Chemical Engineering and Processing 41 (2002) 87 – 98 www.elsevier.com/locate/cep
Fuzzy temperature control of industrial refineries furnaces through combined feedforward/feedback multivariable cascade systems A.G. Abilov a,*, Z. Zeybek b, O. Tuzunalp a, Z. Telatar a a b
Department of Electronic Engineering, Faculty of Sciences, Uni6ersity of Ankara, Tandogan, Ankara, 06100, Turkey Department of Chemical Engineering, Faculty of Sciences, Uni6ersity of Ankara, Tandogan, Ankara, 06100, Turkey Received 2 March 2000; received in revised form 1 December 2000; accepted 5 January 2001
Abstract The purpose of this paper is to improve and apply a multivariable advanced control structure on the basis of fuzzy logic technique for two flow tubular furnace having widespread applications in petroleum refinery industries. After analyzing the dynamic properties of furnaces, it has been concluded that these furnaces are the MIMO processes which have two inputs and two outputs. There are a lot of reciprocal interactions between input and output variables. For this reason, equivalent system methods were used to investigate and develop advanced control structures for these furnaces. According to this method, symmetric MIMO system was divided into two equivalent separate systems. Finally, the fuzzy model control design of combined multivariable symmetric cascade system and its results were given. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Industrial refineries furnaces; Fuzzy advanced control; Multivariable cascade; Feedback/feedforward control
1. Introduction Furnaces are the basic and the most important industrial units of petroleum refineries and petrochemical process. They are used in heated petroleum flows to the desired temperature through chimney gases obtained from burning processes The furnaces can have single or multiple flows according to their technological structures. They have been made up from the regions of convection and radiation. The heat transformation between chimney gasses and petroleum flows is realized in the regions of convection and radiation. Sufficient quantity of the oxygen required for the control of the burning process in these regions is obtained from the air. Also, an optimal amount of air is desired for economical and ecological clear burning in furnaces. There have been many studies in literature to control the burning process [1 – 7]. These works have shown that furnaces are nonlinear, multivariable, distributed, Abbre6iations: AVF, Atmospheric vacuum furnace; MIMO, Multivariable input– multivariable output. * Corresponding author. Fax: +90-312-2232395. E-mail address:
[email protected] (A.G. Abilov).
complex dynamic control systems. In multiple flow furnaces, there are also large delays in dynamic channels and cross relationship between control parameters. In most applications, a linear model has been used to represent the dynamic behavior of the process (impulse/ step response function, transfer function, state space model). But, because of severe nonlinearities in some real-life processes, a fixed linear model for predictive control might not really result in the required performance. Because of these complexities, it is required to investigate of the more effective control systems working on the basis of fuzzy advanced control algorithms. Fuzzy control is now well known as a method of implementing nonlinear controller. Although most industrial applications of fuzzy controller have been based on rule-based controllers, there is a growing interest in model-based fuzzy designs. Many fuzzy modeling and control methods have been proposed in the literature [8–17]. In this paper, a combined multivariable cascade advanced fuzzy control system has been developed for two flow industrial petroleum refinery furnaces. The following sections present the proposed algorithm in detail. The simulation results are then given. Finally, the conclusion is drawn.
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2. Process description Furnaces are the basic and the most important units in petrochemical processes. In the industrial atmospheric vacuum furnaces (AVF), crude oil is passed through two spiral pipes and divided into two flow and then it enters the burning chambers in the convection and radiation sections. Firstly, the petroleum flow is heated by chimney gases, and then at the outlet, two flow are combined by the flame in the burning chamber. The unit operates between the 700 and 900°C intervals with a duty to bring the crude oil’s outlet to 320°C, and to leave the stack gases at 750°C. The outlet temperature of the petroleum in the left and right pipeline is obtained by supplying sufficient quantity of oxygen and natural gas to right and left fuel chambers. This creates reciprocal relations between these parallel transfer channels. When these properties of furnace are considered, it is quite important to develop and apply the new temperature control algorithms. In a burning process, the air/fuel ratio must be dmin 5d5 dmax. So, the complete combustion affecting
the concentration in the chimney gases that pollute the environment is achieved by adjusting the vacuum required for the furnace. On the other hand, the complete combustion is evaluated by the concentration of the oxygen in the chimney gases. Besides the temperature of the combustion depends significantly on the quantity of the air. The schematic diagram of an industrial petrol refinery furnace is illustrated in Fig. 1.
