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Basic Concepts in Data ReconciliationBasic Concepts in Data Reconciliation

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

Chapter 4:Steady-State Data Reconciliation for

Bilinear Systems



4.1 Bilinear System

In industrial plants, process streams often contain several species or components. Besides the stream flow rates measured, the compositions of some of the streams are also measured. If we wish to simultaneously reconcile flow and composition measurements, then component mass balanceshave to be included as constraints in the data reconciliation problem. Since these constraints contain the products of flow rate and composition variables, the term bilinear data reconciliation is consequently used.

Another bilinear DR problem comes from reconciling flow rates and temperatures, in which the enthalpy of the stream is only a function of the temperature. In this case the energy balancesare the constraints in the optimization.

CHAPTER 4

Steady-State Data Reconciliation for Bilinear Systems



4.1 Bilinear System

To illustrate a typical bilinear data reconciliation problem, consider a simple example of reconciling flows and compositions of a binary distillation column which is commonly used in chemical industries, as shown in Figure 4.1.

Figure 4.1: Schematic diagram of a binary distillation column

CHAPTER 4


Feed flow, F1

Distillate flow, F2

Bottom flow, F3

x1,1x1,2

x2,1x2,2

x3,1x3,2



4.1 Bilinear System

The measurements of the feed, distillate, and bottom flows, along with their compositions are listed in Table 4.1.

Table 4.1: Measurements of a binary distillation column

CHAPTER 4


24.4488.23F3

0.00020.0197x3,1

0.00970.9748x3,2

23.9478.40F2

0.00940.9410x2,1

0.00050.0501x2,2

0.00520.004854.8

Standard deviation

0.5170x1,2

0.4822x1,1

1095.47F1

Raw MeasurementVariables

F1

F2

F3

x1,1x1,2

x2,1x2,2

x3,1x3,2

Figure 4.1



4.2 Solution to Bilinear Data Reconciliation

In Table 4.1, the flow measurements are in kg/h, and the compositions are in mass fraction.

At steady-state, the bilinear data reconciliation for the distillation column can be formulated as:

CHAPTER 4


1 1 1 1 1T T

F x

2 2 2 2 2

ˆ ˆˆ ˆ ˆ( , ) ( ) ( ) ( ) (ˆ ˆ

J − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − − + − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 1x x x x xF F F V F F V

x x x x xC,1ˆ =AF 0C,2ˆ =AF 0

Minimize (4.1)

subject to

1 2ˆ ˆ+ =x x 1

where F is the vector of the measured flow rates, , Vf and Vx are the variance matrices corresponding to the measurements of the flow rates and the compositions, respectively, and x1 and x2 are the composition vectors

[ ]T

1 2 3F F F=F




corresponding to components 1 and 2, and they are defined as:

and are the vectors of the component flow rates, defined as:

,

CHAPTER 4


[ ]1,1 1,2

1 2 2,1 2,2

3,1 3,2

x xx xx x

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

x x1

2

3Component 1 2

Stream

C,1F C,2F

C,1

1 1 1,1

C,1 C,1

2 2 2,1

C,1

3 3 3,1

ˆ ˆ ˆF Fxˆ ˆ ˆ ˆF F x

ˆ ˆ ˆF F x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

F

C,2

1 1 1,2

C,2 C,2

2 2 2,2

C,2

3 3 3,2

ˆ ˆ ˆF Fxˆ ˆ ˆ ˆF F x

ˆ ˆ ˆF F x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

F




and are the component mass balances of components 1 and 2 , respectively. A is the incidence matrix, . are the normalization equations for the compositions in each flow where 1 denotes the vector, .

The above formulation of the DR problem is in terms of flows rates and composition variables. Alternatively, we can also formulate it in terms of overall flow and component flow variables. In this case, the normalization equations can be written as:

It should be noted that the constraints are linear in terms of the flow variables, .

