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Model Reference Control Based on Compensatory Fuzzy Neural Network for Gas Collectors of Coke Oven
3rd IFAC International Conference on Intelligent Control and Automation Science. September 2-4, 2013. Chengdu, China
Model Reference Control Based on Compensatory Fuzzy Neural Network for Gas Collectors of Coke Oven Hongxing Li* and Xiangling Kong** * Institute of Automation, Beijing Union University, Beijing, 100101 China (Tel: 086-010-64900502; e-mail: [email protected]). ** Institute of Information, Beijing Union University Beijing, 100101 China (e-mail: [email protected])} Abstract: The pressure system of gas collectors of coke oven is a multivariable non-linear process. In this paper, a model reference adaptive control using the compensatory fuzzy neural network for the pressure system of gas collectors of coke oven is presented. The dynamics model of the fuzzy neural network of the system is identified by the adaptive compensatory fuzzy learning algorithm, which can be employed as the identifier of the system. Another fuzzy neural network is trained to learn the inverse dynamics of the pressure system of gas collectors of coke oven so that it can be used as a nonlinear controller. The simulation results testify that the model obtained is satisfied and the control is effective. Keywords: Model reference control, compensatory fuzzy neural network, pressure of gas collectors, multi-variable system. coupling among the pressure of the collectors and the pressure before the blast blower. When the pressure of one collector is changed, the pressure of other collector will certainly be affected. When the pressure before the blast blower has a little change, the pressure of two collectors may have much change. If the change is sudden and strong, the pressure will react so sluggishly that the system oscillates (Yanga et al, 2001).
1. INTRODUCTION The pressure of gas collectors is an important industrial parameter in thermodynamic system of coke ovens. Its stability directly influences the life-time of coke ovens and the quality of coke production. The stabilized pressure in gas collectors can prolong the life-time of coke oven, reduce the leak of coal gas, lighten environment pollution and save energy. When the pressure of gas collectors is negative, air is able to enter into coke oven chamber and result in burning the coke, increasing the ash and debasing the coke quality. Besides, the air is likely to happen chemical reaction with architectural material of the oven, which brings about that the oven is eroded and the life-time is shortened, sometimes it may be even endanger blast blowers. On the other hand, too high pressure in gas collectors will let the gas fire and leak out, which will not only shorten the ovens’ life-time but also bring poisonous pollution and waste energy. It is obvious that the pressure control of the gas collectors of the coke oven is very important in the coke production process. In general, the pressure of the gas collector should be varied between 80 and 120 Pa (Coking experts group, 1978).
There are two coke-ovens 1# and 2# in the plant, and they have single collector pipe. The pressure system of gas collectors is composed of two single collector pipe, initial cooler, blast blower and circumfluence equipment, as shown in Fig.1.
Fig. 1. Collector structure of coke oven.
The pressure system of gas collectors is a strongly intercoupled, slow time-varying, nonlinear multi-variable system where interference is strong and frequent. The reason for varying the flow of coke oven gas has loading coal and unloading coke in batch, variation of coking time and heating system, change in numbers of opened loading doors and ascending pipes and so on, which they are the direct disturbance to the pressure of collectors. At the same time, the influence of the pressure of collectors has variation of attracting power of a blast blower, variations of temperature, flow of cycling ammonia water, change of the load by gas users and so on yet. Furthermore, there exists inevitable 978-3-902823-45-8/2013 © IFAC
The pressure of gas collector pipe of coke oven is strongly coupled and disturbed, nonlinear and slow time-variant. It is difficult to control the pressure of gas collectors with conventional methods. It will cause the lag of response, coupling shaking, and lead the collector pipe pressure to keep running in lower or higher pressure. Nowadays the so-called intelligent techniques for designing control systems offer appropriate solutions, even in the presence of uncertainty and high complexity of physical processes. One of the pioneer intelligent methods is fuzzy logic, which gave us a new and powerful tool for dealing 654
10.3182/20130902-3-CN-3020.00017
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
CFNN is trained to learn the inverse dynamics so that it can be used as a nonlinear controller. Based on the model and inverse model of the CFNN, an effective MRAC method is proposed for the pressure of gas collector pipe.
with these problems in situations where there is essential inaccuracy in models, information, objectives, constraints and control actions. Fuzzy logic control basically consists of three main parts: fuzzification, inference engine, and defuzzification. Firstly, the crisp values of the input variables obtained from actual processes are transformed into the fuzzy numbers by means of fuzzification. These fuzzy numbers are characterized by defining their degrees of belonging to different input linguistic variables based on the user-specified membership functions. Secondly, appropriate fuzzy control rules, described in the form of IF-THEN clauses, are set in such a way that the controller behaviour can be verbally described by human experience and intuition. Finally, based on the degree of belonging to various output linguistic variables, the resulting fuzzy numbers are transformed back to the crisp value via defuzzification for controlling the processes.
