INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int. J. Robust Nonlinear Control (2016)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3632
Robust fault diagnosis and fault-tolerant control for non-Gaussian
uncertain stochastic distribution control systems
Yuancheng Sun and Lina Yao*,†
School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China.
SUMMARY
The purpose of fault diagnosis of stochastic distribution control systems is to use the measured input and
the system output probability density function to obtain the fault estimation information. A fault diagnosis
and sliding mode fault-tolerant control algorithms are proposed for non-Gaussian uncertain stochastic distribution control systems with probability density function approximation error. The unknown input caused
by model uncertainty can be considered as an exogenous disturbance, and the augmented observation error
dynamic system is constructed using the thought of unknown input observer. Stability analysis is performed
for the observation error dynamic system, and the H1 performance is guaranteed. Based on the information of fault estimation and the desired output probability density function, the sliding mode fault-tolerant
controller is designed to make the post-fault output probability density function still track the desired distribution. This method avoids the difficulties of design of fault diagnosis observer caused by the uncertain
input, and fault diagnosis and fault-tolerant control are integrated. Two different illustrated examples are
given to demonstrate the effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons,
Ltd.
Received 4 March 2016; Revised 20 July 2016; Accepted 20 July 2016
KEY WORDS:
Stochastic distribution control systems; Uncertainty; Robust fault diagnosis; Fault-tolerant
control; Linear matrix inequality
1. INTRODUCTION
In order to improve the reliability of practical systems, fault diagnosis and fault-tolerant control
for dynamic systems have long been one of the important areas of control theory and applications
(see [1–3] for surveys). Most of the current fault diagnosis and fault-tolerant control algorithms for
stochastic systems aim at those stochastic systems subjected to Gaussian distribution, in which the
fault, stochastic inputs, and disturbance signals are supposed to be subjected to Gaussian distribution. However, this assumption is not fit for some practical application processes. In addition, the
shape of probability density function (PDF) of process variables needs to be controlled in some practical systems. For instance, the paper evenness controls the process of paper making, high polymer
polymerization process of chemical industry, flame combustion control, etc. The equations of these
systems describe the relationship between the input and output PDF rather than the traditional relationship between input and output. Such kinds of systems are called stochastic distribution control
(SDC) systems [4], which are proposed by professor Hong Wang. In the framework of non-Gaussian
SDC systems, it has very important theoretical significance and deep application prospect for complex industrial processes for the researchers to develop fault diagnosis and fault-tolerant control
technologies, which can be used in practical industrial systems, and the control product quality and
the distribution of indirect indicators are needed, such as paper-making industry.
*Correspondence to: Lina Yao, School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China.
† E-mail: yaoln@zzu.edu.cn
Copyright © 2016 John Wiley & Sons, Ltd.
Y. SUN AND L.YAO
With the study of fault diagnosis of non-Gaussian SDC systems, some fault diagnosis algorithms
have been proposed. For fault diagnosis of SDC systems, in which the information of system output
PDF and other measured information to generate residuals in order to analyze and estimate the
change of fault, observer or filter-based methods are mainly used so far [5–13]. A stable filterbased residual generator is constructed such that the fault can be detected and diagnosed for general
stochastic systems in [5]. In [6], a nonlinear neural network observer is designed for fault diagnosis
in which the adaptive tuning rule for network parameters is determined by the Lyapunov stability
theorem. In [7], a fault diagnosis algorithm is proposed based on iterative learning observer for SDC
systems. In [8], a fault diagnosis method based on robust filters is proposed for time-delayed SDC
systems, and a robust H1 fault diagnosis scheme is presented in [9]. Otherwise, in [12], a high-gain
nonlinear observer-based fault diagnosis approach is proposed for a class of nonlinear uncertain
systems with measurable output PDFs. In [13], a novel fault-estimation observer is designed for
Takagi–Sugeno (T–S) fuzzy systems with actuator faults, and the problem of fault-tolerant control is
addressed. For fault-tolerant control, the active fault-tolerant control is mainly used so far. Combined
with the controller design method without fault, such as optimal control, PI control, sliding mode
control [14], robust control, iterative learning control, model reference adaptive control, etc, the
controller can be reconfigured or reconstructed when fault occurs. In [15], a neural network-based
active fault-tolerant control scheme with fault alarm is proposed for a class of nonlinear systems.
For SDC systems, when the target PDF is known, the purpose of fault-tolerant control is to design a
fault-tolerant controller to make the post-fault output PDF of the systems still track the given PDF
after the fault happened, which has been shown in [4, 6, 7]. A fault-tolerant controller based on PI
tracking control scheme is designed to make the post-fault PDF still track the given distribution in
[6]. In [7], an optimal algorithm is proposed to reconfigure the controller to compensate the effect of
the fault on the system performance. When the target PDF is unknown, we introduce the concept of
entropy to fault-tolerant control of non-Gaussian SDC systems; the purpose of fault-tolerant control
is to minimize the uncertainty of the system output after the fault happened [10, 11].
