Chapter 6 · Chapter 6: Steady-State Data Reconciliation with Model Uncertainties. Basic Concepts in Data Reconciliation ... 1.009 0.9870 ... 0.8765 0.776 0.1007

Basic Concepts in Data ReconciliationBasic Concepts in Data Reconciliation

© North Carolina State University, USA © University of Ottawa, Canada, 2003

Chapter 6:Steady-State Data Reconciliation

with Model Uncertainties



6.1 Models with Uncertainties

In the previous chapters, the models employed in the DR were considered exact. That is to say, the DR algorithms force the reconciled data to satisfy the models, , exactly. However, perfect models rarely exist and the reconciliation of measured values will more likely be a compromise between inaccuracies in both measurements and process models. In such cases, the reconciled values of the process variables should be obtained by:

minimizing (6.1)subject to

where is a C×1 vector of model residuals that are the estimates of random model errors .

CHAPTER 6

Steady-State Data Reconciliationwith Model Uncertainties

ˆ ˆ( , ) =f y z 0

Tˆ ˆ ˆ ˆ( , ) ( ) ( )J −= − −1y z y y V y yˆˆ ˆ ( , ) =f y z δ

ˆ ˆ( , ) ≥g y z 0

δ( , ) =f y z δ




The process models employed in the DR formulated by (6.1) have uncertainties that account for inaccuracies generated by approximations and thus reflect the modeling error. The variance and covariance of the model errors, indicate the accuracy of the models. For linear models, the variance and covariance of the model errors can be obtained analytically. But for nonlinear models, they are calculated either by a Monte Carlo simulation, or, neglecting correlations between the model errors, the variances can be approximated using a first order Taylor’s expansion around the measurement values to give:

(6.2)

where σj is the variance of the measurement yj .

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Cov( ), δ

2

2

1σ

=

⎧ ⎫⎛ ⎞∂⎪ ⎪⎜ ⎟= ⎨ ⎬∂⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑M

ii j

j j

fVy

y




In the Monte Carlo simulation, a series of normally distributed noise for each variable, yj, having the same variance as the measurement, are generated by a computer program. Then the functions, f(y, z), are calculated, along with their variance and covariance.

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1 2 My y ... y0.0567 0.0087 ... 0.12341.009 0.9870 ... 0.87650.776 0.1007 ... 0.5678

−

− −M M M M

1 2 C...0.0897 0.4329 ... 0.0976-0.5429 0.5732 ... 0.87650.0076 0.0019 ... 0.2345

δ δ δ−

−− −

M M M M

δ = f(y, z)

Generated noise vectors Calculated model errors

Calculate V(δ1), V(δ2), Cov(δ1, δ1), ……



6.2 Data Reconciliation Algorithm with Uncertain Models

When there are uncertain models, the reconciled data should be obtained such that it simultaneously minimizes both the measurement and model errors. Therefore, the data reconciliation problem (6.1) becomes:

minimizing (6.3)subject to

where Ω is the covariance matrix of the model errors.

The data reconciliation algorithm formulated by (6.3) considers the reconciled data as being a compromise between the measurements and models. The variance of the models can also be treated as tuning parameters of the algorithm. In other words, if we have a lot of confidence about the models used,

CHAPTER 6


T T 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( ) ( ) ( , ) ( , )J − −= − − + Ω1y z y y V y y f y z f y zˆ ˆ( , ) ≥g y z 0



6.2 Data Reconciliation Algorithm with Uncertain Models

then we can set the variance at a small value, otherwise, we set the variance at a large value. There are two limiting cases:

(i) The variance of the models is very large compared to the variance of the measurements, then the reconciled data found by (6.3) will be the actual measurements;

(ii) The variance of the models is very small compared to the variance of the measurements, then the reconciled data found by (6.3) will behave like the models are exact, which is like using the conventional data reconciliation algorithm formulated by Equation 1.2.

