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Fault detection and identification for a class of nonlinear systems with model uncertainty
Accepted Manuscript
Fault detection and identification for a class of nonlinear systems with model uncertainty Bingyong Yan , Housheng Su , Wei Ma PII: DOI: Reference:
S0307-904X(16)30143-3 10.1016/j.apm.2016.03.018 APM 11086
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
2 March 2015 15 November 2015 7 March 2016
Please cite this article as: Bingyong Yan , Housheng Su , Wei Ma , Fault detection and identification for a class of nonlinear systems with model uncertainty, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.03.018
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ACCEPTED MANUSCRIPT
A novel fault detection and identification approach is proposed based on FTA. The FTA can detect and identify system faults simultaneously. The FTA based FDI approach is applied to a flexible joint robotic system.
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Highlights
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Fault detection and identification for a class of nonlinear systems with model uncertainty
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Bingyong Yan1, Housheng Su2, Wei Ma3 1. School of Automation, Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China. 2. School of Automation,Key Laboratory of Image Processing and Intelligent Control of Ministry of Education of China, Huazhong University of Science and Technology, Wuhan 430074, China 3. Key Laboratory for Advanced Materials & Institute of Fine Chemicals, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China. Correspondence author: Housheng Su. Email: [email protected].
Abstract – In this paper, we present a novel fault detection and identification (FDI) scheme for a class of
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nonlinear systems with model uncertainty. At the heart of this approach is an on-line approximator, referred to as fault tracking approximator (FTA). Differently from the other approximators, the FTA uses iterative algorithms to detect and identify nonlinear system faults, even in the presence of model uncertainty, which is motivated by predictive control theory and iterative learning control theory. The FTA can simultaneously detect and identify the shape and magnitude of the faults. The rigorous stability analysis and fault tracking properties of the FTA are also proved. Finally, two examples are given to illustrate the feasibility and effectiveness of the proposed approach.
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Index Terms –fault detection and identification; predictive control; iterative learning control
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I. INTRODUCTION
In the last two decades, the analysis and design of fault detection and identification (FDI) schemes for dynamic systems have received considerable attention due to the increasing demand for system safety and reliability [1-27]. A large class of engineering
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systems are inherently nonlinear, which include, among others, power systems, chemical processes, and robotic manipulators with homonymic constraints [28-37]. So the study of FDI algorithms for nonlinear systems plays an important role in practical
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applications. In recent years, FDI problems for nonlinear systems have attracted much attention among researchers and many fruitful results have been achieved [1-7]. In [21], the geometric approach is used to design fault detection filters. The reconstruction of the fault signal is formulated as a control problem. For a class of affine nonlinear systems subjected to an
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unknown time-varying fault vector, a fault diagnosis filter is designed to estimate the time-varying fault by using nonlinear observer techniques [23]. For a class of nonaffine nonlinear systems, the adaptive control problems for fault tolerant control (FTC) are investigated in [28] and [29]. The existing approaches for fault detection and diagnosis and fault-tolerant control in a general framework of active fault-tolerant control systems are considered in [22]. Fruitful results on fault diagnosis and fault tolerant control (FTC) can be found in [24-27], and the reference therein. With the development of the processing speed of microprocessor as well as the new technologies which reduce the cost of computer chips, the learning approaches are rapidly becoming an issue of primary importance in automated condition monitoring and fault diagnosis [8]. In [8-10], a general learning methodology for fault diagnosis was developed and its stability properties were also investigated. The learning methods are based
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on adaptive algorithms and on-line approximation structures such as neural network, fuzzy logic networks, etc. However, in these neural network based approaches, there is no predetermined way of finding an optimum network size and architecture, and the online training of the neural network will be time-consuming. Moreover, it is often quit difficult to guarantee the stability of the neural-network based algorithms. All these limit the applicability of these approaches. More recently, a new nonlinear diagonal fault diagnosis observer has been proposed which is valid for a wide class of nonlinear systems based on linear transformation and solutions of matrix equations [11]. Using Lyapunov–Krasovskii functional and a suitable change of variables, Karimi et al. in [12] proposed a new delay-dependent stability criterion for robust fault detection filter problem of linear systems subjected to
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mixed neutral and discrete time-varying delays and some nonlinear perturbations which satisfied the Lipschitz conditions. In [13], Zhang et al. used the adaptive fault diagnosis observer to study the problem of fault estimation and accommodation for a class of nonlinear time-varying delay systems. For nonlinear non-Gaussian systems with multiple faults in the presence of noises, Lei et al. proposed a fault isolation algorithm. By constructing a filter to estimate the states, the fault isolation problem can be reduced to an entropy optimization problem subjected to the non-Gaussian estimation error systems [14]. In [15], Chen proposed an iterative learning observer (ILO) for fault detection, estimation and accommodation. The main characteristic of the ILO is that its states are updated or driven successively by the previous system output errors and the previous control input. Despite these promising
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approaches to addressing the problem of fault diagnosis in a nonlinear framework, there have not been many analytical results on fault detection and identification design schemes, especially in the case of unstructured modelling uncertainty. Developing effective FDI algorithm for uncertain nonlinear systems is consequently a great challenge.
In our previous work [17], a general robust fault tracking approximator (RFTA) based fault detection and identification scheme for a class of nonlinear systems was developed. In this work, we extend the previous results by considering a more
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general class of nonlinear uncertainty systems and provide a rigorous stability analysis and the fault tracking properties of the FTA. Two cases will be considered. The first one is that the initial state is known, the second one is that the initial state is supposed to be immeasurable. The main contributions of the paper are: this paper extends previous works on FTA and derives
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some new theoretical results, including the derivation of the fault detection threshold, the stability analysis of the FTA and the fault tracking properties.
The organization of the paper is as follows. In section Ⅱ, the problem formulation, including the design of the FTA, is
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presented. In section III, the derivation of fault detection threshold, including the stability analysis and fault tracking properties of the FTA, are proved. In section IV, two numerical examples are presented to illustrate the feasibility and effectiveness of the
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proposed approach. Finally, in section V concluding remarks are stated.
II. FTA ALGORITHM
Consider a class of nonlinear uncertain systems described by:
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x (t ) Ax(t ) g ( x(t ), t ) Bu (t ) Bw w(t ) B f f (t ) y(t ) Cx(t )
(1) (2)
where x(t ) R is the system state vector, u(t ) R is the input vector, y(t ) R is the output vector, f (t ) R represents the n
p
q
p
characteristics of system failure, w(t ) R h represents the unstructured modelling uncertainty. A, B, B f , Bw , C are known parameter matrices with appropriate dimensions. Assumption 1[33]: The known nonlinear term g ( x(t ), t ) satisfies Lipschitz condition about x : i.e. For all x1 , x2 R n , g ,
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( x1 , t ) ( x2 , t ) x1 x2 .
where is the known Lipschitz constant. Assumption 2: The function w(t ) in (1), representing the unstructured modeling uncertainty, is bounded by a known constant, i.e., w(t ) Lw . Assumption 3: (A, C) is an observable pair. Assumption 4[33]: Let S denote state mapping: ( x(0), f (t )) x(t ) . Let O denote the output mapping: ( x(0), f (t )) y(t ) . S and O are one-to-one mapping.
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To achieve fault detection and identification for the nonlinear uncertain systems described in (1)(2), constructing a FTA with the following form (3)～(8):
xˆk (t ) Axˆk (t ) g ( xˆk (t ), t ) Buk (t ) B f fˆk (t )
yˆ k (t ) Cxˆk (t ) ek (t ) xk (t ) xˆk (t ) rk (t ) Cek (t )
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P ˆ ˆ f k 1 (t ) f k (t ) rk (t )， rk (t ) ((exp(P)) 1) mP P 0， rk (t ) ((exp(P)) 1) mP
yk (t ) yˆ k (t ) , t [ta , tb ]
( fˆk (0) 0)
q
(4) (5) (6) (7)
(8)
Where xˆk (t ) R , yˆ k (t ) R are estimated system state and output respectively. k is the iteration index, n
(3)
is the given
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performance index, which is used to measure the tracking error. is constant gain matrix, and its elements are within the scope (0,1). fˆk (t ) is a new introduced parameter, named virtual fault, which is an estimate of f (t ) (further discussed below), the value
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of fˆ (t ) is set to zero until a fault is detected. t [t a , tb ] and tb t a P . P is the iterative operation horizon; e(t ) denotes the estimated system state error. ek (t ) denotes the error of e(t ) after kth iterative operation. r (t ) denotes the estimated system
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output error. rk (t ) denotes the error of r (t ) after kth iterative operation. Exp denotes the exponential function. sup C( ) , m sup C( ) Bw , sup ( ) , sup ( ) Bw , Lw , ( t ) is the state transition matrix. t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ ta ,tb ]
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Inequality (7) is the iterative algorithm which is used to update the virtual fault fˆ (t ) . If y(t ) yˆ k (t )
, we use the iterative
algorithm (7) to adjust the virtual fault fˆ (t ) within a specified horizon [t a , t b ] . The same operation will be repeated in time
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interval [t+i*P, t+(i+1)*P] (i=0,1,2…), where P is iteration horizon. Figure 1 shows the flow chart of design procedures of FTA.
Remark 1: Inequality (7) shows the iterative algorithm which is used to update the virtual fault fˆ (t ) . When rk (t ) P ((exp(P)) 1) mP , it indicates there is a fault, then the FTA begin to detect and identify the system
fault.
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Remark 2: Inequality (8) is the operation condition under which the virtual fault is updated according to the iterative algorithm (7). The virtual fault fˆ (t ) is updated successively by the output error and previous control input so long as y(t ) yˆ (t )
holds. In the process of calculations, if the iteration index k N , then updating the virtual fault will be stopped in order to save
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CPU resources. In theory, the maximal iteration times N .
