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IEEE TRANSACTIONS ON CYBERNETICS 1
Saturation-Tolerant Prescribed Control for a Class
of MIMO Nonlinear Systems
Ruihang Ji , Student Member, IEEE, Baoqing Yang, Jie Ma, and Shuzhi Sam Ge ,Fellow, IEEE
Abstract—This article proposes a saturation-tolerant
prescribed control (SPC) for a class of multiinput and multi-
output (MIMO) nonlinear systems simultaneously considering
user-specified performance, unmeasurable system states, and
actuator faults. To simplify the control design and decrease
the conservatism, tunnel prescribed performance (TPP) is
proposed not only with concise form but also smaller overshoot
performance. By introducing non-negative modified signals into
TPP as saturation-tolerant prescribed performance (SPP), we
propose SPC to guarantee tracking errors not to violate SPP
constraints despite the existence of saturation and actuator
faults. Namely, SPP possesses the ability of enlarging or recov-
ering the performance boundaries flexibly when saturations
occur or disappear with the help of these non-negative signals.
A novel auxiliary system is then constructed for these signals,
which bridges the associations between input saturation errors
and performance constraints. Considering nonlinearities and
uncertainties in systems, a fuzzy state observer is utilized to
approximate the unmeasurable system states under saturations
and unknown actuator faults. Dynamic surface control is
employed to avoid tedious computations incurred by the back-
stepping procedures. Furthermore, the closed-loop state errors
are guaranteed to a small neighborhood around the equilibrium
in finite time and evolved within SPP constraints although
input saturations and actuator faults occur. Finally, comparative
simulations are presented to demonstrate the feasibility and
effectiveness of the proposed control scheme.
Index Terms—Actuator faults and saturations, auxiliary
system, finite-time stability, multiinput and multioutput (MIMO)
nonlinear systems, prescribed performance control (PPC).
I. INTRODUCTION
THE STUDIES of complex systems have attracted con-
siderable attention and led to fruitful research over the
Manuscript received December 14, 2020; accepted July 5, 2021. This work
was supported in part by the National Natural Science Foundation of China
under Grant 61427809, and in part by the China Scholarship Council. This
article was recommended by Associate Editor J. Q. Gan. (Corresponding
author: Jie Ma.)
Ruihang Ji is with the Department of Control Science and Engineering,
Harbin Institute of Technology, Harbin 150001, China, and also with the
Department of Electrical and Computer Engineering, National University of
Singapore, Singapore 117576 (e-mail: jiruihang@hit.edu.cn).
Baoqing Yang and Jie Ma are with the Department of Control Science and
Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail:
ybq@hit.edu.cn; majie@hit.edu.cn).
Shuzhi Sam Ge is with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117576 (e-mail:
elegesz@nus.edu.sg).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCYB.2021.3096939.
Digital Object Identifier 10.1109/TCYB.2021.3096939
past decade [1]–[5]. Some elegant research on SISO, strict-
feedback nonlinear systems, and nonstrict-feedback nonlinear
systems can be found in [6]–[10]. As most practical platforms
are composed of a series of interconnected and complicated
subsystems, it is of certainty that control designs for the mul-
tiinput and multioutput (MIMO) nonlinear systems will find
immediate and wide applications [11]–[13]. In [14], an adap-
tive backstepping control is proposed for MIMO nonlinear
systems in the form of parametric strict-feedback, where para-
metric uncertainties are not considered. In [15], the problem
of adaptive control for the strict-feedback MIMO nonlinear
systems with full-state constraints and unknown nonlinear
functions is addressed. In [16], the studies of fuzzy adap-
tive output feedback control for nonstrict-feedback MIMO
nonlinear systems with errors constraint and unknown dead
zone are presented. Generally, each subsystem of MIMO non-
linear systems, in practicality, is very complex due to the
couplings among system states. Considering unknown non-
linearities and uncertainties in systems making the nonlinear
functions, in general, it is very difficult to comply with the
growth or linear conditions in [17]–[19]. It thus becomes more
essential in the study of the more general MIMO nonlinear
systems. In the literature of some adaptive control designs,
fuzzy-logic systems (FLSs) [20]–[26] and neural networks
(NNs) [27]–[33] are mostly utilized for unknown nonlinear-
ities due to their excellent approximation abilities. In [34],
by introducing a joint switching mechanism into the adja-
cent instants, observer-based discrete-time fuzzy systems are
addressed, which provide a chance to reduce the design con-
servatism. However, all approaches in [21]–[25] and [27]–[31]
are only concerned with infinite-time stabilization. Namely,
the desired tracking performance is achieved when time goes
infinity, which may lead to a long transient response. Note
that the convergence performance is an essential index of the
systems, and the control objective is expected to be achieved
within finite time, which can further enhance the application
efficiency in practical engineering. From both theoretical and
practical viewpoints, finite-time control is receiving tremen-
dous attention as it can enjoy better transient performance,
robustness, as well as disturbance rejection abilities.
Following the pioneering works on finite-time theorems,
there have been considerable strides made both in theory and
practical applications. The finite-time stability theory is first
established in [35] for the double integrator without unwinding.
Although finite-time stability is achieved, the fast convergence
rate cannot be guaranteed when the tracking errors are far
away from the equilibrium. Modified theorems are thereby
2168-2267 c
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2IEEE TRANSACTIONS ON CYBERNETICS
given in [36] and [37], in which the linear term and double
fractional power terms are, respectively, added to guarantee fast
tracking performance both at close and far distance of the origin.
Considering uncertainties and unknown external disturbance,
it is difficult to obtain full knowledge about systems and sat-
isfy theorem requirements. This stimulates the development of
the semiglobal finite-time stability theory to relax these restric-
tions [38]. Moreover, notable results are reported by combining
the finite-time theory with FLSs [39] or NNs [40]. An adaptive
NNs finite-time control is developed for MIMO systems by
adopting iterative Lyapunov functions in [41]. State constraints
and dead zone are addressed in [42] with the support of barrier
Lyapunov functions. However, it is worth noting that all of the
aforementioned literature follows healthy conditions or without
system constraints. Unfortunately, in practical engineering, the
systems’ performances may be severely degraded, instable, or
may even be a catastrophic accident when the inevitable actu-
ator faults or constraints occur. The above nonlinear effects
pose an unexpected challenge on the control design due to
their nonlinear properties [43]–[46].
There are various constraints in almost all practical appli-
cations, which, in general, can be classified into two types:
1) system abilities related (e.g., input saturations and unmea-
surable system states) and 2) the performance related (e.g.,
specified convergence rate and overshoot). To compensate
for the nonlinearities caused by input saturations, adaptive
tracking control is proposed in [47] for MIMO nonlinear
systems, where an auxiliary system is first introduced to ana-
lyze the constraints effect. In [48], the finite-time tracking
control problem is addressed for the SISO nonlinear systems
with asymmetric input saturations. Smooth hyperbolic tangent
functions are utilized to approximate the nonlinear saturation
functions. In [49], a novel auxiliary dynamic system is con-
structed to tackle saturations where additional auxiliary states
can be integrated into the control scheme, and a nonlinear
finite-time observer is developed to estimate the unknown dis-
turbance. In [50] and [51], the problem of actuator faults
including the loss of effectiveness, bias faults, and stuck con-
ditions is further addressed for the constrained systems. Note
that the above approaches are strictly confined within the state
feedback control field. Namely, all system states should be
measurable, which may not always be met in practical engi-
neering. Output feedback control provides a strong tool by
developing various observers to estimate unmeasurable system
states. In [52], a command-filtered output feedback control
is investigated for uncertain nonlinear systems with full-state
constraints, where only one adaptive law is needed. In [53],
a finite-time output feedback control is developed for MIMO
nonlinear systems, where a FLSs-based observer is constructed
to estimate unmeasurable system states despite saturations and
actuator faults that simultaneously occur.