2. CFNN 2.1 CFNN structure The CFNN structure is shown in Figure 2.
Artificial neural networks (ANNs) have shown an excellent ability to model any nonlinear function to a desired degree of accuracy. Because of this property, they are suitable for the identification and control of nonlinear plants (Hornik, Stincheombe and White, 1989). Neural network control can be considered as a feed-forward adaptive controller that employs a multilayered neural network to offer learning and optimization capabilities. Basically, a neural network is composed of a number of nodes and layers, and those nodes are connected by links that correspond to so-called weights, and those weights are initialized by small random values. Making use of various learning algorithms such as backpropagation, all weights are adjusted in such a way that the neural network can minimize the error between the actual and expected outputs.
Fig. 2. CFNN structure. In this 5-layered structure, the layer 1 represents the input of the network. The layer 2 shows the fuzzification of the input linguistic variables based on membership functions. The layer 3 represents the fuzzy logic rule base. The layer 4 shows compensatory fuzzy computation. The layer 5 represents the defuzzification of output linguistic variables. 2.2 Learning algorithm of CFNN
Fuzzy logic incorporates human-like thinking and expert knowledge. In contrast, neural networks offer learning and optimization capabilities. Based on such a fusion of ideas from fuzzy logic and neural networks, fuzzy neural network control (FNNC) possesses the advantages of both schemes. It brings the low-level learning and computational power of neural networks to fuzzy logic while providing high-level, human-like IF-THEN rule thinking and reasoning of fuzzy logic to network networks (Goldberg, 1989). A compensatory fuzzy neural network (CFNN) is used in this paper, which combines compensatory fuzzy logic with neural network. In the network, the compensatory neurons are employed, so its stability is better. Besides, fuzzy computation of CFNN is dynamic and global optimized, and CFNN is optimized dynamically in the learning algorithm. Therefore, the network is good adaptive and convergent (Li, Lu and Yan, 2005).
The fuzzy rules of the compensatory fuzzy logic system are described as FR(k):
k
k
IF x1 is A1 , …, and x n is An k
THEN y1 is B1 , …, and y m is where
Bmk
xi (i =1,…, n) and y j (j =1,…, m) are the input and
the output of the network respectively. Further,
Aik is the
linguistic label of the ith input variable appearing in the rule k
of stage k and B j is the fuzzy term of the jth output in the rule of stage k, and k =1,…, m. The nodes of the layer 1 denote the crisp input states of the system, and transmit directly them to the layer 2. The nodes in the layer 2 are input term nodes representing the input linguistic variables. They act as membership functions to transform those crisp values into fuzzy numbers, so-called fuzzification. The Gaussian function is adopted as the membership function.
There are several schemes that have been proposed for the neural control of nonlinear systems. The model reference adaptive control (MRAC) strategy is considered in this paper, due to its excellent robustness and stability (Wang, Hong and Su, 2003a; Teng, Shieh and Chen, 2003b; Sharma, McLoone and Irwin, 2005a; Hsu, Lin and Chen, 2005b). The MRAC strategy based on CFNN consists in training a network to learn the process dynamics of the pressure of gas collector pipe so that it can be used as a system identifier. Another
⎛ ( xip − aik ) 2 ⎞ ⎟ ⎟ σ ik ⎝ ⎠
μ Ak (xip ) = exp⎜⎜ − i
655
(1)
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
where
aik and σ ik are, respectively, the mean and the
Ep =
standard deviation of the Gaussian function in the kth term of
xip in pth sample to the node
the ith input linguistic variable
The layer 3 represents the fuzzy logic rule. Each node in the layer 3 is a rule node that represents one fuzzy logic rule. Hence, all nodes of layer 3 form a fuzzy rule base. The two nodes are linked by every rule based on the compensatory principle, which one is negative neuron. It can map the input onto the worst output and make a conservative decision. n
u = ∏μ x =1
k Ai
2
(7)
Step 1 training the center of the output membership function: b k (t + 1) = b k (t ) − η
[( )
]
f x p − y p δ k zk ∂E p k ( ) b t η = − m ∂b k t ∑δ k zk k =1
(x ) p i
(2)
δ k (t + 1) = δ k (t ) − η
1
( )
( )]
(9) t
Step 3 training the center of the input membership function:
1− r
k n
[
bk − f x p z k ∂E p = δ k (t ) − η m k ∂δ t ∑δ k zk
(3) ∂E p = 2 f x p − y p bk − f x p ∂σ ik
( ( )
The layer 4 shows compensatory fuzzy computation. A good trade-off decision is given by the compensatory computation when the worst input and the best input are mapped onto the output.