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Figure 1. The flow chart of iteration by FTA
Remark 3: The FTA based approach is designed to reconstruct the fault signal by using an iterative algorithm within a specified horizon, which is motivated by iterative learning control theory and predictive control theory. One of the main characteristics of
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the FTA is that the same operation will be repeated in time interval [t+i*P, t+(i+1)*P] (i=0,1,2…) until inequality (8) holds, where P is iteration horizon. From inequality (8), we can clearly see that tb t a P . The value of P will have great influence on
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the fault detection time. Generally speaking, the smaller the value of P is, the faster the fault can be detected.
III. MAIN RESULTS
In this section, we provide a rigorous stability analysis and fault tracking properties of the FTA. Two cases will be
considered. The first one is that the initial state is known, the second one is that the initial state is supposed to be immeasurable. For the convenience of analysis, we introduce a definition of norm and a lemma. Definition 1 [16] : let the function f : [0, T ] R n , then the norm is defined as: f
sup f (t ) e t . 0t T
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ak satisfies： pak qak 1 dk
Lemma 1 [33]: Assuming
(k 1,2,) . Where, d k is real array with disturbances,
d0 . pq Lemma 2[38]: Let I denote an interval of the real line of the form [a,∞) or [a,b] or [a,b) with a < b. Let α, β and u be realand d k d 0 . If p q 0 , then: lim sup ak k
valued functions defined on I. Assume that β and u are continuous and that the negative part of α is integrable on every closed and bounded subinterval of I.
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(a) If β is non-negative and if u satisfies the integral inequality then (b) If, in addition, the function α is non-decreasing, then
A. Threshold selection for fault detection
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Subtracting equation (3) from system equation (1) and subtracting equation (4) from system equation (2) result in the estimation error dynamics:
~ e(t ) Ae(t ) B f f (t ) g ( x(t ), t ) g ( xˆ(t ), t ) Bw w(t )
r (t ) Ce(t )
(9) (10)
is system state estimation error, r y (t ) yˆ (t ) is system output estimation error,
Where e(t ) x(t ) xˆ (t )
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~ f k (t ) f k (t ) fˆk (t ) .
Theorem 1 Consider the system (1)(2) which meet the assumptions 1-4, and the FTA described by (3)～(8) , the initial state and output of the FTA are: xˆ k (0) x(0) , yˆ k (0) y(0) ( k 1,2, ), if the following condition is satisfied,
P ((exp(P)) 1) mP
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rk (t )
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then there is fault occurring in the system (1)(2).
Where sup C( ) , m sup C( ) B , sup ( ) , sup ( ) B , Lw . w w t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
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Proof: without loss of generality, we assume: t [ta , tb ] and tb t a P . From equation (9) and (10), we can obtain:
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t ~ ek (t ) (t , ta )ek (ta ) (t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(11)
ta
t ~ rk (t ) C(t , ta )ek (ta ) C(t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(12)
ta
where ( t ) L1[(sI A) 1 ] ， L1 denotes inverse Laplacian transform. From xˆ k (0) x(0), yˆ k (0) y(0) ( k 1,2, ), we have:
yˆ k 1 (t a ) Cxˆ k 1 (t a ) Cxˆ k (t a ) yˆ k (t a ) ， rk (t a ) 0 , ek (ta ) 0 Substituting (13) into (12) yields:
(13)
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t ~ rk (t ) C(t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(14)
ta
According to assumption 1 and 2, the time weighted norm of (14) is: t t t ~ rk (t ) C(t , ) B f f ( ) d C(t , )( g k ( x( ), ) g k ( xˆ ( ), )) d C(t , ) Bw w( ) d ta
ta
ta
In the fault-free case fˆ (t ) 0 , we have: t
t
ta
ta
rk (t ) C(t , ) ek ( ) d C(t , ) Bw w( ) d t
t
ta
ta
ek (t ) (t , ) ek ( ) d (t , ) Bw w( ) d t
(t , ) ek ( ) d P ta
Using Gronwall-Bellman inequality, the above equation can be simplified as:
where
sup (t , ) t , [ ta ,tb ]
， sup (t , ) Bw . t , [ ta ,tb ]
Substituting (16) into (15) yields: t
t
(16)
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ek (t ) P exp( (t t a ))
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According to equation (11), we have:
(15)
rk (t ) C(t , ) ek ( ) d C(t , ) Bw w( ) d ta
ta
t
t
C(t , ) P exp ( ta )d C(t , ) Bw w( ) d ta
t
t
ta
ta
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ta
P exp( ( t a ))d md
P ((exp( (t t a ))) 1) mP P ((exp(P)) 1) mP
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■ Remark 4: As it is well known in the fault diagnosis literature, there is a trade-off between robustness and fault detectability. In the presence of the unstructured modeling uncertainty, Theorem 1 provides an upper bound of the residual signal. It can be used as a threshold for fault detection. Once the residual signal exceeds the threshold, it indicates that system fault occurred.
B. The stability analysis
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Theorem 2 Consider the system (1)(2) and the FTA described by (3)～(8). If the following two conditions hold: (1) xˆ k (0) x(0) k 0,1,2, (2) I CB f 1
then lim y (t ) yˆ k (t ) k t[ 0,tb ]
L 1 exp(b1 )t b (b c 2 b1
1 exp(b1 )t b (b4 br t b b7 bw ) Lw b1 bw L w t b ) 1 exp( t b ) 1 exp(b8 )t b 1 (b3 b4 b5 b8 ) b8
(b4 b6t b b7 )bw Lw t b
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where ( t ) is the states transition matrix, b1 A ， b2 B f ， bw Bw ， b3 ( I CB f ) ， b4
CA ，
b5 sup (t , ) B f ， b6 sup (t , ) ， b7 C ， br sup (t , ) Bw ， t[ 0,tb ]
t[ 0,tb ]
t[ 0,tb ]
b8 max{( b4b6tb b7 )b2 , b1 } ， Lc C . Proof: Without loss of generality, we assume: t [ta , tb ] , t a 0 and tb t a P .
xk (t ) xˆ k (t ) xk (0) xˆ k (0)
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Based on equation (1) (3), we can obtain:
t
Axk ( s) Bu ( s) g ( xk ( s), s) B f f k ( s) Bw wk ( s) ds 0
t
Axˆk ( s) Bu ( s) g ( xˆk ( s), s) B f fˆk ( s) ds 0
t
t
0
0
b2 f k fˆk ds bw wk ds t
t
0
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ˆ ˆ x（ xˆ（ k 0） k 0） b1 xk xk ds g ( xk ( s ), s ) g ( xk ( s ), s ) ds 0
t t ˆ ˆ x（ xˆ（ k 0） k 0） (b1 ) xk xk ds b2 f k f k ds bw wk ds t
0
0
0
(b1 ) xk xˆk ds b2 f k fˆk ds bw Lwtb t
0
0
Where b1 A , b2 B f , bw Bw .
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t
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Using the Bellman-Gronwall inequality, the above equation can be simplified as: t xk (t ) xˆ k (t ) exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
(17)
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~ According to equation (7) and the definition f k (t ) f k (t ) fˆk (t ) , we have：
~ ~ fˆk 1 (t ) fˆk (t ) f k fˆk 1 (t ) f k 1 fˆk (t ) f k (t ) f k1 (t ) Cek (t ) The above equation can be rewritten as:
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~ ~ ~ f k1 (t ) f k (t ) C[ Aek (t ) B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
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t ~ ~ f k (t ) C[ A((t )ek (0) (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
From the condition (1), we have
xˆ k (0) x(0) , therefore ek (0) 0 。
Substituting ek (0) 0 into (18) yields:
t ~ ~ ~ f k1 (t ) f k (t ) C[ A( (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
(18)
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t ~ ~ ( I CB f ) f k (t ) C[ A( (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0
Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
(19)
By taking norms of (19), we have: t ~ t ~ ~ f k1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 xk ( ) xˆk ( ) d b7 xk (t ) xˆk (t ) 0
0
(b4br tb b7 bw ) Lw
(20)
Where b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , b7 C , t[ 0,tb ]
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t[ 0,tb ]
br sup (t , ) Bw . t[ 0,tb ]
For the purpose of convenient, assume (t ) exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds . t
0
We can easily get: for any s [0, t ] , ( s) (t ) . Then, by equation (20), we have: 0
0
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t ~ t ~ ~ f k 1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 ( )d b7(t ) (b4br tb b7 bw ) Lw t ~ ~ b3 f k (t ) b4b5 f k ( ) d b4 b6tb (t ) b7(t ) (b4br tb b7 bw ) Lw 0
t ~ t ~ b3 f k (t ) b4b5 f k ( ) d (b4b6tb b7 )( exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds) 0
0
(b4br tb b7 bw ) Lw
(21)
1 exp( tb ) ~ 1 exp(b8 )tb ~ ~ b3 f k (t ) b4 b5 f k (t ) b8 f k (t ) b8
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~ f k 1 (t )
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Let b8 max{( b4b6tb b7 )b2 , b1 } , by taking norm of (21), we have:
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
(b4b6tb b7 )bw Lwtb 1 exp( tb )
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(b3 b4 b5
b8
(b4b6tb b7 )bw Lwtb
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1 exp(b8 )tb ~ ) f k (t ) b8
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
According to condition (2), we have:
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For which is large enough, ( I CB f ) b4b5
1 exp( tb )
b8
1 exp(b8 )tb 1. b8
According to Lemma 1, we have:
lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
By substituting the above inequality into (17):
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xk (t ) xˆ k (t )
1 exp(b1 )tb (b2 f k fˆk b1
bw Lwtb )
By substituting the above inequality into equation (2) and (4), we can obtain: lim y (t ) yˆ k (t ) Lc
k t[ 0,tb ]
1 exp(b1 )t b (b2 b1
1 exp(b1 )t b (b4 br t b b7 bw ) Lw b1 bw Lwt b ) 1 exp( t b ) 1 exp(b8 )t b 1 (b3 b4 b5 b8 ) b8
(b4 b6t b b7 )bw Lwt b
C .