Recently, more studies are concerned with some crucial
performance-related constraints to obtain user-specified char-
acteristics [54]–[58]. Motivated by these requirements, the
prescribed performance control (PPC) is first proposed in [59],
where the tracking errors are regulated within prescribed func-
tions boundaries. In [60], the finite-time control strategy is
introduced in an event-triggered robust control to guarantee the
tracking errors into a small residual set in finite time. In [61], a
modified PPC is developed toward the nonlinear systems sub-
ject to external disturbance and input saturations, where an
explicit prespecified terminal time can be given. Furthermore,
a robust fault-tolerant control (FTC) is synthesized in [62]
by conjunction of a low-pass filter with an auxiliary dynamic
system in spite of the fact that exogenous disturbances, actu-
ator faults, and input saturation occur. Up to now, the PPC
scheme has become an interesting topic in various applica-
tions [63]–[65]. However, it is worth noting that there exist
three aspects, which may limit the PPC development to some
extend: 1) all the prescribed performance functions (PPFs),
control design, as well as stability analysis should be redevel-
oped according to the different initial errors conditions, which
impose undesirable complexities in control design; 2) the tra-
ditional PPFs boundaries are distributed on both sides of the
origin and converge to the prescribed terminal values, where
no restriction on the overshoots performance is in a positive
manner; and 3) most references analyze input saturations and
performance constraints independently and assume that both
can be achieved simultaneously. However, the performance-
related constraints always conflict with input saturations as
they may not be satisfied simultaneously. The unknown actu-
ator faults, input saturations, and external disturbance may
lead PPFs infeasible or to be violated. If we over enlarge the
final performance constraints, the advantages of PPFs may be
degraded and the conservatism of control increases. To accom-
plish a tradeoff between input saturations and performance
related constraints, the development of control schemes is of
significance from both theoretical and practical viewpoints.
Motivated by the approaches for systems with input constraints
in [49]–[51] and [61]–[71], an auxiliary system may be a
promising method to construct the associations between input
saturations and performance constraints. We, thereby, propose
a novel saturation-tolerant prescribed control (SPC) scheme for
a class of MIMO nonlinear systems simultaneously consider-
ing unmeasurable systems states and actuator faults. Compared
with the existing results, the main contributions are highlighted
in the following.
1) Compared with the traditional PPFs designed in
[54]–[66], we propose a novel tunnel prescribed
performance (TPP) in a concise form and extend it
to a more general concept, where the performance
functions and control design no longer need to be rede-
veloped according to different initial tracking errors. The
performance boundaries of TPP are on the same side
with tight space, which means that smaller overshoot
performance can be achieved in a positive manner.
2) By introducing non-negative modified signals into TPP
as SPP, SPP features the ability of flexibly enlarging or
recovering the performance boundaries when saturations
occur or disappear. Compared with [61]–[66], we tact-
fully construct an auxiliary system for these signals to
bridge the associations between input saturations errors
and performance constraints instead of neglecting their
interconnection and conflict.
3) We develop SPC for MIMO nonlinear systems
simultaneously considering specified performance,
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 3
unmeasurable states, and actuator faults. SPC can not
only achieve finite-time stability but also the tracking
errors evolve within SPP boundaries despite of input
saturations and actuator faults. Moreover, a fuzzy state
observer is constructed to approximate unmeasurable
states although saturations and unknown actuator faults
occur. Compared with [6]–[9], [14]–[19], [43]–[47],
and [72]–[83], the control design is more charming and
challenging, which can pave the way for more general
conditions.
The remainder of this article is organized as follows.
Section II describes the MIMO nonlinear systems, SPP, and
observer design. Section III presents the SPC. Simulation
example is provided in Section IV. Finally, Section V draws a
conclusion.
Notations: Throughout this article, we follow the notations
as follows. Ii:jrepresents the set {i,i+1,...,j−1,j}with
integers j>i.ijdenotes the corresponding components in
the jth subsystem and ρjis the jth subsystem order. Define
G+(g1,¯g1)=[(1/2)sign(g1−¯g1)+(1/2)](g1−¯g1)and
G−(g2,g2)=[(1/2)sign(g2−g2)−(1/2)](g2−g2).λmin(·)and
λmax(·), represents the minimum and maximum eigenvalues,
respectively.
II. PROBLEM FORMULATION AND PRELIMINARIES
A. Problem Formulation
Consider the following MIMO nonlinear nonstrict feedback
system:
⎧
⎨
⎩
˙xj,ij=fj,ijxj+xj,ij+1
˙xj,ρj=fj,ρj(¯x)+uj(t)
yj=xj,1
(1)
where j∈I1:m,xj=[xj,1,...,xj,ρj]T,¯x=[x1,1,...,
x1,ρ1,...,xm,1,...,xm,ρm]T,yjis the only measurable state,
fj,ij(·)represents the unknown nonlinear smooth function, and
uj(t)represents the control input suffering from actuator faults
and saturations as
uj(t)=gj(t,t0)hjvj+hj,s(t,t1)(2)
where gj(t,t0)and hj,s(t,t1)denote the actuator effectiveness
and additive actuation faults where t0and t1being instants
when these effects are stimulated, and hj(vj)is
hj(vj)=⎧
⎨
⎩
vj,max,vj>vj,max
vj,vj,min ≤vj≤vj,max
vj,min,vj<vj,min
(3)
where vj,max and vj,min are known constants, and vjis the actual
control input to be designed.
The control objective is to design a control scheme for the
MIMO nonlinear systems such that: 1) TPP is first designed in
a concise form to simplify the control design, where smaller
overshoot performance can be achieved in a positive man-
ner; 2) To accomplish the tradeoff between input and output
constrains, SPP is then introduced to possess the ability of
enlarging or recovering the performance boundaries when
saturations occur or disappear; and 3) SPC is proposed to guar-
antee the closed-loop system is not only finite-time stable but
also the tracking errors evolve within the specified constraints
although input saturations and actuator faults occur.
We first give the following definition, lemmas, and
assumptions.
Definition 1 [84]: Considering the system ˙x=f(x), for any
initial condition x(0),x≤holds for ∀t>t0+T(x(t0), ),
where >0 and T(x(t0), ) is the settling time. Then, the
nonlinear system is said semiglobal practical finite-time stable
(SGPFS).
Lemma 1 [85]: Considering any real number ykwith k∈
I1:nand 0 <β <1, we have
n
k=1
|yk|β
≤
n
k=1
|yk|β≤n1−βn
k=1
|yk|β
.(4)
Lemma 2 [86]: Let p,q, and rbe positive constants, and
the following inequality holds:
|y1|p|y2|q≤p
p+qr|y1|p+q+q
p+qr−p
q|y2|q+r(5)
where y1and y2are real variables.
Lemma 3 [36]: Considering the nonlinear system in defi-
nition, V(x)is a positive-definite function with
˙
V(x)≤−c1V(x)−c2Vβ(x)+(6)
where c1,c2, and are positive constants and 0 <β<1.
Then, the nonlinear system is said SGPFS and the settling
time can be expressed as
T=t0+1
c1(1−β) lnc2c3+c1V1−β(t0)
c2c3.(7)
Assumption 1: The actuator effectiveness gj(t,t0)and
the additive actuation faults hj,s(t,t1)are bounded:
ρj≤gj(t,t0)≤1 and |hj,s(t,t1)|≤¯
hswith ρjand ¯
hs
being positive constants.
Remark 1: The discussed MIMO nonlinear systems (1) are
very general and it is of certainty that their control design can
find immediate and wide applications, for example, unmanned
aerial vehicles, servo motor, unmanned surface vehicle, etc.
As most of the practical systems are composed of a series
of interconnected subsystems with couplings in the forms
of unknown nonlinearities, it imposes an unexpected control
design burden for MIMO nonlinear systems compared with
pure-feedback, strict-feedback, and SISO nonlinear systems
in [6]–[10] since the stability analysis does not follow the stan-
dard backstepping design. To capture more general conditions,
no linear parameterization or growth conditions in [17]–[19],
[39], [72]–[74], and [77] are imposed on the unknown non-
linear functions in this article. Besides, it is worth noting that
each subsystem can be different in order which can describe
the interactions among multiple platforms.
Remark 2: Actuator faults and saturations are inevitable
factors in almost all practical engineering, which may lead
the systems to instability or even a catastrophic accident. In
the previous literature [8], [11], [30], [43]–[52], [61]–[65],
[77], [78], either input saturations are taken into the control
designs, or actuator faults. We, in this article, simultaneously
consider both effects to obtain favorable tracking performances
although both occur. The control design can pave the way for
more general conditions.
