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IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 1
Low-complexity Fuzzy Neural Control Of
Constrained Waverider Vehicles Via Fragility-free
Prescribed Performance Approach
Xiangwei Bu, Baoxu Jiang, Humin Lei
Abstract—In this article, we propose a concise fuzzy neural
control framework for Waverider Vehicles with input constraints,
while the spurred prescribed performance can be guaranteed,
and the challenging fragility problem associated with the existing
prescribed performance control (PPC) is avoided. Unlike the ex-
isting control protocols without considering computational costs,
in this study, the low-complexity fuzzy neural approximation
is combined with simple performance functions, which reduces
the complexity burden and improves the practicability. Then,
in order to handle the adverse effect of the actuator saturation
on the control performance, bounded-input-bounded-state stable
systems are developed to stabilize the closed-loop control system
based on bounded compensations. Specially, flexible adjustment
terms are exploited to modify the developed simple performance
functions, while fragility-free prescribed performance is achieved
for tracking errors, and moreover the fragility defect of the
existing PPC is remedied. Finally, the efficiency and superiority
of the design are verified via compared simulations.
Index Terms—Waverider Vehicles, Fuzzy neural control,
Fragility problem, Actuator saturation, Fragility-free prescribed
performance.
I. INTRODUCTION
THE Waverider is the most commonly used aerodynamic
configuration for hypersonic flight vehicles since it con-
tributes to achieve hypersonic flight in near-space [1]-[3]. Re-
cently, the related technologies of Waverider Vehicles (WVs)
have achieved remarkable developments. Among them, the
flight control design is the most significant aspect that plays a
key role in ensuring safe flight and completing complex tasks
[4]-[10]. However, the flight control design is facing various
problems owing to unique dynamic peculiarities and compli-
cated flight circumstances. The control system should meet
comprehensive index requirements. For the control system of
traditional aircraft, only the steady-state accuracy needs to be
guaranteed. However, as to WVs, it is far from enough to
ensure the steady-state accuracy, but also to pursue excel-
lent transient performance for the control system to realize
maneuvering flight. In addition, the control complexity and
computational load should be as low as possible, for the sake
of avoiding the control delay.
Manuscript received XX, 2022; revised XX X, 202X; accepted XX XX,
201X. This journal was supported by Young Talent Support Project for
Science and Technology (Grant No. 18-JCJQ-QT-007). This article has
been accepted for publication in IEEE Transactions on Fuzzy Systems,
doi:10.1109/TFUZZ.2022.3217378 (Corresponding author: Xiangwei Bu).
X. Bu, B. Jiang and H. Lei are with the Air and Missile Defense
College, Air Force Engineering University, Xi’an 710051, China (e-mail:
buxiangwei1987@126.com; 13124285328@163.com; hmleinet@126.com).
Before the emergence of prescribed performance control
(PPC) [11]-[14], it was not easy to simultaneously ensure
the control system with desired transient performance and
steady-state accuracy. The feedback control protocols [7], [9]
derived from the traditional Lyapunov-synthesis approaches
are only able to ultimately adjust the system output to a
compact set whose radius may be arbitrarily small by choosing
suitable design parameters. Although this guarantees desired
steady-state accuracy, the transient performance cannot be
satisfied due to the short of suitable tools. In 2008, a new
control methodology called PPC was proposed in [15]. The
PPC provides a control design framework that pursues the
balance between the transient performance and the steady-
state performance, by developing a type constraint envelope
composed of performance functions to limit the convergence
trajectories of tracking errors. Then, both the transient per-
formance and the steady-state performance, being collectively
referred to as the prescribed performance, can be satisfied
by setting appropriate shapes for the constraint envelope. The
initial version of PPC addressed in [15] took the signs of the
initial tracking errors as the priori information that is necessary
for the subsequent design of constraint envelopes and control
protocols. However, this may not be practical for engineering
practice, in which the initial value of tracking error is not
always certain for many cases. To improve the practicability,
an improved PPC [16] was proposed for WVs to guarantee
the velocity tracking error and the altitude tracking error with
prescribed performance, while it is no longer required to know
the initial value of tracking error in advance. Furthermore,
the uncertain dynamics of WVs was approximated by neural
networks [16], and the approximation errors were stabilized
by devising regulation strategies for the weights of neural
systems. In [17], the back-stepping design was introduced to
develop both the virtual control laws and the actual one for the
strict feedback model of WVs. On this basis, the asymptotic
tracking controller with prescribed performance was addressed
for a class of WVs. Though the presented simulation results
infer that the pursued prescribed performance is able to be
achieved, the fuzzy estimation that is adopted to ensure the
robustness performance, still produces a certain computation
burden caused by online learning parameters. In another study
[18], the back-stepping was combined with the projection
algorithm to design adaptive PPC protocols for WVs with
faulted and hysteretic actuators. At each step of the recursive
design procedure derived from back-stepping, considerable
adaptive parameters were defined and online updated via
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 2
the projection algorithm. Although tracking errors satisfy
the prescribed performance, the structural and computational
complexity need to be further reduced. Such problem of com-
putational load also exists in [19] due to the online regulation
parameters constructed for Type-2 fuzzy neural networks.
Another representative PPC methodology was presented in
[20] for the sake of achieving the finite convergence that ex-
hibits more practical significance for engineering applications.
However, this requires performance functions with switching
formulations [20]. Noticing that the above PPC approaches
can only guarantee the qualitative prescribed performance,
the quantitative PPC (QPPC) scheme [21] was exploited and
applied to WVs, and numerical simulations performed in those
studies showed the proposed QPPC is able to achieve finite
time convergences of the velocity and the altitude tracking
errors with almost zero overshoots and quantitative given
steady-state values. In addition, other researchers [22], [23]
have investigated performance-enhanced PPC protocols for
WVs via reinforcement learning and adaptive critic design.
Despite the achievements of the existing literature, it is
still necessary to discuss what should be further improved.
(1) The current researches focus on the improvement of
performance functions, from which several new versions have
been derived. Even though the performance is enhanced such
that small overshoot and finite time convergence [20], [21]
can be achieved, the constraint envelopes of the existing PPC
become quite complex. Moreover, by combing back-stepping
and fuzzy/neural approximations [17]-[20] to devise PPC
protocols, both the control complexity and computational load
are too high. This isn’t conducive to ensuring the practicality
and fast response of the control system in maneuvering flight.
