I n d . Eng. C h e m . Res. 1991, 30, 898-908
898
Fault Detection and Diagnosis in a Closed-Loop Nonlinear Distillation Process: Application of Extended Kalman Filters Ruokang Li and Jon H.Olson* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
A strategy for fault detection and diagnosis in a closed-loop nonlinear system is described. An extended Kalman filter (EKF) is applied inside the control loop. The EKF recovers information from noisy measurement signals, providing estimates of state variables and unknown parameters of the process. The state estimates produced by the EKF are the inputs to the controller. Since the noise in the measurements is reduced significantly, the control quality is much better than that achieved with application of exponential filters. The estimates of unknown process parameters, where the deviations from normal values are considered faults, are the basis to make alarm decisions. The EKF successfully tracks and distinguishes faults occurring simultaneously. The influences of correlated noises, a special issue of estimation in closed-loop systems, model errors, and number of measurements have been studied. The proposed strategy is applied to a 30-stage binary distillation column with a partial condenser and a reboiler. The simulation results are presented to show the effectiveness of the approach. 1. Introduction
Fault detection and diagnosis are receiving substantial technical interest as a method for increasing reliability and safety in chemical plants. A fault implies a certain degradation of performance or some type of process deviation. Failure, on the other hand, refers to complete inoperability of equipment or the process. Faults can occur in the process, the sensors, the actuators, and the controllers and may lead to failure of the whole process. The task of fault detection is to determine the existence of faults in the systems, while that of fault diagnosis is to find the root causes of the faults. In most plants, fault detection and diagnosis are left to the process operators. With the increasing complexity of plants and the growing popularity of computers, it is desirable and feasible to utilize computer-based fault detection and diagnosis. Past efforts of fault detection and diagnosis can be divided into two categories: qualitative and quantitative approaches. The qualitative approaches involve fault trees (Lees, 1983),signed directed graph (Kramer and Palowitch, 1987),functional decomposition (Finch and Kramer, 1988), fuzzy logic (Vaija, 1985), neural networks (Venkatasubramanian and Chan, 1989),and expert systems (Dhurjati et al., 1987). The quantitative approaches are basically modeling, filtering, and estimation methods. A wide variety of methods and applications have been reviewed by Willsky (1976), Isermann (1984),and Himmelblau (1986). The quantitative approaches include state and parameter estimation, generalized likelihood ratio, x2 test, and other probability ratios (e.g., Narasimhan and Mah, 1988), and analytical redundancy (Tylee, 1983). Many estimation techniques are available (Ljung, 1987; Sinha and Kuszta, 1983; Willsky, 1976; Isermann, 1984), and a special issue of Automatica (Vol. 17, January 1981) focused on estimation. Two of the most frequently used methods are the Kalman filter and the extended Kalman filter (EKF) (Gelb, 1974; Catlin, 1989). Though not guaranteed to converge in nonlinear filtering problems (Yoshimura et al., 1980),the EKF has been found to yield accurate estimates in a number of important practical applications. Some examples in the chemical engineering area follow. Park and Himmelblau (1983) applied an EKF to detect and diagnose faults in a CSTR coupled with a heat exchanger. Dalle Molle and Himmelblau (1987) also
* Author to whom correspondence
should be addressed.
0888-5885/91/2630-0898~02.50/0
applied an EKF in a single-stage evaporator and showed how to adjust the tuning parameters in the filter. Watanabe and Himmelblau (1983a,b, 1984) proposed a twolevel identification strategy for fault detection and diagnosis of a chemical reactor. The states were reconstructed by an observer, and then parameters were estimated by an EKF. The process models they used were nonlinear but had some restriction in the form. It was claimed that this two-level strategy gave more accurate estimates than a straightforward usage of an EKF. Morari and Stephanopoulos (1980) demonstrated the superiority of a Kalman filter employing a nonstationary noise model to represent persistent disturbances commonly occurring in the chemical engineering environment. Their method can only detect the existence of faults but cannot discriminate different sources of faults. Nonlinear and adaptive Kalman filter was used to estimate catalyst activity and heat-transfer coefficient for a tubular reactor assumed in pseudosteady state (Rutzler, 1987). Recently Griffin et al. (1988) presented an example of industrial use of a Kalman filter on a reagent analyzer. It should be pointed out that all the systems mentioned in this paragraph are open-loop systems. Most chemical engineering processes operate as a part of a control configuration, and the control action will correct small changes of the states caused by faults. Hence the changes of some output variables may not be noticeable. Though there are a few papers dealing with closedloop systems, it is not clear whether fault detection and diagnosis can be successfully performed based on closedloop performance. Hamilton et al. (1973) applied the Kalman filter for state estimation in a closed-loop pilot plant evaporator. The filter estimates were sensitive to unmeasured process disturbances (faults), since they were not accounted for by the linear process model. Gilles (1986) used the EKF to estimate unmeasured states in several closed-loop systems, including a fixed-bed reactor, a polymerization reactor, and an extractive distillation column. The experiments were successful except for a biased estimate (see Figure 32 of his paper). The purpose of this work is to evaluate the effectiveness of the EKF for fault detection and diagnosis in closed-loop systems. A distillation column is used to illustrate the procedure. The following questions are to be investigated 1. What is the structure of the whole system? 2. Is the colored noise important to the performance of the filter? 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 899 Table I. Summary of Test Problem of a Distillation Column number of components 2 1.6 relative volatility (a) feed composition (X,) 0.5-0.55 feed enthalpy ( q ) 0.5-0.6 feed rate (F) 24.0 kmol/h reflux ratio -4 -0.5 distillate/feed ratio equilibrium condenser 15 rectification trays 15 stripping trays equilibrium reboiler holdups condenser 0.50 kmol each tray 0.25 kmol reboiler 1.00 kmol composition distillate (XD) 0.96 0.05 bottom product (X,)
3. What is the impact of modeling errors? 4. How many measurements are needed? The strategy for fault detection and diagnosis in the closed-loop system is described in section 2. The process models and dynamic analysis of accuracy of the models are presented in section 3. In section 4, the implementation of EKF is briefly described. Simulation results in section 5 show that two faults occurring simultaneously in the closed-loop system can be successfully detected and diagnosed and that reduction of measurement noise by the filter leads to better process control. Finally, the conclusions from this work are presented in section 6.