Because of the complexity of practice systems, the difference between the model and the practical object caused by simplified system modeling, environmental changes, or parameter drift is
called model uncertainty. This uncertainty makes the application of fault diagnosis and fault-tolerant
control technique more difficult and reduces the sensitivity and accuracy of fault diagnosis and faulttolerant control. The main control method for the uncertain system is robust control [16–20]. In
[21], the problem of fault-tolerant control for a class of uncertain nonlinear systems with actuator
fault is discussed, and an observer-based fault-tolerant control scheme is proposed. The uncertainty
considered in [21] is in the unknown control gain functions rather than in the system function. There
are very few literatures about fault diagnosis and fault-tolerant control of uncertain SDC systems.
In [22], a robust tracking control method is proposed for uncertain SDC systems to eliminate the
influence of uncertainty and B-spline approximation error, but fault diagnosis is not involved. A
robust nonlinear adaptive observer-based fault diagnosis algorithm for SDC systems approximated
by square-root model is presented in [23], but a pre-designed tracking controller is needed to deal
with the uncertain input, and fault-tolerant control is not considered.
From the aforementioned literatures, it can be shown that the existing literatures on uncertain
systems are aimed at either fault diagnosis or design of controller. They can not integrate the two
parts, and a pre-specified input is always needed. Besides, SDC is a new research field; only few
works were focused on the study of uncertain SDC systems. The application of sliding mode technology to fault-tolerant control for uncertain SDC systems is not reported. In this paper, robust fault
diagnosis and fault-tolerant control algorithms are proposed to diagnose the fault and eliminate the
impact of model uncertainty and PDF approximation error on uncertain SDC systems. First, a fault
diagnosis observer is constructed, and the uncertain input in the observation error can be regarded
as an unknown exogenous disturbance in order to design an augmented observation error dynamic
system. In this way, the integration of fault diagnosis and fault-tolerant control can be achieved.
Then, the augmented observation error dynamic system is proved stable, and H1 performance is
guaranteed. The desired gain matrices are obtained by solving linear matrix inequalities (LMIs).
Because sliding mode control technology has good robustness to uncertain systems, a sliding mode
fault-tolerant controller is designed to compensate the performance degeneration caused by fault,
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
uncertainty, and PDF approximation error making the post-fault output PDF still track the desired
distribution. Finally, the simulation results show the effectiveness of the algorithm to time-varying
fault. The main contributions of this work can be summarized as follows: (1) for the uncertain
SDC system, integrated fault diagnosis and fault-tolerant control are achieved; (2) the whole uncertainty including B is considered in fault diagnosis and tolerant control; and (3) the impact of PDF
approximation error is considered.
The rest of this paper is organized as follows. The model description is given in Section 2. In
Section 3, a fault diagnosis algorithm is proposed. A fault-tolerant controller based on sliding mode
technology is designed in Section 4. Simulation results and discussion are included in Section 5.
Finally, some concluding remarks are shown in Section 6.
2. MODEL DESCRIPTION
In the paper-making machine system, fibers can be regarded as major components. Because the
fiber network is randomly formed, the pore sizes are also random and should obey a stochastic
distribution. Experimentally, it has been found that the radius of the pores can be approximated
by a truncated distribution. In this case, the following truncated type PDF can be used to
approximate the flocculated size distribution [4]
ˇ ˇ 1
y
ˇ
.y; ; ˇ/ D
e yˇ = 8y 2 Œa; b;
.ˇ/
where y 2 Œa; b represents the flocculation size, a and b are the minimum and maximum flocculation sizes, respectively. In practice, a and b can be determined experimentally, and .ˇ/ stands for
the well-known Gamma function. It can thenıbe shown that the mean value is expressed as yN D ,
and the variance is denoted by Var.y/ D 2 ˇ. This indicates that controls the mean of the distribution while the spread and shape of the PDF are controlled by 1=ˇ. The system state vector
can be selected as and ˇ, and system input u.t / is either the flow rate or the concentration of the
retention polymer.
Denote y.t / 2 Œa; b as the output of a paper-making machine system at time instant t and assume
that it is uniformly bounded, and then, y.t / can be characterized by its conditional PDF .y; u.t //,
which is defined by
Z b
P .a 6 y.t / < &ju.t // D
.y; u.t //dy;
a
where P .a 6 y.t / < & ju.t // is the probability of the output y.t / lying inside the interval Œa; & /
when u.t / is applied to the system. In practice, it is difficult to obtain the exact expression of the
PDF albeit the distribution sample can be obtained easily. Therefore, for the general non-Gaussian
SDC systems, the linear B-spline model is adopted to approximate the PDF .y; u.t // of the system.