(2) As a problem often encountered by WVs, the actuator
saturation may lead to the risk that the spurred prescribed
performance of the existing methods fails to realize. This
is owing to that the traditional compensation strategy [24]
isn’t able to guarantee the boundedness of transformed errors.
In this case, the tracking errors cannot be kept within the
constraint envelopes to satisfy the prescribed performance. (3)
The actuator saturation will further lead to control singularity
due to the fragility [21] associated with the existing PPC
methodologies.
Motivated by theoretical and technical defects of the exist-
ing PPC, this paper focuses on avoiding the fragility of PPC
by proposing a fragility-free guaranteed performance approach
for WVs with actuator saturation based on low-complexity
fuzzy-neural approximations. The main contributions include:
1) Unlike the existing PPC methodologies [16]-[23], this
article proposes a concise framework in which simple
performance functions are developed and combined with
the low-computational fuzzy-neural approximation strat-
egy, while the computation burden and structure com-
plexity are recued, and the practicability is improved.
2) Bounded-input-bounded-state (BIBS) stable systems are
devised to modify the transformed errors and com-
pensate the PPC protocols, which guarantees that the
spurred prescribed performance still is satisfied in the
presence of actuator saturation.
3) Novel sensing systems are designed to modify the con-
straint envelopes and endow them with the ability to
predict the error fluctuations and then actively adjust the
envelopes, such that the tracking errors are always kept
within the constraint envelopes to satisfy the prescribed
performance. As a result, the well-known fragility prob-
lem associated with the existing PPC is tackled.
The rest parts of this paper are given as follows. The
considered problems are stated in Section II. The obtained
main results are presented in Section III. In Section IV, the
addressed method is compared with some existing ones via
simulation. Finally, the conclusions are given in Section V.
Notations:ℜis the set of real numbers. ℜ≥0is the set
of non-negative real numbers. ℜ>0=ℜ≥0/{0}.:= denotes
“defined as”. ∈means “belong to”, →denotes “converge to”.
7→ means “map to”.
II. PRO BL EM S TATEM EN T
A. Vehicle model
The motion equations of WVs are described as [25]:
˙
V=ψV+ Φ (1)
˙
h=Vsin γ(2)
˙γ=ψγ+θ(3)
˙
θ=Q(4)
˙
Q=ψQ+δe(5)
k1¨η1=−2ζ1ω1˙η1−ω2
1η1+N1
−˜
ψ1
M
Iyy
−˜
ψ1˜
ψ2¨η2
Iyy
(6)
k2¨η2=−2ζ2ω2˙η2−ω2
2η2+N2
−˜
ψ2
M
Iyy
−˜
ψ2˜
ψ1¨η1
Iyy
(7)
with
ψV:= Tcos (θ−γ)
m−D
m−gsin γ−Φ
ψγ:= L
mV +Tsin (θ−γ)
mV −g
Vcos γ−θ
ψQ:= M
Iyy
+˜
ψ1¨η1
Iyy
+˜
ψ2¨η2
Iyy
−δe
where ψV,ψγ, and ψQare unknown but continuously
differentiable functions.
The above vehicle model of WVs has five rigid-body states
(i.e., the velocity V∈ ℜ>0, the altitude h∈ ℜ>0, the flight-
path angle γ∈ ℜ , the pitch angle θ∈ ℜ and the pitch rate
Q∈ ℜ ), two flexible states (i.e., η1∈ ℜ and η2∈ ℜ), and
two control inputs (i.e., the fuel equivalence ratio Φ∈ ℜ>0
and the elevator angular deflection δe∈ ℜ ) that are implicit
in the trust force T, the drag force D, the lift force L, the
pitching moment M, and the generalized forces N1and N2,
given by:
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 3
Fig. 1. Geometry and force map of WVs.
T≈Φβ1(h, ¯q)α3+α3β2(h, ¯q)
+Φβ3(h, ¯q)α2+α2β4(h, ¯q)
+Φβ5(h, ¯q)α+αβ6(h, ¯q)
+Φβ7(h, ¯q) + β8(h, ¯q)
D≈Cα2
D¯qSα2+Cα
D¯qSα +Cδ2
e
D¯qSδ2
e
+Cδe
D¯qSδe+C0
D¯qS
M≈Cα2
M,α ¯qS ¯cα2+Cα
M,α ¯qS ¯cα
+C0
M,α ¯qS ¯c+δe¯qS¯cce+T zT
L≈Cα
L¯qSα +Cδe
L¯qSδe+C0
L¯qS
N1=α2Nα2
1+αNα
1+N0
1
N2=α2Nα2
2+αNα
2+δeNδe
2+N0
2
¯q=1
2¯ρV 2,¯ρ= ¯ρ0exp −h−h0
hs.
Remark 1. It is known that the motion equations of WVs
are controllable [1]-[10]. In Eqs. (1), (3), and (5), we add and
subtract Φ,θ, and δeto handle the nonaffine property of
WVs. This is a commonly selected approach in the existing
study [4], and the Implicit Function Theorem guarantees the
rationality of such approach.
The geometry and force map of WVs are clearly shown
in Fig. 1, and all the other definitions of the coefficients and
variables in the above equations can be seen in [25].
The control inputs Φand δehave their physical execution
scopes. The execution efficiency of Φand δedecrease signif-
icantly due to the high altitude flight of WVs. Thereby, when
WVs are in large maneuvering flight, the required control
quantities are very easy to exceed their physical executable
ranges, resulting in the actuator saturation problem that is
described as:
Φ =
ΦU,Φd>ΦU
Φd,ΦL≤Φd≤ΦU
ΦL,Φd<ΦL
(8)
δe=
δe,U ,δe,d > δe,U
δe,d,δe,L ≤δe,d ≤δe,U
δe,L,δe,d < δe,L
(9)
where Φd∈ ℜ and δe,d ∈ ℜ are the desired values of Φand
δe,ΦU∈ ℜ>0and ΦL∈ ℜ>0are the upper bound and the
lower bound of Φ, and δe,U ∈ ℜ>0and δe,L ∈ ℜ>0are the
upper bound and the lower bound of δe. With such definition,
we obtain: Φ∈[ΦL,ΦU]and δe∈[δe,L, δe,U ].