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6.3 Solutions to Data Reconciliation with Uncertain Models

Note that the functions for the uncertain models in Equation 6.3 can be linear or nonlinear. For the simplest case, all the variables are measured and the model functions are linear; thus the data reconciliation problem is formulated as:

minimizing (6.4)

The conditions for the minimum of (6.4) are Rewriting the objective function gives:

Taking the first derivative, we have:

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T T 1ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )J − −= − − + Ω1y y y V y y Ay Ay

.ˆJ∂ =

∂0

y

-1 T 1ˆ ˆ2 ( ) 2ˆJ −∂ = − − + Ω =

∂V y y A Ay 0

y

T T 1ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )J − −= − − + Ω1y y y V y y y A A y

(6.5)




Rearranging Equation 6.5 gives:

which is the solution to the DR with linear uncertain models.♣For the cooling water network illustrated in Figure 1.3, suppose the true values of the flows satisfy mass imbalances for each node because of fluctuations in the plant. The variance of the imbalances are obtained as:

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T 1 1 1ˆ ( )− − − −= Ω + 1y A A V V y (6.6)

1.165 0 0 00 0.785 0 0

.0 0 0.414 00 0 0 2.147

⎡ ⎤⎢ ⎥⎢ ⎥Ω =⎢ ⎥⎢ ⎥⎣ ⎦




Using Equation 6.6, the reconciled data constrained by the mass imbalances can be obtained by the following MATLAB code:**********************************************

y=[110.5;60.8;35.0;68.9;38.6;101.4];

V=[0.6724 0 0 0 0 0;0 0.2809 0 0 0 0;0 0 0.2116 0 0 0;0 0 0 0.5041 0 0;0 0 0 0 0.2025 0;0 0 0 0 0 1.44];

A=[1 -1 -1 0 0 0;0 1 0 -1 0 0;0 0 1 0 -1 0;0 0 0 1 1 -1];

O=[1.165 0 0 0;0 0.785 0 0;0 0 0.414 0;0 0 0 2.147];

yhat=inv(A'*inv(O)*A+inv(V))*inv(V)*y

*********************************************

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The calculation results are listed here in Table 6.1.

Table 6.1: Results of data reconciliation with imbalanced models for a cooling water network

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102.60

37.88

66.52

36.76

63.46

106.73

Reconciled(kt/h)

1.20

-0.72

-2.38

1.76

2.66

-3.77

Adjustment(kt/h)

101.46

38.65

68.94

35.03

60.82

110.51

Raw(kt/h)

Stream No.

FI FI FI FI

FI FI1 3

2 4

5 6CWS CWR

Figure 1.3




It is worth noting that the reconciled data for the flows in Table 6.1 don’t satisfy the mass balances for each plant. However, if we want to force the reconciled data to exactly satisfy the mass balances, we can artificially use very small values for thevariance of the imbalances, and then using Equation 6.6, the reconciled flows will exactly match mass balances for each plant. On the other hand, if we do not have any confidence in the mass balances, we can use very large values for the variance of the imbalances, and then the reconciled data will be equal to the raw measurements.

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The variances of the model errors are treated as tuning parameters of the DR algorithm. The change in the reconciled data with the changing of the tuning parameters is illustrated in Figure 6.1.

Figure 6.1: Change in reconciled data with the change of the model variance

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Raw measurements Reconciled by exact models

Reconciled data

+∞ 0Variance of models




If there exist unmeasured variables in the constraints or if theuncertain models used in the DR are nonlinear, the problem can be formulated by Equation 6.3 and solved by such as the quasi-Newton method. As mentioned in Chapter 1, data reconciliation techniques can also be used to estimate model parameters. The unknowns in the vector, , in (6.3) can be unmeasured variables or model parameters.

It is worth noting that the number of model equations used in (6.3) must be larger than the number of unmeasured variables and model parameters, so that the DR algorithm has sufficient redundancies in the estimation.

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z



6.4 Quiz

Question 1:The reconciled data for DR constrained by uncertain models(a) will satisfy exact models.(b) will be equal to the raw measurements.(c) is a compromise between the measurements and models.(d) is all of the above.