2. Strategy of Fault Detection and Process Control A binary distillation column used by Stewart et al. (1985) and Benallou et al. (1986) is chosen in our study. This system, involving a rectification module, a stripping module, a condenser, and a reboiler, is listed in Table I. Figure 1 shows the basic idea of applying the EKF to fault detection and diagnosis as well as process control in the closed-loop distillation process. The functions of every block are briefly presented below. The full-order model attempts to simulate the actual dynamic behavior of the binary distillation process. The total order of the model is 32. Perfect level control and 100% tray efficiency are assumed, and tray hydraulics and energy dynamics are neglected. The feed rate, F, is assumed known or measurable. The faults, or disturbances, are abrupt changes in the feed composition, X F ,and the feed enthalpy, q. They were chosen for clarity and ease of implementation. The ratio of distillate rate to vapor rate in the rectification section, D/VR, and the ratio of vapor rate in the stripping section to the bottom-product flow, V s / B ,are two manipulated variables controlled by two composition controllers. Four or five states (depending on the order of the reduced model used in the EKF) of the 32-order model, compositions of the distillate, the bottom product, and two or three stages in between are defined as a vector X. The measurement variable, Z, is part or whole of X plus measurement noise, v. Z and the control signals D / VR and V s / B are the inputs to the EKF. The process model in the filter is a reduced-order model which attempts to represent the behavior of X. Using the process model, the EKF recovers the information provided by Z, D/VR, and Vs/B. The filter serves two purposes-estimation of unknown timevariant parameters, the feed composition X F and the enthalpy 6,and furnishing the controllers with more precise information about the composition of distillate XD and
I
Top-Comp.
I
Controller DNR
kDl
II ” Extended
Kalman Filter
I I
vs/ B
I
1 I
II I I ^xs
I
Bottom-Comp. Controller
d
900 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 Table 11. Stage Numbers (from Top Down)and the Sensitive Stage (in Parentheses) of the Compartmental Models fourth order fifth order compartment 1 1 (1) 1-3 (1) compartment 2 2-16 (9) 4-16 (9) compartment 3 17-31 (24) 17-22 (21) compartment 4 32 (32) 23-29 (27) compartment 5 30-32 (32)
-5
-3
-4
-2
0
where the vector X of length 4 or 5 is the compositions of the compartment sensitive stages. The feed composition XF and the feed enthalpy q are the unmeasurable fault parameters
e = (XF,q)T
(2)
The input variables are the output of the controllers
u = (D/VR,Vs/mT
-5
-3
-4
log 10
-2 rad&
--
Figure 2. Bode diagram of transfer functions X D ’ / ( D /VR)’for the fourth and fifth compartmental and full order models.