Control Objective: Utilizing the fuzzy-neural approximation
we devise PPC protocols for Φ∈ ℜ>0and δe∈ ℜ that satisfy
actuator saturation (8) and (9), such that V→Vc∈ ℜ>0and
h→hc∈ ℜ>0, and furthermore the modified prescribed
performance (17) and (32) are guaranteed for the tracking
errors, where Vc∈ ℜ>0and hc∈ ℜ>0are given bounded
and differentiable functions.
B. BIBS system
The compensated system developed in [24] is the commonly
used strategy in the existing studies to deal with actuator
saturation. However, this compensation approach is only able
to stabilize the closed-loop system, but can not guarantee that
the transformed errors of PPC are always bounded in the case
of actuator saturation. As a result, there exists a risk that the
prescribed performance cannot be achieved because of the
nonconvergence of transformed errors. To avoid such risk, we
give the following BIBS system that will be used to provide
bounded compensations on the actuator saturation.
˙ιi=ιi+1 , i = 1,2,· · · , n −1
˙ιn=−
n
X
i=1
li
ιi
abs(ιi)+∆ιi
+uι(10)
where ιi∈ ℜ, i = 1,2,· · · , n are the system states, li∈
ℜ>0,∆ιi∈ ℜ>0, i = 1,2,· · · , n are design parameters, and
uι∈ ℜ is the bounded control input.
Theorem 1. We assume that the bounded control input uι
satisfies |uι| ≤ ¯uι∈ ℜ>0. Then, (10) is BIBS stable, that
is, all the states ιi∈ ℜ, i = 1,2,· · · , n are convergent and
bounded.
Proof. We will prove Theorem 1 utilizing LaSalle’s Invari-
ance Principle (LIP). The trim point of (10) is (uι∆ι1/(l1−
uι),0,· · · ,0
| {z }
n−1
). Then we define Wι(LIP does not require Wι
to be positive definite)
Wι=ln−1Zι1
0
τ1
abs(τ1)+∆ι1
dτ1
+ln−1Zι2
ι1
τ2
abs(τ2)+∆ι2
dτ2
+· · · +ln−1Zιn−1
ιn−2
τn−1
abs(τn−1)+∆ιn−1
dτn−1
+ι2
n
2.(11)
Using (10), ˙
Wιis given by
˙
Wι=ln−1
ι1ι2
abs(ι1)+∆ι1
+ln−1ι2ι3
abs(ι2)+∆ι2
−ι1ι2
abs(ι1)+∆ι1+· · · +
+ln−1ιn−1ιn
abs(ιn−1)+∆ιn−1
−ιn−2ιn−1
abs(ιn−2)+∆ιn−2
+ιn˙ιn
=−
n−2
X
i=1
li
ιiιn
abs(ιi)+∆ιi
−lnabs(ιn)
abs(ιn)+∆ιn
−¯uιabs(ιn).(12)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 4
If ln> lnabs(ιn)
abs(ιn)+∆ιn>¯uι,l1>¯uι, and li>0, i =
2,3,· · · , n −1, we know ˙
Wι≤0and ˙
Wι= 0 only when
ιn= 0 . Moreover, the case of “ ιn= 0” is instable since ιn=
0if ιi= 0, i = 1,2,· · · , n −1. In other words, the solution
of “ ˙
Wι= 0 ” has no any other whole trajectory except its
trim point. In addition, Wι→ ∞ as ιi→ ∞, i = 1,2,· · · , n .
Invoking the Krasovskii Theorem [26] we conclude that (10)
is BIBS stable, that is, in the case that the control input uιis
bounded, then ιi, i = 1,2,· · · , n are convergent and bounded.
This is the end of the proof.
Remark 2. Unlike the compensated systems addressed in
[24], the BIBS stability of (10) will facilitate the subsequent
stability proof of the closed-loop system and contribute to
the guarantee of prescribed performance in the presence of
actuator saturation.
C. Prescribed performance
By prescribed performance, it denotes that the tracking error
e(t)∈ ℜ satisfies [15]
−deβe(t)< e(t)< βe(t), e(0) >0
−βe(t)< e(t)< deβe(t), e(0) <0(13)
where de∈(0,1) is a constant and the performance function
(PF) βe(t) : ℜ≥07→ ℜ>0is a type of strictly monotonically
decreasing continuous function.
By setting specific shapes for βe(t)to keep e(t)within the
constraint envelope (13), we can enforce desired transient and
stead-state prescribed specifications (e.g. overshoot, conver-
gence rate, and steady-state error) on the tracking error e(t)
.
Recently, several improved PFs have been developed to
pursue some special performance specifications such as finite-
time convergence [20], small overshoot [27], and quantitative
prescribed performance [21]. However, this makes the mathe-
matical formulations of the PFs quite complex (See Table 1),
being harmful to the engineering implementation of the PPC
algorithms. For this reason, in this article, we propose a new
PF βe(t)∈ ℜ>0with simple formulation to enforce envelope
constraint on the tracking error e(t)
−βe(t)< e(t)< βe(t)(14)
with
βe(t) = 1
tαe+ςe
+βf,e (15)
where αe∈ ℜ>0, ςe∈ ℜ>0,1
ςe> βf,e ∈ ℜ>0are design
parameters.
Lemma 1 ([28]). Define the transformed error Le(t) =
1
2ln e(t)/βe(t)+1
1−e(t)/βe(t)=1
2ln e(t)+βe(t)
βe(t)−e(t)∈ ℜ . Then the
boundedness of Le(t)is equivalent to that e(t)satisfies the
prescribed performance.
As indicated by Lemma 1, by devising suitable control
protocols to stabilize the transformed error Le(t), the desired
prescribed performance (14) for e(t)can be achieved, as
shown in Fig. 2(a). However, the actuator saturation, defined
by (8) and (9), is a scenario that aircraft often encounter.
In the case that the actuator is saturated, the tracking error
e(t)will inevitably increase significantly since the actual
actuator cannot provide its desired value, this even leads to
the instability of the closed-loop system. Moreover, once the
actuator saturation causes that e(t)escapes from the constraint
envelope (14) as depicted in Fig. 2 (b), then the feedback
controller composed of Le(t)∈ ℜ becomes singular. This
is due to that if e(t)→βe(t), we get e(t)+βe(t)
βe(t)−e(t)→ ∞ ⇒
Le(t) = 1
2ln e(t)+βe(t)
βe(t)−e(t)→ ∞, and if e(t)→ −βe(t), we
obtain e(t)+βe(t)
βe(t)−e(t)→0⇒Le(t) = 1
2ln e(t)+βe(t)
βe(t)−e(t)→ −∞.
