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Citation: Qiao, M.-Y.; Chang, X.-H.
Adaptive Fault-Tolerant Tracking
Control for Continuous-Time Interval
Type-2 Fuzzy Systems. Mathematics
2024,12, 3682. https://doi.org/
10.3390/math12233682
Academic Editor: Quanxin Zhu
Received: 29 October 2024
Revised: 18 November 2024
Accepted: 19 November 2024
Published: 24 November 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Adaptive Fault-Tolerant Tracking Control for Continuous-Time
Interval Type-2 Fuzzy Systems
Ming-Yang Qiao and Xiao-Heng Chang *
School of Information Science and Engineering, Wuhan University of Science and Technology,
Wuhan 430081, China; qiaomingyang@wust.edu.cn
*Correspondence: changxiaoheng@wust.edu.cn
Abstract: This paper investigated the tracking problem of mixed
H∞
and
L2−L∞
adaptive fault-
tolerant control (FTC) for continuous-time interval type-2 fuzzy systems (IT2FSs). For the membership
function mismatch and uncertainty between the modules of the nonlinear system, the IT2 fuzzy
model is applied to linearly approximate it. The observer can sensitively estimate the system state,
and the adaptive fault estimation functions can estimate adaptively the fault signals, which enables
the designed adaptive FTC scheme to ensure the asymptotic stability of the closed-loop control
system and achieve the desired mixed
H∞
and
L2−L∞
tracking performance. The designed adaptive
control law can achieve the purpose of dynamic compensation for faults and disturbances, and
the introduced lemmas further reduce the design conservatism by adjusting the slack parameters
and matrices. Finally, a mass-spring-damping system is used to illustrate the effectiveness of the
designed method.
Keywords: interval type-2 fuzzy systems (IT2FSs); adaptive fault-tolerant control; mixed
H∞
and
L2−L∞performance; tracking control
MSC: 93C40
1. Introduction
As the automatic control systems tend to be large scale, randomized and complicated,
the probability of faults caused by the long-term operation of system components or
improper human operation is increasing greatly. Therefore, how to quickly detect a fault
and repair it, or design a fault tolerance scheme to make the practical system able to tolerate
a certain degree and type of fault, to achieve the purpose of fault tolerance control, has
always been a hot research spot in the control field [
1
]. Among these, model-based fault
detection makes full use of the deep knowledge inside the system and determines whether
the fault occurs by residual error [
2
]. Then, to ensure the automatic control system is stable
and reduce the impact of faults on the system performance, some targeted FTC strategies
are studied [
3
–
10
]. Considering that active FTC relies too much on the fault detection
module, it is easy to cause delay when reconstructing a controller according to detection
results, which affects system performance [
4
–
6
]. Passive FTC is widely used because of its
simple design, easy implementation and good real-time performance [
7
–
10
]. However, due
to its conservative design and low tolerance to unknown faults, adaptive FTC can adjust its
own characteristic feedback control system in real time and intelligently according to the
specific fault affecting the system so that the system can work in the optimal state according
to some set standards [
11
–
19
]. For example, ref. [
13
] proposed a kind of robust adaptive
FTC circuit design and converted to analog control circuit implementation. The adaptive
FTC strategy for an actuator fault in [
12
,
18
] has a good fault-tolerant effect. For unexpected
fault situations in multi-agent systems, ref. [
15
] designed an compensation protocol and
H∞
resilient control scheme to adaptively achieve optimal control results. By designing
the tracking controller, the adaptive optimal tracking problem with the FTC method of a
Mathematics 2024,12, 3682. https://doi.org/10.3390/math12233682 https://www.mdpi.com/journal/mathematics
Mathematics 2024,12, 3682 2 of 18
multi-agent system and active FTC tracking problem with input constraint are embodied
in [
19
] and [
20
], respectively. However, there are few achievements on adaptive FTC for
IT2 nonlinear systems with sensor and actuator faults.
Tracking control is used to achieve the purpose of tracking the desired trajectory or
path through the control system [
19
–
25
]. At present, attitude tracking in aerospace [
22
],
trajectory tracking in robotics [
23
] and path tracking in the field of automatic driving [
24
,
25
]
have a wide range of applications. Therefore, in these real-environmental applications,
when the sensor or actuator failure occurs, the tracking control effect will be greatly reduced
and the system may even be unstable or collapse. The adaptive fault-tolerant tracking
control can realize the tracking control under the fault condition, and the control signal
can be adjusted adaptively to achieve a better tracking effect. In order to achieve a better
tracking effect, we generally define some quantitative indicators to describe the system
performance and give their characterization and calculation methods. Examples include
peak-to-peak, energy-to-energy (
H∞
), energy-to-peak (
L2−L∞
[
26
,
27
]), and so on. Among
them,
H∞
can ensure the robust stability of the system, and
L2−L∞
is suitable for the
scenario where the energy of the external interference signal is bounded and the peak value
of the practical system output signal is bounded. These two kinds of performance indicators
have their own application background, so if the tracking controller is designed to meet the
performance requirements at the same time, it will have a wider application prospect [
28
,
29
].
For a memory neural networks system, there exist state estimation problems caused by
delay and bounded perturbation. Based on the protocol, ref. [
28
] proposed a finite horizon
mixed performance estimation method. Ref. [
29
] studied the hybrid control problem
of measuring outliers in observer-based IT2FSs. But at present, there exist few such
achievements, especially on the fuzzy system of fault-tolerant control, which is the original
intention of this paper.
Note that system nonlinearity is inevitable in practical engineering in addition to other
forms of nonlinear description ([
30
,
31
]). The T–S fuzzy model [
32
], as an effective tool, can
approximate nonlinear systems through some local linear time-invariant systems, thus
introducing traditional linear system theory into the study of nonlinear systems [
33
]. How-
ever, the type-1 T–S fuzzy model only has a good effect on dealing with system nonlinearity;
if the membership function in the T–S fuzzy model contains uncertain information [
9
], the
type-1 fuzzy model will be overwhelmed.
