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Nonlinear State Estimation Using Fuzzy Kalman Filter R. Senthil, K. Janarthanan, and J. Prakash* Department of Instrumentation Engineering, Madras Institute of Technology, Anna UniVersity, Chennai 44, India
In this paper, the authors have presented an approach for designing a nonlinear observer to estimate the states of a noisy dynamic system. The nonlinear observer design procedure involves representation of the nonlinear system as a family of local linear state space models; the state estimator for each linear local state space model uses standard Kalman filter theory and then a global state estimator is developed that combines the local state estimators. The effectiveness of the proposed fuzzy Kalman filter (nonlinear observer) has been demonstrated on a continuously stirred tank reactor (CSTR) process. The performances of the fuzzy Kalman filter (FKF) and the extended Kalman filter (EKF) have been compared in the presence of initial model/plant mismatch and input and output disturbances. Simulation studies also include an estimation of reactor concentration (inferential measurement), based only on the measured variable temperature of the reactor. 1. Introduction State estimation of dynamic systems is an important prerequisite for safe and economical process operations. It is an integral part in applications such as process monitoring, fault detection and diagnosis, process optimization, and model-based control. Because all the process variables are generally not measured, an observer can be designed to generate an estimate xˆ (k) of the state x(k) by making use of the relevant process inputs, outputs, and process knowledge, in the form of a mathematical model. On the other hand, a conventional technique that is used to obtain direct measurement of the process variables (such as product composition and reactor concentration, etc.) provides delayed (or) inaccurate measurement and involves high investment and maintenance cost. The general, state estimation problem in a stochastic linear system is solved by the well-known Kalman filter.1 For linear systems, Kalman filters generate optimal estimates of state from observations. For a linear system with unknown parameters, interval Kalman filtering (IKF) and IKF with fuzzy inference have been proposed.2,3 Fuzzy principles have been used for adaptive tuning of the measurement noise (R) and state noise covariance (Q) matrices in the standard Kalman filter algorithm.4 The Kalman filter has become more useful, even for very complicated real-time applications, and has attracted the attention of the chemical engineering community, because of the recursive nature of its computational scheme. For nonlinear systems, an extended Kalman filter (EKF) is a natural extension of the linear theory to the nonlinear domain through local linearization. There are several variants of the basic EKF which have been evaluated by various researchers.5 Many works in observer design for nonlinear system are based on the EKF approach, which leads to a complex nonlinear algorithm. Despite good results, there is no a priori guarantee for convergence and stability for the algorithms. Hence, there is a need to design a computationally nonintensive, nonlinear observer with convergence being guaranteed. It is well-known that the design of any good state estimator must be based on a good model of the plant. To incorporate more plant information into the design of the observer, a nonlinear process model must be used and a plethora of dynamic models have been proposed in system identification literature * To whom all correspondence should be addressed. Tel.: 9444860188. E-mail address:
[email protected].
to describe a nonlinear dynamic system. Several modeling approaches have been proposed that use local models to approximate nonlinear systems. For a detailed review, see the work of Johansen and co-workers.6-8 The different approaches can be distinguished by the choice of the model weights. In Aufderheide and Bequette,9 model weights have been computed using the recursive Bayesian algorithm. That is, given a set of models, the Bayesian algorithm recursively determines the likelihood that the ith model is the true model of the plant, based on the present residuals and the previous probabilities for each model. In Banerjee et al.,10 the multiple local models are combined into a single-linear-parameter-varying global model. The parameters of the global models are selected to be model probabilities that are estimated using the Bayesian approach. However, a simple way to describe a nonlinear dynamic system using multiple linear models has been proposed by Takagi and Sugeno11 (referenced hereafter as T-S models) and it is being used in this paper to develop a nonlinear observer for a dynamic system. The main advantage of this framework is its transparency. That is, the model structure can be interpreted not only in terms of operating regimes, but also quantitatively, in terms of individual local models. Furthermore, the operating regimes can be represented as a fuzzy set. This representation is appealing, because many systems change behavior smoothly as a function of the operating point, and the soft transition between the regimes introduced by the fuzzy set representation capture this feature in an elegant fashion. Fuzzy state estimation is a topic that has received very little attention. There have been a few papers published recently on fuzzy observer design; however, these papers usually involve only the noise-free case.12-14 That is, fuzzy observers are designed for systems that are not affected by noise. Recently, Simon15 proposed a state estimator for noisy discrete-time dynamic T-S fuzzy models and has shown that the state estimator generates accurate state estimates, in the absence of an unknown input. However, the unknown input to the process, such as feed temperature, feed concentration, feed flow rate, process parameters, etc., are likely to change from their nominal values. Hence, there is a need to develop a nonlinear observer for noisy dynamic systems, which will generate accurate state estimates in the presence of an unknown input. The main contributions of the paper are as follows. First, the nonlinear system is represented as a family of local linear state space models. Second, the state estimator is designed for each local linear state space model, using standard Kalman filter
10.1021/ie0601753 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/08/2006
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theory. Third, a global state estimator is developed by combining the local state estimators. Finally, augmented-state fuzzy Kalman filter (ASFKF) has been developed to generate accurate state estimates in the presence of a steplike disturbance at the output, slow drifts, or a steplike disturbance at the inputs or states. The organization of paper is as follows. Section 2 presents the design of the T-S fuzzy model, followed by the design of a fuzzy Kalman filter (FKF) (section 3). The details of the ASFKF and EKF appear in sections 4 and 5, respectively. The process considered for simulation study has been discussed in section 6. Simulation results are presented in section 7, and the conclusions are presented in section 8.