Otherwise, when e(t)> βe(t)>0or e(t)<−βe(t)<0, we
have e(t)+βe(t)
βe(t)−e(t)<0⇒Le(t)/∈ ℜ. All of those will result in
the control singularity. This is the so-called fragility problem
[21] inherent to the existing PPC, that is, the existing PPC
methodologies exhibit obvious fragility to actuator saturation.
D. Fragility-free prescribed performance
Almost all actual control systems suffer from the problem
of control saturation. However, the fragility defects of the
existing PPC methods [16]-[23] will lead to low reliability
and failure risk in their applications. To avoid this defect, in
this subsection, we propose an improved version of prescribed
performance namely the fragility-free prescribed performance
(FPP), given by
−βe(t)−fe(t)< e(t)< βe(t) + fe(t)(16)
where fe(t)∈ ℜ≥0is the flexible adjustment term (FAT) that
satisfies the features: 1) fe(t)>0if the actuator is saturated,
and 2) fe(t) = 0 if the actuator isn’t saturated. Because of such
features, the FAT fe(t)∈ ℜ≥0is able to increase the upper
envelope of FPP (16) and decrease the lower envelope of FPP
(16) depicted in Fig. 2(c), which enforces desired prescribed
performance (16) on e(t)in the presence of actuator saturation.
As a result, this fragility defect associated with the existing
PPC [16]-[23] is expected to be avoided. In what follows,
the detail formulation of FAT fe(t)∈ ℜ≥0will be designed
according to the actuator saturation.
E. Fuzzy-neural approximation
In the article, the vehicle dynamics is assumed to be
unknown. To guarantee the robust performance, the neural
network [2], [14], [29] is a commonly used approximator
that is able to effectively estimate such unknown dynamics.
To further improve the estimation performance, fuzzy logic
rules have been combined with neural networks for the sake
of constructing a type of fuzzy rule-based neural networks
(FNNs) that exhibit more powerful approximation ability in
comparison with neural networks and fuzzy systems. Repre-
sentative FNNs include Type-2 FNN [19], [30], [31] and fuzzy
wavelet neural network (FWNN) [21], [27]. In this study, to
approximate the unknown vehicle dynamics, we will adopt the
FWNN whose strength for function approximation has been
shown in the research [21], [27].
The structure of FWNN can be described by a set of fuzzy
rules: Ri:if x1is Ai
1, and x2is Ai
2, , and xnis An
1;
then ψi= Ψi
n
P
j=1
ϑij (xj), where xj∈ ℜ, j = 1,2,· · · , n
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 5
TABLE I
THE F ORM UL ATION S OF PFS.
PFs Mathematical expressions Design parameters
Finite-time PF [20] βe(t) = (csch βe0+ret
Te−t+βeT ,0≤t≤Te
βeT , t > 0βe0∈ ℜ>0, re∈ ℜ>0, βeT ∈ ℜ>0, Te∈ ℜ>0
Small overshoot PF [27]
βe(t)=[sign (e(0)) −δe1]ρe(t)−ρe∞sign (e(0))
or βe(t)=[sign (e(0)) + δe2]ρe(t)−ρe∞sign (e(0))
withρe(t) = csch (ρe0+ret) + ρe∞
δe1∈(0,1), δe2∈(0,1), ρe0∈ ℜ>0,
re∈ ℜ>0, ρe∞∈ ℜ>0
Quantitative PF [21]
βe(t) =
Tef −t
Tef pe[sign(e(0)) −ae]×
(ρe0−ρeT,f )−aeρeT ,f ,if t∈0, Tef
−aeρeT,f ,if t∈Tef ,∞
or
βe(t) =
Tef −t
Tef pe[sign(e(0)) + be]×
(ρe0−ρeT,f ) + beρeT ,f ,if t∈0, Tef
beρeT,f ,if t∈Tef ,∞
δe1∈(0,1), δe2∈(0,1), ρe0∈ ℜ>0,
re∈ ℜ>0, ρe∞∈ ℜ>0
Proposed PF βe(t) = 1
tαe+ςe+βf,e αe∈ ℜ>0, ςe∈ ℜ>0, βf,e ∈ ℜ>0
Fig. 2. Prescribed performance. (a) the proposed simplified prescribed performance (14); (b) the fragility defect, that is, actuator saturation causes control
singularity; (c) the proposed FPP that avoids the fragility defect.
is the jth input, ψi∈ ℜ, i = 1,2,· · · , m is the output
of the local model for the rule Ri,Ψi, i = 1,2,· · · , m is
the weight coefficient between the inputs and the ith output,
ϑij , i = 1,2,· · · , m;j= 1,2,· · · , n is a wavelet family
that is defined as: ϑij (xj) = ϑxj−cij
δij , δij = 0, where
cij , i = 1,2,· · · , m;j= 1,2,· · · , n is the dilation parameter
and δij , i = 1,2,· · · , m;j= 1,2,· · · , n is the translation
parameter. Then, from a single mother wavelet function by
the dilations and the translations (c, δ), the output of FWNN
can be given by
ψ(x) =
m
P
i=1
µi(x)ψi
m
P
i=1
µi(x)
with µi(x) =
n
Q
j=1
Ai
j(xj), i = 1,2,· · · , m;j= 1,2,· · · , n,
where mmeans the number of fuzzy rules, ndenotes the
dimension of input vector, and Ai
j(xj), i = 1,2,· · · , m;j=
1,2,· · · , n is the Gaussian membership function
Ai
j(xj) = exp
− xj−bi
j
σi
j!2
where bi
j, i = 1,2,· · · , m;j= 1,2,· · · , n and σi
j, i =
1,2,· · · , m;j= 1,2,· · · , n are the center and the half-width
of Ai
j(xj), i = 1,2,· · · , m;j= 1,2,· · · , n, respectively. In
this article, we use the following Mexican hat wavelet function
ϑ(x) = 1
p|δ|1−2x2exp −1
2x2.
For any unknown but continuous function ψ(x), by utilizing
the singleton fuzzifier, the product inference and the weighted
average defuzzifier, there exists an optimal weight vector Ψ∈
ℜNsuch that [21], [27]
ψ(x)=ΨTϑ(x) + eψ
where x∈ ℜmis the input vector, ϑ(x)∈ ℜNis the basis
function vector, and the approximation error eψ∈ ℜ satisfies
|eψ| ≤ ¯eψ∈ ℜ≥0[21], [27].
