Backstepping control for periodically time-varying systems using high-order neural network and Fourier series expansion.

Abstract

An adaptive backstepping tracking scheme is developed for a class of strict-feedback systems with unknown periodically time-varying parameters and unknown control gain functions. High-order neural network (HONN) and Fourier series expansion (FSE) are combined into a new function approximator to model each uncertain term in the system. The dynamic surface control (DSC) approach is used to solve the problem of 'explosion of complexity' in the backstepping design procedure. Nussbaum gain function (NGF) is employed to deal with the unknown control gain functions. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. Two simulation examples are provided to demonstrate the effectiveness of the control scheme designed in this paper.

Full text

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ISA Transactions 49 (2010) 283–292
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Backstepping control for periodically time-varying systems using high-order
neural network and Fourier series expansion
Weisheng Chena,∗, Wei Lia, Qiguang Miaob
aDepartment of Applied Mathematics, Xidian University, Xi’an 710071, PR China
bSchool of Computer Science, Xidian University, Xi’an 710071, PR China
a r t i c l ei n f o
Article history:
Received 15 May 2009
Received in revised form
3 January 2010
Accepted 15 March 2010
Available online 8 April 2010
Keywords:
Backstepping
Periodically time-varying parameter
High-order neural network
Fourier series expansion
Nussbaum gain function
Dynamic surface control
a b s t r a c t
An adaptive backstepping tracking scheme is developed for a class of strict-feedback systems with
unknown periodically time-varying parameters and unknown control gain functions. High-order neural
network (HONN) and Fourier series expansion (FSE) are combined into a new function approximator to
model each uncertain term in the system. The dynamic surface control (DSC) approach is used to solve
the problem of ‘explosion of complexity’ in the backstepping design procedure. Nussbaum gain function
(NGF) is employed to deal with the unknown control gain functions. The uniform boundedness of all
closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around
the origin. Two simulation examples are provided to demonstrate the effectiveness of the control scheme
designed in this paper.
© 2010 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In the past two decades, many theoretical and practical works
have been reported in the field of approximator-based control
[1–8]. In particular, approximator-based adaptive backstepping
control (ABABC) has been proved to be very useful for solving
the control problem of uncertain lower-triangular systems with
the strict-feedback form [9–16], or pure-feedback form [17–19],
or output-feedback form [20–22]. In ABABC, the unknown mis-
matched system functions are usually modeled online by the
several commonly used function approximators such as radial ba-
sis function neural network (RBFNN) [10,11], or multi-layer neu-
ral network (MLNN) [12], or fuzzy neural network (FNN) [13,14],
or wavelet neural network (WNN) [15], etc. Recently, ABABC ap-
proaches are further extended to several more general nonlinear
systems such as time-delay systems [23–25], discrete-time sys-
tems [26,27], decentralized control systems [28,29], etc. In the de-
velopment of ABABC, some important techniques are proposed
such as adaptive bounding technique [11], which is often used
to deal with the approximation errors, and integral Lyapunov
function (ILF) [12], which is utilized to avoid the possible con-
troller singularity problem, and dynamic surface control (DSC) ap-
proach [16], which is employed to solve the problem of ‘explosion
∗Corresponding author. Tel.: +86 029 88202860; fax: +86 029 88202861.
E-mail address: wshchen@126.com (W. Chen).
of complexity’ in the backstepping design procedure, and Nuss-
baum gain function (NGF) approach [19,23], which is employed
to deal with the unknown control gain functions. However, in all
these ABABC approaches, function approximators are only used
to approximate unknown time-invariant functions. For systems
with unknown time-varying functions, especially with unknown
nonlinearly parameterized and time-varying functions, to the best
of the authors’ knowledge, no results have been reported at the
present stage. The main obstacle is that there are no good function
approximators to model the unknown time-varying functions.
The control problem of systems with unknown time-varying
parametersordisturbancesiswellknowntobeachallengingprob-
lem in the area of control. In early works, robust control strate-
gies are often employed to solve these kinds of problems, for
example, disturbance attenuation [30,31], almost disturbance de-
coupling [32,33], and disturbance rejection [34,35], etc. However,
most of these control approaches (e.g. [30–35]) in essence em-
ploy the high frequency (or gain) feedback control idea, which can
cause the poor transient performance of closed-loop systems. In
order to overcome this drawback of robust control, adaptive con-
trol strategies are also introduced to handle the unknown time-
varying parameters, where unknown time-varying parameters of
systems are estimated in various ways. For example, the parame-
ter adaptive laws are designed for slowly time-varying systems in
a way similar to the constant setup [36–39]. Under the assumption
that the bounds of time-varying parameters are known, adaptive
lawsbasedonprojectionalgorithmarepresentedin[40–43].Espe-
cially in the case when unknown rapidly time-varying parameters
0019-0578/$ – see front matter © 2010 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2010.03.002