Question 2:In DR which has infinite variance for its model uncertainties, the reconciled data then(a) will satisfy exact models.(b) will be equal to the raw measurements.(c) is a compromise between the measurements and models.(d) is all of the above.

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6.4 Quiz

Question 3:For the model, y - x = δ, the variance for y and x are 2.0 and 2.2 respectively. The variance of the model errors δ is(a) 2.2 - 2.0 = 0.2.(b) 2.0 - 2.2 = –0.2.(c) 2.2.(d) 2.0 +2.2 = 4.2.

Question 3:For the model, y2 - x = δ, the variance for y and x are 2.0 and 2.2 respectively. The variance of δ is(a) 2.02 + 2.2 = 6.2.(b) 2.02 - 2.2 = 1.8.(c) calculated by Monte Carlo simulation.(d) approximated by linearizing the function y2 - x.

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6.5 Suggested Readings

Kao, E.P.C. (1997). “An Introduction to Stochastic Processes”. Duxbury Press, Belmont, Calif., USA.

Maquin, D.; Adrot, O. and Ragot, J. (2000). Data reconciliation with uncertain models, ISA Transactions, 39, 35-45.

Mandel, D. ; Abdollahzadeh, A.; Maquin, D. and Ragot, J. (1998). Data reconciliation by inequality balance equilibration: a LMI approach. International Journal of Mineral Processing, 53, 157-171.

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Chapter 7:Dynamic Data Reconciliation



7.1 Formulation of Dynamic Data Reconciliation

In the previous chapters, the data reconciliation problem was discussed for processes at steady state. However, many chemical processes are intrinsically dynamic and disturbances frequently occur. Consequently, it is desirable to develop dynamic data reconciliation (DDR) strategies for dynamic processes, so that at every sampling instant more accurate and reliable process data is available for real-time control and optimization.

In steady-state data reconciliation, only one set of data is used at the current sampling time. Steady-state data reconciliation uses the spatial redundancy of the measurements. There is no previous information used to estimate the current state of the process.

CHAPTER 7

Dynamic Data Reconciliation




Along with the spatial redundancy of measurements, when a process is continuously sampled at discrete points in time, we also have temporal redundancy. The temporal redundancy of measurements is usually used for three types of state estimation as shown in Figure 7.1.

Figure 7.1: Types of estimation using temporal redundancy of measurements.

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tData used for estimation

Filtering

Data used for estimationPrediction

Data used for estimation

Smoothing




When the measurements prior to time t, including the measurement at time t, are used to estimate the current state of the process, the estimation is called filtering.

When the measurements prior to time t are used to predict the process variables at time t, t+1, t+2, etc., the estimation is called prediction.

When the measurements prior to time t, and after time t, are used to estimate the process variables at time t, the estimationis called smoothing.

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Filtering and prediction can be applied on-line, whereas smoothing can only be used off-line.

♣

Dynamic data reconciliation uses both spatial and temporal redundancy to estimate the state of a process, so that at every sampling instant, more accurate and reliable process data are available. For discretely sampled measurements, the DDR problem is typically formulated as:

minimizing (7.1)

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pt

t t t t t tt=0

ˆ ˆ ˆ ˆ( , ) [( ) ( )]TJ −= − −∑ 1y z y y V y y




subject to

where yt is an M×1 vector of observed values for measured process variables at time t ( t = 0, 1, 2, …, tp), is the vector of reconciled values for the measured process variables at time t, is an N×1 vector of estimates for unmeasured variables or model parameters at time t, V is an M×Mcovariance matrix for the measurements, f is a C ×1 vector of differential model equality constraints, g is an S ×1 vector of algebraic model equality constraints, and h is an E ×1 vector of inequality model constraints including simple upper and lower bounds.

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ˆd (t) ˆ ˆ[ , (t), (t)]dt

=yf y z 0

ˆ ˆ[ (t), (t)] =g y z 0

ˆ ˆ[ (t), (t)] ≥h y z 0

ty

tz




The dynamic data reconciliation formulated by Equation 7.1makes the reconciled data satisfy exact models. It applies a moving window approach where tp represents the window width. At each sampling time, only the measurements within the window width, t = 0, 1, 2, …, tp, are reconciled, and only the reconciled values for the current time measurements are used for on-line monitoring and control. Other reconciled values of measurements prior to the current time are discarded. Then the moving window moves forward for the next time step.