In what follows, the FWNN will be used to approximate the
unknown dynamics of the velocity subsystem and the altitude
subsystem.
III. MAI N RE SU LTS
A. Velocity control protocol
In this subsection, we will devise a PPC protocol based
on fuzzy-neural approximation for (1) under the actuator
saturation (8), such that the velocity tracking error ˜
V=V−Vc
satisfies the following FPP
−βV(t)−fV(t)<˜
V < βV(t) + fV(t)(17)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 6
where the simplified PF βV(t)∈ ℜ>0and the FAT fV(t)∈
ℜ≥0are designed as
βV(t) = 1
tαV+ςV
+βf,V (18)
fV(t) = lV,Φ,2tanh (abs(ιV ,Φ)) (19)
where αV∈ ℜ>0, ςV∈ ℜ>0,1
ςV> βf,V ∈ ℜ>0, lV,Φ,2∈ ℜ>0
are user defined constants, and ιV,Φ∈ ℜ is the state of the
BIBS system (22).
To transform the “constrained” condition (17) into an “un-
constrained” one, we define the transformed error LV(t)∈ ℜ
.
LV(t) = 1
2ln ˜
V / (βV(t) + fV(t)) + 1
1−˜
V / (βV(t) + fV(t)) !.(20)
We will use LV(t)to devise the feedback control protocol
for (1) such that the velocity subsystem is stable and LV(t)
is bounded, which guarantees the FPP (17) on ˜
V.
˙
LV(t) = σV
ψV+ Φ −˙
Vc−X
L,V
(21)
with
σV=1
2βV(t)+2fV(t)
×1
˜
V / (βV(t) + fV(t)) + 1
+1
1−˜
V / (βV(t) + fV(t)) >0
˙
βV(t) = −αVtαV−1(tαV+ςV)−2
X
L,V
:= ˙
βV(t) + ˙
fV(t)
βV(t) + fV(t)˜
V
˙
fV(t) = lV,Φ,21−tanh2(abs(ιV ,Φ))
×sign(ιV,Φ) ˙ιV,Φ.
To handle the saturation on Φ, by utilizing Theorem 1, we
develop the following BIBS system
˙ιV ,Φ=σV−lV,Φ,1ιV ,Φ
abs(ιV,Φ)+∆ιV
+ Φ −Φd(22)
where lV,Φ,1∈ ℜ>0and ∆ιV∈(0,1) are constants, and the
state ιV,Φ∈ ℜ is used to modify LV(t).
˜
LV(t) = LV(t)−ιV,Φ.(23)
Utilizing (21) and (22), ˙
˜
LV(t)is given by
˙
˜
LV(t) = ˙
LV(t)−˙ιV ,Φ
=σV
ψV−˙
Vc−X
L,V
+lV,Φ,1
abs(ιV,Φ)−1
abs(ιV,Φ)+1 + Φd.(24)
The unknown function ψVin (24) is approximated by one
FWNN [21], [27]
ψV= ΨT
VϑV(V) + eψ,V (25)
where Vis the input, ΨV∈ ℜN1is the weight vector,
ϑV(V)=[ϑV,1(V), ϑV ,2(V),· · · , ϑV ,N1(V)]T∈ ℜN1is
the basis function vector, and the approximation error eψ,V
satisfies |eψ,V | ≤ ¯eψ,V ∈ ℜ≥0[21], [27].
We define Φdas
Φd=−kV,1˜
LV(t)−kV,2Zt
0
˜
LV(τ)dτ
−lV,Φ,1
ιV,Φ
abs(ιV,Φ)+∆ιV
+X
L,V
−1
2˜
LV(t) ˆϖVϑT
V(V)ϑV(V) + ˙
Vc(26)
where kV,1∈ ℜ>0and kV ,2∈ ℜ>0are user-defined constants,
and ˆϖVdenotes the estimate of ϖV:= ∥ΨV∥2, which is
updated by
˙
ˆϖV=σVλV
2˜
L2
V(t)ϑT
V(V)ϑV(V)−2 ˆϖV(27)
with λV∈ ℜ>0.
Next, I will analyze the stability of the velocity subsystem.
By using (24)-(26), we obtain
˙
˜
LV(t) = σV−kV,1˜
LV(t)−kV,2Zt
0
˜
LV(τ)dτ +eψ,V
+ ΨT
VϑV(V)−˜
LV(t) ˆϖVϑT
V(V)ϑV(V)
2#.(28)
Define ˜ϖV= ˆϖV−ϖV, and select the Lyapunov function
WV
WV=˜
L2
V(t)
2+σVkV,2
2Zt
0
˜
LV(τ)dτ2
+˜ϖ2
V
2λV
.(29)
Employing (27)-(29), we get ˙
WV
˙
WV=˜
LV(t)˙
˜
LV(t) + σVkV,2˜
LV(t)Zt
0
˜
LV(τ)dτ
+˜ϖV˙
˜ϖV
λV
=σVh−kV,1˜
L2
V(t) + eψ,V ˜
LV(t)
+˜
LV(t)ΨT
VϑV(V)−2
λV
˜ϖVˆϖV
−1
2˜
L2
V(t)ϖVϑT
V(V)ϑV(V).(30)
Note the fact that
2 ˜ϖVˆϖV≥˜ϖ2
V−ϖ2
V
2˜
LV(t)ΨT
VϑV(V)≤˜
L2
V(t)ϖVϑT
V(V)ϑV(V)+1
2eψ,V ˜
LV(t)≤¯e2
ψ,V ˜
L2
V(t)+1.
Then (30) becomes
˙
WV≤σV−kV,1−1
2¯e2
ψ,V ˜
L2
V(t)−1
λV
˜ϖ2
V
+1
λV
ϖ2
V+ 1.(31)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 7
Define compact sets:
Ξ˜
LV(t)=(˜
LV(t)˜
LV(t)≤sϖ2
V/λV+ 1
kV,1−1
2¯e2
ψ,V )
Ξ˜ϖV=˜ϖV|˜ϖV| ≤ qϖ2
V+λV.