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W. Chen et al. / ISA Transactions 49 (2010) 283–292
with known period appear in systems, a kind of so-called periodic
adaptationlawsareestablishedtoestimateunknownrapidlytime-
varying parameters [44–47]. However, all these adaptive control
approaches are only used to deal with unknown time-varying pa-
rameters which are allowed to enter the systems only in a linear
fashion.
Very recently, a new adaptive learning control scheme has
been proposed for a class of nonlinear systems [47], where the
periodically time-varying ideal control input is approximated by
Fourier series expansion (FSE), and the adaptive laws are designed
forunknownFouriercoefficientsinsteadofunknowntime-varying
parameters. However, this kind of control strategy [47] only
considers unknown time-varying parameters instead of unknown
functions.
Based on the discussion above, we consider a more challenging
problem in this paper, i.e., ABABC design for a class of unknown
nonlinearly parameterized systems with unmeasured and period-
ically time-varying parameters. One of main obstacles is how to
design a novel function approximator to approximate the nonlin-
early parameterized and periodically time-varying functions. It is
shown in this paper that this obstacle can be overcome. The main
contributions are listed as follows:
(1) From the viewpoint of function approximation theory, FSE
and HONN are combined into a new approximator to model
each time-varying uncertainty, where FSE is used to estimate the
time-varying parameters, and the estimated values are further
used as the HONN inputs to approximate the unknown time-
varying functions, which is different from all existing function
approximators that are employed to only model either pure
time-invariantfunctions[11–29]orpureperiodicallytime-varying
input [47].
(2) We combine the DSC approach and the NGF technique to
design our ABABC scheme in this paper. The former is used to solve
the problem of ‘explosion of complexity’ in backstepping design
procedure, and the latter is employed to deal with the unknown
control gain functions, which is also different from all the existing
results where either only DSC [17], or only NGF [40] is solely
used. To the best of authors’ knowledge, no works are reported to
simultaneously solve the above two problems at present stage.
The remainder of this paper is organized as follows. System
description, problem formulation, neural network approximation
and NGF property are given in Section 2. In Section 3, we present
the design procedure of ABABC. Section 4 shows the stability
analysis of closed-loop system and the main result of this paper.
In Section 5, two simulation examples are given to demonstrate
the feasibility of the proposed control scheme. In Section 6, we
conclude the work of this paper.
Throughout this paper, ? · ? denotes Euclidean norm of a
vector or its induced matrix norm; |ν|1 =
[a1,a2,...,am]T∈ Rm; λmax(B) and λmin(B) denote the largest
and smallest eigenvalues of a square matrix B, respectively; ˆ ?
denotes the estimate of an unknown parameter/or vector ? with
the estimation error denoted by ˜ ? = ? − ˆ ?.
2. System description and preliminaries
?m
i=1|ai| with ν =
2.1. System description and problem formulation
Consider a class of periodically time-varying and nonlinearly
parameterized systems with the following strict-feedback form