CHAPTER 4


C,1ˆ =AF 0 C,2ˆ =AF 0

[ ]1 1 1= − −A1 2

ˆ ˆ+ =x x 1

[ ]T1 1 1=1

C,1 C,2ˆ ˆ ˆ− − =F F F 0

C,1 C,2ˆ ˆ, and F F F




Although, the constraints are linear in terms of the flow variables, the objective function (4.1) contains composition variables. In order to overcome this problem, the objective function is modified to minimize the sum of squares of the adjustments made to flows and component flow variables, by:

minimizing (4.2)

subject to

where V is the variance matrix of the “measurements” having the form:

CHAPTER 4


T

C,1 C,1 C,1 -1 C,1 C,1

C,2 C,2 C,2 C,2 C,2

ˆ ˆ ˆˆ ˆ ˆ( )ˆ ˆ ˆ

J⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

F F F F FF F F V F FF F F F F

C,1ˆ =AF 0C,2ˆ =AF 0

C,1 C,2ˆ ˆ ˆ− − =F F F 0




(4.3)

In (4.3), is the variance matrix of the “measured” component flows. However, the elements in can’t be directly obtained from the raw measurements.

The estimation of the variance of the component flows in (4.3) can be obtained by linearizing the term, Fk xi,j , using a first order Taylor’s series:

(4.4)

CHAPTER 4


CFV

CFV

* * * * * *

k i,j k i,j i,j k k k i,j i,jF x F x x (F F ) F (x x )≈ + − + −* 2 * 2

k i,j i,j k k i,jVar(F x ) (x ) Var(F ) (F ) Var(x )≈ +

C

F

F

⎡ ⎤= ⎢ ⎥⎣ ⎦

V 0V

0 V




where the superscript, * , denotes the value of the measurements.

The modified objective function (4.2), subject to the linear constraints in the flow variables, results in a linear DR problem. The estimates of the flows can be obtained analytically by Equation (2.7).

After the data reconciliation of the flow estimates, the values of the compositions can be obtained by:

(4.5)

CHAPTER 4


C,1

1,1 1 1

C,1

2,1 2 2

C,1

3,1 3 3

ˆx F /Fˆx F /F ,ˆx F /F

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

C,2

1,2 1 1

C,2

2,2 2 2

C,2

3,2 3 3

ˆx F /Fˆx F /F .ˆx F /F

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦




For the illustrative example of the distillation column, the vector of the stream flows is given by:

The “raw measurements” of the component flows are given by:

Using Equation (4.4), the variance matrix is calculated as:

CHAPTER 4


1095.47478.40488.23

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

F

1 1,1

C,1

2 2,1

3 3,1

Fx 528.24F x 450.17F x 9.62

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

F1 1,2

C,2

2 2,2

3 3,2

Fx 566.36F x 23.97F x 475.93

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

F




Write the linear constraint equations in Equation (4.2) in the matrix form:

CHAPTER 4


⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

3003.0571.2 0

595.4725.0

V 526.00.241

0 835.11.491

588.2

C,1

C,2

,− −⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

I I I F0 A 0 F 00 0 A F




Using Equation (2.7), the reconciled flows by MATLAB are:

CHAPTER 4


C,1

C,2

1 0 0 1 0 0 1 0 00 1 0 0 1 0 0 1 0

.0 0 1 0 0 1 0 0 10 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 1

− −⎡ ⎤⎢ ⎥− − ⎡ ⎤⎢ ⎥ ⎢ ⎥ =− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎢ ⎥⎢ ⎥− −⎣ ⎦

FF 0F

1

2

3

1 1,1

2 2,1

3 3,1

1 1,2

2 2,2

3 3,2

F

F

Fˆ ˆFxˆ .ˆF xˆ ˆF x

Fx

F x

F x

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

1010.2500.1510.1485.7

= 476.09.7

524.524.0

500.5




It can be justified that the reconciled flows satisfy overall mass balances and component mass balances around the distillation column.

Using Equation (4.5), the calculated compositions are:

Note that the reconciled compositions satisfy the normalization equations. For comparison, the reconciled values along with their raw measurements are listed in Table 4.2.

CHAPTER 4


1,1

2,1

3,1

x 0.4808x 0.9518x 0.0190

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

1,2

2,2

3,2

x 0.5192x 0.0480x 0.9810

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦




Table 4.2: Results of bilinear data reconciliation for a binarydistillation column.

