Noname manuscript No.
(will be inserted by the editor)
Finite-Time Performance Guaranteed Event-Triggered
Adaptive Control for Nonlinear Systems with Unknown
Control Direction
Min Wang · Lixue Wang
Received: date / Accepted: date
Abstract This paper studies the issue of finite-time
performance guaranteed event-triggered (ET) adaptive
neural tracking control for strict-feedback nonlinear sys-
tems with unknown control direction. A novel finite-
time performance function is first constructed to de-
scribe the prescribed tracking performance, and then
a new lemma is given to show the differentiability and
boundedness for the performance function, which is im-
portant for the verification of the closed-loop stability.
Furthermore, with the help of the error transformation
technique, the origin constrained tracking error is trans-
formed into an equivalent unconstrained one. By utiliz-
ing the first-order sliding mode differentiator, the issue
of “explosion of complexity” caused by the backstep-
ping design is adequately addressed. Subsequently, an
ingenious adaptive updated law is given to co-design the
controller and the ET mechanism by the combination of
the Nussbaum-type function, thus effectively handling
the influences of the measurement error resulted from
the ET mechanism and the challenge of the controller
design caused by the unknown control direction. The
presented event-triggered control scheme can not only
guarantee the prescribed tracking performance, but al-
so alleviate the communication burden simultaneously.
Finally, numerical and practical examples are provid-
ed to demonstrate the validity of the proposed control
strategy.
M. Wang, L. Wang
School of Automation Science and Engineering, Guangdong
Provincial Key Laboratory of Technique and Equipment for
Macromolecular Advanced Manufacturing, South China Uni-
versity of Technology, Guangzhou, 510641, China.
Tel.: +8620-87111258
Fax: +8620-87111258
E-mail: auwangmin@scut.edu.cn
Keywords Event-triggered control · Finite-time
prescribed performancel · First-order sliding mode
differentiator · Strict-feedback systems · Unknown
control direction
1 Introduction
In the past decades, strict-feedback nonlinear systems
(SFNSs), as a special kind of nonlinear systems, have
evoked widespread attention because of their power-
ful capability to model various kinds of practical sys-
tems, such as chemical stirred tank reactor [1], flexible
joint robotic system [2], hypersonic flight vehicles [3]
and so on. As a consequence, for the control problem-
s of SFNSs, many scholars have carried out in-depth
research owing to the huge needs in engineering appli-
cations. The adaptive backstepping method [4], as a
breakthrough method in the field of nonlinear control,
has become an effective way to deal with the control
issues of SFNSs. With the aid of neural network (NN)
[5] or fuzzy logic approximator, the adaptive backstep-
ping method has been widely used to control the SFNSs
with unknown nonlinear dynamics [6–9]. However, the
results mentioned above exist the issue of “explosion
of complexity” resulted from repeated derivations of
the virtual controller in backstepping design. To han-
dle such a problem, many useful approaches have been
developed. For example, the dynamic surface control
(DSC) method was employed to estimate the deriva-
tive of the virtual controller in [10,11]. The authors in
[12] use a more efficient technique namely first-order
sliding mode differentiator (FOSMD) to avert tedious
calculations.
Noting that in some practical systems, the track-
ing error often needs to meet some specified transient
2 Min Wang, Lixue Wang
and steady state performance indicators, such as smal-
l overshoot, fast convergence speed and small steady-
state error, so as to ensure the control performance of
the system. In view of such situation, the authors in
[13] firstly presented a performance function transfor-
mation method, whose basic idea was to convert the
constrained tracking error into the unconstrained one.
Subsequently, such a method was not only applied to
more general systems with miscellaneous engineering-
oriented phenomena, such as unmodeled dynamics [14],
input saturation [15], actuator failures [16] and input
quantization [17], but also widely employed to a great
deal of practical systems [18,19]. Unfortunately, the
performance functions in the aforementioned works are
not concerned about finite time convergence, which lim-
its their engineering applications with high-accuracy
control. The prescribed finite time convergence issue
is extremely difficult for controller design of nonlinear
systems. More recently, this issue were gained some con-
cern, and a few preliminary and valuable results were
developed in [20–24]. Specifically, the authors in [20]
constructed a finite-time performance function, whose
convergence time can be set arbitrarily. This result [20]
has been extended to a finite-time fuzzy tracking con-
trol scheme for SFNSs with dynamic disturbances [22].