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4IEEE TRANSACTIONS ON CYBERNETICS
Fig. 1. Performance of PPFs and TPP. (a) Performance of PPFs under
ej(0)≥0. (b) Performance of PPFs under ej(0)<0. (c) Performance of TPP
under ej(0)≥0. (d) Performance of TPP under ej(0)<0.
B. Saturation-Tolerant Prescribed Performance
Review the conventional PPFs, where the tracking error
ej(t)=yj−yj,dwith yj,dbeing desired trajectories is
constrained by the PPFs as
−δj(t)<ej(t)<
j(t), ej(0)≥0
−j(t)<ej(t)<δ
j(t), ej(0)<0(8)
where 0 ≤δ≤1 and j(t)=(0−∞)e−κt+∞with 0>
∞>0 and κ>0. Based on the approaches in [54]–[60], the
traditional PPC can indeed guarantee the tracking errors to sat-
isfy the user-specified performance constraints (8). However,
it is worth noting that PPFs are separately defined according to
initial tracking errors. Namely, the control design and stability
analysis should be redeveloped due to different initial condi-
tions. This mechanism imposes undesirable complexities in the
control design. Supposing double integrators system ¨x=u,the
boundaries constraints of PPFs are distributed on both sides
of the origin and gradually converge to the residual sets as
shown in Fig. 1(a) and (c). It is easy to find that the initial
shape of PPFs is similar to a “horn,” which cannot have a
positive constraint on the transient performance and obvious
overshoot performance can be seen.
Motivated by the above constraints of the traditional PPFs,
we, in this article, propose TPP to handle these problems as
−ej,l(t)<ej(t)<ej,u(t)(9)
with
ej,l(t)=δj,l−signej(0)j(t)+∞signej(0)
ej,u(t)=δj,u+signej(0)j(t)−∞signej(0)(10)
where 0 ≤δj,l≤1 and 0 ≤δj,u≤1. According
to (9) and (10), TPP is defined with a concise form, where
performance boundaries are with only one group expression.
Thus, TPP can dramatically reduce complexities of the control
design and stability analysis. The performance boundaries of
TPP are described in Fig. 1(b) and (d). Compared with the
Fig. 2. Performance functions of ej,uand ej,l.(a)ej,uand ej,lunder ej(0)≥0.
(b) ej,uand −ej,lunder ej(t)≥0. (c) ej,uand ej,lunder ej(0)<0. (d) ej,u
and −ej,lunder ej(0)<0.
traditional PPFs, there is a significant shape change, where
TPP is exactly similar to a “tunnel” with more tight space.
The boundaries of TPP are distributed on the same side and
converge to the final residual sets, which can make TPP have
an ability of achieving the transient performance with smaller
overshoot by appropriate selecting TPP.
To better show the TPP design mechanism and extend it to
a more general concept, the performance functions of ej,uand
ej,lin (10) are exhibited in Fig. 2, where the corresponding
areas of functions are shaded. Here, we give detail analysis of
ej(0)>0 conditions and ej(0)<0 follows the same process.
First, according to (10) under ej(0)>0, the trajectories of
j(t),ej,land ej,uare described in Fig. 2(a). The areas less
than ej,land ej,uare shaded by black and gray, respectively.
Then, according to (9), the area of −ej,l(t)<ej(t)<ej,u(t)
is exhibited in Fig. 2 (b). The boundary constraints are dis-
tributed at the same side of the origin with more tight space,
and gradually converge to the prescribed set. The correspond-
ing designed area is shaped like a Tunnel. From a more general
perspective, the objective of TPP is to design a prescribed
performance similar to Tunnel, but not limit to (10). We can
develop more modified TPP, such as finite-time convergence,
small overshoot or even no overshoot, fast convergence rate,
etc. Thus, we extend our TPP design to a more general concept
for our future research.
Note that all the above PPFs and TPP are user prespecified,
where the boundaries are rigid without adaptive adjustment
ability. However, almost all practical systems are subject to
input saturations, actuator faults, and external disturbance. The
above uncertainties and nonlinearities may lead the TPP design
infeasible or over conservative. To illustrate this problem, con-
sider ¨x=usystem with ufollowing (2) with gj(t,0)=1 and
hj,s(t,10)=0.75 tan(t−10)−0.75 tan(t−11). Referring to (9)
and (10), we select two boundaries, TPP1 and TPP2, where
TPP2 has more relaxed constraints than TPP1. According
to simulation results, control inputs are subject to saturation
effects during 0–2 s and 10–13 s. The tracking performance
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 5
Fig. 3. System response and user-specified TPP.
is effected by saturations nonlinearities as shown in Fig. 3.
Especially when 10–13 s, there has obvious performance
degradation. If we select TPP1 as user-specified performance
constraints, the tracking errors approach or even cross the
boundaries once saturations occur. Unfortunately, the control
design becomes singular when the tracking errors violate the
specified constraints. Then, we select other relaxed conditions
such as TPP2. Although the tracking errors can be guaran-
teed within the specified regions, there exist two problems:
1) almost all practical systems are subject to uncertainties
and nonlinearities, which we cannot obtain full knowledges
about in advance. Namely, we cannot select TPP2 such that
tracking errors always satisfy the specified constraints and
2) over-relaxed TPP2 not only makes the control design too
conservative to some extend but also leads to the superiority
of TPP weak.
Motivated by the above discussion, we aim to develop a
flexible performance boundaries related to the saturations. We
thus introduce non-negative modified signals into TPP as SPP,
which can be formulated as
¯ej,l(t)=ej,l(t)+η1,l(t), ¯ej,u(t)=ej,u(t)+η1,u(t)(11)
where η1,l(t)and η1,u(t)are non-negative modified signals.
When input saturations occur, the positive modified signals
can enlarge the prescribed boundaries of SPP, and then fast
converge to zero when saturations disappear. The proposed
SPP has flexible ability to the uncertainties and nonlinearities
in systems, and reduce the conservatism of the control design.
The magnitudes of these signals are associated with input sat-
urations errors, and their values can be obtained by the later
designed auxiliary system.
Remark 3: Compared with the traditional PPFs designed
in [54]–[66], we propose TPP with concise form, which can
not only simplify the control design but also small overshoot
performance can be achieved in a positive manner. We further
extend the TPP design mechanism to a more general con-
cept, which can pave the way for more expressions of TPP.
To compensate for saturations effects, SPP is proposed fea-
turing the ability of enlarging or recovering the performance
boundaries when saturations occur or disappear, which can
guarantee the tracking errors within SPP constraints despite of
saturations and actuator faults. These non-negative modified
signals construct the associations between input saturations
and performance constraints.
C. Fuzzy-Logic Systems
The basic configuration of the FLSs consists of a knowledge
base, a fuzzy inference, a fuzzifier, and a defuzzifier. Construct
the lth IF-THEN rules as follows.
RULE(l):Ifx1is Bl
1,x2is Bl
2,...,xnis Bl
n
THEN: yis Cl,l=1,...,L
where Lrepresents the rules number, xis the FLSs input,
and yis the corresponding output variable. μBl
iand μCl
iare
membership functions of the fuzzy set Bl
iand Cl
i. Therefore,
the expression of FLSs can be
y(x)=L
l=1¯yln
k=1μBl
k(xk)
L
l=1n
k=1μBl
k(xk)(12)
where x=[x1,...,xn]Tand ¯yl=maxμCl(yl). For the fuzzy
basic function, we have
ϕl(x)=n
k=1μBl
k(xk)
L
l=1n
k=1μBl
k(xk).(13)
Then, the output of the FLSs (12) can be described as
y=θTϕ(x)(14)
where θ=[θ1,...,θ
L]Tand ϕ(x)=[ϕ1,...,ϕ
L]T.