■
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Where Lc
Theorem 3 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold:
x(0) xˆ (0) C0 k 0,1,2,
(1)
(2) I CB f 1
k t[ 0,tb ]
Lc (C0
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
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then lim y (t ) yˆ k (t )
where
b1 A , b2 B f , bw Bw , b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , Lc C , t[ 0,tb ]
,
b8 sup CA(t , )
,
t[ 0,tb ]
br sup (t , ) Bw
m2 (b4b6tb b7 b8 )C0 , m1 (b4b6tb b7 )bw Lwtb
b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw , b1
1 exp(b9 )tb , ( t ) is the states transition matrix. b9
Without loss of generality, we assume: t [ta , tb ] , t a 0 and tb t a P .
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Proof:
1 exp( tb )
b9 max{( b4b6tb b7 )b2 , b1 }
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m3 b3 b4b5
,
t[ 0,tb ]
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b7 C
Based on equation (1)(3), we can obtain:
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xk (t ) xˆ k (t ) xk (0) xˆ k (0)
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t
Axk ( s) Bu ( s) g ( xk ( s), s) B f f k ( s) Bw wk ( s) ds 0
t
Axˆk ( s) Bu ( s) g ( xˆk ( s), s) B f fˆk ( s) ds 0
t
t
ˆ ˆ x（ xˆ（ k 0） k 0） b1 xk xk ds g ( xk ( s ), s ) g ( xk ( s ), s ) ds 0
0
b2 f k fˆk ds bw wk ds t
t
0
0
t t ˆ ˆ x（ xˆ（ k 0） k 0） (b1 ) xk xk ds b2 f k f k ds bw wk ds t
0
t[ 0,tb ]
0
0
,
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t t C0 (b1 ) xk xˆ k ds b2 f k fˆk ds bw Lwtb 0
0
Where b1 A , b2 B f , bw Bw . By using Bellman-Gronwall inequality, the previous inequality can be rewritten as: t xk (t ) xˆ k (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
(22)
~ According to definition f k (t ) f k (t ) fˆk (t ) , we have:
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~ ~ f k fˆk 1 (t ) f k 1 fˆk (t ) f k (t ) f k1 (t ) Cek (t ) ~ ~ ~ f k1 (t ) f k (t ) C[ Aek (t ) B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] t ~ ~ f k (t ) C[ A((t )ek (0) (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] Therefore the previous equality can be rewritten as:
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From the given condition (2), we have ek (0) C0 .
t ~ ~ ~ f k1 (t ) f k (t ) C[ A((t )C0 (t )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] t ~ ~ ( I CB f ) f k (t ) C[ A((t )C0 (t )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0
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Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] By taking norm, we obtain: 0
(b4br tb b7 bw ) Lw
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t ~ t ~ ~ f k1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 xk ( ) xˆk ( ) d b7 xk (t ) xˆk (t ) b8C0 0
(23)
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Where b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , t[ 0,tb ]
t[ 0,tb ]
b7 C , b8 sup CA(t , ) , br sup (t , ) Bw . t[ 0,tb ]
t[ 0,tb ]
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t Let (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds , for any s [0, t ] , we have ( s) (t ) . 0
Then, by (23), we have:
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t ~ t ~ ~ f k 1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 ( )d b7(t ) b8C0 (b4br tb b7 bw ) Lw 0
0
t ~ ~ b3 f k (t ) b4b5 f k ( ) d b4b6tb (t ) b7(t ) b8C0 (b4br tb b7 bw ) Lw 0
t ~ ~ b3 f k (t ) b4 b5 f k ( ) d (b4 b6tb b7 )(C0 exp(b1 )t 0
b2 exp((b1 )(t s)) f k fˆk ds) b8C0 (b4br tb b7 bw ) Lw t
0
let b9 max{( b4b6tb b7 )b2 , b1 } ，by taking norm, we have:
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1 exp( tb ) ~ 1 exp(b9 )tb ~ ) f k (t ) b9 f k (t ) (b4 b6tb b7 b8 )C0 b9
(b3 b4 b5
1 exp(b1 )tb (b4br tb b7bw ) Lw b1
(b4b6tb b7 )bw Lwtb (b3 b4 b5
1 exp( tb )
(b4b6tb b7 )bw Lwtb
b9
1 exp(b9 )tb ~ ) f k (t ) (b4 b6tb b7 b8 )C0 b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
By using Lemma 1, we have： lim f (t ) fˆk (t ) k t[ 0,tb ]
m3 b3 b4b5
m1 m2 1 m3
(24)
1 exp(b1 )tb (b4br tb b7 bw ) Lw , m2 (b4b6tb b7 b8 )C0 , b1
1 exp( tb )
b9
1 exp(b9 )tb b9
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Where m1 (b4b6tb b7 )bw Lwt b
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~ f k 1 (t )
By taking norm of (22), we have:
t xk (t ) xˆ k (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
C0
1 exp(b1 )tb (b2 f k fˆk b1
Combining (24) with (25), we obtain:
lim xk (t ) xˆ k (t ) C0
bw Lwt b )
(25)
1 exp(b1 )tb m m2 (b2 1 bw Lwtb ) b1 1 m3
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k t[ 0,tb ]
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xk (t ) xˆ k (t )
By substituting the above inequality into (2), we can obtain:
C
.
■
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Where Lc
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
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lim y(t ) yˆ k (t ) Lc (C0
k t[ 0,tb ]
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Remark 5: In addition to providing an upper bound on the system outputs estimation error in the term of norm, Theorem 2 and Theorem 3 also give a relationship between the system outputs estimation error in different initial conditions. If the system initial conditions are known, the system outputs estimation error:
lim y(t ) yˆ k (t ) Lc
k t[ 0,tb ]
1 exp(b1 )tb m m2 (b2 1 bw Lwtb ) b1 1 m3
If the system initial conditions are unknown, the system outputs estimation error:
lim y(t ) yˆ k (t ) Lc (C0
k t[ 0,tb ]
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
It is worth noting that, in Theorem 3, if with Theorem 2.
x(0) xˆ (0) , i.e., C0 0 , then the system outputs estimation error will be the same
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C. The fault tracking properties Theorem 4 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold: (1)
xˆk (0) x(0) k 0,1,2,
(2) I CB f 1
lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
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then
Theorem 5 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold:
x(0) xˆ (0) C0 k 0,1,2,
(2) I CB f 1 then
m m2 . lim f (t ) fˆk (t ) 1 k 1 m 3 t[ 0,t ] b
Where m1 (b4b6tb b7 )bw Lwt b
1 exp( tb )
b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw , m2 (b4b6tb b7 b8 )C0 , b1
1 exp(b9 )tb . b9
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m3 b3 b4b5
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(1)
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From the proof of Theorem 2 and Theorem 3, we can easily get the proof of Theorem 4 and Theorem 5, omitted. Remark 6: In Theorem 5, if C0 0 , we can obtain:
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lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
It is obvious that the term
m2 (b4b6tb b7 b8 )C0 is generated by the unknown initial systems states. Comparing
Theorem 4 with Theorem 5, we can find interesting results:
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1. If the initial system states are known, i.e., C0 0 , the tracking error m2 (b4b6tb b7 b8 )C0 generated by the unknown initial systems states will be eliminated.
2. If w(t ) 0 , i.e., Lw 0 , then :
1 exp(b1 )tb (b4br tb b7bw ) Lw b1 0 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4b5 b8 ) b8
(b4b6tb b7 )bw Lwtb
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So we have: lim f (t ) fˆk (t ) k t[ 0 ,tb ]
0 . This means that if the modelling uncertainty or disturbance w(t ) 0 , the virtual fault
fˆk (t ) can estimate the system fault f (t ) in the term of norm. Remark 7: Compared with existing results on FDI for nonlinear uncertainty systems, our proposed method has three advantages. One is that it can detect and identify the system faults simultaneously. The other is that it can be used for systems with uncertainties that cannot be decoupled from faults or residuals. Third, avoid to do complex online calculations and the training of
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neural network compared with neural network based approach. In neural network based approaches, guaranteeing the stability may be often quite difficult. However, in Theorem 2-5, we provide a rigorous stability analysis and the fault tracking properties of the FTA.