And then, the finite-time consensus tracking control
was handled in [24] for multi agent systems with pre-
scribed performance and mismatched uncertainties.
It is worth pointing out that the results mentioned
above can only be obtained when the control direction
of the system is known in advance. Once the control di-
rection is unknown, these control schemes are no longer
feasible. In view of such a case, the authors in [25] first-
ly developed a Nussbaum-type function (NTF), which
effectively handled the difficulty in controller design
caused by the unknown control direction. Subsequent-
ly, on the basis of NTF, a great number of interesting
works have been reported. To mention a few, in com-
bination with NTF and Barrier Lyapunov function, an
adaptive tracking control strategy has been construct-
ed for a class of state-constrained SFNSs with unknown
control direction [26]. Based on the NTF and fuzzy log-
ic approximator, the DSC fuzzy controller presented in
[27] has successfully guaranteed the stability of uncer-
tain non-strict-feedback systems with unknown virtual
control coefficients. These NTF-based results does not
concern two hot directions: the prescribed finite time
convergence and the limited network resources.
Recently, networked control has been developed rapid-
ly due to its peculiarities of low cost, good flexibility,
reliable operation, convenient installation, etc. Howev-
er, the network bandwidth is limited, which inevitably
brings some problems including transmission delay [28],
packet disorder [29], and so on. To deal with such prob-
lems, the authors in [30] presented an event-triggered
control (ETC) approach, which efficaciously economizes
the network resource. On the basis of the ETC, a se-
ries of valuable results have been reported [31–37]. For
instance, a relative threshold ETC method has been
developed for SFNSs with unknown parameters in [33],
which achieves a good balance between the system p er-
formance and the limited network resources. In combi-
nation with the relative threshold strategy, some fruit-
ful ETC schemes have been presented for more gener-
al nonlinear systems with prescribed performance [34],
input saturation [36] and unmeasured state [37]. Un-
fortunately, these aforementioned ETC methods can
not be directly extended to prescribed performance-
guaranteed SFNSs with unknown control direction. The
main reason lies in the unknown control direction is cou-
pled in the compensation process of the measurement
error between the controller and the actuator. In this
case, it becomes significantly challenging to construct a
suitable ETC scheme to weaken the effect of the mea-
surement error on system stability. As a consequence,
it is significant and necessary to investigate the ETC
for SFNSs subject to both prescribed performance and
unknown control direction.
According to the aforementioned discussion so far,
the main objective of this paper is to construct a finite-
time performance guaranteed ETC scheme for SFNSs
with unknown control direction. Firstly, the finite-time
performance function is constructed to guarantee the
tracking performance constraint by the aid of the error
transformation approach. Secondly, the FOSMD is em-
bedded in the backstepping procedure to cope with the
issue of “explosion of complexity”, and then an inge-
nious adaptive law is given to facilitate the co-design of
controller and ET mechanism. Finally, a finite-time per-
formance guaranteed ETC scheme is developed based
on the novel adaptive law and the NTF, which guaran-
tees the prescribed tracking performance, alleviates the
communication burden and compensates the measure-
ment error at the same time. The main contributions
are listed as follows:
1) A novel finite-time performance function is devel-
oped such that the prescribed tracking performance
is achieved in a predetermined finite time instead of
the infinite time. Moreover, a new lemma is derived
to show the differentiability and boundedness of the
constructed performance function, which plays an
important role for the system stability;
2) A novel adaptive law is given to estimate the up-
per bound of the actual control gain, and then the
controller and the ET mechanism are co-designed
Title Suppressed Due to Excessive Length 3
to compensate successfully the measurement error
resulted from the ET mechanism;
3) In combination with such an adaptive law and the
hyperbolic tangent function, a novel ET actuator
under the relative threshold strategy is proposed,
thereby achieving the prescribed finite-time track-
ing performance and saving the communication re-
source.
