Robust fault diagnosis algorithm for a class of Lipschitz system with unknown exogenous disturbances

Measurement 46 (2013) 2324–2334

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Robust fault diagnosis algorithm for a class of Lipschitz system with unknown exogenous disturbances q Hua-Ming Qian a, Zhen-Duo Fu a,⇑, Jun-Bao Li b, Lei-Lei Yu a,c a

College of Automation, Harbin Engineering University, Harbin 150001, PR China Department of Automatic Test and Control, Harbin Institute of Technology, Harbin 150001, PR China c The fifty-fourth Research Institute of China Electronic Technology Group Corporation, Shijiazhuang 050081, PR China b

a r t i c l e

i n f o

Article history: Received 23 November 2012 Received in revised form 28 February 2013 Accepted 5 April 2013 Available online 18 April 2013 Keywords: Fault diagnosis Nonlinear system robustness Unknown input disturbances

a b s t r a c t A robust fault diagnosis scheme for nonlinear system is designed and a novel algorithm for a robust fault diagnosis observer is proposed in this paper. The robustness performance index is defined to ensure the robustness of the observer designed. The norm of most unknown input disturbances are assumed bounded at present. However, some systems are proved unstable under traditional assumptions. In the proposed algorithm, the external disturbances constraint condition that satisfies the system stability is derived based on Gronwall Lemma. The design procedure of the observer proposed is implemented by pole assignment. Adaptive threshold is generated using the designed observer. Simulations are performed on continuous stirred tank reactor (CSTR) and the results show the effectiveness and superiority of the proposed algorithm.  2013 Elsevier Ltd. All rights reserved.

1. Introduction The continuous stirred tank reactor (CSTR) is used widely in polymerization chemical reactions [1–4]. Polymerization reactions are described by complex nonlinear kinetic mechanisms and polymerization reactors exhibit highly nonlinear behavior. Polymer processing is an important sector of the chemical process industry. CSTR is important in the chemical reactions process, e.g. in the application of chemical dye, medical reagent, synthetic material and so on. Therefore, the improvement of the reliability in the CSTR process is a challenging and meaningful work [5–8]. The fault diagnosis technology plays a key role in enhancing the system reliability among the multitude methods. Model-based fault diagnosis has been an active field of research over the past decades because of the ever q Supported by ‘‘National Natural Science Foundation of China’’ (Grant No. 61102107). ⇑ Corresponding author. Tel.: +86 13684505176; fax: +86 451 82518741. E-mail address: [email protected] (Z.-D. Fu).

0263-2241/$ - see front matter  2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2013.04.012

increasing demand for higher performance, higher safety and reliability standards [9–11], therefore, the probabilities of the system occurring fault is increasing as well. The reliable operation (or decrease the performance index) for plants is a guarantee to avoid the system paralysis and personal injury, especially for the safety–critical systems, such as, aeronautic and marine system [12–14]. The fault diagnosis and tolerant control technology applied to construct the on-line fault diagnosis observer is a promising approach. The observer can track and detect the system state. In order to make the system function well in the fault mode, we should reduce the plant performance index appropriately and then switch to the control laws pre-designed when the system fault is detected. Hence, the fault diagnosis and tolerant control technology are deserving of researching deeply [15–17,32]. Existing fault diagnosis approaches can be roughly classified into model-free and model-based. In the former methods, P. M. Frank professor considers it can be classified into knowledge-based and data-based fault diagnosis methods [23–27]. These methods do not depend on the model of the systems, but the historical data is needed. However, the main disadvantage

H.-M. Qian et al. / Measurement 46 (2013) 2324–2334

of these model-free methods in that they need a large amount of the plant data. The model-based methods use the system model of the monitored process and then extract the features from the residuals constructed. At present, model-based methods have attracted remarkable attention, e.g., robust fault diagnosis, sliding-mode fault diagnosis and unknown input observer (UIO) fault diagnosis methods [18–22]. Because of the model uncertainties and unknown exogenous disturbances, it is impractical to achieve the system model accurately. Improvement of the false rates is an essential prerequisite for the fault diagnosis algorithm design; meanwhile, the algorithm should be robust to the unknown exogenous disturbances. Consequently, the most important issues are the algorithm robustness to the disturbances in the model-based methods. It is the key factor whether the algorithm applies with the actual system successfully or not. Considerable works have been done to overcome limitations of the fault diagnosis algorithm presented in connection with the progress in nonlinear control theory. Jafar Zarei and Javad Poshtan designed a robust fault detection scheme for nonlinear systems. In the proposed method, the linear UIO design algorithm is extended to nonlinear system [5]. However, the effect of disturbances on the system is not considered. Additionally, the algorithm assumes that the disturbances distribution matrix E must be known a prior, in actual system, the assumption is not always existence. Srinivasan Rajaraman and Juergen Hahn present a novel methodology for systematically designing a fault detection, isolation, and identification algorithm for nonlinear systems with known model structure but uncertainty in parameters [28]. However, the robustness of the detection filter designed was not investigated clearly. In reference [6], a strategy for fault detection and diagnosis in a closed-loop nonlinear system was described and then the state estimates produced by the EKF were the inputs to the controller. Besides the computational effort, the algorithm was based on linearization. In paper [33], multi-sensor data fusion techniques are used to diagnose sensor and process faults based on the EKF estimation algorithm, the new adaptive modified EKF algorithm proposed present the filter divergence and improve the robustness of the algorithm. Hui Luo et al. [34] proposed a novel module level fault diagnosis method for analog circuits based on system identification and genetic algorithm. By comparing the estimated parameters to the normal, the fault of the circuits can be detected by the threshold pre-designed. Motivated by these papers, in this paper, we present an asymptotic convergence robust fault diagnosis observer design method for a class of Lipschitz nonlinear uncertainty system with external disturbances. To the author’s best knowledge, it is the first time to study the stability character when the system suffers from the unknown exogenous disturbances under the nonlinear system fault diagnosis background. Under the Lipschitz condition, we proof that the fault diagnosis observer system can form an operator set for purpose of generating an asymptotic convergence linear semi-group. The system analytic solution about stable linear semi-group is obtained on the assumption that the system state initial value is available.