CHAPTER 4


0.98100.0190510.10.04800.9518500.10.51920.48081010.2

Reconciled

4.29488.23F3

3.680.0197x3,1

0.630.9748x3,2

4.34478.40F2

1.130.9410x2,1

-4.370.0501x2,2

0.42-0.29-8.44

Adjustment, %

0.5170x1,2

0.4822x1,1

1095.47F1

RawVariables

F1

F2

F3

x1,1x1,2

x2,1x2,2

x3,1x3,2

Figure 4.1




To summarize, the constraints of a bilinear data reconciliation problem contain products of two variables (flow and composition, or flow and temperature). With all variables measured, the bilinear DR problem can be reduced to a linear DR problem by introducing the “measured” compound flows (component flows or energy flows).

♣

Often, there are some flows or compositions that are unmeasured in a process. The presence of unmeasured flow or composition variables introduces complications in the bilinear data reconciliation problem. Crowe (1989) proposed a method, a two-stage projection matrix method, to handle this problem. This method is summarized in the following steps:

CHAPTER 4





Step 1: Unmeasured composition variables are eliminated though Q-R factorization.

Step 2: Reconcile flow and composition measurements and estimate unmeasured total flow rates.

Step 3: Estimate unmeasured compositions.

Because the model constraints in bilinear DR are intrinsically nonlinear, an alternative method to solve a bilinear data reconciliation problem is to use Nonlinear Programming (NLP)techniques that can give the estimates for the measured and unmeasured variables simultaneously . Details of nonlinear data reconciliation problems will be discussed in the next chapter.

CHAPTER 4




4.3 Quiz

Consider a heat-exchanger process as shown in Figure 4.2. All the flows and temperatures for the streams are measured, and the heat capacity, Cp, for all streams is constant.

Figure 4.2: A heat-exchanger process.

CHAPTER 4

Steady-State Data Reconciliation for Bilinear Systems Hot stream

Hot stream return Cold stream

Cold stream return

F1, T1

F2, T2

F3, T3

F4, T4



4.3 Quiz

Question 1:The mass balances used in the reconciliation of the eight raw measurements are

(a)(b)

(c)

(d)

CHAPTER 4


1 2 4 3ˆ ˆ ˆ ˆF - F + F - F = 0

1 2ˆ ˆF - F = 0

4 3ˆ ˆF - F = 0

1 3ˆ ˆF - F = 0

4 2ˆ ˆF - F = 0

1 2 4 3ˆ ˆ ˆ ˆF - F + F - F = 0

1 3ˆ ˆF - F = 0

4 2ˆ ˆF - F = 0

F1, T1

F2, T2

F3, T3

F4, T4

Figure 4.2



4.3 Quiz

Question 2:The heat balances used in the reconciliation of the eight raw measurements are

(a)(b)

(c)

(d) either (a), (b), or (c)

CHAPTER 4


F1, T1

F2, T2

F3, T3

F4, T4

Figure 4.2

1 1 2 2 4 4 3 3ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆFT - F T + F T - F T = 0

1 1 2 2ˆ ˆ ˆ ˆFT - F T = 0

4 4 3 3ˆ ˆ ˆ ˆF T - F T = 0

1 1 3 3ˆ ˆ ˆ ˆFT - F T = 0

4 4 2 2ˆ ˆ ˆ ˆF T - F T = 0



4.3 Quiz

Question 3:The eight raw measurements and their standard deviations are listed as in the table. Calculate the reconciled values for each variable.

CHAPTER 4


F1, T1

F2, T2

F3, T3

F4, T4

Figure 4.25.7 t/h450.0 t/hF4

1.1 oC30.0 oCT4

6.8 t/h460.6 t/hF3

1.8 oC44.8 oCT3

3.0 t/h153.66 t/hF2

1.2 oC45.4 oCT2

1.5 oC4.5 t/h

Standard deviation

90.7 oCT1

150.89 t/hF1

Raw measurementsVariables



4.4 Suggested Readings

Crowe, C.M. (1989). Reconciliation of process flow rates by matrix projection, II: Nonlinear case. AIChE J., 32, 616-623.

Schraa, O.J. and Crowe, C. M. (1996). The numerical solution of bilinear data reconciliation problems using unconstrained optimization methods. Comp. and Chem. Engng, 20 , Supp. A, S727.

