This paper proposes a backstepping control system that uses a tracking error constraint and recurrent fuzzy neural networks (RFNNs) to achieve a prescribed tracking performance for a strict-feedback nonlinear dynamic system. A new constraint variable was defined to generate the virtual control that forces the tracking error to fall within prescribed boundaries. An adaptive RFNN was also used to obtain the required improvement on the approximation performances in order to avoid calculating the explosive number of terms generated by the recursive steps of traditional backstepping control. The boundedness and convergence of the closed-loop system was confirmed based on the Lyapunov stability theory. The prescribed performance of the proposed control scheme was validated by using it to control the prescribed error of a nonlinear system and a robot manipulator.

1. Research Article
Recurrent fuzzy neural network backstepping control for the
prescribed output tracking performance of nonlinear dynamic systems
Seong-Ik Han, Jang-Myung Lee n
School of Electrical Engineering, Pusan National University, Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea
a r t i c l e i n f o
Article history:
Received 4 July 2013
Received in revised form
13 August 2013
Accepted 28 August 2013
Available online 20 September 2013
This paper was recommended for
publication by Jeff Pieper
Keywords:
Prescribed tracking performance
Error constraint variable
Backstepping control
Recurrent fuzzy neural networks
a b s t r a c t
This paper proposes a backstepping control system that uses a tracking error constraint and recurrent
fuzzy neural networks (RFNNs) to achieve a prescribed tracking performance for a strict-feedback
nonlinear dynamic system. A new constraint variable was defined to generate the virtual control that
forces the tracking error to fall within prescribed boundaries. An adaptive RFNN was also used to obtain
the required improvement on the approximation performances in order to avoid calculating the
explosive number of terms generated by the recursive steps of traditional backstepping control. The
boundedness and convergence of the closed-loop system was confirmed based on the Lyapunov stability
theory. The prescribed performance of the proposed control scheme was validated by using it to control
the prescribed error of a nonlinear system and a robot manipulator.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
For many industrial control systems, the performance con-
straints of control systems have received much attention, in terms
of the type of physical stoppage, saturation, performance, and
safety specifications to prevent performance degradation, hazards,
or system damage. Some notable constraint techniques for posi-
tion, velocity, and force constraints that exist in servo dynamic
systems are the barrier Lyapunov function (BLF) technique [1–3]
inspired by Brunovsky-type systems [4], the performance trans-
formation function technique developed by Rovithakis et al. [5–7],
Na et al. [8,9], and the funnel control technique proposed by
Ilchman et al. [10,11] and Hackl et al. [12,13].
The BLF technique uses the logarithmic function in the Lyapunov
function, and the state variable of the control system can be
constrained by the symmetric, or asymmetric and time-invariant
[1,2] or time-varying constraint [3] of the state variable. Therefore,
the tracking errors can be indirectly constrained. On the other hand,
the BLF controller must be redesigned to accommodate the change in
the Lyapunov function and to establish the stability of the closed-
loop system and bound condition parameters. A piecewise smooth
BLF was also adopted in the asymmetric BLF design. Consequently,
extra effort is needed to ensure the continuity and differentiability of
the piecewise smooth stabilizing functions.
The approach proposed by Rovithakis et al. [5–7] and Na et al.
[8,9] is to construct a prescribed performance function, and
subsequently, to provide the inverse of a transformation function
that converts the tracking error of an original nonlinear system
into a new error in the transformed system. Therefore, the tracking
performance of the transient property and the steady-state error
can be characterized by a prescribed constraint function. However,
a tangent hyperbolic function has been adopted as the transforma-
tion function that is combined with a prescribed smooth function
to transform the tracking error. Therefore, the inverse transforma-
tion function, which would inevitably include a partial differential
terms in the controller, may cause the singularity problem in
controllers when implemented.
On the other hand, as a non-model-based (memoryless) con-
straint technique, the funnel control proposed by Ilchman et al. and
Hackl et al. also guarantees the prescribed transient behavior, and
asymptotic tracking of the system. This technique bypasses the
difficulties of identification and estimation of traditional high-gain
adaptive control. However, funnel control is limited, because it can
only be applied to a class S of a linear or nonlinear system with a
relative degree of one or two stable zero-dynamics (minimum-phase
in the LTI case) and with a known positive high-frequency gain.
In this paper, we propose a new error-constraint variable for
transient and asymptotic tracking that does not use the afore-
mentioned complex transformation function and is not limited by
a class S like the conventional funnel control. The error constraint
variable is used as a virtual control variable in the backstepping
design to ensure a prescribed transient and steady-state perfor-
mance. A backstepping control provides guaranteed global or
Contents lists available at ScienceDirect
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ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2013.08.012
n
Corresponding author. Tel.: þ82 51 510 2378; fax: þ82 51 515 5190.
E-mail address: jmlee@pusan.ac.kr (J.-M. Lee).
ISA Transactions 53 (2014) 33–43