Lemma 4 [76]: Let f(x)be a continuous function on a com-
pact set . For any positive constant ε, there exist FLSs (14)
such that
sup
x∈f(x)−θTϕ(x)≤ε. (15)
D. Fuzzy State Observer Design
The only measurable state of the nonlinear system (1) is the
output yj. Utilizing FLSs, a fuzzy state observer is expected to
be constructed to estimate the system state xj,ij,ij∈I2:ρjfor
the control design. For the unknown function fj,ij,wehave
ˆ
fj,ijˆxj|ˆ
θj,ij=ˆ
θT
j,ijϕj,ijˆxj(16)
where ˆxjand ˆ
θj,ijare the estimated vectors of xjand θj,ij. Then,
the nonlinear system (1) can be rewritten as
˙xj=Ajxj+Kjyj+Fjxj+Bj,ρjuj
yj=Cjxj(17)
where Fj(xj)=ρj
k=1Bj,kfj,k(xj),Bj,k=[0,..., 1,...,0]T,
Bj,ρj=[0,...,1]T, and Cj=[1,...,0]. Ajand Kjare with
Aj=⎡
⎢
⎣
−kj,1
.
.
.Iρj−1
−kj,ρj0··· 0⎤
⎥
⎦,Kj=⎡
⎢
⎣
kj,1
.
.
.
kj,ρj
⎤
⎥
⎦(18)
where kj,ij>0, ij∈I1:ρj. The state observer, hereby, can be
constructed as
˙
ˆxj=Ajˆxj+Kjyj+ˆ
Fjˆxj+Bj,ρjhjvj
yj=Cjxj(19)
where ˆ
Fj(ˆxj)=ρj
k=1Bj,kˆ
fj,k(ˆxj). For the sake of simplicity,
the variables are left out in the brackets in the following design
process. Combining (17) and (19), we can obtain
˙ej,0=Ajej,0+Fj−ˆ
Fj+Bj,ρjuj−hjvj (20)
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where ej,0=xj−ˆxjwith ej,0=[e1
j,0,...,eρj
j,0]T. We adopt
the candidate Lyapunov function as Vj,0=eT
j,0Pjej,0with Pj
being the positive-definite matrix. Considering Lemma 4 and
substituting (2), (16), and (20) into the time derivative of Vj,0
yield
˙
Vj,0=2eT
jPjρj
k=1
Bj,kεj,k+˜
θj,kϕj,kˆxj
+Bj,ρjhj,s−hj+gjhj
+eT
j,0AT
jPj+PjAjej,0(21)
where ˜
θj,k=θ∗
j,k−ˆ
θj,kwith θ∗
j,kbeing the optimal parameter.
From Young’s inequality, we have
2eT
j,0PjBj,ρjhj,s−1−gjhj
≤2
ej,0
2+
Pj
2¯
h2
s+1−gj2h2
j(22)
2eT
j,0Pj
ρj
k=1
Bj,k˜
θj,kϕj,kˆxj+εj,k
≤2ρj
ej,0
2+
Pj
2
ρj
k=1˜
θT
j,k˜
θj,k+ε2
j,k.(23)
We can design the parameter in zto make it be a strict Hurwitz
matrix. Namely, there exists a positive-definite matrix Qjsuch
that AT
jPj+PjAj=−Qj. Taking (22) and (23) into (21), we
have
˙
Vj,0≤−eT
j,0Qjej,0+2+2ρj
ej,0
2+
Pj
2
ׯ
h2
s+
ρj
k=1
ε2
j,k+1−gj2h2
j+
Pj
2
ρj
k=1
˜
θT
j,k˜
θj,k
≤−λj,0
ej,0
2+
Pj
2
ρj
k=1
˜
θT
j,k˜
θj,k+j,0(24)
where λj,0=λmin(Qj)−2−2ρjand j,0=Pj2(¯
h2
s+
ρj
k=1ε2
j,k+(1−gj)2h2
j).
Remark 4: The fuzzy state observer is developed to approx-
imate the unmeasurable system states, in which the actuator
faults and saturations are synthesized. However, unlike the
assumptions in [79]–[83], the knowledge of actuator faults
is difficult to be obtained when the systems are in opera-
tions. Therefore, the state observer is constructed without the
requirement of actuator faults restrictions. From the practi-
cal point of view, the proposed state observer design can be
adopted to a wide class of general conditions.
III. SATURAT I O N -TOLERANT PRESCRIBED CONTROL
A. Auxiliary System Design
An auxiliary system is developed to provide non-negative
modified signals η1,l(t)and η1,u(t)in (11). To make SPP
feature the ability of enlarging or recovering the prescribed
performance constraints when saturations occur or disappear,
developing the associations between input saturations errors
(errors between the actual and desired control inputs) and the
modified signals is significant. To accomplish the non-negative
signals, we introduce the following lemma about the positive
systems.
Lemma 4 [87]: Consider the system ˙x(t)=Kx(t)+(t),
where Kis the Metzler matrix and (t)is a non-negative value.
There exists a unique solution x(t)if for any non-negative x(0)
such that the inequality x(t)≥0 holds for ∀t≥0.
Following ETF T(·), we can transform the constraint error
ejinto an equivalent unconstrained variable zj,1:
ej=¯ej,u+¯ej,l
2Tzj,1+¯ej,u−¯ej,l
2(25)
where T(·)follows the properties in [59] that: 1) −1<
T(zj,1)<for −∞ <zj,1<∞and 2) limzj,1→+∞ T(zj,1)=1,
limzj,1→−∞ T(zj,1)=−1. Thus, the bounded equivalent vari-
able zj,1can guarantee that the tracking errors ej(t)can satisfy
the inequality (9) for all time.
Motivated by the previous systems with input saturations
in [47], [49], [61], [62], and [66]–[71], an auxiliary system
is a promising approach to tackle input saturations and
performance constraints without neglecting their interaction
and conflict. To establish the bridge between saturations errors
and performance constraints, an auxiliary system is developed,
where non-negative signals ηij,uand ηij,lare provided for the
modified signals η1,uand η1,l. We first define =∂zj,1/∂ ej,
u=∂zj,1/∂ ¯ej,u, and l=∂zj,1/∂ ¯ej,l. The auxiliary system
can be designed as
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙η1,u=−inv
u−1η1,u+η2,u
˙ηij,u=−ijηij,u+ηij+1,u,ij∈I2:ρj
˙η1,l=inv
l−1η1,l+η2,l
˙ηij,l=−ijηij,l+ηij+1,l,ij∈I2:ρj
(26)
where ijrepresents the designed positive matrix, ηρj+1,u=
G−(vj,vj,min),ηρj+1,l=G+(vj,vj,max), and the initial condi-
tions for ηij,uand ηij,lare set to zeros. The auxiliary system
has the following property.
Theorem 1: Consider ETF (25) and SPP (11) with the
modified signals updated by (26). If |vj−uj|≤¯
hand
−¯ej,l(t)≤ej≤¯ej,u(t)hold for ∀t≥0, then 0 ≤ηij,u,η
ij,l≤
kul ¯
h,ij∈I1:ρjwith kul being a positive constant.
Proof: Following Lemma 5 and some simple derivations,
we can easily obtain the above theorem. The proof of this
theorem is thus omitted for clarity and conciseness.
Actually, the auxiliary system (26) establishes a mecha-
nism between saturations errors and performance boundaries.
According to the definition of ηρj+1,uand ηρj+1,l, the absolute
values of saturations errors are tactfully extracted, where the
unreachable increasing or decreasing additional signals ηij,u
or ηij,lcan be obtained for xj,ij. Referring to (26), the modi-
fied signals η1,uand η1,lare obtained by gradually integrating,
which will be further provided for SPP (11). These signals are
utilized to enlarge the performance boundaries once the sat-
urations are stimulated. Moreover, the positive matrix ijis
significant to guarantee the stability of the auxiliary system,
which can make the modified signals fast converge to zero
when saturations are disappear. Namely, SPP will recover to
the user-specified performance boundaries.