2.1. Transfer functions of the process Our industrial research on the dynamic characteristics of furnaces reveal that furnaces consist of a MIMO system with two inputs and two outputs and that there are symmetric reciprocal interactions between input– output variables. A block diagram showing the dynamic channels is given in Fig. 2. Where x1, x2; consumption of the natural gas given to the left and right fuel chambers. xf: consumption of the petroleum flow given to furnace. y1,y2: the outlet temperatures of the petroleum in the left and right sides. ycg1, ycg2: the temperature of the chimney gases which enter the left and right sections of the convection chamber. The transfer function of the dynamic channels was determined from the reaction curve of the process obtained by giving9 step disturbances to the fuel inlet of the furnace. Since the furnace with two flow is a symmetric process, the transfer functions are as follows: W11(s)= W22(s), W12(s)= W22(s), K11(s)= K22(s) The Laplace domain transfer function of the symmetric multivariable system is: W11(s) = [5×102/(3.56×106 s3 + 7.96× 104 s2 + 4.71×102 s + 1)]e-180s
Fig. 1. Schematic diagram of the refineries furnace.
(1)
°C/% maximum fuel consumption W12(s) = [4×102/(8.74×106 s3 + 10.9× 104 s2 + 5.31×102 s + 1)]e − 180s
(2)
°C/% maximum fuel consumption Wf1(s) =[1.6× 102/(5.22×106 s3 + 8.68× 104 s2 + 4.88×102 s + 1)]e − 180s
(3)
°C/% maximum fuel consumption K11(s)= [10×102/(3680 s2 + 280 s+1)]e − 180s Fig. 2. Block diagram of the dynamic channels.
°C/% maximum fuel consumption
(4)
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Here, W0(s)is the transfer function of the main control channel; W1(s) is the transfer function of the inside cross dynamic channel and Wf(s) is the transfer function of the petroleum flow as a disturbance. Since the system is symmetric: W0(s)= W11(s)= W22(s)
(7)
W1(s)= W12(s)= W21(s) Wf1(s)= Wf2(s) Fig. 3. Block diagram of the symmetric control systems with two inputs– outputs.
Due to the symmetry of the system, the transfer function of the main and cross channels are identical and in accordance with one another. Differential equation of the controller u1 = WR(s)(x1 − y1)
(8)
u2 = WR(s)(x2 − y2) The form of transfer function matrices Y=[W0(s)E + W1(s)A]u+ Wf(s)Ef Ã0 A= Ã Ã1
1Ã Ã 0Ã
Ã1 E= Ã Ã0
0Ã Ã 1Ã
Ãf1 Ã f= Ã Ã Ãf2 Ã
(9) (10)
U= WR(s)E(x− y)
2.3. Equi6alent decoupled control system
Fig. 4. (a) Block diagram of control of the equivalent transfer function. (b) Block diagram of coupled equivalent separate system.
Equivalent block diagram of the relevant control system is given in Fig. 4a and 4b. In this system, because the transfer function of the equivalent channels is symmetric, it is identified, as follows. Weq1(s)= Weq2(s)= W0(s)9 W1(s)
(11)
2.4. Control system in multi6ariable cascade
Fig. 5. Block diagram of cascade equivalent separate system.
