Design of Nonlinear Unknown Input Observer for Process Fault

Ind. Eng. Chem. Res. 2010, 49, 11443–11452

11443

Design of Nonlinear Unknown Input Observer for Process Fault Detection Jafar Zarei† and Javad Poshtan* School of Electrical Engineering, Iran UniVersity of Science and Technology, Tehran 16846-13114, Iran

The objective of this paper is to design a robust fault detection scheme for nonlinear systems. A novel procedure for the design of a nonlinear unknown input observer (UIO) is proposed. The basic principle of unknown input observers is to decouple disturbances from the state estimation error. In the proposed method, the linear UIO design algorithm is extended to nonlinear systems and then observer gain is obtained using unscented transformation (UT). To illustrate the efficiency of the proposed method, it is adopted to detect sensor faults of a highly nonlinear dynamic system. The faulty behavior of output sensors in a jacketed continuous stirred tank reactor (CSTR) is investigated. A single full-order observer is designed to detect sensor faults in the presence of unknown inputs (disturbances). Simulation results show that disturbances and, therefore, a certain degree of model uncertainties can be distinguished from a response to a sensor fault. 1. Introduction In the real world, no system can work perfectly at all times under all conditions. In chemical plants faulty sensors may cause process performance degradation (e.g., lower product quality) or fatal accidents (e.g., temperature run away).1 A report estimates that the loss to petrochemical industries in the U.S. alone is $20 billion/year.2 While petrochemical plants are becoming larger, loss and maintenance costs will increase. Besides the economic loss, irreparable damage to human operators should be considered. Moreover, researchers have been recently attracted to designing fault-tolerant systems for the chemical process.3-8 One of the methods to design such systems is first detecting faults by appropriate methods and then choosing a proper switching policy to orchestrate activation/deactivation of the constituent control configurations for each faulty condition. The most important part of designing these systems is the correct diagnosis of the faults. Therefore, it is essential that a fault detection scheme can be developed so as to be able to detect and identify possible faults in the system as early as possible.9 Then, the system can be maintained and kept reliable by means of this early warning, enabling repair or replacement to take place at the earliest or the most convenient time, with the minimum loss of time or productivity. Today, fault detection and diagnosis have become inseparable parts of modern complex systems. Existing fault detection approaches can be roughly classified into model-free approaches, i.e., approaches based on statistical analysis, neural networks, and/or expert systems, and approaches based on the analytical redundancy, resorting to the available mathematical model of the process.10 Statistical techniques do not require a model of the system, but only a good database of historical data regarding normal operating conditions is needed, since statistical tests on the measured data are used to detect any abnormal behavior. This is, of course, the main disadvantage of these methods in that they need a large amount of plant data that are collected along a quite large window of operating time and are used to construct a statistical model of the process.11 Model-based fault detection and isolation (FDI) techniques use mathematical models of the monitored process and extract features from measured signals, to generate a fault-indicating * To whom correspondence should be addressed. Phone: +98 21 77240492. Fax: +98 21 77240490. E-mail: [email protected]. † E-mail: [email protected].

signal, which is called a residual. Whereas mathematical models are necessary for control purposes, model-based FDI technology has attracted remarkable attention in modern complex systems during the past 3 decades.9 There are a great variety of modelbased FDI methods in the literature.9,10,12-15 Among these, observer-based schemes have been successfully adopted in a variety of application fields. The extended Kalman filter (EKF) is one of the most popular model-based techniques used for fault detection and diagnosis in chemical processes.16-19 It is obtained by first-order linearization of nonlinear models so that the traditional linear Kalman filter can be applied. Although successful applications of this tool have been reported in the literature for fault detection and diagnosis in chemical processes, the EKF contains several drawbacks that may seriously affect its performance.20 For instance, although a priori state mean may be estimated by propagating the nonlinear system directly, the corresponding covariance estimate calls for the calculation of the state transition matrix, which, in turn, requires linearization of the system model.21 This first-order linearization may introduce significant errors if the error in the state estimate is not sufficiently small.22 This leads to bias estimates of the EKF. The possibility of unpredictable bias in estimations is regarded as the main reason which has prevented the use of EKF in many realistic fault monitoring systems.23 Therefore, practical applications of EKF are still very limited. Moreover, the derivation of Jacobian matrices is nontrivial in many applications. Therefore, researchers have been looking for alternative methods to improve the EKF. An enhanced filtering technique that effectively addresses the aforementioned issues was proposed in the literature. This technique is called the unscented Kalman filter (UKF) method.24 Theoretical treatment of the UKF formalism may be found elsewhere.22,25,26 Many applications utilizing the UKF have been reported since that time. Note that most of them are in the area of aerospace navigation and tracking, where one frequently encounters severe nonlinearity and fast dynamics and where the UKF was developed.27 Some of EKF’s inconvenience can be overwhelmed by UKF.28 However, model-based FDI is built upon a number of idealized assumptions, one of which is that the mathematical model used is a faithful replica of the plant dynamics.12 There are, of course, unavoidable disturbances and model uncertainties for any practical system, which lead to bias in the residual. This bias can be misinterpreted as a response to a fault in the sensor.