by Horton et al. (1986), and experimentally evaluated by Kumar et al. (1985). This modeling approach describes the dynamics of a staged separation process with fewer differential equations than does the stage-to-stage model. This low-order modeling technique was chosen because it has some attractive features that are required by fault detection and diagnosis. They are as follows: 1. It is guaranteed that the steady-state of this model is identical with the full-order model. 2. All the state variables and parameters are physically significant. Besides the assumptions made in the full-order model, further postulations are made: 1. All the stages including condenser and reboiler can be arbitrarily grouped into a number of compartments. 2. The dynamic behavior of a compartment can be represented by that of one of its stages, the sensitive stage, with the holdup of this compartment equal to the sum of holdup of individual stages in this compartment. 3. The composition of inlets and outlets of a compartment can be related to the sensitive stage and the top or bottom compositions via steady-state balances. This relationship is defined as separation functions. The selection of the order of the compartmental model is a trade-off between accuracy and computation load. The higher the order, the more accurate the model but the heavier the computation load. A fourth- and a modified fifth-order (for simplicity, it is called fifth-order in the remainder of this work) models, tested by Benallou et al. (1986) and Horton et al. (1986) separately, are considered. Table 11 shows the parameters of these models. The sensitive stages of the first and the last compartments are the condenser and the reboiler, respectively. Thus the reduced-order distillation model is described by the nonlinear differential equation (see Appendix for details) x = f(X@,U) (1)
(3)
The object of the rest of this section is to investigate the modeling error of the fourth and the fifth models with respect to the full-order model and select an appropriate one. First the nonlinear model, (l), is linearized, and then the analysis is applied both in frequency domain and in time domain. This analysis is inspired by the work of Bonvin et al. (1989) and De Valliere and Bonvin (1989), who used the techniques to validate accuracy of a calorimeter model and to determine the sensitivity frequency region for estimation. Linearizing the distillation model (1)yields X’ = AX’ + BU’ De’ (4)
+
where the superscript denotes the change in the variable from the linearization point, and the constant matrices A, B,and D are evaluated at the linearization point. The frequency response of the transfer function =/ (D/VR)’ is shown in Figure 2. (The Bode diagrams of other functions have similar results and are omitted here.) Both the fourth and the fifth models are in good agreement at low frequencies with the full-order model, which shows the preservation of steady state by the compartmental model. The amplitude ratio (AR) of the compartmental models diverges from the full-order model at frequency 7 X lo4 rad/s (Figure 2a), and the phase shift diverges at 2 X rad/s (Figure 2b). Hence the compartmental models are accurate up to frequency of 2 X 1V rad/s. For the purpose of controller specification,this accuracy is not sufficient. However, in fault detection and diagnosis, this accuracy may be acceptable since the fault model used in estimation is relatively slow (see the next section). When the frequency is larger than 2 X rad/s, the absolute value of phase shift for the reduced models is smaller than that for the full model. Consequently, a lead in the estimation is anticipated. It is expected that as the order of the model increases, the frequency-response curves will approach to the full-order model. The eigenvalues of matrix A serve as another criterion of the system dynamics. Table I11 lists all the eigenvalues of the reduced-order models and part of the eigenvalues of the full-order model that are larger than -0.0500. The omitted eigenvalues of the full-order model are in between -0.2693 and -0.0579. From Table 111, one can see that, except for two eigenvalues of the fifth-order model that are located near the real axis, all the eigenvalues are real. Since all the eigenvalues are on the left side of the s-plane, the reduced models as well as the full model are stable. The largest eigenvalue of the full model is -0.0003, corresponding to the time constant of the distillation column.
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 901 Table 111. Eigenvalues of the Linearized Models" fourth fifth thirty-secondb
-0.0002
-0.0002
-0.0040
-0.0044
-0.0003 -0.0008 -0.037 -0.0054 -0.0093 -0.0127 -0.0185
-0.0228
.......
..................
-0.0240
.......
*
-0.0318 -0.00368 k 0.0007i -0.0424 -0.0475
-0.0393 -0.0485
" ..., Nyquist
cutoff. bTwenty-one eigenvalues, between -0.2693 and 0.0579,are omitted.
--
Table IV. Zeros of the Transfer Function Xb/(V s / B ) ' of the Fourth and the Fifth Linearized Models fourth fifth
0.0003 -0.0375
-0.0041 -0.0233 -0.0349 -0.0382
The second largest eigenvalue is -0.0008. These two dominant eigenvalues are represented by the eigenvalue -0.0002 of the reduced models. Again, the two eigenvalues (-0.0037 and -0.0054),which are 1 order of magnitude larger than the dominant eigenvalues, are lumped to 4.0040 of the fourth model and -0.0044 of the fifth model. Because the sampling rate is 100 s, the Nyquist frequency is */loo, or 0.0314, rad/s. The fastest eigenvalues of the reduced models correspond to 0.0475 rad/s for the fourth model and 0.0368 rad/s for the fifth model. These roots and the omitted values of the complete model are outside of the range of the control system, and consequently their contribution to the closed-loop system is lost. From the discussion of the frequency response and the eigenvalues of the reduced model, one concludes that these two models are almost the same. However, Horten et al. (1986) claimed that the fifth model is better than the fourth in terms of the inverse response to a step input. The major disadvantage of the compartmental model is the presence of an initial inverse response that does not appear in the full-order models. Benallou et ai. (1986) attributed the inverse response characteristics to the assumption of steady state inside the compartments. Horton et al. suggested some guidelines for constructing the reduced-order compartmental models in order to avoid the inverse response and increase the transition accuracy. A system that has transfer function zeros in the right s-plane will exhibit inverse response. The calculation of zeros of the fourth and the fifth models backs the suggestion of Horton et al. Table IV lists the zeros of the transfer function of %/ ( Vs/B)' of the fourth and the fifth models. Clearly, the fourth model has a positive zero very close to the origin. According to Bonvin et al. (1989), estimation for this kind of process model is more difficult than the model with large positive zeros. On the other hand, the fifth model has no positive zeros. (A similar situation exists for the transfer -function X B ' / ( D /VR)' and is omitted here.) I t is worthwhile to mention that the positive zeros are not found in the transfer functions of G1/G2 - of both reduced models, where G1 represents XD' or XB'and G2represents or 9'.
