; n, are continuously differentiable to n-order with respect to the state variables xiand the input v(u)
Since the unknown functions fi, i¼ 1; ; n, are con-tinuously differentiable with respect to xi, we apply the first-order Taylor expansion for fi, i¼ 1; ; n as
fiðxi;xiþ1Þ ¼ fiðxi;x0
iþ1Þ þ@fiðxi;xiþ1Þ
@xiþ1
x iþ1 ¼xai þ1
ðxiþ1 x0
iþ1Þ ; 1 i n 1
fnðxn;vÞ ¼ fnðxn;v0Þ þ@fnðxn;vÞ
@u jv¼van ðv v0Þ
(3)
where xai
iþ1¼ aixiþ1þ ð1 aiÞx0
iþ1, with 0 < ai<1,
1 i n 1, and va n¼ anvþ ð1 anÞv0, with 0 <
an<1 By choosing x0iþ1¼ 0 and v0¼ 0, equation (3) can
be rewritten as
fiðxi;xiþ1Þ ¼ fiðxi;0Þ þ@fiðxi;xiþ1Þ
@xiþ1
x iþ1 ¼xai þ1
xiþ1;1 i n 1
fnðxn;vÞ ¼ fnðxn;0Þ þ@fnðxn;vÞ
@v
v¼van v
(4)
For the analysis convenience, it is defined that
giðxi;xai
iþ1Þ ¼@fiðxi;xiþ1Þ
@xiþ1
x iþ1 ¼xai þ1;1 i n 1
gnðxn;vanÞ ¼@fnðxn;vÞ
@v
v¼van
(5)
Trang 3which are unknown nonlinear functions From equations
(4) and (5), equation (1) can be re-expressed as
_xi¼ fiðxi;0Þ þ giðxi;xai
iþ1Þ xiþ1 ; 1 i n 1 _xn¼ fnðxn;0Þ þ gnðxn;va iÞ v
y¼ x1
8
>
>
(6)
System coordinate transformation
In the following, it will be shown that the original system
(1) can be transformed into the canonical form with respect
to the newly defined state variables.16
Let
z1¼ y ¼ x1
z2¼ _z1¼ f1ðx1Þ þ g1ðx1;xa1
2Þx2
(7) The time derivative of z2 is derived as
_z2¼@f1ðx1Þ
@x1
_x1þ @ 1ðx1;x
a 1
2Þ
@x1
_x1þ@ 1ðx1;x
a 1
2Þ
@x2
_x2
0
@
1
Ax2
þ g1ðx1;xa1
2Þ _x2
¼ @f1
@x1
þ@ 1
@x1
x2
0
@
1 Aðf1þ g1x2Þ þ @ 1
@x2
x2þ g1
0
@
1 Aðf2þ g2x3Þ
¼D a2ðx2Þ þ b2ðx2;xa2
3Þx3
(8) where a2ðx2Þ ¼ @f1
@x 1þ@g1
@x 1x2
ðf1þ g1x2Þþ @g1
@x 2x2þ g1
f2
and b2ðx2;xa2
3Þ ¼ @g 1
@x 2x2þ g1
g2 Again, let z3¼ a2þ
b2x3, and its time derivative is
_z3¼X2
j¼1
@a2
@xj
_xjþX3 j¼1
@b2
@xj
_
xjx3þ b2_x3
¼X2
j¼1
@a2
@xj þ@b2
@xj
0
@
1 Aðfjþ gjxjþ1Þ
þ @b2
@x3
x3þb2
0
@
1 Aðf3þ g3x4Þ
¼D a3ðx3Þ þ b3ðx3;x 3
4 Þx4
(9)
where a3ðx3Þ ¼P2
j¼1
@a 2
@x jþ@b 2
@x j
ðfjþ gjxjþ1Þþ @b 2
@x 3x3þb2
f3
and b3ðx3;xa3
4 Þ ¼ @b 2
@x 3x3þb2
g3 When defining ai1and
bi1, i¼ 2; ; n, we can obtain
zi¼Dai1ðxi1Þ þ bi1ðxi1;xai1
i Þxi
_zi¼ aiðxiÞ þ biðxiÞxiþ1
(10)
where
aiðxiÞ ¼D Xi1
j¼1
@ai1
@xj
þ@bi1
@xj
xi
0
@
1 Aðfjþ gjxjþ1Þ
þ @bi1
@xi
xiþbi1
0
@
1
Afi
biðxi;xai
iþ1Þ ¼D @bi1
@xi
xiþbi1
0
@
1
Agi
(11)
Thus, the pure feedback system (6) can be rewritten in the canonical form with respect to the newly defined state variables as
_zi¼ ziþ1; i¼ 1; ; n 1 _zn¼ anðxnÞ þ bnðxn;vanÞ v
y¼ z1
8
>
To proceed the design procedure, the control function
bnðxn;va nÞ in equation (12) is assumed to be positive and bounded satisfying 0 < b1<bnðxn;vanÞ < b2, where b1
and b2 are positive constants It is pointed out that this condition has been widely used in the literature17–20as a necessary condition for the controllability of equation (1) The control objective of this article is to design a dynamic surface sliding-mode controller vðtÞ for the system (12), such that the system output y can track the desired reference signal