Denote that 1 .y/; 2 .y/; ; n .y/ are n pre-specified B-spline basis functions on the interval
Œa; b, and !1 ; !2 ; ; !n are the corresponding weights associated with the input u.t /, then
.y; u.t // D
n
P
!i .u.t //i .y/ C e0 .y; t /;
(1)
i D1
where e0 .y; t / is the PDF approximate error. As such, the changes of the output PDF .y; u.t //
can be regarded as being caused by the corresponding changes in the weights !i .u.t //. Notice that
.y; u.t // is a PDF, and its integral on the interval Œa; b is 1. Consequently, the following equation
holds
!1 b1 C !2 b2 C C !n bn D 1;
(2)
Rb
where bi D a i .y/dy .i D 1; 2; ; n/ are positive constants when the basis functions are
selected. Therefore, there are only n 1 independent weights, and the linear B-spline model can be
simplified as follows:
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
.y; u.t // D C.y/V .t / C T .y/ C e0 .y; t /;
(3)
where,
T .y/ D nb.y/
2 R11 ;
n
V .t / D Œ!1 ; !2 ; ; !n1 T 2 R.n1/1 ;
1
2
; 2 .y/ n .y/b
; ; n1 .y/ C.y/ D Œ1 .y/ n .y/b
bn
bn
n .y/bn1
bn
2 R1.n1/ :
Then the continuous time model of the uncertain SDC system can be described as follows:
x.t
P / D .A C A/x.t / C .B C B/u.t / C GF .t /
V .t / D Dx.t /
.y; u.t // D C.y/V .t / C T .y/ C e0 .y; t /;
(4)
where x.t / 2 Rm1 is the system state vector (such as and ˇ in the paper-making machine
system), u.t / 2 Rm1 is the control input vector, and F .t / 2 Rr1 is the fault vector. ¹A; B; D; Gº
are proper constant matrices, and A and B are the model uncertainties that can be described by
the following equation:
ŒA; B D EN.t /ŒH1 ; H2 ;
(5)
where E, H1 , and H2 are proper constant matrices. N.t / is a time-varying matrix and satisfies
N T .t /N.t / 6 I , where I is an identity matrix.
Remark 1
Generally, in the SDC systems, the output PDF .y; u.t // is the control object, which can be
measured or estimated by using instruments (for example, digital camera) or Bayesian estimation
technique. The first equation of (4) is the inherent characteristic of this dynamic system and independent of the choice of the basis functions. The second one describes the relationship between system
state vector and weight vector, such that the input u.t / can decide the output PDF .y; u.t //. The
third one is the static model of the output PDF. The SDC system dynamic and static models in (4)
are modeled in Section 2.4 of [4] by Wang.
Assumption 1
Suppose that the fault and its first-order derivative are bounded, and the upper bound of the fault
size is M =2, that is, kF k 6 M
, where M > 0 is a positive constant.
2
Lemma 1
Q N; HQ are real matrices with appropriate dimension and kN k 6 1, and then for any
Assume that E;
scalar ı > 0, the following inequality holds.
Q HQ 6 ı EQ EQ T C ı 1 HQ T HQ :
HQ T N T EQ T C EN
3. FAULT DIAGNOSIS ALGORITHM
The purpose of fault diagnosis is to determine the time when fault occurs and estimate the size of
fault in order to provide the information for design of the subsequent fault-tolerant control. A fault
diagnosis observer is constructed as follows:
PO / D Ax.t
x.t
O / C Bu.t / C G FO .t / C Kd ".t /
VO .t / D D x.t
O /
O .y; u.t // D C.y/VO .t / C T .y/
Rb
".t / D a .y/.O .y; u.t // .y; u.t ///dy;
Copyright © 2016 John Wiley & Sons, Ltd.
(6)
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
where x.t
O / is the estimated system state vector, FO .t / is the estimated fault vector, ".t / is the residual signal, and Kd is the observer gain matrix with appropriate dimensions to be determined later.
Denote the observation error vector as ed .t / D x.t
O / x.t /, and let the fault estimation error be
denoted as FQ .t / D FO .t / F .t /. The residual signal can be described as
Rb
".t / D a .y/.O .y; u.t // .y; u.t ///dy
Rb
Rb
D a .y/C.y/dy.VO .t / V .t // a .y/e0 .y; t /dy
D †Ded .t / .t /;
(7)
Rb
Rb
where † D a .y/C.y/dy, .t / D a .y/e0 .y; t /dy, and .y/ is the pre-specified adjustment
factor defined on Œa; b. The observation error dynamic system can be formulated as
ePd .t / D .A C Kd †D/ed .t / C G FQ .t / Ax.t / Bu.t / Kd .t /:
(8)
For the traditional adaptive observer-based fault diagnosis method, it is usually assumed that
the fault is not time varying, that is, FP .t / D 0. Thus the first-order derivative of fault estimation
error can be calculated as FPQ .t / D FPO .t /. Besides, when dealing with uncertain systems, the common approach is to denote the output or state feedback as the system input. This brings a series of
difficulties of designing the fault-tolerant controller. In this paper, an augmented observation error
dynamic system is obtained, and the uncertain input and PDF approximation error can be considered
as unknown disturbances. The adaptive fault estimation algorithm can be given as follows:
FPO .t / D ".t /;
(9)
where is the gain matrix with appropriate dimensions to be determined later. Then it can be
obtained that
FPQ .t / D †Ded .t / .t / FP .t /:
(10)
The augmented observation error dynamic system can be constructed as
N /
PN / D ANe.t
e.t
N / C Bv.t
Q
N
F .t / D I e.t
N /;
(11)
