=Paper=
{{Paper
|id=Vol-1623/paperapp7
|storemode=property
|title=Application of Active Set Method for Soft Sensor Model Identification of Crude Oil Distillation Process
|pdfUrl=https://ceur-ws.org/Vol-1623/paperapp7.pdf
|volume=Vol-1623
|authors=Anton Goncharov,Andrey Torgashov
|dblpUrl=https://dblp.org/rec/conf/door/GoncharovT16
}}
==Application of Active Set Method for Soft Sensor Model Identification of Crude Oil Distillation Process==
https://ceur-ws.org/Vol-1623/paperapp7.pdf
Application of Active Set Method for Soft Sensor Model
Identification of Crude Oil Distillation Process
Anton Goncharov 1, Andrei Torgashov 1,2
1
Institute of Automation and Control Processes FEB RAS, Vladivostok, Russia,
antalg@mail.ru
2
Far Eastern Federal University, Vladivostok, Russia,
torgashov@iacp.dvo.ru
Abstract. The problem of identification of the soft sensors is considered. Using
the active set method the problem of least squares with inequality constraints on
the variables has been solved. As a result of using soft sensors, obtained taking
into account constraints on the model coefficients, their efficiency as compared
with soft sensors obtained without constraints is shown.
Keywords: parametric identification, optimization, soft sensor, constraints, dis-
tillation column.
1 Introduction and statement of the problem
The approach to solve the soft sensor model obtaining problem, taking into account
inequality constraints on the model coefficients, is described [1]. The obtained results
are tested on industrial oil fractionation process for atmospheric distillation column.
The soft sensor model coefficients for prediction of the key product quality are
identified.
The technological plant with several measured inputs u1 , u 2 ,, u N and one output
y ( ) is considered. The measured technological parameters (pressure, temperature,
flow) are utilized as inputs. In practice, the amount of available process measured
parameters to predict the quality of a product is much more than the required number
of parameters. A priori knowledge of technological process allows to select necessary
parameters.
The problem of the soft sensor (SS) evaluation which is best predicting quality of
products of crude distillation technological process is considered.
The model of the soft sensor is obtained in the form of linear regression model for
solution of the problem [2]:
y( ) b0 b1u1 b2u2 ... bN uN
, (1)
Copyright © by the paper's authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
724 A. Goncharov and A. Torgashov
where b j – j-th model coefficient, j 0,1,..., N , b0 – constant term, N – the number
of input variables, - irregular timepoints of output measurement: 1 , 2 , 3 ,... ,
i i 1 0 , i 2 ; 1 0 ; 0 - constant term; - random component is
limited by specific range.
The determination coefficient (the part of explained deviations variance of the
dependent variable from its mean value):
R 2 1 i( yi yi ) 2 i( yi y a ) 2 (2)
and root mean squared error (RMSE):
RMSE i 1 ( y i y i ) 2 / M
M
,
1/ 2
(3)
are used as criteria of identification on a given time interval,
where y i - the measured value of the output variable, y i - the value is obtained
based on the SS, y a - the mean value of the measured output variable, M - the
number of output measurement. The model is more consistent if the closer to unity the
value of the coefficient of determination R 2 , or the closer to zero the value of the
RMSE.
2 The proposed algorithm problem solution
Let u [1, u1 ( ), u2 ( ),..., uN ( )]T be a combined vector of measured input
variables, b = [b0 , b1 ,..., bN ]T - vector of coefficients of the same dimension, the
components of which reflect the contributions of the respective input variables. Then
the equation (1) takes the form:
y uT b .
We form the vector Y of dimension q from the output value y:
Y [ y( 1 ), y( 2 ),..., y( q )]T
and the matrix U, containing the measured inputs uj, corresponding to output value y
from (1):
1 u1 ( 1 ) u2 ( 1 ) ....uN ( 1 )
1 u1 ( 2 ) u2 ( 2 ) ....uN ( 2 )
U ,
1 u1 ( q ) u2 ( q ) ...u N ( q )
Application of Active Set Method 725
and we write the matrix equation:
Y Ub .