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 7
Remark 5: It is remarkable that the auxiliary system estab-
lishes a mechanism between saturations errors and modified
signals, which first excavate the interconnections between
performance functions and saturations errors. The modified
signals ηρj+1,uand ηρj+1,lrepresent the unreachable increas-
ing or decreasing rates for xj,ij.η1,uand η1,lare utilized as
additional signals for SPP, which can make SPP have the abil-
ity to elastically enlarge performance boundaries according to
the auxiliary system when saturations occur, and then recover
the user-specified performance when saturations disappear to
degrade the user-specified tracking performance to achieve a
balance among various constraints in a positive manner. In
addition, SPP can not only guarantee the tracking errors not
to violate the performance constraints but also accomplish a
tradeoff between input saturations and performance related
constraints.
B. Control Design
In this section, we propose an SPC for the MIMO nonlinear
system simultaneously considering actuator faults, satura-
tions, and unmeasurable states. We first define the following
auxiliary variables as:
ej=¯ej,u+¯ej,l
2Tzj,1+¯ej,u−¯ej,l
2(27)
zj,ij=ˆxj,ij+ηij,l−ηij,u−αf
j,ij,ij∈I2:ρj(28)
where αf
j,ijis the first-order filter output by directly passing
through αj,ijto the filter. The virtual and actual control inputs
can be developed as
αj,1=−
ˆ
θT
j,1ϕˆxj,1+˙yj,d−invu˙ej,u+l˙ej,l
−1η1,u−η1,l−invcj,1zj,1+bj,1zβ
j,1(29)
αj,2=−kj,2e1
j,0−2η2,u−η2,l−ωj,2
δj,2−zj,1
−ˆ
θT
j,2ϕj,2ˆ
¯xj,2−cj,2zj,2−bj,2zβ
j,2(30)
αj,ij=−kj,ije1
j,0−ijηij,u−ηij,l−ωj,ij
δj,ij
−zj,ij−1−ˆ
θT
j,ijϕj,ijˆ
¯xj,ij−cj,ijzj,ij
−bj,ijzβ
j,ij,i∈I3:ρj−1(31)
vj=ρjηρj,l−ηρj,u−zj,ρj−1−ωj,ρj
δj,ρj
−cj,ρjzj,ρj−bj,ρjzβ
j,ρj−kj,ρje1
j,0(32)
with
˙
ˆ
θj,1=γj,1zj,1ϕj,1ˆxj,1−σj,1ˆ
θj,1(33)
˙
ˆ
θj,2=γj,2zj,2ϕj,2ˆ
¯xj,2−σj,2ˆ
θj,2(34)
˙
ˆ
θj,ij=γj,ijzj,ijϕj,ijˆ
¯xj,ij−σj,ijˆ
θj,ij,ij∈I3:ρj−1(35)
˙
ˆ
θj,ρj=−γj,ρjzj,ρjϕj,ρjˆxj,ρj+σj,ρjˆ
θj,ρj(36)
where cj,ij,bj,ij,δj,ij,γj,ij, and σj,ijare positive constants,
ωj,ij+1=αf
j,ij+1−αj,ij,ˆ
¯xj,ij=[ˆxj,1,...,ˆxj,ij]T, and 0 <β <1.
According to Lemma 3, we have the following main result
of this article.
Theorem 2: Consider the closed-loop system consisting of
MIMO nonlinear system (1) suffering from actuator faults (2)
and saturations (3), the state observer (19), virtual con-
trol inputs (29)–(31), actuate control input (32), adaptive
laws (33)–(36), and SPP (11) associated with the auxiliary
system (26). Then, all the system states can be regulated to
a small neighborhood around the desired trajectories within
finite time, and the tracking errors are guaranteed within SPP
even saturations and faults occur.
Proof: From (88) and Lemma 3, the following inequality
holds:
˙
Vj,ρj≤−c1Vj,ρj−c2c3V¯
β
j,ρj+¯−c2(1−c3)V¯
β
j,ρj(37)
where c3is a positive constant. We define V={ξ|V¯
β
j,ρj≤
(¯/[c2(1−c3)])}and ¯
V={ξ|V¯
β
j,ρj>(¯/[c2(1−c3)])}.
Case 1: If ξ∈¯
V, the following inequality holds:
˙
Vj,ρj≤−c1Vj,ρj−c2c3V¯
β
j,ρj.(38)
Integrating the above inequality over [t0,T], we have
T
t0
V−¯
β
j,ρj
c2c3+c1V1−¯
β
j,ρj
dV ≤−T
t0
dt.(39)
Then, the settling time can be described as
T=t0+1
c11−¯
βln⎛
⎝
c2c3+c1V1−¯
β
j,ρj(t0)
c2c3⎞
⎠.(40)
Namely, ξ∈Vfor ∀t≥T.
Case 2: Considering the system state ξin the region V,
from the case 1 above, ξcannot escape the set V. This com-
pletes the proof. In addition, the proposed control design is
shown in Fig. 4 to make the designed approach in a clear
manner.
For more details, see the Appendix.
Remark 6: Compared with the existing control designs
in [61]–[66], SPC establishes the bridge between input sat-
urations and performance constraints instead of assuming
both constraints being satisfied simultaneously. Compared
with [54]–[66], the proposed SPC can not only guarantee
finite-time stability but also guarantee the tracking errors to
evolve within performance constraints despite of the existence
of saturations and actuator faults. Namely, SPC can flexibly
enlarge or recover the performance boundaries according to
saturation signals, which can dramatically decrease the con-
trol conservatism and achieve favorable tracking performance
although various constraints.
IV. SIMULATION EXAMPLES
In this section, simulation examples are conducted to further
demonstrate the feasibility and effectiveness of the proposed
control scheme. We first consider the dynamics of the tilting
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8IEEE TRANSACTIONS ON CYBERNETICS
Fig. 4. Block diagram of the control scheme.
quadcopter in [88]
⎧
⎨
⎩
˙xj,1=fj,1xj+xj,2
˙xj,2=fj,2(¯x)+uj(t)
yj=xj,1
(41)
where j∈I1:3,xj=[xj,1,xj,2]T,¯x=[xT
1,xT
2,xT
3]T,fj,1(xj)=0,
f1,2(¯x),f2,2(¯x), and f3,2(¯x)are the components of the vec-
tor −M0(¯x)−1(C0(¯x)χ +D0(¯x)χ) with χ=[x1,2,x2,2,x3,2]T.
For the terms M0(¯x),C0(¯x), and D0(¯x), refer to [88] for
details and we omit here for brevity. Following the proposed
control scheme, the control inputs and adaptive law can be
designed as:
αj,1=˙yj,d−invu˙ej,u+l˙ej,l−1
×η1,u−η1,l−invcj,1zj,1+bj,1zβ
j,1(42)
vj=2η2,l−η2,u−zj,1−ωj,2
δj,2−cj,2zj,2
−bj,2zβ
j,2−kj,2e1
j,0(43)
˙
θj,2=−γj,2zj,2ϕj,2ˆxj,2+σj,2ˆ
θj(44)
TAB L E I
PARAMETERS FOR SIMULATIONS
and the auxiliary system is
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙η1,u=−inv
u−1η1,u+η2,u
˙η2,u=−2η2,u+η3,u
˙η1,l=inv
l−1η1,l+η2,l
˙η2,l=−2η2,l+η3,l
(45)
where 1=2=2 and all the other parameters for
simulations can be found in Table I.
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 9
Fig. 5. Tracking error trajectories of e1,1,e2,1,ande3,1.
Fig. 6. Control inputs of SPC and TPC.
To showcase the effectiveness and flexibility of the proposed
control scheme, comparative simulations are carried out to
explore the merit of SPC. We design the comparative control
without the auxiliary system, where the performance con-
straints are always TPP and the simulation results are denoted
as tunnel prescribed control (TPC). Simulation results can be
found in Figs. 5–8. The tracking errors trajectories of SPC
and TPC are exhibited in Fig. 5. The tracking errors of SPC
can be guaranteed to converge to a small neighborhood around
the equilibrium within finite time and the tracking errors are
evolved within the specified performance constraints of SPP
despite of the existence of input saturations and actuator faults.
Fig. 7. Fuzzy state observer tracking trajectories.
Fig. 8. Auxiliary system states.