2. regional regulations and tracking properties and avoids the unneces-
sary cancellation of useful nonlinearities, unlike the feedback linear-
ization technique. It also provides a systematic procedure for
designing a stabilizing controller for a nonlinear system by following
a step-by-step recursive algorithm [14,15]. However, this control
undergoes the complex term of the controller because of the
repeated differentiations of virtual control functions in recursive
design procedures. A robust adaptive backstepping controller is then
designed to ensure that the tracking error falls within the prescribed
performance bounds and that all other signals of the closed-loop
systems are continuously bounded in some small residual set. The
RFNN system, which provides the benefits of both recurrent neural
networks (RNNs) [16,17] and fuzzy logic systems [18,19], was
considered to compensate for a large number of complex terms
and to avoid the complexity of the controller.
The main contributions of this paper are as follows: (1) a proposed
error performance variable to constrain the performance within the
prescribed bounds by using a virtual backstepping control for a strict
feedback nonlinear system; (2) a proposed error performance con-
straint that can handle the time-invariant and time-varying error
constraints without using the complex barrier Lyapunov function, as
used in [1–3]; (3) a proposed backstepping method for nonlinear
systems that uses the RFNNs approximation to solve the problem of
controller complexity, which is inherent in conventional backstep-
ping control; (4) an evaluation of the proposed constrained back-
stepping method by simulation and experiment in a nonlinear
system and a robot manipulator. The backstepping BLF-based con-
straint schemes [1–4], Transformation-function-based constraint
scheme [5–7,8], and the funnel control [10,12,13] have been evalu-
ated using only numerical demonstrations, and experiment was
executed only in [9] and [11].
The rest of this paper is organized as follows. Section 2 outlines
the nonlinear system dynamics, error constraint, and RFNN
system. The formulation of the recursive backstepping controller
with a prescribed error constraint and its stability analysis are
discussed in Section 3. The controller′s performance is verified
through simulation of a nonlinear system and experiment of the
Scorbot robot manipulator. The results are illustrated and analyzed
in Section 4. In Section 5, conclusions are finally summarized.
2. Motivating problem
2.1. Dynamics of a strict-feedback nonlinear system
Consider a strict-feedback nonlinear system whose dynamics
can be described using the following equations:
_xi ¼ f iðxiÞþgiðxiÞxiþ 1 þdi; 1rirnÀ1;
_xn ¼ f nðxnÞþgnðxnÞuþdn;
y ¼ x1; ð1Þ
where xi is the state of the ith subsystem, xi ¼ ½x1; …; xiŠT
ARi
represent the vector of the partial state variables in the nonlinear
system, uAR and yAR denote the control input and output of the
system, respectively, f iðxiÞ; i ¼ 1; :::; n; are unknown bounded
smooth functions, and di; i ¼ 1; :::; n; are the uncertainties
belonging to the compact set and are composed of the unmodeled
dynamics and external disturbances. giðxiÞ; i ¼ 1; …; n; are the
smooth control gain functions and assumed to be bounded within
the compact set ΩARn
, where Ω can be made as large and as
desired.
Assumption 1. The signs of gi; i ¼ 1; …; n, are strictly positive or
negative. Without loss of generality, there exist positive constants
0ogi min rgi max such that gi min rgiðxÞr gi max, 8xn AΩ & Rn
.
If the signs of gi are unknown, the Nussubaum function technique
can be introduced in the backstepping design procedure [20].
Assumption 2. The desired trajectory signal ydðtÞ is available with
its n time derivatives, piecewise continuous and the vector
y′d ¼ ½yd; yð1Þ
d
; :::; yðiÞ
d
ŠT
is bounded.
The control objectives are:
1. Determine a state feedback control system such that the output
x1 of the system can track a desired trajectory yd, while
ensuring that all the solutions of the closed-loop system are
semi-globally, uniformly, and ultimately bounded.
2. Ensure that the prescribed funnel performance bound for
tracking error eðtÞ ¼ yðtÞÀydðtÞ is always satisfied.
2.2. Funnel control and the new error constraint virtual variable
Funnel control is a strategy that employs a time-varying gain
τðtÞ to control systems of class S with a relative degree r ¼ 1 or 2,
stable zero dynamics, and known high-frequency gains. The
system S is governed by the funnel controller with the control
input
uðtÞ ¼ τðFφðtÞ; ψðtÞ; ‖eðtÞ‖Þ Â eðtÞ ð2Þ
by evaluating the vertical distance at the actual time, as shown in
Fig. 1, between the funnel boundary FφðtÞ and the Euclidian norm
‖eðtÞ‖ of error, as follows
dvðtÞ ¼ FφðtÞÀ‖eðtÞ‖ ð3Þ
The funnel boundary is given by the reciprocal of an arbitrarily
chosen bounded, continuous and positive function φðtÞ40 for all
tZ0 with supt Z 0φðtÞo1. The funnel is defined as the set
Fφ : t-feARm
φðtÞ Â ‖e‖o1

4. g ð4Þ
The control gain of (2) is adjusted to ensure that the error eðtÞ
evolves inside the funnel FφðtÞ, as follows:
τðtÞ ¼
ψðtÞ
FφðtÞÀ‖eðtÞ‖
; ð5Þ
where ψðtÞ denotes the scaling factor. Thus, as the error eðtÞ
approaches the boundary FφðtÞ, the gain τðtÞ increases, and as
the error eðtÞ becomes small, the gain τðtÞ decreases conversely.
A proper funnel boundary to prescribe the performance is selected
by the following:
FφðtÞ ¼ ξ0expðÀatÞþξ1; ð6Þ
where ξ0 Z ξ1 40; ξ1 ¼ lim
t-1
inf FφðtÞ, and eð0Þ

8. oFφð0Þ.
However, this funnel control is limited in class S, which
requires the aforementioned conditions; therefore, we propose
a new error constraint variable combined with virtual control of
the backstepping scheme described in Section 3. In the back-
stepping scheme, any limitation required by class S of the funnel
control is not imposed on the control system. In addition, a new
Fig. 1. Basic concept of funnel control.
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–4334