R and u ∈ R are the system state vector, system output and

˙ xi= gi(¯ xi)xi+1+ fi(¯ xi,θi(t))
i = 1,...,n − 1
˙ xn= gn(x)u + f2(x,θn(t))
y = x1
where ¯ xi = [x1,...,xi]T∈ Ri; x = [x1,...,xn]T∈ Rn, y ∈
(1)
control input, respectively; fi : Ri+1→ R and gi : Ri+1→ R
(i = 1,...,n) are unknown continuous functions; θi(t) ∈ R
are unknown continuously time-varying parameters with known
periods Ti,i = 1,...,n (for simplicity, we only consider the case
whenθi(t)arescalars,butthedesignideaisalsoappliedtothecase
when θi(t) are vectors).
Remark 1. Compared with the existing results [11–29], the main
property of system (1) is that the uncertainties fi(¯ xi,θi(t)) include
the unknown time-varying parameters θi(t) as their inputs. Since
the unknown parameters θi(t), are not used as neural network
inputs, the main design difficulty focuses on the construction of
new function approximator which can approximate fi(¯ xi,θi(t)).
Moreover, system (1) includes some previous results as its special
cases. For example, when fi(¯ xi,θi(t)) = fi(¯ xi), system (1) becomes
ones studied in [12,13]; When fi(¯ xi,θi(t)) = θi(t)fi(¯ xi) and gi(¯ xi) =
1, system (1) becomes ones studied in [46].
Remark 2. Whydoweconsideronlytheperiodicallytime-varying
parameters instead of the more general ones? Main reasons lie in:
(1) As pointed out in [48], periodic parameters (also called dis-
turbances) often exist in many mechanical control systems such as
industrial robots and numerical control machines or parameters
depending on the frequency of the power supply. Very recently,
Tomizuka [49] presented some fundamental issues and new chal-
lenges in dealing with periodic disturbances along with applica-
tions to mechanical systems. Also, there are indeed some physical
systems that can be described by the model (1), for example, van
der Pol oscillator [46] and the controlled Brusselator model [50];
(2) Theoretically, it would be extremely difficult to find a suit-
able method to solve the tracking problem of system (1) with the
more general time-varying parameters. As stated in [45], a more
realistic way is first to classify the parameters into subsets, e.g. pe-
riodic parameters vs non-periodic ones, and then look for an ap-
propriate feasible method for each subclass;
(3) In fact, many works on the rejection or estimate of peri-
odic parameters or disturbances have been widely reported, e.g.
[44–49], where, however, the periodic parameters or disturbances
are allowed to enter the controlled systems only linearly instead of
nonlinearly.
The control objective of this paper is formulated as follows. For
a given reference signal yr(t) find a dynamic adaptive control law
as follows
˙ˆ χ = ν(x,yr, ˙ yr, ˆ χ,t)
where ˆ χ usually denotes the filter signals and the estimates of
parameters, such that, while maintaining all closed-loop signals
semi-globally uniformly ultimately bounded (SGUUB), the output
tracking error y(t) − yr(t) satisfies
1
t
0
where ? ≥ 0 is a constant which can be designed as small as
desired.
To achieve the above objective, we make the following
assumptions on system (1).
u = u(x,yr, ˙ yr, ˆ χ,t),
lim
t→∞
?t
[y(τ) − yr(τ)]2dτ ≤ ?
Assumption 1. Functions gi(·), i = 1,...,n and their signs are
unknown, and there exist constants giand gisuch that gi
|gi(·)| ≤ gi, where 0 < gi≤ gi.
≤
Assumption 2. The reference signal yr(t) and its derivative ˙ yr(t)
are continuous and bounded.