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Moving window

Timetp0

Window width=10



7.2 Linear Dynamic Data Reconciliation

Often, process dynamics can be described or approximated by linear differential equations. For a discretely sampled system, it is convenient to use differential equations. The dynamics ofa linear time invariant (LTI) process can be written as:

where xt is an M×1 vector of the true values of state variables at time t, ut-1 is a P×1 vector of manipulated inputs or disturbances to the process at time t-1, and w is an M×1vector of random variables that represent model errors (accounting for model mismatch and unknown disturbances corrupting the process). A and B are matrices of appropriate dimensions whose coefficients are known at all times.

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t t-1 t-1 + + =x Ax Bu w (7.2)




In dynamic DR, at each sampling time, both model predicted values found by Equation 7.2 and measurements are available. Under the assumption that measurement and model errors are normally distributed with zero mean and known variance, the estimates (reconciled or adjusted values) of the measured process variables can be obtained by solving an optimization problem that simultaneously minimizes the weighted sum of square measurements and model errors:

minimizingsubject to

(7.3)

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T 1 T 1t t t t t m,t t m,t tˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )J − −= − − + − Ω −y y y V y y x y x y

l t uˆy y y≤ ≤m,t t-1 t-1ˆ + =x Ay Bu




where xm,t denotes the M×1 vector of model predicted values.

The manipulated inputs or disturbances ut-1 in (7.2) are assumed to always be exactly known. In Equation 7.3, Ω is an M ×M variance-covariance matrix of the model predictions.

The reconciled values at each sampling time, t, can be obtained analytically by Equation 7.3. Taking partial derivatives of the objective function with respect to and setting it equal to zero results in:

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ty

-1 -1t t m,t t

t

ˆ ˆ2 ( ) 2 ( )ˆJ V y y x y 0y

∂ = − − − Ω − =∂




Rearranging the above equation gives:

In Equation 7.4, the variance and covariance of the measurements in V can be obtained from the raw measurements. However, it is sometimes difficult to obtain the variance and covariance of the model predictions in Ω. This is not only because of model inadequacies, but also because of various kinds of unknown disturbances corrupting the process. As a result, we can treat the ratio of the variance of the modelpredictions, σm

2, to that of the process measurements, σ2, as tuning parameters in the DDR algorithm.

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-1 1 1 1 -1t t m,tˆ ( ) ( )y V V y x− − −= + Ω + Ω (7.4)




When σm2/ σ2 is used as a tuning parameter for the DDR

algorithm, the locus of the reconciled values typically follows an asymptotic curve as a function of σm

2/ σ2 . Figure 7.2, for example, illustrates the locus of a reconciled value with a change in σm

2/ σ2. The measured value for the liquid level of a tank is 60.0 cm, and the model predicted value of the liquid level is 60.5 cm, at time t. The reconciled value is a point onthe curve whose location depends on the values of σm

2/ σ2 , and is bounded by the measured and predicted values. There are two limiting cases: i) σm

2/ σ2 = 0 which corresponds to an exact model, making the reconciled value equal to the model predicted value; and ii) σm

2/ σ2 = ∝ indicating that the level of confidence on model predictions is poor, and the reconciled values approach the measured values.

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Figure 7.2: Locus of reconciled data as a function of the ratioof model to measurement uncertainties in the DDR algorithm

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0 2 4 61 3 5σ2

m/σ2

59.8

60

60.2

60.4

60.6

60.8

Liqu

id le

vel,

cm

Model predicted

Measured

Reconciled




In process control, the presence of measurement noise will often deteriorate controller performance. It is therefore necessary to attenuate the measurement noise before it reaches the controllers. The linear DDR strategy introduced can be applied inside a control loop as a “filter” to reduce the propagation of measurement noise.

