Data Reconciliation and Bias Estimation in On-Line Optimisation

Moufid Mansour

Faculty of Electronics and Computer Science, University of Sciences and Technology

BP. 32, El-Allia, Algiers, Algeria M.Mansour@city.ac.uk

Abstract— The reliability of measured data, which can be

subject to both gross and random errors, is of great importance for the monitoring and evaluation of process performance and the determination of control action. This paper assesses bias estimation (as a type of gross error) technique and data reconciliation methods for the detection, estimation and elimination of iases and random errors respectively. It is shown how these methods can be successfully employed within an on-line Integrated System Optimisation and Parameter Estimation (ISOPE) scheme for the determination of the process optimum, despite the existence of model-reality differences. The performance of the resulting scheme is demonstrated by application to a two tank CSTR system.

Keywords— Data Reconciliation; Gross Error Detection; Bias Estimation; Optimization

I. INTRODUCTION In recent times, established techniques of Integrated

System Optimisation and Parameter Estimation (ISOPE) have been seen to be successfully applied in the on-line process optimization situation when model-reality differences exist (see for example [1] and [2]). The ISOPE approach includes process measurements as part of the procedure and has been seen to perform well, and obtain the real process optimum, when employed with faithful measurements. However, in many practical situations, errors in process measurements may exist, and it is this setting of the ISOPE algorithm that is being considered.

Measurements from process plants are seldom error free, as they are prone to contain both random and gross errors. Gross errors are caused by non-random events such as process leaks, biases in instrument measurements, malfunction of instruments and inadequate accounting of departures from steady state conditions. The random errors arise from chance occurrences are generally normally distributed.

The aim here is to investigate a technique for Bias Estimation (BE) to tackle biases as a type of gross error and methods of Data Reconciliation (DR), to deal with random errors, and incorporate these techniques within an ISOPE scheme. This is the first time that the effects of such types of errors, and dealing with them by techniques of DR and BE, on the performance of the ISOPE algorithm, have been assessed.

II. BIAS ESTIMATION When both random and gross errors are present on process

measurements, Gross Error Detection techniques are firstly applied to the measurements to eliminate, or reduce, the non-random errors. Gross errors are caused by non-random events and are the result of different effects such as systematic errors or biases [3].

In the special case where the locations of the biased variables are known a priori, bias can be estimated as a parameter [4].

This methodology is appropriate here and the procedure is to solve the following non-linear programming (NLP) problem:

),(Min

∧

byJ (1) subject to:

0)( =yf ,,, iyyy iuiil ∀≤≤ (2)

,,, ibbb iuiil ∀≤≤∧∧∧

Where :

2

111

1

2

222

2

2

( , )

m

m

imii

i

y y bJ y b

y y b

y y b

σ

σ

σ

∧

∧

∧

∧

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= +

⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ +

⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟+

⎜ ⎟⎜ ⎟⎝ ⎠

(3)

where miy is the ith measured variable, iy is the ith estimate, iσ is the measurement noise standard deviation of

the ith measured variable and ib∧

is the estimate of bias on the

ith measured variable. It should be noted that the bias, ib∧

, is also included in the inequality constraints. This allows for physical limits on the range of admissible biases.

III. DATA RECONCILIATION

Generally, once the Gross Errors on the process measurements have been attended to, Data Reconciliation (DR) techniques are required to deal with random errors.

Data Reconciliation (DR) is a necessary operation for obtaining accurate and consistent data in process plants by forcing them to obey natural laws such as material and energy balances so that, ultimately, the material and, if considered, the energy balances are satisfied exactly[5].

Generally speaking, the data reconciliation problem can be formulated as a constrained optimization problem. That is, as a least squares estimation problem if the measurements contain random errors only. This will be the case here as any prior biases have already been removed.

Let ε be a vector of random measurement errors:

truem yy −=ε (4)

where my is the vector of measured process variables, and truey denotes the vector of true values of measured variables.

If these errors are considered to be normally distributed with zero mean, and a covariance matrix, V , the data reconciliation problem can be defined as a least squares estimation problem:

1

Minimise: ( , )1( , ) ( ) ( )2

subject to: ( ) 0

m true

Tm true m true m true

true

F y y

F y y y y V y y

h y

−= − −

=

(5)

where h is a set of algebraic equality constraint equations, and V is the variance-covariance matrix, where each element iiV is

2iσ (i =1, m), and is assumed to be the same for all data sets.

If the equality constraints can be considered to be linear, about the region of measurement, then the above optimisation problem, (5), can be reduced to an unconstrained Quadratic Programming Problem (QP) that can be solved analytically [6].