Examination of the linearized full-order model also shows that, if the linearized model represents the process correctly, the distillation process is controllable and observable. In this section we have examined the model errors in frequency domain and time domain and found that the fifth model is superior to the fourth in that the former does not have inverse response and the latter does. Unless indicated otherwise, the fifth-order model is used as the process model in EKF in the remainder of this paper. 4. Extended Kalman Filter In this section the EKF algorithm (Gelb, 1974) is briefly summarized and its features are discussed. The stochastic partner of (1) is x = f(x,e,u) w (5) where w is a white noise with covariance matrix Q. The EKF attempts to estimate the state X from the sampled measurements of the form = hk(X(tk))+ Vk, k = 1, 2, ... (6) where vk is a white noise with covariance matrix, Rk,and, in our problem, hkis a linear function. Equations 5 and 6 represent a nonlinear estimation problem where the process is continuous and the measurement is discrete. The process model in our work takes the form as same as (5) and (6) except that parameters 0 first are assumed known (this assumption will be released later). For simplicity, (5) is rewritten as follows: X ( t ) = f(X(t),t)+ w(t) (7) It is assuqed that the measurement Zk-l and the corresponding X(tk-,) have been obtained. The recursive algorithm of the filter involves two phases: 1. Propagation of state estimation and error covariance. Integration of (8) and (9) in the sampling interval times I ,t < tk) when the measurement has not been made yields X(tk),and estimation error covariance matrix P(tk), defined as XJ-) and Pk(-). act, = f(X(t),t) (8)
+
P(t) = F(X(t),t)P(t) + P(t) FT(X(t),t)+ Q ( t ) (9) where F is the Jacobian matrix of f
2. Update of state estimate and error covariance. The qeasuremcnt is taken at time tk and is used to modify Xk(-) and Pk(-) by minimizing the estimgtion error, The improved estimates at t k are denoted as &(+) and Pk(+). X k ( + ) = X k ( - ) + Kk[Zk - hk(Xk(-))I (11) Pk(+) = [I - KkHk(Xk(-))]Pk(-) (12) where
and the gain matrix
Kb = (14) If the parameters 8 are time-variant and unknown, they can be considered as additional states. Most faults in chemical processes are either slow degradation (e.g., catalyst activity) or abrupt change to a biased value (e.g., leak in a pressure vessel). Either way, there are low-frequency
902 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
components involved in the fault information. Since the nature of the time variation of fault parameters cannot be known exactly, it is usually assumed that the time deviation of unknown parameters changes more slowly than the system and is driven by random noise e ( t ) = w&t) (15) Then the states X and parameters 8 are combined into an augmented-state vector, and hence the EKF can be applied to estimate both the state variables and unknown parameters. In this work, the noise covariance matrices Q and Rk, the measure of uncertainty in the model and measurements, were assumed diagonal and were properly adjusted via simulations. 5. Results and Discussions This section investigates the impact of colored noise, compares two different designs for the EKF, examines the effects of model mismatch, and determines the trade-off for reducing the number of input sensors for the system. 5.1. Colored Noise. In this subsection the concept of the whiteness of noise is briefly introduced, the reason why the noise becomes colored in closed-loop systems is explained, and the simulation results are presented. White noise is defined as a random process with a mean of zero and with constant power spectral density (Friedland, 1986). Though only a theoretical abstraction, the concept of white noise has proved to be very useful in many applications. The correlated noise is called colored. The “Whiteness”of a stationary random discrete series ([(k), k = 0, f l , f2, ...Iwith mean, m, is measured quantitatively by the normalized autocorrelation function 1 N
r,(k) =
lim - - C [ f ( k + l ) - m ] [ [ ( l-) m] N1~1
N--..