ydand all signals in the closed-loop system are bounded
Nonlinear saturation model
As shown in Figure 1, the control input vðtÞ 2 R is the output of the following nonlinear input saturation and uðtÞ 2 R is the input of the saturation (practical control signal) The saturation is approximated by a smooth non-affine function defined as
Figure 1 Saturation satðuÞ (solid line) and smooth function gðuÞ (dot line)
Trang 4gðuÞ ¼ vmax tanh
u
vmax
¼ vmaxe
u=v max eu=v max
eu=v maxþ eu=v max
(13)
Then, vðuÞ ¼ satðuÞ in equation (2) can be expressed as
vðuÞ ¼ satðuÞ ¼ gðuÞ þ dðuÞ (14)
where dðuÞ ¼ satðuÞ gðuÞ is a bounded function and its
bound is
jdðuÞj ¼ j satðuÞ gðuÞj vmax
1 tanhð1Þ
¼ D (15) where D is the upper bound ofjdðuÞj
According to the mean value theorem,8there exists a
constant x with 0 < x < 1, such that
gðuÞ ¼ gðu0Þ þ guxðu u0Þ (16)
where gu x ¼@gðuÞ@u
u¼ux >0; ux¼ xu þ ð1 xÞu0 and
u02 ½0 ;u By choosing u0 ¼ 0, equation (16) can be
rewritten in the following affine form
Substituting equations (17) and (14) into equation (12),
we can obtain
_zi¼ ziþ1; i¼ 1; ; n 1 _zn¼ aðxnÞ þ bðxn;vanÞ u
y¼ z1
8
>
where aðxnÞ ¼ anðxnÞ þ d, bðxn;va nÞ ¼ bnðxn;va nÞgu x
NN approximation
Due to good capabilities in function approximation, NNs
are usually used for the approximation of nonlinear
func-tions.8,9The following NN will be used to approximate the
continuous function
hðX Þ ¼ WTjðX Þ þ e (19) where W2 Rn 1 n 2 is the ideal weight matrix,
jðX Þ 2 Rn 1 1 is the basis function of the NN,e is the
NN approximation error satisfyingjej eN, and jðX Þ can
be chosen as the commonly used sigmoid function, which is
in the following form
r2þ expðX =r3Þþ r4 (20) where r1, r2, r3, and r4are appropriate parameters, and exp
is an exponential function
Remark 1 The employed NN with sigmoid function
repre-sents a class of linearly parameterized approximation
methods and can be replaced by any other approximation approaches such as spline functions, RBF functions, or fuzzy systems However, the structure of the employed
NN in this article is simpler than the other NNs that are commonly used in other works There is no hidden layer in the employed NN, in which five inputs and one output are included and the corresponding weight matrix is 5 1
Controller design
In this section, we will incorporate the DSC and integral sliding mode techniques into a NN-based adaptive control design scheme for the nth-order system described by equa-tion (12) Similar to the tradiequa-tional backstepping design, the recursive design procedure contains n steps From step 1 to step n 1, virtual control ziþ1, i¼ 1; n 1, is designed at each step, and the integral sliding mode surface