3
x.t /
6 u.t / 7
ed .t /
A B 0 Kd
A C Kd †D G
6
7
N
N
where e.t
N /D
,B D
,
, v.t / D 4 P 5, A D
†D
0
0
0 I FQ
F .t /
.t /
AG
N
N
N
N
N
N
N
I D Œ0 I . A can be decomposed as A D A1 C LC , where A1 D
2 R.mCr/.mCr/ ,
0 0
Kd
LN D
2 R.mCr/r , CN D †D 0 2 Rr.mCr/ .
2
Lemma 2
(Bounded real lemma for continuous-time systems [24]): The continuous-time linear system
²
Copyright © 2016 John Wiley & Sons, Ltd.
x.t
P / D Ax.t / C B!.t /
y.t / D C x.t / C D!.t /
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
satisfies H1 performance ky.t /k2 < k!.t /k2 , if and only if, there exists a positive definite
symmetric matrix P satisfying the following LMI.
2
3
PA C AT P PB C T
4
I D T 5 < 0
I
Lemma 3
(Region poles assignment lemma of ˛ stability margin [25]): The eigenvalues of a given matrix
A 2 Rnn is assigned in the region D˛ , if and only if, there exists a positive definite symmetric
matrix P 2 Rnn satisfying the following LMI.
AT P C PA C 2˛P < 0
Theorem 1
For the SDC system (4), it is supposed that Assumption 1 is satisfied. For the parameters
; ˛ > 0
P12
P
11
2
and given constant ı > 0, there exist a positive definite symmetric matrix PN D
P
P22
21
YN
R.mCr/.mCr/ and matrix YN D N1 2 R.mCr/r to make the following LMI hold. Then with the
Y2
gain matrices Kd and , the augmented observation error dynamic system (11) is stable, and the
N /k2 < kv.t /k2 is achieved.
H1 performance ke.t
2
'11
6 6
6 6
6 ˆD6
6 6
6 4 '12
'22
0
0
'33
3
0 P12 YN1 I P11 E
0 P22 YN2 I P21 E 7
7
0
0
0
0
0 7
7
'44 0
0
0
0 7
7<0
I 0
0
0 7
7
I 0
0 7
I 0 5
ıI
PN AN1 C ANT1 PN C YN CN C CN T YN T C 2˛ PN < 0;
(12)
(13)
where
N '11 D P11 A C AT P11 C YN1 †D C .YN1 †D/T , '12 D P11 G C AT P12 C .YN2 †D/T ,
YN D PN L,
'22 D P21 G C G T P12 , '33 D I C ıH1T H1 , '44 D I C ıH2T H2 .
Proof
It is supposed that Lemmas 2–3 are satisfied; then, the following LMI can be obtained from (11):
3
'1 PN BN I
ˆ1 D 4 I 0 5 < 0 ;
I
(14)
PN AN1 C ANT1 PN C YN CN C CN T YN T C 2˛ PN < 0 ;
(15)
2
N Substitute AN1 , B,
N CN , PN and YN into (14), and
where '1 D PN AN1 C ANT1 PN C YN CN C CN T YN T ; YN D PN L.
then, it can be calculated that
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
2
'11
6 6
6 6
ˆ1 D 6 6
6 4 '12 P11 A P11 B P12 YN1 I
'22 P21 A P21 B P22 YN2 I
I
0
0
0
0
I
0
0
0
I 0
0
I 0
I
3
7
7
7
7
7 < 0:
7
7
5
ˆ1 can be decomposed as ˆ1 D ˆ11 C ˆ12 , where
2
ˆ11
'11
6 6
6 6
D6 6
6 4 '12 0
0 P12 YN1 I
0 P22 YN2 I
'22 0
I 0
0
0
0
I 0
0
0
I 0
0
I 0
I
3
7
7
7
7
7:
7
7
5
From ŒA; B D EN.t /ŒH1 ; H2 , it can be obtained that
3
3
2
2
0
P11 E
6 P21 E 7
6 0 7
7
6
6 T7
6 0 7
6 H1 7 h
i
7
7
6
6
ˆ12 D 6 0 7 N Œ0 0 H1 H2 0 0 0 C 6 H2T 7 N .P11 E/T .P21 E/T 0 0 0 0 0 :
7
6 0 7
6
7
6
6 0 7
4 0 5
4 0 5
0
0
From Lemma 1, it can be formulated that
3
2
2
0
P11 E
6 P21 E 7
6 0
7
6
6 T
6 0 7h
6 H1
i
7
6
6
T
T
ˆ12 6 ı 1 6 0 7 .P11 E/ .P21 E/ 0 0 0 0 0 C ı 6 H2T
6 0 7
6
7
6
6 0
4 0 5
4 0
0
0
D ˆ13 :
3
7
7
7
7
7 Œ0 0 H1 H2 0 0 0
7
7
5
From the Schur complement lemma, ˆ D ˆ11 C ˆ13 holds. Then ˆ < 0 is equivalent to ˆ1 < 0.