μ A ×L× A (x p ) = (u k )
t
k =1
⎛ ⎞n v k = ⎜⎜ ∏ μ Aki xip ⎟⎟ ⎝ ⎠
(8)
Step 2 training the width of the output membership function:
The other one is positive neuron, which can map the input onto the best output and make an optimum decision.
k 1
)
The learning algorithm based on the back propagation method is described below.
of layer 2.
k
( ( )
1 f xp − yp 2
⎛ ⎞ = ⎜⎜ ∏ μ Aki xip ⎟⎟ ⎝ ⎠
(v )
( )
k r
r 1− r + n
(x
p i
)(
)
(10)
( ) ∑δ
− a ik (1 − r + r n )δ k z k σ ik
a ki (t + 1) = a ki (t ) − η
(4)
( ))
∂E p ∂a ki
m
2
k
zk
k =1
t
(11) t
where r is compensatory degree, r∈[0, 1].
Step 4 training the width of the input membership function:
The layer 5 represents the output of the network. The nodes in this layer convert the resulting fuzzy numbers into the crisp output control signals, so-called defuzzification. Also the Gaussian function is adopted as the membership function.
∂E p = 2 f x p − y p bk − f x p ∂σ ik
μ
k Bj
(y ) p j
⎛ ( y jp − b kj ) 2 = exp⎜ − ⎜ σ kj ⎝
⎞ ⎟ ⎟ ⎠
( ( )
(x
p i
− aik
)(
) (1 − r + r n)δ 2
σ k (t + 1) = σ k (t ) − η i
The output of the defuzzification is given as m
k =1
m
∑δ
n
x =1
k Ai
zk
p i
r= r 1− r + n
( ) ∑δ
z k σ ik
3
k
k =1
zk t
k
, b and
δk
∂E p ∂σ ik
(13) t
Step 5 training the compensatory degree is defined as:
(6)
k =1
⎞ μ (x )⎟⎟ ∏ ⎠ ⎝ ⎛
where z k = ⎜⎜
k
k
(12) m
(5) i
f ( x p ) = ∑ bkδ k z k
( ))
are center and
c2 , r ∈ [0 ,1] c2 + d 2
(14)
∂E p ⎛1 ⎞ = f x p − y p b k − f x p ⎜ − 1⎟ ∂r ⎝n ⎠
(( )
width of the output membership function when kth rule is activated.
)(
( ))
⎛ ⎞ δ k z k ln⎜⎜ ∏ μ Ak i x ip ⎟⎟ ⎝ ⎠
( ) ∑δ
To describe the online learning algorithm of the CFNN using the supervised gradient descent method, first the energy function E is defined as:
656
m
k =1
k
(15)
zk t
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
⎧⎪ 2c(t )d 2 (t ) c(t + 1) = c(t ) − η ⎨ ⎪⎩ c 2 (t ) + d 2 (t )
(
⎫⎪ ∂E p 2 ⎬ ⎪⎭ ∂r
)
⎧⎪ 2d (t )c 2 (t ) ⎫⎪ ∂E p d (t + 1) = d (t ) − η ⎨ 2⎬ ⎪⎩ c 2 (t ) + d 2 (t ) ⎪⎭ ∂r
(
r (t + 1) =
)
c 2 (t + 1) c 2 (t + 1) + d 2 (t + 1)
Once the model structure has been defined, the next step is to train this CFNN by the proposed method. In experiment, the structure of CFNN is 8×18×49×8×2, and the data of inputs and outputs are acquired from the actual locale data of the pressure of gas collector pipe, and they are pretreated before training. Our method can automatically choose the best weights and thresholds. The goal of error is 0.001, learning rate is 0.005 and the largest training time is 5000.
(16) t
(17) t
(18)
where η is a constant (learning rate). 3. IDENTIFICATION 3.1 Identification of the mode based on CFNN Artificial neural networks have been increasingly used in many aspects of controlling and modeling in the industry. The traditional use of neural network modeling is a black-box approach; i.e. a neural network is trained on the available process data. However, in the real world, quite often the available process data are not sufficient to develop a good neural network model. The main difficulties arise from lack of excitation in the training data, uneven distribution of the data samples, significant noise in the modeling data, etc. They will result in the inaccurate neural network model and the not converged process of learning algorithm. The CFNN is trained to learn the dynamics of the pressure of gas collector pipe using the learning algorithm, as shown in Fig.3.
Fig. 4. Flowchart of neural network training by the learning algorithm. After the CFNN model training algorithm is implemented, the error curve is obtained in the Fig.5, the dotted line is shown as the error-goal and the solid line is shown as errortrained. From the Fig.5, we can see that about 1100 epochs the sum-squared error reached error-goal.
Fig. 3. CFNN is trained to learn the dynamics of the pressure of gas collector pipe. In the training scheme of the CFNN, the determination of parameters usually involves the minimization of an error function between the actual outputs and the targets for the whole training set. For a multi-output system, E is calculated using all the training examples over each network output by using eqn.7. Flowchart of CFNN training by the learning algorithm is shown as Fig.4.
