MODEL-BASED MULTIPLE FAULT DETECTION AND ISOLATION FOR NONLINEAR SYSTEMS

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1 MODEL-BASED MULIPLE FAUL DEECION AND ISOLAION FOR NONLINEAR SYSEMS Ivan Castillo, and homas F. Edgar he University of exas at Austin Austin, X David Hill Chemstations Houston, X Abstract A model-based detection and isolation (FDI) system based on nonlinear state estimation and high filtering conditions is roosed. A better understanding of the residual trends, calculated from the difference between measurements and the extended Kalman filter (EKF) estimates, can be obtained when a fault occurs by develoing a model that is able to redict the behavior of the residuals. his residuals model is utilized as the basis for detection and isolation of multile faults, having the advantage of distinguishing single and multile faults from a diverse array of ossible faults, a common occurrence in comlex rocesses. he roosed aroach is validated using a CSR, which is simulated using unit oeration software CHEMCAD. Keywords Nonlinear fault detection and isolation (FDI), Residuals modeling, FDI of Multile faults, CHEMCAD CSR simulations. Introduction Nonlinear model-based FDI systems (Venatasubramanian et al., 200 and Castillo et al., 2010) use dynamic models that are hysically-based or emirically-defined. A tyical model-based aroach utilizes nonlinear state estimators and the concet of analytical redundancy, where residuals are derived by calculating the difference between the actual oututs of the monitored system with the oututs obtained from a mathematical model and the state estimator. o detect faults, the residuals are evaluated by using either threshold values or statistical decisions. o isolate faults, a signature matrix can be defined in which residuals that fall outside of the threshold values are matched with different faults that could occur in the system. Parameter estimation is another alternative for isolation (Isermann, 2005), whereby variations of arameters of the nonlinear model, from their nominal oeration values, are associated with different faults. he latter isolation technique is nown as a arametric aroach. he disadvantage of these modelbased aroaches lies in the isolation of multile faults in which distinguishing one from another becomes a formidable tas. In this aer, multile fault cases are analyzed when different inds of faults (such as sensor, actuator or rocess faults) are included under a restrictive amount of available measurements. With this restriction, there is no guarantee of successful identification, excet for extracting major information from the residuals. herefore, the main contribution of this aer is to develo an FDI system based on develoing a high fidelity model that is able to redict and understand residual dynamics in the event fault occurs in a nonlinear system. he roosed architecture, based on residuals modeling to detect and isolate faults, is illustrated in Figure 1. he right ortion of Figure 1 illustrates how the FDI system is designed. A state estimator, which has different characteristics than the one used for control uroses (shown on the left hand side), is utilized to generate the residuals. hese residual trends R es can be

2 redicted using the residuals model bloc at each time ste. Additionally, using both the residuals R es and redicted residual trends r, the mechanism of detection is designed whereby the detection of the fault is erformed at each time ste. Once a fault is detected, the residuals model and the residual signals are utilized to isolate the faults at each time ste. At the outut of this isolation bloc, different modes are generated, such as Faulti or False Alarm. hese four comonents (the state estimator, residuals model, the detection and isolation mechanisms) are briefly resented in the next section. Figure 1: Fault Detection and Isolation (FDI) System his aer is comrised of six sections. Section 2 formulates the dynamic residuals model as well as the FDI mechanism. Next, Section resents the CSR system that is simulated by using CHEMCAD in which control loos and faults are created using this unit oeration software. he multile fault case is resented in Section 4, followed by the validation of the residuals modeling aroach in Section 5. Finally, closing remars are resented in Section 6. FDI Mechanism Based on Residuals Modeling he residuals model is derived by considering the following assumtions: (1) the rocess is formulated as a nonlinear stochastic differential equation shown in Eqs. (1) and (2); (2) the nonlinear system is observable; () the effect of the noises w and v is considered, even though these vectors are unnown; and (4) a state estimator is needed. hen, the extended Kalman filter (EKF) will be utilized for the formulation of this function. x = f x,u, w (1) y = h x, v (2) n m where: x is the vector of state variables, u is the vector of system inuts, y is the vector of system oututs, f is the nonlinear state equation function, h is the nonlinear outut function, w and v are gaussian white noises with covariance matrices Q and R resectively. hese covariances satisfy the following conditions: E w w j Q- j, v v j R - j E v w 0. j E, and he extended Kalman filter (EKF), which is an imortant comonent to determine the residuals model, is a widely used algorithm for nonlinear state estimation (Simon, 2006). he EKF linearizes the model about the current estimated state. he rediction of the state variables n xˆ is obtained using Eq. (), which is in terms of the Kalman gain K and the measurement vector z, xˆ = f xˆ -1,u -1,0 Kz - hf xˆ -1,u -1,0,0 () note that deending uon the value of the Kalman gain which is in terms of the noise covariances. Additional filtering results from giving more riority to the model redictions (first term on the right hand side of Eq. ()). herefore, small values of the Kalman gain can be obtained when the norm of the ratio R Q is more than 1. he residuals Re s, Eq. (4), are calculated from the difference between measurement vector z and the EKF oututs' estimates ŷ h( xˆ,0). he redicted residuals r refer to the values obtained from the residuals model, given by Eq. (5), Res = z - ŷ r (4) + H I - KH - HK I - K H A r - r + (5) H -1-1 where: the vectors, and result from noise effects xn xn and errors in the modeling. H and H are the Jacobian matrices given by Eqs. (6) and (7) resectively: H i, j (6) i,j H h i xj x f xˆ, u,0 1 1,0 h i (7) xj ˆ,0 x for the non-square case, which alies for the case study, is calculated using Eq. (8). 1 H H H, n (8) 1 H H H, n Note that the residuals model, given by Eq. (5), is in terms of the Kalman gain K which ermits to control less or more filtering in the state estimations. Under conditions of normal oeration and low filtering, the magnitude of the residuals is close to zero. However, a larger magnitude of the residuals can be obtained when more filtering is designed in the Kalman filter, maing it ossible to analyze its dynamic for uroses of detection and isolation of faults. he detection of a fault is evaluated in the sace of the residuals, whereby multile saces can be generated roviding redundancy and increasing sensitivity for detection. hree different inds of residual saces can be generated: (1) saces that include residuals (Eq. (4))

3 versus redicted residuals (Eq. (5)); (2) residual saces that are comrised of two different residuals obtained from Eq. (4); and () residual saces that are derived from the redictive residuals formulated in Eq. (5). hese residual trends can be enclosed by defining a trajectory that best reresents the normal oerating behavior of the system. Ellitical trajectories are defined for each residual sace. herefore, a fault is detected once any of these residual trajectories surass the ellitical trajectories in any residual sace. hree stes are considered to isolate faults. First, multile modes are defined in the isolation system: (1) a false alarm mode; (2) different single and multile fault modes f i that include sensor, actuator or rocess faults; and () unnown fault mode. Second, for each of the single and multile fault cases f i, arameters of the residuals model Eq. (5) are associated with each fault mode and defined as P f i. Note that the arameters that consider the effect of noise and model mismatch, such as, and, can be considered in addition to the arameters of the nonlinear model. hird, the objective function of Eq. (9) is utilized to calculate the arameters Pf i of each single and multile fault case at each time ste, J = minp W fi f i f i (9) lb Pf ub i x where the matrix W is a weighting constant. he lower and uer bounds of the arameters to be estimated are given by lb and ub resectively. he function f i, which is in terms of the arameters Pf i associated with each fault case f i, is given by Eq. (10). Once a fault is detected, every f i is calculated. hen, a false alarm is diagnosed when NO, which is obtained by calculating the difference between the values of the redicted residuals and residual values in normal oeration, is smaller than all the f i cases