2.2. Symetric multi6ariable control system of the process Block diagram of symmetric control system with two input –output for two flow furnaces is given in Fig. 3. The differential equations of this system can be formulated as follows. y1 = W0(s)u1 +W1(s)u2 +Wf(s)f1
(5)
y2 = W1(s)u1 +W0(s)u2 +Wf(s)f1
(6)
A block diagram of the multivariable cascade control of the equivalent system is shown in Fig. 5. In this separated system, the object of the control was also divided into two stages [1,2]. The transfer function of the internal chimney gases is K11(s) and the transfer function of the equivalent channel is Weq(s). In this system, the same controller could be used for both of two control loops. The main feature of the system is that the optimum parameters of the stabilizator and the regulator controller are found in two control loops with different frequencies. For this purpose, firstly the parameters of the stabilizator loop are calculated and then the parameters of the regulator are evaluated within the framework of the calculated optimum parameters. In this stage, for the evaluation of the parameters of the regulator loop, a transfer function is obtained in the
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form of a transfer function including the closed stabilizator loop and the transfer function of the equivalent channel. So the transfer function of the controller object is as follows. W*eqv(s) = K11(s)/1 + WR(s)K11(s) +Weqv(s)
(12)
2.5. Combined feedback/feedforward control system in multi6ariable cascade of process As known, these systems consist of two loops. One of them is open loop based on compensating the exterior effects (disturbances) and the other is closed loop based on the feedback control principle. Combined open and closed loop multivariable cascade control structure of the furnace is given in Fig. 6. u = x1 −WR(s)E;
(13)
yeqv1 =Weqv(s)u =(W0(s) −W1(s)) u;
(14)
h= Wf(s)xf1;
(15)
y*eqv1 = yeqv1 +h;
(16)
ycg1 =K11(s).u;
(17)
u*= ycg1 +WR2(s)y*eqv1 − Wc(s)xf1;
(18)
The transfer function of the whole system between the inputs (x1, xf1) and the output (y) are described as follows. y*eqv1 =
Weqv1(s) x + 1+ weqv1(s)WR1(s)WR2(s) +WR1(s)K11(s) 1 (19)
n
Wf1(s)+ WR1(s)K11(s)Wf1(s) + WR1(s)We(s) xf 1 Weqv1(s)
From this statement, according to provision of the absolute invariant, the transfer function of the compensator is defined as follows. Wc(s)= − [WR2(s)+ WR2(s)WK11(s)WRf1(s)]/Weqv1(s) × WR2(s);
Fig. 6. Combined feedforvard/feedback multivariable cascade.
(20)
When the numerical values of the transfer functions that are WR2(s),K11(s),Weqv12(s),Wf1(s), are put on the equation (We(s)), the order of numerator becomes 9, and the order of denominator becomes 8. Mostly, in this type of system, because the transfer function determined by the principle of the absolute invariant consists of a higher degree of fraction, realization of this statement can be difficult. This feature can practically be expressed by showing the compensatory as a simplified differential block, Wc(s)= K
Ts Ts + 1
(21)
Naturally, the compensator block expressed in this way cannot completely compensate the external disturbances. But, by selecting the appropriate parameters of the K and t(T), the outlet value of y*eqv1 could be made independent of external disturbance. Since the block of the compensatory cannot affect the stability of the system there is more freedom to determine its parameters. When the system was designed, optimum parameters of the closed loop system were selected and the parameters of the compensator were calculated in a way to reduce the dynamic deviations to minimum.
3. Fuzzy model identification and control Most fuzzy controllers have been designed on the basis of the experience of a human operator and/or the knowledge of a control engineer. It is, however, often the case that an operator cannot tell linguistically what kind of action he takes in a particular situation. In this respect it is quite useful to model his control actions using numerical data. Further, we have to develop model-based control just as in ordinary control theory if there is no reason to believe that an operator’s control is optimal. For this purpose, it is necessary to consider for fuzzy modeling of a system [11]. For this purpose, the fuzzy relational model with a new approximation was improved to control the outlet temperature of the chimney gases and petroleum in furnace with two input–two output, utilizing the algorithm which is based on fuzzy adaptive model in literature [12]. In this paper, the fuzzy relational matrix with three dimensions has been done, utilizing the pseudo random binary sequence (PRBS), before the fuzzy control of the system was done. PRBS signal is tuned according to structure of the system and given to the natural gas /air which is the manipulated variable. Therefore the reference sets and scaling factors are used for the outlet temperature petroleum and chimney gases and also natural gas/air ratio deviation variables from steady state values. Required data was generated
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by applying the PRBS uniform effects in every 10 min. The deviations from steady state of quantity of the natural gas/air ratio are distributed between 9 5%. The adaptive fuzzy models discussed so far have described systems with zero order dynamics i.e. the current output depends only on the current input.