If ˜
LV(t)/∈Ξ˜
LV(t)or ˜ϖV/∈Ξ˜ϖVthen ˙
WV<0. We further
have ˜
LV(t)→Ξ˜
LV(t)and ˜ϖV→Ξ˜ϖVas t→ ∞. This
indicates the boundedness of ˜
LV(t)and ˜ϖV. Moreover, the
input of the BIBS system (22) (i.e., Φ−Φd) also is bounded
since Φdis bounded. Thereby the state ιV,Φof (22) is bounded,
and the transformed error LV(t)also is bounded. We finally
know that the spurred FPP (17) can be achieved.
B. Altitude control protocol
In this subsection, we will develop a PPC protocol based
on fuzzy-neural approximation for (2)-(5) under the actuator
saturation (9), such that the altitude tracking error ˜
h=h−hc
satisfies the following FPP
−βh(t)−fh(t)<˜
h < βh(t) + fh(t)(32)
where the simplified PF βh(t)∈ ℜ>0and the FAT fh(t)∈
ℜ≥0are designed as
βh(t) = 1
tαh+ςh
+βf,h (33)
fh(t) = lh,δe,2tanh (abs(ιh,δe,1) + abs(ιh,δe,2)
+ abs(ιh,δe,3)) (34)
where αh∈ ℜ>0, ςh∈ ℜ>0,1
ςh> βf,h ∈ ℜ>0, lh,δe,2∈ ℜ>0
are design parameters, ιh,δe,1∈ ℜ, ιh,δe,2∈ ℜ, ιh,δe,3∈ ℜ are
the states of the BIBS system (39).
Define the transformed error
Lh(t) = 1
2ln ˜
h/ (βh(t) + fh(t)) + 1
1−˜
h/ (βh(t) + fh(t))!.(35)
Invoking (2), ˙
Lh(t)is given by
˙
Lh(t) = σh
Vsin γ−˙
hc−X
L,h
(36)
with
σh=1
2βh(t)+2fh(t)
×1
˜
h/ (βh(t) + fh(t)) + 1
+1
1−˜
h/ (βh(t) + fh(t))>0
˙
βh(t) = −αhtαh−1(tαh+ςh)−2
X
L,h
:= ˙
βh(t) + ˙
fh(t)
βh(t) + fh(t)˜
h
˙
fh(t) = lh,δe,21−tanh2(abs(ιh,δe,1) + abs(ιh,δe,2)
+ abs(ιh,δe,3))] ×[sign(ιh,δe,1)˙ιh,δe,1
+ sign(ιh,δe,2)˙ιh,δe,2+ sign(ιh,δe,3) ˙ιh,δe,3].
From (36), we design γc(i.e., the reference command of γ
) as
γc= arcsin −kh,1Lh(t)−kh,2Rt
0Lh(τ)dτ
V
+
˙
hc+PL,h
V!(37)
with kh,1∈ ℜ>0and kh,2∈ ℜ>0.
If γ→γc, we get kh,2σhRt
0Lh(τ)dτ +kh,1σhLh(t) +
˙
Lh(t)=0. Noting that kh,1σhand kh,2σhare positive, thus
Lhis bounded and lim
t→∞ Lh= 0. This guarantees the spurred
FPP (32).
The next design objective is to devise a suitable controller
δefor (3)-(5) such that γ→γc. For the model (3)-(5) of
strict feed-back formulation, the back-stepping is a common
selection to design control protocols [2], [6], [24], which
however results in the complicatedly recursive procedure. In
this article, inspired by [5], we equivalently transform (3)-(5)
to the following form
˙sγ=sθ
˙sθ=sQ
˙sQ=ψH+δe(38)
where sγ=γ∈ ℜ, sθ∈ ℜ and sQ∈ ℜ are system states, and
ψHis an unknown but continuous function.
Remark 3. Based on the model transformation, the unknown
functions ψγand ψQare normalized as the lumped unknown
one ψHsuch that only one FWNN is needed, which reduces
the computational burden. Furthermore, unlike the existing
back-stepping-based control methodologies [2], [6], [24], in
this study, the recursive procedure is avoided and the structure
complexity is reduced.
To compensate the saturation on δe, we devise the following
BIBS system
˙ιh,δe,1=ιh,δe,2
˙ιh,δe,2=ιh,δe,3
˙ιh,δe,3=−lh,δe,11
ιh,δe,1
abs(ιh,δe,1)+∆ιh1
−lh,δe,12
ιh,δe,2
abs(ιh,δe,2)+∆ιh2
−lh,δe,13
ιh,δe,3
abs(ιh,δe,3)+∆ιh3
+δe−δe,d (39)
where ιh,δe,1∈ ℜ, ιh,δe,2∈ ℜ, ιh,δe,3∈ ℜ are system states,
and lh,δe,11 ∈ ℜ>0, lh,δe,12 ∈ ℜ>0, lh,δe,13 ∈ ℜ>0,∆ιh1∈
(0,1),∆ιh2∈(0,1),∆ιh3∈(0,1) are design parameters.
Define
eγ=γ−γc=sγ−γc.(40)
By using ιh,δe,1to modify eγ, we obtain
˜eγ=eγ−ιh,δe,1.(41)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 8
Invoking (38)-(40), we have
˙
˜eγ= ˙eγ−˙ιh,δe,1= ˙eγ−ιh,δe,2
˜e(2)
γ=e(2)
γ−˙ιh,δe,2=e(2)
γ−ιh,δe,3
˜e(3)
γ=e(3)
γ−˙ιh,δe,3=s(3)
γ−γ(3)
c−˙ιh,δe,3
=ψH+lh,δe,11
ιh,δe,1
abs(ιh,δe,1)+∆ιh1
+lh,δe,12
ιh,δe,2
abs(ιh,δe,2)+∆ιh2
+lh,δe,13
ιh,δe,3
abs(ιh,δe,3)+∆ιh3
+δe,d −γ(3)
c.
Define
Eγ=d
dt +µγ3Zt
0
˜eγdτ
= ˜e(2)
γ+ 3µγ˙
˜eγ+ 3µ2
γ˜eγ+µ3
γZt
0
˜eγdτ (42)
with µγ∈ ℜ>0.
Employing (38)-(41), ˙
Eγis derived as
˙
Eγ= ˜e(3)
γ+ 3µγ˜e(2)
γ+ 3µ2
γ˙
˜eγ+µ3
γ˜eγ
=ψH+δe,d +X
γ
(43)
with
X
γ
:= lh,δe,11ιh,δe,1
abs(ιh,δe,1)+∆ιh1
+lh,δe,12ιh,δe,2
abs(ιh,δe,2)+∆ιh2
+lh,δe,13ιh,δe,3
abs(ιh,δe,3)+∆ιh3
+ 3µγ˜e(2)
γ+ 3µ2
γ˙
˜eγ
+µ3
γ˜eγ−γ(3)
c.