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2.2. FSE-HONN-based approximator for unknown time-varying func-
tions
In this paper, the main design obstacle is that the unknown
parameters θi(t) are not used as the HONN inputs. Consider the
period property of θi(t), we first employ FSE to approximate θi(t),
and then employ the state ¯ xiand the estimate ofθi(t) as the HONN
inputs to approximate the unknown function fi(¯ xi,θi(t)).
Without loss of generality, we consider an unknown function
f(χ,θ(t)), where χ ∈ Rl⊆ Ωχis a measured signal with Ωχa
compact set, andθ(t) ⊆ Ωθ∈ R is an unmeasured continuous and
periodically time-varying parameter with Ωθa compact set.
On the one hand, the continuous and periodically time-varying
parameter θ(t) can be also modeled by a linearly parameterized
FSE as follows [47]
θ(t) = VTΦ(t) + δθ(t),
where V = [v1,...,vq]T∈ Rqis a vector consisting of the
first q coefficients of the FSE of θ(t) (q is an odd integer); δθ(t)
is the truncation error with the minimum upper bound¯δq >
0, which can be arbitrarily decreased by increasing q; Φ(t) =
[φ1(t),...,φq(t)]Twith φ1(t) = 1, φ2j(t) =
and φ2j+1(t) =
derivatives up to n order are smooth and bounded.
Ontheotherhand,ifθ(t)ismeasured,thecontinuousfunctions
f(χ,θ(t)) can be approximated over the compact set Ω = Ωχ×
Ωθby a HONN as follows [26–28]
|δθ(t)| ≤¯δθ,
(2)
√
2sin(2πjt/T)
√
2cos(2πjt/T), j = 1,...,(q − 1)/2, whose
f(χ,θ(t)) = WTS(χ,θ(t)) + δf(χ,θ(t)),
with
S(z) = [s1(z),...,sp(z)]T
si(z) =
j∈Ii
where z = [z1,...,zl+1]T=: [χT,θ(t)]T; the positive number p
denotes the neural node number;{I1,I2,...,Ip} is a collection of p
not-ordered subsets of {1,2,...,l + 1} and dj(i) are nonnegative
integers; W is an adjustable synaptic weight vector defined by
?
and s(zj) is chosen as a hyperbolic tangent function s(zj) = (ezj−
e−zj)/(ezj+ ezj); δf(χ,θ(t)) is the approximation error with the
minimumupperbound¯δf> 0,whichcanbereducedbyincreasing
the number of the adjustable weights. Universal approximation
result for HONN indicates that, if neural network node number
p is sufficiently large, then¯δf can be made arbitrarily small on a
compact region.
Based on (2) and (3), we construct a novel FSE-HONN-based
approximator
|δf(χ,θ(t))| ≤¯δf(3)
?
[s(zj)]dj(i),
i = 1,...,p
W := arg min
ˆ W∈Rp
sup
z∈Ω
???f(z) −ˆ WTS(z)
???
?
G(χ,t) = WTS(χ,VTΦ(t))
to model the unknown function f(χ,θ(t)) as follows
(4)
f(χ,θ(t)) = WTS(χ,VTΦ(t)) + δ(χ,t)
where
δ(χ,t) = δf+ WTS(χ,VTΦ(t) + δθ)
−WTS(χ,VTΦ(t)).
It can be seen from (6) that the approximation error δ(χ,t) is
bounded. Without loss of generality, we assume
(5)
(6)
|δ(χ,t)| ≤¯δ
(7)
Fig. 1. Structure of FSE-HONN-based approximator.
where¯δ denotes the minimum upper bound of δ(χ,t), which can
be arbitrarily decreased by increasing the values of p and q, which
implies that the new approximator (4) has good approximation
capability. The novel FSE-HONN-based approximator is presented
in Fig. 1.
In general, the parameters W and V are unknown and need to
be estimated in controller design. Letˆ W andˆV be the estimates
of W and V, respectively, and the weight estimation errors are
˜ W = W −ˆ W and˜V = V −ˆV. The following lemma shows the
bound of the estimation error of the approximator (4).
Lemma 1. For approximator (4), the estimation error can be
expressed as
WTS(χ,VTΦ(t)) −ˆ WTS(χ,ˆVTΦ(t))
=˜ WT(ˆS −ˆS?ˆVTΦ(t)) +˜VTΦ(t)ˆ WTˆS?+ d
whereˆS
∂si(χ,θ)
∂θ
|θ=ˆVTΦ(t), i = 1,...,p, and the residual term d is bounded
by
(8)
=
S(χ,ˆVTΦ(t)),ˆS?
= [ˆ s?
1,ˆ s?
2,...,ˆ s?
p]Twith ˆ s?
i
=
|d| ≤ ?V??Φ(t)ˆ WTˆS?? + ?W??ˆS?ˆVTΦ(t)? + 2|W|1
Proof. The proof is similar to that of Lemma 2.1 in [51] and is
omitted here.
?
(9)
2.3. NGF Properties
In order to deal with the gain functions gi(·) with unknown
signs, the Nussbaum gain technique is employed in this paper.
A function N(ζ) is called a Nussbaum-type function if it has the
following properties:
?s
lim
s
0
The commonly used Nussbaum functions include: ζ2cosζ,
ζ2sinζ, and eζ2cos[(π/2)ζ]. For clarity, the even Nussbaum func-
tion, N(ζ) = eζ2cos[(π/2)ζ], is used throughout this paper.
Lemma 2 ([23]). Let V(·) and ζ(·) be smooth functions defined on
[0,tf) with V(t) ≥ 0, ∀t ∈ [0,tf), and N(·) be an even smooth
Nussbaum-type function. If the following inequality holds:
?t
∀t ∈ [0,tf)
where constant c1 > 0, g(x(t)) is a time-varying parameter which
takes values in the unknown closed interval [l−,l+] or [−l+,−l−]
with 0 < l−< l+< +∞, and c0represents some suitable constant,
0g(x(τ))N(ζ)˙ζdτ must be bounded [0,tf).
lim
s→+∞sup1
s→+∞inf1
s
0
N(ζ)dζ = +∞
?s
N(ζ)dζ = −∞.
V(t) ≤ c0+ e−c1t
0
?
g(x(τ))N(ζ)˙ζ +˙ζ
?
ec1τdτ,
(10)
then V(t), ζ(t) and?t
3. Adaptive backstepping DSC design
In this section, we present the backstepping control design
steps for system (1) by combining the DSC approach [17] with the