3.1. Fuzzy models for dynamic system components For the discrete-time representation of our third-order process Eqs. (1)– (4) obtained by the third order difference approximation of the derivative. Also, the digital transfer function (z-transform) of the process can be written as in third or second order, and e -st in the transfer function of upstairs can be calculated by the separation to the first degree Pade series. This functional relationship is easily described by matrix form as follows, y’(k) = R $ y(k − 1) $ u1(k − ~i ) $···$ un(k − ~n)
(22)
where y and u are possibility vectors for model output and input respectively and R is the model relation matrix. Subscript k represents current time, k-1 one sampling time in the past and k-ti the dead time of d sampling periods in the past. If a process has n inputs, it can be represented by a relational fuzzy model according to the Eq. (22) where, y’ : array of memberships of the output reference sets representing the predicted value of the process output, R: relational array (an array which represents the fuzzy relationships between the inputs and outputs), uI : array of memberships of the input reference sets representing the manipulated variable i, ~i : a ‘dead time’ which represents the delay between a particular input changing and its effects being observed at the process output, °: fuzzy compositional operator. Relation matrix could be written as a function of the current error which references the outlet temperature of the petroleum and chimney gases, and the current process input, referring the natural gas/air in the furnace. R = u k − d $ ek − 1 $ e k k
(23)
R = u R( k or
(24)
R(i*, j *,k) = aR( (i*, j *,k), k =1,N
(25)
k=1
R(i*, j *,k)=max(R( (i, j,k),R(i, j,k)) for all i, j,k except i =i*, j =j * (26) e(k) = R(i*, j *,k) $ e(k − 1) $ X(k −~1)
(27)
where a is a scalar constant between 0.5 (good noise rejection) and 1.0 (fastest update), i* and j * are the positions of the maximum membership values in the possibility vectors uk and yk − 1 respectively. Early model-based fuzzy controllers shown in Fig. 8 have been reported in the literature by Graham and
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Newell. Suggested controller is the same as Graham and Newell’s controller [11,12], but there are differences about learning of the process model in this paper. The model is used to predict what the output would be for each of these actions. The decision maker then selects the most favorable action to take, for example, the one which results in the smallest error. The selected control action is then applied to the process and the whole procedure repeats itself at each control interval. There are two factors, which require careful engineering: 1. the number and range of the fuzzy reference sets must be selected in such a way that the model is a fair representation of the process over the expected range of operating conditions; 2. the set of possible control actions must be selected to give sufficient, yet fine enough, control. There are as yet no guidelines to assist with the selection of these factors.
3.2. Fuzzy relation matrix for the fuzzy identification and control of furnace To use the fuzzy controller, the first step was to produce a relational fuzzy model of the process. The model used in this case was developed with structure. DTcg(k)= R $ DTcg(k−1) $ DX(k− ~1)
(28)
DT(k)=R $ DT(k− 1) $ DX(k− ~1)
(29)
where DTcg(k+1); The measurement temperature of the chimney gases mines set point (720°C) and (DT(k + 1)); the outlet temperature of the petroleum mines set point (320°C) at the current sample k are related to both the temperature and inlet natural gas/air ratio (DX(k− t1)) at the previous sample. Five fuzzy reference sets were used to describe each of the variables and firstly, the model was identified from I/O data generated by making open-loop variations in the inlet natural gas/air ratio. The sampling time was set to 10 s (the same as that used in [11,12], and the controller set up with a prediction horizon of 1. This manipulation did not succeed, because the process was a third order and consisted of a big dead time. Therefore, we considered the fuzzy identification method suggested previously for applying to this furnace [15]. According to this method, the reaction curve of the process is separated into two regions. First region has a dead time effect, and a mean slope of 0.5, the second region has a mean slope of 1.4 which is bigger than 1.0. However, two different fuzzy reference sets describe two regions of the process. But relational matrix (R) obtained from I/O data generated by making open loop variations is modified according to fuzzy membership degree of the central value of the fuzzy sets. Because, the memberships of the center of the fuzzy sets are
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Fig. 7. Surface representation of the fuzzy relation between input and output variables of furnace.