Based on this, the IT2 fuzzy model proposed in [
34
] effectively solves the above
problems by defining upper and lower bound membership functions. Inspired by this,
many interesting and meaningful research results have appeared successively, enriching
the relevant achievements of fuzzy control and fuzzy filtering [
9
,
25
,
35
–
40
]. For example,
by establishing the IT2 fuzzy model, ref. [
25
] effectively approximated the tire dynamic
nonlinearity and speed variation in the path tracking control system and studied the path
tracking control problem of autonomous ground vehicles under handover trigger and
sensor attack. Ref. [
36
] studied the sampling exponential stability and nonlinear control
of IT2 fuzzy systems. Based on event-driven faults, ref. [
37
] discussed the FD problem
for IT2 fuzzy systems. Ref. [
38
] proposed an event-based control method for IT2FSs with
fading channels. Ref. [
39
] studied a multistep model predictive control problem for IT2FSs
subject to event-triggered faults. In this kind of research, FTC is indispensable in the study
of fault handling, especially the membership function mismatch caused by many modules
in adaptive FTC. In an IT2 fuzzy system, refs. [
9
,
40
] obtained appropriate FTC schemes
for finite-time dynamic event-triggered and adaptive sliding mode control, respectively.
However, the research content of other FTC problems of IT2 fuzzy systems is relatively
small, which is the driving force of this paper.
Driven by the above considerations, this paper studies the tracking problem of mixed
H∞
and
L2−L∞
adaptive FTC for continuous-time IT2FSs. Fully considering the mismatch
and uncertainty of the membership function between systems, the faults estimate functions
in observer and adaptive law in the controller, ensuring stricter system stability and
performance requirements. The main innovations are summarized:
Mathematics 2024,12, 3682 3 of 18
•
Considering the membership function mismatch and uncertainty of each part in the
practical system, the proposed design scheme is aimed at the tracking problem of a
mixed H∞and L2−L∞adaptive FTC for continuous-time IT2FSs.
•
Compared with the general adaptive FTC scheme in [
11
,
13
], the adaptive control
function is improved in this paper, which is simultaneously tolerant of the sensor
and the actuator faults. Based on the adaptive signal, the dynamic parameters in
the disturbance compensation term can be dynamically adjusted to achieve a better
FTC effect.
•
Mixed
H∞
and
L2−L∞
performance is considered in the design of the fuzzy tracking
controller and observer to meet a wider range of practical requirements. Based on
the matrix inequality transformation technique in the lemmas, the designed algo-
rithm reduces conservatism by introducing suitable slack variables and matrices in
the theorem.
The rest of this paper is organized as follows. The problem statement and preliminary
is formulated in Section 2. The main results about adaptive FTC system performance
analysis and the design control strategy are presented in Sections 3and 4, respectively.
In Section 5, a mass-spring-damping system is used to illustrate the effectiveness of the
designed method. Finally, Section 6summarizes this paper.
2. Problem Statement and Preliminarie
In the real-environment application, due to the membership function mismatch and
uncertainty between the modules of the fuzzy system, as well as the possible sensor and
actuator faults in the system, this paper uses the IT2FSs to model the system, observer and
controller. Moreover, in order to realize tracking control, a reference system is introduced.
2.1. Continuous-Time IT2FSs
In this section, similar to [
9
,
37
], the continuous-time IT2FSs with
m
fuzzy rules are
expressed as
Rule i: If ϕ1(x(t)) is Mi
1,· · · , and ϕı(x(t)) is Mi
ı, then
˙
x(t) = Aix(t) + Biuf(t) + Eiw(t) + Fifa(t)
y(t) = Cix(t) + Gifs(t)(1)
where
Mi
α
is an IT2 fuzzy set of the
i
-th fuzzy rule with the function
ϕα(x(t))
where
i=
1, 2,
· · ·
,
m
and
α=
1, 2,
· · ·
,
ı
.
m
and
ı
are the positive integers;
x(t)∈Rnx
,
uf(t)∈
Rnu
,
y(t)∈Rny
are the state variable, the control input after the system faults, and the
measurement output, respectively.
w(t)∈Rnw
is the external disturbance signal in
L2[
0,
∞)
.
fa(t)∈Rnf
and
fs(t)∈Rnf
are the stuck fault signals from the actuator and sensor,
respectively; The matrices Ai,Bi,Ei,Fi,Ci,Giare known proper dimensional matrices.
The firing strength of the i-th fuzzy rule satisfies
˜
α= [αi,αi]
where
αi=
ı
∏
α=1
ϵMi
α(ϕα),ϵMi
α(ϕα)∈[0, 1]
αi=
ı
∏
α=1
ϵMi
α(ϕα),ϵMi
α(ϕα)∈[0, 1]
ϵMi
α(ϕα)≤ϵMi
α(ϕα),αi≤αi
with
ϵMi
α(ϕα)
and
αi
representing the lower grade and function of membership,
ϵMi
α(ϕα)
and
αi
representing the upper grade and function of membership, respectively. The global
dynamic fuzzy model can be described as
Mathematics 2024,12, 3682 4 of 18
˙
x(t) =
m
∑
i=1
gi[Aix(t) + Biuf(t) + Eiw(t) + Fifa(t)]
y(t) =
m
∑
i=1
gi[Cix(t) + Gifs(t)]
(2)
where
gi=piαi+piαi
,
m
∑
i=1
gi=
1, while the nonlinear functions have 0
≤pi≤pi≤
1 and
satisfy pi+pi=1.
