Fuzzy Model Based Control: Application to an Oil Production Separator

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Fuzzy Model Based Control: Application to an Oil Production Separator Conference Paper · October 2008 DOI: 10.1109/HIS.2008.100 · Source: IEEE Xplore

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Eighth International Conference on Hybrid Intelligent Systems

FUZZY MODEL BASED CONTROL: APPLICATION TO AN OIL PRODUCTION SEPARATOR MIGUEL ANGEL RAMIREZ CANELON ELIEZER COLINA MORLES Universidad de los Andes: Facultad de Ingeniería Complejo La Hechicera, Mérida, Venezuela E-mail: [email protected], [email protected]

Abstract Inverting systems is an important issue in engineering applications, especially in linear and nonlinear control problems [2][3]. The underlying principle of inverse control is based on the following remark: Since a plant model can be view as a mapping from control inputs to future outputs according to the process history, the inverse mapping can be used from the desired outputs to the inputs as a methodology of control [2]. An inversion mechanism is often included in nonlinear control structures and two major approaches are known: Feedback linearizing control [4][5] and internal model control [6][7][8].

Classical fuzzy controllers have been of the rulebased type, where the rules in the controller attempt to model the response of an operator to a particular process situation. Based on the ability of fuzzy systems to approximate any nonlinear mapping, the nonlinear plant is represented by a zeroth-order Takagi Sugeno fuzzy model and it is analytically inverted for designing a fuzzy controller. In order to minimize the steady-state-error due to model-plant mismatch, an Internal Model Control (IMC) will be considered. The proposed fuzzy controller is applied to control the oil level in a production separator and is shown to be capable of providing good overall system performance.

Both control strategies are promising approaches for controlling input-output stable nonlinear systems but their applicability is constrained to the use of an accurate representation of the nonlinear plant under consideration. As control problems arising in a large variety of complex industrial processes are characterized by uncertain environments and nonlinearities, the identification of accurate processes models according to first principles is a difficult and time-consuming task. In order to overpass the limitation of having an accurate representation of a nonlinear plant, the use of fuzzy models has become in a very useful alternative and have been successfully implemented in [9][10].

1. Introduction Many industrial processes are characterized by frequent changes in the operating conditions, such as the ones caused by increasing the production task, varying quality of materials, changing product mix and varying process throughput. In order to keep a similar performance in a new operating point, the process control system must be able to cope with frequent changes in the process parameters and structure [1]. The implementation of classical fuzzy controllers for this sort of processes, where the controller is designed based only on the knowledge of an expert operator, may cause in some cases unacceptable steady-state errors and undesired oscillations. A different approach consists of determining the fuzzy model of the plant and obtaining its respective inversion to derive the control-input. Therefore, the obtained inverse model is used as a controller, and under special conditions stable control can be guaranteed for minimum phase systems [1].

978-0-7695-3326-1/08 $25.00 © 2008 IEEE DOI 10.1109/HIS.2008.100

The paper is organized as follows. In section 2, the fuzzy models inversion, specifically zero order Takagi Sugeno fuzzy models will be revised in details. Section 3 will describe relevant aspects of fuzzy model-based control. Section 4 will present a case of study related to the oil level control in a production separator, where a white-box modeling and the fuzzy modelling of the production separator will be described. Conclusions are given in section 5.

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process is available. Such a model can be constructed directly from process measurements. In that case, a general fuzzy rule Rk may be defined as follows:

2. Fuzzy Models Inversion The inverse of a dynamical process model is often used as a feedforward controller, which computes the control input such that the process output obtained is closed to the desired reference [11].

R k : If x(τ ) is X and u (τ ) is U Then y (τ + 1) = c k (1)

Where x(τ) is a state vector containing the m-1 past inputs and p-1 past outputs (m and p are the orders of the inputs and outputs) and the current output y(τ) that can be expressed as follows:

When an ideal model M mapping the control actions u to the system outputs y is considered, the control actions may be simply given by u = M-1r, where r are the references to be followed, as depicted figure 1.

x(τ ) = [ y(τ ),...y(τ − p + 1),u(τ −1),...u(τ − m + 1)]

T

(2)

X is a multidimensional fuzzy set for x(τ) defined on , the Cartesian product of the individual ℜ universes of discourse. When conjunctive aggregation is used, the multidimensional fuzzy set X is given by: p+m-1

Figure 1: Mapping and perfect system inversion

Certain fuzzy model structures can be exactly inverted and these inversions may be used for control purposes [9]. In general, there are two ways of inverting a fuzzy model: global inversion and partial inversion. In the global model inversion all states become outputs of the inverted model, the output of the original model becomes the state variable of the inverted model (see figure 2-A) and normally has a non-unique solution, therefore a family of solutions is usually given [10].