region on reaction curve) is also chosen for the change in error. This approximately corresponds to the magnitude of change in error over one sampling time that results from the maximum change in control with the process (W11(s) initially at steady state at temperature 320°C) (Fig. 9). The developed model is used to predict the output for possible control actions. Eleven allowable control changes in natural gas/air ratio are taken as the values of 0.0, 9 0.01, 9 0.02, 90.03, 9 0.04, 9 0.05. These were added in turn to the current values of x1(natural gas/air ratio) and fed to the model together with the current values of outlet temperature of the petroleum and chimney gases. The model calculates the expected value of the outlet temperature of the petroleum at the reset control interval. The decision maker then selects the most favorable action to take the one, which results in the smallest error. The selected control action is then applied to the process and this procedure is repeated at each control interval.
4. Application results
Fig. 8. Fuzzy adaptive-model based controller.
approximately 1.0, there are five center values and five fuzzy memberships of the fuzzy sets for each of DT(k+ 1), DT(k), Dx(k). Then : 1.0 put on the part (co-ordinates) of the relational matrix of 5×5 × 5 which equals to the membership degree of each of center of the DT(k+1), DT(k), Dx(k), fuzzy sets. In this stage, we have (obtained) a standard relation matrix which will be used in two regions. When the modified fuzzy relation matrix is used, the process model could be identified with all conditions. The relation matrix with 5×5 × 5 dimensions is given in Fig. 7. However, the fuzzy model identification must be taken into consideration when and which sampling intervals of the fuzzy reference sets should be activated. The fuzzy controller system block diagram in Fig. 8 was given for describing the multivariable cascade control and combined feedback/feedforward control of the furnace. For the K11(s) and Weqv(s), which are described in Section 3.1, a scaling factor of 0.01 is chosen for the control action. This is large enough to provide adequate control action away from set point, yet small enough for good control close to set point. A scaling factor of 2.4 and 7.3 (for fuzzy identification of second
In this Section, the results obtained from application of the MIMO cascade fuzzy advanced control structure in industrial refineries are given. The performance of the suggested fuzzy advanced control structure was shown and compared with the performance of the MIMO cascade of the classical controller design. The dynamic model of the channels of the furnace with initial condition y(0)= 0 is used in the simulation. To compare the effectiveness of a fuzzy controller with a PI controller, two kinds of disturbance are introduced into the system. The results are summarized below. In Fig. 10 (a–d), the reaction curve of the W11(s), Weq(s), K11(s) and Wf(s) that are the dynamic channels of the process and their comparisons with the results of fuzzy dynamic simulation are given. Furthermore, in Fig. 10, how the developed model learns the identification process was used to predict the output for possible disturbance of natural gas/air ratio. It showed good performance for all dynamic channels. The e -st in the transfer function of upstairs was calculated, separating to first degree Pade series. The results for the application of procedure explained in Section 3.1 show how the fuzzy relation converges on the process function. During these learning processes, the integral of squared error was evaluated (the error between the predicted output and the real function). The four values are (4.95, 0.61, 0.169, 0.148) for the W11(s), Weqv(s), K11(s) and Wf(s) application of the learning procedure respectively. The simulation results of the outlet temperature of petroleum on main control channel W11(s) and the
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temperature of the chimney gases on interval channel K11(s) to the input disturbance (unit step function) are shown in Fig. 10(a) and Fig. 12(a). With the use of fuzzy controller, the outlet temperature of the petroleum is kept at the desired temperature. Practically, all the time adjusting the outlet temperature of the petroleum is insensitive to this kind of disturbance. From Fig. 12(a), it is clear that both rising time and overshoot in the case of a fuzzy controller are less than PI controller (shown in Fig. 19). In general, a shorter rise time yields a larger overshoot. For better system performance, we need to reduce both of them, as we have done here. Since the outlet temperature of the petroleum and the temperature of the chimney gases are related to each
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other, the overall control system performance is satisfactory only if both of them are controlled properly. In Fig. 14(a), the same perturbation of the input was given on the equivalent channel Weqv(s), the same performance was observed in Fig. 15 for fuzzy controller in Fig. 12(a). The fuzzy simulation result was better than the PI control which had waited about 50 min (3000 s) to reach its desired value (in Fig. 19(2)). Furthermore, Figs. 11, 13 and 15 show that using a fuzzy controller for the outlet temperature of the petroleum can be maintained at its desired value even with disturbance of 5% of the petroleum flow (xf) which is outside the effect of the furnace.