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W. Chen et al. / ISA Transactions 49 (2010) 283–292
NGF technique [23], but the new function approximator (4) is uti-
lized to approximate the unknown and nonlinearly parameterized
time-varying functions fi(¯ xi,θi(t)), i.e.,
fi(¯ xi,θi(t)) = WT
|δi(¯ xi,t)| ≤¯δi
where (¯ xi,θi(t)) ∈ Ωi⊂ Ri+1with Ωibeing compact sets, and ad-
ditional inherent approximation errors and the estimation errors
will be considered.
Step 1. Define s1= x1− yr, and using the expression (11) for
i = 1, we have
˙ s1= g1(¯ x1)x2+ WT
|δ1(¯ x1,t)| ≤¯δ1.
Choose the first virtual control α1as follows
?
where c1> 0 is a design constant called the control gain;ˆ Ψ1is the
estimate of unknown vector
?
?
with
?
in which ς1is a positive design constant. The adaptive laws are
designed as


0 is a small design parameter called σ-modification coefficient.
Introduce a new state variable z2and let α1pass through a first-
order adaptive filter z2with time constant ?2
iSi(¯ xi,VT
iΦi(t)) + δi(¯ xi,t),
(11)
1S1(¯ x1,VT
1Φ1(t)) + δ1(¯ x1,t) − ˙ yr,
(12)
α1= N(k1)
c1s1+ˆ WT
1S1(¯ x1,ˆVT
1Φ1(t)) +ˆ ΨT
1Q1(s1,β1) − ˙ yr
?
(13)
Ψ1=
¯ g2
1,
?
?W1?2+ ?V1?2+ (2?W1?1+¯δ1)2
?β1s1
?T
∈ R2
Q1(s1,β1) =
s1,β1tanh
ς1
??T
β1=|Φ1(t)ˆ WT
1ˆS?
1|2+ |ˆS?
1ˆVT
1Φ1(t)|2+ 1

˙k1= s1
˙ˆ W1= Γw1[(ˆS1−ˆS?
˙ˆV1= Γv1[Φ1(t)ˆ WT
˙ˆ Ψ1= Γψ1[Q1(s1,β1)s1− σ1ˆ Ψ1]
where Γw1,Γv1,Γψ1 > 0 are adaptive gain matrices, and σ1 >
?
c1s1+ˆ WT
1S1(¯ x1,ˆVT
1ˆVT
1ˆS?
1Φ1(t)) +ˆ ΨT
1Φ1(t))s1− σ1ˆ W1]
1s1− σ1ˆV1]
1Q1(s1,β1) − ˙ yr
?
(14)
?2˙ z2= (−z2+ α1),
Step i, (i = 2,...,n − 1). Define si = xi− zi, and using the
expression (11), we obtain
˙ si= gi(¯ xi)xi+1+ WT
− ˙ zi,
Choose the ith virtual control αias follows
?
+ˆ ΨT
z2(0) = α1(0).
(15)
iSi(¯ xi,VT
iΦi(t)) + δi(¯ xi,t)
|δi(¯ xi,t)| ≤¯δi.
(16)
αi= N(ki)
cisi+ˆ WT
iSi(¯ xi,ˆVT
?
iis a design constant with c0
iΦi(t))
iQi(si,βi) − ˙ zi
(17)
where ci =
estimate of unknown vector
?
Qi(si,βi) =
1
2+ c0
i> 0; ˆ Ψiis the
Ψi= [¯ g2
i,
?Wi?2+ ?Vi?2+ (2?Wi?1+¯δi)2]T∈ R2
?
si,βitanh
?βisi
ςi
??T
with
βi=
in which ςiis a positive design constant. The adaptive laws are
designed as