2 Problem formulation and preliminaries
2.1 System description
This paper considers a class of SFNSs as follows:
˙x
i
= f
i
(x
i
) + g
i
(x
i
)x
i+1
, i = 1, 2, . . . , n − 1
˙x
n
= f
n
(x
n
) + g
n
(x
n
)u
y = x
1
(1)
where x
i
= [x
1
, x
2
, . . . , x
i
]
T
∈ R
i
(i = 1, 2, . . . , n),
u ∈ R and y ∈ R denote, respectively, the state vector,
the system control input and output. f
i
(x
i
) and g
i
(x
i
)
stand for the unknown smooth nonlinear functions.
Assumption 1 [38] The control coefficients g
i
(x
i
), i =
1, 2, . . . , n with unknown signs and g
i
(x
i
) = 0. More-
over, g
i
(x
i
) satisfy g
i1
≤ |g
i
(x
i
)| ≤ g
i2
, where g
i1
and
g
i2
are unknown positive constants.
Assumption 2 The reference trajectory y
d
and its time
derivatives ˙y
d
and ¨y
d
are bounded.
2.2 Radial basis function NNs
It has been shown in [39] that the radial basis func-
tion (RBF) NN has a powerful ability to approximate
any unknown smooth nonlinear function T (Z) over a
compact set Ω
Z
as
T (Z) = W
∗T
S(Z) + δ(Z)
where Z ∈ Ω
Z
⊂ R
o
sands for the input vector of NN,
S(Z) = [S
1
(Z), . . . , S
l
(Z)]
T
∈ R
l
denotes the Gaus-
sian basis function vector, where S
i
(Z) = exp[−(Z −
Θ
i
)
T
(Z − Θ
i
)/△
i
] with Θ
i
= [Θ
i1
, . . . , Θ
io
]
T
and △
i
are the center and width of NN, respectively. W
∗
∈ R
l
represents the optimal neural weight vector with l > 1
being the number of neural node, δ(Z) denotes the ap-
proximation error satisfying |δ(Z)| ≤
¯
δ, where
¯
δ > 0 is
an arbitrarily small constant.
2.3 Novel finite-time performance function
A novel finite-time performance function is constructed
as follows:
ϕ(t) =
− tanh
ϕ
1
+
t
T
c
−t
+ ϕ
2
+ 1, 0 ≤ t < T
c
ϕ
2
, t > T
c
(2)
where ϕ
1
, ϕ
2
and T
c
are positive constants, tanh(·) rep-
resents the hyperbolic tangent function.
Lemma 1 ϕ(t) and
˙
ϕ(t) are continuously differentiable
and bounded on [0, +∞) and
¨
ϕ(t) is continuous and
bounded on [0, +∞).
Proof : See the Appendix.
Remark 1 Notice that the finite-time performance con-
trol problem has been considered in the existing re-
sults in [20–22], [24]. These results mentioned above
require some restrictions. For example, the initial con-
dition ϕ(0) is required to satisfy ϕ(0) ≤ 1 in [20–22];
and ϕ(t) depends on the order n of the controlled sys-
tem [24], which makes the computational complexity of
ϕ(t) greatly increase for high-order nonlinear systems.
Compared with the existing works [20–22], [24], it is ob-
vious from (2) that the finite-time performance function
developed in this paper is easy-to-implement due to the
mild initial condition ϕ(0) = − tanh(ϕ
1
) + ϕ
2
+ 1 > 0
and the independence of system order n.
In this paper, the tracking error e
1
= y − y
d
should
remain within the predefined performance constraint as
follows
−ς
1
ϕ(t) < e
1
(t) < ς
2
ϕ(t) (3)
where ς
1
and ς
2
are both positive design parameters.
It can be concluded from (2) and (3) that −ς
1
ϕ(0)
and ς
2
ϕ(0) with ϕ(0) = − tanh(ϕ
1
) + ϕ
2
+ 1 denote,
respectively, the minimum value of the transient un-
dershoot and the maximum value of the transient over-
shoot of e
1
. −ς
1
ϕ
2
and ς
2
ϕ
2
represent the low bound
and upper bound of the steady-state tracking error e
1
,
respectively. Besides, T
c
stands for the time when track-
ing error e
1
decays to the steady-state value ϕ
2
.