around different operating steady-state values. The global system behavior is described by a fuzzy fusion of all linear model outputs. For a given input vector u(k), the global state and output of fuzzy model are inferred as follows:
x(k) ) N
hi(z(k))[Φi(x(k - 1) - xji) + Γi(u(k - 1) - uji) + xji] ∑ i)1 y(k) ) Cx(k)
(7) (8)
where the membership grades hi(z(k)) are defined as
2. Takagi-Sugeno (T-S) Fuzzy Model Consider a nonlinear system represented by the following nonlinear differential equations:
x˘ ) hf (x,u,d)
(1)
y ) gj(x,u,d)
(2)
Equation 1 describes a deterministic system evolution and can be obtained from the material and energy balances of the process under consideration. Equation 2 describes the relationships between the measurements and the state variables. To describe a discrete nonlinear system, eqs 1 and 2 can also be functionally represented in discrete form as
x(k) ) f(x(k - 1),u(k - 1),d(k - 1)) + w(k - 1)
(3)
y(k) ) g(x(k - 1),u(k - 1)) + V(k)
(4)
where x(k) is the system state vector (x(k) ∈ Rn), u(k) the system input/known deterministic input (u(k) ∈ Rm), d(k) the unmeasured disturbance/unknown input (d(k) ∈ Rq), w(k) the state noise (w(k) ∈ Rq), y(k) the measured variable (y(k) ∈ Rr), and V(k) the measurement noise (V(k) ∈ Rr). The parameter k represents the sampling instant, and the symbol f represents an ndimensional function vector. The random state noises can either represent random disturbances in the input variables or inaccuracies in the system model. We assume that measurements are made at discrete sampling instants with a sampling period T. Note that the d(t) term described in eq 1 is assumed to be piecewise constant for kT e t < (k + 1)T. A T-S fuzzy model has been proposed to represent a nonlinear system using locally linearized dynamic models.11 Two different methods for developing a T-S fuzzy model have been suggested in the literature, namely (i) the black box identification via fuzzy clustering technique16 and (ii) linearization of an existing nonlinear system around the centers of the fuzzy regions partitioning the state space.17 The T-S fuzzy model is nothing but a piecewise interpolation of local linear models through the membership function. The T-S fuzzy model is described by IF-THEN rules, which represent the local linear relations of the nonlinear system. The rule to describe the nonlinear system around an operating point (xji,uji) is as follows.11 Rule i (for i ) 1:N): If z1(k) is Mi,1 and ... zg(k) is Mi,g, then
hi(z(k)) )
µi(z(k)) µ(k)
(9)
g
Mij ∏ j)1
(10)
µi(z(k)) ∑ i)1
(11)
µi(z(k)) ) N
µ(k) )
N Note that hi(z(k)) ∈ [0,1] and ∑i)1 hi(z(k)) ) 1.
3. Fuzzy Kalman Filter (FKF) For a nonlinear dynamic system that is described by the T-S fuzzy model, a FKF can be designed to estimate the system state vector. For the FKF design, it is assumed that the linearized models are locally observable, (i.e., all (Ci, Φi) (i ) 1, ..., N) pairs are observable). The local linear models are uncorrelated and each of the N local model parameters is time invariant. We have assumed that the initial state and the sequence {w(k)} and {V(k)}are white, Gaussian, and independent of each other.