whereΓwi,Γvi,Γψi> 0 are adaptive gain matrices, andσi> 0 is a
small design parameter. Introduce a new state variable zi+1and let
αipass through a first-order adaptive filter zi+1with time constant
?i+1
?i+1˙ zi+1= (−zi+1+ αi),
Step n. Define sn= xn− zn, and using the expression (11) for
i = n, we get
˙ sn= gn(x)u + WT
− ˙ zn,
Choose the control law u as follows
?
where cn =
estimate of unknown vector
?
?
with
?
in which ςnis a positive design constant. The adaptive laws are
designed as


where Γwn,Γvn,Γψn> 0 are adaptive gain matrices, and σn> 0
is a small design parameter.
The design procedure for the second-order system (i.e., system
(1) with n = 2) is shown in Fig. 2.
?
|Φi(t)ˆ WT
iˆS?
i|2+ |ˆS?
iˆVT
iΦi(t)|2+ 1

˙ki= si
˙ˆ Wi= Γwi[(ˆSi−ˆS?
˙ˆVi= Γvi[Φi(t)ˆ WT
˙ˆ Ψi= Γψi[Qi(si,βi)si− σiˆ Ψi]
?
cisi+ˆ WT
iSi(¯ xi,ˆVT
iˆVT
iˆS?
iΦi(t)) +ˆ ΨT
iΦi(t))si− σiˆ Wi]
isi− σiˆVi]
iQi(si,βi) − ˙ zi
?
(18)
zi+1(0) = αi(0).
(19)
nSn(x,VT
|δn(x,t)| ≤¯δn.
nΦn(t)) + δn(x,t)
(20)
u = N(kn)
cnsn+ˆ WT
nSn(x,ˆVT
nΦn(t)) +ˆ ΨT
nQn(sn,βn) − ˙ zn
?
(21)
1
2+ c0
nis a design constant with c0
n> 0; ˆ Ψnis the
Ψn=
0,
?
?Wn?2+ ?Vn?2+ (2?Wn?1+¯δn)2
?βnsn
?T
∈ R2
Qn(sn,βn) =
0,βntanh
ςn
??T
βn=|Φn(t)ˆ WT
nˆS?
n|2+ |ˆS?
nˆVT
nΦn(t)|2+ 1,

˙kn= sn
˙ˆ Wn= Γwn[(ˆSn−ˆS?
˙ˆVn= Γvn[Φn(t)ˆ WT
˙ˆ Ψn= Γψn[Q(sn,βn)sn− σnˆ Ψn]
?
cnsn+ˆ WT
nSn(x,ˆVT
nˆVT
nˆS?
nΦn(t)) +ˆ ΨT
nΦn(t))sn− σnˆ Wn]
nsn− σnˆVn]
nQn(sn,βn) − ˙ zn
?
(22)
Remark 3. Intheabovedesignsteps,thedesignparametersc1and
c0
are,thesmallerthetrackingerroris.However,itiswellknownthat
high gain control can deteriorate the the transient performance of
closed-loopsystem.Therefore,inpracticalapplicationsthedesign-
ers should choose the appropriate c1and c0
satisfactory transient performance and the ideal tracking error.
iarecalled‘controlgains’.Generallyspeaking,thelargerc1andc0
i
iin order to obtain the