We introduce error function:
E Y Y Y Ub ,
where Y is the actual measurement of output, and minimize the objective function:
Ψ E 2 (Y Ub ) 2 . (4)
We obtain estimates of the parameters b by least squares method:
b (UT U)1 UT Y . (5)
The multicollinearity case, which occurs when there is almost a linear relationship
between inputs, is considered. In this case the matrix C U T U is close to singular,
so it is the smallest eigenvalue min 0 and the condition number is infinitely
increased and causing the instability of the solution (5). If the min 0 then it
corresponds the strict multicollinearity [3]. In order to obtain a stable solution of the
equation (5) it is necessary to reduce the condition number of the matrix C, for
example, by adding thereto a diagonal matrix B kI (k> 0). Then the solution is
found in a class of ridge parameter estimates:
b (UT U kI)1 UT Y . (6)
The quality, obtained by (5-6) models, depends on the number of available output
measurements. The length of training sample is often insufficient to obtain reliable
results. Also, the available data contain significant measurement error of inputs and
outputs, unmeasured influences. Taking into account constraints on the model
coefficients b j allows to avoid these problems. When taking into account constraints
on coefficients at input, the problem of least squares with simple constraints on the
variables is solved:
min Y Ub
2
(7)
b min b b max .
The solution of the problem (7) is obtained by the active set numerical method [4].
The given constraints are reduced to the form:
Ab bˆ ,
726 A. Goncharov and A. Torgashov
1 0 0
0 1 0
bˆ b
min
where
0 0 1 , max .
A b
1 0 0
0 1 0
0 0 1
Constraint aTi b bˆi call active in acceptable point b if aTi b bˆi , and inactive if
aTi b bˆi , aTi - i-th row of A .
The sufficient minimum conditions for simple constraints are as follows:
FR b FR b FR
1. b min b * b max , b min * max
2. UTFR (Y Ub* ) 0
3. λ
min
UTmin (Y Ub * ) , λ imin 0 , i 1,, t min , (8)
λ max UTmax (Y Ub* ) , λ max 0 , i 1,, t max
i
4. U TFR U FR is positive definite,
where b* - the minimum point of the solution of problem (7); subscript FR indicates
that in the vector and matrices the elements and columns with index numbers
corresponding to the index numbers of b elements, that have not met the boundary
values (7), are used; subscript min, max, indicates that the in matrix only the columns
with index numbers corresponding to the index numbers of b elements, taking the
appropriate minimum or maximum boundary value, are used. tmin,, tmax.- number of
active upper and lower limits respectively; λ min , λ max - vectors of Lagrange
multipliers corresponding to the lower and upper active constraints.
To start the method of the active set it is necessary to determine the starting point
using (6).
The minimum point b* for the search algorithm for iteration k is:
1. Performance verification of the stop conditions. (Reaching the performance errors
of conditions (8), constraints on the number of iterations).
2. Selection of a logic branch. Does it make sense to remove any constraint of the set
of active constraints list. The condition of performance of a condition 3 in (8) is
checked. If the condition is not satisfied for some of the vector element, constraint
is excluded from the list of active constraints.
3. The calculation of the search direction p k . Like (4) solves the problem
min Y Ub k U FRp FR . Calculate the non-zero N 1 tk - dimensional vector
2
Application of Active Set Method 727
p FR and the direction of search p k AT FR p FR , where t k - the number of active
constraints on k iteration.
b FR p
4. Calculate the step length k . From Ψ FR bˆ FR diagonal matrix
b FR p FR
Ψ is calculated, b̂ FR - consists of the elements b̂ which aren't active constraints,
elements b̂ , opposite boundary values in (7) for constraints in the active set, are
excluded from b̂ FR . The k min Ψii is an available minimum positive step
from b k along p k . The index j of minimum positive diagonal element Ψ is
remembered. If k 1 , then k 1 , otherwise k k .
5. Constraint is added in the list of active constraints. If k , then j constraint
bˆ FR becomes active, it is necessary to add to the list.
6. Recalculation approximation. bk 1 bk k pk is calculated, and return to step 1
of the algorithm is carried out.