E[w(k)] ) 0
(12)
E[V(k)] ) 0
(13)
Cov[w(k)] ) Q(k)
(14)
Cov[V(k)] ) R(k)
(15)
Cov[w(k),w(j)] ) 0
(for j * k)
(16)
Cov[V(k),V(j)] ) 0
(for j * k)
(17)
Cov[w(k),V(j)] ) 0
(18)
xi(k) ) Φi(x(k - 1) - xji) + Γi(u(k - 1) - uji) + Ψiw(k - 1) (5) yi(k) ) Cixi(k) (6)
Equations 12 and 13 imply that the random variables w(k) and V(k) have a zero mean. The respective covariance matrices are given by eq 14 and 15. Equations 16 and 17 imply that, the disturbances at different times are not correlated, and, similarly, the measurement errors at different time instants are not correlated. Equation 18 stipulates that the disturbances and measurement errors are not cross-correlated.
where zj(k) are premise variables and Mij(k) are fuzzy sets. Φi, Γi, Ci, and Ψi are known time-invariant matrices of appropriate dimensions. In this paper, we have assumed that the local linear models can be developed by linearizing the nonlinear system
A local linear observer can be designed for each local linear dynamic model using Kalman filter theory. At an operating point, the local observer is associated with each fuzzy rule as given below.
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Rule i (i ) 1:N): If z1(k) is Mi,1and .... zg(k) is Mi,g, then
xˆ i(k|k - 1) ) Φi(xˆ (k - 1|k - 1) - xji) + Γi(u(k - 1) - uji) (19) yˆ i(k|k - 1) ) Cixˆ i(k|k - 1)
(20)
xˆ i(k|k) ) xˆ i(k|k - 1) + Ki(k)[(y(k) - yji) - yˆ i(k|k - 1)] (21)
x′i(k) ) Φ′i(x′(k - 1) - xji′) + Γ′i(u(k - 1) - uji) + Ψ′iw(k 1) (31)
Φ′i )
Vi(k) ) Ci(k)Pi(k|k - 1)CTi + R(k)
(23)
Ki(k) ) Pi(k|k - 1)CTi Vi-1(k)
(24)
Pi(k|k) ) (I - Ki(k)Ci)Pi(k|k - 1)
(25)
where Pi(k|k - 1) and Pi(k|k) are the covariance matrices of errors in predicted and updated state estimates of the ith local observer, respectively. The overall state estimation is a nonlinear combination of individual local observer outputs. The overall observer dynamics will then be a weighted sum of individual linear observers, given by
]
[]
(32)
[ ]
Φ i Υi Γ Ψi , Γ′i ) i , C′ ) [CCη ], Ψ′i ) 0 I 0 0
where
[ ]
xi(k) x′i(k) ) βi(k) ηi(k)
The Kalman gain matrix (Ki(k)) in eq 21 can be calculated from the following set of equations:
Pi(k|k - 1) ) ΦiPi(k - 1|k - 1)ΦTi + ΨiQ(k - 1)ΨTi (22)
[
yi(k) ) C′ix′i(k) + Vi(k)
and
[ ]
w(k) w(k) ) wβ(k) wη(k)
This augmented-state space model can used to design an ASFKF, using eqs 19-26. Note that the number of extra states that can be augmented should be less than or equal to the number of outputs, for the detectability of the augmented system.5 5. Extended Kalman Filter (EKF) The standard approach to estimate the system state of a nonlinear system (refer to eqs 3 and 4), using the well-known EKF, is as follows:
N
xˆ (k|k) )
hi(z(k)){xˆ i(k|k - 1) + ∑ i)1
xˆ (k|k - 1) ) f(xˆ (k - 1),u(k - 1))
(Ki(k)[(y(k) - yji ) - yˆ i(k|k - 1)]) + xji} (26)
4. Augmented-State Fuzzy Kalman Filter (ASFKF) The FKF described in the previous section will produce accurate state estimates, but only in the absence of a steplike disturbance at the output, slow drifts, or steplike disturbances at the input or states. To achieve offset free tracking in the presence of input and output disturbances, the process model must be augmented with artificially introduced input and/or output variables (β and η), which behave similar to integrated white noise. The resulting augmented model for each fuzzy rule can be written as
xi(k) ) Φi(x(k - 1) - xji) + Γi(u(k - 1) - uji) + Ψiw(k - 1) + Υiβi(k - 1) (27) βi(k) ) βi(k - 1) + wβ(k - 1)
(28)
ηi(k) ) ηi(k - 1) + wη(k - 1)
(29)
yi(k) ) Cixi(k) + Cηηi(k) + Vi(k)
(30)
where β and η are newly introduced additional linear states to describe the input and output disturbances (β ∈ Rs and η ∈ Rt). The vectors wβ and wη are zero mean white noise sequences with covariances Qβ and Qη, respectively (wβ ∈ Rs and wη ∈ Rt). Note that the model coefficient matrices [Υ, Cη] and noise covariance matrices [Qβ, Qη] are tuning parameters. The aforementioned set of equations can be combined into an augmented-state space model and is given as follows:
(33)
xˆ (k|k) ) xˆ (k|k - 1) + K(k)[y(k) - g(xˆ (k|k - 1))] (34) P(k|k - 1) ) F(k - 1)P(k - 1|k - 1)FT(k - 1) + Q(k - 1) (35) K(k) ) P(k|k - 1)CT(k)(C(k)P(k|k - 1)CT(k) + R(k))-1 (36) P(k|k) ) (I - K(k)C(k))P(k|k - 1)