Narasimhan, S. and Jordache, C. (2000). “Data Reconciliation & Gross Error Detection, An Intelligent Use of Process Data”. Gulf Publishing, Houston, Texas.

CHAPTER 4




Chapter 5:Steady-State Nonlinear Data

Reconciliation



5.1 Formulation of Nonlinear Steady-State Data reconciliation

In most cases, the steady-state equality constraints that are used to describe the physical system of a process are nonlinear. For example, in the bilinear case, the constraints of component balances are nonlinear.

Moreover, if we take thermodynamic equilibrium relationships and complex correlations for thermodynamic and physical properties as constraints, then nonlinear data reconciliationtechniques have to be used.

If we impose bounds on the estimates, for example, the mass fraction of a component must be in the interval [0, 1]. This gives rise to inequality constraints in the data reconciliation problem which can only be solved using nonlinear data reconciliation techniques.

CHAPTER 5

Steady-State Nonlinear Data Reconciliation



5.1 Formulation of Nonlinear Steady-State Data reconciliation

The nonlinear data reconciliation problem can be formulated as an optimization problem by:

minimizing (5.1)subject to

where y and z are the vectors of measured and unmeasured variables, and f and g are the function vectors of equality and inequality model constraints including simple lower and upper bounds.

CHAPTER 5


Tˆ ˆ ˆ ˆ( , ) ( ) ( )J −= − −1y z y y V y yˆ ˆ( , ) =f y z 0ˆ ˆ( , ) ≥g y z 0



5.2 Successive Linearization

The method developed for solving the linear data reconciliation problem introduced in Chapters 2 and 3 can be extended to handle the nonlinear constraints. The idea is that the nonlinear constraints, , can be linearized using a first-order Taylor’s series around an estimation of the variables.

The linearized system of the constraint equations obtained can be written in the form:

(5.2)

where

CHAPTER 5


ˆ ˆ( , )f y z

y zˆ ˆ+ =A y A z b



5.2 Successive LinearizationCHAPTER 5


i i

1 1 1

1 2 M

2 2 2

1 2 My i i

ˆ ˆ,

C C C

1 2 M

f f f...ˆ ˆ ˆy y yf f f...ˆ ˆ ˆy y y ˆ ˆ ˆat and , (5.3)

ˆ...

f f f...ˆ ˆ ˆy y y

∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥

∂ ∂ ∂⎢ ⎥∂ ⎢ ⎥∂ ∂ ∂= = = =⎢ ⎥∂

⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

y z

fA y y z zy

M M M

i i

1 1 1

1 2 N

2 2 2

1 2 Nz i i

ˆ ˆ,

C C C

1 2 N

f f f...ˆ ˆ ˆz z zf f f...ˆ ˆ ˆz z z ˆ ˆ ˆat and , (5.4)

ˆ...

f f f...ˆ ˆ ˆz z z

∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥

∂ ∂ ∂⎢ ⎥∂ ⎢ ⎥∂ ∂ ∂= = = =⎢ ⎥∂

⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

y z

fA y y z zz

M M M




(5.5)

and are the vectors of estimates corresponding to measured and unmeasured variables.

It should be noted that, in Equation 5.3 and Equation 5.4, M is the number of measured variables, N is the number of unmeasured variables, C is the total number of equality constraints, and i is the ith iteration during the successive linearization.

The matrices Ay and Az are also known as Jacobian matrices.

CHAPTER 5


y i z i i iˆ ˆ ˆ ˆ( , ),= + −b A y A z f y z

ˆ ˆ, y z




In general, the raw measurements are used as initial estimates for the measured variables in the first iteration. At each iteration, the estimates of process variables are obtained by:


Following the procedures to solve the linear DR problems described in Chapter 3, the unmeasured variables, , in Equation 5.6 are first eliminated using a projection matrix via Q-R factorization of Az. Then, the subset of constraint equations contains only measured variables, and the problem becomes:

CHAPTER 5


Tˆ ˆ ˆ ˆ( , ) ( ) ( )J −= − −1y z y y V y yy zˆ ˆ+ =A y A z b

z





where Q2T is the projection matrix. The solution to (5.7) can be

given as:

(5.8)

and the solution to the unmeasured variables can be given as:

(5.9)

In (5.7), (5.8), and (5.9), the matrices Q1, Q2, R1, and R2 are the corresponding submatrices of the Q-R factorization of Az .