2325

As a result, we conclude the analytic relationship between the nonlinear semi-group and stable linear semi-group. And then, the robust fault diagnosis observer constructed is proved that it is asymptotic convergence. In the decision-making unit of fault diagnosis process, the residuals analytic expression in connection with the state estimation errors initial value and unknown exogenous disturbances is deduced from the residuals and state estimation errors system. By applying for the Growall lemma, the supremum of the residuals is derived from the fault-free mode. Therefore, the threshold derivation algorithm proposed avoids the blind of the traditional methods which assume the threshold is known a prior [29]. We can infer whether the system is malfunction or not by comparing the residuals with the threshold designed.

1.1. Contributions of this paper The contributions of this paper are shown as follows: (1) Unlike the previous works on assumption that the disturbances distribution matrix is known a prior. The algorithm proposed avoids the deficiency. Compared with the latest ref [5], the algorithm proposed is novel. (2) In Section 3.2, the unknown input disturbances constraint condition that satisfies the system stability is derived based on Gronwall Lemma. Unlike the previous works on assumption that the unknown input disturbances are norm bounded, for example kgk < 6. We proof that a class of nonlinear system is not stable when the external is norm bounded. More details are shown in Section 3.2 and Remark 2. To our best knowledge, the external disturbances constraint condition derived is novel. (3) The robustness performance index concerned with the fault diagnosis observer is defined. Using the performance index defined, an asymptotic convergence fault diagnosis observer is designed. In order to make the observer design procedures convenient for engineering practice, the design procedures are approached by the pole assignment. We address this novel method in Section 3.2. Furthermore, the adaptive threshold is designed in Section 4.

1.2. Structure of this paper The rest of this paper is organized as follows: a brief review of the preliminaries and problem formulation are described in Section 2. In Section 3, the fault diagnosis approach based on robust fault detection observer for a class of Lipschitz nonlinear system with unknown exogenous disturbances is proposed and the stability of system is also introduced. The threshold is designed and the decision-making unit of fault diagnosis is described in detail in Section 4. Experiments are designed and the simulations results are analyzed and compared in Section 5. Finally, conclusions from this research as well as the future works are discussed in Section 6.

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2. Preliminaries and problem formulation Consider the nonlinear system with the additive model uncertainties and the unknown exogenous disturbances, the nonlinear system dynamic equation can be written in the form:

_ xðtÞ ¼ ðA þ DAÞxðtÞ þ ðB þ DBÞuðtÞ þ hðxðtÞ; uðtÞÞ þ g 0 ðxðtÞ; uðtÞ; dðtÞ; tÞ þ f ðtÞ yðtÞ ¼ CxðtÞ

ð1Þ ð2Þ

Where, E1, E2, F1, F2 are known matrices. DA, DB are modeling errors given by:

(

X1 ¼ fDAjDA ¼ E1 R1 F 1 ; RT1 fR1 6 Ig X2 ¼ fDBjDB ¼ E2 R2 F 2 ; RT2 R2 6 Ig

Based on the above discussion, nonlinear time-varying uncertain model (1) can be further rewritten as:

_ xðtÞ ¼ AxðtÞ þ BuðtÞ þ hðxðtÞ; uðtÞÞ þ gðxðtÞ; uðtÞ; dðtÞ; tÞ þ f ðtÞ

ð3Þ

where x(0) = x0 is the state initial value, x(t) 2 Rn is the state vector, u(t) 2 Rp is the control input vector, y(t) 2 Rq is the measurable output vector, g 2 Rn is the unknown exogenous disturbances, h 2 Rn and g are sufficiently smooth nonlinear functions. f(t) 2 Rn is the fault to be detected and diagnosed. A, B, C are real constant matrices of appropriate dimensions. Remark 1. It is worth pointing out that the Eq. (2) is a generic form when the measure output is a linear procedure. Let us consider the following nonlinear system to demonstrate that.



_ xðtÞ ¼ f ðxðtÞ; dðtÞ; uðtÞ; f a ðtÞÞ yðtÞ ¼ CxðtÞ

The system initial condition x0 = x(0), the state equation of the system with fault can be expressed by:

xðtÞ ¼ xð0Þ þ

Z

algorithm of the actuator fault diagnosis is not loss of generality. Prior to fault diagnosis observer design, the following assumptions are used throughout: Assumption. (A1) (C, A) is detectable and A is Hurwitz matrix; (A2) The supremum of the external disturbances denotes as follows:

k0 ¼ sup kgðxðtÞ; uðtÞ; dðtÞ; tÞk t2½0;T

It should be noted that this traditional norm bounded assumption method to external disturbances will be checked in Section 3.2. h(x(t), u(t)) is the system nonlinearity verifying the Lipschitz conditions:

^ðtÞ; uðtÞÞk 6 k1 kxðtÞ  x ^ðtÞk khðxðtÞ; uðtÞÞ  hðx with k1 > 0 called the Lipschitz constant. Before introducing to the fault detection algorithm, this section presents some preliminary results and definitions for the paper’s succession and readability. Definition 1 (Bounded linear operator semi-group [30]). Define X as a Banach space, {T(t), t P 0} represents the bounded linear operator family X ? X. There exists the following conditions such that {T(t), t P 0} is bounded linear operator semi-group. If the following equations hold: 1 T(0) = I; 2 T(t + s) = T(t)T(s) = T(s)T(t)(t, s P 0) 3 limt!0þ kTðtÞx  xk ¼ 0; x 2 X Lemma 1 (Gronwall Lemma [31]). @ 1(t), @ 2(t) and @ 3(t) denote as positive semi-define continuous formula, @ 0 2 R+ If:

@ 1 ðtÞ 6 @ 0 þ

Z

t

½@ 2 ðsÞ@ 1 ðsÞ þ @ 3 ðsÞds

0

t

f ðxðsÞ; dðsÞ; uðsÞ; f a ðsÞÞds

0

Hence, the measure output is described as:

yðtÞ ¼ Cxð0Þ þ C

Z

t

f ðxðsÞ; dðsÞ; uðsÞ; f a ðsÞÞds

0

Consider the integral term stated above, there exists F1(d(t)), F2(fa(t)), F3(x(t), u(t)) such that the following equation holds:

F 1 ðdðtÞÞ þ F 2 ðf a ðtÞÞ þ F 3 ðxðtÞ; uðtÞÞ Z t ¼ f ðxðsÞ; dðsÞ; uðsÞ; f a ðsÞÞds 0

And therefore:

yðtÞ ¼ Cxð0Þ þ C½F 1 ðdðtÞÞ þ F 2 ðf a ðtÞÞ þ F 3 ðxðtÞ; uðtÞÞ That is to say, the external disturbances and actuator fault can be mapped into the measure output by an operator. Compared with the sensor fault diagnosis, the

hR i t then: @ 1 ðtÞ 6 @ 0 exp 0 ½@ 2 ðsÞ þ @ 3 ðsÞ=@ 0 ds Using the group algebra theory and the Gronwall Lemma, the system state can be obtained. In the next two sections, we approach the nonlinear Lipschitz system with the unknown exogenous disturbances from two aspects: stability and robustness. 3. Stability analysis and robust fault diagnosis observer design This section focuses on robust fault diagnosis observer design for a class of Lipschitz nonlinear system with unknown exogenous disturbances. Firstly, the effect of external disturbances acting on the system will be discussed. And then we conclude the external disturbances constraint condition in Theorem 1 that satisfies the system stability under the disturbances influence. Secondly, robustness performance index of the fault diagnosis observer is defined, the convergence of the observer proposed is guaranteed by the following Theorem 2.

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3.1. Derived of g(x(t), u(t), d(t),t) constraint condition

kxðtÞk 6 Ma expðxtÞ þ

Z

t

kM exp½xðt  sÞNkds

ð9Þ

0

Generally speaking, it is impractical to construct analytical model precisely due to the universal existence of nonlinearities and model uncertainties in practice. Scholars generally add the model uncertainties g(x(t), u(t), d(t), t) into the systems to match with the uncertainties, just like formula (3) illustrated. The unknown exogenous disturbances play an important role on the systems stability, how the disturbances effect on the system deserves to research in depth. Unfortunately, there are not any papers to study this problem under the nonlinear system fault diagnosis background. In this section we derive the unknown exogenous disturbances constraint conditions which make the system hold stable under the disturbances interfere.

In order to get the same mode as in the Lemma, divide (9) by exp (xt) for both sides:

kxðtÞk expðxtÞ 6 Ma þ

Z

t

kM expðxsÞNkds

ð10Þ

0

kxðtÞk expðxtÞ 6 Ma þ

Z

t

kMxðsÞ

0

 expðxsÞNk=kxðsÞkds

ð11Þ

Apply for the Lemma:

kxðtÞk 6 Ma exp

Z

t

ðx þ kMNk=kxðsÞkÞds

 ð12Þ

0

Theorem 1. Consider the Lipschitz nonlinear system (2) and (3). Its global stability is ensured by a constraint condition as follows. And then, $M P 1, x < 0, t P 0, consequently, the constraint condition satisfies:

with definitions:

b0 ðtÞ ¼ kBuðtÞk; b1 ðtÞ ¼ khðxðtÞ; uðtÞÞk Proof. Consider the system model described in (2) and (3). From the Assumption stated above, we can know that the system matrix A is Hurwitz matrix. Therefore, the matrix A can form an operator set for purpose of generating an asymptotic convergence linear semi-group ft, and then $M P 1, x < 0, t P 0 such that the following inequation holds:

ð4Þ

The state representation of the nonlinear system (2) and (3) with the unknown exogenous disturbances under the fault free case is as follows:

xðtÞ ¼ xð0Þ þ

Z

#ðtÞ ¼

Z

t

ðx þ kM Nk=kxðsÞkÞds

ð13Þ

0

Consequently, if there exists a finite constant jej and jej < 1, for "t ? + 1 can realize limt?+1#(t) < jej < + 1. And then, we regard the system with external disturbances as stable system.