9. error constraint variable would be continuously differentiable in
order to be used as a virtual control of the recursive backstepping
design, because ‖eðtÞ‖ is discontinuous at eðtÞ ¼ 0. By setting
the tracking error as e ¼ x1Àyd, we define the following virtual
error surface z1ðtÞ as the first error variable of the backstepping
control
z1ðtÞ ¼
eðtÞ
FφðtÞÀeðtÞ
qðeðtÞÞÀ
eðtÞ
FφðtÞþeðtÞ
ð1ÀqðeðtÞÞ; ð7Þ
qðeðtÞÞ ¼
1 if eðtÞZ0
0 if eðtÞo0
(
ð8Þ
where the funnel boundary FφðtÞ satisfies the condition given in (6).
This variable will be employed to ensure the prescribed output
performance of the backstepping control scheme.
2.3. Function approximation using RFNN system
An approximation method was developed by combining a
fuzzy system with RNNs to obtain better approximation perfor-
mance for uncertain nonlinear functions. A four-layer RFNN, as
shown in Fig. 2, which is comprised of the input, membership,
rule, and output layers, was adopted to implement the function of
online gain tuning. The output of the second layer is repeatedly
entered into the input layer through a time interval. The signal
propagation and the basic function in each layer of the RFNN are
defined with several components. The net inputs and the net
outputs in the input layer are represented as follows:
y1
i ¼ u1
i ; i ¼ 1; :::; na; ð9Þ
where u1
i represents the inputs and na is the number of inputs.
In the membership layer, each node performs the Gaussian
function, which was adopted as the membership function. The
jth node input and output are represented as follows:
y2
j ¼ exp À
½y1
i þw2
ijy2
j ðtÀTÞÀmijŠ2
s2
ij
!
; j ¼ 1; :::; nb; ð10Þ
where m2
ij and s2
ij are the mean and the standard deviation,
respectively, of the Gaussian function in the jth term of the ith
term input linguistic variable u2
ij to the node of the membership
layer, nb is the total number of the linguistic variables with respect
to the input nodes, and T is the recurrent time interval. In the rule
layer, the kth rule node can be represented as follows:
y3
k ¼ ϕ ¼ ∏
j
y2
j ; k ¼ 1; :::; nc; ð11Þ
where nc is the number of rules with complete rule connection, if
the same linguistic variables are selected for each input node. For
the output layer, the overall output is the summation of all input
signals as follows:
y4
o ¼ ∑
k
w4
kou4
k þ∑
i
wiou1
i ; o ¼ 1; ð12Þ
where the connecting weight wio is the output action strength of the
ith input associated with the output of the fuzzy neural network, w4
ko
is the output action strength of the oth output associated with the
kth rule; u4
k represents the kth input to the node of the output layer,
and y4
o is the output of the fuzzy neural network.
The output equation of the output layer can be expressed as
follows:
y4
o ¼ ½y4
i ; :::; y4
i ; :::; y4
nd
Š ¼ WT
wϕþWf u ¼ WT
o χ; ð13Þ
where Wo ¼ ½WT
w WT
f ŠT
ARnd
, Ww ¼ ½Wk1; :::; Wknd
ŠT
, Wf ¼ ½W1o; :::;
WnaoŠT
ARna
,χ ¼ ½ϕ uŠT
ARne
, and ne ¼ na þ1. For any given real
continuous function f ðUÞ : Rn
-R on a sufficiently large compact
set Ω & R and arbitrary εm 40, there exists a RFNN system yðxÞ in
the form of (10), such that
supx A Ωjf ðxÞÀy4
oðxÞjrεm ð14Þ
The function f ðxÞ can then be expressed as follows:
f ðxÞ ¼ WnT
o χðxÞþεn
; 8xAΩ & Rn
; ð15Þ
where jεn
jrεm, εn
is the error of the RFNN approximation and
Wn
o is the optimal value of Wo that minimizes the RFNN
Fig. 2. Schematic diagram of RFNNs.
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–43 35