is proposed in the first step Finally, the control law u is obtained at step n
Step 1 In this step, we consider the first equation of equa-tion (12), that is, _z1¼ z2
Define the tracking error and its sliding surface as
e¼ y yd
s1 ¼ e þ l
ð
e dt
8
<
where yd is the desired reference signal and l is a positive constant
The derivatives of e and s1are
_e¼ _y _yd ¼ z2 _yd _s1¼ _e þ le ¼ z2 _ydþ l _e
(22) Choose a virtual control z2 as
z2¼ k1s1þ _yd le (23) where k1 is a positive constant
Introduce a new state variable b2and let z2pass through
a first-order filter with time constant t2>0, and we have
t2b_2þ b2¼ z2; b2ð0Þ ¼ z2ð0Þ (24) Define
Substituting equation (25) into equation (24), we can obtain
_
b2 ¼z2 b2
t2
¼ y2
t2
(26)
Step 2 Consider
Let
Trang 5s2¼ z2 b2 (28) which is called the second error surface Then, we have
_s2¼ z3 _b2 (29) Choose a virtual control z3as
z3¼ k2s2 s1þ _b2 (30) where k2is a positive constant
Again, introducing a new state variable b3 and let z3
pass through a first-order filter with time constant t3>0,
we have
t3b_3þ b3¼ z3; b3ð0Þ ¼ z3ð0Þ (31)
Define
Substituting equation (32) into equation (31), we can obtain
_
b3¼z3 b3
t3 ¼ y3
Step i Consider
Let
which is called the ith error surface Then, we have
_si¼ ziþ1 _bi (36) Choose a virtual control ziþ1as
ziþ1¼ kisi si1þ _bi (37) where kiis a positive constant
Introduce a new state variable biþ1 and let ziþ1 pass
through a first-order filter with time constant tiþ1>0, and
we have
tiþ1b_iþ1þ biþ1¼ ziþ1; biþ1ð0Þ ¼ ziþ1ð0Þ (38)
Define
yiþ1¼ biþ1 ziþ1 (39) Substituting equation (39) into equation (38), we can
obtain
_biþ1¼ziþ1 biþ1
tiþ! ¼ yiþ1
Step n The final control law will be derived in this step Consider
_zn¼ aðxnÞ þ bðxn;vanÞ u (41)
Given a compact set Ozn2 Rn, let Wand ebe such that for anyðz1; ;znÞ 2 Ozn
H ¼aðxnÞ _bn
bðxn;vanÞ ¼ W
TfðznÞ þ e (42) withjej eN Let
which is called the nth error surface From equations (41) and (43), we have
_sn¼ aðxnÞ þ bðxn;vanÞu _bn (44) Finally, design the final law u as
u¼ knsn sn1 ^WTfðznÞ ^eNtanh sn
d
(45) where ^W is the estimation of Wand ^eNis the estimation of the upper limit forjej The adaptive learning laws of ^W and ^eN are given by
_^
W ¼W_~¼ G½jðz
nÞsn s ^W _^eN ¼ _~eN ¼ veN
h
sntanh sn
d
i
8
>
where s and d are positive small constants and G¼ GT >0
is a constant matrix
Stability analysis
In this section, a theorem is provided to show the bounded-ness of all signals in system (12) and convergence of track-ing error e as well as slidtrack-ing surface s
Theorem 1 Consider the nonlinear system (12) with unknown input saturation (14), the integral sliding mode surface (21), control law (45), and adaptive learning laws (46) Given any positive number, for all initial conditions satisfying P
n1 j¼1
ðs2
i þ y2 iþ1Þþ1
bs2
nþ ~WTG1W~þ 1
veN~eN2
!