(37)
where
F(k) )
δf |x(k) ) xˆ (k|k) δx
C(k) )
δg |x(k) ) xˆ (k|k) δx
Note that the calculation of the covariances and the gain of the EKF are same as those of the linear Kalman filter. Also, we have assumed that the initial state and the sequence {w(k)} and {V(k)} are white, Gaussian, and independent of each other (refer to eqs 12-18). In the EKF previously described, it is assumed that all the nonrandom inputs and parameters of the process are precisely known. However, the inputs to the process such as feed temperature, feed concentration, and feed flow rate are likely to change from their nominal values. Hence, to generate accurate state estimates, the state equations must be augmented with an additional differential equation, of the form
β˙ ) 0
(38)
where β is the vector of unknown inputs/parameters that must be estimated. Because there is no a priori knowledge about the aforementioned variables, it is assumed that these variables do
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Figure 1. Schematic of the continuously stirred tank reactor (CSTR) system. Table 1. Steady-State Operating Data process variable
(41)
y(t) ) Cx(t)
(42)
The state vector x(t) and input vector u(t) are given by x(t) ) [CA;T] and u(t) ) [qc]. The continuous linear state space model is obtained by linearizing the differential equations (eqs 39 and 40) around the nominal operating point (C h A and T h ). 7. Simulation Studies
normal operating condition
measured product concentration, CA reactor temperature, T coolant flow rate, qc process flow rate, q feed concentration, CA0 feed temperature, T0 inlet coolant temperature, Tc0 CSTR volume, V heat-transfer term, hA reaction rate constant, k0 activation energy term, E/R heat of reaction, -∆H liquid density, F, Fc specific heats, Cp, Cpc
In all the simulation runs, the process is simulated using the nonlinear first-principles model (eqs 39 and 40) and the true state variables are computed by solving the nonlinear differential equations using the differential equation solver in Matlab 6.5. The entire simulation has been performed with the following initial conditions:
0.0989 mol/L 438.7763 K 103 L/min 100.0 L/min 1 mol/L 350.0 K 350.0 K 100 L 7 × 105 cal/(min K) 7.2 × 1010 min-1 1 × 104 K -2 × 105 cal/mol 1 × 103 g/L 1 cal/(g K)
CA ) 0.09 and C ˆ A ) 0.1 T ) 437.44 and Tˆ ) 438.44
Table 2. Damping Factor and Undamped Natural Frequency at Different Operating Points operating point
damping factor
frequency (rad/s)
at qc ) 97, C h A ) 0.0795, T h ) 443.4566 at qc ) 100, C h A ) 0.0885, T h ) 441.1475 at qc ) 103, C h A ) 0.0989, T h ) 438.7763 at qc ) 106, C h A ) 0.1110, T ) 436.3091 at qc ) 109, C h A ) 0.1254, T h ) 433.6921
0.661 0.540 0.416 0.285 0.141
3.93 3.64 3.34 3.03 2.71
not change with time, which is the implication of the aforementioned equation. 6. Continuous Stirred Tank Reactor (CSTR) The first-principles model of the continuous stirred tank system and the operating point data (see Table 1) as specified in the paper by Huang et al.18 has been used in the simulation studies. The highly nonlinear CSTR process is very common in chemical and petrochemical plants. In the process considered for simulation study (as shown in Figure 1), an irreversible, exothermic reaction A f B occurs in a constant-volume reactor that is cooled by a single coolant stream. The CSTR system has two state variables, namely, the reactor temperature and the reactor concentration. The process is modeled by the following equations:
( )
dCA(t) q(t) -E ) (C (t) - CA(t)) - k0CA(t) exp dt V A0 RT(t)
( )
(39)
(-∆H)k0CA(t) dT(t) q(t) -E exp + ) (T (t) - T(t)) dt V 0 FCp RT(t) FcCpc -hA (Tc0(t) - T(t)) (40) q (t) 1 - exp FCpV c qc(t)FCp
{
x˘ ) f(x(t),u(t))
(
)}
The state space equation of the system is as follows:
The introduced variation in the coolant flow rate (see Figure 2) is same as that introduced in the paper by Huang et al.18 for validating the fuzzy model and the rigorous model. The dynamic behavior of the CSTR process is not the same at different operating points, and the process is, indeed, nonlinear. To verify this fact, the nonlinear system has been linearized at different operating points (see Table 2). The damping factor and undamped natural frequency have been obtained at different operating points and are reported in Table 2. From this table, it can be inferred that the process is highly nonlinear, because there is significant variation in the damping factor and undamped natural frequency. Figure 3 shows a comparison of the open-loop responses of the linear model developed around a nominal operating point values reported in Table 1 with the rigorous model. From the response, it can be concluded that the linear model developed around a nominal operating point is not be able to capture the dynamics (oscillatory behavior) of the CSTR process adequately. Hence, it is necessary to represent the nonlinear system using the T-S fuzzy model. 7.1. Fuzzy Model for the CSTR Process. The T-S fuzzy model is based on multiple local linear state space models that are weighted using a fuzzy membership function. In the T-S fuzzy model, the rule premises can be considered as an input space partition and the rule consequences can be considered as local models, valid in the rule’s partition. To combine multiple local linear models, one must devise a method for partitioning the operating state space. The choice of variables to be used to characterize the operating regimes are highly dependent on the problem.6 We observed that the dynamic behavior of the CSTR system changes significantly, depending on the operating point. Therefore, the coolant flow rate (premise variable) has been selected to partition the operating space of the CSTR system. Furthermore, fuzzy sets described by triangular membership functions on the domain of coolant flow rate are used to partition the operating space of the system into overlapping regions. As suggested by Schott and Bequette,19 we have selected the number of local models to be equal to that of the number of operating regimes over which the system is expected to operate. Furthermore, each of the local model parameters (and, consequently, part of the T-S fuzzy model) is determined by linearizing the nonlinear differential equation at different operating points.
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Figure 2. Variation in the coolant flow rate.
Figure 3. Comparison of the open-loop responses of the rigorous model and the linear model: (a) concentration and (b) temperature.
Figure 4. Membership function.
In the next step for developing a multiple model system, one must formulate a method for weighting the independent local
models. To express smooth transitions between adjacent regimes, the domain of each operating regime is characterized by a fuzzy
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Figure 5. Comparison of the open-loop responses of the rigorous model and the fuzzy model: (a) reactor concentration and (b) reactor temperature.
Figure 6. Evolution of true and estimated value of states: (a) reactor concentration and (b) reactor temperature. Table 3. Estimation Error for 25 Monte Carlo Simulation Runs Fuzzy Kalman Filter, FKF
Extended Kalman Filter, EKF
CA measurement CA and T T only
µ
T σ
10-4
2.7768 × 2.7845 × 10-4
µ 10-6
2.6025 × 2.6134 × 10-6
11.6628 11.6897
set membership function. The shape of the membership function has been selected in such a way that the weight for model i will be equal to 1 if operated exactly at the point at which the model has been devised. If the input value is between two linearization points, the output will consider only the two
CA σ 0.1472 0.1475
µ
T σ
10-4
1.9127 × 1.9214 × 10-4
10-5
3.577 × 3.5931 × 10-5
µ
σ
9.8652 9.8943
1.9190 1.9246
associated linear models, which implies that the remainder of the membership function must have a value of zero. In this work, we have intended to interpolate five models that have been generated at five different operating points. Because the operating space has been partitioned on a single
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Figure 7. Histogram of computation time per sampling instant: (a) fuzzy Kalman filter (FKF) and (b) extended Kalman filter (EKF).
parameter (coolant flow rate), there are only five rules in the rule base. The universe of discourse is divided into five intervals, which are defined by the linguistic variables Very low, low, medium, high, and Very high, with operating points.