10. approximation error εn
. Therefore,
Wn
o ¼ arg min
Wo A RNÂ1
fsupx AΩjf ðxÞÀ ^W
T
o χðxÞjg ð16Þ
because Wn
o was unknown, it was replaced by ^Wo, which is an
estimation of Wn
o.
3. Design of the controller and adaptive estimators
The backstepping controller is derived from n recursive steps.
In the first step, a funnel and virtual error variables are defined.
Next, at each step, a virtual control αiðdÞ and control input u are
constructed to stabilize the ith order subsystem using an appro-
priate Lyapunov function candidate Vi. Adaptive laws are derived
to update the output weight of RFNNs for closed-loop stability.
Finally, the stability condition for the whole closed-loop system
with respect to an overall Lyapunov function V is analyzed.
We define the error variables as follows:
zi ¼ xiÀαiÀ1ðxiÀ1; ^WoiÀ1; ^ρiÀ1; yðiÀ1Þ
d
Þ; i ¼ 2; :::; n; ð17Þ
where ^WoiÀ1 and ^ρiÀ1 are defined later. The time derivative of the
funnel and virtual error variables becomes
_z1 ¼
1
ðFφÀeÞ2
½_eðFφÀeÞÀeð_FφÀ_eÞŠqÀ
1
ðFφ þeÞ2
½_eðFφ þeÞÀeð_Fφ þ _eÞŠð1ÀqÞ
¼ Fφ
q
ðFφÀeÞ2
À
1Àq
ðFφ þeÞ2
!
_eÀ_Fφ
q
ðFφÀeÞ2
À
1Àq
ðFφ þeÞ2
!
e
¼ FφΦF ½f 1ðx1Þþg1ðx1Þx2 þd1À_ydŠÀ_FφΦF e; ð18Þ
_zi ¼ f iðxiÞþgiðxiÞxiþ 1 þdiÀ_αiÀ1; i ¼ 2; :::; nÀ1; ð19Þ
_zn ¼ f ′nðxn; ydðnÀ1ÞÞþgnðxnÞuþdnÀ_αnÀ1; ð20Þ
where ΦF ¼ q
ðFφÀeÞ2À 1Àq
ðFφ þ eÞ2.
The time derivative of the virtual control becomes
_αi ¼ ∑
i
j ¼ 1
∂αi
∂xj
ðf jðxjÞþgjðxjÞxjþ 1 þdjÞ
þ ∑
i
j ¼ 1
∂αi
∂ ^Woj
_^Woj þ ∑
i
j ¼ 1
∂αi
∂^ρj
_^ρj þ ∑
i
j ¼ 1
∂αi
∂yðjÀ1Þ
d
yðjÞ
d
¼ f ðiÞðiþ 1Þðzi; xi þ1Þþ ∑
i
j ¼ 1
∂αi
∂xj
dj þ ∑
i
j ¼ 1
∂αi
∂yðjÀ1Þ
d
yðjÞ
d
; i ¼ 2; :::; nÀ1; ð21Þ
_αnÀ1 ¼ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂x1
ðf jðxjÞþgjðxjÞxjþ 1 þdjÞ
þ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂ ^Woj
_^Woj þ ∑
nÀ1
j ¼ 1
∂αi
∂^ρj
_^ρj þ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂yðjÀ1Þ
d
yðjÞ
d
¼ f ðnÀ1ÞnðznÀ1; xnÞþ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂xj
dj þ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂yðjÀ1Þ
d
yðjÞ
d
; ð22Þ
f ðiÀ1ÞiðziÀ1; xiÞ ¼ ∑
iÀ1
j ¼ 1
∂αiÀ1
∂xj
ðf jðxjÞþgjðxjÞxjþ 1Þþ ∑
iÀ1
j ¼ 1
∂αiÀ1
∂ ^Woj
_^Woj þ ∑
i
j ¼ 1
∂αi
∂^ρj
_^ρj;
f ðnÀ1ÞnðznÀ1; xnÞ ¼ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂xj
ðf jðxjÞþgjðxjÞxjþ 1Þþ ∑
nÀ1
j ¼ 1
∂αnÀ1
∂ ^Woj
_^Woj þ ∑
nÀ1
j ¼ 1
∂αi
∂^ρj
_^ρj:
Therefore, we obtain
_zi ¼ f iðxiÞþgiðxiÞxiþ 1 þdiÀ_αiÀ1
¼ f ′iðxi; ydðiÀ1ÞÞþgiðxiÞxiþ 1 þdiÀ ∑
iÀ1
k ¼ 1
∂αiÀ1
∂xk
dkÀ ∑
iÀ1
k ¼ 1
∂αiÀ1
∂yðkÀ1Þ
d
yðkÞ
d
¼ f ′iðxi; ydðiÀ1ÞÞþgiðxiÞxiþ 1 þd′i; i ¼ 2; :::; nÀ1; ð23Þ
_zn ¼ f ′nðxn; ydðnÀ1ÞÞþgnðxnÞuþd′n; ð24Þ
where
d′i ¼ diÀ ∑
iÀ1
k ¼ 1
∂αiÀ1
∂xk
dkÀ ∑
iÀ1
k ¼ 1
∂αiÀ1
∂yðkÀ1Þ
d
yðkÞ
d
; ð25Þ
f ′iðxi; ydðiÀ1ÞÞ ¼ f iðxiÞÀf ðiÀ1ÞiðziÀ1; xiÞ; ð26Þ
d′n ¼ dnÀ ∑
nÀ1
k ¼ 1
∂αnÀ1
∂xk
dkÀ ∑
nÀ1
k ¼ 1
∂αnÀ1
∂yðkÀ1Þ
d
yðkÞ
d
; ð27Þ
f ′nðxn; ydðnÀ1ÞÞ ¼ f nðxnÞÀf ðnÀ1ÞnðznÀ1; xnÞ ð28Þ
We consider the Lyapunov function candidate as follows:
V ¼ ∑
n
i ¼ 1
Vi; ð29Þ
where
V1 ¼
1
2
z2
1 þ
1
2
~W
T
o1ΓÀ1
1
~W o1 þ
1
2ηρ1
~ρ2
1; ð30Þ
Vi ¼
1
2
z2
i þ
1
2
~W
T
oiΓÀ1
i
~Woi þ
1
2ηρi
~ρ2
i ; i ¼ 2; :::; n: ð31Þ
By taking the time derivation of (29) using (18), (23) and (24)
yield the following:
_V1 ¼ z1 _z1 þ ~W
T
o1ΓÀ1
1
_~W o1 þ
1
ηρ1
~ρ1
_~ρ1
¼ z1FφΦF ½WnT
o1χ1ðx1Þþεn
1 þg1ðx1Þx2 þd1À_ydŠÀz1
_FφΦF e
À ~W
T
o1ΓÀ1
1
_^WoiÀ
1
ηρ1
~ρ1
_^ρ1
rz1FφΦF ½ ^W
T
o1χ1ðx1Þþg1ðx1Þx2 þ ^ρ1À_ydŠÀz1
_FφΦF e
þz1FφΦF
~W
T
o1χ1ðx1Þþz1FφΦF ~ρ1À ~W
T
o1ΓÀ1
1
_^W
T
o1À
1
ηρ1
~ρ1
_^ρ1
¼ z1FφΦF ½ ^W
T
o1χ1ðx1Þþg1ðx1Þx2 þ ^ρ1À_ydŠÀz1
_FφΦF e
þ ~W
T
o1ðFφΦF χ1ðx1Þz1ÀΓÀ1
1
_^Wo1Þþ ~ρ1 z1FφΦF À
1
ηρ1
_^ρ1
!
; ð32Þ
_Vi ¼ zi _zi þ ~W
T
oiΓÀ1
i
_~W oi þ
1
ηρi
~ρi
_~ρi
¼ ziðf ′iðz1; xi; ydiÞþgiðxÞxiþ 1 þd′iÞÀ ~W
T
oiΓÀ1
i
_^WoisÀ
1
ηρi
~ρi
_^ρi
¼ ziðWnT
oi χiðxiÞþεn
i þgiðxiÞxiþ 1 þd′iÞÀ ~W
T
oiΓÀ1
i
_^WoiÀ
1
ηρi
~ρi
_^ρi
rzið ^W
T
oiχiðxiÞþgiðxiÞxiþ1 þ ^ρiÞþ ~W
T
oiðχiðxiÞziÀΓÀ1
i
_^W
T
oiÞ
þ ~ρi ziÀ
1
ηρi
_^ρi
!
; i ¼ 2; :::; nÀ1; ð33Þ
_Vn ¼ znðf ′nðxn; ydðnÀ1ÞÞþgnðxnÞuþd′nÞþ ~W
T
onΓÀ1
n
~W
T
on þ
1
ηρn
~ρn
_~ρn
rznð ^W
T
oiχnðxnÞþgnðxnÞuþ ^ρnÞþ ~W
T
onðχnðxnÞzn
ÀΓÀ1
n
_^WonÞþ ~ρn znÀ
1
ηρn
_^ρn
!
; ð34Þ
where ~W oi ¼ Wn
oiÀ ^Woi, with the assumptions that jεn
i þd
′
ijrρn
i ,
that ρn
i is a positive constant, that ~ρi ¼ ρn
i À^ρi, ^ρi is an estimate of ρn
i ,
that ηðdÞ and η′ðdÞ are positive constants, and that Γi ¼ diagðηwiÞ40
is a constant matrix. We specify the virtual control laws as follows:
α1 ¼
1
g1ðx1Þ
Àk1ΞF z1À ^W
T
o1χ1ðx1ÞÀ^ρ1 þ _yd þ
_Fφ
Fφ
e
!
; ð35Þ
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–4336