2p, all the closed-loop signals are semi-global uniformly ulti-mately bounded, and the tracking error can be made arbitrarily small by properly choosing the design parameters
Proof Firstly, define the estimation error as
~
W ¼ ^W W
~e¼ ^e e
(
(47) Then, the closed-loop system in the new coordinates, si,
bi, and ~Wi, can be expressed as follows
Trang 6_s1¼ s2þ b2þ le _yd
_s2¼ s3þ b3 _b2
_
si¼ siþ1þ biþ1 _bi; i¼ 3; n 1
_sn
b ¼ knsn sn1 ~W fðznÞ þ eN ^eNtanh sn
d
(48)
From equation (48), we have biþ1¼ yiþ1þ ziþ1;
i¼ 1; n 1 Then, we can obtain
_
s1¼ s2þ y2 k1s1
_
s2¼ s3þ y3 k2s2 s1 _b2
_
si¼ siþ1þ yiþ1 kisi si1 _bi; i¼ 3; n 1
_sn
b ¼ knsn sn1 ~W fðznÞ þ eN ^eNtanhðsn= Þ:
(49)
Besides, we know the fact that _y2¼ _b2 _z2¼ y2
t2
ðk1_s1þ ¨yd l _eÞ
¼ y2
t2
þ B2ðs1;s2;y2;yd; _yd; ¨ydÞ (50)
where B2ðs1;s2;y2;yd; _yd; ¨ydÞ ¼ ðk1_s1þ ¨yd l _eÞ, which is a continuous function
Similarly, for i¼ 2; n 1, we have _yiþ1¼ yiþ1
tiþ1 ki_si _si1 ¼ yiþ1
tiþ1
þ Biþ1ðs1; ;siþ1;y2; yi;yd; _yd; ¨ydÞ:
(51)
Consider the Lyapunov function candidate
V ¼1 2
Xn1 i¼1
ðs2
i þ y2 iþ1Þþ 1 2bs
2
2W~
T
G1W~ þ 1
2veN
~e2N (52) The derivative of the Lyapunov function is
_
V ¼Xn1
i¼1
ðsi_siþ yiþ1_yiþ1Þ þ1
bsn_snþ ~WTG1W_^þ 1
veN
~eN_^eN ¼Xn
i¼1
ðkisi2Þ þXn1
i¼1
siyiþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ sn ~W fðznÞ þ e
N ^eNtanh sn
d
þ ~WTG1W_^þ 1
veN
~eN_^eN ¼Xn
i¼1
ðkis2iÞ þXn1
i¼1
siyiþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ sn eN ^eNtanh sn
d
s ~WTW^þ 1
veN
~eN_^eN Xn
i¼1
ðkisi2Þ þXn1
i¼1
siyiþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ eN jsnj sntanh sn
d
þ eNsntanh sn
d
sn^eNtanh sn
d
s ~WTW^ þ 1
veN
~eN_^eN Xn
i¼1
ðkisi2Þ
þXn1
i¼1
siyiþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ eN jsnj sntanh sn
d
s ~WTW^
(53)
Using the following property with regard to function
tanh(.), we have
0 jxj x tanh x
d
0:2785 d (54) Using the fact
s ~WTW^ s ~WTð ~W þ WÞ
sk ~Wk2 þ sk ~Wk kWk
sk ~Wk2 þs
2k ~Wk2 þs
2 kWk2
s
2k ~Wk2 þs
2 kWk2
s
2kWk2
(55)
and substituting equations (54) and (55) into equation (53),
we can obtain _
V Xn i¼1
ðkis2iÞ þXn1
i¼1
siyiþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ 0:2785e
Nds
Using the fact
si2þ1
4y
2
we have _
V Xn i¼1
ðkis2iÞ þXn1
i¼1
si2þ1
4y
2 iþ1y
2 iþ1
tiþ1þ Biþ1yiþ1
þ 0:2785eNds
2kWk2
(58)
Trang 7ki¼ 1 þ a0; ¼ 1; ; n 1
kn¼ a0
1
tiþ1¼1
4þM
2 iþ1
where a0and Z are positive constants andjBiþ1j Miþ1
Remark 2 As pointed out in the study by Li et al.15for any
B0>0 and p > 0, the sets Q
:¼ fðyd; _yd; ¨ydÞ ¨y2þ _y2
dþ ¨y2
j¼1ðs2
i þ y2 iþ1Þþ
1
bs2
nþ
~
WTG1W~þ 1
veN~eN2Þ 2p, i ¼ 2; ; n, are compact in
R3 and R
Pi j¼1iþ 2
, respectively.QQ
i is also com-pact in R
Pi j¼1iþ 5
Therefore, jBiþ1j has a maximum
Miþ1onQQ
i Noting that, for any positive number Z
y2 iþ1B2 iþ1
2Z þZ
2 jBiþ1yiþ1j (60)
we can obtain
_
V Xn
i¼1
ða0si2Þ þXn1
i¼1
1
4y
2
1
4þM
2 iþ1
2Z þ a0
y2iþ1þM
2 iþ1y2 iþ1B2 iþ1
2ZMiþ12 þZ
2
þ 0:2785eNds
2kWk2
Xn
i¼1
ða0s2iÞ þXn1
i¼1
a0yiþ12
1B
2 iþ1
M2 iþ1
M2 iþ1y2 iþ1
2Z
þ 0:2785e
Nds
2kWk2
Xn
i¼1
ða0s2iÞ þ 0:2785e
Nds
2kWk2
(61)
Hence, we can conclude _V 0 if
j sij
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:2785e
Ndþ sk ~Wk kWk
a0
s
(62)
Then, the ultimate boundedness of siis guaranteed, and