In addition to the additive bounded faults existing in the system, the system may also
have partially degenerate multiplicative faults expressed as
ςf(t) = dςς(t)(3)
where dς∈(0, 1]with ς=y,uare the fault parameters of a partially degenerate incident.
For brevity,
˜
α
,
αi
,
αi
,
ϵMi
α(ϕα)
,
ϵMi
α(ϕα)
,
pi
,
pi
, and
gi
are used to stand for
˜
α(x(t))
,
αi(x(t))
,
αi(x(t))
,
ϵMi
α(ϕα(x(t)))
,
ϵMi
α(ϕα(x(t)))
,
pi(x(t))
,
pi(x(t))
, and
gi(x(t))
in the IT2
fuzzy system. By the same token, the following symbols
˜
ϑ
,
ϑl
,
ϑl
,
υHl
ϑ(ψϑ)
,
υHl
ϑ(ψϑ)
,
sl
,
rl
,
rl
,
˜
β
,
βj
,
βj
,
εNj
β
(φβ)
,
εNj
β
(φβ)
,
hj
,
qj
, and
qj
are also abbreviations in the IT2 fuzzy observer
and controller.
2.2. Reference Model
To reflect the control effect under the fault condition and to realize the tracking control,
the following reference system is adopted in this paper
˙
xr(t) = Arxr(t) + Brir(t)
yr(t) = Crxr(t)(4)
where
xr(t)∈Rnr
,
yr(t)∈Rny
, and
ir(t)∈Rni
are the state, the reference output and
reference input of the reference model, respectively. To verify the tracking effect of the
controller,
Ar
is designed as a Hurwitz matrix. Moreover,
Br
and
Cr
are known proper
dimensional matrices.
2.3. IT2 Fuzzy Observer
Considering the practical engineering, the observer is nonlinear and does not match the
membership function of other parts in the system. Similar to [
9
], to design the appropriate
adaptive controller, the sensor/actuator fault values and system state are necessary to be
estimated, so an IT2 fuzzy observer in the following form is designed as
Rule l: If ψ1(ˆ
x(t)) is Hl
1,· · · , and ψℓ(ˆ
x(t)) is Hl
ℓ, then
˙
ˆ
x(t) =
m
∑
i=1
m
∑
l=1
gisl[Aiˆ
x(t) + Biuf(t) + Eiw(t)
+Fiˆ
fa(t) + Llr(t)]
ˆ
y(t) =
m
∑
i=1
gi[Ciˆ
x(t) + Giˆ
fs(t)]
(5)
where
Hl
ϑ
is an IT2 fuzzy set of the
l
-th fuzzy rule with the function
ψϑ(ˆ
x(t))
where
l=
1, 2,
· · ·
,
m
and
ϑ=
1, 2,
· · ·
,
ℓ
.
m
and
ℓ
are the positive integers;
ˆ
x(t)∈Rnx
,
ˆ
y(t)∈Rny
,
and
r(t)∈Rny
are the state, the output, and the residual vector estimated by the IT2 fuzzy
observer, respectively.
Ll
is the designed observer gain matrix. In a similar way, the firing
strength of the l-th rule satisfies
˜
ϑ= [ϑl,ϑl]
Mathematics 2024,12, 3682 5 of 18
where
ϑl=
ℓ
∏
ϑ=1
υHl
ϑ(ψϑ),υHl
ϑ(ψϑ)∈[0, 1]
ϑl=
ℓ
∏
ϑ=1
υHl
ϑ(ψϑ),υHl
ϑ(ψϑ)∈[0, 1]
υHl
ϑ(ψϑ)≤υHl
ϑ(ψϑ),ϑl≤ϑl
where
υHl
ϑ(ψϑ)
and
ϑl
are the lower grade and function of membership, and
υHl
ϑ(ψϑ)
and
ϑl
are the upper grade and function of membership, respectively;
sl=rlϑl+rlϑl
m
∑
j=1
(rlϑl+rlϑl)
,
m
∑
l=1
sl=
1,
rl∈[0, 1]and rl∈[0, 1]are the nonlinear functions and satisfy rl+rl=1.
Moreover, the adaptive fault information estimation functions are
˙
ˆ
fs(t) = s∥ξT(t)PL∥
˙
ˆ
fa(t) = a∥ζ(t)∥(6)
where
ζ(t) = ξT(t)PB
and
r(t) = yf(t)−ˆ
y(t)
, while the estimated values of state and
fault information are represented by
ˆ
x(t)
,
ˆ
fs(t)
, and
ˆ
fa(t)
, respectively.
s
,
a
are any positive
constants.
2.4. IT2 Fuzzy Adaptive Tracking Controller
Based on this, an observer-based adaptive tracking controller can be proposed.
Rule j: If φ1(ˆ
x(t)) is Nj
1,· · · , and φȷ(ˆ
x(t)) is Nj
ȷ, then
u(t) =
o
∑
j=1
hj[Kjˆ
y(t) + Krj yr(t) + Ka(t)] (7)
where
Nj
β
is an IT2 fuzzy set of the
j
-th fuzzy rule with the function
φβ(ˆ
x(t))
where
j=
1, 2,
· · ·
,
o
and
β=
1, 2,
· · ·
,
ȷ
.
m
and
ȷ
are the positive integers;
Ka(t)
is the adaptive
control function.