2.4 Useful Definition and Lemmas
Definition 1 [25] Any smooth even function N (φ) can
be called as a function of Nussbaum-type when it satis-
fies lim
s→∞
sup
1
s
s
0
N(φ)dφ = +∞ and lim
s→∞
inf
1
s
s
0
N(φ)dφ = −∞.
4 Min Wang, Lixue Wang
Lemma 2 [25] Let V (t) ≥ 0 and φ(t) be smooth func-
tions defined on [0, t
f
), and N(φ) = φ
2
cos(φ). For
∀t ∈ [0, t
f
), if
V (t) ≤ r
1
+ e
−r
2
t
t
0
[g(x(τ))N(φ) + 1] ˙φe
r
2
τ
dτ
where r
1
and r
2
stand for, respectively, a suitable con-
stant and a positive constant. g(x(t)) denotes a un-
known smooth function which takes values in the un-
known closed interval D = [d
−
, d
+
] with 0 ∈ D. Then,
V (t),
t
0
g(x(τ))N(φ) ˙φdτ and φ(t) must be bounded on
[0, t
f
).
The FOSMD [40–42] is described as
˙
ϑ
1
= ϱ
1
ϱ
1
= −ω
1
|ϑ
1
− υ|
1
2
sign(ϑ
1
− υ) + ϑ
2
˙
ϑ
2
= −ω
2
sign(ϑ
2
− ϱ
1
)
where ϑ
1
, ϑ
2
and ϱ
1
represent the system states, ω
1
and
ω
2
stand for the positive design constants, υ denotes the
input signal of the FOSMD. Then, we can obtain the
following lemmas.
Lemma 3 [41,42] By selecting suitable design constants,
the following equalities can be obtained after a finite
time in the absence of input noises:
ϑ
1
= υ
0
, ϱ
1
= ˙υ
0
Notice that Lemma 3 is derived under the case of no
input noises, i.e. υ = υ
0
, when the input noise exists,
the following Lemma holds.
Lemma 4 [41,42] If the input noise meets |υ − υ
0
| ≤
ι, the following inequalities can be obtained in a finite
time:
|ϑ
1
− υ
0
| ≤ a
1
ι =
¯
ℓ
|ϱ
1
− ˙υ
0
| ≤ b
1
ι
1
2
= ¯ε
where
¯
ℓ and ¯ε are both positive constants exclusively
depended on the design constants of the FOSMD.
Lemma 5 [43] For any π > 0 and ~ ∈ R, the inequal-
ity 0 < |~| − ~tanh(~/π) ≤ 0.2785π holds.
3 Event-triggered adaptive NN controller
design
Define the following error variables:
z
i
= x
i
− α
i−1
, i = 2, 3, . . . , n (4)
where virtual law α
j
(j = 1, 2, . . . , n − 1) will be de-
signed later.
Step 1: For the purpose of transforming the con-
strained tracking error e
1
(3) into the equivalent uncon-
strained variable z
1
, we introduce a smooth and strictly
increasing transformation function Υ (z
1
), which satis-
fies
−ς
1
< Υ (z
1
) < ς
2
lim
z
1
→+∞
Υ (z
1
) = ς
2
, lim
z
1
→−∞
Υ (z
1
) = −ς
1
.
With the help of Υ (z
1
), (3) can be expressed as
e
1
(t) = ϕ(t)Υ (z
1
) (5)
where Υ (z
1
) is constructed as follows
Υ (z
1
) =
ς
2
e
z
1
− ς
1
e
−z
1
e
z
1
+ e
−z
1
. (6)
Then, it follows from (6) that
z
1
= Υ
−1
e
1
(t)
ϕ(t)
=
1
2
ln
ς
1
+ e
1
(t)/ϕ(t)
ς
2
− e
1
(t)/ϕ(t)
. (7)
Based on (1) and (7), one obtains
˙z
1
=ξ
˙e
1
(t) − e
1
(t)
˙
ϕ(t)
ϕ(t)
=ξ
f
1
(x
1
) + g
1
(x
1
)x
2
− ˙y
d
− e
1
(t)
˙
ϕ(t)
ϕ(t)
(8)
where ξ = (1/2ϕ(t))[1/(ς
1
+e
1
(t)/ϕ(t))+1/(ς
2
−e
1
(t)/ϕ(t))].