11. α2 ¼
1
g2ðx2Þ
ðÀk2z2À ^W
T
o2χ2ðx2ÞÀ^ρ2Àg1ðx1Þz1FφΦF Þ; ð36Þ
αi ¼
1
giðxiÞ
ðÀkiziÀgiÀ1ðxiÀ1ÞziÀ1À ^W
T
oiχiðxiÞÀ^ρiÞ; i ¼ 3; :::; nÀ1; ð37Þ
u ¼ αn; ð38Þ
where
ΞF ¼
ðF2
φÀe2
Þ2
Fφ½ðFφ þeÞ2
qÀðFφÀeÞ2
ð1ÀqÞŠ
The adaptive laws are specified as follows:
_^W
T
o1 ¼ Γ1ðFφΦF χ1ðx1Þz1Àη′w1
^Wo1Þ; ð39Þ
_^Woi ¼ ΓiðχiðxiÞziÀη′wi
^WoiÞ; i ¼ 2; :::; n; ð40Þ
_^ρ1 ¼ ηρ1ðz1FφΦF Àη′ρ1 ^ρ1Þ; ð41Þ
_^ρi ¼ ηρiðziÀη′ρi ^ρiÞ; i ¼ 2; :::; n ð42Þ
Lemma 1. The virtual control α1ðz1; ^θ1; ^ρ1; _ydÞ is piecewise continu-
ously differentiable with respect to z1 over the initial condition
jeð0ÞjoFφð0Þ.
Proof. To show that the virtual control, α1, is continuously diffe-
rentiable, a proof is needed to show that lim
z-0
∂α1=∂z1 is identical
from both directions. For 0reð0ÞoFφð0Þ and q ¼ 1,
lim
z1-0þ
∂α1=∂z1 ¼
1
g1ðx1Þ
Àk1ðF2
φÀe2
Þ2
Fφ½ðFφ þeÞ2
Š
o0 ð43Þ
Similarly, for ÀFφð0Þoeð0Þo0 and q ¼ 0, we obtain
lim
z1-0À
∂α1=∂z1 ¼
1
g1ðx1Þ
k1ðF2
φÀe2
Þ2
ðÀFφÞ½ðÀFφÀeÞ2
Š
¼
1
g1ðx1Þ
Àk1ðF2
φÀe2
Þ2
Fφ½ðFφ þeÞ2
Š
o0
ð44Þ
Therefore, lim
z1-0þ
∂α1=∂z1 ¼ lim
z1-0À
∂α1=∂z1, where ∂α1=∂z1 is con-
tinuous and not ill-defined at z1 ¼ 0, which means that α1 is
continuously differentiable with respect to z1 over the initial
condition jeð0ÞjoFφð0Þ.
In (35), the virtual control α1 contains the term, Àk1ΞF z1, which
forces the error surface to stay within the boundary of the
prescribed funnel function. The behavior of the virtual control
corresponds to the control input of the conventional funnel control
system. Therefore, the proposed funnel variable adopted in the
proposed backstepping scheme provides the same constraint
effect as the funnel control system.
From the relations of (35)–(42), we obtain
_V1 rÀk1z2
1 þg1ðx1Þz1z2FφΦF þη′w1
~W
T
o1
^Wo1 þη′ρ1 ~ρ1 ^ρ1; ð45Þ
_V2 rÀk2z2
2Àg1ðx1Þz1z2FφΦF þg2ðx2Þz2z3 þη′w2
~W
T
o2
~W
T
o2 þη′ρ2 ~ρ2 ^ρ2;
ð46Þ
_Vi rÀkiz2
i ÀgiÀ1ðxiÀ1ÞziÀ1zi þgiðxiÞziziþ 1 þη′wi
~W
T
oi
^Woi þη′ρi ~ρi ^ρi;
i ¼ 3; :::; nÀ1; ð47Þ
_Vn rÀknz2
nÀgnÀ1ðxnÀ1ÞznÀ1zn þη′wn
~W
T
on
^Won þη′ρn ~ρn ^ρn ð48Þ
By using the inequalities
η′wk
~W
T
ok
^Wok rÀ
η′wk
~W
T
ok
~Wok
2
þ
η′wkWnT
ok Wn
ok
2
; k ¼ 1; :::; n; ð49Þ
η′ρk ~ρk ^ρk rÀ
η′ρk ~ρ2
k
2
þ
η′ρkρn2
k
2
; k ¼ 1; :::; n; ð50Þ
_V can be written as
_V rÀ ∑
n
j ¼ 1
kjz2
j À ∑
n
j ¼ 1
η′wj
~W
T
oj
~W oj
2
À ∑
n
j ¼ 1
η′ρj ~ρ2
j
2
þ ∑
n
j ¼ 1
η′wjWnT
oj Wn
oj
2
þ ∑
n
j ¼ 1
η′ρjρn2
j
2
rÀ
1
2
∑
n
j ¼ 1
kjz2
j À ∑
n
j ¼ 1
η′wj
~W
T
oj
~Woj
2
À ∑
n
j ¼ 1
η′ρj ~ρ2
j
2
þΔ; ð51Þ
where
Δ ¼ ∑
n
j ¼ 1
η′wjWnT
oj Wn
oj
2
þ ∑
n
j ¼ 1
η′ρjρn2
j
2
Next, the stability of the closed-loop system could be proven by
the Lyapunov theory. □
Theorem 1. Under Assumptions 1 and 2 (Section 2.1), consider
a closed-loop funnel constrained strict-feedback system consisting
of the plant (1), the virtual control functions (35)–(37),the control
law (38), and the adaption laws (39)–(42). If the initial condi-
tion satisfies eð0ÞAΩ0 : ¼ fz1 AR : jeð0ÞjoFφg, then the following
properties hold:
(i) The overall closed-loop signals are semi-globally, uniformly, and
ultimately bounded.
(ii) The output tracking errors are smaller than the prescribed funnel
error bounds, and the sizes of the tracking errors can be
arbitrarily decreased by the appropriate selection of design
parameters.
Proof. Let Λ ¼ min ½ki; η′wi; η′ρiŠ, i ¼ 1; :::; n. Then, (51) becomes
_V rÀΛV þΔ ð52Þ
Solving inequality (52) yields
0rVðtÞr Vð0ÞÀ
Δ
Λ
eÀΛt
þ
Δ
Λ
rVð0Þþ
Δ
Λ
; 8t ð53Þ
From (53), VðtÞ is eventually bounded by Δ=Λ, which can be
made arbitrarily small by controlling the design parameters.
Therefore, all error signals are semi-globally, uniformly, and
ultimately bounded.
ii) From V and (53), we can obtain the following expression:
1
2z2
1 ¼ 1
2
e
FφÀeqÀ e
Fφ þ eð1ÀqÞ
2
¼
1
2
e2
ðFφÀjejÞ2
r Vð0ÞÀ
Δ
Λ
eÀΛt
þ
Δ
Λ
; ð54Þ
because the following relations are easily obtained:
2
eq
ðFφÀeÞ
Â
eð1ÀqÞ
ðFφ þeÞ
¼ 0; ð55Þ
e2
ðFφÀeÞ2
qþ
e2
ðFφ þeÞ2
ð1ÀqÞ ¼
e2
ðFφÀjejÞ2
ð56Þ
Eq. (54) can then be written as the following inequality
eðtÞ

15. r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðFφÀjejÞ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Vð0ÞÀ
Δ
Λ
eÀΛt þ
Δ
Λ
s
ð57Þ
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–43 37