Fig. 5. Error curve of the CFNN model. 3.2 Identification of the inverse mode based on CFNN
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IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
In the inverse model control strategy, another CFNN needs to be trained to learn the inverse dynamics of the gas collector pipe so that it can be used as a nonlinear controller. The CFNN is trained to learn the inverse dynamics of the pressure system of the gas collector pipe, as shown in Fig.6. The input vector of the neural network is denoted by
X (k ) = [ y1m (k ), L , y1m (k − n), r1 (k + 1), u1 (k − 1), L , u1 (k − m), y2m (k ), L , y2m (k − n), r2 (k + 1),
4. THE MODEL REFERENCE ADAPTIVE CONTROL The model and the inverse model using the CFNN have been identified by above section. A model reference adaptive control based on the CFNN is presented in this paper. 4.1 MRAC for the pressure of gas collector pipe
(19)
In the MRAC strategy, the model of the neural network is used as the estimator of the pressure system of gas collector of coke oven, and the inverse model is employed as the controller. The configuration of MRAC for the pressure system of gas collector of coke oven is shown as Fig.8. The d is disturbance of system. The reference model is selected as
u 2 (k − 1), L , u 2 (k − m)]
T
where rt(k+1) (t=1,2) is the input of the controller, replace with ytm (k + 1) because it can be not measured in the actual plant. The other pressure is the same with the aforementioned those. The CFNN inverse model is trained by using the proposed learning algorithm.
y m (k + 1) = ay m (k ) + r ( k )
(20)
where a is constant.
Fig. 8. The configuration of MRAC for the pressure system of gas collector. In Fig.8, y1(k), y2(k) , u1(k), and u2(k) are the two output and input of the pressure system of gas collector of coke oven respectively. The r1(k) and r2(k) are the input of the control system. 4.2 Simulation results Fig. 6. CFNN is trained to learn the inverse dynamics of the pressure of the gas collector pipe.
The traditional PID control is employed in order to compare with the proposed method MRAC in the paper. In all disturbances d=0, the parameter of the reference model is selected as a=0.56. The input of the pressure is 100 Pa, a plot of the response of the system appears in Fig.9. The green line and purple line are the output of the MRAC, which denote the pressure curves of the coke-ovens 1# and 2# respectively. The red line and blue line are the output of the PID control. It is clearly seen that the transient and steady-state response of the system has the advantage of the MRAC method over the traditional PID control. Thus, the response parameters of the system, as the overshoot, the settling time and the steadystate value is very satisfied. We now consider that the system has disturbance. Assume that the disturbance signal d is 10% of the system input, i.e. d = 10, the plot of the response of the system is shown in Fig.10. It is clearly seen that the MRAC system proposed has a good capability of resisting disturbance.
After the CFNN model training algorithm is implemented, the error curve of the inverse model is obtained in the Fig.7, the dotted line is shown as the error-goal and the solid line is shown as error-trained.
Fig. 7. Error curve of the CFNN inverse model. 658
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
5. CONCLUSIONS The pressure of gas collector pipe of coke oven is strongly coupled and disturbed, nonlinear and slow time-variant. It is difficult to control the pressure of gas collectors with conventional methods. The model reference adaptive control strategy based on the CFNN is considered to control the pressure of gas collectors in this paper. The model of the neural network of the system is identified by using the CFNN, which is employed as the system identifier. Another CFNN is trained to learn the inverse dynamics of the system by using the adaptive compensatory fuzzy learning algorithm, which is used as the controller. A powerful control method, i.e. the model reference adaptive control, for the pressure of gas collectors is proposed in this paper. The simulation results show that the method proposed for controlling the pressure of gas collectors of the coke oven is effective.
Fig.9. The response of the pressure system.
REFERENCES Yanga, C. H., Wu, M., Shen, D. Y. and Deconinck, G. (2001). Hybrid intelligent control of gas collectors of coke ovens, Control Engineering Practice, 9, pp.725-733. Hornik, K., Stincheombe, M. and White, H. (1989). Multilayer feedforward networks are universal approximators, Neur Ntwks, 2, pp.359-366. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning, chapter 5. AddisonWesley, Reading, Massachusetts. Li, H. X., Lu, J. S. and Yan, H. S. (2005). Model reduction using the genetic algorithm and Routh approximations, Journal of Systems Engineering and Electronics, 16 (3), pp.632-639. Wang, X. S., Hong, H. and Su, C. Y. (2003). Model reference adaptive control of continuous-time systems with an unknown input dead-zone, IEE Proc.-Control Theory Appl., 150 (3), pp.261-266. Teng, T. K., Shieh, J. S. and Chen, C. S. (2003). Genetic algorithm applied in online autotuning PID parameters of a liquid-level control system, Transactions of the institute of Measurement and Control, 25 (5), pp.433450. Sharma, S. K., McLoone, S. F. and Irwin, G. W. (2005). Genetic algorithms for local controller network construction, IEE Proc.-Control Theory Appl., 152 (5), pp.587-597. Hsu, C.F., Lin, C.M. and Chen, T.Y. (2005). Neural network identification based adaptive control of wing rock motions, IEE Proc.-Control Theory Appl., 152 (1), pp.65-71.