IV. SIMULATION EXAMPLE
In this section, we present two examples to illustrate the effectiveness of the proposed fault detection and identification approach. The first example is based on a simple nonlinear uncertain system, and aims at showing a complete application of the
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analytical results of the FTA. The second example is a flexible joint robotic example. This application is particularly important in order to point out both the practical significance and the applicability of the FDI architecture. A. a numerical example Consider the nonlinear uncertain system as follows:
9 0 1 1 1 0.1 x (t ) x(t ) u sin( x(t )) f (t ) w(t ) 0 8 0 0 1 0.1
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y(t ) 1 1x(t )
The parameters are as follows: w(t ) is assumed to be zero-mean white noises with power 0.1, acceptable tracking
0.5 . Assuming the faults are: f ( t ) :
0.5 exp (0.2 *（t / 100))
50s t 350s . 100s t 300s
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are
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errors: 0.1 , maximal iteration times: N 10 , time interval length: P 5 , the elements of the iterative algorithm gain matrix
In this simulation study, two kinds of faults are considered. The first fault is f (t ) 0.5 which occurs at time T 50
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second. The second fault is f (t ) exp(0.2 *（t / 100)) which occurs at time T 100 second. The two kinds of faults mentioned above aim at showing a complete application of the analytical results of the FTA, and the simulation results are given in Fig. 2 and Fig. 3 respectively. In all the simulation cases, the output of the FTA can be used as an indicator of a system failure. It can be
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seen from Fig. 2 and Fig. 3 that after the system fault occurs, the output of the FTA (solid line) rapidly tracks the system fault (dotted line). Therefore, the output of the FTA can be used not only for fault detection, but also for fault identification. Fig. 2 and Fig. 3 also show that, in the presence of system uncertainty, the output of the FTA is not identical to system fault. This is due to the fact that the output of the FTA estimates the lumped value of the system uncertainty and system fault, i.e., after the occurrence of the fault, the output of the FTA can estimate the system uncertainty and system fault simultaneously. Fig. 4 and Fig. 5 show the time evolution of the actual system fault (dotted line) and the estimated system fault (solid line) by using FTA based approach and neural-network based approaches respectively. The simulation study mentioned above is to demonstrate the improvement in the effectiveness and accuracy of fault detection and identification due to the use of the FTA-
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based approach. We use a radial basis function network as the on-line approximation model. Specifically, a Gaussian RBF network, which is described by [6]: fˆ ( x1 ,ˆ)
N
ˆ exp( x i 1
i
1
ci / 2 ) . The uniform width 0.6 for the basis function, 2
and 11 fixed centres evenly distributed in the interval [-2, 2]. At time t, the system states variable is sent to the RBF neural
network, which acts as the input vector. Then according to a learning algorithm: ˆ e to update the weights ˆ of the on-line
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approximation model. Where e is the states estimation error and is the learning algorithm to update the parameter ˆ . Comparing Fig. 4 with Fig. 5, it can be clearly seen that the FTA-based fault detection and identification scheme detect and identify the system fault more quickly and accurately compared with the neural network based FDI scheme. Moreover, it does not need train the network online, and can calculate in real time. 0.6 0.5
1.5
0.4
fault
fault
0.3
0.2 0.1
1
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0.5
0 actual fault estimated fault 0
100
200
300 time(s)
400
500
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-0.1
actual fault estimated fault
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2
0
0
200
300
400
Fig.3 The exponential function fault
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0.2
actual fault estimated fault
-0.2
actual fault output of OLA 0
-0.2
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-0.4
-0.8
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-1
-0.4
faults
faults
-0.6
-0.6
-0.8
-1.2
-1.4 -2
0
2
4
500
time(s)
Fig2. The step function fault 0
100
-1
6
8 10 time(secs)
12
14
16
Fig4. FDI by using FTA based approach
18
-1.2 -2
0
2
4
6
8 10 time(secs)
12
14
16
18
Fig5. FDI by using neural-network based approach
B. a flexible joint robotic example
Consider a one-link manipulator presented in Jiang et al.[18-19]. In this system, the states are the angular positions and
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velocities of the motor and of the link. The control u is the torque delivered by the motor. The state-space model is [19]:
m m k ( ) b k u m 1 m m Jm Jm Jm 1 1 k mgh 1 (1 m ) sin(1 ) w(t ) J J1 1
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Where J m denote the inertia of the motor, J1 denote the inertia of the link. k is the elasticity constant, b is the related viscous friction coefficient, and k is the amplifier gain. w(t ) denotes the modeling uncertainty [19]. If we select the vector x1 [ m ,1 ] and x2 [ m , 1 ] T
T
as system states, y x1 x 2 is the system output, where () is the T
transposition. The state-space model of the one-link manipulator can be written as:
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x1 x2 x2 ( x1, x2 ) Bu Bw w(t ) B f f (t ) y x1 x 2
Where ( x1 , x2 ) [1 ( x1 , x2 ), 2 ( x1 , x2 )]T , 1 ( x1 , x2 )
k b k mgh A1 x1 A2 x2 , 2 ( x1 , x2 ) A3 x1 sin( A4 x1 ) Jm Jm Jm J1 k ,0]T 。 Jm
k mgh k k b 1.95 , 0.333 . The modeling uncertainty w(t ) is 48.6 , 1.25 , 21.6 , Jm J1 J1 Jm Jm
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The parameters are:
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Bw [0,1]T , B f [0,1]T , A1 [1,1] , A2 [1,0] , A3 [1,1] , A4 [0,1] , B [
0, t 1s assumed to be white noises. Assuming the occurred fault is: f (t ) . The simulation results 0.4 0.2 cos(2t ) 1s t 8s
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are shown in Fig. 6 and Fig. 7. 0.7
0.6
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0.6
0.7
0.4
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fault
0.4
0.5
fault
0.5
0.3
0.3
0.2
0.2
0.1
0
0
1
0.1
actual fault estimated fault 2
3
4 time(secs)
5
6
Fig6. The neural network-based FDI
7
8
0
actual fault estimated fault 0
1
2
3
4 time(secs)
5
6
7
8
Fig.7 The FTA-based FDI
Fig.6 shows the simulation study by using the neural network based approach. In this simulation study, we use a radial basis
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function network as the on-line approximation model. Fig 7 shows the time-domain wave shapes of the virtual fault fˆ (t ) by using FTA-based method. In the simulation study, we use the FTA to detect and identify the system faults. The virtual fault fˆ (t ) is used as an indicator of a system failure. It can be seen from Fig. 7 that after a system fault occurs, the virtual fault fˆ (t ) rapidly jumps to a value above within one second. Comparing Fig. 6 with Fig. 7, we can clearly see that the FTA
based FDI approach provide perfect tracking of the actual faults compared with the neural network based approach. Remark 8: Different from the traditional fault diagnosis approaches, the outstanding feature of this scheme is that it can
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simultaneously perform fault detection and identification in real time without training the network compared with the neural network based FDI schemes. Moreover, the FTA can identify different kinds of faults, even slowly developing faults, known as incipient faults. Compared with the traditional neural network or support vector machine based fault diagnosis methods, the merits of the FTA based scheme are as follows:
① It can simultaneously detect and identify the shape and magnitude of the faults, which are very useful for fault accommodation study.
inequality or matrix equality, even nonlinear equations.
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② The fault is estimated by using iterative algorithms. Therefore, there is no need to solve the complex linear matrix
③ Provide a new FDI methodology for nonlinear uncertain systems.
V. CONCLUSION
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This paper has proposed a new fault detection and identification algorithm for a class of nonlinear uncertain systems. At the heart of this approach is the fault tracking approximator with adjustable parameters. A crucial parameter here is the virtual fault, which can estimate the unknown fault function and system uncertainty occurred in the system. The parameter—virtual fault can
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be used to detect and identify as well as to accommodate system failures. Since the proposed approach is capable of identifying the shape and magnitudes of the faults, it can distinguish between a momentary fault that clears itself, and a persistent one. Therefore, the approach can reduce the number of false alarms. The simulation results show the effectiveness of the method
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developed in this paper. It is well known that the online fault identification and parameter estimation play an important role in the reconfigurable controller design. One of the challenges in the fault tolerant control area is how to obtain accurate parameter
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estimation online and real-time in the presence of poor input excitation [22]. Therefore, the FTA based fault tolerant control problem will be the further research work. ACKNOWLEDGEMENT
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This work is supported by National Natural Science Foundation of China (Nos.51407078 and 61473129). Special Fund of East China University of Science and Technology for Basic Scientific Research (WH1514049, WJ1313004-1, H200-4-13192), the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, the Fundamental Research Funds for the Central Universities (HUST: Grant No. 2015TS025), the Fundamental Research Funds for the Central Universities (WUT: Grant No. 2015VI015), the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1245. REFERENCES [1] L. Zhen, H. Fang, Fault detection for nonlinear systems with unknown input. Asian Journal of Control. 2013, 15(5):1503-1509. [2] F. Zhu, J. Yang, Fault detection and isolation design for uncertain nonlinear systems based on full-order,reduced-order and high-order high-gain slidingmode observer. International Journal of Control.2013, 86(10):1800-1812. [3] N. Meskin, K. Khorasani, Robust fault detection and isolation of time-delay systems using a geometric approach, Automatica 2009, 45: 1567–1573.
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[4] Jiang B., M. Staroswiecki, and V. Cocquempot. Fault accommodation for nonlinear dynamic systems. IEEE Transactions on Automatic Control. 2006, 51 (9): 1578–1583. [5] X.D. Zhang, T. Parisini, and M.M Polycarpou. Sensor bias fault isolation in a class of nonlinear systems. IEEE Trans. on Automatic Control. 2005, 50: 370– 376. [6] M. M. Polycarpou and A. B. Trunov. Learning Approach to nonlinear fault diagnosis: delectability analysis. IEEE Trans. on Automatic Control. 2000, 45: 806–812. [7] X.D. Zhang, M.M Polycarpou and T. Parisini. A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans. on Automatic Control. 2002, 47: 576–593. [8] M. M. Polycarpou and A. B. Trunov. Learning Approach to nonlinear fault diagnosis: delectability analysis. IEEE Trans. on Automatic Control. 2000, 45(4): 806–812. [9] X.D. Zhang, M.M Polycarpou and T. Parisini. A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans. on Automatic Control. 2002, 47: 576–593. [10] X.D. Zhang, T. Parisini, and M.M Polycarpou. Sensor bias fault isolation in a class of nonlinear systems. IEEE Trans. on Automatic Control. 2005, 50: 370– 376. [11] S. Narasimhan, P. Vachhani, R. Rengaswamy. New nonlinear residual feedback observer for fault diagnosis in nonlinear systems. Automatica. 2008, 44: 2222–2229. [12] H.R Karimii, M. Zapateiro, N. Luo. A linear matrix inequality approach to robust fault detection filter design of linear systems with mixed time-varying delays and nonlinear perturbations. Journal of the Franklin Institute. 2010, 347: 957–973. [13] K. Zhang, B. Jiang, V. Cocquempot. Fast adaptive fault estimation and accommodation for nonlinear time-varying delay systems. Asian Journal of Control. 2009, 11(6): 643–652. [14] G. Lei, LP Yin, H. Wang, et al. Entropy optimization filtering for fault isolation of nonlinear non-Gaussian stochastic systems. IEEE Trans. on Automatic Control. 2009, 54(4): 804–810. [15] W. Chen, M Saif. An iterative learning observer for fault detection and accommodation in nonlinear time-delay systems. Int. J. robust and nonlinear conl. 2006, 16: 1-19. [16] S. Xie, S. Tian, Y. Fu. A new algorithm of iterative learning control with forgetting factors. IFAC Control Eng. Prac, 2004, 5: 625–630. [17] B. Yan, Z. Tian, S. Shi. A novel distributed approach to robust fault detection and identification. Electrical Power and Energy Systems. 2008, 30(1): 343– 360. [18] B. jiang, M. staroswiecki, V. Cocquempot. Fault accommodation for nonlinear dynamic systems. IEEE Trans. on Automatic Control, 2006, 51(9): 1578– 1583.