Fig. 9. Fuzzy set definition.
Fig. 10. Relation-based identification from process models which are (a) K11(s), (b) W11(s), (c) weqv(s), (d) Wf1(s), (e) Graham’s globally relational-based identification from process model, weqv(s).
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Fig. 11. (a) Fuzzy adaptive control of outlet temperature of petroleum on W11(s) channel which is based on input perturbation of 5%. (b) Manipulated.
Fig. 12. (a) Fuzzy control of W11(s) channel with external disturbance of 5%. (b) Manipulated.
Fig. 13. (a) Fuzzy temperature control of K11(s) with input perturbation of 5%. (b) Manipulated.
Fig. 16 shows the performance of a fuzzy cascade controller where the manipulated variable affects the two controlled variables that are the outlet temperature of the petroleum (y °C) and the temperature of
the chimney gases (ycg °C) in parallel. An important fuzzy control application was applied in the furnace temperature control where natural gas /air ratio affects both of ycg and y. the process has a parallel
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structure and this leads to a parallel cascade control system (Fig. 5). It can be seen from the results of fuzzy– fuzzy control (Fig. 16a) that, secondary loop being relatively fast, the
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inner fuzzy controller acts immediately (curve ycg) upon to the input disturbance entering at increase of 5%. When the study of cascade is compared with fuzzy– fuzzy versus PI–PI control (Fig. 19(3)), the results
Fig. 14. (a) Fuzzy control of K11(s) channel with external disturbance of 5%. (b) Manipulated.
Fig. 15. (a) Fuzzy adaptive control outlet temperature of petroleum on Weqv(s) channel which is based on input perturbation of 5%. (b) Manipulated.
Fig. 16. (a) Fuzzy adaptive control of outlet temperature of the petroleum on Weqv(s) channel which based on external disturbance of 5%. (b) Manipulated.
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Fig. 17. (a) Fuzzy cascade control of the outlet temperature of the petroleum with external disturbance of 5%. (b) Manipulated.
Fig. 18. (a) Combined feedback/feedforward fuzzy cascade control of temperatures in furnace. (b) Manipulated.
show that, performance of the fuzzy cascade control was better than the PI cascade control of this furnace. In addition, Fig. 17(a) shows that using a fuzzy cascade controller the outlet temperature can be maintained at its desired value with the disturbance of the petroleum flow which is an outside load on the furnace. But the set value of the temperature of the chimney gases was dropped by time in primary control loop (inner), while the secondary loop (outer) was controlled. Fig. 17(b) is like ‘on – off’ type control since the output is either a positive deviation (increase the quantity of air) or a negative deviation (increase the quantity of the natural gas). One benefit of this is that we can start the controller up ‘naively’; the controller will take in process data on-line, and update the weights to proper ones in the control of the process. The system requires no prior tuning; just plug it in and turn it on. Process profiles of the combined multivariable fuzzy cascade control system for the outlet temperature of the chimney gases and petroleum are shown in the form of feedback/feedforward in Fig. 18.