16. As t-1, eðtÞ

20. ¼ jyÀydjr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðFφÀjejÞΔ
Λ
q
.
If the design parameters are selected as ΛZΔ, then
eðtÞ

24. ¼ jyÀydjr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðFφÀjejÞΔ
Λ
r
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
FφÀjej
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fφð1Àjej=FφÞ
q
o
ffiffiffiffiffiffi
Fφ
q
;
ð58Þ
where 0ojejoFφ yields 0ojej=Fφ o1. Therefore, the output
tracking errors are smaller than the prescribed error bounds and
the sizes of jeðtÞj can be arbitrarily decreased into small values by
controlling the design parameters. The proof of Theorem 1 is
therefore complete. □
Remark 1. For the constrained method proposed by Rovithakis
et al. [5–7], the error transformation is given by
eðtÞ ¼ FφTðz1ðtÞÞ; ð59Þ
z1ðtÞ ¼ TÀ1 eðtÞ
Fφ
¼ R
eðtÞ
Fφ
; ð60Þ
where Tðz1ðtÞÞ and its inverse Rðz1ðtÞÞ are given as
Tðz1ðtÞÞ ¼
expðz1ðtÞÞÀδexpðÀz1ðtÞÞ
expðz1ðtÞÞ þ expðÀz1ðtÞÞ if eð0ÞZ0
δexpðz1ðtÞÞÀexpðÀz1ðtÞÞ
expðz1ðtÞÞ þ expðÀz1ðtÞÞ if eð0Þo0
8
:
ð61Þ
Fig. 3. Simulated responses for case I: (a) tracking output for eð0Þ40, RPB_WO (dashed line), RPB_W (solid line), (b) tracking errors for eð0Þ40, RPB_WO (dashed line), RRB
(dash dotted line), RPB_W (solid line), (c) tracking errors for eð0Þo0, RPB_WO (dashed line), RRB (dash dotted line), RNB (dotted line), RPB_W (solid line), (d) tracking error
(dashed line) and Àk1ΞF z1 Â 10 (solid line) for eð0Þ40, (e) tracking error (dashed) and Àk1ΞF z1 Â 5 (solid line) for eð0Þo0.
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–4338

25. Then, differentiating z1ðtÞ with respect to time yields
_z1 ¼
1
Fφ
∂Rðf 1ðx1Þþg1ðx1Þx2 þd1À_ydÞ; ð62Þ
where
∂R ¼
1þ δ
2
1
ðe=Fφ þ δÞð1Àe=FφÞ if eð0ÞZ0
1þ δ
2
1
ðe=Fφ þ 1ÞðδÀe=FφÞ if eð0Þo0:
8
:
ð63Þ
A virtual control law can be then specified as follows:
α1 ¼
1
g1ðx1Þ
Àc1∂RÀ1
Fφz1À ^W
T
o1χ1ðx1ÞÀ^ρ1 þ _yd þ
_Fφ
Fφ
e
!
ð64Þ
Remark 2. Next, if the constrained method proposed by Na
et al. [8,9] is used instead of the our proposed method, the error
transformation is given by
z1ðtÞ ¼ TÀ1 eðtÞ
FφðtÞ
; ð65Þ
where
Tðz1ðtÞÞ ¼
δexpðz1ðtÞþr1ÞÀδexpðÀðz1ðtÞþr1ÞÞ
expðz1ðtÞþr1ÞþexpðÀðz1ðtÞþr1ÞÞ
; ð66Þ
r1 ¼
1
2
ln
δ
δ
ð67Þ
Therefore, the error transformation can be expressed as
z1ðtÞ ¼ TÀ1 eðtÞ
FφðtÞ
¼ ΘðtÞ
¼
1
2
lnðδðeðtÞ=FφðtÞÞþδδÞÀ
1
2
lnðδδÀδðeðtÞ=FφðtÞÞ ð68Þ
Then, time differentiating z1ðtÞ yields
_z1 ¼
1
FφðtÞ
∂Θðf 1ðx1Þþg1ðx1Þx2 þd1À_ydÞ; ð69Þ
where
∂Θ ¼
1
2
1
eðtÞ=FφðtÞþδ
À
1
eðtÞ=FφðtÞÀδ
ð70Þ
A virtual control law can be then specified as follows:
α1 ¼
1
g1ðx1Þ
Àc1∂ΘÀ1
Fφz1À ^W
T
o1χ1ðx1ÞÀ^ρ1 þ _yd þ
_Fφ
Fφ
e
!
: ð71Þ
Remark 3. In a prescribed constraint performance function such
as (6), which was almost same as the constraint functions used by
Rovithakis et al. [5–7] and Na et al. [8,9], a larger exponential
decreasing rate a causes a fast approach of FφðtÞ to ξ1. This makes
the values of ξ1 and error eðtÞ closer or same, and thus, the
possibility such that jξ1j ¼ jeðtÞj at an earlier time increases
substantively. In particular, in (63) and (70), if the values of the
constraint parameter, ξ0, ξ1, δ, and δ are selected as eðtÞ=FφðtÞ
-71 or eðtÞ=FφðtÞ -7δ, and eðtÞ=FφðtÞ-δ or eðtÞ=FφðtÞ -Àδ,
then ∂R-71 and ∂Θ-71, and the resulting error increases,
which can gives rise to the closed-loop system stability problem
and to violation of the prescribed constraint conditions.
Remark 4. On the contrary, for our system, if jeðtÞj- approaches
FφðtÞ, the virtual control input of (35) increases in the opposite
manner (by virtue of Àk1ΞF z1) to suppress the error decrease.
Thus, the constraint condition and stability are satisfied, and our
prescribing method is more robust than Rovithakis′s and Na′s
transformation methods. This was demonstrated in Section 4.
Fig. 4. Simulated responses for case II: (a) tracking errors for eð0Þo0, RNB (dashed line), RRB (dash dotted line), RPB_W (solid line), (b) tracking errors for eð0Þ40, RNB
(dashed line), RRB (dash dotted line), RPB_W (solid line), (c) ∂Θ (dashed line) of RNB and ∂R of RRB (solid line) for eð0Þ40, (c) control inputs, RNB (dashed line), RRB (dash
dotted line), RPB_W (solid line).
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–43 39