10.1021/ie100477m  2010 American Chemical Society Published on Web 10/05/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

Therefore, it is essential in the design of any fault diagnosis system to take these effects into consideration, so that fault diagnosis can be done reliably and robustly. Undoubtedly, the most common approach that can be used to tackle such a problem is to employ the unknown input observers (UIOs),12,14,29-31 in which the residual is designed in such a way to be insensitive to unknown disturbances, while sensitive to faults. In spite of the importance of robust FDI, the issue of robustness has not been sufficiently addressed for chemical processes fault detection and isolation in the literature.1,11,32-35 In ref 1 a robust observer is used for sensor fault detection and isolation in chemical batch reactors, while in ref 32 the robust approach is compared with an adaptive observer for actuator fault diagnosis. However, these approaches require too much computational effort. In refs 33-35 a methodology for systematically designing a fault detection, isolation, and identification system for nonlinear systems with known model structure but uncertainty in parameters is presented. Besides the computational effort, the algorithm is also based on linearization. In ref 11 a robust FDI scheme was designed on the basis of UIO for two chemical processes. However, robustness evaluation of the designed detection filters is not investigated clearly. Besides, the UIOs were designed for a linearized system around an operating point. Linearization-based strategy only works well when the linearization does not cause a large mismatch between the linear model and nonlinear behavior, and the system closely operates around the operating point specified. However, for a system with high nonlinearity and a wide dynamic operating range, the linearized approach fails to give satisfactory results. The problem of designing nonlinear UIOs can be summarized in three distinct categories:36 (i) nonlinear state-transformationbased techniques; (ii) linearization-based techniques; (iii) observers for particular classes of nonlinear systems. Unfortunately, the existing nonlinear extensions of the UIO12,37-39 require a relatively complex design procedure, even for simple laboratory systems.40 Witczak et al. proposed an alternative UIO for nonlinear systems, whose design procedure is used the EKF.29 Convergence of the proposed method is then improved in ref 36. However, it is well-known that such a solution works well only when there is no large mismatch between the model linearized around the current state estimate and the nonlinear behavior of the system. This can be a severe limitation for highly nonlinear systems such as many chemical plants. Some important recent works related to robust fault detection in nonlinear process systems include refs 3, 4, 7, 41, and 42, where the emphasis has been on fault-tolerant control (FTC) of plants described by distributed parameter systems. In ref 3 an integrated FD and FTC architecture is designed for a spatially distributed process described by quasi-linear parabolic partial differential equations (PDEs). In ref 42 an adaptive detection observer with a time-varying threshold is proposed for particular classes of nonlinear systems in which the nonlinear dynamics is locally Lipschitz. This method minimizes the fault detection time, regardless of increasing the computational burden. These recent studies emphasize the importance of robust fault detection in nonlinear systems for control purposes. Motivated by these considerations, in this work we propose a modified version of UIO that uses unscented transformation (UT) in the design procedure. As discussed earlier, UT has apparent advantages over the linearization algorithms and performs better than EKF since the nonlinear behavior becomes more profound in this case. The proposed extended UIO (EUIO) consists of first extending linear UIO to a nonlinear framework, and then employing UT to obtain the observer gain. This method

Figure 1. Unknown input filter and residual.

is applicable to a wide class of nonlinear systems. To evaluate the ability of the proposed method, a nonlinear unknown input observer is designed for model of a highly nonlinear chemical process. The ability and performance of the designed nonlinear unknown input observer (NUIO) is investigated for abrupt fault detection in a jacketed continuous stirred tank reactor (CSTR). It is shown that the modified version of UIO can discriminate disturbance and hence a certain degree of model uncertainty from faulty sensors. 2. Preliminaries The main objective of this section is to show how to derive a modified version of the well-known unknown input filter (UIF), which can be applied to linear stochastic systems, to construct a nonlinear deterministic observer.29 2.1. Extension of UIO. Consider a system with additive disturbances described by the following equations: xk+1 ) Akxk + Bkuk + Ekdk + L1k fk + wk

(1)

yk ) Ckxk + L2k fk + vk

(2)

where xk ∈ Rn is the state vector, yk ∈ Rm is the output vector, uk ∈ Rr is the known input vector, and dk ∈ Rq is an unknown input vector representing the disturbance. wk and vk are independent zero mean white noise sequences. The terms L1k f and L2kf represent actuators and sensors fault, respectively. Note that Ak, Bk, Ck, and Ek are known matrices with appropriate dimensions corresponding to the state-space description of the linear, time-invariant system. Consider an observer in the form zk+1 ) Fk+1zk + Tk+1Bkuk + Kk+1yk