X = A1 • A2 • ... • Ap • B2 • ... • Bm (3)

Where • represents a t-norm operator and A1,…,Ap and B2,…,Bm are fuzzy sets assigned for the different elements of the state vector. Instead of B1, U is used in equation (1) for notational of clarity. Let N denote the number of different fuzzy sets Xi defined for the state x(τ) and M the number of different fuzzy sets Uj for the input . If the rule base consists of all possible combinations of Xi and Uj, the rule base is complete and the total number of rules is given by K = N*M. The entire rule base may be represented by the next table:

u (τ )

Figure 2: A) Global model Inversion. B) Partial model inversion.

In case of partial inversion, only one of the states (For instance, x1 in Figure 2-B) of the original model becomes the output of the inverted model and other states along with the original output are the inputs of the inverted model. A partially inverted model has often a unique solution, which is a significant advantage compared to global inversion. Also, a partially inverted model can also be more embedded into the control system than global inverted models [10].

x (τ ) A1 A2 .

U1 c11 c 21 .

U2 c12 c 22 .

... U M ... c1M ... c 2 M . .

. AN

. cN1

. cN 2

. . ... c NM

( 4)

The logical and connective is assumed to be represented by the product t-norm operator, since it is a necessary condition to perform the inversion and the degree of fulfillment of the rule antecedent is calculated as:

β ij (τ ) = µ Xi ( x(τ )) • µUj (u (τ ))

(5)

Where µXi(x(τ)) is the membership degree of a particular state x(τ) in the set Ai and µUj(u(τ)) is the

Let us assume that a singleton fuzzy model of a certain

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membership degree of an input u(τ) in the fuzzy set Uj. The predicted output y(τ+1) of the model is computed by the fuzzy-mean defuzzification, where an average of the consequents cij is weighted by the degrees of fulfillment βij expressed as follows:

y (τ + 1) =

N

M

i =1

j =1 N M

∑∑β

ij

∑ i =1

(6) ij

∑

Xi

j =1

Uj

At a certain time τ the process is at the state x(τ), and the inverse of the singleton model is given by the fuzzy rules: If r (τ + 1) is C j (τ ) then u (τ ) is U j , j = 1,2,..., M , (9)

(τ ) c ij

∑∑β i =1

Theorem: Let the process be represented by the singleton fuzzy model of equation (1) with the weighted-mean defuzzification method from equation N M (6). Further, let . µ ( x) = 1 and µ (u ) = 1

(τ )

j =1

Where Cj are fuzzy sets that form a partition as indicated in figure 4. The cores cj of the fuzzy sets Cj are given by:

The rule-based model of equation (1) corresponds to the non-linear regression model shown schematically in figure 3-A, which can be written as: y (τ + 1) = f ( x(τ ), u (τ ))

N

cj =

(7)

∑

i =1 N

µ Xi ( x (τ )) c ij

∑

i =1

The model inputs are the current state x(τ) and the current input u(τ) and the output is the system’s predicted output at the next sampling instant y(τ+1). Given the current system state x(τ) and the desired output reference at the next sampling time r(τ+1), the inversion consists of finding u(τ) such that the system output y(τ+1) is as close as possible to the desired output r(τ+1).

µ Xi ( x (τ ))

, j = 1, 2 ,..., M . (10)

The defuzzification of the rules in equation (9) is accomplished by the fuzzy-mean method: u (τ ) =

M

∑µ

Cj

( r (τ + 1)).core(U j ) (11)

j =1

Figure 3: A) System’s model B) Derived controller Figure 4: Partition of fuzzy sets Cj using the cores cj.

This can be achieved by inverting the plant model, as indicated in figure 3-B, substituting the reference r(τ+1) for y(τ+1) in the next static function: u (τ ) = f

−1

( x(τ ), r (τ + 1))

The inversion is made for a given state x(τ). The degree of fulfillment for this sate given in the proposition, ‘x(τ) is Xi’, is denoted by µXi(x(τ)). Then the N consequents of the rules containing a particular Uj can be aggregated by equation (10).