1 N
(16)
lim - X [ [ ( k + l ) - m ] [ [ ( k + l )- m]
N--
When r,(k,k#O) from -1 to 1. r,(O) always equals 1. When r,(k,k#O) is 0, [ is totally uncorrelated, or white. When Ir,(k,k#O)l is 1,[ is totally correlated. When Ir,(k,kfO)l is in between 0 and 1, 6 is somewhat correlated. If N in (16) is sufficiently large instead of infinity, the estimate of r,(k), ?,(k), is obtained. The appearance of colored noise is one of the problems of identification in a closed-loop system. In open-loop systems the assumption of white noise is usually valid. In fact whiteness of the innovation, the difference between an observed output and the predicted output based on the system model and previous observations,is even one of the typical tests to determine the existence of faults (Himmelblau, 1986; Narasimhan and Mah, 1988). In closed-loop systems, the input noise to the filter is the combination of the measurement noise, which may be white, and the system noise caused by the noise in the control signal which, in turn, is affected by the input noise of the filter. Thus the input noise of the filter becomes somewhat “colorful”. For example, when in Figure 1 the measurement noise, v, is white at a noise level of 0.01, which is defined as the ratio of the standard deviation of noise to the expectation value of measurement, the whiteness of the filter input, Z,becomes about 0.5 due to the feedback of the control signal. In the model of EKF, (5) and (61, it is assumed that the process noise, w, and the measurement noise, vk, are perfectly white. If the noises are nonwhite (colored),they are usually modeled as output of a linear filter driven by white noise. This either leads to the requirement of
(a) Exponential Filter + EKF
1 measurement noise Process
-2 t
Controller
Controller
+
knowledge of the filter or an additional computation load when the state vectors are augmented with the noise variables (Sorenson, 1985). In order to investigate the effect of the whiteness of the noise on the behavior of the filter, the two control loops were removed and the measurement noise level was kept constant (0.01). The EKF had four measurements, i.e., the compositions of stages 1 (distillate), 9,27, and 32 (bottom product). The disturbances (faults), occurring simultaneously, are a step change in the feed composition from 0.50 to 0.55 at time 100oO s and a ramp change in the feed enthalpy from 0.5 to 0.6 during the time interval 1000012 OOO s. Unless indicated otherwise, the noise level (0.01), the four measurements, and the faults just specified are used in the remainder of this work. Two simulations for different noises were examined, the first being perfectly white and the whiteness (r,) of the second being 0.5, which was obtained by passing white noise through an exponential filter. The differences of the estimation results between the two simulations are so small it is not necessary to show them here. Because of the minimal effects of the colored noise on filter performance, which is in agreement with the observation by Litchfield et al. (1979) and by Goldmann and Sargent (19711, in the following tests no special measures are taken to deal with them. 5.2. Reduction of Measurement Noise. This subsection presents two ways of using the EKF, i.e., EKF outside or inside the control loop. Then the stability and dynamics of the later system are discussed. Finally the performance of the two systems is compared. The first design, called structure A and shown on Figure 3a, places the EKF outside the control loop to monitor the unknown process parameters. This system also uses an exponential filter in the control loop to reduce the effect of sensor and system noise on the control system and the estimator. One of the key difficulties in using the EKF is that the filter may be unstable (see the following stability analysis). However, structure A excludes the possible unstability from the control loop. This is the reason why the EKF is not used in the feedback loop even though it
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 903 is already in the system. The exponential filter equation is i(k+1) = @Z(k+l)+ (1 - P ) i ( k ) (17) where 2 and 2 are the measurement and the estimated value, respectively, and /3 is a weighting constant normally assigned a value between zero and one. A lower value of /3 increases the filtering function but also increases the phase lag introduced by the filter. Hence choosing the filter constant involves a trade-off between noise transmitted and dynamic lag introduced. In this work 0 is chosen as 0.5. An alternative, the one emphasized in this work, places the EKF in the control loop. This system is structure B on Figure 3b. The EKF inside the control loop provides the controllers with information of the control variables as well as estimation of unknown parameters. The issue of the combined stability and dynamics of the whole system will be discussed before comparing the performance of the two systems. The stability of the control system of the configuration as shown in Figure 3b can be determined by the separation theorem (Safonov, 1980). The separation theorem shows that nondivergent estimates can, unconditionally, be substituted for true values in otherwise stable feedback systems without ever causing instability. In other words, stable output feedback controllers can be designed, even for nonlinear multiloop feedback systems, by designing separately (1) a stable state feedback controller and (2) a nondivergent filter. Safonov further proves that the stability and the nondivergence can be ensured if the the gain of the EKF is constant and the EKF incorporates an accurate internal model of the nonlinear process dynamics. Though there are always modeling errors in practice and the convergence of the EKF cannot be guaranteed (Yoshimura et al., 1980), the separation theorem does give us some guidelines: the controller and the filter do not interact with each other in terms of stability and therefore these components can be designed separately. Because of the nonlinearity of the plant and the filter and the time-variant gain matrix of the filter, the analysis of the dynamics of the closed-loop system is difficult. Some approximation must be made. If it is assumed that the dynamics of EKF can be represented by the dynamics of the linear, constant-gain Kalman filter, the results of the linear quadratic Gaussian (LQG; see Astrom and Wittenmark, 1984) control can be used in analyzing the properties of the closed-loop system. The LQG consists of two parts: one linear Kalman filter, which gives estimates from the measurements, and one linear-feedback law from the estimated states. The design of LQG is based on the discretized linear model: X(k+l) = @X(k)+ l'U(k) + w(k) (18) Z(k) = CX(k) + v(k) (19) The dynamics of the closed-loop system are determined by the matrix (20)
where L is the control law. The eigenvalues of M are the eigenvalues of the matrices @ - l'L and @ - KC. Notice that the eigenvalues of @ - I'L are the desired closed-loop poles obtained by designing the controller without considering the Kalman filter and the eigenvalues of @ - KC are the poles of the filter. In other words, the controller and the Kalman filter can be designed separately. The poles of the resulting system are simply the poles of the
Top Composition 0.99,
,
0.56,
..