(3)

xˆk+1 ) zk+1 + Hk+1yk+1

(4)

Here xˆk+1 ∈ Rn is the estimated state vector and zk+1 ∈ Rn is the state of the observer. The observer state-space matrices Fk+1, Tk+1, Kk+1, and Hk+1 will be designed to decouple the disturbance from the state estimation error, ek+1 ) xk+1 - xˆk+1. The block diagram to illustrate this filter is shown in Figure 1. The state estimation error of the fault-free system is governed by the following equation:


ek+1 ) xk+1 - xˆk+1 ) xk+1 - (zk+1 + Hk+1yk+1) ) (I - Hk+1Ck+1)xk+1 - zk+1 - Hk+1vk+1 ) (I - Hk+1Ck+1)xk+1 - Hk+1vk+1 -

Loosely speaking, if the matrix Fk+1 is stable, ek+1 will vanish asymptotically. Therefore, according to the definition, an unknown input observer is designed by first selecting a stable F and then solving eqs 6-9. The gain matrix K1k+1 is obtained such that the state estimation error, ek+1, has the minimum variance. It can be shown that the gain matrix should be determined by43

1 2 [Fk+1zk + Tk+1Bkuk + (Kk+1 + Kk+1 )yk]

) (I - Hk+1Ck+1)xk+1 - Hk+1vk+1 Tk+1Bkuk - Fk+1(xk - ek - Hkyk) -

1 Kk+1 ) A1,k+1PkCTk [CkPkCTk + Rk]-1

1 2 Kk+1 (Ckxk + vk) - Kk+1 yk

+ v k) -

(13)

The corresponding covariance matrix is given by T T T Pk+1 ) A1,k+1Pk+1|kA1,k+1 + Tk+1QkTk+1 + Hk+1Rk+1Hk+1 (14)

) (I - Hk+1Ck+1)(Akxk + Bkuk + Ekdk + wk) Tk+1Bkuk - Fk+1(xk - ek - Hkyk) 1 (Ckxk Kk+1

11445

1 T Pk+1|k ) Pk - Kk+1 CkPkA1,k+1

2 Kk+1 yk

(15)

Indeed, the UIF can be transformed to the Kalman filter like form as 1 ) -[Fk+1 - (I - Hk+1Ck+1)Ak + Kk+1Ck+1]xk +

xˆk+1 ) Akxˆk + Bkuk - Hk+1Ck+1[Akxˆk + Bkuk] 1 1 Kk+1 + Fk+1Hk+1]yk + Hk+1yk+1 Ckxˆk - Fk+1Hk+1yk + [Kk+1 (16)

2 - Fk+1Hk+1]yk Fk+1ek - [Kk+1

[Tk+1 - (I - Hk+1Ck+1)]Bkuk + (I - Hk+1Ck+1)Ekdk - Hk+1vk+1 +

Substituting the solution of (10) in (16), factorization, and omitting the identical terms yield

1 (I - Hk+1Ck+1)wk - Kk+1 vk

(5) 1 2 where Kk+1 ) Kk+1 + Kk+1 . By definition, an observer is defined as UIO for the system defined by (1) and (2) if its state estimation error vector ek+1 vanishes asymptotically, regardless of the presence of the unknown inputs (disturbances) in the system.43 To synthesize UIF, the following relationships must hold for the observer matrices:

Ek ) Hk+1Ck+1Ek

(6)

Tk+1 ) I - Hk+1Ck+1

(7)

Fk+1 ) Tk+1Ak 2 Kk+1

1 Kk+1 Ck+1

) Fk+1Hk+1

(8) (9)

Theorem 1: Necessary and sufficient conditions for the obserVer (3) and (4) to be a UIO for the defined system in (1) and (2) are12 condition i:

1 xˆk+1 ) [I - EkHk+1 Ck+1][Axˆk + Buk] 1 1 Kk+1 Ckxˆk + Kk+1 yk + Hk+1yk+1 1 ) [I - EkHk+1Ck+1][Axˆk + Buk] + 1 1 EkHk+1 yk+1 + Kk+1 (yk - Ckxˆk)

or equivalently 1 xˆk+1 ) xˆk+1|k + Kk+1 (yk - Ckxˆk)

(18)

where 1 1 xˆk+1|k ) [I - EkHk+1 Ck+1][Axˆk + Buk] + EkHk+1 yk+1 (19)

In another way (the second approach), the system is transformed into a form without disturbance.44 For a deterministic fault free system, i.e., when f ) 0, by 1 and then inserting in (1), it is multiplying (2) by Hk+1 straightforward to show that

rank(Ck+1Ek) ) rank(Ek)