Specifically, we here take the tracking error e1,1of SPC as
an example. According to the control signals in Fig. 6, the
control input h1of SPC stimulates the saturation effect dur-
ing 0–5 s, 6–11 s, and 18–24 s caused by nonlinearities and
uncertainties in systems. Therefore, the tracking error e1,1of
SPC shows obvious hills or volatility, respectively, during the
saturations periods. To compensate for these tracking degrada-
tions, the auxiliary system is introduced to generate modified
signals as the red dotted line in Fig. 8. The signals η2,1,u
and η2,1,lrepresent unreachable increasing or decreasing rates
for x1,2, and η1,1,uand η1,1,lare utilized into SPP for adjust-
ing performance constraints. With the help of these signals,
SPC has the ability of enlarging SPP constraints to guarantee
favorable performance, where the tracking error e1,1does not
violate SPP boundaries as shown in Fig. 5 although under input
saturations and actuator faults conditions. As input saturations
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10 IEEE TRANSACTIONS ON CYBERNETICS
disappear, these modified signals η2,1,u,η2,1,l,η1,1,u, and η1,1,l
shown in Fig. 8 fast converge to zero, which can further lead
the enlarged SPP to fast recover the specified performance
as shown in Fig. 5. Namely, this mechanism accomplishes a
tradeoff between the system abilities related constraints (e.g.,
input saturations and unmeasurable system states) and the
performance related constraints (e.g., specified convergence
rate and overshoot), and avoids the conservative control design.
Moreover, due to the tight space shaped by the SPP, SPP
can achieve smaller overshoot performance in a positive man-
ner and the tracking errors trajectories in Fig. 5 can verify
its superiority significantly. The other tracking errors analy-
sis of SPC is omitted here for brevity. However, TPC cannot
guarantee the favorable tracking performance under the same
conditions. From Figs. 5 and 6, it can be seen that the tracking
error e3,1of TPC approaches the boundary and violates the
performance after 6.71 s resulting in the singularity of the con-
trol design. It means we cannot assume that input saturations
and performance constraints are satisfied simultaneously and
handle these constraints with neglecting their interconnections
and conflict. The user-specified performance constraints of
TPC may be not achieved once saturations and actuator faults,
and the closed-loop systems may be severely degraded or even
instability. According to the simulation results, the effective-
ness and superiorities of SPC have been verified. In addition,
Fig. 7 describes the estimated states of fuzzy state observer
of SPC, and the observer errors can achieve finite-time stabil-
ity. Fig. 8 is the auxiliary system states of SPC, which bridge
the associations between input saturations and performance
constraints. When input saturations are stimulated, the auxil-
iary system can generate modified signals for enlarging SPP
boundaries. When input saturations disappear, the modified
signals fast converge to zero to recover the enlarged SPP to
the user-specified performance.
V. CONCLUSION
We addressed the tracking control problem of MIMO non-
linear systems simultaneously considering input saturations,
actuator faults, and unmeasurable system states. We proposed
TPP with concise form to simplify the control design, and
smaller overshoot performance can be achieved due to the
tight space shaped. To accomplish a tradeoff among var-
ious constraints, SPP is designed by introducing modified
non-negative signals into TPP, which features the ability of
enlarging performance boundaries when saturations occur and
recover to the specified performance when saturations disap-
pear. This novel mechanism can significantly achieve reducing
conservative control design and better tracking performance
under constraints. An auxiliary system is developed for these
modified signals, which constructs the associations between
input saturations and performance constraints. SPC is then
proposed not only to guarantee the closed-loop system with
finite-time stability but also the tracking errors are evolved
within SPP despite of the existence of input saturations and
actuator faults. To approximate the unmeasurable states, a
fuzzy state observer is constructed without the knowledge
of actuator faults. Finally, comparative simulations have been
provided to further illustrate the effectiveness and superiori-
ties of the proposed SPC. In the future, we will concentrate on
extending our work to MIMO nonlinear systems with unknown
control coefficients.
APPENDIX
PROOF OF THEOREM 2
Step 1: From (27), the time derivative of zj,1is
˙zj,1=xj,2+η2,l−η2,u+fj,1xj−˙yj,d
+u˙ej,u+l˙ej,l+1η1,u−η1,l.(46)
Let zj,2=ˆxj,2+η2,l−η2,u−αf
j,2and ωj,2=αf
j,2−αj,1, where
αf
j,2is the first-order filter output with the designed virtual
control αj,1as input. Due to e2
j,0=xj,2−ˆxj,2,wehave
˙zj,1=e2
j,0+zj,2+ωj,2+αj,1+fj,1xj
+u˙ej,u+l˙ej,l+1η1,u−η1,l−˙yj,d.(47)
Consider the candidate Lyapunov function as
Vj,1=Vj,0+1
2z2
j,1+1
2γj,1
˜
θT
j,1˜
θj,1.(48)
Substituting (29), (33), and (47) into the time derivative of
Vj,1yields
˙
Vj,1=˙
Vj,0+zj,1˙zj,1−1
γj,1
˜
θT
j,1˙
ˆ
θj,1
=˙
Vj,0+zj,1θT
j,1ϕj,1ˆxj,1−ˆ
θT
j,1ϕj,1ˆxj,1
+ωj,2+θT
j,1ϕj,1ˆxj−θT
j,1ϕj,1ˆxj,1
−˜
θT
j,1zj,1ϕj,1ˆxj,1−σj,1ˆ
θj,1−cj,1z2
j,1
+zj,1e2
j,0+zj,2+εj,1−bj,1zj,1
β+1.(49)
According to Young’s inequality, we can obtain
zj,1θ∗T
j,1ϕj,1ˆxj−θ*T
j,1ϕj,1ˆxj,1≤z2
j,1+2
θ∗
j,1
2(50)
zj,1e2
j,0+εj,1≤z2
j,1+1
22ej,02+1
22ε2
j,1.(51)
Due to ˜
θj,1=θ∗
j,1−ˆ
θj,1, we further have
σj,1˜
θT
j,1ˆ
θj,1=σj,1˜
θT
j,1θ∗
j,1−˜
θj,1
≤−
σj,1
2
˜
θj,1
2+σj,1
2
θ∗
j,1
2
.(52)
Substituting (50)–(52) into (49), we can obtain
˙
Vj,1≤−λj,1
ej,0
2+
Pj
2
ρj
k=1
˜
θj,k
2−¯cj,1z2
j,1
−bj,1zj,1
β+1+zj,1zj,2+zj,1ωj,2
−σj,1
2
˜
θj,1
2+j,1(53)
where λj,1=λj,0−(1/2)2,¯cj,1=cj,1−2 and j,1=
(σj,1/2)θ∗
j,12+j,0+2θ∗
j,12+(1/2)2ε2
j,1. A first-order
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 11
filter is introduced to avoid the prohibitively complicated
computation in the backstepping process
δj,2˙αf
j,2+αf
j,2=αj,1,α
f
j,2(0)=αj,1(0). (54)
Therefore, we have
˙ωj,2=−
ωj,2
δj,2−˙αj,1.(55)
Step 2: According to (28), the time derivative of zj,2is
˙zj,2=˙
ˆxj,2+˙η2,l−˙η2,u−˙αf
j,2
=kj,2e1
j,0+ˆxj,3+ˆ
θT
j,2ϕj,2ˆxj
+2η2,u−η2,l+η3,l−η3,u+ωj,2
δj,2.(56)
Let zj,3=ˆxj,3+η3,l−η3,u−αf
j,3be the tracking error and
ωj,3=αf
j,3−αj,2, where αf
j,3is obtained by through the virtual
control αj,2into the filter. We then have
˙zj,2=zj,3+ωj,3+αj,2+kj,2e1
j,0
+2η2,u−η2,l+ωj,2
δj,2+ˆ
θT
j,2ϕj,2ˆxj.(57)
The virtual control αj,2and the adaptive law for ˆ
θj,2are
designed as (30) and (34). Construct the Lyapunov function
Vj,2=Vj,1+1
2z2
j,2+1
2γj,2
˜
θT
j,2˜
θj,2+1
2ω2
j,2.(58)
Substituting (30), (34), and (55)–(57) into the time derivative
of Vj,2,wehave
˙
Vj,2=˙
Vj,1+zj,2zj,3+ωj,3−zj,1−cj,2zj,2
+θ∗T
j,2ϕj,2ˆ
¯xj,2−ˆ
θT
j,2ϕj,2ˆ
¯xj,2
+θ∗T
j,2ϕj,2ˆxj−θ∗T
j,2ϕj,2ˆ
¯xj,2
−˜
θT
j,2ϕj,2ˆxj−bj,2zβ
j,2
−ωj,2ωj,2
δj,2+˙αj,1−˜
θT
j,2zj,2ϕj,2ˆ
¯xj,2−σj,2ˆ
θj,2.