Theorem 1 is proved. When PN and YN are calculated from Theorem 1, the gain matrices Kd and can be obtained as LN D PN 1 YN , leading to robust fault diagnosis of SDC system (4).
Remark 2
Theorem 1 can guarantee the stability of the augmented observation error dynamic system. At the
same time, state observation error and fault estimation error converge as the stability margin index
˛ and H1 performance is achieved. Bounded real lemma can ensure the stability of the observation
error dynamic system, and region poles assignment lemma can make the eigenvalues locate in the
left region of ˛ in complex plane to improve the reliability of fault estimation error system.
4. FAULT-TOLERANT CONTROL
After estimating the fault using the proposed fault diagnosis algorithm, it is necessary to design a
fault-tolerant controller to make the post-fault output PDF still track the desired PDF. Because of
the existence of model uncertainty and PDF approximation error, the perfect tracking is impossible
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
to be achieved. As sliding mode technology has good robustness to uncertain systems [18], a sliding
mode fault-tolerant controller is designed to make the distribution tracking error converge to the
sliding mode in finite time, and the tracking error dynamic system is stable. The desired target PDF
is expressed as follows:
g .y/ D C.y/Vg C T .y/;
(16)
where Vg represents the desired weight vector. Denote the weight tracking error as ev .t / D V .t / Vg , where Vg D Dxg . Then, xg D D C Vg is the desired state variable, where D C is the pseudoinverse of D. Then, the weight tracking error dynamic system can be obtained as follows:
ePx .t /D x.t
P / xP g
D x.t
P /
D .A C A/x.t / C .B C B/u.t / C GF .t /
D .A C A/x.t / .A C A/xg C
.A C A/xg C .B C B/u.t / C GF .t /
D .A C A/ex .t / C .A C A/xg C .B C B/u.t / C GF .t /;
(17)
where ex .t / D x.t / xg .
For sliding mode control, two requirements need to be satisfied: the reachability of the system
state and the asymptotic stability of the sliding mode. By designing a sliding mode control law,
the moving point that started from any position can reach the switching surface in limited time.
Therefore, the asymptotical stability of sliding mode can be realized by designing an appropriate
switching function. The sliding mode control law is designed as
u.t / D ueq .t / C un .t /;
(18)
where ueq .t / is the equivalent control part that guarantees the existence of sliding mode domain,
and un .t / reflects the discontinuity of the sliding mode control that makes the system state tend to
the sliding mode domain from any position.
When the switching function is designed, the available information is the certain system matrices.
Denote A D 0 and B D 0. The tracking error dynamic system can be obtained as
ePx .t / D Aex .t / C Bu.t / C Axg C GF .t /:
(19)
According to dynamic system (19), the following integral switching function can be designed:
s.t / D Lex .t / Rt
0 L.A
C BK/ex . /d ;
(20)
where the selection of matrix K can guarantee that matrix A C BK is a Hurwitz matrix, and L is
chosen such that LB is nonsingular.
Remark 3
If matrix B is a nonsingular matrix, L can be an identity matrix.
By the equivalent control method, the equivalent control part can be obtained when sP .t / D 0
ueq .t / D Kex .t / L Axg L GF .t /;
(21)
where L D .LB/1 L. Substituting (21) into (4) and (17), the post-fault closed-loop dynamic
system and the tracking error dynamic system can be formulated as follows, respectively:
x.t
P / D .A C A/x.t / C .B C B/Kex .t / .B C B/L Axg
C.I BL BL /GF .t /
Copyright © 2016 John Wiley & Sons, Ltd.
(22)
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
ePx .t / D .A C A C BK C BK/ex .t / C .A C A BL A BL A/xg
C.I BL BL /GF .t /:
(23)
A new state vector can be defined as ´.t / D ŒexT .t / x T .t /T , and then,
Ṕ .t / D A´ ´.t / C Ax xg C Af F .t /;
where A´ D
(24)
Q 0
Q A
Q G
AQ C BK
AQ C BL
G BL
Q
, Ax D
Q A , Af D G BL
Q G , A D A C A,
Q
BL
BK
AQ
BQ D B C B.