The unknown function ψHin (43) is approximated by one
FWNN [21], [27]
ψH= ΨT
HϑH(xH) + eψ,H (44)
where xH= [V, γ , θ, Q]is the input, ΨH∈ ℜN2is the weight
vector, ϑH(xH)=[ϑh,1(xH), ϑh,2(xH),· · · , ϑh,N2(xH)]T∈
ℜN2is the basis function vector, and the approximation error
eψ,H satisfies |eψ,H | ≤ ¯eψ,H ∈ ℜ≥0[21], [27].
We design δe,d as
δe,d =−kγEγ−1
2EγˆϖHϑT
H(xH)ϑH(xH)−X
γ
(45)
where kγ∈ ℜ>0is design parameter, and ˆϖHmeans the
estimate of ϖH:= ∥ΨH∥2, which is regulated by
˙
ˆϖH=1
2λHEγϑT
H(xH)ϑH(xH)−2 ˆϖH(46)
with λH∈ ℜ>0.
Next, I will analyze the stability of the altitude subsystem.
Substituting (44) and (45) into (43), it leads to
˙
Eγ=−kγEγ+ ΨT
HϑH(xH) + eψ,H
−1
2EγˆϖHϑT
H(xH)ϑH(xH).(47)
Define ˜ϖH= ˆϖH−ϖHand choose the Lyapunov function
Wh=1
2E2
γ+˜ϖ2
H
2λH
.(48)
Utilizing (46) and (47), ˙
Whis given by
˙
Wh=Eγ˙
Eγ+˜ϖH˙
˜ϖH
λH
=−kγE2
γ+EγΨT
HϑH(xH) + Eγeψ,H
−1
2E2
γϖHϑT
H(xH)ϑH(xH)−2 ˜ϖHˆϖH
λH
.(49)
Note that
2 ˜ϖHˆϖH≥˜ϖ2
H−ϖ2
H
EγΨT
HϑH(xH)≤1
2E2
γϖHϑT
H(xH)ϑH(xH) + 1
2
Eγeψ,H ≤1
2E2
γ¯e2
ψ,H +1
2.
Then (49) becomes
˙
Wh≤ − kγ−1
2¯e2
ψ,H E2
γ−˜ϖ2
H
λH
+1+ϖ2
H
λH
.(50)
Define compact sets:
ΞEγ=(Eγ
|Eγ| ≤ s1 + ϖ2
H/λH
kγ−¯e2
ψ,H /2)
Ξ˜ϖH=˜ϖH|˜ϖH| ≤ qλH+ϖ2
H.
If Eγ/∈ΞEγor ˜ϖH/∈Ξ˜ϖH, then ˙
Wh<0. Thus, we have
Eγ→ΞEγand ˜ϖH→Ξ˜ϖHas t→ ∞. This infers that Eh
and ˜
ϑhare bounded, and γ→γc. As a result, the spurred FPP
(32) can be guaranteed.
This is the end of the control design, and the structure of
the proposed control system is clearly shown in Fig. 3.
Remark 4. The existing PPC methodologies are fragile to
actuator saturation that will lead to that the tracking errors ˜
V
and ˜
htend to the upper/lower bounds of constraint envelopes,
resulting in the control singularity problem. In this article,
two novel FATs fV(t)∈ ℜ≥0and fh(t)∈ ℜ≥0are devised to
modify the constraint envelopes (17) and (32) for the sake of
tackling the fragility problem. When the actuators Φand δeare
saturated, the developed FATs fV(t)∈ ℜ≥0and fh(t)∈ ℜ≥0
are able to increase the upper envelope of FPP (17) and (32),
and meanwhile decrease the lower envelope of FPP (17) and
(32). As a result, the well-known fragility problem associated
with the existing PPC is tackled in this study.
IV. NUMERICAL SIMULATION
To validate the effectiveness, in this section, the addressed
control protocols (26) and (45) with adaptive laws (27), (46),
and BIBS systems (22) and (39) are applied to the vehicle
mode (1)-(7) via numerical simulation utilizing the Matlab
software. The values of design parameters are chosen as: αV=
1.5,ςV= 0.5,βf,V = 0.6,lV,Φ,2= 2.5,lV,Φ,1= 1.5,∆ιV=
0.8,kV,1= 0.2,kV ,2= 0.8,λV= 0.05,αh= 1.8,ςh= 0.4,
βf,h = 0.5,lh,δe,2= 1000,kh,1= 2,kh,2= 0.8,lh,δe,11 =
0.5,lh,δe,12 = 1,lh,δe,13 = 1,∆ιh1= ∆ιh2= ∆ιh3= 0.8,
µγ= 5,kγ= 40,λH= 0.05.
We consider the following three examples.
Example 1. In this example, we test the prescribed perfor-
mance of the exploited control protocol that is compared with
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 9
c
V
V
V
()
V
Lt
,V
()
V
Lt
d
U
L
d
ˆV
c
h
h
h
c
V
V
E
e
,
ed
,eU
,eL
,eed
,V
,,1 ,,2 ,,3
,,
eee
hhh
ˆ
H
,,1
e
h
()
h
Lt
Q
Fig. 3. Structure of the proposed control method.
Velocity [ft/s]
Fig. 4. Velocity tracking performance in Example 1.
0 1020304050
Time [s]
-4
-2
0
2
4
0 1020304050
8.5
8.55 104
h by proposed control
hc
h by MNBC
Fig. 5. Altitude tracking performance in Example 1.
a modified neural back-stepping control (MNBC) addressed in
[29], while we assume that there is no actuator saturation.
The simulation results can be seen in Figs. 4-7. Figs.
4 and 5 infer that the proposed control is able to limit
Φ
δe [deg]
Fig. 6. Control inputs in Example 1.
0 1020304050
0
0.05 proposed control
0 1020304050
Time [s]
0
2
4
610-7
proposed control
Fig. 7. The reposes of ˆϖVand ˆϖHin Example 1.