The results of related figure are signed that the stability of the system was not affected from the external disturbance in the form of feedback/feedforward. In practice, many feedforward control systems are implemented by using ratio control systems. Most feedforward control systems are installed as combined feed-
Fig. 19. Responses of PI control to the (1)-W11(s), (2)-Weqv(s), (3)-Cascade control of Weqv(s) and 4-K11(s).
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Table 1 The optimal PI parameters of the control systems in furnace [5] Control systems
For W11(s) For Weqv(s) Multivariable cascade control Combined multivariable cascade
Loop of regulaton
Loop of stabilization
Loop of compensation
K×104
T×104
K×104
T×104
K×104
T×104
−5.85 44 5.5 5.5
0.05 0.01 0.013 0.013
70 70
1.12 1.12
0.1
20
forward–feedback systems. The feedforward controller takes care of the large and frequent measurable disturbances. The feedback controller takes care of any errors that come through the process because of inaccuracies in the feedforward controller or other unmeasured disturbances. The comparison of our study based on cascade fuzzy control of the furnace with PI cascade control of the furnace and PI control of other channels were shown in Fig. 19. PI controller response of these systems that are W11(s), Weqv(s) and K11(s) and cascade controller were performed in Fig. 19 and its parameters were listed in Table 1. When the results of PI controller are compared with the fuzzy controller design, Plots of the Fig. 19 are not better than the design of fuzzy controller. As it is seen in the figures, the fuzzy control algorithm applied to a furnace with two flow in the form of a dynamic matrix cascade temperature control, gives a better result than other classical control algorithms [1– 5].
5. Discussion and conclusions In this paper, the application of the multivariable cascade fuzzy advanced control structure for an industrial AVT furnace is investigated. In this respect, fuzzy relational model of the process is determined by considering all the disturbance effect of the input –output variables on all dynamic channels of the furnace with two flow. When the results of the fuzzy control system designed on the basis of fuzzy relational model are compared with those of the classical control systems, it is seen that, the fuzzy controller reaches damping in a shorter time and yields a better performance. In addition, the feedforward controller has no effect on the closed loop stability of the system for linear systems. The denominators of the closed loop transfer functions are unchanged. In a nonlinear system, the addition of a feedforward fuzzy controller often permits tighter tuning of the feedback controller because the magnitude of the disturbances is reduced. Finally, experimental tests of the control of tempera-
ture in industrial furnaces using fuzzy advanced control system has also verified the conclusion that fuzzy control system give considerable performance.
Appendix A. Nomenclature unit matrix of the process; unit matrix of the error; the feed control; temperature of feed; the gain value of the compensator; the transfer function of the interval channel in left side; K22(s) the transfer function of the interval channel in right side; PC control of pressure; R relation matrix with three dimensions; RC ratio control; TC1 temperature control of the chimney gasses; TC2 the outlet temperature of the petroleum; t dead time; T time constant; U manipulated variable of fuzzy process; W11(s) the transfer function of the main channel in left side; W12(s),W21(s) the transfer functions of the inside cross dynamic channel of the furnace; W22(s) the transfer function of the main channel in right side; Wc(s) the transfer function of the compensator; Weqv(s) the transfer function of the controller object for equivalent channel; Wf1(s) the transfer function of the petroleum, given to the left side as a disturbance; Wf2(s) the transfer function of the petroleum flow in right side as a disturbance; the transfer function of the PI control WR(s) or fuzzy control block;
A E FC FT K K11(s)
98
WR1 (s)
WR2(s)
X x1 x2 xf1 xf2 y1 y2 ycg1 ycg2 y *eqv
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the transfer function of the PI control or fuzzy control block in cascade control of furnace; the transfer function of the PI control or fuzzy control block in cascade control of furnace; manipulated variable; consumption of the natural gas/air ratio, given to the left chambers; consumption of the natural gas/air ratio, given to the right chambers; consumption of the petroleum flow, given to the right chambers; consumption of the petroleum flow, given to the right chambers; the outlet temperature of the petroleum in left side; the outlet temperature of the petroleum in right side; the temperature of the chimney gasses in left side; the temperature of the chimney gasses in right side; output of the equivalent channel which is independent of the external disturbance.
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