Theorem 2
For the SDC system (4), the integral switching
function
(20) is chosen. It is supposed that there exist
P1 0
a positive definite symmetric matrix P D
and K such that the following LMI holds;
0 P2
then, the post-fault closed-loop dynamic system (22) and the sliding mode tracking error dynamic
system (23) are stable, where 1 and 1 are given constants,
2
3
M1 Y1 B T M2
M3 .H1 X1 C H2 Y1 /T .H2 Y1 /T
6 M4 BL A M3
0
.H1 X2 /T 7
6
7
T
T 7
6 2
I
0
.H
H
L
A/
.H
L
A/
1
2
2
6
7 < 0;
1
ˆD6
22 I
.H2 L G/T
.H2 L G/T 7
6 7
4 5
ıI
0
ıI
(25)
where M1 D X1 AT C AX1 C Y1T B T C BY1 CıEE T C I , M2 D A BL A, M3 D G BL G,
X1 0
, X1 D P11 , X2 D P21 , Y1 D KX1 .
M4 D AX2 C X2 AT C ıEE T , X D P 1 D
0 X2
Proof
A Lyapunov function is selected as follows:
…1 .t / D ´T .t /P ´.t /:
Then, the first-order derivative of …1 .t / can be obtained as follows:
P 1 .t / D ´T .t / AT´ P C P A´ ´.t / C 2´T .t /P Ax xg C 2´T .t /P Af F .t /
…
6 ´T .t /ˆ0 ´.t / C
where ˆ0 D AT´ P C P A´ C
2 T
1 xg xg
1
P Ax ATx P
12
C
Pre-multiplying and post-multiplying X D P
can be transformed as ˆ0 D ˆ1 C ˆ2 , where
C
2 T
2 F .t /F .t /;
1
P Af
22
1
ATf P .
to the left and the right side of ˆ0 , and then, ˆ0
3
M11
Y1T B T
M12 M13
6 AX2 C X2 AT BL A M13 7
7;
ˆ1 D 6
4 21 I
0 5
21 I
2
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
where M11 D AX1 CX1 AT CY1T B T CBY1 , M12 D ABL A, M13 D G BL G, Y1 D KP11 ;
3
Y1T B T
M22 M22
M21
6 AX2 C X2 AT BL A M22 7
7;
ˆ2 D 6
4 0
0 5
0
2
where M21 D X1 AT C Y1T B T C AX1 C BY1 , M22 D A BL A, M23 D BL G.
From Lemma 1, it can be obtained that
2
3
.H1 X1 C H2 Y1 /T .H2 Y1 /T
E 0 T
6
0
.H1 X2 /T
0 00
6 0 E7 E
ˆ2 6 ı 4
C ı 1 6
5
T
T
4 .H1 H2 L A/ .H2 L A/T
0 0
0 E 00
0 0
.H2 L G/T
.H2 L G/T
H1 X 1 C H 2 Y
0 H1 H2 L A H 2 L G
:
H2 Y1
H1 X2
H 2 L A
H 2 L G
2
3
7
7
5
From the Schur complement lemma, it can be fomulated that
2
3
M31
Y1 B T
M12 M13 .H1 X1 C H2 Y1 /T .H2 Y1 /T
6 AX2 C X2 AT BL A M3
0
.H1 X2 /T 7
6
7
T
T 7
6 2
I
0
.H
H
L
A/
.H
L
A/
1
2
2
6
7;
1
ˆ0 6 ˆ3 D 6
T
T 7
2
I
.H
L
G/
.H
L
G/
6
7
2
2
2
4 5
ıI
0
ıI
where M31 D X1 AT C Y1T B T C AX1 C BY1 C ıEE T . Adding I on M31 , we have M1 D
M31 C I . Then ˆ is obtained, where X D P 1 and Y1 D KX1 . When ˆ < 0 holds, it can be
acquired that ˆ0 < I , so that the following inequality
P 1 .t / < ´T .t /´.t / C
…
is satisfied. Thus, when k´.t /k2 >
T x C 2 M 2
412 xg
g
2
2 T
1 xg xg
T x C 2 M 2
412 xg
g
2
4
C
2 T
2 F .t /F .t /
P 1 .t / < 0, leading to k´.t /k2 6
holds, …
. Then post-fault closed-loop dynamic system (22) and the sliding mode tracking
4
error dynamic system (23) are stable, simultaneously. Theorem 2 is proved.
In order to ensure that the state trajectory starting from any position can reach the switching
surface in limited time, the following sliding mode switching control law is designed:
´
un D
where ˇ > 0 is a small constant.
ˇ s.t / 2 ; s.t / ¤ 0
ks.t /k
0
; s.t / D 0;
(26)
Theorem 3
For the SDC system (4), the integral switching function (20) is chosen. The sliding model control
law (24) can ensure that the system state trajectory can reach the switching surface s.t / D 0 in
limited time.
Proof
N 1 s.t /, then the first-order derivative of
A Lyapunov function is selected as …2 .t / D 12 s T .t /.LB/
…2 .t / can
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
P 2 .t / D s T .t /.LB/1 sP .t /
…
N 1 L.A C BK/ex .t /
D s T .t /.LB/1 L.Aex .t / C Bu.t / C G FO .t / C Axg / s T .t /.LB/
D s T .t /.LB/1 LBun .t /
D ˇ
s T .t /s.t /
ks.t /k2
D ˇ < 0:
Therefore, the system state trajectory can reach the switching surface in limited time. Theorem 3 is
proved.