CHAPTER 5


Tˆ ˆ ˆ( ) ( ) ( )J −= − −1y y y V y yT T

2 y 2ˆ =Q A y Q b

1T T T T T T T

2 y 2 y 2 y 2 y 2ˆ ( ) ( ) ( ) ( )

−

= − −⎡ ⎤⎣ ⎦y y V Q A Q A V Q A Q A y Q b

-1 T -1 T -1

1 1 1 1 y 1 2 N-rˆ ˆ= − −z R Q b R Q A y R R z




When the new estimates for measured and unmeasured variables are obtained, the next iteration begins. The functions

and the Jacobian matrices, Ay and Az are reevaluated. Equation 5.6 is re-formulated, and solved with Equations 5.8 and 5.9. The iterations continue until

satisfy some small tolerance criteria.

♣

In order to demonstrate the strategies for solving the nonlineardata reconciliation problem, let’s take the distillation column discussed in Chapter 4 as an illustrative example.

CHAPTER 5


ˆ ˆ( , )f y z

ˆ ˆ ˆ ˆ(n) - (n-1) and (n) - (n-1)y y z z




Suppose the compositions of component 2 in the distillate and bottom product streams, x2,2 and x3,2, are unmeasured. For convenience, the distillation column is reproduced in Figure 5.1.

Figure 5.1: Distillation column with unmeasured variables

CHAPTER 5


Feed flow, F1

Distillate flow, F2

Bottom flow, F3

x1,1x1,2

x2,1x2,2

x3,1x3,2




For the distillation column, the vectors of measured and unmeasured variable are defined as:

The data reconciliation problem for this column, then, can be formulated as:

CHAPTER 5


11

22

33

1,14

1,25

2,16

3,17

FyFyFyx ,yxyxyxy

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

y 2,21

3,22

xz.

xz⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦z




minimizing (5.10)

subject to

For this problem, using Equations 5.3 and 5.4, the Jacobianmatrices corresponding to the measured and unmeasured variables can be evaluated as:

CHAPTER 5


Tˆ ˆ ˆ( ) ( ) ( )J −= − −1y y y V y y

1 4 2 6 3 7ˆ ˆ ˆ ˆ ˆ ˆy y y y y y 0− − =

1 5 2 1 3 2ˆ ˆ ˆ ˆ ˆ ˆy y y z y z 0− − =

4 5ˆ ˆy y 1 0+ − =

6 1ˆ ˆy z 1 0+ − =

7 2ˆ ˆy z 1 0+ − =

Component balances

Normalization equationsF1

F2

F3

x1,1x1,2

x2,1x2,2

x3,1x3,2

Figure 5.1




The functions of the constraint equations can be given as:

CHAPTER 5


4 6 7 1 2 3

5 1 2 1

ˆ ˆ ˆ ˆ ˆ ˆy -y -y y 0 -y -yˆ ˆ ˆ ˆy -z -z 0 y 0 0

,0 0 0 1 1 0 0ˆ

0 0 0 0 0 1 00 0 0 0 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥∂ = ⎢ ⎥

∂ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

fy

2 3

0 0ˆ ˆy y

.0 0ˆ

1 00 1

⎡ ⎤⎢ ⎥− −⎢ ⎥∂ = ⎢ ⎥

∂ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

fz

1 4 2 6 3 7

1 5 2 1 3 2

4 5

6 1

7 2

ˆ ˆ ˆ ˆ ˆ ˆy y y y y yˆ ˆ ˆ ˆ ˆ ˆy y y z y z

ˆ ˆ ˆ ˆ( , ) .y y 1ˆ ˆy z 1ˆ ˆy z 1

− −⎡ ⎤⎢ ⎥− −⎢ ⎥

= + −⎢ ⎥⎢ ⎥+ −⎢ ⎥⎢ ⎥+ −⎣ ⎦

f y z

(5.11)

(5.12)




In order to bring the measurements of flows and compositions into the same order of magnitude for the DR, the flow measurements are scaled down by dividing by 1000. Referring to Table 4.1, the initial estimates for the reconciled values are given as:

CHAPTER 5


0

1.095470.4784

0.48823, and 0.4822

0.5170.941

0.0197

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y0

0.05.