26. 4. Verification examples
4.1. Simulated example
As the first example of the proposed control, we present a
simulation study for the following second-order nonlinear output
feedback system
_x1 ¼ 0:1x2
1 þx2;
_x2 ¼ f 2ðx2Þþð1þx2
1Þu;
y ¼ x1; ð72Þ
where f 2ðx2Þ ¼ 0:1x1x2À0:2x1. The control objective is for yðtÞ to
track a desired command, i.e., ydðtÞ ¼ 0:5 sin 7:54tU sin 3:77tÀ0:2
for eð0Þ40 and ydðtÞ ¼ 0:5 sin 7:54tU sin 3:77tþ0:2 for eð0Þo0,
which are subject to the error performance functions as follows:
Case I
FφðtÞ ¼ 0:5eÀt
þ0:005; ð73Þ
Case II
FφðtÞ ¼ 0:5eÀ31t
þ0:005 if eð0Þ40; ð74Þ
FφðtÞ ¼ 0:5eÀ60t
þ0:005 if eð0Þo0: ð75Þ
The initial points of each state are selected as xð0Þ ¼ ð0; 0Þ. Four
control systems were designed to compare two controllers: the
proposed RFNN-based backstepping controller with the prescribed
constraint (RPB_W), the RFNN-based conventional backstepping
controller without the prescribed constraint (RPB_WO), the RFNN-
based backstepping controller with the Rovisthaki′s prescribed
constraint (RRB) [5–7], and the RFNN-based backstepping con-
troller with the Na′s prescribed constraint (RNB) [8,9]. The para-
meters of the controller were selected as k1 ¼ 1; k2 ¼ 1; ηw2 ¼ 5;
ηs ¼ 0:5; ηm ¼ 0:5; ηwij ¼ 0:5; ηw2 ¼ 5; η′w2 ¼ 0:05; ηρ2 ¼ 2, and η′ρ2 ¼
0:01. Fig. 3 shows the simulated responses for case I. The tracking
output for eð0Þ40 of RPB_W and RPB_WO systems is presented in
Fig. 3(a), and the tracking errors for four systems are shown in
Fig. 3(b) ðeð0Þ40Þ and Fig. 3(c) ðeð0Þo0Þ, where the RPB_W, RRB,
RNB systems satisfies the prescribed performance constraints for
both initial conditions, whereas the RPB_WO system violates the
prescribed performance constraints. In Fig. 3(d) and (e), it is
known that Àk1ΞF z1 acts to suppresses the tracking error, so that
the given prescribed performance condition could be satisfied.
For case II, simulated results are shown in Fig. 4. In Fig. 4(a)
when eð0Þo0, the tracking errors of three systems remained
within the prescribed boundary. However, when eð0Þ40, Fig. 4(b)
shows that the tracking errors of RRB and RNB systems violated the
prescribed error performance owing to aforementioned singularity
problem in Remarks 1–3, while the proposed control system still
satisfied the prescribed performance in spite of a large value of a
with the moderate control input shown in Fig. 4(d). The sudden
increases of ∂R and ∂Ξ were described in Fig. 4(c) and these caused
the large control inputs of RRB and RNB as shown in Fig. 4(d).
4.2. Experimental example
As an experimental example, the control system for the Scorbot
robot ER VII manipulator shown in Fig. 5 was designed. The
dynamic equation for its upper arm (link1) is described as the
follows:
M€qþGðqÞþFf ðq; _qÞþdL ¼ τ;
τ ¼ nkti;
Lm
di
dt
þRmiþkb _q ¼ V; ð76Þ
where
M ¼ ðm1 þm2ÞL2
=3 ¼ Mn þΔM;
GðqÞ ¼ ðm1 þm2ÞLgcosq ¼ GnðqÞþΔGðqÞ;
Ff ¼ ½Fc þðFsÀFcÞexpðÀðj_q=vsjÞ2
Þ Šsgnð_qÞ ¼ Ff n þΔFf
q; _q; €qAR denote the joint position, velocity, and acceleration
vectors, respectively, Ff ðq; _qÞAR represents the friction torque,
dL AR represents the external disturbance, τAR is the control
torque applied by the joint actuators, n is the gear ratio of the
harmonic drive, kt is the torque constant, i is the motor current;
V is the voltage applied into the motor drive, Lm and Rm are the
motor inductance and resistance, respectively, and kb is the back
electromotive force (emf) constant of the motor. ΔM, ΔGðqÞ and
ΔFf represent the unknown uncertainties of M, GðqÞ, and Ff ,
respectively. The values of the robot manipulator and actuator
parameters are presented in Table 1, where M is sum of the
upperarm m1ð12:1 kgÞ and forearm m2ð3:6 kgÞ, the friction para-
meters were estimated by experimental identification, Rm, Lm, kt,
kb are the parameters of the DC servo motor drive. The e state
equations are as follows:
_x1 ¼ x2;
_x2 ¼ g2x3 þf 2ðx2ÞþFd2;
_x3 ¼ f 3ðx3Þþg3u; ð77Þ
where x1 ¼ q, x2 ¼ _q, x3 ¼ i, Fd2 ¼ ÀMÀ1
n ðΔM€qþΔGþΔFf þdLÞ,
g2 ¼ MÀ1
n nkt, f 2 ¼ ÀMÀ1
n ½GnðqÞþFf nðq; _qÞŠ, f 3ðx3Þ ¼ ÀLÀ1
m ðRmx3 þ
kbx2Þ, g3 ¼ LÀ1
m , and u ¼ V. The parameters of the controller were
selected as k1 ¼ 10; k2 ¼ 0:1; k3 ¼ 0:1; ηw2 ¼ 1; η′
w2 ¼ 0:001; ηs ¼ 0:5;
ηm ¼ 0:5; ηwij ¼ 0:5; ηρ2 ¼ 0:5 and η′ρ2 ¼ 0:005. The parameters of the
prescribed performance functions were selected as follows:
FφðtÞ ¼ 0:2eÀt
þ0:005 ðradÞ ð78Þ
The chosen sine-wave position input command was ydðtÞ ¼
0:1 sin ð1:675tÞÀ0:05 ðradÞ for eð0Þ40 and ydðtÞ ¼ 0:1 sin
ð1:675tÞþ0:05 ðradÞ for eð0Þo0, under the initial condition.
Fig. 5. Structure of the Scorbot robot control system.
Table 1
Manipulator and actuator parameters.
Symbol Parameter Value
M Total mass of link1 and 2 15:7 kg
L Total length of link1 and 2 0:71 m
Fs Stiction level of joint1 0.063 Nm
Fc Coulomb friction of joint1 0.061 Nms/rad
vs Stribeck velocity of joint 1 0.00075 rad/s
n Gear ratio of reduction gear 65.5
Lm Inductance of motor 0.6292 mH
Rm Resistance of motor 0.8294 Ω
kt Torque constant 0.0182 Nm/A
kb Back emf constant 0.0182 V/rad/s
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–4340