Fig.10. The response of the pressure system. When the parameters of the pressure model of the gas collector pipes of the coke oven 1# and the coke oven 2# are increased by 25%, the response of the system of the two control methods is shown in Fig.11. After the parameters of the pressure model of the gas collector pipes of the coke oven1# and the coke oven2# are decreased by 25%, the response of the system of the two control methods is shown in Fig.12. It is clearly seen that the MRAC is robust.
Fig.11. The response of the pressure system.
Appendix A. ACKNOWLEDGMENT This work was supported by the project of science and technology development plan of Beijing Municipal Commission of Education under Grant KM201011417015, Beijing, P. R. China.
Fig.12. The response of the system in parameters of the gas collector pipes are decreased by 25%. 659
Model Reference Control Based on Compensatory Fuzzy Neural Network for Gas Collectors of Coke Oven Hongxing Li* and Xiangling Kong** * Institute of Automation, Beijing Union University, Beijing, 100101 China (Tel: 086-010-64900502; e-mail: [email protected]). ** Institute of Information, Beijing Union University Beijing, 100101 China (e-mail: [email protected])} Abstract: The pressure system of gas collectors of coke oven is a multivariable non-linear process. In this paper, a model reference adaptive control using the compensatory fuzzy neural network for the pressure system of gas collectors of coke oven is presented. The dynamics model of the fuzzy neural network of the system is identified by the adaptive compensatory fuzzy learning algorithm, which can be employed as the identifier of the system. Another fuzzy neural network is trained to learn the inverse dynamics of the pressure system of gas collectors of coke oven so that it can be used as a nonlinear controller. The simulation results testify that the model obtained is satisfied and the control is effective. Keywords: Model reference control, compensatory fuzzy neural network, pressure of gas collectors, multi-variable system. coupling among the pressure of the collectors and the pressure before the blast blower. When the pressure of one collector is changed, the pressure of other collector will certainly be affected. When the pressure before the blast blower has a little change, the pressure of two collectors may have much change. If the change is sudden and strong, the pressure will react so sluggishly that the system oscillates (Yanga et al, 2001).
1. INTRODUCTION The pressure of gas collectors is an important industrial parameter in thermodynamic system of coke ovens. Its stability directly influences the life-time of coke ovens and the quality of coke production. The stabilized pressure in gas collectors can prolong the life-time of coke oven, reduce the leak of coal gas, lighten environment pollution and save energy. When the pressure of gas collectors is negative, air is able to enter into coke oven chamber and result in burning the coke, increasing the ash and debasing the coke quality. Besides, the air is likely to happen chemical reaction with architectural material of the oven, which brings about that the oven is eroded and the life-time is shortened, sometimes it may be even endanger blast blowers. On the other hand, too high pressure in gas collectors will let the gas fire and leak out, which will not only shorten the ovens’ life-time but also bring poisonous pollution and waste energy. It is obvious that the pressure control of the gas collectors of the coke oven is very important in the coke production process. In general, the pressure of the gas collector should be varied between 80 and 120 Pa (Coking experts group, 1978).
There are two coke-ovens 1# and 2# in the plant, and they have single collector pipe. The pressure system of gas collectors is composed of two single collector pipe, initial cooler, blast blower and circumfluence equipment, as shown in Fig.1.
Fig. 1. Collector structure of coke oven.
The pressure system of gas collectors is a strongly intercoupled, slow time-varying, nonlinear multi-variable system where interference is strong and frequent. The reason for varying the flow of coke oven gas has loading coal and unloading coke in batch, variation of coking time and heating system, change in numbers of opened loading doors and ascending pipes and so on, which they are the direct disturbance to the pressure of collectors. At the same time, the influence of the pressure of collectors has variation of attracting power of a blast blower, variations of temperature, flow of cycling ammonia water, change of the load by gas users and so on yet. Furthermore, there exists inevitable 978-3-902823-45-8/2013 © IFAC
The pressure of gas collector pipe of coke oven is strongly coupled and disturbed, nonlinear and slow time-variant. It is difficult to control the pressure of gas collectors with conventional methods. It will cause the lag of response, coupling shaking, and lead the collector pipe pressure to keep running in lower or higher pressure. Nowadays the so-called intelligent techniques for designing control systems offer appropriate solutions, even in the presence of uncertainty and high complexity of physical processes. One of the pioneer intelligent methods is fuzzy logic, which gave us a new and powerful tool for dealing 654
10.3182/20130902-3-CN-3020.00017
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
CFNN is trained to learn the inverse dynamics so that it can be used as a nonlinear controller. Based on the model and inverse model of the CFNN, an effective MRAC method is proposed for the pressure of gas collector pipe.
with these problems in situations where there is essential inaccuracy in models, information, objectives, constraints and control actions. Fuzzy logic control basically consists of three main parts: fuzzification, inference engine, and defuzzification. Firstly, the crisp values of the input variables obtained from actual processes are transformed into the fuzzy numbers by means of fuzzification. These fuzzy numbers are characterized by defining their degrees of belonging to different input linguistic variables based on the user-specified membership functions. Secondly, appropriate fuzzy control rules, described in the form of IF-THEN clauses, are set in such a way that the controller behaviour can be verbally described by human experience and intuition. Finally, based on the degree of belonging to various output linguistic variables, the resulting fuzzy numbers are transformed back to the crisp value via defuzzification for controlling the processes.