Moreover, by (3), it can be concluded that ξ > 0.
Define the Lyapunov function candidate as follows
V
1
=
1
2
z
2
1
+
1
2
˜
W
1
Γ
−1
1
˜
W
1
(9)
where Γ
1
= Γ
T
1
> 0 is a constant matrix,
˜
W
1
=
ˆ
W
1
−
W
∗
1
stands for the weight estimation error with
ˆ
W
1
be-
ing the estimation of W
∗
1
. Then, the dynamic of V
1
a-
long (8) is
˙
V
1
=z
1
ξ
g
1
(x
1
)x
2
+ f
1
(x
1
) − ˙y
d
− e
1
(t)
˙
ϕ(t)
ϕ(t)
+
˜
W
T
1
Γ
−1
1
˙
ˆ
W
1
.
(10)
Let unknown function
T
1
(Z
1
) = ξ[f
1
(x
1
) − ˙y
d
]
where Z
1
= [x
1
, y
d
, ˙y
d
, ϕ(t)]
T
∈ R
4
.
We utilize the RBF NN to approximate T
1
(Z
1
) as
T
1
(Z
1
) = W
∗T
1
S
1
(Z
1
) + δ
1
(Z
1
) (11)
where |δ
1
(Z
1
)| ≤
¯
δ
1
.
Title Suppressed Due to Excessive Length 5
By substituting (11) into (10) yields
˙
V
1
=z
1
ξ
g
1
(x
1
)x
2
− e
1
(t)
˙
ϕ(t)
ϕ(t)
+ z
1
W
∗T
1
S
1
(Z
1
)
+ z
1
δ
1
(Z
1
) +
˜
W
T
1
Γ
−1
1
˙
ˆ
W
1
.
(12)
According to Young’s inequality, we have
z
1
δ
1
(Z
1
) ≤
λ
1
2
z
2
1
+
1
2λ
1
¯
δ
2
1
(13)
where λ
1
> 0 is a design constant.
Substituting (13) into (12), one has
˙
V
1
≤z
1
ξ
g
1
(x
1
)x
2
− e
1
(t)
˙
ϕ(t)
ϕ(t)
+ z
1
W
∗T
1
S
1
(Z
1
)
+
˜
W
T
1
Γ
−1
1
˙
ˆ
W
1
+
λ
1
2
z
2
1
+
1
2λ
1
¯
δ
2
1
.
(14)
The virtual law α
1
and neural weight updated law
˙
ˆ
W
1
are constructed as follows
α
1
=
1
ξ
N(φ
1
)ψ
1
(15)
ψ
1
= k
1
z
1
+
ˆ
W
T
1
S
1
(Z
1
) +
λ
1
2
+
ξ
2
2
z
1
− ξe
1
(t)
˙
ϕ(t)
ϕ(t)
(16)
˙φ
1
= z
1
ψ
1
(17)
˙
ˆ
W
1
= Γ
1
z
1
S
1
(Z
1
) − σ
1
ˆ
W
1
(18)
where k
1
> 0 and σ
1
> 0 are both design parameters.
Based on (15)-(17) and x
2
= z
2
+ α
1
, we obtain
z
1
ξg
1
(x
1
)x
2
=z
1
ξ
g
1
(x
1
)z
2
+ e
1
(t)
˙
ϕ(t)
ϕ(t)
−
k
1
+
λ
1
2
+
ξ
2
2
z
2
1
− z
1
ˆ
W
T
1
S
1
(Z
1
)
+ (g
1
(x
1
)N(φ
1
) + 1) ˙φ
1
.