0
.
0.93
0.95p0 oo 0
. 5
0.48
,
"8 9
Feed Composition
oo
oo
3
1
10 15 Time (k s)
(b)
0
.
1
0 . 5 0 p
11-184.0 20
5
15 Time (k s) (d)
10
20
Figure 4. Effect of structures A and B on quality of control and estimation: the true, measured, and estimated values of the top composition for (a) structure A and (b) structure B and the true and estimated values of the feed composition for (C)structure A and (d) structure B. -, true value; +, estimated by the exponential filter; X, estimated by the EKF; 0,measured.
closed-loop poles assigned by designing the controller and the poles of the filter. The controllers have been designed through pole-placement method. The poles are placed at 0.75 f 0.31i on the z-plane. If it is assumed that the dynamics of EKF can be represented by the dynamics of the linear Kalman filter and that the gain matrix is fixed, the poles of the EKF can then be obtained. The eigenvalues of @ - KC are 0.0075,0.0202,0.2689,and 0.1338 f 0.2223i. Thus all poles of M are inside the unit circle, and the closed-loop system is stable. Further, all the poles are in the region that corresponds to modes with sufficient damping ( k t r o m and Wittenmark, 1984) and the dynamics of the whole system are adequate. These results are a specific example of the separation theorem. Having established the details of the two configurations, the effectiveness of the design was compared by simulation. A portion of these results are given on Figure 4. The upper row is for structure A, and the lower is for structure B. The left column is the distillate composition, one of the controlled variables. The right column is for feed composition, one of the estimated forcing functions. In both systems the EKF's were tuned properly. Figure 4a shows the behavior of the distillate when structure A was used. Three items are shown on this graph, the true signal as a heavy line, the measurement including noise, and the output of the exponential filter, which is the input to the controller and to the EKF estimator. The figure shows that the overhead composition meanders about the 0.96 set point. Figure 4b parallels Figure 4a in structure except that the EKF was used in the control loops both as a noise filter and as an estimator. It shows that the feed disturbances produce a small blip in the overhead composition. Structure B follows this change very closely even though the raw signal (open dots) shows no structure. There is a large reduction in the standard deviation of the signal as well. Using the EKF the noise is reduced by 50 times, compared to 1.7 times using the exponential filter. Better estimation leads to better process control. Comparison of (a) and (b) of Figure 4 indicates a substantial improvement in the system when the EKF is included in the control loop. The standard
904 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
deviation of the true value of the top composition by using the EKF is almost zero (O.oooO), compared to 0.0040 using the exponential filter. Placing the K h a n filter inside the control loop clearly leads to better control. Comparison of the bottom behavior gives the same conclusion and is omitted here. Figure 4c,d shows the disturbance of feed composition and its estimate by EKF. Comparison of these figures reveals a vast improvement in the tracking of the disturbances and a large reduction in the background noise when structure B is used. The standard deviation of the estimate in structure B is 4 times lower than that in structure A, even though the input noise in structure B is higher than that in structure A. (Recall that the input of structure B is directly from the measurements while in structure A the input is from the exponential filter.) There are two reasons to explain this phenomenon. First, the lower noise reduction by the exponential filter yields greater fluctuations of the outputs of the controllers, which in turn are the inputs to the plant. These fluctuations, though measurable, affect the estimates of the states and the unknown parameters. Second, the dynamics of the exponential filters was not considered in the process model of EKF and poorer performance results. Inclusion of the dynamics of the exponential filters will substantially increase the order of the EKF and is less feasible and desirable. This subsection has shown that prudent design would include the EKF in the control loop even when the major function of the EKF code is to estimate feed composition and enthalpy. In addition, this configuration (structure A) not only reduces the measurement noise but also recovers unmeasurable states from noisy measurements (see section 5.4). 5.3. Model Mismatch. In section 3, the fourth- and the fifth-order models have been compared with the full-order model in both frequency domain and time domain. Here, the effect of modeling errors on the performance of the EKF is investigated by simulations. Three models, the fourth, the fifth and the full, were used in the EKF. The number of measurements in all the cases was four. The measurements were the compositions in stages 1, 9, 24, and 32 for the four and the fifth models and in stages 1,9,27, and 32 for the fifth model. Other conditions were as same as those mentioned before. Results are shown in Figure 5 (state variables) and Figure 6 (parameters). In Figure 5, the left column is for the distillate composition and the right for the bottom composition. The rows from top down are the fourth, the fifth, and the full models. For the top composition, all the models work equally well in the steady state (in a statistical sense). In the disturbance period, thanks to the smaller phase shift of both reduced models (section 31, the reduced models lead in estimation (cross marks), and hence the overshoot and the settling time of the true value (solid line) for these models are smaller than those of the full model. Although in this special case it seems that the modeling errors improve the control quality, due to the unpredictable behavior under other disturbances, one still wants the modeling errors as small as possible. Examining the right-hand column of Figure 5, specifically for the bottom composition a t steady state, shows the fluctuation of true value and estimate decrease as the model order increases. A great improvement can be seen as the model order increased from fourth to fifth. The true values of the bottom composition for the fifth and the full models have a peak during the disturbance period, while no observable structure is seen for the fourth model. This result probably
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Figure 6. Effect of model error on parameter estimations: estimates of the feed composition (left column) and the feed enthalpy (right column) with the fourth (first row), the fifth (second row), and the full (third row) models. -, true value; X, estimated.