considered. Otherwise, the fault f i is identified by comaring the multile f i against each other. In the case there are inconsistent comarisons, an unnown fault is diagnosed. fi P f - Res = r (10) i Non-isothermal Chemical Reactor he hydrolysis of roylene oxide to roylene glycol (Baosova et al., 2009), where the reaction is defined by Eq. (11), is simulated using CHEMCAD through a nonisothermal CSR reactor. CH6O H 2O CH8O2 (11) Figure 2 shows the P&ID of the system. he reactor has two inuts: (1) the flow of roylene oxide m / min (2) the flow of water m / min F PO which is reresented by feed stream 1; and F W, given by feed stream 7, which is roortional to the oening of the control valve given by unit oeration number 5. he outut F m / min r of the reactor is given by roduct stream 6. he jacet of F c m / min, the reactor is fed with cooling water flow given by feed stream 5, which is roortional to the oening of the control valve denoted by unit oeration number. Figure 2: CSR P&ID Diagram Using CHEMCAD he deendence of a reaction rate constant on the temerature is described by the Arrhenius equation, given by Eq. (12) and the reaction inetics, which is in terms of the concentration of the roylene oxide C PO mol / m and water C W mol / m, and is of the second order, given by Eq. (1). E R r - = 0 e (12) PO E - R r W0e r = C C (1) Figure : Concentration Proylene Glycol Control Loo rajectory wo PI controllers maintain the temerature of the reactor r K and the concentration of the roylene glycol C PG mol / m within a desired range. he temerature in the reactor is controlled by maniulating the jacet cooling flow, which is roortional to the

4 oening of the valve with unit oeration number. On the other hand, the concentration of glycol is controlled by maniulating the water flow through the oening of the control valve with unit oeration number 5. Figures and 4 show the erformance of the controllers whereby the blac (dashed) lines corresond to the set oint trajectories and the red (thin) lines reresent the rocess variables. and actuator fault cases, maing the tas of isolation a formidable one. Fault Detection and Isolation Results he arameters of the residuals model, given by Eq. (5), are obtained and listed in able 1. he ellitical trajectories are calculated from the normal oeration data in which nine residual saces were created. Figure 7 shows an examle of the residual saces created in normal oeration. Figure 4: Reactor emerature Control Loo rajectory Fault Case Scenario he multile fault case, simulated using CHEMCAD (Massey, 2002), is the combination of both sensor and actuator faults. he faults are simulated at t= min. he sensor fault is created through inserting a constant bias in the reactor temerature sensor. his fault generates an instantaneous change in the reactor temerature and a fast resonse in the controller temerature, therefore causing changes in the temerature of the jacet. Figure 5: Reactor emerature under Sensor and Actuator Fault he actuator fault is created by modifying the coefficient of the control valve of the jacet. he flow of the cooling jacet decreases and creates an increment in the reactor and jacet temeratures. Figures 5 and 6 show the reactor and jacet temerature trajectories under these two faults. hese trends are similar to the single sensor Parameters Figure 6: Jacet emerature under Sensor and Actuator Fault able 1. Parameters of the Residuals Model CSR α η γ e Figure 8 shows the isolation results for the multile fault case considered. he results are summarized in able 2 in which good isolation results were obtained. During the first 0 seconds, some incorrect detection and isolation statements are generated. hese errors are obtained because both the CSR and the Kalman state estimator start at different initial conditions, consequently the magnitude of the residuals is big enough to generate these FDI errors, until the state estimator reaches convergence. Some false alarms are confirmed in the isolation statement given by the blue bars, the multile fault is indicated by the cyan bar. -7-7