The DDR algorithm as an integral part of conventional PID control loops is illustrated in Figure 7.3. The raw measurements of controlled variables are first reconciled by the DDR algorithm, and then used by the controller to calculate the control moves.

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Figure 7.3: Application of a DDR algorithm inside a PID control loop

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Process

Noise

DigitalControllers

Raw measurements

Dynamic DataReconciliation

Reconciled data

Control Valve

Manipulated variables

Controlled variables




To illustrate dynamic DR strategies inside a control loop, consider a liquid storage tank process with a PI controller represented in Figure 7.3.

Figure 7.3: Schematic diagram of a storage tank process

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LC

FI

Feed

Outlet




The diameter and height of the tank are 1.0 m and 1.2 m, respectively. A PI controller is used to regulate the liquid level of the tank by manipulating the outlet flow. The feed flow to the tank is measured but not controlled. The sampling interval is 1 minute. The measurement errors are assumed to be Gaussian white noise, and the standard deviations for the measurements of the feed flow rate and the liquid level are 0.036 m3/h and 0.012 m, respectively. The PI controller parameters are Kc = -1.87m2/h and τI = 30.0 min.

The dynamics of the storage tank process corresponding to a 20% step change in feed flow rate was simulated, and the simulation results are presented in Figure 7.4.

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Figure 7.4: Dynamics of a storage tank for a 20% step change in feed flow at time t = 90 min

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0 30 60 90 120 150 180 210 240 270 300 330 360Time, min

60

70

80

55

65

75

Tank

leve

l, h,

cm

1.8

2

2.2

1.7

1.9

2.1

Out

let f

low

, Fo,

m3 /

h

PI controllerKC = -1.87 m2.h-1

τΙ = 30.0 min




As shown in Figure 7.4, the measurements of the liquid level have noise. The manipulated variable, the outlet flow, exhibitsconsiderable variations because the noisy measurements affect the calculated control moves.

To attenuate the impact of the measurement noise, a DDR algorithm was embedded into the control loop to reconcile the liquid level before calculating the control moves. The reconciled liquid level at each sampling time is obtained by:

minimizing (7.5)

subject to

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2 2 2t i,t-1 i,t-1 t t m,t t2 2 2

F h m

1 1 1ˆ ˆ ˆ ˆJ(h )= (F -F ) (h -h ) + (h -h )σ σ σ

+

m,t t-1 i,t-1 o,t-1Δtˆ ˆh =h + (F -F )A

t maxˆ0<h <h




where the model constraint is the mass balance around the tank in discrete form, A is the cross-section of the tank, and Δt is the discrete sampling time.

The ratio of the variance of the model predicted values to the variance of the measurements, σm

2/ σ2, is given as 0.25 in testing the performance of the DDR algorithm. The same feed flow perturbation is used to evaluate the controller performance with the DDR algorithm embedded in the control loop.

As shown in Figure 7.5, the reconciled data for the liquid level was less noisy than the raw measurements and closer to their true values. The manipulated variables displayed much smaller variations when the reconciled data was used to calculate the controller action.

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Figure 7.5: Dynamics of storage tank with embedded DDR inside a PI control loop, for a 20% step change in feed flow at time, t = 90 min

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0 30 60 90 120 150 180 210 240 270 300 330 360Time, min

60

70

80

55

65

75

Tank

leve

l, h,

cm

1.8

2

2.2

1.7

1.9

2.1

Out

let f

low

, Fo,

m3 /

h

PI controllerKC = -1.87 m2.h-1

τI = 30.0 min

Without DDRWith DDRTrue




This example demonstrates that the DDR algorithm developed can efficiently attenuate measurement noise, resulting in enhanced controller performance.

♣In the application of DDR, dynamic models have to be developed. For the simple process of a storage tank, a fundamental model, the mass balance, was used. But for complex processes, for example, a distillation column, it is often difficult or impractical to develop fundamental dynamic models. Therefore, empirical input-output models have to be identified. Different techniques are available for developing an empirical dynamic model. These techniques are described under model identification in several textbooks (Ljung, 1987).