Kj
and
Krj
are the controller gain matrices. In a similar way, the firing
strength of the j-th rule satisfies
˜
β= [βj,βj]
where
βj=
ȷ
∏
β=1
εNj
β
(φβ),εNj
β
(φβ)∈[0, 1]
βj=
ȷ
∏
β=1
εNj
β
(φβ),εNj
β
(φβ)∈[0, 1]
εNj
β
(φβ)≤εNj
β
(φβ),βj≤βj
where
εNj
β
(φβ)
and
βj
are the lower grade and function of membership, while
εNj
β
(φβ)
and
βj
are the upper grade and function of membership, respectively;
hj=qjβj+qjβj
m
∑
j=1
(qjβj+qjβj)
,
o
∑
j=1
hj=1, qj∈[0, 1]and qj∈[0, 1]are the nonlinear functions and satisfy qj+qj=1.
The adaptive control function is defined as
Ka(t) = −ζT(t)a(t)
∥ζ(t)∥2dmin
Mathematics 2024,12, 3682 6 of 18
where dmin =min{min{dy}, min{du}}
a(t) = ∥ξT(t)PL∥ ˆ
fs(t) + ∥ζ(t)∥ˆ
fa(t) + ∥ζ(t)∥ρ(t)Zt
t0
∥ξ(τ)∥dτ.
There has
ρ(t)≥
0 in the compensation term
ρ(t)Rt
t0∥ξ(τ)∥dτ
to achieve the compensation
effect to the system fault and disturbance.
Assumption 1. Similar to [
13
], sensor/actuator faults and disturbances can be expressed in the
relevant segmented bounded functions, and it can be assumed that there exist matrices
¯
Ei
,
¯
Fi
,
¯
Gi
and unknown constants ¯
ϖ1,¯
ϖ2such that the following inequality conditions were established
∥ϖ1(t)∥=∥¯
Eiw(t) + ¯
Fifa(t)∥ ≤ ∥ξ(t)∥α+¯
ϖ1
∥ϖ2(t)∥=∥¯
Gifs(t)∥ ≤ ¯
ϖ2
where Ei=Bi¯
Ei, Fi=Bi¯
Fiand Gi=Ci¯
Gi.
Remark 1. Different from other papers ([
11
,
13
]), the adaptive control strategy for the faults from
the sensor and actuator is considered in the adaptive function, and in the compensation part of
external disturbance and faults, the parameter
ρ(t) = ρwhen ξ(t)=0
0when ξ(t) = 0
with a large enough constant
ρ
, which will be changed according to the compensation situation.
When the influence of faults and disturbances on the IT2FSs ends, the compensation can be quickly
compensated, and the compensation item is 0 after the influence ends. In this way, better control can
be achieved.
2.5. Adaptive FTC System
Based on the above description and definition, the constructed augmented closed-loop
adaptive FTC system (in Figure 1) gives
˙
ξ(t) =
m
∑
i=1
m
∑
l=1
o
∑
j=1
gislhj[(A+duBK)ξ(t) + Eses(t)
+Eaea(t) + duBKa(t) + Bϖ1(t) + Lϖ2(t)
+Brir(t)]
ey(t) =
m
∑
i=1
gi[Cξ(t) + Gies(t)]
(8)
where
A=
Ai−¯
dyLlCi−dyLlCi0
−¯
dyLlCiAi−dyLlCi0
0 0 Ar
,B=
Bi
0
0
C=Ci0−Cr,K=KjCi0KrjCr,¯
dy=1−dy
L=
duBiKjCi−¯
dyLlCi
−¯
dyLlCi
0
,Es=
duBiKjGi−LlGi
−LlGi
0
Ea=
0
Fi
0
,Br=
0
0
Br
,ξ(t) =
ˆ
x(t)
ex(t)
xr(t)
with
ey(t) = ˆ
y(t)−yr(t)
,
ex(t) = ˆ
x(t)−x(t)
,
es(t) = ˆ
fs(t)−fs(t)
, and
ea(t) = ˆ
fa(t)−fa(t)
.
Mathematics 2024,12, 3682 7 of 18
ᤷԔ⭏ᡀಘᢗ㹼ᵪᶴ㻛᧗ሩ䊑Րᝏಘ৽侸᧗ࡦಘᢗ㹼ᵪᶴ᭵䳌㌫㔏㔃ᶴ᭵䳌Րᝏಘ᭵䳌ᢠࣘPlant Sensor Partially
Degenerate Fault
IT2 Fuzzy
Observer
IT2 Adaptive
Controller
( )w t
( )y t
( )
s
f t
( )
a
f t
( )
y
e t
( )
f
y t
( )
r
y t
( )u t
( )
r
x t
( )
f
u t
ˆ( )x t
-
Actuator Partially
Degenerate Fault
Reference
Model
( )
r
i t
Figure 1. Framework of closed-loop adaptive FTC system.
Definition 1 ([
29
]).The adaptive FTC system (8) is said to be asymptotically stable when
¯
w(t) =
0,
and it has the mixed
H∞
and
L2−L∞
performances if under the zero initial condition, with the
existed matrices Ξ1≤0, Ξ2>0, Ξ3≥0, the following inequality holds
Zta
0J(t)dt ≥sup
0≤t≤ta
{eT
y(t)Ξ3ey(t)}(9)
with
J(t) = eT
y(t)Ξ1ey(t) + ¯
wT(t)Ξ2¯
w(t)
. With the different selection of matrices, system (8) has
different performance:
(1)
The adaptive FTC system (8) has the H∞performance if Ξ1=−I,Ξ2=γ2I,Ξ3=0.
(2)
The adaptive FTC system (8) has the L2−L∞performance if Ξ1=0, Ξ2=γ2I,Ξ3=I.
Lemma 1 ([
41
]).For the real matrices
T0
,
T1
,
T2
and
T3
with the proper dimensions and scalar
β
,
the inequality T0+TT
3T1T3<0holds if there exists the following situation
T0∗
T2T3−He{βT2}+β2T1<0.