(59)
According to Young’s inequality and ˜
θj,2=θ∗
j,2−ˆ
θj,2,wehave
zj,2θ∗T
j,2ϕj,2ˆxj−θ∗T
j,2ϕj,2ˆ
¯xj,2≤z2
j,2+
θ∗
j,2
2(60)
−zj,2˜
θT
j,2ϕj,2ˆxj≤1
2z2
j,2+1
2
˜
θj,2
2(61)
σj,2˜
θT
j,2ˆ
θj,2=σj,2˜
θT
j,2θ∗
j,2−˜
θj,2
≤−
σj,2
2
˜
θj,2
2+σj,2
2
θ∗
j,2
2
.(62)
Taking (53) and (60)–(62) into (59), we can obtain
˙
Vj,2≤−λj,1
ej,0
2−¯cj,1z2
j,1−bj,1|zj,1|β+1
−cj,2−3
2z2
j,2−bj,2zj,2
β+1+zj,1ωj,2
−σj,1
2
˜
θj,1
2−σj,2−1
2
˜
θj,2
2+j,1
+
Pj
2
ρj
k=1
˜
θj,k
2+zj,2zj,3+ωj,3
−ωj,2ωj,2
δj,2+˙αj,1+σj,2+2
2
θ∗
j,2
2
≤−λj,1
ej,0
2−
2
k=1
¯cj,kz2
j,k−
2
k=1
bj,kzj,k
β+1
−
2
k=1
¯σj,k
˜
θj,k
2+
Pj
2
ρj
k=1
˜
θj,k
2
+zj,2zj,3+ωj,3+zj,1ωj,2+j,2
−ωj,2ωj,2
δj,2+˙αj,1(63)
where ¯cj,2=cj,2−(3/2),¯σj,1=(σj,1/2),¯σj,2=(σj,2−1/2),
and j,2=j,1+(σj,2+2/2)θ∗
j,22.Letαj,2pass through the
following first-order filter:
δj,3˙αf
j,3+αf
j,3=αj,2,α
f
j,3(0)=αj,3(0). (64)
The time derivative of ωj,3can be described as
˙ωj,3=−
ωj,3
δj,3−˙αj,2.(65)
Step ij(3 ≤ij≤ρj−1): We define the tracking error zj,ij
as zj,ij=ˆxj,ij+ηij,l−ηij,u−αf
j,ij. Then, the time derivative of
zj,ijcan be formulated as
˙zj,ij=˙
ˆxj,ij+˙ηij,l−˙ηij,u−˙αf
j,ij
=kj,ije1
j,0+ˆxj,ij+1+ˆ
θT
j,ijϕj,ijˆxj
+ijηij,u−ηij,l+ηij+1,l−ηij+1,u+ωj,ij
δj,ij
.(66)
Let zj,ij+1=ˆxj,ij+1+ηij+1,l−ηij+1,u−αf
j,ij+1and ωj,ij+1=
αf
j,ij+1−αj,ij. The tracking error zj,ijcan be rewritten as
˙zj,ij=zj,ij+1+ωj,ij+1+αj,ij+kj,ije1
j,0
+ijηij,u−ηij,l+ωj,ij
δj,ij
+ˆ
θT
j,ijϕj,ijˆxj.(67)
The virtual control input αj,ijand the adaptive law ˆ
θj,ijcan
be designed as (31) and (35). Then, the Lyapunov function is
constructed as
Vj,ij=Vj,ij−1+1
2z2
j,ij+1
2γj,ij
˜
θT
j,ij˜
θj,ij+1
2ω2
j,ij.(68)
Taking (31), (35), (65), and (67) into the time derivative of
Vj,ijyields
˙
Vj,ij=˙
Vj,ij−1
+zj,ijzj,ij+1+ωj,ij+1−zj,ij−1
+θ∗T
j,ijϕj,ijˆ
¯xj,ij−ˆ
θT
j,ijϕj,ijˆ
¯xj,ij−bj,ijzβ
j,ij
+θ∗T
j,ijϕj,ijˆxj−θ∗T
j,ijϕj,ijˆ
¯xj,ij−cj,ijzj,ij
−˜
θT
j,ijϕj,ijˆxj
−˜
θT
j,ijzj,ijϕj,ijˆ
¯xj,ij−σj,ijˆ
θj,ij
−ωj,ijωj,ij
δj,ij
+˙αj,ij−1.(69)
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12 IEEE TRANSACTIONS ON CYBERNETICS
According to Young’s inequality and the definition ˜
θj,ij=
θ∗
j,ij−ˆ
θj,ij, we can have
zj,ijθ∗T
j,ijϕj,ijˆxj−θ∗T
j,ijϕj,ijˆ
¯xj,ij≤z2
j,ij+
θ∗
j,ij
2(70)
−zj,ij˜
θT
j,ijϕj,ijˆxj≤1
2z2
j,ij+1
2
˜
θj,ij
2(71)
σj,i˜
θT
j,ijˆ
θj,ij=σj,ij˜
θT
j,ijθ∗
j,ij−˜
θj,ij
≤−
σj,ij
2
˜
θj,ij
2+σj,ij
2
θ∗
j,ij
2
.(72)
Taking (70)–(72) into (69), we have
˙
Vj,ij≤˙
Vj,ij−1+zj,ijzj,ij+1+ωj,ij+1−zj,ij−1
+˜
θT
j,ijϕj,ijˆ
¯xj,ij−bj,ijzβ
j,ij−cj,ijzj,ij
+1
2z2
j,ij+1
2
˜
θj,ij
2+z2
j,ij+
θ∗
j,ij
2
−˜
θT
j,ijzj,ijϕj,ijˆ
¯xj,ij−σj,ij
2
˜
θj,ij
2
+σj,ij
2
θ∗
j,ij
2−ωj,ijωj,ij
δj,ij
+˙αj,ij−1
≤−λj,1
ej,0
2−
ij
k=1
¯cj,kz2
j,k−
ij
k=1
bj,k|zj,k|β+1
−
ij
k=1
¯σj,k
˜
θj,k
2+
Pj
2
ρj
k=1
˜
θj,k
2
+
ij
k=2
zj,kωj,k+1+zj,1ωj,2+zj,ijzj,ij+1
−
ij
k=2
ωj,kωj,k
δj,k+˙αj,k−1+j,ij(73)
where ¯cj,ij=cj,ij−(3/2),j,ij=j,ij−1+([σj,ij+2]/2)θ∗
j,ij2,
and ¯σj,ij=([σj,ij−1]/2).Letaj,ijthrough the first-order filter
δj,ij+1˙αf
j,ij+1+αf
j,ij+1=αj,ij,α
f
j,ij+1(0)=αj,ij(0)(74)
where δj,ij+1is a positive constant. Therefore, we have
˙ωj,ij+1=−
ωj,ij+1
δj,ij+1−˙αj,ij.(75)
Step ρj:Let zj,ρj=ˆxj,ρj+ηρj,l−ηρj,u−αf
j,ρj. According
to the definition ωj,ρj=αf
j,ρj−αj,ρj−1, we obtain the time
derivative of zj,ρjas
˙zj,ρj=˙
ˆxj,ρj+˙ηρj,l−˙ηρj,u−˙αf
j,ρj
=hjvj+ˆ
θT
j,ρjϕj,ρjˆxj+ηρj+1,l−ηρj+1,u
+ρjηρj,u−ηρj,l+ωj,ρj
δj,ρj
+kj,ρje1
j,0.(76)
Considering the auxiliary system designed in (26), the term
hj(vj)+ηρj+1,l−ηρj+1,u=vjholds. The actual control input
vjand the adaptive law ˆ
θj,ρjcan be described as (32) and (36).