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[26] R. Isermann. Fault-Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance. Springer-Verlag, November 2005. [27] S.X. Ding, Model-based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools. Springer, Berlin Heidelberg, 1st edition. ISBN: 9783540763031.
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[33] M X Sun and B.J Huang. Iterative learning control. National Defence Industry Press -Beijing, May 1995. [34] X.D. Zhang, M.M Polycarpou and T. Parisini. Fault diagnosis of a class of nonlinear uncertain systems with Lipschitz nonlinearities using adaptive estimation. Automatica. 2010, 46: 290–299. [35] Bokor, J and Szabo. Fault detection and isolation in nonlinear systems. Annual Reviews in Control. 2009, 33: 113-123. [36] X.Wan,H.Fang,S.Fu. Observer-based fault detection for networked discrete-time infinite-distributed delay systems with packet dropouts. Applied Mathematical Modelling.2012, 36(1):270-278. [37] B.Boulaid,H.Souheil,Z.Ali. H-/H-∞ fault detection filter for a class of nonlinear descriptor systems. International Journal of Control.2013, 86(2):253-262 [38] J. Wang. The application of Inequality [M]. Hunan education publisher. 1993.
Fault detection and identification for a class of nonlinear systems with model uncertainty Bingyong Yan , Housheng Su , Wei Ma PII: DOI: Reference:
S0307-904X(16)30143-3 10.1016/j.apm.2016.03.018 APM 11086
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
2 March 2015 15 November 2015 7 March 2016
Please cite this article as: Bingyong Yan , Housheng Su , Wei Ma , Fault detection and identification for a class of nonlinear systems with model uncertainty, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.03.018
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A novel fault detection and identification approach is proposed based on FTA. The FTA can detect and identify system faults simultaneously. The FTA based FDI approach is applied to a flexible joint robotic system.
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Highlights
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Fault detection and identification for a class of nonlinear systems with model uncertainty
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Bingyong Yan1, Housheng Su2, Wei Ma3 1. School of Automation, Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China. 2. School of Automation,Key Laboratory of Image Processing and Intelligent Control of Ministry of Education of China, Huazhong University of Science and Technology, Wuhan 430074, China 3. Key Laboratory for Advanced Materials & Institute of Fine Chemicals, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China. Correspondence author: Housheng Su. Email: [email protected].
Abstract – In this paper, we present a novel fault detection and identification (FDI) scheme for a class of
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nonlinear systems with model uncertainty. At the heart of this approach is an on-line approximator, referred to as fault tracking approximator (FTA). Differently from the other approximators, the FTA uses iterative algorithms to detect and identify nonlinear system faults, even in the presence of model uncertainty, which is motivated by predictive control theory and iterative learning control theory. The FTA can simultaneously detect and identify the shape and magnitude of the faults. The rigorous stability analysis and fault tracking properties of the FTA are also proved. Finally, two examples are given to illustrate the feasibility and effectiveness of the proposed approach.
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Index Terms –fault detection and identification; predictive control; iterative learning control
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I. INTRODUCTION
In the last two decades, the analysis and design of fault detection and identification (FDI) schemes for dynamic systems have received considerable attention due to the increasing demand for system safety and reliability [1-27]. A large class of engineering
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systems are inherently nonlinear, which include, among others, power systems, chemical processes, and robotic manipulators with homonymic constraints [28-37]. So the study of FDI algorithms for nonlinear systems plays an important role in practical
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applications. In recent years, FDI problems for nonlinear systems have attracted much attention among researchers and many fruitful results have been achieved [1-7]. In [21], the geometric approach is used to design fault detection filters. The reconstruction of the fault signal is formulated as a control problem. For a class of affine nonlinear systems subjected to an
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unknown time-varying fault vector, a fault diagnosis filter is designed to estimate the time-varying fault by using nonlinear observer techniques [23]. For a class of nonaffine nonlinear systems, the adaptive control problems for fault tolerant control (FTC) are investigated in [28] and [29]. The existing approaches for fault detection and diagnosis and fault-tolerant control in a general framework of active fault-tolerant control systems are considered in [22]. Fruitful results on fault diagnosis and fault tolerant control (FTC) can be found in [24-27], and the reference therein. With the development of the processing speed of microprocessor as well as the new technologies which reduce the cost of computer chips, the learning approaches are rapidly becoming an issue of primary importance in automated condition monitoring and fault diagnosis [8]. In [8-10], a general learning methodology for fault diagnosis was developed and its stability properties were also investigated. The learning methods are based
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on adaptive algorithms and on-line approximation structures such as neural network, fuzzy logic networks, etc. However, in these neural network based approaches, there is no predetermined way of finding an optimum network size and architecture, and the online training of the neural network will be time-consuming. Moreover, it is often quit difficult to guarantee the stability of the neural-network based algorithms. All these limit the applicability of these approaches. More recently, a new nonlinear diagonal fault diagnosis observer has been proposed which is valid for a wide class of nonlinear systems based on linear transformation and solutions of matrix equations [11]. Using Lyapunov–Krasovskii functional and a suitable change of variables, Karimi et al. in [12] proposed a new delay-dependent stability criterion for robust fault detection filter problem of linear systems subjected to
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mixed neutral and discrete time-varying delays and some nonlinear perturbations which satisfied the Lipschitz conditions. In [13], Zhang et al. used the adaptive fault diagnosis observer to study the problem of fault estimation and accommodation for a class of nonlinear time-varying delay systems. For nonlinear non-Gaussian systems with multiple faults in the presence of noises, Lei et al. proposed a fault isolation algorithm. By constructing a filter to estimate the states, the fault isolation problem can be reduced to an entropy optimization problem subjected to the non-Gaussian estimation error systems [14]. In [15], Chen proposed an iterative learning observer (ILO) for fault detection, estimation and accommodation. The main characteristic of the ILO is that its states are updated or driven successively by the previous system output errors and the previous control input. Despite these promising
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approaches to addressing the problem of fault diagnosis in a nonlinear framework, there have not been many analytical results on fault detection and identification design schemes, especially in the case of unstructured modelling uncertainty. Developing effective FDI algorithm for uncertain nonlinear systems is consequently a great challenge.
In our previous work [17], a general robust fault tracking approximator (RFTA) based fault detection and identification scheme for a class of nonlinear systems was developed. In this work, we extend the previous results by considering a more
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general class of nonlinear uncertainty systems and provide a rigorous stability analysis and the fault tracking properties of the FTA. Two cases will be considered. The first one is that the initial state is known, the second one is that the initial state is supposed to be immeasurable. The main contributions of the paper are: this paper extends previous works on FTA and derives
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some new theoretical results, including the derivation of the fault detection threshold, the stability analysis of the FTA and the fault tracking properties.
The organization of the paper is as follows. In section Ⅱ, the problem formulation, including the design of the FTA, is
PT
presented. In section III, the derivation of fault detection threshold, including the stability analysis and fault tracking properties of the FTA, are proved. In section IV, two numerical examples are presented to illustrate the feasibility and effectiveness of the
CE
proposed approach. Finally, in section V concluding remarks are stated.
II. FTA ALGORITHM
Consider a class of nonlinear uncertain systems described by:
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x (t ) Ax(t ) g ( x(t ), t ) Bu (t ) Bw w(t ) B f f (t ) y(t ) Cx(t )
(1) (2)
where x(t ) R is the system state vector, u(t ) R is the input vector, y(t ) R is the output vector, f (t ) R represents the n
p
q
p
characteristics of system failure, w(t ) R h represents the unstructured modelling uncertainty. A, B, B f , Bw , C are known parameter matrices with appropriate dimensions. Assumption 1[33]: The known nonlinear term g ( x(t ), t ) satisfies Lipschitz condition about x : i.e. For all x1 , x2 R n , g ,
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( x1 , t ) ( x2 , t ) x1 x2 .
where is the known Lipschitz constant. Assumption 2: The function w(t ) in (1), representing the unstructured modeling uncertainty, is bounded by a known constant, i.e., w(t ) Lw . Assumption 3: (A, C) is an observable pair. Assumption 4[33]: Let S denote state mapping: ( x(0), f (t )) x(t ) . Let O denote the output mapping: ( x(0), f (t )) y(t ) . S and O are one-to-one mapping.
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To achieve fault detection and identification for the nonlinear uncertain systems described in (1)(2), constructing a FTA with the following form (3)～(8):
xˆk (t ) Axˆk (t ) g ( xˆk (t ), t ) Buk (t ) B f fˆk (t )
yˆ k (t ) Cxˆk (t ) ek (t ) xk (t ) xˆk (t ) rk (t ) Cek (t )
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P ˆ ˆ f k 1 (t ) f k (t ) rk (t )， rk (t ) ((exp(P)) 1) mP P 0， rk (t ) ((exp(P)) 1) mP
yk (t ) yˆ k (t ) , t [ta , tb ]
( fˆk (0) 0)
q
(4) (5) (6) (7)
(8)
Where xˆk (t ) R , yˆ k (t ) R are estimated system state and output respectively. k is the iteration index, n
(3)
is the given
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performance index, which is used to measure the tracking error. is constant gain matrix, and its elements are within the scope (0,1). fˆk (t ) is a new introduced parameter, named virtual fault, which is an estimate of f (t ) (further discussed below), the value
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of fˆ (t ) is set to zero until a fault is detected. t [t a , tb ] and tb t a P . P is the iterative operation horizon; e(t ) denotes the estimated system state error. ek (t ) denotes the error of e(t ) after kth iterative operation. r (t ) denotes the estimated system
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output error. rk (t ) denotes the error of r (t ) after kth iterative operation. Exp denotes the exponential function. sup C( ) , m sup C( ) Bw , sup ( ) , sup ( ) Bw , Lw , ( t ) is the state transition matrix. t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ ta ,tb ]
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Inequality (7) is the iterative algorithm which is used to update the virtual fault fˆ (t ) . If y(t ) yˆ k (t )
, we use the iterative
algorithm (7) to adjust the virtual fault fˆ (t ) within a specified horizon [t a , t b ] . The same operation will be repeated in time
AC
interval [t+i*P, t+(i+1)*P] (i=0,1,2…), where P is iteration horizon. Figure 1 shows the flow chart of design procedures of FTA.