Lemma 2 ([
33
]).For the real matrices
X1
,
X2
, and real symmetric matrix
X0
with the proper
dimension and scalar
℘
, the inequality
X0+He{X1X2}<
0holds if there exists matrix
S
satisfying
X0∗
℘XT
1+SX2−He{℘S}<0.
3. Adaptive FTC System Performance Analysis
In the following, the sufficient conditions of asymptotic stability with the mixed
H∞
and L2−L∞performances for the adaptive FTC system (8) are provided.
Theorem 1. Consider the adaptive FTC system (8). If there exist matrices
P>
0and
Ψ=ΨT
with the given performance index
γ
and parameters
ηi
such that the inequalities hold with
i=1, 2, · · · ,m;l=1, 2, · · · ,m;j=1, 2, · · · ,o
Λilj −Ψ<0 (10)
ηiΛili −ηiΨ+Ψ<0 (11)
ηjΛilj +ηiΛjli −ηjΨ−ηiΨ+2Ψ<0 (12)
m
∑
i=1
giΓ≤0 (13)
Mathematics 2024,12, 3682 8 of 18
where
Λilj =Λ1+CTC ∗
Λ2+GT
1CΛ3+GT
1G1
Γ=CTC − P∗
GT
2C −diag{s−1I,a−1I}+GT
2G2
Λ1= (A+duBK)TP+P(A+duBK)
ΛT
2=PEsPEaPBr,G2=Gi0
Λ3=−diag{γ2I,γ2I,γ2I},G1=Gi0 0.
Then, the fuzzy observer (5) and controller (7) can ensure the adaptive FTC system (8) is asymptoti-
cally stable and meets the mixed H∞and L2−L∞performances.
Proof. Consider a Lyapunov function as
V(t) = ξT(t)Pξ(t) + s−1e2
s(t) + a−1e2
a(t). (14)
Then, since
fs(t)
and
fa(t)
are the stuck faults, i.e., there has
˙
fs(t) = ˙
fa(t) =
0. The
difference equation of V(t)can be obtained
˙
V(t) = ˙
ξT(t)Pξ(t) + ξT(t)P˙
ξ(t) + 2s−1es(t)˙
es(t)
+2a−1ea(t)˙
ea(t)
=ξT(t)[(A+duBK)TP+P(A+duBK)]ξ(t)
+2ξT(t)PEses(t) + 2ξT(t)PEaea(t) + 2ξT(t)PBϖ1(t)
+2ξT(t)PLϖ2(t) + 2ξT(t)PBrir(t)
+2duζ(t)Ka(t) + 2s−1es(t)˙
es(t) + 2a−1ea(t)˙
ea(t)
=ξT(t)[(A+duBK)TP+P(A+duBK)]ξ(t)
+2ξT(t)PEses(t) + 2ξT(t)PEaea(t) + 2ξT(t)PBϖ1(t)
+2ξT(t)PLϖ2(t) + 2ξT(t)PBrir(t)
+2s−1es(t)˙
es(t) + 2a−1ea(t)˙
ea(t)
−2duζ(t)ζT(t)a(t)
∥ζ(t)∥2dmin
≤ξT(t)[(A+duBK)TP+P(A+duBK)]ξ(t)
+2ξT(t)PEses(t) + 2ξT(t)PEaea(t) + 2ξT(t)PBϖ1(t)
+2ξT(t)PLϖ2(t) + 2ξT(t)PBrir(t)
+2s−1es(t)˙
es(t) + 2a−1ea(t)˙
ea(t)−2∥ξT(t)PL∥ ˆ
fs(t)
−2∥ζ(t)∥ˆ
fa(t)−2∥ζ(t)∥ρ(t)Zt
t0
∥ξ(τ)∥dτ.
(15)
By recalling the conditions in Assumption 1, (15) is reduced as
˙
V(t)≤ξT(t)[(A+duBK)TP+P(A+duBK)]ξ(t)
+2ξT(t)PEses(t) + 2ξT(t)PEaea(t) + 2ξT(t)PBrir(t)
+2s−1es(t)˙
es(t) + 2a−1ea(t)˙
ea(t)−2∥ξT(t)PL∥ ˆ
fs(t)
−2∥ζ(t)∥ˆ
fa(t)−2∥ζ(t)∥ρ(t)Zt
t0
∥ξ(τ)∥dτ
+2∥ζ(t)∥∥ξ(t)∥α+2∥ζ(t)∥¯
ϖ1+2∥ξT(t)PL∥ ¯
ϖ2.
(16)
Mathematics 2024,12, 3682 9 of 18
Similar to [13], the following inequality will hold
∥ζT(t)∥¯
ϖ1≤ ∥ζT(t)∥fa(t)
∥ξT(t)PL∥ ¯
ϖ2≤ ∥ξT(t)PL∥ fs(t).(17)
The nonlinear function
ρ(t)Rt
t0∥ξ(τ)∥dτ
is obviously a monotonically increasing func-
tion of the augmented vector
ξ(t)
. Therefore, when
ξ(t)>
0 holds, if the parameter
ρ
is
large enough, and if the parameter α≤2 exists, one obtains
∥ξ(t)∥α≤ρ(t)Zt
t0
∥ξ(τ)∥dτ.(18)
Then, combining (6), (17), and (18), the inequality (16) is reduced as
˙
V(t)≤ξT(t)[(A+duBK)TP+P(A+duBK)]ξ(t)
+2ξT(t)PEses(t) + 2ξT(t)PEaea(t) + 2ξT(t)PBrir(t).(19)
Similar to [
9
], by introducing
¯
ξ(t) = ξT(t)¯
wT(t)T
with
¯
w(t) = eT
s(t)eT
a(t)iT
r(t)T
and the slack matrix
Ψ=ΨT
, based on the membership function information and its characters,
there exists
m
∑
i=1
m
∑
l=1
o
∑
j=1
gisl×(sj−hj)Ψ=m
∑
i=1
m
∑
l=1
(o
∑
j=1
sj−o
∑
j=1
hj)Ψ=m
∑
i=1
m
∑
l=1
(
1
−
1
)Ψ=
0.