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287
Fig. 2. Schematic representation of the adaptive backstepping design procedure for system (1) with n = 2.
4. Stability analysis
Noting
xi= si+ zi= si+ ?i+ αi−1,
here we denote the filter error signals as ?i = zi− αi−1, i =
2,...,n − 1. According to (15) and (19), ?ican be arbitrarily
decreased by decreasing the design parameters ?i. So, without
loss of generality, in this paper we made the following additional
assumption on ?i.
(23)
Assumption 3. For i
bounded, i.e., there exists constants ¯ ?i, such that |?i(t)| ≤ ¯ ?i.
Using (23), we rewrite (12), (16) and (20) as follows
˙ s1 = g1(¯ x1)s2+ g1(¯ x1)?2+ g1(¯ x1)α1
+WT
˙ si = gi(¯ xi)si+1+ g1(¯ xi)?i+1+ gi(¯ xi)αi
+WT
˙ sn= gn(x)u + WT
Substituting (13), (17) and (21) into (24)–(26) respectively, and
noting the fact (see Lemma 2) that
WT
=˜ WT
we have
˙ s1= g1(¯ x1)s2+ g1(¯ x1)?2+ [g1(¯ x1)N(k1) + 1]
×[c1s1+ˆ WT
+ˆ ΨT
+˜ WT
˙ si = gi(¯ xi)si+1+ gi(¯ xi)?i+1+ [gi(¯ xi)N(ki) + 1]
×[cisi+ˆ WT
−cisi−ˆ ΨT
+˜VT
˙ sn= [gn(x)N(kn) + 1][cnsn+ˆ WT
+ˆ ΨT
+˜ WT
=
2,...,n, the filter errors ?i(t) are
1S1(¯ x1,VT
1Φ1(t)) + δ1− ˙ yr
(24)
iSi(¯ xi,VT
iΦi(t)) + δi− ˙ zi
nS(x,VT
i = 2,...,n − 1(25)
nΦn(t)) + δn− ˙ zn.
(26)
iSi(¯ xi,VT
iΦi(t)) −ˆ WT
i(ˆSi−ˆS?
iSi(¯ xi,ˆVT
iΦi(t))
iΦi(t)ˆ WT
iˆVT
iΦi(t)) +˜VT
iˆS?
i+ di,
(27)
1S1(¯ x1,ˆVT
1Φ1(t))
1Q1(s1,β1) − ˙ yr] − c1s1−ˆ ΨT
1(ˆS1−ˆS?
1Q1(s1,β1)
1Φ1(t)ˆ WT
1ˆVT
1Φ1(t)) +˜VT
1ˆS?
1+ d1+ δ1,
(28)
iSi(¯ xi,ˆVT
iQi(si,βi) +˜ WT
iΦi(t)ˆ WT
iΦi(t)) +ˆ ΨT
i(ˆSi−ˆS?
i+ di+ δi,
iQi(si,βi) − ˙ zi]
iˆVT
i = 2,...,n − 1,
nSn(x,ˆVT
nQ(sn,βn)
nΦn(t)ˆ WT
iΦi(t))
iˆS?
(29)
nΦn(t))
nQ(sn,βn) − ˙ zn] − cnsn−ˆ ΨT
n(ˆSn−ˆS?
nˆVT
nΦn(t)) +˜VT
nˆS?
n+ dn+ δn.
(30)
The main result is summarized as follows.
Theorem 1. Under Assumptions 1–3, consider the closed-loop adap-
tive system consisting of the plant (1), the virtual control func-
tions (13), (17), the filters (15), (19), and the control law (21) with
the adaptive laws (14), (18), (22), for bounded initial conditions, the
following properties hold:
(I) All the closed-loop signals are SGUUB.
(II) The tracking error signal s1(t) = y(t) − yr(t) will eventually
converge to a residual set around zero in the mean square sense,
i.e.,
?t
where ? =
an adjusted constant, which can be designed as small as desired
by tuning the design parameters ci,ςi,σi,?iand increasing the
values of piand qi.
lim
t→∞
1
t
0
[y(ζ) − yr(ζ)]2dζ ≤ ?
?n
(31)
1
c1
i=1[σi
2(WT
iWi+ VT
iVi) + ψiκς + ¯ ?2
i+1] is
Proof. Consider the ith Lyapunov function
Vsi=
1
2s2
+1
i+
1
2
˜ WT
iΓ−1
wi˜ Wi+
1
2
˜VT
iΓ−1
vi
˜Vi
2
˜ ΨT
iΓ−1
ψi˜ Ψi,
i = 1,...,n
(32)
whose derivative along (14), (18), (22) and (28)–(30) is given by
˙Vsi= gi(¯ xi)sisi+1+ gi(¯ xi)si?i+1
+[gi(¯ xi)N(ki) + 1]˙ki−
−ΨT
−σi(˜ WT
where for convenience of denotation, we introduce sn+1= ?n+1=
0. Using the following inequalities
?
ci+
1
2
?
s2
i
iQi(si,βi) + si(di+ δi)
iˆ Wi+˜VT
iˆVi+˜ ΨT
iˆ Ψi)
(33)
gi(¯ xi)sisi+1 ≤
1
2g2
1
2¯ g2
i(¯ xi)s2
is2
i+
1
2s2
1
2s2
i+1
≤
i+
i+1
(34)