1 dk ) Hk+1 [yk+1 - Ck+1(Akxk + Bkuk)]

condition ii: (Ck+1, Tk+1Ak) is a detectable pair

(17)

(20)

Substituting (20) into (1) gives

Proof: See page 74 in ref 12. A special solution of (6) is12

¯ xk + B¯uk + E¯kyk+1 xk+1 ) A

(21)

1 Hk+1 ) (Ck+1Ek+1)+ ) [(Ck+1Ek+1)TCk+1Ek+1]-1(Ck+1Ek+1)T (11)

¯ kB ¯k ) G ¯ k Ak , B¯k ) G A 1 1 ¯ ¯ Gk ) I - EkHk+1Ck+1, Ek ) EkHk+1

(22)

Given these relationships, the state estimation error reduces to

Thus, the unknown input observer for (1) and (2) can be given as follows:

1 ek+1 ) Fk+1ek - Kk+1 vk - Hk+1vk+1 + Tk+1wk

1 xˆk+1 ) xˆk+1|k + Kk+1 (yk - Ckxˆk)

1 Hk+1 ) Ek+1Hk+1

(10)

(12)

where

(23)

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¯ xˆk + B¯uk + E¯kyk+1 xˆk+1|k ) A

(24)

Comparing these with (18), it can be seen that the observer structures being considered are identical. 2.2. Unscented Transformation. The unscented transformation is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The motivation for the UT is that it is easier to approximate a Gaussian distribution than a nonlinear function.22 Consider propagating a random variable x (dimension L) through a nonlinear function, j and covariance Px. To calculate y ) g(x). Assume x has mean X the statistics of y, we form a matrix χ of 2L + 1 sigma vectors χi (with corresponding weights Wi), according to the following:26 χi,k ) xˆk i)0 χi,k ) xˆk + (√(L + λ)Px)i i ) 1, ..., L χi,k ) xˆk - (√(L + λ)Px)i-L i ) L + 1, ..., 2L

(25)

where λ ) R2(L + κ) - L is a scaling parameter, R determines j and is usually set to a the spread of the sigma points around X small positive value (e.g., 1e-3), κ is a secondary scaling parameter which is usually set to 0, and β is used to incorporate prior knowledge of the distribution of x (for Gaussian distributions, β ) 2 is optimal). ([(L + λ)Px]1/2)i is the ith row of the matrix square root. These sigma vectors are propagated through the nonlinear function: yi ) g(χi) i ) 0, ..., 2L

(26)

Figure 2. Example of the UT for mean and covariance propagation: (a) actual, (b) first-order linearization (EKF), and (c) UT.

the state random variable is redefined as the concatenation of the original state and noise variables: xak ) [xT wT vT ]T 3. Extended Unknown Input Observer Based on UT Consider the following nonlinear discrete-time system with unknown input:

and the mean and covariance are approximated using a weighted sample mean and covariance of the posterior sigma points by

xk+1 ) g(xk, uk) + Ekdk + L1k fk

(30)

yk+1 ) h(xk+1) + L2k fk

(31)

2L

jy ≈

∑W

(m) i yi

(27)

i)0

The output equation is often linear for many industrial systems. In a more general case, however, it can be linearized around a point of operation as

2L

Py ≈

∑W

(c) i {yi

- jy}{yi - jy}T

yk+1 = Ck+1xk+1 + L2k fk

(28)

(32)

i)0

where with weights Wi given by W(m) 0 W(c) 0 W(m) i

Ck+1 )

) λ/(L + λ) ) λ/(L + λ) + (1 - R2 + β) ) W(c) i ) 1, ..., 2L i ) 1/{2(L + λ)}

∂h(xk+1) ∂xk+1

|

(33) xk+1)xˆk+1|k

Consider a nonlinear filter in the form

(29) The deceptively simple approach taken with the UT results in approximations that are accurate to the third-order for Gaussian inputs for all nonlinearities. For non-Gaussian inputs, approximations are accurate to at least the second-order, with the accuracy of third- and higher-order moments determined by the choice of R and β (see refs 25 and 26 for a detailed discussion of the UT). A simple example is shown in Figure 2 for a twodimensional system: the left plot shows the true mean and covariance propagation using Monte Carlo sampling; the center plots show the results using a linearization approach as would be done in the EKF; the right plots show the performance of the UT (note only five sigma points are required). The superior performance of the UT is clear.22 The unscented Kalman filter (UKF) is a straightforward extension of the UT to the recursive estimation in eq 2, where

zk+1 ) (I - Hk+1Ck+1)g(zk + Hkyk, uk) + Kk+1(yk - yˆk) (34) xˆk+1 ) zk+1 + Hk+1yk+1

(35)