In this case,

0)( == truetrue Ayyh (6)

where A is the Jacobian of the constraint equations, and the

solution is obtained by the use of Lagrange multipliers and is given by [7]:

δ1)( −−= TTmtrue AVAVAyy (7)

where δ is the residual of the unsatisfied balances and is

given by:

mAyA == εδ (8)

However, it should be noted that the technique has wider

applications as, even if the constraints cannot be considered to be linear, Non Linear Programming (NLP) techniques are available for the problem solution.

IV. ISOPE AND THE INCLUSION OF DATA RECONCILIATION AND BIAS ESTIMATION

The ISOPE algorithm is a model-based system optimisation technique that was developed to overcome model-reality differences and generate the system optimum [8]. The basic form of ISOPE is discussed here in connection with BE and DR but extended versions can readily be employed if the situation demands [1].

As process measurements are being used within the algorithm, there are, inevitably, difficulties in that measurements are likely to be contaminated by various types of errors. By applying BE and DR techniques to the measurements, it would be hoped that the performance of the ISOPE algorithm could be improved.

The ISOPE algorithm addresses the general non-linear programming problem [9, 10], where * refers to the real process:

Identify applicable sponsor/s here. (sponsors)

),( *ycQMinc

(9)

subject to: )(** cFy = (10) 0)( * ≤yg (11) maxmin ccc ≤≤ (12) where, c and y* are the controls and outputs, respectively,

of the process. The general form of the ISOPE algorithm, with error free

output measurements can be seen in [1] and [8].

With the inclusion of Bias Estimation and DR, the ISOPE algorithm takes the following form:

(i)Apply the current control, kc , to the real process and,

after an appropriate settling time, obtain steady-state measurements, ky*

~

. Where, k is the iteration and ky*

~

is the error burdened output measurement. (ii) Apply BE and DR techniques as required to the measured

outputs, ky*

~

, to yield the error free outputs, ky* .

(iii) The process model is given by: ),( αcFy = (13)

where, α are free model parameters. Assuming that measurements of all outputs are available, (10) and (13) can be used in a simple estimation procedure to determine the model parameters, α:

)(),( * cFcFy == α (14) This estimation procedure also has the benefit of satisfying

one of the necessary system optimality conditions [1]. (iv) Solve the modified model-based optimisation problem

given by:

0),(),(

)),((

≤=

−

αα

λα

cgcFy

ccQMin T

c

(15)

Where,

⎥⎦⎤

⎢⎣⎡∂∂

⎥⎦

⎤⎢⎣

⎡∂

∂⎥⎦

⎤⎢⎣

⎡ −∂

∂=−

ααλ QF

dcdF

cF TT 1

* (16)

λ is termed a modifier and arises from the necessary optimality conditions, of the system optimisation problem [1, 2, 8].

(v) In order to control convergence of the algorithm, the new control kc

∧

, obtained from the model-based problem of (15), is not directly applied to the system. Instead, the following under-relaxation scheme is used to provide updated controls, 1+

∧

kc , for the process : )(1 kkkk ccKcc −+=

∧

+ (17) where K is a relaxation gain matrix, and governs the actual

changes made to the real process inputs from one iteration to another. Its purpose is to ensure that excessive alterations are not made.

The above steps are repeated until satisfactory convergence is obtained. Convergence occurs when no further improvement is observed. In other words, when the new control is no longer a better candidate than the previous one.

V. APPLICATION TO A CONTINUOUS STIRRED TANK REACTOR SYSTEM

The ISOPE algorithm, using the Bias Estimation (BE) and Data Reconciliation (DR) schemes, is now applied, under simulation, to a Continuous Stirred Tank Reactors (CSTR) system [11] which has two tanks connected in cascade (Figure 1). An exothermic autocatalytic reaction takes place in the reactors with interaction taking place in the units in both directions due to a recycle of 50% of the product stream into the first reactor. Regulatory controllers are used to control the temperature in both reactors.

The reaction is: 2k

kA B B+

−⎯⎯→+ ←⎯⎯ (18)

The system has four outputs which are the concentrations

of the two components A and B in the two tanks, i.e.: 1 1 2 2( , , , )a b a by C C C C= . In our example, the concentrations of

species B in both tanks 1bC and 2bC are to be monitored for steady-state identification. Temperatures in the two tanks,

1T and 2T are the set-points.