Remark 1: Inequality (7) shows the iterative algorithm which is used to update the virtual fault fˆ (t ) . When rk (t ) P ((exp(P)) 1) mP , it indicates there is a fault, then the FTA begin to detect and identify the system
fault.
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Remark 2: Inequality (8) is the operation condition under which the virtual fault is updated according to the iterative algorithm (7). The virtual fault fˆ (t ) is updated successively by the output error and previous control input so long as y(t ) yˆ (t )
holds. In the process of calculations, if the iteration index k N , then updating the virtual fault will be stopped in order to save
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CPU resources. In theory, the maximal iteration times N .
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Figure 1. The flow chart of iteration by FTA
Remark 3: The FTA based approach is designed to reconstruct the fault signal by using an iterative algorithm within a specified horizon, which is motivated by iterative learning control theory and predictive control theory. One of the main characteristics of
CE
the FTA is that the same operation will be repeated in time interval [t+i*P, t+(i+1)*P] (i=0,1,2…) until inequality (8) holds, where P is iteration horizon. From inequality (8), we can clearly see that tb t a P . The value of P will have great influence on
AC
the fault detection time. Generally speaking, the smaller the value of P is, the faster the fault can be detected.
III. MAIN RESULTS
In this section, we provide a rigorous stability analysis and fault tracking properties of the FTA. Two cases will be
considered. The first one is that the initial state is known, the second one is that the initial state is supposed to be immeasurable. For the convenience of analysis, we introduce a definition of norm and a lemma. Definition 1 [16] : let the function f : [0, T ] R n , then the norm is defined as: f
sup f (t ) e t . 0t T
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ak satisfies： pak qak 1 dk
Lemma 1 [33]: Assuming
(k 1,2,) . Where, d k is real array with disturbances,
d0 . pq Lemma 2[38]: Let I denote an interval of the real line of the form [a,∞) or [a,b] or [a,b) with a < b. Let α, β and u be realand d k d 0 . If p q 0 , then: lim sup ak k
valued functions defined on I. Assume that β and u are continuous and that the negative part of α is integrable on every closed and bounded subinterval of I.
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(a) If β is non-negative and if u satisfies the integral inequality then (b) If, in addition, the function α is non-decreasing, then
A. Threshold selection for fault detection
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Subtracting equation (3) from system equation (1) and subtracting equation (4) from system equation (2) result in the estimation error dynamics:
~ e(t ) Ae(t ) B f f (t ) g ( x(t ), t ) g ( xˆ(t ), t ) Bw w(t )
r (t ) Ce(t )
(9) (10)
is system state estimation error, r y (t ) yˆ (t ) is system output estimation error,
Where e(t ) x(t ) xˆ (t )
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~ f k (t ) f k (t ) fˆk (t ) .
Theorem 1 Consider the system (1)(2) which meet the assumptions 1-4, and the FTA described by (3)～(8) , the initial state and output of the FTA are: xˆ k (0) x(0) , yˆ k (0) y(0) ( k 1,2, ), if the following condition is satisfied,
P ((exp(P)) 1) mP
ED
rk (t )
PT
then there is fault occurring in the system (1)(2).
Where sup C( ) , m sup C( ) B , sup ( ) , sup ( ) B , Lw . w w t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
t , [ t a ,tb ]
CE
Proof: without loss of generality, we assume: t [ta , tb ] and tb t a P . From equation (9) and (10), we can obtain:
AC
t ~ ek (t ) (t , ta )ek (ta ) (t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(11)
ta
t ~ rk (t ) C(t , ta )ek (ta ) C(t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(12)
ta
where ( t ) L1[(sI A) 1 ] ， L1 denotes inverse Laplacian transform. From xˆ k (0) x(0), yˆ k (0) y(0) ( k 1,2, ), we have:
yˆ k 1 (t a ) Cxˆ k 1 (t a ) Cxˆ k (t a ) yˆ k (t a ) ， rk (t a ) 0 , ek (ta ) 0 Substituting (13) into (12) yields:
(13)
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t ~ rk (t ) C(t , )( B f f k ( ) g k ( x( ), ) g k ( xˆ ( ), ) Bw wk ( ))d
(14)
ta
According to assumption 1 and 2, the time weighted norm of (14) is: t t t ~ rk (t ) C(t , ) B f f ( ) d C(t , )( g k ( x( ), ) g k ( xˆ ( ), )) d C(t , ) Bw w( ) d ta
ta
ta
In the fault-free case fˆ (t ) 0 , we have: t
t
ta
ta
rk (t ) C(t , ) ek ( ) d C(t , ) Bw w( ) d t
t
ta
ta
ek (t ) (t , ) ek ( ) d (t , ) Bw w( ) d t
(t , ) ek ( ) d P ta
Using Gronwall-Bellman inequality, the above equation can be simplified as:
where
sup (t , ) t , [ ta ,tb ]
， sup (t , ) Bw . t , [ ta ,tb ]
Substituting (16) into (15) yields: t
t
(16)
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ek (t ) P exp( (t t a ))
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According to equation (11), we have:
(15)
rk (t ) C(t , ) ek ( ) d C(t , ) Bw w( ) d ta
ta
t
t
C(t , ) P exp ( ta )d C(t , ) Bw w( ) d ta
t
t
ta
ta
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ta
P exp( ( t a ))d md
P ((exp( (t t a ))) 1) mP P ((exp(P)) 1) mP
ED
CE
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■ Remark 4: As it is well known in the fault diagnosis literature, there is a trade-off between robustness and fault detectability. In the presence of the unstructured modeling uncertainty, Theorem 1 provides an upper bound of the residual signal. It can be used as a threshold for fault detection. Once the residual signal exceeds the threshold, it indicates that system fault occurred.
B. The stability analysis
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Theorem 2 Consider the system (1)(2) and the FTA described by (3)～(8). If the following two conditions hold: (1) xˆ k (0) x(0) k 0,1,2, (2) I CB f 1
then lim y (t ) yˆ k (t ) k t[ 0,tb ]
L 1 exp(b1 )t b (b c 2 b1
1 exp(b1 )t b (b4 br t b b7 bw ) Lw b1 bw L w t b ) 1 exp( t b ) 1 exp(b8 )t b 1 (b3 b4 b5 b8 ) b8
(b4 b6t b b7 )bw Lw t b
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where ( t ) is the states transition matrix, b1 A ， b2 B f ， bw Bw ， b3 ( I CB f ) ， b4
CA ，
b5 sup (t , ) B f ， b6 sup (t , ) ， b7 C ， br sup (t , ) Bw ， t[ 0,tb ]
t[ 0,tb ]
t[ 0,tb ]
b8 max{( b4b6tb b7 )b2 , b1 } ， Lc C . Proof: Without loss of generality, we assume: t [ta , tb ] , t a 0 and tb t a P .
xk (t ) xˆ k (t ) xk (0) xˆ k (0)
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Based on equation (1) (3), we can obtain:
t
Axk ( s) Bu ( s) g ( xk ( s), s) B f f k ( s) Bw wk ( s) ds 0
t
Axˆk ( s) Bu ( s) g ( xˆk ( s), s) B f fˆk ( s) ds 0
t
t
0
0
b2 f k fˆk ds bw wk ds t
t
0
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ˆ ˆ x（ xˆ（ k 0） k 0） b1 xk xk ds g ( xk ( s ), s ) g ( xk ( s ), s ) ds 0
t t ˆ ˆ x（ xˆ（ k 0） k 0） (b1 ) xk xk ds b2 f k f k ds bw wk ds t
0
0
0
(b1 ) xk xˆk ds b2 f k fˆk ds bw Lwtb t
0
0
Where b1 A , b2 B f , bw Bw .
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t
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Using the Bellman-Gronwall inequality, the above equation can be simplified as: t xk (t ) xˆ k (t ) exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
(17)
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~ According to equation (7) and the definition f k (t ) f k (t ) fˆk (t ) , we have：
~ ~ fˆk 1 (t ) fˆk (t ) f k fˆk 1 (t ) f k 1 fˆk (t ) f k (t ) f k1 (t ) Cek (t ) The above equation can be rewritten as:
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~ ~ ~ f k1 (t ) f k (t ) C[ Aek (t ) B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
AC
t ~ ~ f k (t ) C[ A((t )ek (0) (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
From the condition (1), we have
xˆ k (0) x(0) , therefore ek (0) 0 。
Substituting ek (0) 0 into (18) yields:
t ~ ~ ~ f k1 (t ) f k (t ) C[ A( (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
(18)
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t ~ ~ ( I CB f ) f k (t ) C[ A( (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0
Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )]
(19)
By taking norms of (19), we have: t ~ t ~ ~ f k1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 xk ( ) xˆk ( ) d b7 xk (t ) xˆk (t ) 0
0
(b4br tb b7 bw ) Lw
(20)
Where b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , b7 C , t[ 0,tb ]
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t[ 0,tb ]
br sup (t , ) Bw . t[ 0,tb ]
For the purpose of convenient, assume (t ) exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds . t
0
We can easily get: for any s [0, t ] , ( s) (t ) . Then, by equation (20), we have: 0
0
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t ~ t ~ ~ f k 1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 ( )d b7(t ) (b4br tb b7 bw ) Lw t ~ ~ b3 f k (t ) b4b5 f k ( ) d b4 b6tb (t ) b7(t ) (b4br tb b7 bw ) Lw 0
t ~ t ~ b3 f k (t ) b4b5 f k ( ) d (b4b6tb b7 )( exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds) 0
0
(b4br tb b7 bw ) Lw
(21)
1 exp( tb ) ~ 1 exp(b8 )tb ~ ~ b3 f k (t ) b4 b5 f k (t ) b8 f k (t ) b8
ED
~ f k 1 (t )
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Let b8 max{( b4b6tb b7 )b2 , b1 } , by taking norm of (21), we have:
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
(b4b6tb b7 )bw Lwtb 1 exp( tb )
PT
(b3 b4 b5
b8
(b4b6tb b7 )bw Lwtb
CE
1 exp(b8 )tb ~ ) f k (t ) b8
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
According to condition (2), we have:
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For which is large enough, ( I CB f ) b4b5
1 exp( tb )
b8
1 exp(b8 )tb 1. b8
According to Lemma 1, we have:
lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
By substituting the above inequality into (17):
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xk (t ) xˆ k (t )
1 exp(b1 )tb (b2 f k fˆk b1
bw Lwtb )
By substituting the above inequality into equation (2) and (4), we can obtain: lim y (t ) yˆ k (t ) Lc
k t[ 0,tb ]
1 exp(b1 )t b (b2 b1
1 exp(b1 )t b (b4 br t b b7 bw ) Lw b1 bw Lwt b ) 1 exp( t b ) 1 exp(b8 )t b 1 (b3 b4 b5 b8 ) b8
(b4 b6t b b7 )bw Lwt b
C .