The state estimation error of the fault-free system is governed by the following equations: ek+1 ) xk+1 - xˆk+1 ) xk+1 - (zk+1 + Hk+1yk+1) ) (I - Hk+1Ck+1)xk+1 - g¯(xˆk, uk) - Kk+1(yk - yˆk) ) (I - Hk+1Ck+1)(g(xk, uk) + Ekdk) g¯(xˆk, uk) - Kk+1(yk - yˆk) ) (I - Hk+1Ck+1)(g(xk, uk) - g(xˆk, uk)) + (I - Hk+1Ck+1)Ekdk - Kk+1(yk - yˆk)

(36)


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In order for the observer defined by (34) and (35) to be as UIO for the system defined by (30) and (31), the state estimation error vector ek+1 must vanish asymptotically, regardless of the presence of the unknown inputs (disturbances) in the system. To synthesize UIF, the following relationship must hold for the observer matrices:

defined in a way that it is almost zero in a fault-free steady state and it is deviated from zero when a fault occurs. If the observer is stable, the state error, x˜k ) [xk - xˆk], vanishes asymptotically. Hence,

Ek ) Hk+1Ck+1Ek

Therefore, the output estimation error (or innovation in the stochastic case) can be defined as a fault signature. 3.2. Residual Evaluation. The residual is examined in terms of the likelihood of a fault, and a logical decision-making process is then applied, with the aim of deciding if the fault has occurred and avoiding wrong decisions, such as a false alarm and a fault ignored.45 There are several decision-making methods, but they all end in a simple binary decision variable Sr, where a fixed or adaptive threshold Tk is used on a residual evaluation function J(rk):

(37)

and Kk+1 should be designed in a way to stabilize the UIF. To demonstrate that the EUIO structure is selected correctly, following a similar procedure as in the previous section, by multiplying 1 and then inserting into (30), it can be shown that (32) by Hk+1 1 dk ) Hk+1 [yk+1 - Ck+1g(xk, uk)]

(38)

xk+1 ) g¯(xk, uk) + E¯kyk+1

(39)

¯ kg(...), gj(...) ) G

1 ¯ k ) I - EkHk+1 G Ck+1,

1 ¯ k ) EkHk+1 E

(40) ¯ kyk+1 + Kk+1(yk - yˆk) xˆk+1 ) g¯(xˆk, uk) + E

(41)

¯ kyk+1 xˆk+1|k ) g¯(xˆk, uk) + E

(42)

Comparing these with (34) and (35), it can be seen that the observer structures being considered are identical. Therefore, the defined structure is selected correctly. Theorem 2: Necessary and sufficient conditions for the obserVer (34) and (35) to be a UIO for the defined system in (30) and (32) are condition i:

rank(Ck+1Ek) ) rank(Ek)

lim rk ) lim (yk - yˆk) ) 0

kf∞

(45)

kf∞

J(rk) e Tk for f ) 0 then Sr ) 0 (fault-free case) J(rk) > Tk for f * 0 then Sr ) 1 (faulty case) 3.3. Design Procedure. From the above derivation and theorem, the computational procedure for the filtering algorithm using unscented transform can be listed as the following. (1) initialize with xˆ0 ) E[x0] P0 ) E[(x0 - xˆ0)(x0 - xˆ0)T]

(46)

xâ0 ) E[xa] ) [xˆT0 0 0 ] H0 ) 0 Pa0 ) E[(xa0 - xâ0)(xa0 - xâ0)T] )

condition ii: existence of Kk+1 in such a way thatstabilizes the UIF

For k ∈ {1, ..., ∞},

Proof. Sufficiency: Equation 42 is solvable when condition i holds true.12 A special solution for H is obtained from eqs 10 and 11, which leads to decoupling of the effect of disturbance. Since Kk+1 exists in a way that stabilizes the UIF, the estimation error vanishes asymptotically, regardless of the presence of the unknown inputs in the system. Thus, the observer (39) and (40) is a UIO for the system (30) and (31). Necessity: Since (39) and (40) are a UIF for (30) and (31), eq 37 is solvable. This leads to the fact that condition i holds true (according to Lemma 3.1 in ref 12). The general solution of the matrix H for eq 42 can be calculated as

(2) calculate Ck+1 Ck+1 )

∂h(xk+1) ∂xk+1

[

P0 0 0 0 Pw 0 0 0 PV

]

|

(47)

(48) xk+1)xˆk+1|k

(3) check the rank condition i in theorem 2; if it holds true, 1 compute Hk+1 1 Hk+1 ) (Ck+1Ek)+ (49) otherwise, the UIF cannot be designed

Hk+1 ) Ek+1(Ck+1Ek+1)+ + H0[Ck+1Ek+1(Ck+1Ek+1)+]

(43)

(4) calculate sigma points χak ) [xâk

+

where H0 ∈ R is an arbitrary matrix and (Ck+1Ek+1) is the left inverse of (Ck+1Ek+1), which is n×m