kgðxðtÞ; uðtÞ; dðtÞ; tÞk < ðb1 ðtÞ  b0 ðtÞÞ  xkxðsÞk=M

kft k 6 M expðxtÞ

For simplification, denote as follows:

From the formula (7), the system exists a nonlinear semi-group pt such that the system state is represented as follows:

xðtÞ ¼ pt xð0Þ

ð14Þ

From expressions (12) and (13):

kxðtÞk ¼ kpt xð0Þk 6 Ma expð#ðtÞÞ

ð15Þ

Therefore, the nonlinear semi-group pt is stable when #(t) < 0, i.e., the system (2) and (3) with the unknown exogenous disturbances is stable. That is to say, the system is stable under x + kMNk/kx(s)k < 0 condition:

kNk < xkxðsÞk=M Substitute into formula (8):

t

½AxðsÞ þ BuðsÞ þ hðxðsÞ; uðsÞÞ

kBuðtÞ þ hðxðtÞ; uðtÞÞ þ gðxðtÞ; uðtÞ; dðtÞ; tÞk

0

þ gðxðsÞ; uðsÞ; dðsÞ; sÞds

ð5Þ

$a stable linear semi-group ft such that the formula (6) comes into existence:

xðtÞ ¼ ft xð0Þ þ fts

Z

½BuðsÞ þ hðxðsÞ; uðsÞÞ ð6Þ

Approach the 2-norm form to Eq. (6) for both sides:

Z

t

½BuðsÞ þ hðxðsÞ; uðsÞÞ ð7Þ

For simplification:

N ¼ BuðsÞ þ hðxðsÞ; uðsÞÞ þ gðxðsÞ; uðsÞ; dðsÞ; sÞ

b0 ðtÞ ¼ kBuðtÞk; b1 ðtÞ ¼ khðxðtÞ; uðtÞÞk According to formula (16) and (17), one can further obtain that:

b0 ðtÞ  b1 ðtÞ þ kgðxðtÞ; uðtÞ; dðtÞ; tÞk < xkxðsÞk=M

0

þ gðxðsÞ; uðsÞ; dðsÞ; sÞdsk

ð17Þ

u(t) and h(x(t), u(t)) are definite, denote as follows:

0

þ gðxðsÞ; uðsÞ; dðsÞ; sÞds

ð16Þ

kBuðtÞ þ hðxðtÞ; uðtÞÞ þ gðxðtÞ; uðtÞ; dðtÞ; tÞk > kBuðtÞk  khðxðtÞ; uðtÞÞk þ kgðxðtÞ; uðtÞ; dðtÞ; tÞk

t

kxðtÞk ¼ kft xð0Þ þ fts

< xkxðsÞk=M

ð8Þ

The state initial value is x0 = x(0), and therefore, a = kx(0)k, kftk 6 M exp (xt), as a result, Eq. (7) satisfies the inequation:

kgðxðtÞ; uðtÞ; dðtÞ; tÞk < ðb1 ðtÞ  b0 ðtÞÞ  xkxðsÞk=M

ð18Þ

Based on the statement above, we can conclude that the system (2) and (3) is stable when the unknown exogenous disturbances satisfy the constraint condition as the formula (18) stated. This completes the proof. h Furthermore, if there exists b3 2 R+ such that the following inequation holds:

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ðb1 ðtÞ  b0 ðtÞÞ 6 b3 kxðsÞk

^ðtÞ þ BuðtÞ þ hðx ^ðtÞ; uðtÞÞ þ G½yðtÞ  y ^ðtÞ þ f ðtÞ ^_ ðtÞ ¼ Ax x ð19Þ

Therefore, the Eq. (18) can be rewritten as follows:

kgðxðtÞ; uðtÞ; dðtÞ; tÞk < ðb3  x=MÞkxðsÞk kgðxðtÞ; uðtÞ; dðtÞ; tÞk=kxðsÞk < b3  x=M We denote RðAÞ ¼ x=M as the robustness of the fault diagnosis observer to the external disturbances. Remark 2. The nonlinear system considered this paper holds stable under the external disturbances affect if the external disturbances satisfy the constraint condition as formula (18) represented. It is clear that the system cannot always hold stable under some form disturbances interfere. So far as that is concerned, in most papers, the authors suppose that the external disturbances satisfy norm bounded. However, whether the system is stable or not deserve to analyze when the disturbances are in the norm bounded constraint condition. As described the system in this paper, the following assumptions are made for the system considered: We define the form of the external disturbances as represented in Assumption (A2)

k0 ¼ sup kgðxðtÞ; uðtÞ; dðtÞ; tÞk t2½0;T

Hence, when the inequation k0 P (b1(t)  b0 (t))  xkx(s)k/M holds, it can be seen obviously that the system (2) and (3) is not stable on this norm bounded constraint condition. Prior to design fault diagnosis algorithm, we all suppose that the system holds stable under the disturbances influence. Therefore, it is meaningful to analyze the effect of the disturbances injecting into the system in the system control laws design and fault diagnosis algorithm design. The analysis methods to the system with the external disturbances of this section possess the referential sense for prior period analysis of the fault diagnosis algorithm design.