2. CFNN 2.1 CFNN structure The CFNN structure is shown in Figure 2.
Artificial neural networks (ANNs) have shown an excellent ability to model any nonlinear function to a desired degree of accuracy. Because of this property, they are suitable for the identification and control of nonlinear plants (Hornik, Stincheombe and White, 1989). Neural network control can be considered as a feed-forward adaptive controller that employs a multilayered neural network to offer learning and optimization capabilities. Basically, a neural network is composed of a number of nodes and layers, and those nodes are connected by links that correspond to so-called weights, and those weights are initialized by small random values. Making use of various learning algorithms such as backpropagation, all weights are adjusted in such a way that the neural network can minimize the error between the actual and expected outputs.
Fig. 2. CFNN structure. In this 5-layered structure, the layer 1 represents the input of the network. The layer 2 shows the fuzzification of the input linguistic variables based on membership functions. The layer 3 represents the fuzzy logic rule base. The layer 4 shows compensatory fuzzy computation. The layer 5 represents the defuzzification of output linguistic variables. 2.2 Learning algorithm of CFNN
Fuzzy logic incorporates human-like thinking and expert knowledge. In contrast, neural networks offer learning and optimization capabilities. Based on such a fusion of ideas from fuzzy logic and neural networks, fuzzy neural network control (FNNC) possesses the advantages of both schemes. It brings the low-level learning and computational power of neural networks to fuzzy logic while providing high-level, human-like IF-THEN rule thinking and reasoning of fuzzy logic to network networks (Goldberg, 1989). A compensatory fuzzy neural network (CFNN) is used in this paper, which combines compensatory fuzzy logic with neural network. In the network, the compensatory neurons are employed, so its stability is better. Besides, fuzzy computation of CFNN is dynamic and global optimized, and CFNN is optimized dynamically in the learning algorithm. Therefore, the network is good adaptive and convergent (Li, Lu and Yan, 2005).
The fuzzy rules of the compensatory fuzzy logic system are described as FR(k):
k
k
IF x1 is A1 , …, and x n is An k
THEN y1 is B1 , …, and y m is where
Bmk
xi (i =1,…, n) and y j (j =1,…, m) are the input and
the output of the network respectively. Further,
Aik is the
linguistic label of the ith input variable appearing in the rule k
of stage k and B j is the fuzzy term of the jth output in the rule of stage k, and k =1,…, m. The nodes of the layer 1 denote the crisp input states of the system, and transmit directly them to the layer 2. The nodes in the layer 2 are input term nodes representing the input linguistic variables. They act as membership functions to transform those crisp values into fuzzy numbers, so-called fuzzification. The Gaussian function is adopted as the membership function.
There are several schemes that have been proposed for the neural control of nonlinear systems. The model reference adaptive control (MRAC) strategy is considered in this paper, due to its excellent robustness and stability (Wang, Hong and Su, 2003a; Teng, Shieh and Chen, 2003b; Sharma, McLoone and Irwin, 2005a; Hsu, Lin and Chen, 2005b). The MRAC strategy based on CFNN consists in training a network to learn the process dynamics of the pressure of gas collector pipe so that it can be used as a system identifier. Another
⎛ ( xip − aik ) 2 ⎞ ⎟ ⎟ σ ik ⎝ ⎠
μ Ak (xip ) = exp⎜⎜ − i
655
(1)
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
where
aik and σ ik are, respectively, the mean and the
Ep =
standard deviation of the Gaussian function in the kth term of
xip in pth sample to the node
the ith input linguistic variable
The layer 3 represents the fuzzy logic rule. Each node in the layer 3 is a rule node that represents one fuzzy logic rule. Hence, all nodes of layer 3 form a fuzzy rule base. The two nodes are linked by every rule based on the compensatory principle, which one is negative neuron. It can map the input onto the worst output and make a conservative decision. n
u = ∏μ x =1
k Ai
2
(7)
Step 1 training the center of the output membership function: b k (t + 1) = b k (t ) − η
[( )
]
f x p − y p δ k zk ∂E p k ( ) b t η = − m ∂b k t ∑δ k zk k =1
(x ) p i
(2)
δ k (t + 1) = δ k (t ) − η
1
( )
( )]
(9) t
Step 3 training the center of the input membership function:
1− r
k n
[
bk − f x p z k ∂E p = δ k (t ) − η m k ∂δ t ∑δ k zk
(3) ∂E p = 2 f x p − y p bk − f x p ∂σ ik
( ( )
The layer 4 shows compensatory fuzzy computation. A good trade-off decision is given by the compensatory computation when the worst input and the best input are mapped onto the output.