The simulations were started from the same initial operating point given by 1 2307 and 302 ,T K T K= = yielding the following steady-state output values of the concentration of

Figure 1. Continuous Stirred Tank Reactor System

CONC 1

FR 1

CW1 CW2

CSTR1 CSTR2 TRC 1 2

TRC

FR 2

CONC 2

FRC 1

product B in the two tanks 1 and 2, 3

1 (0) 0.0516 [ / ]bC kmol m= , and (0) 0.0586 [ ]3

b2C = kmol/m .

Measurement noise was simulated as normally distributed with zero mean. The value of the variance-covariance matrix was chosen to be:

21

22

00

Vσ

σ⎡ ⎤

= ⎢ ⎥⎣ ⎦

(19)

where 1σ is the standard deviation for the variable b1C and was chosen to be 5% of the nominal value, 2σ is the standard deviation for b2C and was of a value of 5%. These values were chosen as they represent typical values in many realistic situations.

The aim here is to maximize the concentration of component B in tank 2, giving the objective function, (9), as:

* 2( , ) bQ c y C= − (20)

The controls, at the real process optimum, obtained directly from the real process equations, are: 1T =312 K and

2T =310.2K. With corresponding output concentration

measurements 0.0644 3b1C = kmol/m and

0.0725 .3b2C = kmol/m

The model adopted is of the approximate linear form [1]:

1

2

111 12 1

221 22 2

b

b

C a a Ty

a a TCαα

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

(21)

where 1α and 2α are the free model parameters to be

estimated and ija {i, j = 1,2} are also model parameters which are assigned the values of the process derivatives.

The real process derivatives, df/dc , may be estimated by

such techniques as applying control perturbations or, as is the case here, by the use of Broydon’s method [12]. The estimation of the process derivatives may be made by such techniques for the calculation of the modifier, λ, in (16).

The procedure is implemented, using a MATLAB/SIMULINK software platform, following the steps described in Section IV.

Initially, to illustrate the performance of the ISOPE algorithm, without any BE or DR being implemented, both measurements were subject to 5% additive noise. Figures 2(a) shows the measurements, while Figure 2(b) shows the controls not converging to the correct real optimum. This is due to flawed data measurements producing erroneous model parameters.

The sample extreme case of gross error, represented here is in the form of measurement biases. With noise present on the measurements as well, it is observed with BE and DR applied, as described in Sections 2 and 3, to the real process measurements. Figure 3(a) shows, due to the introduction of GED and DR, error free measurements being obtained.

Having error free measurements available, the ISOPE algorithm is now able to function correctly, as can be seen in Figure 3(b), where the controls converge the correct real process optimum.

Figures 3(a) and 3(b) demonstrate the effectiveness of the BE and DR procedures to enable the ISOPE algorithm to perform successfully. Less extreme situations, such as when only noise is present on the measurements, have also been seen to be handled satisfactorily by the BE and DR procedures [10].

VI. CONCLUSIONS In the on-line process optimization problem, when

measurements are subject to gross errors and/or noise, it has been seen how techniques of Bias Estimation (BE) and Data Reconciliation (DR) can be employed in conjunction with an algorithm for Integrated System Optimisation and Parameter Estimation (ISOPE) to enable the real process optimum to be found. This is in the situation when there also exists model-reality differences.

These techniques have been demonstrated for the on-line optimization of a two tank CSTR system. Despite the presence of multiple biases and noise on the process measurements, together with an unfaithful model, the real process optimum was seen to be achieved.

Thus, these techniques of BE and DR are therefore eminently suitable for the often encountered on-line optimization situation, when measurements are subject to errors, yet still enable the process optimum to be achieved.

Figure 2(a). ISOPE Process Outputs and Measurements: Noise applied without BE and DR

0 50000 100000 150000 200000 250000 3000000.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

Con

cent

ratio

n

Time (S)

Cb1:

__: Real process output

---: Noisy measurement

Cb2:

__: Real process output

Cb1:

__: Real process output

: Reconciled measurement

Figure 3(a). ISOPE Process Outputs and Measurements with Implementation Of BE and DR: Multiple Biases and Noise Present

0 50000 100000 150000 200000 250000 300000

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Con

cent

ratio

n

Time (S)

Cb2:

__: Real process output

…: Reconciled measurement

0 50000 100000 150000 200000 250000 300000300

305

310

315

Tem

pera

ture

Time (S)

Figure 2(b). ISOPE Control Trajectories: Noise applied without GED and DR

T1

T2

0 50000 100000 150000 200000 250000 300000300

305

310

315

Tem

pera

ture

Time (S)

Figure 3(b). ISOPE Control Trajectories with Implementation of BE and DR: Multiple Biases and Noise Present

T1

T2

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