0.98⎡ ⎤= ⎢ ⎥⎣ ⎦

z

24.4488.23F3

0.00020.0197x3,1

N/Aunmeasuredx3,2

23.9478.40F2

0.00940.9410x2,1

N/Aunmeasuredx2,2

0.0052

0.0048

54.8

Standard deviation

0.5170x1,2

0.4822x1,1

1095.47F1

Raw Measurement

Variables

Table 4.1




Iteration 1:With these initial values, using Equation 5.11, the linearizedmatrices can be given as:

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0 0y ,

,

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y z

0.4822 -0.941 -0.0197 1.09547 0 -0.4784 -0.488230.517 -0.0588 -0.9803 0 1.09547 0 0

A 0 0 0 1 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0 0z ,

0 00.4784 0.48823

.0 01 00 1

⎡ ⎤⎢ ⎥− −⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y zA




Using Equation 5.12, the values of the constraint functions are evaluated as:

.

Using Equation 5.5, the right hand vector of the linearized equations is evaluated as:

.

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0 0

0.06840.0640

ˆ ˆ( , ) 0.00080.00900.0003

⎡ ⎤⎢ ⎥⎢ ⎥

= −⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

f y z

0 0 0 00 , 0 , 0 0 0

0.06840.064

ˆ ˆ ˆ ˆ( , ) 111

⎡ ⎤⎢ ⎥⎢ ⎥

= + − = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y z y zb A y A z f y z




The results of the Q-R factorization of the matrix are

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0 0z ,y zA

0 0 0 -0.9571 0.28960.4316 -0.3636 0 0.2391 0.7902

0 0 1 0 00.9021 0.1739 0 0.1144 0.378

0 0.9152 0 0.1167 0.3858

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎣ ⎦

Q

1.1085 -0.21070 1.09270 00 00 0

−⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

R

Q1 Q2

R1

0




Note that the matrix R only contains R1, implying that all the unmeasured variables are observable. Applying Equation 5.8, the estimated measured variables for the first iteration are:

Applying Equation 5.9, the estimates for the unmeasured variables for the first iteration are:

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1

1.00220.50260.4999

ˆ .0.48220.51780.94210.0197

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y




Iteration 2:Repeating the procedure above using and as the new initial values results in:

Note that there are only very small changes for the estimates in the second iteration.

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1

0.0579ˆ .

0.9803⎡ ⎤= ⎢ ⎥⎣ ⎦

z

1y 1z

2

1.00240.50260.4997

ˆ ,0.48220.51780.94210.0197

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y2

0.0579ˆ .

0.9803⎡ ⎤= ⎢ ⎥⎣ ⎦

z




Iteration 3 converges the estimates at the values shown in Table 5.1.

Table 5.1: Results of nonlinear DR for a distillation column

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0.98030.0197499.70.05790.9421502.60.51780.48221002.4

Reconciled (Estimated)

488.23F3

0.0197x3,1

Unmeasuredx3,2

478.40F2

0.9410x2,1

Unmeasuredx2,2

0.5170x1,2

0.4822x1,1

1095.47F1

RawVariables

F1

F2

F3

x1,1x1,2

x2,1x2,2

x3,1x3,2

Figure 5.1




The MATLAB code used for the successive linearizationmethod is listed below.