One has m
∑
i=1
m
∑
l=1
o
∑
j=1
gislhj¯
ξT(t)Λilj ¯
ξ(t)
+
m
∑
i=1
m
∑
l=1
o
∑
j=1
gisl×(sj−hj)Ψ
≤
m
∑
i=1
m
∑
l=1
gigi(ˆ
x(t))sl¯
ξT(t)(ηiΛil i −ηiΨ+Ψ)¯
ξ(t)
+
m
∑
i=1
m
∑
l=1
o
∑
j=1
gisl(hj−ηigj)¯
ξ(t)(Λil j −Ψ)¯
ξT(t)
+
m
∑
i=1
m
∑
l=1
∑
i<j
gislhjׯ
ξT(t)(ηjΛil j +ηiΛjli −ηjΨ
−ηiΨ+2Ψ)¯
ξ(t).
(20)
Furthermore, using congruence properties in (10)–(12) yields
˙
V(t)−J(t)<0. (21)
If the given matrices Ξ1=−I,Ξ2=γ2I, the above inequality is reduced to
˙
V(t) + eT
y(t)ey(t)−γ2¯
wT(t)¯
w(t)<0. (22)
When external input variable
¯
w(t) =
0, with the situation of
eT
y(t)ey(t)>
0, that means
˙
V(t)<0, so the adaptive FTC system (8) is asymptotically stable.
When external input variable
¯
w(t)=
0, under zero initial conditions, integral from 0
to taat both ends of the inequality (22), one obtains
Zta
0J(τ)dτ>V(ta)−V(0) = V(ta)(23)
with the condition
V(ta)≥
0 and
Ξ3=
0, the
H∞
performance condition
Rta
0eT
y(t)ey(t)dτ<
γ2Rta
0¯
wT(t)¯
w(t)dτis obtained.
Mathematics 2024,12, 3682 10 of 18
In addition, using congruence properties with
Ξ1=
0,
Ξ2=γ2I
,
Ξ4=I
, and
ˆ
ξ(t) = ξT(t)eT
s(t)eT
a(t)Tto inequality (13) yields
eT
y(t)ey(t)−[ξT(t)Pξ(t) + s−1eT
s(t)es(t)
+a−1eT
a(t)ea(t)]
=eT
y(t)ey(t)−V(t)≤0.
(24)
Furthermore, with the situation 0 ≤t≤ta, one has
eT
y(ta)ey(ta)−V(ta)≤0. (25)
By combining inequality (23), we can obtain the result
sup
0≤t≤ta
{eT
y(t)ey(t)} ≤ γ2Zta
0¯
wT(t)¯
w(t)dτ
which implies that the system (8) satisfies
L2−L∞
performance. The proof is completed.
4. Fuzzy Observer and Controller Design
The nonlinear coupling terms in Theorem 1 mean that the unknown observer and
controller gain matrices cannot be solved directly, so they need to be designed and obtained
through the following theorem obtained by some lemmas.
Theorem 2. Consider the adaptive FTC system (8). For given performance index
γ
, positive
parameters
β1
,
β2
,
℘
,
ηi
, partial degradation failure coefficients
dy
,
du
, if there exist matrices
P1>
0,
P2>
0,
¯
Ll
,
¯
Kj
,
¯
Krj
,
S
,
T1
2
,
T2
2
, and
¯
Ψ=¯
ΨT
such that the inequalities hold with
i=1, 2, · · · ,m;l=1, 2, · · · ,m;j=1, 2, · · · ,o
ˆ
Λilj −¯
Ψ<0 (26)
ηiˆ
Λili −ηi¯
Ψ+¯
Ψ<0 (27)
ηjˆ
Λilj +ηiˆ
Λjli −ηj¯
Ψ−ηi¯
Ψ+2¯
Ψ<0 (28)
m
∑
i=1
gi¯
Γ≤0 (29)
where
ˆ
Λilj =¯
X0∗
℘¯
XT
1+S¯
X2−He{℘S},ˆ
X0=¯
X01 ∗
¯
X02 Λ3
¯
X0=ˆ
X0∗
T1
2T1
3−He{β1T1
2}+β2
1I
¯
X01 =
¯
X1∗ ∗
¯
X2¯
X3∗
CT
r¯
KT
rj BT
i0He{P2Ar}
¯
X1=He{P1Ai−¯
dy¯
LlCi+Bi¯
KjCi}
¯
X2=−¯
dy¯
LlCi−dyCT
i¯
LT
l,¯
X3=He{P1Ai−dy¯
LlCi}
¯
X2=S−TduBT
iPT
1−STBT
i06,S=ST
¯
X1=¯
KjCi0¯
Krj Cr¯
KjGi03T,T1
3=C G1
¯
X02 =
GT
i¯
KT
jBT
i−GT
i¯
LT
l−GT
i¯
LT
l0
0FT
iP10
0 0 BT
rP2
Mathematics 2024,12, 3682 11 of 18
¯
Γ=T2
0∗
T2
2T2
3−He{β2T2
2}+β2
2I
T2
0=−diag{P1,P1,P2,s−1I,a−1I},T2
3=C G2.
Then, the fuzzy observer (5) and controller (7) can ensure the adaptive FTC system (8) is asymptoti-
cally stable and has the mixed
H∞
and
L2−L∞
performances. Accordingly, the gains of a fuzzy
tracking controller and observer are obtained as follows
Ll=P−1
1¯
Ll,Kj=S−1¯
Kj,Krj =S−1¯
Krj . (30)
Proof.