■
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Where Lc
Theorem 3 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold:
x(0) xˆ (0) C0 k 0,1,2,
(1)
(2) I CB f 1
k t[ 0,tb ]
Lc (C0
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
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then lim y (t ) yˆ k (t )
where
b1 A , b2 B f , bw Bw , b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , Lc C , t[ 0,tb ]
,
b8 sup CA(t , )
,
t[ 0,tb ]
br sup (t , ) Bw
m2 (b4b6tb b7 b8 )C0 , m1 (b4b6tb b7 )bw Lwtb
b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw , b1
1 exp(b9 )tb , ( t ) is the states transition matrix. b9
Without loss of generality, we assume: t [ta , tb ] , t a 0 and tb t a P .
PT
Proof:
1 exp( tb )
b9 max{( b4b6tb b7 )b2 , b1 }
ED
m3 b3 b4b5
,
t[ 0,tb ]
M
b7 C
Based on equation (1)(3), we can obtain:
CE
xk (t ) xˆ k (t ) xk (0) xˆ k (0)
AC
t
Axk ( s) Bu ( s) g ( xk ( s), s) B f f k ( s) Bw wk ( s) ds 0
t
Axˆk ( s) Bu ( s) g ( xˆk ( s), s) B f fˆk ( s) ds 0
t
t
ˆ ˆ x（ xˆ（ k 0） k 0） b1 xk xk ds g ( xk ( s ), s ) g ( xk ( s ), s ) ds 0
0
b2 f k fˆk ds bw wk ds t
t
0
0
t t ˆ ˆ x（ xˆ（ k 0） k 0） (b1 ) xk xk ds b2 f k f k ds bw wk ds t
0
t[ 0,tb ]
0
0
,
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t t C0 (b1 ) xk xˆ k ds b2 f k fˆk ds bw Lwtb 0
0
Where b1 A , b2 B f , bw Bw . By using Bellman-Gronwall inequality, the previous inequality can be rewritten as: t xk (t ) xˆ k (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
(22)
~ According to definition f k (t ) f k (t ) fˆk (t ) , we have:
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~ ~ f k fˆk 1 (t ) f k 1 fˆk (t ) f k (t ) f k1 (t ) Cek (t ) ~ ~ ~ f k1 (t ) f k (t ) C[ Aek (t ) B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] t ~ ~ f k (t ) C[ A((t )ek (0) (t , )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] Therefore the previous equality can be rewritten as:
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From the given condition (2), we have ek (0) C0 .
t ~ ~ ~ f k1 (t ) f k (t ) C[ A((t )C0 (t )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0 ~ B f f k (t ) Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] t ~ ~ ( I CB f ) f k (t ) C[ A((t )C0 (t )( B f f k ( ) Bw wk (t ) g ( xk ( ), ) g ( xˆk ( ), ))d ) 0
M
Bw wk (t ) g ( xk (t ), t ) g ( xˆk (t ), t )] By taking norm, we obtain: 0
(b4br tb b7 bw ) Lw
ED
t ~ t ~ ~ f k1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 xk ( ) xˆk ( ) d b7 xk (t ) xˆk (t ) b8C0 0
(23)
PT
Where b3 ( I CB f ) , b4 CA , b5 sup (t , ) B f , b6 sup (t , ) , t[ 0,tb ]
t[ 0,tb ]
b7 C , b8 sup CA(t , ) , br sup (t , ) Bw . t[ 0,tb ]
t[ 0,tb ]
CE
t Let (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds , for any s [0, t ] , we have ( s) (t ) . 0
Then, by (23), we have:
AC
t ~ t ~ ~ f k 1 (t ) b3 f k (t ) b4b5 f k ( ) d b4b6 ( )d b7(t ) b8C0 (b4br tb b7 bw ) Lw 0
0
t ~ ~ b3 f k (t ) b4b5 f k ( ) d b4b6tb (t ) b7(t ) b8C0 (b4br tb b7 bw ) Lw 0
t ~ ~ b3 f k (t ) b4 b5 f k ( ) d (b4 b6tb b7 )(C0 exp(b1 )t 0
b2 exp((b1 )(t s)) f k fˆk ds) b8C0 (b4br tb b7 bw ) Lw t
0
let b9 max{( b4b6tb b7 )b2 , b1 } ，by taking norm, we have:
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1 exp( tb ) ~ 1 exp(b9 )tb ~ ) f k (t ) b9 f k (t ) (b4 b6tb b7 b8 )C0 b9
(b3 b4 b5
1 exp(b1 )tb (b4br tb b7bw ) Lw b1
(b4b6tb b7 )bw Lwtb (b3 b4 b5
1 exp( tb )
(b4b6tb b7 )bw Lwtb
b9
1 exp(b9 )tb ~ ) f k (t ) (b4 b6tb b7 b8 )C0 b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw b1
By using Lemma 1, we have： lim f (t ) fˆk (t ) k t[ 0,tb ]
m3 b3 b4b5
m1 m2 1 m3
(24)
1 exp(b1 )tb (b4br tb b7 bw ) Lw , m2 (b4b6tb b7 b8 )C0 , b1
1 exp( tb )
b9
1 exp(b9 )tb b9
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Where m1 (b4b6tb b7 )bw Lwt b
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~ f k 1 (t )
By taking norm of (22), we have:
t xk (t ) xˆ k (t ) C0 exp(b1 )t exp((b1 )(t s))(b2 f k fˆk bw Lwtb )ds 0
C0
1 exp(b1 )tb (b2 f k fˆk b1
Combining (24) with (25), we obtain:
lim xk (t ) xˆ k (t ) C0
bw Lwt b )
(25)
1 exp(b1 )tb m m2 (b2 1 bw Lwtb ) b1 1 m3
ED
k t[ 0,tb ]
M
xk (t ) xˆ k (t )
By substituting the above inequality into (2), we can obtain:
C
.
■
CE
Where Lc
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
PT
lim y(t ) yˆ k (t ) Lc (C0
k t[ 0,tb ]
AC
Remark 5: In addition to providing an upper bound on the system outputs estimation error in the term of norm, Theorem 2 and Theorem 3 also give a relationship between the system outputs estimation error in different initial conditions. If the system initial conditions are known, the system outputs estimation error:
lim y(t ) yˆ k (t ) Lc
k t[ 0,tb ]
1 exp(b1 )tb m m2 (b2 1 bw Lwtb ) b1 1 m3
If the system initial conditions are unknown, the system outputs estimation error:
lim y(t ) yˆ k (t ) Lc (C0
k t[ 0,tb ]
1 exp(b1 )tb m m2 (b2 1 bw Lwtb )) b1 1 m3
It is worth noting that, in Theorem 3, if with Theorem 2.
x(0) xˆ (0) , i.e., C0 0 , then the system outputs estimation error will be the same
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C. The fault tracking properties Theorem 4 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold: (1)
xˆk (0) x(0) k 0,1,2,
(2) I CB f 1
lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
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then
Theorem 5 Consider the system (1)(2) and the FTA described by (3)～(8). If the following conditions hold:
x(0) xˆ (0) C0 k 0,1,2,
(2) I CB f 1 then
m m2 . lim f (t ) fˆk (t ) 1 k 1 m 3 t[ 0,t ] b
Where m1 (b4b6tb b7 )bw Lwt b
1 exp( tb )
b9
1 exp(b1 )tb (b4br tb b7 bw ) Lw , m2 (b4b6tb b7 b8 )C0 , b1
1 exp(b9 )tb . b9
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m3 b3 b4b5
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(1)
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From the proof of Theorem 2 and Theorem 3, we can easily get the proof of Theorem 4 and Theorem 5, omitted. Remark 6: In Theorem 5, if C0 0 , we can obtain:
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lim f (t ) fˆk (t )
k t[ 0 ,tb ]
1 exp(b1 )tb (b4 br tb b7 bw ) Lw b1 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4 b5 b8 ) b8
(b4 b6tb b7 )bw Lwtb
It is obvious that the term
m2 (b4b6tb b7 b8 )C0 is generated by the unknown initial systems states. Comparing
Theorem 4 with Theorem 5, we can find interesting results:
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1. If the initial system states are known, i.e., C0 0 , the tracking error m2 (b4b6tb b7 b8 )C0 generated by the unknown initial systems states will be eliminated.