μ A ×L× A (x p ) = (u k )
t
k =1
⎛ ⎞n v k = ⎜⎜ ∏ μ Aki xip ⎟⎟ ⎝ ⎠
(8)
Step 2 training the width of the output membership function:
The other one is positive neuron, which can map the input onto the best output and make an optimum decision.
k 1
)
The learning algorithm based on the back propagation method is described below.
of layer 2.
k
( ( )
1 f xp − yp 2
⎛ ⎞ = ⎜⎜ ∏ μ Aki xip ⎟⎟ ⎝ ⎠
(v )
( )
k r
r 1− r + n
(x
p i
)(
)
(10)
( ) ∑δ
− a ik (1 − r + r n )δ k z k σ ik
a ki (t + 1) = a ki (t ) − η
(4)
( ))
∂E p ∂a ki
m
2
k
zk
k =1
t
(11) t
where r is compensatory degree, r∈[0, 1].
Step 4 training the width of the input membership function:
The layer 5 represents the output of the network. The nodes in this layer convert the resulting fuzzy numbers into the crisp output control signals, so-called defuzzification. Also the Gaussian function is adopted as the membership function.
∂E p = 2 f x p − y p bk − f x p ∂σ ik
μ
k Bj
(y ) p j
⎛ ( y jp − b kj ) 2 = exp⎜ − ⎜ σ kj ⎝
⎞ ⎟ ⎟ ⎠
( ( )
(x
p i
− aik
)(
) (1 − r + r n)δ 2
σ k (t + 1) = σ k (t ) − η i
The output of the defuzzification is given as m
k =1
m
∑δ
n
x =1
k Ai
zk
p i
r= r 1− r + n
( ) ∑δ
z k σ ik
3
k
k =1
zk t
k
, b and
δk
∂E p ∂σ ik
(13) t
Step 5 training the compensatory degree is defined as:
(6)
k =1
⎞ μ (x )⎟⎟ ∏ ⎠ ⎝ ⎛
where z k = ⎜⎜
k
k
(12) m
(5) i
f ( x p ) = ∑ bkδ k z k
( ))
are center and
c2 , r ∈ [0 ,1] c2 + d 2
(14)
∂E p ⎛1 ⎞ = f x p − y p b k − f x p ⎜ − 1⎟ ∂r ⎝n ⎠
(( )
width of the output membership function when kth rule is activated.
)(
( ))
⎛ ⎞ δ k z k ln⎜⎜ ∏ μ Ak i x ip ⎟⎟ ⎝ ⎠
( ) ∑δ
To describe the online learning algorithm of the CFNN using the supervised gradient descent method, first the energy function E is defined as:
656
m
k =1
k
(15)
zk t
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
⎧⎪ 2c(t )d 2 (t ) c(t + 1) = c(t ) − η ⎨ ⎪⎩ c 2 (t ) + d 2 (t )
(
⎫⎪ ∂E p 2 ⎬ ⎪⎭ ∂r
)
⎧⎪ 2d (t )c 2 (t ) ⎫⎪ ∂E p d (t + 1) = d (t ) − η ⎨ 2⎬ ⎪⎩ c 2 (t ) + d 2 (t ) ⎪⎭ ∂r
(
r (t + 1) =
)
c 2 (t + 1) c 2 (t + 1) + d 2 (t + 1)
Once the model structure has been defined, the next step is to train this CFNN by the proposed method. In experiment, the structure of CFNN is 8×18×49×8×2, and the data of inputs and outputs are acquired from the actual locale data of the pressure of gas collector pipe, and they are pretreated before training. Our method can automatically choose the best weights and thresholds. The goal of error is 0.001, learning rate is 0.005 and the largest training time is 5000.
(16) t
(17) t
(18)
where η is a constant (learning rate). 3. IDENTIFICATION 3.1 Identification of the mode based on CFNN Artificial neural networks have been increasingly used in many aspects of controlling and modeling in the industry. The traditional use of neural network modeling is a black-box approach; i.e. a neural network is trained on the available process data. However, in the real world, quite often the available process data are not sufficient to develop a good neural network model. The main difficulties arise from lack of excitation in the training data, uneven distribution of the data samples, significant noise in the modeling data, etc. They will result in the inaccurate neural network model and the not converged process of learning algorithm. The CFNN is trained to learn the dynamics of the pressure of gas collector pipe using the learning algorithm, as shown in Fig.3.
Fig. 4. Flowchart of neural network training by the learning algorithm. After the CFNN model training algorithm is implemented, the error curve is obtained in the Fig.5, the dotted line is shown as the error-goal and the solid line is shown as errortrained. From the Fig.5, we can see that about 1100 epochs the sum-squared error reached error-goal.