(Ck+1Ek+1)+ ) [(Ck+1Ek+1)TCk+1Ek+1]-1(Ck+1Ek+1)T

xâk ( √(L + λ)Pak ]

(50)

(44)

Since the defined observer is a UIO, the estimation error vanishes asymptotically, regardless of the presence of the unknown inputs in the system. Therefore, Kk+1 exists in such a way that stabilizes the UIF. It is worth mentioning that convergence analysis of this filter while condition i is satisfied is similar to the standard UKF. Convergence analysis of the standard UKF is also evaluated in ref 20. 3.1. Residual Generation. In general, faults can occur either in actuators or in components inside the system, or in sensors. To provide useful information for fault diagnosis, the residual should be

(5) compute the periodic mean x χk+1|k ) g¯(χxk, uk) + E¯kyk+1

(51)

2L

xˆk+1|k )

∑W

(m) x i χi,k|k-1

(52)

i)0

and the predicted covariance 2L

Pk+1|k )

∑W

(c) x x ˆ k+1|k][χi,k|k-1 i [χi,k|k-1-x

- xˆk+1|k]T

i)0

(53)

11448


(6) compute the predicted observation x γk+1|k ) h(χk+1 )

FR dCA ) (C - CA) - k0e-Ea/RTRCA dt VR A0 FR (hAT)i dTR ) (T0 - TR) + (T - TW) + dt VR VR(FCp)f R

(54)

2L

yˆk+1 )

∑W

(m) i γi,k+1|k

(55)

-(∆HR) -Ea/RTR ke CA (FCp)f 0 (hAT)i (hAT)e dTW ) (TR - TW) (T - TJ) dt VW(FCp)W VW(FCp)W W FJ (hAT)e dTJ ) (TJ0 - TJ) + (T - TJ) dt VJ VJ(FCp)J W

i)0

2L

Py˜ky˜k )

∑W

(c) i [γi,k+1|k

- yˆk+1][γi,k+1|k - yˆk+1]T

(56)

- xˆk+1|k][γi,k|+1k - yˆk+1]T

(57)

i)0

2L

Pxkyk )

∑W

(c) i [χi,k+1|k

i)0

(62)

Kk+1 ) PxkykPy-1 ˜ ky˜k

(58)

xˆk+1 ) xˆk+1|k + Kk+1(yk+1 - yˆk+1)

(59)

T T Pk+1 ) Pk+1|k - Kk+1Py˜ky˜kKk+1 ) Pk+1|k - PxkykPy-1 ˜ ky˜kPxkyk

The schematic of this CSTR is illustrated in Figure 3. These equations can be rewritten in normalized dimensionless form as follows:

{

(60) (7) compute the residuals rk+1 ) yk+1 - yˆk+1

(61)

where xa ) [xT wT vT ]T

χa ) [(χx)T (χw)T (χV)T ]T

dx1 ) p1u1 + p1u1u2 - p1u2x1 dt p2e-p3/(1+x2)(1 + x1) - p1x1 dx2 ) p1u3 + p1u2u3 - p1x2 - p1u2x2 - p4x2 + dt p4x3 + p5p2e-p3/(1+x2)(1 + x1) dx3 ) p6x2 - p6x3 - p7x3 + p7x4 dt dx4 ) p8u4 + p8u4u5 - p8x4 dt p8u5x4 + p9x3 - p9x4

with the system state vector defined as xT )

λ ) composite scaling parameter, L ) dimension of augmented state, Pw ) process noise covariance, Pn ) measurement noise covariance, and Wi ) weights calculated in eq 26. Note that no explicit calculations of Jacobians or Hessians are necessary to implement this algorithm. Furthermore, the overall number of computations is the same order as for the EKF. It is important to note that the filtering algorithm proposed in this section is equivalent to a standard unscented Kalman filter for systems without unknown disturbances, by setting the matrices Hk+1 ) 0 when there is no disturbance, i.e., when E ) 0. 4. Simulation Results In this section, the simulation results of the proposed nonlinear unknown input observer (NUIO) will be verified. To evaluate the performance of the proposed method in comparison to the observers without unknown input, it is applied to a highly nonlinear CSTR model explained in ref 27. 4.1. Process Description. This is a highly nonlinear dynamic system describing the behavior of a nonadiabatic CSTR in which an irreversible highly exothermic chemical reaction (AfB) takes place. The reactor wall significantly affects the system dynamics and then has also been taken into account. 4.2. Dynamic Process Model. The corresponding model leads to the following set of ODEs:27

(63)

[

]

CA - CA,ref TR - TR,ref TW - TW,ref TJ - TJ,ref CA,ref TR,ref TW,ref TJ,ref (64)

where CA is the concentration of reactant A, TR the reactor temperature, TW the wall temperature, and TJ the jacket temperature. The unit of CA is mol/m3, and the unit of TR, TW, and TJ is K. The input vector is uT ) CA0 - CA,ref FR - FR,ref T0 - T0,ref TJ0 - TJ0,ref FJ - FJ,ref CA,ref FR,ref T0,ref TJ0,ref FJ,ref