(8)

This technique was proposed in [12]. The multivariable mapping of the fuzzy model in equation (7) can be reduced to the univariate mapping y(τ+1) = fx(u(τ)), where, the subscript x denotes that fx is obtained for the particular state x(τ). If the model is invertible, the inverse mapping u(τ) = fx-1(r(τ+1)) can be obtained. The inverse of the singleton fuzzy model can be formulated in the following theorem.

3. Theoretical Considerations in Fuzzy Model Based Control Given a process to be controlled, the process knowledge is often inadequate in sense that an accurate inverse model is not available at the time of implementation. To overcome some of the problems

752

introduced by the model plant mismatch, the fuzzy model system is placed in parallel with the real process. This feedback approach called Internal Model Control (IMC) was presented in [13]. The key feature of this control arrangement is the way in which it handles differences between the model and the real process. The difference between the system and model output represents the modeling error and / or unmodelled process disturbances [14]. Figure 5 shows the adaptive IMC architecture.

generally chosen as follows [2][14]: Fo ( z

−1

) =

Tz z − (1 − T )

(12)

By comparing linear and nonlinear IMC cases, it can be stated that the filter F0 plays the same role as the robustness filter in the linear IMC.

4. Case of Study: Design of a fuzzy controller based on a fuzzy model for a production separator Figure 6 shows the drawing of a standard production separator. A separator for petroleum production is a large drum designed to separate production fluids into their constitute components of oil, gas and water. It operates on the principle that the three components have different densities, which allows them to stratify when moving slowly with gas on top, water on the bottom and oil in the middle [16].

Figure 5: The IMC Architecture

A general ICM consists of three elements: a model to predict the effect of the control action on the system, a controller based on the inverse of the process model and a filter to increase robustness to model plant mismatch and disturbances.

4.1 First Principle Modeling When the hydrocarbon fluid stream enters a threephase separator, two different phenomena take place. The first phenomenon is fluid dynamics, which is characterized by the gravity separation of oil and water droplets entrained in the aqueous and the oil phases respectively, the gravity separation of gas bubbles entrained in the stream and the gravity separation of liquid droplets which are dispersed in the gas phase [16].

The principle of ICM scheme is based on the inclusion of a nonlinear plant model within the control structure. Indeed, when a plant model is available, the nonlinear controller can be directly obtained by model inversion. By doing so, an offset-free response can be achieved even in the presence of constant disturbances acting on the plant output. A robustness filter is generally designed to alleviate sensitivity problems. The advantages of the IMC scheme, which are the motivation factors for using this technique in control applications, are summarized in [14].

The second phenomenon is thermodynamics in the sense that some light hydrocarbon and gas solution flash out the oil phase and reach a state of equilibrium due to the pressure drop in the separator [17]. Due to the complexity of such phenomena, we are going to focus only on the hydrodynamic separation of oil droplets entrained in the oil phase. A detailed analysis is done in [16] based on figure 6, and it can be demonstrated that the oil volume (Vs1 as indicated in figure 9) is given by:

Based on this control principle, a fuzzy version of IMC scheme can be obtained as depicted in figure 5, where the analytical model is replaced by a fuzzy model. By applying the inverse mechanism, the control input u(k) is then determined in the presence of the input v(k). Under the assumptions that the fuzzy model and its inverse are stable, the effect of constant disturbances and the model mismatch are compensated in the steady state.

 3sinθ − 3θ cos θ − sin3θ  Vs1 = R 2 L1 θ − 0.5sin(2θ) −  (13) 3(1 − cos θ)   Where θ is given by:

The filter F0, which guarantees the robustness of stability with respect to the plant-model mismatch, is

θ = cos −1 (1 −

753

h ) R

(14)

fuzzy clustering. In this paper, we will determine a singleton fuzzy model instead of the TS fuzzy model because the inversion is easier to calculate. Following the same methodology developed in [6][18], a singleton fuzzy model was calculated from process measurement. A set of 2000 data for each variable was collected. The next table shows the resultant singleton fuzzy model: Rule Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Figure 6: Oil Separation Hydrodynamics under Normal Operation Conditions