0.1110, and T h ) 436.3091], [qc ) 109, C h A ) 0.1254, and T h) 433.6921]) have been obtained by discretizing the continuous state space model equations with a sampling time equal to 0.083 min.
To reduce the number of decision variables in the optimization scheme, each triangular membership function is designed symmetrical with a maximum value of one (1) at the associated linearization points. A numerical optimization technique has been adopted to obtain the optimal values of the fuzzy membership function parameters by minimizing the sum of the squares of the error differences between the process output and the fuzzy model output. Figure 4 shows the triangular membership functions that are used to partition the input space qc. The linear time-invariant discrete state space models (eqs 5 and 6) for five different operating points ([qc ) 97, C h A ) 0.0795, and T h ) 443.4566], [qc ) 100, C h A ) 0.0885, and T h ) 441.1475], [qc ) 103, C h A ) 0.0989, and T h ) 438.7763], [ qc ) 106, C hA )
Figure 5 shows the rigorous model and fuzzy model responses for step changes in the coolant flow rate qc and in the presence of an initial condition mismatch in both concentration and temperature variables. From the response, it is evident that the fuzzy model is able to capture the dynamics of CSTR process completely. 7.2. Fuzzy Kalman Filter for the CSTR Process. 7.2.1. Inferential Measurement. The performance of the FKF, when the reactor temperature alone is measured, is shown in Figure 6. We have assumed that the random errors are present in the measurement (T) as well as in the coolant flow rate (qc). The covariance matrices of measurement noise (state noise) are assumed as
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R ) ((0.05)2) and
Q ) [(0.05)2] The initial value of the error covariance matrix P(0/0) is assumed to be
P(0/0) )
(
(0.05)2 0 0 (0.05)2
)
for both the FKF and EKF. From Figure 6, it can be concluded that reasonably good estimates of the reactor concentration and
reactor temperature are obtained using FKF. Also, the performance of the FKF is similar to that of the EKF. 7.3. Performance Assessment. The performance of the proposed nonlinear state estimation scheme must be assessed through simulation, because stochastic systems are involved in these studies. For each case that is being analyzed, a simulation run that consists of NT trials (with the length of each simulation trail being equal to L) is conducted. In all the simulation trails, the sum of the squares of the estimation errors, which is nothing but the difference between the true value of the state variables and the estimated value of the state variables, has been obtained. The mean and standard deviation of the estimation errors, based on 25 Monte Carlo simulations for the FKF and EKF,
Figure 8. Evolution of true and estimated value of states in the presence of an input disturbance: (a) reactor concentration and (b) reactor temperature.
Figure 9. Evolution of true and estimated feed temperature of CSTR.
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Figure 10. Evolution of true and estimated value of states in the presence of an output disturbance: (a) reactor concentration (b) reactor temperature.
Figure 11. Evolution of sensor bias estimates for the CSTR.
are reported in Table 3. For FKF and EKF, identical realization of state and measurement noises have been used in all the simulation trials. From Table 3, it can be observed that the estimates obtained by the FKF are as good as those obtained by the EKF in all cases. Also, the FKF helps to reduce the number of computations needed, compared to the conventional EKF. That is, in the EKF, for each time k, all the system matrices must be calculated using the related Jacobians, as well as the updated state estimates at time k. Also, in EKF, the nonlinear differential equations must be numerically integrated to obtain the predicted estimates of the state variables. On the other hand, in the FKF, although more matrices are needed, all of them have constant values, which limits the calculation to (i) the determination of weights, which will be provided by the operating region membership functions, and (ii) state propagation calculations of each model
using the appropriate matrices and a weighted average of the local linear model outputs. It is not necessary to calculate the Jacobians and the numerical integration of nonlinear differential equations; therefore, the proposed FKF approach has better implementation capabilities than the EKF approach. To test the efficacy of the EKF and FKF algorithms, the computation time per sampling instant in a single simulation trial (the length of the simulation trial is 900) has been presented in the form of a histogram. From Figure 7a, it can be concluded that the computation time per sampling instant of the FKF algorithm is always