[

]

(65)

where CA0 is the feed concentration of reactant A, FR the flow rate, T0 the feed temperature, TJ0 the inlet jacket temperature, and FJ the coolant flow rate. The unit of CA0 is mol/m3, the unit of T0 and TJ0 is K, and the unit of FR and FJ is m3/s. The corresponding reference values are CA0,ref ) CA,ref ) 3 mol/m3,

FR,ref ) 60 × 10-5 m3/s,

FJ,ref ) 15 × 10-4 m3/s, FR,ref ) 60 × 10-5 m3/s, TR,ref ) Tw,ref ) TJ,ref ) T0,ref ) TJ0,ref ) 298 K

The measurement model is assumed to be

[

1 0 0 0 y ) Cx with C ) 0 1 0 0 0 0 0 1

]

(66)

that is, measurements of the wall temperature x3 are not available.


11449

Figure 3. Schematic of a nonadiabatic CSTR. Table 1. Model Parameters parameter

expression

value

p1 p2 p3 p4 p5 p6 p7 p8 p9

(FR,ref)/(VR) k0 (Ea)/(RTR,ref) ((hAT)i)/(VR(FCp)f) (-(∆HR)CA,ref)/((FCp)fTR,ref) ((hAT)i)/(VW(FCp)W) ((hAT)e)/(VW(FCp)W) (FJ,ref)/(VJ) ((hAT)e)/(VJ(FCp)J)

3.333 × 10-2 s-1 4.08 × 107 s-1 25.347 6.63 × 10-1 s-1 1.45 5.97 s-1 5.97 s-1 1.67 × 10-1 s-1 1.33 s-1

Table 2. Steady States

x1 x2 x3 x4

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

Table 1 summarizes the model parameters used in the present work. The model has three steady states, which are presented in Table 2. 4.3. Unknown Input Observer Design. In this section, a EUIO is designed for the described CSTR using the proposed algorithm. The robustness of the EUIO in disturbance decoupling is also evaluated. The primary requirement for disturbance decoupling-based approaches is that the disturbance distribution matrix E must be known a priori, although the actual unknown input itself does not need to be known.12 However, this is not the case for many applications.37 One of the ways to tackle this problem is to represent modeling errors as a disturbance term with an estimated distribution matrix. This estimated distribution matrix is then used to design disturbance decoupled residuals; in this way, optimal robust FDI solutions are achievable. In this work, the initial condition is adjusted so that the reactor works in high-temperature steady states. It is also assumed that all states are affected by disturbance by the same factor. Therefore, the disturbance distribution matrix will be Ek ) [1 1 1 1 ]

T

(67)

It should be noted that although in the UIO procedure design the matrix Ek is assumed to be constant and independent of the state, the term Ekdk, which is generally a nonlinear function, constructs the overall uncertainty. However, in this algorithm as well as the usual robust observer design problem, all uncertainties, whether disturbances or parametric uncertainties, are usually considered as exogenous inputs. It has been shown in ref 46 a certain degree of robustness can be achieved in the case of parametric uncertainties.

Figure 4. Residuals for an observer without unknown input decoupling.

At first, a discrete model is obtained from eq 10 using Runge-Kutta method with ∆t ) 0.1 s. In all simulations, the parameters of the UKF are selected as follows: PV ) diag{10-6, 10-6, 10-6, 10-6}

(68)

Pn ) diag{10-8, 10-8, 10-8}

(69)

xˆ0 ) x0 + [10-3, -10-3, 10-3, 10-3]T ) [-0.97540, 0.47245, 0.42690, 0.37934]T Pˆ0 ) diag{10-6, 10-6, 10-6, 10-6}

(70) (71)

Additionally, parameter κ of the UF is set to -5 on the basis of the dimension of the state. To start the design procedure, condition i of theorem 2 is first verified by observing that rank(Ck+1Ek) ) rank(Ek) in each step, and then the full-order EUIO is designed by computing the appropriate Kk+1 that satisfies condition ii. To evaluate the disturbance decoupling ability of the designed EUIO, it is assumed that the unknown input dk acts on the system according to dk ) 0.005 sin(0.5πtk) cos(0.5πtk)

(72)

Considering the disturbance distribution matrix as defined in (67), this disturbance can be interpreted as deviations in CA, TR, Tw, and TJ. The mainspring of selecting such a signal is that the sinusoid is an orthogonal basis; thus, on the basis of Fourier theorem all signals can be rewritten in the form of the summation of the sinusoids functions. The defined residuals as output errors are obtained from eq 61. Figure 4 shows the residual signals for an observer without unknown input decoupling. It can be seen that the unknown input influences the residual signals, and hence it may be misinterpreted as a

11450


{

0, t < 20 s f3,k ) 0.1y , t g 20 s 3,k fj,k ) 0,

j ) 1, 2

(77) (78)

Finally (case D), to study the effect of multiple faults, it is assumed that two faults occur simultaneously. In this scenario, a sudden fault occurs in the CA sensor between t ) 20 s and t ) 60 s, while a sudden fault occurs in the TR sensor at t ) 40 s, and then another occurs in the TJ sensor at t ) 70 s.