0 1020304050
7700
7800
Velocity [ft/s]
V
Vc
0 1020304050
Time [s]
-2
-1
0
1
2
12 14 16 18
-1
-0.5
0
Fig. 8. Velocity tracking performance in Example 2.
tracking errors within the constraint envelopes such that the
spurred prescribed performance is guaranteed for tracking
errors. However, for comparison, when utilizing the MNBC,
both the velocity tracking error and the altitude tracking error
exceed the constraint envelopes. As a result, in comparison
with the MNBC, the proposed method can guarantee tracking
errors with better prescribed performance. Moreover, Fig. 6
shows that the control inputs of the proposed control have
no high frequency chattering, though the control inputs of the
MNBC are smoother. Finally, Fig. 7 reveals that ˆϖVand ˆϖH
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 10
0 1020304050
8.5
8.55
Altitude [ft]
104
h
hc
0 1020304050
Time [s]
-4
-2
0
2
4
Fig. 9. Altitude tracking performance in Example 2.
0 1020304050
0
0.2
0.4
0.6
Φ
0 1020304050
Time [s]
0
10
20
δe [
d
eg]
10 15 20
0.4
0.6
0.8
14 14.5 15 15.5 16
14
16
18
Fig. 10. Control inputs in Example 2.
0 1020304050
0
0.05
0.1
0 1020304050
Time [s]
0
2
4
610-7
Fig. 11. The reposes of ˆϖVand ˆϖHin Example 2.
are bounded.
Example 2. In this example, we test the fragility-free perfor-
mance of the proposed controller by considering the actuator
saturation: Φ∈[0.05,0.8] and δe∈[−18.4 deg,18.4 deg] .
Moreover, the proposed control is compared with one existing
PPC [23], [28] whose PF has no the FAT.
The simulation results, presented in Figs. 8-13, show that the
proposed control guarantees tracking errors with satisfactory
prescribed performance in the present of actuator saturation. In
can be seen from Figs. 8-10 that when the actuator saturation
PF of [23], [28]
PF of [23], [28]
control singularity
Fig. 12. Velocity tracking performance by using the PF of [23], [28] in
Example 2.
PF of [23], [28]
PF of [23], [28]
control singularity
Fig. 13. Altitude tracking performance by using the PF of [23], [28] in
Example 2.
02468101214
Time [s]
7700
7720
7740
7760 V
Vc
02468101214
Time [s]
-2
0
2
Fig. 14. Velocity tracking performance in Example 3.
causes the increase in error (See Fig. 10), the developed FATs
fV(t)∈ ℜ≥0and fh(t)∈ ℜ≥0are capably of re-adjusting
the constraint envelopes (17) and (32). As a result, both
the velocity tracking error and the altitude tracking error are
always within the constraint envelopes (17) and (32). However,
under the same condition, there exists the control singularity
IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2022, DOI: 10.1109/TFUZZ.2022.3217378 11
Altitude [ft]
Fig. 15. Altitude tracking performance in Example 3.
Φ
δe [deg]
Fig. 16. Control inputs in Example 3.
Fig. 17. The reposes of ˆϖVand ˆϖHin Example 3.
problem when using the PFs developed in [23], [28] since the
tracking errors exceed the constraint envelopes, as depicted in
Figs. 12 and 13. This proves that the exploited controllers
are able to predict the error fluctuations and then actively
adjust the constraint envelopes, such that the tracking errors
are always kept within the constraint envelopes to satisfy the
prescribed performance. Meanwhile, the well-known fragility
problem associated with the existing PPC is avoided. Finally,
the responses of ˆϖVand ˆϖHare shown in Fig. 11.
Example 3. In this example, the actuator saturation condition
is the same as Example 2, that is, Φ∈[0.05,0.8] and
δe∈[−18.4 deg,18.4 deg]. In contrast, the developed BIBS
systems (22) and (39) are removed.
The obtained simulation results, presented in Figs. 14-17,
reveal that the proposed control system becomes unstable
owing to actuator saturation if the developed BIBS systems
(22) and (39) are removed. When the velocity control input Φ
is saturated (see Fig. 16), we note that the velocity tracking
error ˜
Vincreases and tends to reach the lower bound of
constraint envelope −βV(t)−fV(t)(See Fig. 14), which
causes the destabilization of the closed-loop system. Besides,
Fig. 17 shows that the adaptive variable ˆϖVtends to diverge.
To sum up, the simulation results obtained in Examples 2 and
3 fully verify the effectiveness of the developed BIBS systems
(22) and (39) in dealing with the actuator saturation problem.
V. CONCLUSIONS
This article considers the challenging fragility problem of
the existing PPC approaches devised for WVs. A new type of
simple PFs are designed and further modified by proposing
novel flexible adjustment terms. On this basis, adaptive fuzzy-
neural control protocols are addressed for constrained WVs,
capable of guaranteeing tracking errors with fragility-free
prescribed performance, and meanwhile avoiding the fragility
defect associate with the existing PPC. Then, to reduce the
computation burden caused by fuzzy-neural approximation,
and also to guarantee satisfactory real-time performance, low-
computational fuzzy neural approximation schemes without
recursive design of back-stepping are used, yielding a concise
control framework with low-complexity structure. Moreover,
BIBS systems are developed to generate bounded compensa-
tions on the actuator saturation, while the spurred prescribed
performance still can be guaranteed in the presence of con-
strained inputs. Finally, compared simulation results show that
the proposed controller is superior to some existing ones.
In our future study, we will further consider both unknown
dynamics and external disturbances.
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Xiangwei Bu received his B.S., M.S. and Ph.D.
degrees in Control Science and Engineering from
Air Force Engineering University in 2010, 2012
and 2016, respectively. He is currently an associate
professor in Air and Missile Defense College, Air
Force Engineering University.
His research interests include advanced flight con-
trol. Dr. Bu is the Associate Editor of Measurement
and Control, and also is the Editorial Board Member
of Advances in Mechanical Engineering.
Baoxu Jiang received his B.S. degrees from
Changchun University of Science and Technology
in 2020. He is currently a graduate student in Air
and Missile Defense College, Air Force Engineering
University.
His research interests include advanced control
with their applications to flight vehicles.
Humin Lei received the M.S. and Ph.D. de-
grees from Northwestern Polytechnical University,
Xi’an, China, in 1989 and 1999, respectively. He is
currently a Professor and a Supervisor for doctors
with Air and Missile Defense College, Air Force
Engineering University, Xi’an, China.
His current research interests include advanced
guidance law design, hypersonic vehicle controller
design, and hypersonic interception strategy.