The influence of the process dynamics on the quality of the products is taken into
account by the dynamic SS. The predictive model is represented as a sum of
convolutions of plant inputs and a finite impulse response (FIR) hi (discrete analogues
of the first degree Volterra kernels):
y( ) h0 k10 h1 (k 1)u1 ( k ) k 20 h2 (k 1)u 2 ( k ) ... k N0 hN (k 1)u N ( k ), (9)
n 1 n 1 n 1
where h0 – constant term, - irregular timepoints of output measurement:
1 , 2 , 3 ,... , i i 1 0 , i 2 ; 1 0 ; 0 - constant component; -
random component is limited by the specific range.
Let u [1, u1 ( ),..., u1 ( n1 1),..., u N ( ),..., u N ( n N 1)]T - the combined
vector of measured input variables of dynamic SS (DSS) with dimensionality
q 1 k 1 nk where nk - is a number of values of k-th input,
N
h h0 , h1 (1), ..., h1 (n1 ), ..., hN (1), ..., hN (nN ) - vector FIR of the same dimension, the
т
components of which reflect the contributions of the respective input variables of
DSS. Then the equation (9) takes the form:
y uT h .
We form the vector Y of dimension q from the output value y:
Y [ y( 1 ), y( 2 ),..., y( q )]T
and the matrix U, containing the measured inputs uj, corresponding to output value y
from (9):
728 A. Goncharov and A. Torgashov
1 u1 ( 1 ) u1 ( 1 n1 1) u N ( 1 ) u N ( 1 n N 1)
1 u ( ) u ( n 1) u ( ) u ( n 1)
U .
1 2 1 2 1 N 2 N 2 N
1 u1 ( q ) u1 ( q n1 1) u N ( q ) u N ( q n N 1)
Then write the matrix equation:
Y Uh .
We introduce the error function:
E Y Y Y Uh ,
where Y is the actual measurement of output, and minimize the objective function:
Ψ E2 (Y Uh ) 2 . (10)
The constraints on transient response components are written as:
s min s s max , (11)
where s s1 (1), ..., s1 (n1 ), ..., s N (1), ..., s N (nN )
т
,
s min s1min , ..., s min
N т
,
s max s1max , ..., s max
N
.
т
The transient response components s are related with the components of the
impulse response h by the relations:
s j k i 1 h j (i), j 1, 2, , N , k 1, , n j .
k
(12)
The constraints (11) are reduced to:
~
Ah sˆ , (13)
where
1 0 0
1 1 0 h1 (1)
, , ˆ s
min
h ( n )
0 0 1
1 1
s .
A ~
s max
1 0 0 h
hN (1)
1 1 0
h (n )
N N
0 0 1
The sufficient minimum conditions are as follows:
~ ~
1. Ah sˆ , A ACT h sˆ ACT
Application of Active Set Method 729
2. Z T * UT ( Y Uh * ) 0 (14)
3. λ A ACT A TACT A
1
ACT U T ( Y Uh * ) , λ i 0 , i 1,, t
4. Z T UT U Z is positive definite,
where h* - is the minimum point, the solution of problem (10) with constraints (13);
subscript ACT indicates that in vector, matrix only the elements, rows with index
numbers corresponding to the elements index numbers of active constraint in (13) are
used; t - number of active constraints, λ - the vectors of Lagrange multipliers
corresponding to the active constraints, Z - matrix the columns of which is basis of
the feasible direction of search for equality constraints (13). The matrix Z is formed
by the variable-reduction technique. [4].
In order to start the method of the active set it is necessary to determine the starting
point (using a solution of the problem (10) without any constraints, with subsequent
correction of coefficients h i that does not fall under the constraints (13)).
The search algorithm of minimum point h* for iteration k is:
1. Performance verification of the stop conditions (reaching the performance errors of
conditions (14), constraints on the number of iterations).
2. Selection of a logic branch. Does it make sense to remove any constraint of the set
of active constraints list. The condition of performance of a condition 3 in (14) is
checked. If the condition is not satisfied for some of the vector element, constraint
is excluded from the list of active constraints and it is need to recalculate Z k .