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7.3 Dynamic Data Reconciliation with Kalman Filter

For convenience, we rewrite the linear time invariant, discrete,dynamic model (7.2) as:

where xt is the M×1 vector of the true values of process variables at time, t, ut-1 is the P×1 vector of process input variables (manipulated and disturbances) that are assumed to be exactly known at time, t-1, A is an M×M matrix whose coefficients are known at all times, B is an M×P matrix whose coefficients are also known at all times, and w is an M×1 vector of random variables which are assumed to be Gaussianwhite noise with zero mean and known variance.

Equation 7.6 describes the dynamic evolution of the stochastic process variables.

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t t-1 t-1= + +x Ax Bu w (7.6)




The random variable vector, w, in this equation considers the following factors that affect the process:(i) random fluctuations of external disturbances, (ii) random errors in manipulated inputs resulting from electrical noise during control signal transmission and imprecise actuator positioning, and(iii) modeling error during model identification.

If all the state variables are measured, the measurement model can be written as:

where ε is the M×1 vector of random measurement error which is also assumed to be Gaussian white noise with zero mean and known variance.

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t t= +y x ε (7.7)




The two random vectors, w and ε, are assumed to uncorrelated. The statistical properties of these two random vectors are summarized as:

Given Equations 7.6 and 7.7, the optimal estimation of the process variables can be given by the Kalman Filter.

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[ ] [ ]E E= =w ε 0

[ ]Cov =w R

[ ]Cov =ε V

i j i jCov w ,w Cov , 0 i jε ε= = ≠⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

i jCov w , 0ε =⎡ ⎤⎣ ⎦




The detailed derivations of the Kalman filter can be found in the textbook “Introduction to Optimal Estimation” (Kamen and Su, 1999). The Kalman filter has a recursive form, and its equations are given by:

where is the model predicted values by the deterministic model:

is called the Kalman gain and is given as:

The matrix in (7.10) is the covariance matrix of . Thematrix is given by:

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- -

t t t t tˆ ˆ ˆ( )= + −x x K y x

-

tx

-

t t-1 t-1ˆ ˆ .= +x Ax Bu

- - 1

t t t( ) .−= +K P P VtK

(7.8)

(7.9)

(7.10)

-

tP -

tx-

tP




where is the covariance matrix of , which is given by:

The calculations of the recursive Equations 7.8~12 can be summarized as:

Step 1 Filter Initialization:

Set t=1

Step 2 Time update:

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- T

t t-1= +P AP A R

t-1P t-1x-

t-1 t-1 t-1( )= −P I K P

(7.11)

(7.12)

0 0 0ˆ ˆE( ), guess of =x x x

T

0 0 0 0 0ˆ ˆ( >0), or guess of E[( )( ) ]λ λ= − −P I x x x x




Step 3 Measurement update:

Step 4 Time increment:Increment t and repeat step 2.

It is important to note that the estimates in the LTI case generated by a Kalman filter have been proven to be unbiased and have a minimum mean squared error.

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-

t t-1 t-1ˆ ˆ .= +x Ax Bu

- T

t t-1= +P AP A R

- - 1

t t t( ) .−= +K P P V- -

t t t t tˆ ˆ ˆ( )= + −x x K y x

-

t t t( )= −P I K P




In addition, the matrices , , and are independent of the measurements, , but are dependant upon the increment of time, t. They will become constants with the increasing of time, therefore, they can be calculated off-line.

Recall the example of the storage tank process. If we neglect the reconciling of the feed flow, and solve the optimization problem (7.5), the reconciled liquid level, at each sampling time, t, can be obtained by the explicit equation:

Equation 7.13 can be written in the form:

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-

tP tP tKty

2 2m t h m,t

t 2 2h m

σ h σ hh

σ σ+

=+ (7.13)

2m

t m,t t m,t2 2h m

σh h (h h )σ σ

= + −+

(7.14)




Compare the form of Equation 7.14 to Equation 7.8. The term in (7.14) is equivalent to the Kalman gain in (7.8).

