(285b) Fault Detection and Identification in Nonlinear Systems Using Generalized Likelihood Ratio Approach

FAULT DETECTION AND IDENTIFICATION IN NONLINEAR SYSTEMS USING GENERALIZED LIKELIHOOD RATIO APPROACH

J. Prakash^(*), Sirish L. Shah ^{(**) ,}Shankar Narasimhan(***)

^(*)Department of Instrumentation Engineering,

Madras
Institute of
Technology,
Anna University, Chennai, 600044,
India; prakaiit@rediffmail.com

^(**)Department of Chemical and Materials Engineering,

University of
Alberta,
Edmonton,
T6G 2G6Canada; slshah@ualberta.ca

(***) Department of Chemical Engineering,

Indian Institute of Technology, Madras, Chennai, 600036, India; naras@che.iitm.ac.in

ABSTRACT

Significant research has been carried out over the past four decades in the area of fault detection and identification. Most methods available in the chemical engineering literature are capable of detecting, identifying and estimating faults for linear processes. This work is aimed at developing a method for diagnosing faults in a nonlinear dynamic system, by representing it as a weighted sum of locally linear state space models. The weights for the local linear models are determined using a fuzzy membership function which depends on a chosen scheduling variable. This representation can also be viewed as a linear time varying state space model, which allows the standard Kalman filter to be used for state estimation, and the generalized likelihood ratio (GLR) method for fault diagnosis. However, biases in the scheduling variables can lead to biased estimates of states due to incorrect choice of the weights. In this work, we propose a simple extension of the GLR method to diagnose and compensate for biases in scheduled variables, which gives unbiased state estimates. The online fault diagnosis and accommodation scheme described by Prakash et al. [2] for linear systems is thus naturally extended to nonlinear systems using the proposed method. The efficacy of the proposed fault detection and identification scheme is demonstrated by conducting simulation studies on a Continuous Stirred tank reactor. Analysis of the simulation results reveals that the proposed method is able to generate unbiased state estimates and is capable of identifying multiple faults that occur sequentially in time.

FAULT DIAGNOSIS AND ACCOMMODATION SCHEME

The T-S fuzzy model is nothing but a piecewise interpolation of local linear models through user-chosen membership functions [1]. The rule to describe the nonlinear system around an operating point is as follows:

Rule i (i =1: N)

(1)

(2)

Where
and
are the scheduling variables (or premise variables) and the fuzzy sets respectively.
,
,
and
are known time invariant matrices of appropriate dimensions. We have assumed that
and
are mutually independent white, Gaussian noise sequences. In this work it is assumed that the process model can be developed from first principles by linearizing them around different operating steady state values. For a given input vector
the global state and output of fuzzy model are inferred as follows:

(4)

In equation (3),
represents the grade of membership. It can be observed that Equation (3) is a time varying linear model of the nonlinear process. Thus, the Kalman Filter (KF) can be applied to estimate the system state vector [2]. Fault detection can be carried out by testing whether the mean of the innovations deviate significantly from zero. An online procedure was proposed by Prakash et al. [2] for detecting and identifying faults in linear dynamical systems using the GLR method. The steps in the on-line fault detection and identification method are as follows. At each instant, a fault detection test (FDT) is applied on the innovations to detect whether a fault has occurred at that instant. If FDT rejects the null hypothesis, at say time t, then we wait for N sampling instants and apply a fault confirmatory test at
, using all innovations in the time interval [t,
]. If FCT rejects null hypothesis (confirms that a fault has occurred at time t), then we apply the GLR method using the innovations in the time interval to identify the fault that has occurred and to estimate its magnitude. The information provided by the FDI scheme is used to obtain compensated or corrected estimates of the state variables, measured values, and manipulated inputs to enable subsequent faults that may occur to be detected and diagnosed.