5 Figure 7: Residual Values versus Predicted Residual Values Figure 9: Residual Values and Predicted Residual Values versus ime Figure 8: Isolation Results Sensor and Actuator (Valve) Fault able 2. Fault Detection and Isolation Results CSR Case Study Criteria Actuator-Sensor Fault otal Number of Detections 207 Pct. of Correct Isolation of the Fault [%] 94.7 Pct. of Correct Detection [%] 72.9 Pct. of Correct Isolation of False Alarms [%] 80.4 o better understand how the isolation mechanism is erformed, Fig. 9 shows the residuals trajectories for the multile fault case before the fault is isolated. he green line corresonds to the residual values and the blue line illustrates the redicted residual values. Notice that the difference between the redicted residuals and the residual values, defined as f i in Eq. (10), is catured by the detection mechanism. Figure 10 shows the results after the arameter estimation is calculated, in which the faulty residual values are matched using the residuals model and the estimation of the arameters associated with each fault. Figure 10: Residual Values and Predicted Residual Values versus ime After Isolation Conclusions A model that redicts the residuals dynamic behavior is formulated in this aer and used with the urose of fault detection and isolation. he aroach is based on a nonlinear state estimator and the estimation goals are based on erforming high filtering over the measurements. In this way, the magnitude of the residuals will be imortant enough to be analyzed. In using this residuals model, a better understanding regarding the residual trends, when a fault occurs, can be studied and further utilized to isolate faults efficiently. he detection is erformed by analyzing the residual trends of both residuals signals and redicted residuals. Multile fault modes are defined and validated through arameter estimation for the isolation mechanism. he aroach has the advantage of verifying false alarms. Having a restricted number of measurements available maes the objectives of detection and isolation a formidable tas. However, it has been demonstrated that the residuals modeling based aroach can deal with this restriction. A nonlinear rocess, simulated using CHEMCAD, was utilized to successfully validate the roosed aroach, showing accetable erformance under both closed-loo and oen-loo situations. hese

6 results serve as imortant evidence to extend the FDI formulation to other nonlinear alications. Acnowledgments he authors want to than the Process Science & echnology Center (PSC) at the University of exas at Austin and the Roberto Rocca Foundation for the suort of this roject. Aendix: CSR Model Parameters he CSR simulation arameters are: 1 he reexonential factor or frequency factor, 0min, 11 is ; which is utilized in the Arrhenius equation (given by Eq. 11). he activation energy, E J mol, is R J mol K, is 8.14 he universal gas constant, min 1 r K is the reactor temerature. r mol min - m consumtion of reactants. he volume of the reactor, V, is., defined in Eq. 11, is the secific reaction rate., defined by Eq. 12, is the rate of r m hree molar concentrations are considered: (1) concentration of the roylene oxide, C PO ; (2) concentration of the water, C W ; and () the concentration of the roylene glycol, C. PG where the valve osition, U cc %, is obtained from the reactor temerature controller. he coefficient of the valve,, is cv References Baoov, M., Puna D., Dostl P., and Zvac J. (2009). Robust stabilization of a chemical reactor. Chemical Paers, 6(5), Castillo, I., Edgar,.F., and Dunia R. (2010). Nonlinear modelbased fault detection with fuzzy set fault isolation. IECON th Annual Conference on IEEE Industrial Electronics Society, Nov., Isermann, R. (2005). Model-based fault-detection and diagnosis - status and alications. Annual Reviews in Control, 29(1), Massey, N. (2002). Incororating Reality Into Process Simulation. htt:// /echnical_articles/nmreality.df Simon, D. (2006). Otimal state estimation: Kalman, H infinite and nonlinear aroaches. John Wiley & Sons. Venatasubramanian V., Rengaswamy R., Yin K., and Kavuri S. (200). A review of rocess fault detection and diagnosis: Part I: Quantitative model-based methods. Comuters & Chemical Engineering, 27(), Fr m min is the inlet/outlet flow rate of the reactor and reresented by feed stream 4 (shown in Fig. 1). he roylene oxide flow rate, FPO m min, is held constant at aroximately he water flow, FW m min, is roortional to the oening of the control valve denoted by unit oeration 5. he flow rate through the control valve is given by: Ucr 1 F W 1 cv P Rang where the rangeability, Rang, is 10. he valve osition, %, is obtained from the roylene glycol controller. U cr he differential ressure of the valve, Pbar, is assumed constant at 0.1. he coefficient of the valve, cv5, is Similarly, the coolant flow, Fc m min, is roortional to the oening of the control valve, denoted by unit oeration, and calculated using the following equation: Ucc 1 F c 1 cv P Rang 100