By substituting the state estimation x.t
O / and fault estimation FO .t / for x.t / and F .t / into (21),
the practical sliding mode fault-tolerant controller can be obtained as follows:
u.t / D ´
ueq .t / C un .t /
K.x.t
O / xg / L Axg L G FO .t / ˇ s.t / 2 ; s.t / ¤ 0
ks.t /k
D
K.x.t
O / xg / L Axg L G FO .t /; s.t / D 0:
(27)
5. SIMULATION EXAMPLES
To illustrate the effectiveness of the proposed algorithms in this paper, a typical example in a papermaking machine as shown in Figure 1 is considered [4]. The fiber flocculation and filler distribution
in the white water pit are mainly controlled by the filler input, the fiber input, and the chemical aid
such as retention polymer. For the example of the paper-making wet control system, the flocculation
size distribution in the related system are all subjected to unexpected changes, which are regarded
as fault in the system, such as the shear force variations in the headbox approaching system. The
unexpected environmental changes can make the system parameters different from the identification
ones, which are regarded as model uncertainties. Therefore, fault diagnosis and fault tolerant are
needed to ensure that faulty paper-making system can still work stably, and the post-fault system
performance is made to be close to the desired one as far as possible.
We consider two SDC systems whose output PDFs can be approximated by the following B-spline
functions i .y/; .i D 1; 2; 3; 4/,
1 .y/ D
2 .y/ D
3 .y/ D
4 .y/ D
1
.y
2
1
.y
2
1
.y
2
1
.y
2
C 3/2 I1 C .y 2 3y 1:5/I2 C 12 .y 0/2 I3
C 2/2 I2 C .y 2 y 0:5/I3 C 12 .y 1/2 I4
C 1/2 I3 C .y 2 C y 0:5/I4 C 12 .y 2/2 I5
C 0/2 I4 C .y 2 C 3y 1:5/I5 C 12 .y 3/2 I6 ;
Figure 1. A paper-making machine wet end.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
²
1; y 2 Œi 4; i 3
; i D 1; 2; 3; 4; 5; 6. Denote the PDF approximation error as
0; ot herwise
e0 .y; t / D a sin. y /, y 2 Œ0; 2, where a is a random number defined on Œ0; 0:02. The initial
weight state vector of robust fault diagnosis observer and the desired weight vector are selected as
V0 D Œ0:1 0:1 0:1T and Vg D Œ0:3 0:6 0:1T . H1 performance index is chosen as D 2:8, and
stability margin index is ˛ D 0:2. The sampling time is 0.1 min and total simulation time is 5 min.
where Ii D
5.1. Example 1
The model parameter matrices and vectors are given as follows:
3
3
2
3
2
3
2
2
100
3
0:5 0:3 0
5 2 0
A D 4 0:8 4 0 5 ; B D 4 0 0:9 0:1 5 ; G D 4 1 5 ; D D 4 0 1 0 5 ;
001
1:5
0:5 0:2 0:1
0 2 3
3
2
0:02
E D 4 0:01 5 ; H1 D 0:1 0:1 0:2 ; H2 D 0:1 0:1 0:1 :
0:1
N.t / is a time-varying function subjected to uniform distribution in Œ0:3 0:3. To validate the
algorithm, it is assumed that the time-varying fault for system 1 has the following form:
8
0; t 6 10s
ˆ
ˆ
<
0:5 C 0:1 .t 10/; 10s < t 6 15s
F .t / D
ˆ 2 e 0:55.t 15/ ; 15s < t 6 30s
:̂
1; t > 30s:
From the LMIs of Theorems 1 and 2, the gain matrices are calculated as follows:
3
2
0:5961 0:0516 0:3993
D 4:2498; Kd D Œ0:5554 0:0734 1:2572T ; K D 4 0:2223 0:6177 0:0393 5 ;
0:7705 0:2770 0:2732
and the corresponding positive definite symmetric matrices are given as follows:
2
2:9419
2:5300
6
PN D 4
1:5729
1:2620
2:5300
5:3751
1:9460
0:9499
1:5729
1:9460
2:8944
0:4480
3
1:2620
0:9499 7
0:4480 5
1:8436
3
3
2
0:3448 0:1097 0:0012
0:3710 0:1098 0:0154
P1 D 4 0:1097 0:2975 0:0601 5 P2 D 4 0:1098 0:3356 0:0611 5 :
0:0012 0:0601 0:1910
0:0154 0:0611 0:2094
2
The fault diagnosis result has been presented in Figure 2, and it is seen that the estimation of fault
can track the fault value quickly and accurately, leading to the desired fault diagnosis result. When
fault occurs, the controller is reconfigured, and fault-tolerant control is introduced to eliminate the
impact of fault actively. Figure 3 shows the response of control input u.t / of the whole fault-tolerant
control process. The initial PDF, the desired PDF, and the final PDF with and without fault-tolerant
control are shown in Figures 4 and 5, respectively. It is shown that the impact of fault in Figure 5.