siwill converge to the following positively invariant set
O¼ fjsij gsg (63) with gs
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:2785e
Ndþsk ~ Wk kW k
a0
q
When s1 reaches the positively invariant set gs, it
remains inside thereafter From equation (12), the error
dynamics inside gsis
Due to the boundedness of s1and _s1, the ultimate
bound-edness of the tracking error e can be directly concluded
using the Lyapunov function W¼ ð1=2Þ e2.21
Simulations
In order to show the superior tracking performance of the
proposed scheme, we consider three different control
schemes for comparison: (S1) neural dynamic SMC with
saturation compensation; (S2) neural dynamic SMC
with-out saturation compensation; and (S3) neural DSC withwith-out
saturation compensation.15
The following four indices are adopted to compare the
tracking performance of each control algorithm
1) IAE¼
ð jeðtÞjdt, which is the integrated absolute error to measure the system tracking performance 2) ISDE¼
ð ðeðtÞ e0Þ
2
dt, where e0 is the average error of whole process ISDE is the integrated square error and used to demonstrate the smooth-ness of the profile
3) IAV¼
ð jvðtÞj dt, which is the integrated absolute control and taken as a measurement of the overall amount of control effort
4) ISDV¼
ð ðvðtÞ v0Þ
2
dt, where v0 is the average control input of whole process ISDV is the inte-grated square control and used as a measurement of the fluctuations of control signal around its mean value
In the following, two simulation examples are adopted for the fair comparison of different control schemes
Spring mass and damper system
The considered spring mass and damper system repre-sents a class of widely used second-order electromecha-nical servo systems, such as hydraulic systems, rigid robots, or turntable systems.22–24 In those systems, the number of freedom degrees is always equal to the num-ber of control inputs, and thus the controller design is relatively easier
Trang 8As shown in Figure 2, a second-order system is
described as7
_x1¼ x2
_x2¼ k
mx1c
mx2þ1
mvðuÞ þ EðtÞ (65) where y¼ x1, x1 and x2 are the position and velocity,
respectively; m is the mass of the object; k is the stiffness
constant of the spring; and c is the damping
According to equation (8), equation (65) can be
trans-formed into
_x1¼ x2
_x2¼ aðx2Þ þ bðx2;vÞ vðuÞ (66)
where 2¼ ½x1;x2T aðx2;vÞ ¼ k
mx1c
mx2þ EðtÞ, bðx2;vÞ ¼ 1=m
In the simulation, two different signal waves are adopted
as the desired reference signals, and the system parameters
are fixed for various reference signals The initial states of
the system are½x1;x2T ¼ ½0; 0T The constants of adaptive
leaning laws d¼ 0:5, s ¼ 0:01, ^eN ¼ 0:01, veN ¼ 1
The time constant is t¼ 0:01 The NN parameters are
r1¼ 1, r2¼ 5, r3¼ 5, r4¼ 0:1, and G ¼ diagf5g The
system parameters are set as m¼ 1 kg, c ¼ 2 N s= m, and
k¼ 8 N= m, which are not needed to be known in our
controller design The external disturbance is
EðtÞ ¼ 0:1 sinð2p tÞ The control parameters are k1¼ 10,
k2¼ 8, and l ¼ 5
Case 1 yd ¼ 0:2cosð3p tÞ þ 0:2 is employed as the
ref-erence signal
The input saturation bound is vmax¼ 14 N
Compara-tive tracking performance, tracking errors, and control
input are shown in Figures 3, 4, and 5, respectively From