According to equation (13) and (14), the volume of oil of a production separator has a very strong nonlinear dependence of the h (The level of oil of the production separator). Then, applying the principle of mass balance, the following equation is obtained Fin − Fout =

d (Vs1 ) dt

(15)

Where the term Fin is the flow of the fluid that enters the separator, the term Fout is the flow of fluid at the output of the separator and the term dVs1/dt is the variation of the volume with respect to the time of the fluid inside the separator. It is important to highlight that the term Fout is given by: Fout = CV u

F ∆P ⇒ u = out G Cv

G ∆P

y(τ)

u(τ)

y(τ+1)

Low Normal High Very High Low Normal High Very High Low Normal High Very High Low Normal High Very High

Open Open Open Open Half Open Half Open Half Open Half Open Almost Closed Almost Closed Almost Closed Almost Closed Closed Closed Closed Closed

3,0141 3,1626 3,2151 3,7478 3,0380 3,2040 3,4358 3,6999 3,0001 3,4000 3,5286 3,6725 3,0010 3,5338 3,5807 3,7006

Where y(τ) is the current level of oil in the production separator, u(τ) is the valve position located at the output of the production separator and y(τ+1) is the level of oil at the next sampling instant. Four linguistic terms were defined for the two inputs as is shown in the rule base.

(16)

Where Cv is the coefficient of the valve, u is the aperture of the valve (expressed in percentage), G is the specific gravity of oil and ∆P is the differential pressure in the valve. By applying the Taylor Series (First order approximation) to the nonlinear terms u and Vs1 taking into account that hl = h – h0 and ul = u – u0, where h0 and u0 are the equilibrium points of the variables h and u, substituting the numeric value of R, L1, Cv, ∆P, G and finally applying the Laplace transform, then the next solution is obtained:

hl (s) 0 .0011 = u l (s) 2s + 1

(17)

Figure 7: Validation of the singleton fuzzy model obtained from data measurement

4.2 Fuzzy Modeling In [18] a TS fuzzy model was determined for the production separator through the use of product space

Figure 7 shows the validation made on a different data set from the one used for the identification.

754

Figure 8 shows the membership functions for the two inputs u(τ) and y(τ) respectively.

The inverse fuzzy model and the fuzzy model were developed in MATLAB / Simulink through the use of S-functions. A simple illustration of the inversion implementation is shown in figure 10. Figure 10 shows the inversion algorithm for r(τ+1) = 3.45 and y(τ) = 3.50. When the fuzzy inverse is placed in series its model, it can be demonstrated that the inversion is exact, i.e., r(τ+1) = y(τ+1) = 3.45.

Figure 8: Membership Functions of u(τ) and y(τ)

4.3 Inverse fuzzy model The inversion of the fuzzy model obtained in section 4.2 is calculated through the application of equation (9), (10) and (11). In this particular case, N = 4 and M = 4. For a given state x(τ) = y(τ), the degree of fulfillment of the first antecedent proposition ‘x(τ) is X’, is denoted by µXi(x(τ)). Based on equation (10), the consequents cj(τ) are given by: 4

cj =

∑µ i =1 4

Xi

∑µ i =1

( x (τ )) cij

Xi ( x (τ ))

, j = 1, 2,3, 4 (18)

Figure 10: Fuzzy inverse in series with its model

4.4 Simulations Results The implementation of the Model Based Fuzzy Controller along with the IMC scheme was done in Simulink based on the scheme depicted in figure 5.

Equation (18) generates the following generic base rule: If u(τ) is Uj then y(τ+1) is cj(τ), j=1,2,3,4 (19) An example of membership functions Cj(τ), j = 1,2,3,4 of the fuzzy partition created by using the consequent singletons c1(τ) , c2(τ), c3(τ) y c4(τ) is shown in figure 9. Assuming that the rule base is monotonic, it can be inverted resulting in: i = 1,..,4 (20) If r(τ+1) is Ci(τ) then u(τ) is Ui

Figure 11: Implementation of a fuzzy model based control under the IMC scheme

Figure 9: Fuzzy partitions created from c1(τ) , c2(τ), c3(τ) and c4(τ).

755

Figure 11 shows the detailed connection of the block function in Simulink.

state errors occurred in the other scheme were reduced in the presence of disturbances.