(19)
Substituting (18) and (19) into (14), one can obtain
˙
V
1
≤ z
1
ξg
1
(x
1
)z
2
+ (g
1
(x
1
)N(φ
1
) + 1) ˙φ
1
−
k
1
+
ξ
2
2
z
2
1
− σ
1
˜
W
T
1
ˆ
W
1
+
1
2λ
1
¯
δ
2
1
.
(20)
With the help of Young’s inequality, we have
z
1
ξg
1
(x
1
)z
2
≤
ξ
2
2
z
2
1
+
1
2
g
2
1
(x
1
)z
2
2
−σ
1
˜
W
T
1
ˆ
W
1
≤ −
σ
1
2
∥
˜
W
1
∥
2
+
σ
1
2
∥W
∗
1
∥
2
.
(21)
Substituting (21) into (20), one has
˙
V
1
≤ −k
1
z
2
1
−
σ
1
2
∥
˜
W
1
∥
2
+
σ
1
2
∥W
∗
1
∥
2
+
1
2λ
1
¯
δ
2
1
+ (g
1
(x
1
)N(φ
1
) + 1) ˙φ
1
+
1
2
g
2
1
(x
1
)z
2
2
≤ −p
1
V
1
+ q
1
+ (g
1
(x
1
)N(φ
1
) + 1) ˙φ
1
+
1
2
g
2
1
(x
1
)z
2
2
(22)
where q
1
= σ
1
∥W
∗
1
∥
2
/2 +
¯
δ
2
1
/2λ
1
and p
1
= min{2k
1
,
σ
1
/λ
max
(Γ
−1
1
)}.
Step i (2 ≤ i ≤ n − 1): Noting that z
i
= x
i
− α
i−1
,
its derivative along (1) is
˙z
i
= f
i
(x
i
) + g
i
(x
i
)x
i+1
− ˙α
i−1
. (23)
In order to effectively estimate ˙α
i−1
and overcome the
issue of explosion of complexity, the FOSMD is con-
structed as
˙
ϑ
i1
= ϱ
i1
ϱ
i1
= −ω
i1
|ϑ
i1
− α
i−1
|
1
2
sign(ϑ
i1
− α
i−1
) + ϑ
i2
˙
ϑ
i2
= −ω
i2
sign(ϑ
i2
− ϱ
i1
)
(24)
where ϑ
i1
, ϑ
i2
and ϱ
i1
represent the system states, ω
i1
and ω
i2
are both positive design constants.
Combining with (24) and Lemma 3-4, it follows that
˙α
i−1
= ϱ
i1
+ ε
i
(25)
where ε
i
satisfies |ε
i
| ≤ ¯ε
i
with ¯ε
i
being a positive con-
stant. If the input signal α
i−1
of the FOSMD (24) is not
influenced by the noise, it can be concluded from Lem-
ma 3 that ¯ε
i
= 0. Moreover, if the input signal α
i−1
is
influenced by the bounded noise, we can conclude from
Lemma 4 that |ε
i
| ≤ ¯ε
i
.
Consider the Lyapunov function candidate
V
i
=
1
2
z
2
i
+
1
2
˜
W
T
i
Γ
−1
i
˜
W
i
(26)
where Γ
i
= Γ
T
i
> 0 is a constant matrix,
˜
W
i
=
ˆ
W
i
−W
∗
i
stands for the weight estimation error with
ˆ
W
i
being the
estimation of W
∗
i
.
Based on (23), we have
˙
V
i
=z
i
g
i
(x
i
)x
i+1
+ W
∗T
i
S
i
(Z
i
) + δ
i
(Z
i
) − ˙α
i−1
+
˜
W
T
i
Γ
−1
i
˙
ˆ
W
i
(27)
where W
∗T
i
S
i
(Z
i
) is employed to approximate f
i
(x
i
)
with Z
i
= [x
1
, x
2
, ..., x
i
]
T
∈ R
i
, W
∗
i
denotes the ideal
weights vector and |δ
i
(Z
i
)| ≤
¯
δ
i
.
According to Young’s inequality, one can obtain
z
i
δ
i
(Z
i
) ≤
λ
i
2
z
2
i
+
1
2λ
i
¯
δ
2
i
(28)