^ðtÞ y^ðtÞ ¼ C x

ð20Þ

Denote the state estimation errors and residuals respectively as:

^_ ðtÞ eðtÞ ¼ xðtÞ  x

ð21Þ

rðtÞ ¼ yðtÞ  y^ðtÞ

ð22Þ

Substitute (3) and (19) into (21):

^_ ðtÞ _ _ x eðtÞ ¼ xðtÞ ^ðtÞ þ hðxðtÞ; uðtÞÞ  hðx ^ðtÞ; uðtÞÞ  G½CxðtÞ ¼ AxðtÞ  Ax ^ðtÞ þ gðxðtÞ; uðtÞ; dðtÞ; tÞ þ f ðtÞ  Cx _ ^ðtÞ; uðtÞÞ eðtÞ ¼ ðA  GCÞeðtÞ þ hðxðtÞ; uðtÞÞ  hðx þ gðxðtÞ; uðtÞ; dðtÞ; tÞ þ f ðtÞ

ð23Þ

rðtÞ ¼ yðtÞ  y^ðtÞ ¼ CeðtÞ

ð24Þ

Under the fault-free mode, on the assumption that the initialization value of the residuals and state estimation errors system of the fault diagnosis observers are:

e0 ¼ eð0Þ; b ¼ keð0Þk For simplification, denote as follows:

W ¼ hðxðsÞ; uðsÞÞ  hðx^ðsÞ; uðsÞÞ þ f ðtÞ þ gðxðsÞ; uðsÞ; dðsÞ; sÞ Consequently, the state estimation errors act as:

eðtÞ ¼ exp½ðA  GCÞteð0Þ þ

Z

t

exp½ðA  GCÞðt 0

 sÞWds

ð25Þ

Approach the 2-norm form to Eq. (25) for both sides: 3.2. Derived of RðA  GCÞ constraint condition This section introduces the robust fault diagnosis observer design methods of the nonlinear system (2) and (3). The robustness performance index of the fault diagnosis observer is defined in order to ensure the observer designed asymptotic convergence and robustness to the disturbances. Theorem 2. Consider the fault b2 ¼ sup kf ðtÞk and the t2½0;T system (2) and (3). There exists an asymptotic convergence fault diagnosis observer (19) and (20) such that it can estimate the system state robust. Therefore, the fault diagnosis observer satisfies the robustness performance index as follows:

RðA  GCÞ > k1 þ ðkgðxðsÞ; uðsÞ; dðsÞ; sÞk þ b2 Þ=keðsÞk

keðtÞk ¼ k exp½ðA  GCÞteð0Þ þ

Z

t

exp½ðA  GCÞðt

0

 sÞWdsk

ð26Þ

The system matrix A  GC is Hurwitz matrix when the system (23) and (24) is stable. Therefore, the matrix A  GC can form an operator set for purpose of generating an asymptotic convergence linear semi-group ft, and then $M P 1,x < 0, t P 0 such that the following inequation holds:

kft k 6 M expðxtÞ Hence, one can further obtain that the formula (26) satisfies the inequation:

keðtÞk 6 Mb expðxtÞ þ

Z

t

kM exp½xðt  sÞWkds

0

Proof. For the purpose of residual generation, we construct the following nonlinear robust fault diagnosis observer for the system (2) and (3) with actuator fault as the following forms:

Divide (27) by exp (xt) for both sides:

keðtÞk expðxtÞ 6 Mb þ

Z 0

t

kM expðxsÞWkds

ð27Þ

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keðtÞk expðxtÞ 6 Mb þ

Z

t

4. Fault diagnosis algorithm and the adaptive threshold Jth(r(t)) design

kMeðsÞ

0

 expðxsÞkWk=keðsÞkds Without loss of generality, we assume that the fault injected into the system is norm bounded. That is to say,

b2 ¼ sup kf ðtÞk t2½0;T

^ðsÞ; uðsÞÞk þ kf ðtÞk kWk 6 khðxðsÞ; uðsÞÞ  hðx

(

þ kgðxðsÞ; uðsÞ; dðsÞ; sÞk 6 k1 keðsÞk þ b2 þ kgðxðsÞ; uðsÞ; dðsÞ; sÞk Z

 ½x þ MkWk=keðsÞkds

Fault diagnosis algorithm involves residuals generation, residuals processing and the fault decision making. Generally, comparing the residuals with the threshold to verify whether the systems are malfunction or not. The logical relationship of the fault decision making as follows:

ð28Þ

krðtÞk2;T > J th ðrðtÞÞ ) fault  freecaseðH0 Þ krðtÞk2;T 6 J th ðrðtÞÞ ) faultycaseðH1 Þ

Therefore, there exists an asymptotic stable fault diagnosis observer (19) and (20) such that following inequality holds:

This section avoids the limitations that the traditional methods suppose the threshold is known a prior. The threshold design algorithm is proposed and then the threshold is theoretically obtained. The fault-free case: From the formula (24), we can get the system residuals:

x þ MkWk=keðsÞk < 0

ð30Þ

rðtÞ ¼ CeðtÞ

x=M > k1 þ ðkgðxðsÞ; uðsÞ; dðsÞ; sÞkÞ þ b2 Þ=keðsÞk

ð31Þ

keðtÞk 6 Mb exp

t

ð29Þ

0

And then, the state estimation errors:

From the definition of Section 3.2, the robustness RðA  GCÞ of the fault diagnosis observer (19) and (20) satisfies:

Z

t

^ðsÞ; uðsÞÞ ½hðxðsÞ; uðsÞÞ  hðx

0

þ gðxðsÞ; uðsÞ; dðsÞ; sÞds

ð34Þ

Therefore

RðA  GCÞ > k1 þ ðkgðxðsÞ; uðsÞ; dðsÞ; sÞk þ b2 Þ=keðsÞj

eðtÞ ¼ eð0Þ þ

ð32Þ

Thus, the observer designed is stable and asymptotic convergence. From the statement above, we can conclude that there exists an asymptotic convergence fault diagnosis observer such that it can estimate the system state robust when the robustness performance index is given. This completes the proof. h

rðtÞ ¼ Ceð0Þ þ C

Z

t

^ðsÞ; uðsÞÞ ½hðxðsÞ; uðsÞÞ  hðx

0

þ gðxðsÞ; uðsÞ; dðsÞ; sÞds

ð35Þ

Approach the 2-norm form to Eq. (35) for both sides:

krðtÞk ¼ kCeð0Þ þ C

Z

t

^ðsÞ; uðsÞÞ ½hðxðsÞ; uðsÞÞ  hðx

0

Remark 3. From the analysis above, when the state estimation error and the supremum of the external disturbances as demonstrated in Theorem 1 are given, i.e. kemax(s)k and kgmax(x(s), u(s), d(s), s)k are known. Define the robustness performance index:

1 RðA  GCÞ ,  kmax ½ðA  GCÞ þ ðA  GCÞT  2 Substitute into Eq. (32):

þ gðxðsÞ; uðsÞ; dðsÞ; sÞdsk And then, denote bc = kCk

^ðsÞ; uðsÞÞ dðhÞ ¼ hðxðsÞ; uðsÞÞ  hðx

ð36Þ

The 2-norm condition of residuals fulfills the inequation as following:

krðtÞk 6 bbc þ bc k

Z

t

½dðhÞ 0

ki ½ðA  GCÞ þ ðA  GCÞT  < 2fk1

þ gðxðsÞ; uðsÞ; dðsÞ; sÞdsk

ð37Þ

þ ðkgðxðsÞ; uðsÞ; dðsÞ; sÞk þ b2 Þ=keðsÞkg

ð33Þ

Obviously, the fault diagnosis observer gain matrix constrained in RðA  GCÞ can be obtained by pole assignment when the state estimation error and the supremum of the external disturbances are given. Where, denote kmax() as the matrix () maximal eigenvalue, ki() is arbitrary eigenvalue of the matrix (). When robustness performance index is known, the gain matrix of the robust fault diagnosis observer can be achieved by pole assignment method.

The expression of the formula (36) about residuals:

^ðsÞk ¼ k1 keðsÞk ¼ kdðhÞk 6 k1 kxðsÞ  x

k1 krðsÞk bc

ð38Þ

Hence

krðtÞk 6 bbc þ

Z

t

½k1 krðsÞk

0

þ bc kgðxðsÞ; uðsÞ; dðsÞ; sÞkds

ð39Þ

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H.-M. Qian et al. / Measurement 46 (2013) 2324–2334

From the Theorem 1 demonstrated, we can get the unknown exogenous disturbances restriction condition that makes the system stable, hence, the formula (39) can be described as:

krðtÞk 6 bbc þ

Z

t

k1 bc keðsÞkds þ

0

Z

t

bc ½b1 ðsÞ  b0 ðsÞ

0

 xkxðsÞk=Mds

ð40Þ

where in the formula (40), b1(s) and b0(s) are known scalar as denoted in Section 3.2 (Theorem 1). In the engineering applications, the maximum tolerant state estimation errors and the system state steady values are available, that is to say:

8 dx 1 > ¼ p1 u1 þ p1 u1 u2  p1 u2 x2  p1 x1 > dt > > > > p2 ð1 þ x1 Þ exp½p3 ð1 þ x2 Þ > > > > dx2 > > < dt ¼ p1 u3 þ p1 u2 u3  p1 x2  p1 u2 x2  p4 x2 þ p4 x3 þp5 p2 ð1 þ x1 Þ exp½p3 ð1 þ x2 Þ > > > > dxdt3 ¼ p6 x2  p6 x3  p7 x3 þ p7 x4 > > > > > dx4 ¼ p u4 þ p u4 u5  p x4  p u5 x4 þ p x3  p x4 > > 8 8 8 8 9 9 > : dt ð45Þ where, system state vector, control input vector and parameters are shown in reference [5]. 5.1. Simulation parameters

emax ¼ sup keðtÞk

ð41Þ

t2½0;T

limkxðtÞk ! xmax

ð42Þ

t!T

Therefore, the least upper bound of the in Eq. (40) represents as:

krðtÞk 6 bbc þ ðk1 bc emax  xxmax =MÞt þ bc

Z

t

2

½b1 ðsÞ

0

 b0 ðsÞds

In all simulations, the parameters of the algorithm are selected as follows: 2 3 The1 measure 0 0 0 output is defined as Eq. (2) with C ¼ 4 0 1 0 0 5, and then A,B,h(x(t), u(t)) are explained 0 0 0in 1formula (3). as previously Therefore

ð43Þ

6 0 6 A¼6 4 0

When the system under the fault case:

krðtÞk > bbc þ ðk1 bc emax  xxmax =MÞt þ bc

2

t

p1 60 6 B¼6 40

½b1 ðsÞ

0

 b0 ðsÞds Accordingly, the fault diagnosis adaptive threshold refers to:

J th ðrðtÞÞ ¼ bbc þ ðk1 bc emax  xxmax =MÞt þ bc Z t  ½b1 ðsÞ  b0 ðsÞds

0

0

ðp1 þ p4 Þ

p4

0

p6

ðp6 þ p7 Þ

p7

0

p9

ðp8 þ p9 Þ

0

Z

0

3

0

p1

0

0

0

3 0 07 7 7; 05

0

0

p8

0

0 0 0 p1

0 0

7 7 7 5

3 p1 x1 u2 6 p x u 7 6 1 2 27 hðxðtÞ; uðtÞÞ ¼ 6 7 5 4 0 2

p8 x4 u5

2

3 p1 u1 u2  p2 ð1 þ x1 Þ exp½p3 ð1 þ x2 Þ þ d1 6 p u u þ p p ð1 þ x Þexp½p ð1 þ x Þ þ d 7 2 3 1 2 27 6 5 2 3 gðxðtÞ;uðtÞ; dðtÞ; tÞ ¼ 6 1 7 4 5 d3 p8 u4 u5 þ d4

ð44Þ

0

From the threshold deduced procedures, we can know that the threshold obtained considers the effect of the unknown exogenous disturbances adequately. The threshold demonstrated in Eq. (44) is adaptive. Compared with the traditional methods on the assumption that the threshold is constant, the algorithm proposed of this paper decreases the false alarm rates in the fault diagnosis process. 5. Simulation results In this section, the simulation results of the proposed robust fault diagnosis algorithm will be verified. To evaluate the performance of the proposed algorithm, it is applied to a highly nonlinear dynamic system describing the behavior of a non-adiabatic continues stirred tank reactors (CSTR) in which an irreversible highly exothermic chemical reaction (A ? B) takes place. The reactor’s wall significantly affects the system dynamics and therefore has also been taken into account. The corresponding model leads to the following set of ODEs in a normalized and dimensionless [5]:

where [p1, p2, p3, p4, p5, p6, p7, p8, p9] is defined as: [0.03333, 4.08  107, 25.347, 0.663, 1.45, 5.97, 5.97, 0.167, 1.33]. Table 1 followed shows three steady states of the model respectively. 5.2. Simulation 1: Analysis of system stability In the section, the main works are to analyze the effect that the unknown exogenous disturbances inject into the system and how to select the robustness performance index in order to make the system hold stable under the disturbances interfere. In the simulations, the parameters of the system are selected as follows:

uðtÞ ¼ ½ 0:2 0:2 0:2 0:2 0:2 T di ðtÞ ¼ 0:01 sinðtÞ i ¼ 1; 2 . . . 4 In this section, we select the high-temperature stable as the unknown exogenous disturbances analysis. It is necessary to notice that the key point of the paper research focuses on the design of the fault diagnosis algorithm not

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H.-M. Qian et al. / Measurement 46 (2013) 2324–2334 Table 1 Steady State Low-temperature stable

Unstable

High-temperature stable

0.0140582 0.0068168 0.0061321 0.0054473

0.37748 0.18304 0.16465 0.14627

0.97640 0.47345 0.42590 0.37834

x 10

3 2.5

norm of g (x,u,d)

x1 x2 x3 x4

-4

3.5

2 1.5 1

0 x1 x2 x3 x4

-0.05

0.5

-0.1

0

x1,x2,x3,x4

0

20

40

60

80

100

t/s

-0.15

Fig. 2. The norm of g(x(t), u(t), d(t), t).

-0.2 -0.25 -0.3 -0.35 0

20

40

60

80

100

t/s Fig. 1. The system state characteristic

on the design of the control laws. Therefore, for simplification, we select constant input

uðtÞ ¼ ½ 0:2 0:2 0:2 0:2 0:2 T as the system control input. Fig. 1 shows the system state characteristic. As expected, the state converges asymptotically under the control input stated. When the system is in the high-temperature stable state, the norm of the unknown exogenous disturbances as follows: Fig. 2 shows that the supremum of the g(x(t), u(t),d(t), t) is 3.171  104, when the right side of the Eq. (18) beyond the supremum of the g(x(t), u(t), d(t), t), the system is stable under the unknown exogenous disturbances interfere. The relationship between robustness and disturbances is shown in Fig. 3. The Fig. 3 shows that, when the robustness performance index of the fault diagnosis observer achieves at 0.01667, the both sides of formula (18) are equivalent. That is to say, just like Theorem 1 depicted, if only the robustness performance index selected surpasses 0.01667 the fault diagnosis observer designed acts steady. 5.3. Simulation2: Efficiency of the algorithm designed In this section, a class of Lipschitz nonlinear system fault diagnosis algorithm is designed for the described CSTR using the proposed algorithm. The objective of Section 5.3 is to design the fault diagnosis observer and analyze the efficiency of the fault diagnosis algorithm proposed. The parameters of the system are selected as follows:

g__max (The right side of equation(18))

6

x 10

-3

Robustness performance index Maximum disturbances

4

2

0

-2

-4

-6 0.013

0.014

0.015

0.016

0.017

0.018

0.019

t/s (Robustness performance index) Fig. 3. The relationship g(x(t), u(t), d(t), t).

between

robustness

and

supremum

of

The systems fault: f ðtÞ ¼ ½ 0:2 0 0 0 T Maximum tolerant value of the state estimation errors: emax ¼ ½ 0:1 0:1 0:1 0:1 T Lipschitz constant: k1 = 0.2 The Theorem 2 and Remark 3 demonstrate the asymptotic convergence robust fault diagnosis algorithm design procedures. It is clearly that design of the gain matrix G of system (23) and (24) actually equivalently to the issues of pole assignment. Consider the parameters presented ahead, obviously the fault diagnosis observer we expected should fulfill the inequality as follows:

ki ½ðA  GCÞ þ ðA  GCÞT  < 2:952 That is to say, the poles should be settled at the left semi-plane of -2.952; therefore, the gain matrix G is not unique. Based on the algorithm stated, choose the poles ½ 4:36 þ 6i 4:36  6i 5:66 þ 3i 5:66  3i , and then using the Matlab software, consequently the gain matrix G is:

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H.-M. Qian et al. / Measurement 46 (2013) 2324–2334

4:8438

5:2294

6 4:7387 6 G¼6 4 8:6840

0:0989

3

ðk1 emax  xxmax =MÞ þ ðb1 ðsÞ  b0 Þ:

3:2840 7 7 7 44:0989 5

2:3984 33:6801

Consequently

b1 ðtÞ ¼ khðxðtÞ; uðtÞÞk 6 bmax

0:5626 3:5639 1:4368 Therefore, the fault diagnosis observer system (23) and (24) can be modeled using the Simulink module in the Matlab. Fig. 4 shows the residuals characters and then it can be seen that the residuals converge asymptotically. Additionally, the effect of the unknown input disturbances on the system is weakened, in other words, the robust fault diagnosis algorithm proposed is efficiency. Furthermore, the residuals obtained from the fault diagnosis observer are sufficiently small; therefore, the fault diagnosis observer designed also has excellent sensitivity to the fault to be detected. What follows in the passage, a selected fault scenario is used to illustrate the performance of the proposed fault diagnosis approach. The fault is supposed as demonstrated in Fig. 5.

 fi ¼

0:2 t P 20s 0

t < 20s

J th ðrðtÞÞ 6 J max Therefore, we can conclude whether the system is malfunction or not by the threshold designed from Fig. 6 and no mater how the control input acts constant or timevarying. 5.4. Simulation3: Superiority of the algorithm designed In order to demonstrate the superiority of the algorithm proposed, in this section we select the gain matrix G of system (23) and (24) randomly. The Fig. 7 shows that the residuals have the poor robustness to the unknown input disturbances compared with Fig. 4. And ref [5] assumes

0.3

i ¼ 1; 2; 3

ð46Þ

In order to examine the performance of the fault diagnosis algorithm proposed, we inject into the system fault parameters as described in formula (46) respectively. The simulation results of the Fig. 6 shows that each residual is affected by the corresponding fault, in other words, the fault diagnosis algorithm proposed this paper can isolate faults remarkably effective. The Eq. (44) shows the adaptive threshold when the control input is time-varying. On second thoughts, if the control input is constant like:

0.25

0.2

Fault

2

0.15

0.1

0.05

uðtÞ ¼ ½ 0:2 0:2 0:2 0:2 0:2 T

0 0

20

40

Therefore, b0(t) = kBu(t)k = b0

J th ðrðtÞÞ ¼ b þ

Z

t

80

100

60

80

100

60

80

100

Fig. 5. Fault signal.

½ðk1 emax  xxmax =MÞ þ ðb1 ðsÞ  b0 Þds

0

Fig. 1 shows that the state converges asymptotically under the control input stated. In other words, there exists a least upper bound of the formula as follows:

r1 r2 r3

0.2 0.1 0 -0.1

0.3

0

r1 r2 r3

20

40

t/s Residuals

0.25 0.2

Residuals

60

t/s

0.15

0.2 0.1 0 -0.1

0.1

0

20

40

t/s

0.05 0 0.2 0.1

-0.05

0 -0.1 0

20

40

60

80

100

t/s Fig. 4. Residuals for system (23) and (24) under the fault free mode.

-0.1 0

20

40

60

t/s Fig. 6. Effect of fault on residuals.

H.-M. Qian et al. / Measurement 46 (2013) 2324–2334 0.08

r1 r2 r3

Residuals

0.06

2333

was partially supported by the National Natural Science Foundation of China under Grants 61102107.

0.04

References

0.02

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0 -0.02 -0.04 -0.06 -0.08 0

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that the disturbances distribution matrix is known a prior, we all know that the simulation results are based on this assumption throughout. However, in practical this assumption is not always available. Based on the analysis above, that is to say, the robust fault diagnosis observer design algorithm proposed is superior to ref [5]. Additionally, when the small fault signal is occurred, we can know from the Fig. 7 that it is difficult to differentiate the fault from the residuals because of the poor robustness. As a comparison, the residuals illustrated in Fig. 4 decreases the false alarm rates in recognizing the fault from the residuals.

6. Conclusions and future work The validity of the fault diagnosis methods are tested in the simulations, the analysis of the exogenous disturbances and the algorithm proposed will be benefit for the practical application. Most papers about model-based fault diagnosis algorithm are based on linear system models. In this paper, a new algorithm of robust fault diagnosis for a class of Lipschitz system with unknown exogenous disturbances is proposed. At first, we initially study the effect of the disturbances on the system stability under the fault diagnosis framework (Theorem 1). Then, the robustness of the fault diagnosis algorithm to the perturbations is ensured by the robustness performance index defined. The asymptotic convergence robust observer is designed in Section 3.2. Besides, the solution algorithm of gain matrix G is represented in Theorem 2 and then the adaptive threshold is designed. The proposed algorithm is applied to a CSTR to evaluate its performance. Simulation results confirm the efficiency and superiority of the proposed fault diagnosis scheme. However, we only consider the robustness performance index but for the other performance index are left to the future works. Acknowledgment The authors thank the guest editor and reviewers for their valuable comments and suggestions which have helped to improve the presentation of the paper. This work

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