Figures 3 and 4, we can see that compared with the
pro-posed S1 method, S2 has the larger overshoot, and S2 and
S3 have larger tracking errors From Figure 5(a), compared
with S2 and S3, the control input of S1 is more smooth, and
the compensation effect of input saturation is shown in
Figure 5(b) From the figures, we can clearly observe the
significantly improved performances with the S1
In order to compare the control performance, four indices are given in Table 1 From Table 1, we can obtain that S3 controller gives the largest IAV and ISDV, while S2 controller gives the largest IAE and ISDE The proposed S1 has the smallest IAE, ISDE, IAV, and ISDV, which means
it performs best among three controllers The comparative result from Table 1 is consistent with the Figures 3 to 5 Case 2 yd ¼ 0:5ð sinðtÞ þ sinð0:5tÞÞ is employed as the reference signal
The first reference trajectory is sinusoidal signal, and all parameters are tuned based on this signal In order to show the high robustness of the proposed method, we give the second reference trajectory (sinusoidal signal with harmo-nics) The control parameters are set the same as case 1, and the input saturation bound becomes vmax¼ 6 N, which is more stringent than that of case 1 Comparative tracking
Figure 2 Spring mass and damper system
Figure 3 Tracking performance of yd¼ 0:2cosð3p tÞ þ 0:2
Figure 4 Tracking errors of yd¼ 0:2cosð3p tÞ þ 0:2 (a) Control inputs of three methods (b) The saturated control vðtÞ and the practical control uðtÞ in S2
Trang 9performance, tracking errors, and control inputs are shown
in Figures 6, 7, and 8, respectively From Figures 6 and 7,
we can see that S2 and S3 have larger tracking errors, while
the proposed S1 method achieves the smallest tracking errors
and fastest convergence speed From Figure 8(a), compared
with S2 and S3, the control input of S1 is more smooth, and
the compensation effect of input saturation is shown in Fig-ure 8(b) In conclusion, S1 has the best performance when tracking the sinusoidal signal with harmonics
In addition, the comparative results of the IAE are shown in Table 2 From Table 2, we can see that the pro-posed S1 method has the smallest IAE among all the three control schemes Besides, other three indices (i.e ISDE, IAV, and ISDV) of the proposed S1 method are also smal-lest, which means S1 has the smoothness of tracking error and control signal Therefore, S1 has the best tracking preference, which is consistent with the results given by Figures 6 to 8
Furthermore, the comparative results of IAE for differ-ent input saturation values of case 1 and case 2 are shown in Tables 3 and 4, respectively From Table 3, we can see that the proposed S1 method has the smallest IAE in case 1 compared with S2 and S3, although the input saturation
Figure 5 Control inputs of yd¼ 0:2cosð3p tÞ þ 0:2
Figure 7 Tracking errors of yd¼ 0:5
sinðtÞ þ sinð0:5tÞ
(a) Control inputs of three methods (b) The saturated control vðtÞ and the practical control uðtÞ in S2
Table 1 Comparison for yd¼ 0:2cosð3p tÞ þ 0:2
ISDV 452.2407 688.8545 609.4313
IAE: integrated absolute error; ISDE: integrated square error; ISDV:
inte-grated square control; IAV: inteinte-grated absolute control.