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

f=[0;0;0;0;0];

Q1=[0 0;0 0;0 0;0 0;0 0];

Q2=[0 0 0;0 0 0;0 0 0;0 0 0;0 0 0];

V=[0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0];

V(1,1)=0.0548^2;

V(2,2)=0.0239^2;

V(3,3)=0.0244^2;




V(4,4)=0.0048^2;

V(5,5)=0.0052^2;

V(6,6)=0.0094^2;

V(7,7)=0.0002^2;

y=[1.09547;0.4784;0.48823;0.4822;0.517;0.941;0.0197];

z=[0.05;0.98];

flag=1;

while flag>0,

SSEy=0;

SSEz=0;

Ay=[y(4) -y(6) -y(7) y(1) 0 -y(2) -y(3);y(5) -z(1) -z(2) 0 y(1) 0 0;0 0 0 1 1 0 0;0 0 0 0 0 1 0;0 0 0 0 0 0 1];

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Az=[0 0;-y(2) -y(3);0 0;1 0;0 1];

f(1)=y(1)*y(4)-y(2)*y(6)-y(3)*y(7);

f(2)=y(1)*y(5)-y(2)*z(1)-y(3)*z(2);

f(3)=y(4)+y(5)-1;

f(4)=y(6)+z(1)-1;

f(5)=y(7)+z(2)-1;

b=Ay*y+Az*z-f;

[Q,R]=qr(Az);

for I=1:5,

Q1(I,1)=Q(I,1);

Q1(I,2)=Q(I,2);

end;

for I=1:5,

for J=3:5,

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Q2(I,J-2)=Q(I,J);

end;

end;

for I=1:2,

for J=1:2,

R1(I,J)=R(I,J);

end;

end;

yhat=y-V*(Q2'*Ay)'*inv((Q2'*Ay)*V*(Q2'*Ay)') *(Q2'*Ay*y-Q2'*b);

zhat=inv(R1)*Q1'*b-inv(R1)*Q1'*Ay*yhat;

yhat

zhat

for I=1:7,

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SSEy=SSEy+(y(I)-yhat(I))^2;

end;

for I=1:2,

SSEz=SSEz+(z(I)-zhat(I))^2;

end;

if and(SSEy<1.0e-6,SSEz<1.0e-6),flag=-1;

end;

y=yhat;

z=zhat;

end

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

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5.3 Nonlinear Program

The method of successive linearization introduced in Section 5.2 has the advantage of relative simplicity and fast calculation. However, variable bounds can’t be handled with this method.

Nonlinear programming (NLP) algorithms allow for a general nonlinear objective function, not just a weighted least square objective function. They can explicitly handle nonlinear constraints, inequality constraints, and variable bounds.

Typical algorithms of NLP include sequential quadratic programming (SQP) techniques, such as NPSOL code, and generalized reduced gradient (GRG) methods, such as GRG2code.

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5.3 Nonlinear Program

Nonlinear programming codes are now commercially available, and they have proved to be numerically robust and reliable for large-scale industrial problems.

The disadvantages of these NLP algorithms is the large amount of computation time required relative to the successive linearization algorithm. However, their range of application is wider, and they are able to manage inequality constraints and bounds on variables.

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5.4 Quiz

Question 1:The constraints of a nonlinear data reconciliation problem can contain(a) material balances. (b) energy balances.(c) physical property correlations.(d) inequality constraints.

Question 2:The method of successive linearization(a) linearizes the objective function. (b) linearizes the equality constraints. (c) linearizes the inequality constraints.(d) has less computational time.

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5.4 Quiz

Question 3:Nonlinear data reconciliation problems which contain inequality constraints can be solved by(a) using the successive linearization method. (b) using sequential quadratic programming.(c) using the generalized reduced gradient method.(d) using other nonlinear programming techniques.

Question 4:The estimates for the measured and unmeasured variables in a nonlinear DR problem are(a) definitely biased. (b) definitely unbiased. (c) perhaps biased.(d) perhaps unbiased.

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5.5 Suggested Readings

Edgar, T.F.; Himmelblau D.M. and Lasdon, L. S. (2001). “Optimization of Chemical Processes”, 2nd Ed., McGraw Hill, New York.

Lasdon, LS. And Waren, A.D. (1978). Generalized reduced gradient software for linearly and nonlinearly constrained problems, in Design and Implementation of OptimizationSoftware, Sijthoff and Noordhoff, Holland: H. Greenberg, Ed..

Romagnoli, J.A. and Sanchez, M.C. (2000). “Data Processing and Reconciliation for Chemical Process Operations”. Academic Press, San Diego.

Vasantharajan, S. and Biegler, L.T. (1988). Large-scale decomposition for successive quadratic programming, Computers Chem. Engng., 12, 1087-1101.

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