2. If w(t ) 0 , i.e., Lw 0 , then :
1 exp(b1 )tb (b4br tb b7bw ) Lw b1 0 1 exp( tb ) 1 exp(b8 )tb 1 (b3 b4b5 b8 ) b8
(b4b6tb b7 )bw Lwtb
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So we have: lim f (t ) fˆk (t ) k t[ 0 ,tb ]
0 . This means that if the modelling uncertainty or disturbance w(t ) 0 , the virtual fault
fˆk (t ) can estimate the system fault f (t ) in the term of norm. Remark 7: Compared with existing results on FDI for nonlinear uncertainty systems, our proposed method has three advantages. One is that it can detect and identify the system faults simultaneously. The other is that it can be used for systems with uncertainties that cannot be decoupled from faults or residuals. Third, avoid to do complex online calculations and the training of
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neural network compared with neural network based approach. In neural network based approaches, guaranteeing the stability may be often quite difficult. However, in Theorem 2-5, we provide a rigorous stability analysis and the fault tracking properties of the FTA.
IV. SIMULATION EXAMPLE
In this section, we present two examples to illustrate the effectiveness of the proposed fault detection and identification approach. The first example is based on a simple nonlinear uncertain system, and aims at showing a complete application of the
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analytical results of the FTA. The second example is a flexible joint robotic example. This application is particularly important in order to point out both the practical significance and the applicability of the FDI architecture. A. a numerical example Consider the nonlinear uncertain system as follows:
9 0 1 1 1 0.1 x (t ) x(t ) u sin( x(t )) f (t ) w(t ) 0 8 0 0 1 0.1
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y(t ) 1 1x(t )
The parameters are as follows: w(t ) is assumed to be zero-mean white noises with power 0.1, acceptable tracking
0.5 . Assuming the faults are: f ( t ) :
0.5 exp (0.2 *（t / 100))
50s t 350s . 100s t 300s
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are
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errors: 0.1 , maximal iteration times: N 10 , time interval length: P 5 , the elements of the iterative algorithm gain matrix
In this simulation study, two kinds of faults are considered. The first fault is f (t ) 0.5 which occurs at time T 50
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second. The second fault is f (t ) exp(0.2 *（t / 100)) which occurs at time T 100 second. The two kinds of faults mentioned above aim at showing a complete application of the analytical results of the FTA, and the simulation results are given in Fig. 2 and Fig. 3 respectively. In all the simulation cases, the output of the FTA can be used as an indicator of a system failure. It can be
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seen from Fig. 2 and Fig. 3 that after the system fault occurs, the output of the FTA (solid line) rapidly tracks the system fault (dotted line). Therefore, the output of the FTA can be used not only for fault detection, but also for fault identification. Fig. 2 and Fig. 3 also show that, in the presence of system uncertainty, the output of the FTA is not identical to system fault. This is due to the fact that the output of the FTA estimates the lumped value of the system uncertainty and system fault, i.e., after the occurrence of the fault, the output of the FTA can estimate the system uncertainty and system fault simultaneously. Fig. 4 and Fig. 5 show the time evolution of the actual system fault (dotted line) and the estimated system fault (solid line) by using FTA based approach and neural-network based approaches respectively. The simulation study mentioned above is to demonstrate the improvement in the effectiveness and accuracy of fault detection and identification due to the use of the FTA-
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based approach. We use a radial basis function network as the on-line approximation model. Specifically, a Gaussian RBF network, which is described by [6]: fˆ ( x1 ,ˆ)
N
ˆ exp( x i 1
i
1
ci / 2 ) . The uniform width 0.6 for the basis function, 2
and 11 fixed centres evenly distributed in the interval [-2, 2]. At time t, the system states variable is sent to the RBF neural
network, which acts as the input vector. Then according to a learning algorithm: ˆ e to update the weights ˆ of the on-line
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approximation model. Where e is the states estimation error and is the learning algorithm to update the parameter ˆ . Comparing Fig. 4 with Fig. 5, it can be clearly seen that the FTA-based fault detection and identification scheme detect and identify the system fault more quickly and accurately compared with the neural network based FDI scheme. Moreover, it does not need train the network online, and can calculate in real time. 0.6 0.5
1.5
0.4
fault
fault
0.3
0.2 0.1
1
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0.5
0 actual fault estimated fault 0
100
200
300 time(s)
400
500
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-0.1
actual fault estimated fault
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2
0
0
200
300
400
Fig.3 The exponential function fault
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0.2
actual fault estimated fault
-0.2
actual fault output of OLA 0
-0.2
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-0.4
-0.8
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-1
-0.4
faults
faults
-0.6
-0.6
-0.8
-1.2
-1.4 -2
0
2
4
500
time(s)
Fig2. The step function fault 0
100
-1
6
8 10 time(secs)
12
14
16
Fig4. FDI by using FTA based approach
18
-1.2 -2
0
2
4
6
8 10 time(secs)
12
14
16
18
Fig5. FDI by using neural-network based approach
B. a flexible joint robotic example
Consider a one-link manipulator presented in Jiang et al.[18-19]. In this system, the states are the angular positions and
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velocities of the motor and of the link. The control u is the torque delivered by the motor. The state-space model is [19]:
m m k ( ) b k u m 1 m m Jm Jm Jm 1 1 k mgh 1 (1 m ) sin(1 ) w(t ) J J1 1
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Where J m denote the inertia of the motor, J1 denote the inertia of the link. k is the elasticity constant, b is the related viscous friction coefficient, and k is the amplifier gain. w(t ) denotes the modeling uncertainty [19]. If we select the vector x1 [ m ,1 ] and x2 [ m , 1 ] T
T
as system states, y x1 x 2 is the system output, where () is the T
transposition. The state-space model of the one-link manipulator can be written as:
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x1 x2 x2 ( x1, x2 ) Bu Bw w(t ) B f f (t ) y x1 x 2
Where ( x1 , x2 ) [1 ( x1 , x2 ), 2 ( x1 , x2 )]T , 1 ( x1 , x2 )
k b k mgh A1 x1 A2 x2 , 2 ( x1 , x2 ) A3 x1 sin( A4 x1 ) Jm Jm Jm J1 k ,0]T 。 Jm
k mgh k k b 1.95 , 0.333 . The modeling uncertainty w(t ) is 48.6 , 1.25 , 21.6 , Jm J1 J1 Jm Jm
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The parameters are:
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Bw [0,1]T , B f [0,1]T , A1 [1,1] , A2 [1,0] , A3 [1,1] , A4 [0,1] , B [
0, t 1s assumed to be white noises. Assuming the occurred fault is: f (t ) . The simulation results 0.4 0.2 cos(2t ) 1s t 8s
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are shown in Fig. 6 and Fig. 7. 0.7
0.6
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0.6
0.7
0.4
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fault
0.4
0.5
fault
0.5
0.3
0.3
0.2
0.2
0.1
0
0
1
0.1
actual fault estimated fault 2
3
4 time(secs)
5
6
Fig6. The neural network-based FDI
7
8
0
actual fault estimated fault 0
1
2
3
4 time(secs)
5
6
7
8
Fig.7 The FTA-based FDI
Fig.6 shows the simulation study by using the neural network based approach. In this simulation study, we use a radial basis
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function network as the on-line approximation model. Fig 7 shows the time-domain wave shapes of the virtual fault fˆ (t ) by using FTA-based method. In the simulation study, we use the FTA to detect and identify the system faults. The virtual fault fˆ (t ) is used as an indicator of a system failure. It can be seen from Fig. 7 that after a system fault occurs, the virtual fault fˆ (t ) rapidly jumps to a value above within one second. Comparing Fig. 6 with Fig. 7, we can clearly see that the FTA
based FDI approach provide perfect tracking of the actual faults compared with the neural network based approach. Remark 8: Different from the traditional fault diagnosis approaches, the outstanding feature of this scheme is that it can
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simultaneously perform fault detection and identification in real time without training the network compared with the neural network based FDI schemes. Moreover, the FTA can identify different kinds of faults, even slowly developing faults, known as incipient faults. Compared with the traditional neural network or support vector machine based fault diagnosis methods, the merits of the FTA based scheme are as follows:
① It can simultaneously detect and identify the shape and magnitude of the faults, which are very useful for fault accommodation study.
inequality or matrix equality, even nonlinear equations.
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② The fault is estimated by using iterative algorithms. Therefore, there is no need to solve the complex linear matrix
③ Provide a new FDI methodology for nonlinear uncertain systems.
V. CONCLUSION
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This paper has proposed a new fault detection and identification algorithm for a class of nonlinear uncertain systems. At the heart of this approach is the fault tracking approximator with adjustable parameters. A crucial parameter here is the virtual fault, which can estimate the unknown fault function and system uncertainty occurred in the system. The parameter—virtual fault can
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be used to detect and identify as well as to accommodate system failures. Since the proposed approach is capable of identifying the shape and magnitudes of the faults, it can distinguish between a momentary fault that clears itself, and a persistent one. Therefore, the approach can reduce the number of false alarms. The simulation results show the effectiveness of the method
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developed in this paper. It is well known that the online fault identification and parameter estimation play an important role in the reconfigurable controller design. One of the challenges in the fault tolerant control area is how to obtain accurate parameter
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estimation online and real-time in the presence of poor input excitation [22]. Therefore, the FTA based fault tolerant control problem will be the further research work. ACKNOWLEDGEMENT
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This work is supported by National Natural Science Foundation of China (Nos.51407078 and 61473129). Special Fund of East China University of Science and Technology for Basic Scientific Research (WH1514049, WJ1313004-1, H200-4-13192), the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, the Fundamental Research Funds for the Central Universities (HUST: Grant No. 2015TS025), the Fundamental Research Funds for the Central Universities (WUT: Grant No. 2015VI015), the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1245. REFERENCES [1] L. Zhen, H. Fang, Fault detection for nonlinear systems with unknown input. Asian Journal of Control. 2013, 15(5):1503-1509. [2] F. Zhu, J. Yang, Fault detection and isolation design for uncertain nonlinear systems based on full-order,reduced-order and high-order high-gain slidingmode observer. International Journal of Control.2013, 86(10):1800-1812. [3] N. Meskin, K. Khorasani, Robust fault detection and isolation of time-delay systems using a geometric approach, Automatica 2009, 45: 1567–1573.
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