If we model the dynamics of the storage tank as a stochastic difference equation, we obtain:

where w is Gaussian white noise with zero mean and known variance of R=3.6×10-5 m2 .

The measurement model of the liquid level is:

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2 2 2m h mσ /(σ σ )+

t t-1 i,t-1 o,t-1Δth =h + (F -F ) wA

+

t ty h ε= +

(7.15)

(7.16)




where ε is the white noise of the liquid level measurements with zero mean and known variance V=1.44 ×10-4 m2 .

The optimal estimation for the liquid level of the tank, given the two models, (7.15) and (7.16), can be obtained by the Kalmanfilter:

where is the model predicted value given by:

and K is the Kalman gain. The value of the Kalman gain in this case can be calculated recursively through steps 1 to 4 as shown in Table 7.1.

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t m,t t m,th h K(y h )= + −

m,th

m,t t-1 i,t-1 o,t-1Δtˆh =h + (F -F )A




Table 7.1: Calculation of Kalman gain for the storage tank

The Kalman gain converges at K=0.39, so Equation 7.14 can be written as:

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0.3909.21×10-55.61×10-570.3909.20×10-55.60×10-560.3899.17×10-55.58×10-550.3879.10×10-55.50×10-540.3828.89×10-55.29×10-530.3678.35×10-54.75×10-520.2424.60×10-53.49×10-51

1.0×10-50KtPt

-PtIterationA guess of an initial

value

convergence

t m,t t m,th h 0.39(h h ).= + −



7.4 Nonlinear Dynamic Data Reconciliation

When nonlinear dynamic models are used as constraints for Equation 7.1 in dynamic data reconciliation, it is generally not possible to analytically obtain a discrete form of the differential constraint equations to represent the process. Liebman et al. (1992) proposed an approach, called nonlinear dynamic data reconciliation (NDDR) to solve this problem. The NDDR method discretizes the nonlinear differential equations by the method of orthogonal collocation on finite elements, such that the differential equations are transformed into algebraic equations at each sampling time. After the discretization, the remaining problem is to minimize the least-squares objective function with the constraints of algebraic equalities and inqualities that can be efficiently solved by the nonlinear programming techniques at each sampling time.

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7.4 Nonlinear Dynamic Data Reconciliation

An alternative method to solving the nonlinear dynamic data reconciliation problem is to use the extended Kalman filter (EKF). This extension typically involves replacing the nonlinear model equations with first-order approximations around the current estimates. Detailed descriptions about the EKF method can be found in the textbook “Introduction to Optimal Estimation” (Kaman and Su, 1999).

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7.5 Quiz

Question 1:If a process is described by a stochastic dynamic model, the term, w, in Equation 7.2 considers(a) modeling error.(b) random external perturbations.(c) errors in input variables.(d) all of the above.

Question 2:Measurement noise in a control loop(a) deteriorates the performance of the controller.(b) causes high-frequency oscillations of manipulated variables.(c) masks the true values of controlled variables.(d) all of above.

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7.5 Quiz

Question 3:In the application of Kalman filter, if K → I, then(a) the output of the filter approaches model predicted values.(b) the output of the filter approaches raw measurements.(c) it indicates that the variance of model errors are very very large compared to the measurement variance.(d) all of the above.

Question 4:The Kalman gain in a Kalman filter(a) is calculated recursively.(b) can be calculated off-line.(c) converges after a certain amount of iterations.(d) all of the above.

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7.6 Suggested Readings

Ljung, L. (1987). “System Identification – Theory for the User”. Prentice-Hall, Englewood Cliff, N.J.

Kamen, E.W. and Su, J.K. (1999). “Introduction to OptimalEstimation”, Springer, London.

Bai, S. (2003). “Assessment of Controller Performance with Embedded Dynamic Data Reconciliation”. Master thesis, University of Ottawa, Canada.

Liebman, M.J.; Edgar, T.F. and Lasdon, L.S. (1992). Efficient data reconciliation and estimation for dynamic process using nonlinear programming techniques. Comp. Chem. Engng, 16, 963-986.

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