The Lyapunov function is designed as
Vj,ρj=Vj,ρj−1+1
2z2
j,ρj+
˜
θT
j,ρj˜
θj,ρj
2γj,ρj
+1
2ω2
j,ρj.(77)
According to (32), (36), and (76), we have the time derivative
of Vj,ρjas
˙
Vj,ρj=˙
Vj,ρj+zj,ρj−zj,ρj−1−cj,ρjzj,ρj+θ∗T
j,ρjϕj,ρjˆxj
−˜
θT
j,ρjϕj,ρjˆxj−bj,ρjzβ
j,ρj
+˜
θT
j,ρjσj,ρjˆ
θj,ρj+zj,ρjϕj,ρjˆxj,ρj+ωj,ρj˙ωj,ρj.(78)
Due to Young’s inequality and ˜
θj,ρj=θ∗
j,ρj−ˆ
θj,ρj,wehave
zj,ρjθ∗T
j,ρjϕj,ρjˆxj≤1
2z2
j,ρj+1
2
θ∗
j,ρj
2(79)
σj,ρj˜
θT
j,ρjˆ
θj,ρj=σj,ρj˜
θT
j,ρjθ∗
j,ρj−˜
θj,ρj
≤−
σj,ρj
2
˜
θj,ρj
2+σj,ρj
2
θ∗
j,ρj
2
.(80)
Substituting (73), (75), (79), and (80) into (78), we have
˙
Vj,ρj≤−λj,1
ej,0
2−
ρj
k=1
¯cj,kz2
j,k+zj,1ωj,2
−
ρj
k=1
bj,kzj,k
β+1−
ρj
k=1
¯σj,k
˜
θj,k
2
+
Pj
2
ρj
k=1
˜
θj,k
2+
ρj−1
k=2
zj,kωj,k+1
−
ρj
k=2
ωj,kωj,k
δj,k+˙αj,k−1+j,ρj(81)
where ¯cj,ρj=cj,ρj−(1/2),¯σj,ρj=(σj,ρj/2), and j,ρj=
j,ρj−1+([σj,ρj+1]/2)θ∗
j,ρj2. From Young’s inequality, we
have
zj,1ωj,2≤1
2z2
j,1+1
22ω2
j,2(82)
ρj−1
k=2
zj,kωj,k+1≤
ρj−1
k=2
1
2z2
j,k+
ρj
k=3
1
2ω2
j,k(83)
−
ρj
k=2
ωj,k˙αj,k−1≤
ρj
k=2
1
2ω2
j,k+
ρj
k=2
1
2˙α2
j,k−1.(84)
Taking (82)–(84) into (81) yields
˙
Vj,ρj≤−λj,1
ej,0
2−
ρj
k=1
lj,kz2
j,k−
ρj
k=1
bj,kzj,k
β+1
−
ρj
k=2
¯
δj,kω2
j,k−
ρj
k=1
pj,k
˜
θj,k
2+¯j,ρj(85)
where ¯j,ρj=ρj
k=2(1/2)˙α2
j,k−1+j,ρj,pj,k=¯σj,k−Pj2,
lj,k=¯cj,k−(1/2)for k∈I1:ρj,¯
δj,2=(1/δj,2)−([1 +2]/2),
and ¯
δj,k=(1/δj,k)−1fork∈I3:ρj. From Lemma 2, we can
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JI et al.: SPC FOR CLASS OF MIMO NONLINEAR SYSTEMS 13
obtain
−
˜
θj,k
2=−
˜
θj,k
2+
˜
θj,k
β+1−
˜
θj,k
β+1
≤1−β
2−1−β
2
˜
θj,k
2−
˜
θj,k
β+1
.(86)
The terms −λj,1ej,02and −ρj
k=2¯
δj,kω2
j,kfollow the same
process. Equation (85) can be
˙
Vj,ρj≤−
1−β
2λj,1
ej,0
2−λj,1
ej,0
β+1+¯
−1−β
2
ρj
k=1
pj,k
˜
θj,k
2−
ρj
k=1
pj,k
˜
θj,k
β+1
−1−β
2
ρj
k=1
¯
δj,kω2
j,k−
ρj
k=1
¯
δj,kωβ+1
j,k
−
ρj
k=1
lj,kz2
j,k−
ρj
k=1
bj,kzj,k
β+1(87)
where ¯=¯j,ρj+([λj,1(1−β)]/2)+ρj
k=1([¯
δj,k(1−β)]/2)+
ρj
k=1([pj,k(1−β)]/2). According to Lemma 1, (87) can be
formulated as
˙
Vj,ρj≤−c11
2
ej,0
2+1
2
ρj
k=1
z2
j,k+
ρj
k=2
1
2ω2
j,k+
ρj
k=1
˜
θT
j,k˜
θj,k
2γj,k
−c21
2
ej,0
2+1
2
ρj
k=1
z2
j,k
+
ρj
k=2
1
2ω2
j,k+
ρj
k=1
˜
θT
j,k˜
θj,k
2γj,k¯
β
+¯
≤−c1Vj,ρj−c2Vj,ρj¯
β+¯(88)
where c1=min((1−β)λj,1,2lj,k,(1−β)¯
δj,k,(1−β)pj,kγj,k),
and c2=min(2¯
βλj,1,2¯
βbj,k,2¯
β¯
δj,k,2¯
βpj,k)with ¯
β=
([1 +β]/2).
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Ruihang Ji (Student Member, IEEE) received the
B.S. degree in automation engineering from the
Harbin Institute of Technology, Harbin, China, in
2016, where he is currently pursuing the Ph.D.
degree in control science and engineering.
He is currently an exchange Ph.D. student sup-
ported by the China Scholarship Council with the
Department of Electrical and Computer Engineering,
National University of Singapore, Singapore. His
current research interests include adaptive control,
robust control, and unmanned aerial vehicle.
Baoqing Yang received the B.S., M.S., and Ph.D.
degrees in automatic control from the Harbin
Institute of Technology, Harbin, China, in 2003,
2005, and 2009, respectively.
He is currently an Associate Professor with the
Control and Simulation Center, Harbin Institute of
Technology. His research interests include flight
control and predictive control.
Jie Ma received the B.Sc. and M.Sc. degrees in
mechatronics engineering and the Ph.D. degree in
instrument science and technology from the Harbin
Institute of Technology (HIT), Harbin, China, in
1997 and 2001, respectively.
Since 2001, she has been with the Department of
Control Science and Engineering, HIT, where she
became a Professor in 2009. She has authored more
than 30 articles and four inventions. Her research
interests include motion control, sampled-data con-
trol systems, and robust control and applications.
Shuzhi Sam Ge (Fellow, IEEE) received the B.Sc.
degree from the Beijing University of Aeronautics
and Astronautics, Beijing, China, in 1986, and the
Ph.D. degree from the Imperial College London,
London, U.K., in 1993.
He is the Director of the Social Robotics
Laboratory of Interactive Digital Media Institute,
Singapore, and the Centre for Robotics, Chengdu,
China, and a Professor with the Department of
Electrical and Computer Engineering, National
University of Singapore, Singapore, on leave
from the School of Computer Science and Engineering, University of
Electronic Science and Technology of China, Chengdu. He has coauthored
four books and over 300 international journal and conference papers. His
current research interests include social robotics, adaptive control, intelligent
systems, and artificial intelligence.
Prof. Ge is the Editor-in-Chief of the International Journal of Social
Robotics (Springer). He has served/been serving as an Associate Editor
for a number of flagship journals, including the IEEE TRANSACTIONS
ON AUTOMATION CONTROL, the IEEE TRANSACTIONS ON CONTROL
SYSTEMS TECHNOLOGY, the IEEE TRANSACTIONS ON NEURAL
NETWORKS,andAutomatica. He serves as a Book Editor for the Automation
and Control Engineering Series (Taylor & Francis). He served as the Vice
President for Technical Activities from 2009 to 2010 and Membership
Activities from 2011 to 2012, and a member of the Board of Governors from
2007 to 2009 at the IEEE Control Systems Society. He is a Fellow of the
International Federation of Automatic Control, the Institution of Engineering
and Technology, and the Society of Automotive Engineering.
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