The above procedure can be used for online FDI for a nonlinear process described using Equation (3) as long as the scheduling variable is directly measured and is not computed based on the state estimates or other measurements. Even in this case, biases in the measurement of the scheduling variable can cause the mean of the innovations to deviate from zero and cause misdiagnosis unless included as a possible fault in the hypotheses set. In the section below, we extend the GLR method for detecting and identifying biases in the measurements of scheduling variables, and a suitable compensation scheme to allow for multiple fault diagnosis.

Compensation of biases in the scheduling variables

If a bias is present in the measurement of the scheduling variable, it can lead to improper computation of the weights which in turn leads to incorrect computation of the system matrices. It should be noted that the system matrices in Eq. (3) are functions of the scheduling variable. Strictly, this fault should be treated as a multiplicative fault. However, in the present work, we continue to treat the bias in scheduling variables as additive faults and include them in the hypotheses set of the GLR method. If a bias in the scheduling variable is identified by the GLR method, the scheduling variable is compensated as
. The compensated value of the scheduling variable is used for subsequent calculation of the weights and system matrices in Equation (3). The main concern with the above approach is that the magnitude and the position of the fault may not be accurately estimated. Thus, there is a need to introduce integral action in such a way that the errors in estimation of magnitude of the fault or position can be corrected in the course of time. That is the proposed method may subsequently identify a fault in the previously identified location or a new sensor bias. In either case, the measurement equation is modified using cumulative estimate of the corresponding biases as follows:

(5)

(6)

In equation (13),
represents the estimate of cumulative bias in the j'th sensor. Similarly, in the case of bias in the scheduling variables, we can write

(7)

(8)

SIMULATION STUDIES

The performance of the proposed scheme for nonlinear dynamic system is assessed through simulation studies. To test the performance of the proposed on-line fault diagnosis scheme for the CSTR system, hundred simulation trials were performed, with each trial lasting for 900 sampling instants. Simulation runs were carried out in the presence of multiple faults that occur sequentially in time. Two fault scenarios were considered, namely biases in two sensors occurring sequentially in time and bias in the scheduling variable.

We have simulated the case in which a 0.01 mol/l bias in concentration sensor occurs at the 10^th sampling instant, which is followed by a 0.25 deg. K bias in the temperature sensor at the 300^th sampling instant. The proposed fault identification scheme successfully isolates the multiple sequential faults in each simulation trial. The cumulative bias estimates of bias magnitudes at the end of a simulation trial averaged over all trials are found to be 0.0103 mol/l and 0.1826 deg. K respectively. Thus, the ability of the proposed approach to isolate multiple faults (occurring sequentially in time) is also clearly demonstrated in Figure 1.

We have simulated case in which a 2 LPM, bias in the coolant flow rate (bias in the scheduling variable) occurs in the first sampling instant. The innovation sequence generated by state estimator is shown in Figure 2. From Figure 2, it is evident that the innovation sequence becomes non-zero mean when the bias in the scheduling variable is introduced. Subsequent to the proposed correction, the residuals are driven to acquire the zero mean property again. The evolution of true and estimated state variables is shown in figure 3.

CONCLUSION

In this paper we have extended the GLR method for fault detection and identification in nonlinear dynamical systems, by representing the process as a weighted sum of local linear models. For the method to be successful, biases in the scheduling variables have to be identified and compensated as outlined in this paper. In this work, we have assumed that the scheduling variables are directly measured. The method has to be extended to the case when the scheduling variable is computed as a function of the state estimates or measurements.

REFERENCES

1. Takagi, T.; Sugeno, M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Manual Cybern. 1985, 15, 116-132.

2. Prakash, J., S.C. Patwardhan and S. Narasimhan (2002). A supervisory approach to fault tolerant control of linear multivariable systems, Ind. Eng. Chem. Res. 41, pp. 2270?2281.

Figure 1: Residuals response to multiple sequential faults

Figure 2: Residuals response to bias in the Scheduling variable

Figure 3: Evolution of True and
Estimated State variables in the presence of bias in the Scheduling variable