can be explained as follows: Owing to the inverse response of the fourth model and the small phase shift, the bottom
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 905 Top Composition
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composition is so overestimated and the lead of estimation is so large that the peak of the true value caused by disturbances disappears. Figure 6 shows the true values and the estimates of the feed composition (left column) and the feed enthalpy (right column). The vertical structure is as same as in Figure 5. It is clear that as the model order increases, the effect of noise on the estimation decreases and the fluctuations in the estimates are smaller. Except for the undershoot of the feed enthalpy (obviously caused by the model mismatch of the high-frequency components of the disturbances), the estimation with the reduced models satisfactorily tracks and distinguishes the two faults, while the performance of the full model is perfect. Further simulations show that while a seventh-order model with four measurements improves the estimation overshoot of feed enthalpy a little, it substantially increases the computation time. In summary, model mismatch has considerable effects on the performance of the system. More accurate models within the restriction of computation capacity still are in demand. 5.4. Number of Measurements. Any combinations of the state variables in the filter can be chosen as measurements. Intuitively, the more measurements there are, the more information and the greater accuracy of the estimates. However, it is technically feasible and desirable to have a small set of measurements. The questions arise, then, of how many measurements are sufficient and which measurements will result in the best estimates possible. In the fifth-order model there are up to five measurements, Le., the compositions on the stages 1 (distillate), 9,21, 27, and 32 (bottom product). Four simulations were run to investigate the impact of the number of measurements on the performance of the filter. In the first, all five
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measurements were used. In the second, the four measurements were on stages l , 9,27, and 32. In the third and fourth, only two measurements were used. The third is called direct measurements since the top and bottom compositions (stages 1 and 32) were directly measured, while the fourth is called indirect measurements since the information on two middle trays (9 and 27) was used to estimate the top and bottom compositions as well as the faults. The simulation results are shown in Figures 7 and 8. Figure 7 compares the true values, the measurements (where applicable), and the estimates of the top and the bottom compositions for the direct (first row in the figure), the indirect (second row), and the four (last row) measurements. Figure 8 shows the true values and the estimates of the two faults, namely, the feed composition and the feed enthalpy. The vertical structure in this figure is as same as that in Figure 7. Comparing these experiment results, one observes the following: 1. As the number of measurements increases, the quality of estimates of the top and bottom compositions improves and so does the quality of the control. 2. As the number of measurements increases, the estimates of the two unknown parameters (faults) become better with the smaller fluctuations and the shorter lag. 3. When the number of measurements increases from four to five, the improvement is less significant (not shown here), since there is less new information in the additional measurement (Joseph and Brosilow, 1978). 4. The indirect measurements provide more accurate estimates of the state variables and the unknown parameters than do the direct measurements. The reason why the middle trays are more sensitive to the faults occurring in the feed can be explained as follows. During the dis-
906 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
is less sensitive to high-frequency noises. Consequently,
both the control quality and the estimation of unknown ' I parameters have approved considerably. From the analysis
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turbance period, the response of the middle trays is larger than that of the top and bottom products. In addition, after the faults are overcome by regulators, the top and bottom compositions remain unchanged while the compositions of the two middle trays transit to new steady states. The frequency analysis of the sensitivity also supports these arguments. We begin with the linearized model, (4). The transfer function matrix relates the state variable to the disturbances, or faults. Figure 9 shows amplitude ratio (upper part) and the phase shift (lower of the of four transfer - part) ---Bode diagram functions, Xl'IXF', Xi/XF', Xl'/q', and Xi/q', where X1 and X 2correspond to the compositions on stages 1and 9, respectively. The amplitude ratios of tray 9 are about 6 times larger than those of the top. The phase shift curves for stages 1 and 9 coincide with each other at low frequencies and diverge at 1 X rad/s, the natural frequency of the column. Beyond this frequency, the phase shift of tray 9 is smaller (absolute value) than that of the top because tray 9 is closer to the fault source. In summary, measuring four of the five states is adequate to guarantee the performance of the filter and the whole system. When only two measurements are allowed, the indirect method (conventionalsensor placement) is the better choice. 6. Conclusions In this paper we have proposed and examined the strategy for fault detection and diagnosis in a closed-loop distillation system with random measurement noise. The controllers use the more accurate state variables reconstructed by the EKF as their inputs. The whole system