Figure 5. Residuals for an observer with unknown input decoupling.

response to a fault. In contrast, Figure 5 shows the residual signals for the designed extended unknown input observer, based on the proposed algorithm. In this case, the residuals converge to zero regardless of the presence of the disturbance. This confirms the disturbance decoupling ability of the designed EUIO. 4.4. Simulation Results for Sensor Fault Detection. The objective of this section is to show the performance of the proposed EUIO as a residual generator in the presence of an unknown input. For this purpose, four different faulty scenarios will be considered. In the first case (case A), it is assumed that the sensor measuring the concentration of reactant A (CA) is damaged. To simulate this fault, the measured value is suddenly deviated +10% from the normal measurement after 20 s elapsed from running of the process:

{

0, t < 20 s f1,k ) 0.1y , t g 20 s 1,k fj,k ) 0,

j ) 2, 3

Figure 6. Successful fault detection in sensor measuring CA in the presence of disturbance and comparison between EUIO using UT and EUIO using linearized algorithm.

(73) (74)

In the second case (case B), a similar fault occurs in the second output which is measuring the reactor temperature (TR):

{

0, t < 20 s f2,k ) 0.1y , t g 20 s 2,k fj,k ) 0,

j ) 1, 3

(75) (76)

In third case (case C), similar fault occurs in the sensor measuring jacket temperature (TJ):

Figure 7. Successful fault detection in sensor measuring TR in the presence of disturbance.


f2,k f3,k

{ { {

0.1y1,k, 0, 0, ) 0.1y , 2,k 0, ) 0.1y , 3,k

f1,k )

20 s e t e 60 s otherwise t < 40 s t g 40 s t < 70 s t g 70 s

11451

(79)

Thus, fk ) [f1,k, f1,k, f3,k]T

(80)

L2k ) [1 1 1 ]T

(81)

It should be noted that in all cases faults are introduced after the reactor reaches the steady state. All cases have been simulated and residuals have been achieved. To draw a comparison, an alternative method proposed in refs 29 and 36, which uses a linearized algorithm (specially EKF) to calculate the observer gain, is employed. The simulation result for case A is shown in Figure 6. From this figure, it is clear that better residual is generated using the proposed method in this paper. The simulation results for cases B-D are illustrated in Figure 7, Figure 8, and Figure 9, respectively. It is our desirable aim that each fault affects only one of the residuals. In other words, if each residual was affected by the corresponding sensor fault, i.e., r1 from CA fault, r2 from TR fault, and r3 from TJ fault, the designed detection filter could isolate the faults successfully. However, from simulation results it is clear when a fault occurs all residuals change simultaneously, and after a transient state one of the residuals keeps a bigger constant bias due to the fault, and the others fall back almost to zero. It should be noted that changes in the residuals amplitude is related to the amplitude of the faults. From these figures, it is clear that this scheme is successful in fault detection. One may see that the fault detection scheme is robust to nonlinearity in dk.

Figure 9. Residuals for multiple faults: a sudden fault in the CA sensor between t ) 20 s and t ) 60 s, a sudden fault in the TR sensor at t ) 40 s, and a sudden fault in the TJ sensor at t ) 70 s.

5. Conclusions The majority of robust model-based fault detection methods are based on the linear system models. In this paper, a new algorithm for nonlinear robust fault detection is presented. The goal of a robust FDI is to discriminate between the fault effects and the effects of uncertain signals and perturbations. Robustness of the observers with respect to external disturbances is ensured using unknown input observers. Nonlinear calculations were performed using unscented Kalman filter. This method can be applied to a large class of nonlinear systems. To evaluate the performance of the proposed method, it is applied to a continuous stirred tank reactor (CSTR). Several simulations have been performed in the presence of external disturbances and different classes of faults. The simulation results confirm the robustness and effectiveness of the proposed scheme for fault detection in the presence of external disturbances. It is shown that external disturbances can be distinguished from a response to a fault in the sensor. Acknowledgment We thank the National Elite Foundation of Iran for their support. Literature Cited

Figure 8. Successful fault detection in sensor measuring Tj in the presence of disturbance.

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ReceiVed for reView March 3, 2010 ReVised manuscript receiVed July 20, 2010 Accepted September 8, 2010 IE100477M

Design of Nonlinear Unknown Input Observer for Process Fault

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