By choosing the matrix
P=diag{P1
,
P1
,
P2}
to the (10)–(13) and defining the follow-
ing matrices
T1
0=¯
Λ1∗
¯
Λ2Λ3,T1
1=T2
1=I
¯
Λ1=
X1∗ ∗
X2X3∗
duCT
rKT
rj BT
iPT
10He{P2Ar}
X1=He{P1Ai−¯
dyP1LlCi+duP1BiKjCi}
X2=−¯
dyP1LlCi−dyCT
iLT
lPT
1,X3=He{P1Ai−dyP1LlCi}
¯
Λ2=
duGT
iKT
jBT
iPT
1−GT
iLT
lPT
1−GT
iLT
lPT
10
0FT
iPT
10
0 0 BT
rPT
2
.
Recalling the matrices
T1
2
,
T1
3
,
T2
0
,
T2
2
,
T2
3
and introducing the scalars
β1
,
β2
. By using
Lemma 1to (10)–(13), the condition (29) is obtained and there is
T1
0∗
T1
2T1
3−He{β1T1
2}+β2
1I<0. (31)
Furthermore, recalling the matrices X2,Sand defining the matrices
X0=
ˆ
Λ1∗
ˆ
Λ2Λ3∗
T1
2T1
3−He{β1T1
2}+β2
1I
ˆ
Λ1=
ˆ
X1∗ ∗
X2X3∗
CT
rKT
rj STBT
i0He{P2Ar}
ˆ
X1=He{P1Ai−¯
dyP1LlCi+BiSKjCi}
ˆ
Λ2=
GT
iKT
jSTBT
i−GT
iLT
lPT
1−GT
iLT
lPT
10
0FT
iPT
10
0 0 BT
rPT
2
X1= (SKjCi0Krj CrKjGi03)T.
Then, by using the Lemma 2to (31), by defining the matrices
¯
Ll=P1Ll
,
¯
Kj=SKj
,
¯
Krj =SKrj
, so that the feasibility conditions (26)–(29) are obtained. The proof is completed.
Remark 2. In actual engineering, due to different models, environmental changes, human factors,
aging and other reasons, if the same membership function is used to model the system modules
(observer and controller) and the nonlinear system, an accurate fuzzy model cannot be established.
However, IT2 fuzzy modeling is designed to model each module separately with the upper and
lower membership degree and membership function, and the method in [
9
] is adopted. The slack
Mathematics 2024,12, 3682 12 of 18
matrix and membership function properties are introduced to obtain (20), which further describes
the mismatching and uncertainty of membership functions between system modules.
5. Simulations
A mass-spring-damping system [
35
,
39
] is selected to demonstrate the effectiveness of
the designed method. According to Newton’s law,
m¨
x+Ff+Fs=u(32)
where
Ff=¯
c˙
x
,
Fs=¯
k(
1
+¯
a2x2)x
,
m
, and
u
are the friction force, hardening spring force,
mass, and control input, respectively, and
¯
c>
0 and
¯
k
and
¯
a
are constants. Then, (32) is
rewritten as
m¨
x+¯
c˙
x+¯
kx +¯
k¯
a2x3=u(33)
where
x
is the displacement of mass from the reference point. Defending
x(t) = xT
1(t)xT
2(t)T=xT˙
xTT
and
¯
b(t)=(−¯
k−¯
k¯
a2x2
1(t))/m
, assume
x1(t)∈
[−
1, 1
]
,
m=
2
kg
,
¯
c=
6
N·m/s
,
¯
a=
0.5
m−1
, and
¯
k∈[
5, 8
]
, so that
¯
b(t)∈[¯
bmin
,
¯
bmax ] =
[−2, −1].
Then, the IT2 T–S fuzzy system matrices are
A1,2 =0 1
¯
b(t)−¯
c
m,B1=B2=0
1
m.
Based on the property of membership function
α1+α2=
1, there has
¯
b(t) = α1¯
bmin +
α2¯
bmax
with
α1=¯
bmax −¯
b(t)
¯
bmax −¯
bmin
and
α2=¯
b(t)−¯
bmin
¯
bmax −¯
bmin
. Then, with the parameter
¯
k
, the homologous
upper and lower membership functions are
α1=α1,α2=α2,¯
k=5
α1=α1,α2=α2,¯
k=8.
Define the nonlinear weighting functions
p1=r1=q1=sin2(x1(t))
,
p1=r1=
q1=
1
−sin2(x1(t))
,
p2=r2=q2=cos2(x1(t))
,
p2=r2=q2=
1
−cos2(x1(t))
. The
membership functions of the observer and controller are selected as
ϑ1=β1=e−ˆ
x2
1(t)/0.5
,
ϑ1=β1=e−ˆ
x2
1(t),ϑ2=β2=1−e−ˆ
x2
1(t),ϑ2=β2=1−e−ˆ
x2
1(t)/0.5.
The corresponding matrices are
A1=0 1
−2−3,A2=0 1
−1−3
B1=B2=0
0.5,E1=E2=−0.2
−0.1,F1=F2=−0.5
−0.3
C1=C2=0.1 0.2,G1=G2=0.3.
Assume that the disturbance
w(t) =
0.5
e−0.3tsin(
0.2
t)
; then, the corresponding math-
ematical expression of sensor fault and actuator fault are
fs(t) =
0.5, 100 ≤t≤150,
0.2, 200 ≤t≤250,
0, else.
fa(t) =
0.5, 130 ≤t≤170,
0.3, 230 ≤t≤250,
0, else.
Mathematics 2024,12, 3682 13 of 18
Moreover, the matrices in the reference system are
Ar=0.1 1.6
−0.3 −1,Br=0.2
−0.2
Cr=−0.5 0.8.
with the reference input
ir(t) =
−0.2, t≤150,
0.2 cos(0.2t), 150 ≤t≤350,
0, else.