and simulations the following conclusions are reached: 1. Inclusion of the EKF in the control loop will significantly reduce the effects of the measurement noise on the control system. This strategy in turn results in better estimation of the unknown parameters. 2. The effects of the colored noise, caused by the interaction in the closed-loop system, are small, at least in this distillation example. 3. A fifth-order compartmental model gives satisfactory estimation of the process states and unknown parameters. However, a more precise process model will give lowered sensitivity to measurement noise and better estimation of unknown parameters. 4. The EKF successfully tracks the varying parameters closely, which makes it possible to detect the occurrence of the faults and diagnose the causes of the faults. 5. Measuring four of the five states gets enough information from the distillation process. The middle trays are more sensitive to the feed faults than are the top and bottom stages. This approach can be applied to detect and diagnose more realistic faults in distillation system, for instance, the decreasing heat-transfer coefficients of the reboiler caused by fouling and the internal reflux ratio. Due to the computation limitation and the measurements available, the number of faults selected to monitor is restricted. Hence the filtering techniques for fault diagnosis can only be effective a t the unit level. It is too complex to apply the filtering techniques to groups of units or at the plant level where there are hundreds of possible faults. Furthermore, the filtering techniques cannot distinguish unmodeled disturbances from those included in the model. A hierarchical approach with filtering techniques at lower levels and artificial intelligence techniques at higher levels may be more suitable for fault detection and diagnosis in large or complex processes.
Acknowledgment We acknowledge financial support provided by the University of Delaware.
Nomenclature A = coefficient matrix in the linearized model, (4) AR = function defined in (A4) As = function defined in (A6) B = coefficient matrix in the linearized model, (4) B = bottom flow rate BR = function defined in (A4) Bs = function defined in (A6) C = coefficient matrix in the discrete model, (19) D = coefficient matrix in the linearized model, (4) D = distillate flow rate
DR = function defined in (A4) Ds = function defined in (A6) ER = function defined in (A4) Es = function defined in (A6) F = Jacobian matrix off F = feed flow rate f = process function fi = separation function for the first stage of compartment i G = transfer function matrix defined in (21) gi= separation function for the last stage of compartment i H = Jacobian matrix of h h = measurement function i = imaginary number
Ind. Eng. Chem. Res., Vol. 30, NO. 5, 1991 907
K = gain matrix of the Kalman filter k = number of compartments L = matrix of the LQG-control law L = liquid flow rate in the distillation column M = dynamic matrix of the LQG-control law M c j = holdup of compartment j m = mean N = total number of stages P = estimation error covariance matrix Q = process noise covariance matrix q = feed enthalpy R = measurement noise matrix ra = normalized autocorrelation function ri = number of the first stage in compartment i s = Laplace transform variable si = number of the sensitive stage in compartment i t = time t i = number of the last stage in compartment i U = measured input vector u = manipulated variable V = vapor flow rate in the distillation column v = measurement noise w = process noise X = state variable vector X = liquid composition Z = measurement vector 2 = a scalar variable in (17)
where the total holdup (MJof compartment j is the sum of holdup of individual stages in this compartment, 6 is the Kronecker delta, and fi and gi are separation functions. The separation function for the vapor stream leaving the compartment i in the rectification section is aXri (A2) = 1 (a - l)xri
+
where
(13)
and
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1 a-1
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r = coefficient matrix in the discrete model, (18) 6 = Kronecker delta 8 = nonmeasurable parameter vector E = stationary random discrete series QP. = coefficient matrix in the discrete model, (18)
1
ER = ~ { A -RBR + [(AR - BR)'+ 4&]'/')
(A4)
The separation function for the liquid stream leaving compartment i in the rectification section is gi(Xi) E Xti
Superscripts = estimation - = Laplace transfer = differentiation with respect to time ' = variable change from the linearization point
-
Subscripts B = bottom D = distillate F = feed f = feed stage k = at time tk R = rectification section S = stripping section 0 = parameter
Appendix: Compartmental Distillation Model The mathematical development of the compartmental model that is presented here follows the methods of Benallou et al. (1986). The following assumptions are made: (1) binary system, (2) equimolal overflow, (3) constant molar holdup, (4) constant relative volatility, and (5) 100% stage efficiency. Extensions to complex multicomponent columns have been reported by Benallou (1982). By the compartmental concept, the N-stage (including condenser and reboiler) column is divided into K compartments. A set of differential equations are obtained from dynamic material balance around every compartment and from the steady-state relationship among the distillate or bottom product and the first, the last, and the sensitive stages of every compartment.
The separation functions for the stripping section are obtained from replacing AR, BR, D R , ER, and XI in (A2), (A3), and (A5) by
1 + [ ( A s- BS)2 + 4 0 ~ ] ~ / (A6) ~) 2 and xk, where Xk is the bottom product composition.
Es = -(As - Bs
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