Figure 6 Tracking performance of yd¼ 0:5
sinðtÞ þ sinð0:5tÞ
Trang 10
values are changed from 10 to 18 Table 4 shows that for case
2, the proposed S1 scheme with the fixed parameters can still
achieve the best tracking performance in the case of various
input saturation values It should be noted that with the input
saturation becoming more stringent, the IAE of S2 and S3 will
be changed much larger than that of S1 In conclusion, the
proposed S1 scheme can achieve a satisfactory tracking
per-formance for different input saturation values
A single-link flexible-joint robotic manipulator system
In the following, we give the second example, that is, a
single-link flexible-joint robotic manipulator system.25
Due to the introduction of joint flexibility in the robot
model, the motion equations become more complicated
In particular, the order of the related dynamics becomes
twice that of the rigid robots, and the number of freedom degrees is larger than the number of control inputs, which may lead the control task more difficult
As shown in Figure 9 (figure 1 in the study by Talole25), the mechanical dynamics of the robotic manipulator system can be described as
I ¨qþ Kðq yÞ þ MgL sinðqÞ ¼ 0
J ¨y Kðq yÞ ¼ vðuÞ
(
(67)
where q and y are the position of the link and motor angles, respectively, I is the link inertia, J is the inertia of the motor, K is the spring stiffness, M and L are the mass and length of link, respectively, and vðuÞ 2 R is the plant input subject to saturation nonlinearity
For convenience of the controller design, defining
x1¼ q, x2¼ _q ¼ _x1, x3¼ y, and x4¼ _y ¼ _x3, equation (67) is transformed into
_x1¼ x2
_x2¼ MgL
I sinx1K
I ðx1 x3Þ _x3¼ x4
_x4¼1
JvðuÞ þK
Jðx1 x3Þ
8
>
>
>
>
>
>
(68)
Let z1¼ x1, z2¼ x2, z3¼ MgLI sinx1K
Iðx1 x3Þ and z4¼ x2MgL
I cosx1K
I ðx2 x4Þ, and equation (68) can be rewritten in terms of the new coordinates as
Table 3 Comparison of IAE for case 1
vmax¼ 10 0.0248 0.6036 0.3756
vmax¼ 14 0.0311 0.1190 0.1436
vmax¼ 18 0.0257 0.0289 0.0853
IAE: integrated absolute error.
Table 2 Comparison for yd¼ 0:5
sinðtÞ þ sinð0:5tÞ
ISDV 192.0874 263.9002 223.3383
IAE: integrated absolute error; ISDE: integrated square error; ISDV: inte-grated square control; IAV: inteinte-grated absolute control.
Table 4 Comparison of IAE for case 2
vmax¼ 4 0.0915 3.8849 1.7634
vmax¼ 6 0.0821 0.0430 0.5413
vmax¼ 8 0.0850 0.0924 0.5194
IAE: integrated absolute error.
Figure 8 Control inputs of yd¼ 0:5
sinðtÞ þ sinð0:5tÞ