Set the parameters
η1=
0.4,
η2=
0.7; then, the observer and controller gain matrices
are obtained as
L1=0.2597 −1.0683T
L2=−0.1621 0.6746T
K1=−0.4141, K2=0.5073
Kr1=0.0104, Kr2=−0.0383.
The system and observer initial values are selected as
x(
0
) = ˆ
x(
0
) = [
0 0
]T
.
Figure 2shows the observer’s estimation of the system states, indicating that the ob-
server has good estimation performance. In Figure 3, a comparison is added with the
method proposed in reference [
13
]. Initially, the system output with faults
yf(t)
can be
quickly achieved under the action of disturbance compensation. Next, when the fault
occurs, the tracking effect is better than the method in [
13
], and the reference output
yr(t)
can be tracked as soon as the fault ends. Figures 3and 4illustrate the tracking
effect of the system output on the reference output. It can be seen that the system
can still achieve the tracking effect even under the influence of external disturbances
and faults. Figure 5represents the control signals that vary with the faults and ex-
ternal inputs. The ratio history curves of
rsup
0≤t≤ta
{eT
y(t)ey(t)}/Rta
0¯
wT(τ)¯
w(τ)dτ
and
qRta
0eT
y(τ)ey(τ)dτ/Rta
0¯
wT(τ)¯
w(τ)dτ
(i.e., the trajectories of
L2−L∞
and
H∞
performance)
are shown in Figures 6and 7, which means, under the influence of disturbance
w(t)
and
faults (
fs(t)
and
fa(t)
), the ratio curve can quickly tends to stabilize, and the maximum
value 0.0166 of
L2−L∞
performance and 0.0480 of
H∞
performance are lower than the
performance index
γ=
1. To sum up, these simulation results demonstrate that the FTC
method is effective and can ensure asymptotic stability and desired mixed
H∞
and
L2−L∞
performances for the system (8).
0 50 100 150 200 250 300 350 400 450 500
-0.1
0
0.1
0.2
0.3
0 50 100 150 200 250 300 350 400 450 500
-0.2
-0.15
-0.1
-0.05
0
Figure 2. System states x(t)and observer states ˆ
x(t).
Mathematics 2024,12, 3682 14 of 18
0 50 100 150 200 250 300 350 400 450 500
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Figure 3. System output with faults yf(t)and reference output yr(t).
0 50 100 150 200 250 300 350 400 450 500
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Figure 4. System tracking error ey(t).
0 50 100 150 200 250 300 350 400 450 500
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Figure 5. Adaptive control signal u(t).
Mathematics 2024,12, 3682 15 of 18
0 100 200 300 400 500
0
0.005
0.01
0.015
Figure 6. The trajectory of L2−L∞performance.
0 100 200 300 400 500
0
0.01
0.02
0.03
0.04
0.05
Figure 7. The trajectory of H∞performance.
In order to further verify the tracking effect of the system, the reference input is
changed to
ir(t) =
0.2
sin(
0.3
t)
. Compared with the results without the adaptive sensor
fault compensation term in [
13
], we can see from Figure 8that the system output con-
taining faults has a good tracking effect following the reference output, which shows the
effectiveness of the consequence.
Comparative Explanations:Compared with the design schemes of adaptive FTC for
actuator faults in [
11
,
13
] and overcompensation for external disturbances, that is, com-
pensation item
Rt
t0∥ξ(τ)∥dτ
will always exist after the disturbance and fault disappear.
Therefore, this paper designs an adaptive compensation scalar
ρ(t)
based on this, which
avoids channel congestion caused by too much data and saves communication resources
under the condition of limited bandwidth. Moreover, a more practical application scenario
of FTC is considered. The fault cases under consideration include not only partial degen-
erate faults of multiplicative type but also stuck fault of additive type. Both the observer
and controller of the joint design meet the mixed
H∞
and
L2−L∞
performances, and the
conservative design is reduced by introducing appropriate Lemmas 1and 2.
Mathematics 2024,12, 3682 16 of 18
0 50 100 150 200 250 300 350 400 450 500
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Figure 8. System output with faults yf(t)and reference output yr(t).
6. Conclusions
In this paper, the tracking problem of mixed
H∞
and
L2−L∞
adaptive FTC for
continuous-time IT2FSs is studied. The IT2 fuzzy model was used to figure out the problem
of mismatch and uncertainty of membership functions between system modules, while the
fuzzy observer and adaptive fault estimation functions were used to estimate the system
state and fault signals, in order for the adaptive tracking controller to realize the tracking
control of the reference output under disturbance and fault conditions. Both the jointly
designed observer and controller can satisfy the mixed
H∞
and
L2−L∞
performances
while ensuring the asymptotic stability of the system. In addition, the designed adaptive
control law can achieve the purpose of dynamic compensation for disturbances and faults,
and the conservatism of the observer and controller was further reduced by the slack
parameters in the lemmas. Finally, a mass-spring-damping system effectively validated
the design method. Our future work will improve the adaptive control algorithm on this
basis or adopt an active fault tolerance scheme to further improve the FTC capability of
the system.
Author Contributions: Conceptualization, M.-Y.Q. and X.-H.C.; methodology, M.-Y.Q. and X.-H.C.;
software, M.-Y.Q. and X.-H.C.; formal analysis, M.-Y.Q. and X.-H.C.; investigation, writing—original
draft preparation, and writing—review and editing, M.-Y.Q. and X.-H.C. All authors have read and
agreed to the published version of the manuscript.
Funding: This work was supported in part by the National Natural Science Foundation of China
under Grant 62173261.
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Conflicts of Interest: The authors declare no conflicts of interest.
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