3. The calculation of the search direction p k . Like (4) solves the problem
min Y Uh k UZ k p Z . Calculate the non-zero 1 k 1 nk t k - dimensional
2 N
vector p Z and the direction of search p k Z k p Z , where t k - the number of active
constraints on k iteration.
4. Calculate the step length k . From A h Ψp
~
~ sˆ diagonal matrix Ψ is
k
calculated. Calculated k min Ψ ii - a minimum non-negative available step
from h k along p k , where i is the index number of element, which is not active
constraints in (13) and is not element of opposite boundary values (11) for
~ consists from elements of p without first element.
constraints in the active set, p k k
The index j of minimum positive diagonal element of Ψ ii is remembered. If
k 1 , then k 1 , otherwise k k .
5. Constraint is added to the list of active constraints. If k , then j constraint
sˆ FR becomes active and recalculate Z k .
6. Recalculation approximation. The hk 1 hk k pk is calculated and return to
step 1.
730 A. Goncharov and A. Torgashov
3 The influence of constraints of parameters of soft sensors
model
A priori knowledge about the process and industrial step test (when the value of the
one control variables is changed at fixed others) allow to define the value of
constraints.
In order to investigate the influence of constraints on the quality of the obtained
static soft sensor model we compare solutions of the equations (6) and constrained
optimization problem (7). One and the same value of the ridge coefficient is used.
The Fig. 1 and Table 1 show the results of the performance of the static models on
the test sample for the bubble-point temperatures of the target product (BP) when a
model obtained on the training sample, consisting of a number of measurements
specified in the Table 1. In the verification sample of models the number of
measurements is equal to 420.
Table 1. Results of the performance of the static models
The number of Without use With use
measurements in the constraints constraints
training sample R2 RMSE R2 RMSE
30 0,188 1,790 0,604 1,249
Fig. 1. Comparative study of static soft sensor models performance
Application of Active Set Method 731
In order to investigate the influence of constraints on the quality of the obtained
dynamic soft sensor model we compare solutions of the optimization problem (10)
and optimization problem (10) with constrains (13). One and the same value of the
ridge coefficient is used.
The Fig. 2 and Table 2 show the results of the performance of the dynamic models
on the test sample for the dew-point temperatures of the target product (DP) when a
model obtained on the training sample, consisting of a number of measurements
specified in the Table 2. In the verification sample of models the number of
measurements is equal to 650.
Table 2. Results of the performance of the dynamic models
The number of Without use con- With use con-
measurements in the straints straints
training sample R2 RMSE R2 RMSE
700 0,327 1,456 0,634 1,074
Fig. 2. Comparative study of dynamic soft sensor models performance
The estimation of improvements of the prediction quality by the criterion RMSE of
static model obtained with the constraints on the parameters of SS is 100(1,79 -
1,249) / 1,79 30% compared to the model without constraints. The estimation of
improvements of the prediction quality by the criterion RMSE of dynamic model
obtained with the constraints on the parameters of SS is 100 · (1,456– 1,074)/ 1,456
25 % compared to the model without constraints.
732 A. Goncharov and A. Torgashov
Conclusion
The using the method of the active set, taking into account constraints on the
model coefficients can improve quality of the evaluated SS models.
The test of the proposed approach to solving the problem of obtaining a soft sensor
model for industrial crude oil distillation unit is showed that the decrease root mean
square error on the test sample can be not less than 25%.
References
1. Bakhtagze. N.N.: Virtual Analyzers: Identification Approach. In: Automation and Remote
Control. Vol. 65, issue 11, pp.1691-1709. (2004)
2. Draper N., Smith H.: Applied regression analysis. M .: Finance and Statistics (1986)
3. Bolshakov A.A., Karimov R.N.: Methods of processing of the multidimensional given and
time numbers. M .: Hotline Telecom (2007)
4. Gill, P.E., Murray, W., and Wright, M.H.: Practical Optimization, London: Academic,
1981. Translated under the title Prakticheskaya optimizatsiya, Moscow: Mir (1985)