Figure 6 is the output PDF 3D mesh of the whole process with fault-tolerant control, which shows
the changes of the measured output PDF that is subjected to fault. From these figures, it can be
concluded that the actual output PDF can still track its target PDF, and the performance degradation
is compensated.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
Figure 2. Fault and fault estimation.
Figure 3. The response of control input u.t/.
Figure 4. The output probability density function with fault-tolerant control when fault occurs.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
Figure 5. The output probability density function without fault-tolerant control when fault occurs.
Figure 6. The output probability density function of the whole process with fault-tolerant control.
5.2. Example 2
The model parameter matrices are given as follows:
3
3
2
2
0:12 1:12 0:22
3:8 1:5 0:5
A D 4 0:5 3 1 5 ; B D 4 0:61 0:66 1:73 5 :
0:76 0:6 0:98
0:3 0:7 2:4
Other matrices are the same as those of example 1. It is assumed that the time-varying fault has the
following form:
²
0; t < 15s
F .t / D
0:5 C 0:06 .t 15/; t > 15 s:
From the LMIs of Theorems 1 and 2, the gain matrices are calculated as follows:
3
2
0:0019 0:1186 0:0599
D 8:8114; Kd D Œ0:5134 0:3398 0:8710T ; K D 4 0:2227 0:1006 0:1490 5 ;
0:1338 0:0509 0:1270
and the corresponding positive definite symmetric matrices are given as follows:
3
2
13:3651 13:9891 7:7293 1:0935
6 13:9891 19:4083 9:4826 0:1174 7
PN D 4
7:7293 9:4826 7:5700 0:7219 5
1:0935 0:1174 0:7219 2:0501
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
3
3
2
0:2362 0:0714 0:0203
0:2501 0:0723 0:0268
P1 D 4 0:0714 0:2047 0:0689 5 ; P2 D 4 0:0723 0:2118 0:0613 5 :
0:0203 0:0689 0:1599
0:0268 0:0613 0:1672
2
The fault diagnosis result has been presented in Figure 7. The initial PDF, the desired PDF, and
the final PDF with fault-tolerant control is shown in Figure 8. Figures 9 and 10 show the output PDF
3D mesh of the whole process without and with fault-tolerant control for the example 2, respectively.
When the fault occurred in 15 s, the fault-tolerant controller intervene into the system quickly to
control the shape of output PDF still track the desired one.
From the aforementioned two simulation examples, it can be seen that the fault diagnosis and
fault-tolerant control algorithm is effective to accurately estimate two kinds of complex time-varying
fault for two different uncertain SDC systems with PDF approximation error. The sliding mode faulttolerant controller is designed using the feedback of state variables tracking error, and the accurate
output PDF tracking is almost impossible to be achieved because of the existence of approximation
error. Thus, the purpose of fault-tolerant control is to make the output PDF still track the desired PDF
as close as possible to eliminate the deterioration caused by fault and uncertainty. The simulation
results show the effectiveness of the fault-tolerant control algorithm.
Figure 7. Fault and fault estimation.
Figure 8. The output probability density function with fault-tolerant control when fault occurs.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
Y. SUN AND L.YAO
Figure 9. The output probability density function of the whole process without fault-tolerant control.
Figure 10. The output probability density function of the whole process with fault-tolerant control.
Remark 4
In [22], an augmentation control input is designed based on the feedback of integral of output PDF
tracking error, which is different from this paper in design of the controller , but perfect tracking is
still impossible. The control target in [22] is that the distribution tracking error at each time instant
satisfies a certain upper bound beyond a limited time.
6. CONCLUSIONS
In this paper, robust fault diagnosis and sliding mode fault-tolerant control algorithms of nonGaussian uncertain SDC systems with PDF approximation error are proposed. A fault diagnosis
observer is constructed, and the uncertain input and PDF approximation error in the observation
error can be regarded as unknown exogenous disturbances, such that an augmented observation error
dynamic system is obtained. The stability and H1 performance of the augmented observation error
dynamic system is proved. Using the information of fault estimation, a sliding mode fault-tolerant
controller is constructed to ensure that the distribution tracking error dynamic system is stable. The
gain matrices are obtained by solving LMIs. In this way, the integration of fault diagnosis and faulttolerant control can be achieved. Finally, encouraging results are obtained by the simulation results
for two kinds of complex time-varying fault as well. Further research includes improving the control
performance with PDF approximation error and studying on the fault diagnosis and fault-tolerant
control for singular uncertain SDC systems.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc
ROBUST FAULT DIAGNOSIS AND FAULT-TOLERANT CONTROL FOR SDC SYSTEMS
ACKNOWLEDGEMENTS
This work was supported by Chinese NSFC grant 61374128, the Science and Technology Innovation
Talents 14HASTIT040 in Colleges and Universities in Henan Province, China, and Excellent Young
Scientist Development Fundation 1421319086 of Zhengzhou University, China.
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Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control (2016)
DOI: 10.1002/rnc