6. References [1] Sousa J., Kaymak U. (2002) "Fuzzy Decision Making in Modeling and Control". World Scientific Series in Robotics and Intelligent Systems, pp. 93-208. [2] Boukezzoula R., Galichet S., and Foulloy L. (2007) “Fuzzy Feedback Linearizing Controller And Its Equivalence With Fuzzy Nonlinear Internal Model Control Structure”. Application of Mathematics and Computer Science, University of Savoie, France, pp. 233-248. Figure 12-A: Output y(t) of the Controlled Process Without ICM Scheme

[3] Boaming G., Jingping J., Pengshen S., and Xiangheng W. (2002). “Nonlinear Internal-Model Control For Switched Reluctance Drives”. IEEE Transactions in Power Electronic, Vol 17, No 3, pp. 379-388. [4] Kang J., Kwon C., Lee H. and Park M. (1998). “Robust Stability Analysis and Design Method For The Fuzzy Feedback Linearization Regulator” IEEE Transactions on Fuzzy Systems, Vol 6, No 4, pp. 464472. [5] Kwanghee, N. (1999). “Stabilization Of Feedback Linearizable System Using A Radial Basis Function Network” IEEE Transactions in Automation And Control, Vol 44, No5, pp. 1026-1031.

Figure 12-B: Output y(t) of the Controlled Process With ICM Scheme

Figure 12-A shows the result the output of the process with its respective changes of the signal reference without the ICM scheme and figure 12-B shows the output of the process with the ICM scheme with the addition of a small disturbance signal.

[6] Babuska, R. (1998). “Fuzzy Modeling For Control”. Dordrecht: Kluwer Academic Publishers. [7] Boukezzoula R., Galichet S. and Foulloy L. (2003). “Nonlinear Internal Model Control: Application Of Inverse Model Based Fuzzy Control”. IEEE Transactions on Fuzzy System, Vol 11, No 6, pp. 814829.

5. Conclusions In this paper, we have shown how to implement a fuzzy controller based on a fuzzy model of a complex process. The fuzzy modeling of a production separator obtained had a good generalization during its validation although its original mathematical model is quite complex.

[8] Fang W. and Rad A. (2000). “Fuzzy Adaptive Internal Model Control”. IEEE Transactions on Fuzzy System, Vol 47, No1, pp. 193-202. [9] Abonyi, J., Nagy L. and Szeifert F. (2005) “ Indirect Model Based Control Using Fuzzy Model Inversion”. Department of Chemical Engineering, University of Vezprem, Hungary.

Some computational validations were done in order to demonstrate the correct functioning of this sort of control scheme. The fuzzy controller showed an acceptable performance without the ICM scheme with some steady-state errors, however, with the implementation of the ICM scheme most of the steady-

[10] Riid A., and Rustern E. (2005). “Automatic linguistic Inversion of Fuzzy Model”

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[11] Abonyi J., Babuska R., and Szeifert F. (2002). “Fuzzy Modeling With Multidimensional Memberships: Grey-Box Identification and Control Design”. [12] Abonyi J., Babuska R., Ayala M., Szeifert F. and Nagy L. (2004). “Identification and Control of Nonlinear System Using Fuzzy Hammerstein Models”. University of Vezprem, Department of Process Engineering. [13] Garcia, C. and Morari, M. (1989). “Internal Model Control: A unifying review and some new results”. Automatica, 25, pp. 335-348. [14] Economou C., Morari M. And Palsson B. (1986) “Internal Model Control: Extension to Nonlinear Systems” Institute of Chemical Engineers, Proc. Des. Dev, Vol 25, No. 5. pp. 403-409. [15] Li H-X. and Deng H (2006) “An Approximate Internal Model-Based Neural Control For Unknown Nonlinear Discrete Processes”. IEEE Transactions on Neural Network, Vol 17, No. 3, PP 659-670. [16] Atalla, F and Taylor J. (2004). “Modeling of three-phase gravity Separator in Oil Production Facilities”. Cape Breton University, Canada. [17] Holland, C (1984). “Fundamental and Modeling of Separation Processes: Absorption, Distillation, Evaporation and Extraction”. Englewood Cliffs, N.Y: Prentice-Hall. [18] Ramirez M., Colina E. (2007) "Fuzzy Clustering Based Models For Supervision of Industrial Processes". Proceedings of the 6th WSEAS International Conference on Mathematical and Computational Intelligence, Man-Machine System & Cybernetics, Tenerife, Spain, pp. 272-281.

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