27. Fig. 6. Experimental responses: (a) position tracking outputs for eð0Þ40, command position input (dash–dotted line), RPB_WO (dashed line), RPB_W (solid line), (b) position
tracking outputs for eð0Þo0, command position input (dash dotted line), RPB_WO (dashed line), RPB_W (solid line), (c) position tracking errors for eð0Þ40, RPB_WO (dashed
line), RPB_W (solid line), (d) position tracking errors for eð0Þo0, RPB_WO (dashed line), RPB_W(solid line), (e) tracking error (dashed line) and Àk1ΞF z1 (solid line) for
eð0Þ40, (f) tracking error (dashed line) and Àk1ΞF z1 (solid line) for eð0Þo0, (g)‖ ^Wo2‖ for eð0Þ40 (solid line) and eð0Þo0 (dashed line), (h) ^ρ2

31. for eð0Þ40 (solid line) and
eð0Þo0 (dashed line), (i) control inputs for eð0Þ40, RPB_WO (dashed line), RPB_W (solid line), (j) control inputs for eð0Þo0, RPB_WO (dashed line), RPB_W (solid-line).
S.-I. Han, J.-M. Lee / ISA Transactions 53 (2014) 33–43 41

32. x3ð0Þ ¼ 0. Two controllers were designed to be the same as the
simulation. Only f 2ðx2Þ was approximated using the RFNN system
under the assumption of a known function f 3ðx3Þ.The designed
controllers were implemented using Matlab real-time-interface
(RTI) system and a MF624 board manufactured by Humusoft Co.
(Czech Republic) [21]. Control signals were transferred to the DC
servomotor of the Scorbot robot through a servo drive, and joint
positions were measured using a rotary encoder in each joint as
shown in Fig. 5. The sample frequency was selected as 1 kHz.
In experiment, only RPB_WO and RPB_W control systems were
designed because RRB and RNB control systems has a singularity
problem in the specific prescribing condition. Fig. 6(a) and (b)
show the tracking outputs for eð0Þ40 and eð0Þo0, where the
tracking performance is much improved in RPB_W system. The
position tracking errors for eð0Þ40 and eð0Þo0 are represented in
Fig. 6(c) and (d), where the prescribed performance constraints of
the RPB_W were satisfied under the same control gain conditions.
However, the tracking errors of the RPB_WO system violate the
prescribed tracking constraints. In Fig. 6(d) and (e), Àk1ΞF z1
suppresses the tracking error as same as in the simulation, and
the prescribed performance condition was satisfied in the experi-
ment. Therefore, it is shown that the statements of Remark 4 are
proven. Fig. 6(f) and (g) show the estimated results for an
unknown function and the uncertainty of the RFNN system and
the adaptive tuning law. The control inputs of both systems are
shown in Fig. 6(i) and (j), where the control inputs of the proposed
control system are rather smaller than those of RPB_WO.
5. Conclusion
In this study, a backstepping control scheme combined with
RFNNs was developed to constrain tracking errors and improve the
position tracking performance of a strict-feedback nonlinear
dynamic system. An adaptive RFNN system was also introduced
to compensate for unknown complex terms that are inherently
associated with backstepping design procedures. A new defined
constraint variable that ensures a prescribed tracking performance
condition was adopted as the virtual control of the backstepping
design. The unknown nonlinear functions and uncertainties asso-
ciated with each recursive step of backstepping control were
approximated using RFNNs and a disturbance compensator. For
a given constraint condition, more favorable position tracking
performance of the proposed control scheme compared with other
prescribing methods was achieved by simulation and by experi-
mental applications using a second order nonlinear system and
a Scorbot robot manipulator.
Appendix A
From the above RFNN, the approximation of uncertainty can be
rewritten as
y4
o ¼ ^W
T
oiχi: ðA1Þ
To train the RFNN effectively via the gradient descent method,
online parameter training methodology is proposed. The adapta-
tion law of Eq. (40) can be rewritten as
_^Woi ¼ ÀΓi
∂Vi
∂ ^Woi
¼ ÀΓi
∂Vi
∂y4
o
∂y4
o
^Woi
¼ ÀΓi
∂Vi
∂y4
o
χiðxiÞ ¼ ΓiðχiðxiÞziÀη′wi
^WoiÞ: ðA2Þ
From the above equation, the Jacobian of the controlled system
is as follows:
∂Vi
∂y4
o
¼ Àzi þη′wi
^WoiχÀ1
i ðxiÞ ðA3Þ
The error term of the output layer to be propagated is com-
puted as follows:
ζ4
o ¼ À
∂Vi
∂y4
o
¼ ziÀη′wi
^WoiχÀ1
i ðxiÞ ðA4Þ
The error term of the rule layer to be propagated is computed
as follows:
ζ3
k ¼ À
∂Vi
∂y3
k
¼ À
∂Vi
∂y4
o
∂y4
o
∂y3
k
¼ ζ4
o
^Woi ðA5Þ
The error term of the membership layer is computed as follows:
ζ2
j ¼ À
∂Vi
∂y2
j
¼ À
∂Vi
∂y4
o
∂y4
o
∂y3
k
∂y3
k
∂y2
j
¼ ζ4
o
^Woi∑
k
y2
j y2
j þ1 ðA6Þ
Further, the update law of m2
ij is
_m2
ij ¼ Àηm
∂Vi
∂m2
ij
¼ ηmζ2
j ½2s2
ijðy1
i þW2
ijy2
j ðtÀTÞÀm2
ijÞŠ; ðA7Þ
where ηm 40 is the learning-rate parameter of the mean of the
Gaussian functions. The update law of s2
ij is
_s2
ij ¼ Àηs
∂Vi
∂s2
ij
¼ ηsζ2
j ½2s2
ijðm2
ijÀy1
i ÀW2
ijy2
j ðtÀTÞÞŠ; ðA8Þ
where ηs 40 is the learning-rate parameter of the standard
deviation of the Gaussian functions. The update law of W2
ij is
_W
2
ij ¼ Àηwij
∂Vi
∂W2
ij
¼ ηwijζ2
j ½2s2
ijðm2
ijÀy1
i ÀW2
ijy2
j ðtÀTÞÞŠy2
j ðtÀTÞ: ðA9Þ
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