Fig. 3. CFNN is trained to learn the dynamics of the pressure of gas collector pipe. In the training scheme of the CFNN, the determination of parameters usually involves the minimization of an error function between the actual outputs and the targets for the whole training set. For a multi-output system, E is calculated using all the training examples over each network output by using eqn.7. Flowchart of CFNN training by the learning algorithm is shown as Fig.4.
Fig. 5. Error curve of the CFNN model. 3.2 Identification of the inverse mode based on CFNN
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IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
In the inverse model control strategy, another CFNN needs to be trained to learn the inverse dynamics of the gas collector pipe so that it can be used as a nonlinear controller. The CFNN is trained to learn the inverse dynamics of the pressure system of the gas collector pipe, as shown in Fig.6. The input vector of the neural network is denoted by
X (k ) = [ y1m (k ), L , y1m (k − n), r1 (k + 1), u1 (k − 1), L , u1 (k − m), y2m (k ), L , y2m (k − n), r2 (k + 1),
4. THE MODEL REFERENCE ADAPTIVE CONTROL The model and the inverse model using the CFNN have been identified by above section. A model reference adaptive control based on the CFNN is presented in this paper. 4.1 MRAC for the pressure of gas collector pipe
(19)
In the MRAC strategy, the model of the neural network is used as the estimator of the pressure system of gas collector of coke oven, and the inverse model is employed as the controller. The configuration of MRAC for the pressure system of gas collector of coke oven is shown as Fig.8. The d is disturbance of system. The reference model is selected as
u 2 (k − 1), L , u 2 (k − m)]
T
where rt(k+1) (t=1,2) is the input of the controller, replace with ytm (k + 1) because it can be not measured in the actual plant. The other pressure is the same with the aforementioned those. The CFNN inverse model is trained by using the proposed learning algorithm.
y m (k + 1) = ay m (k ) + r ( k )
(20)
where a is constant.
Fig. 8. The configuration of MRAC for the pressure system of gas collector. In Fig.8, y1(k), y2(k) , u1(k), and u2(k) are the two output and input of the pressure system of gas collector of coke oven respectively. The r1(k) and r2(k) are the input of the control system. 4.2 Simulation results Fig. 6. CFNN is trained to learn the inverse dynamics of the pressure of the gas collector pipe.
The traditional PID control is employed in order to compare with the proposed method MRAC in the paper. In all disturbances d=0, the parameter of the reference model is selected as a=0.56. The input of the pressure is 100 Pa, a plot of the response of the system appears in Fig.9. The green line and purple line are the output of the MRAC, which denote the pressure curves of the coke-ovens 1# and 2# respectively. The red line and blue line are the output of the PID control. It is clearly seen that the transient and steady-state response of the system has the advantage of the MRAC method over the traditional PID control. Thus, the response parameters of the system, as the overshoot, the settling time and the steadystate value is very satisfied. We now consider that the system has disturbance. Assume that the disturbance signal d is 10% of the system input, i.e. d = 10, the plot of the response of the system is shown in Fig.10. It is clearly seen that the MRAC system proposed has a good capability of resisting disturbance.
After the CFNN model training algorithm is implemented, the error curve of the inverse model is obtained in the Fig.7, the dotted line is shown as the error-goal and the solid line is shown as error-trained.
Fig. 7. Error curve of the CFNN inverse model. 658
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
5. CONCLUSIONS The pressure of gas collector pipe of coke oven is strongly coupled and disturbed, nonlinear and slow time-variant. It is difficult to control the pressure of gas collectors with conventional methods. The model reference adaptive control strategy based on the CFNN is considered to control the pressure of gas collectors in this paper. The model of the neural network of the system is identified by using the CFNN, which is employed as the system identifier. Another CFNN is trained to learn the inverse dynamics of the system by using the adaptive compensatory fuzzy learning algorithm, which is used as the controller. A powerful control method, i.e. the model reference adaptive control, for the pressure of gas collectors is proposed in this paper. The simulation results show that the method proposed for controlling the pressure of gas collectors of the coke oven is effective.
Fig.9. The response of the pressure system.
REFERENCES Yanga, C. H., Wu, M., Shen, D. Y. and Deconinck, G. (2001). Hybrid intelligent control of gas collectors of coke ovens, Control Engineering Practice, 9, pp.725-733. Hornik, K., Stincheombe, M. and White, H. (1989). Multilayer feedforward networks are universal approximators, Neur Ntwks, 2, pp.359-366. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning, chapter 5. AddisonWesley, Reading, Massachusetts. Li, H. X., Lu, J. S. and Yan, H. S. (2005). Model reduction using the genetic algorithm and Routh approximations, Journal of Systems Engineering and Electronics, 16 (3), pp.632-639. Wang, X. S., Hong, H. and Su, C. Y. (2003). Model reference adaptive control of continuous-time systems with an unknown input dead-zone, IEE Proc.-Control Theory Appl., 150 (3), pp.261-266. Teng, T. K., Shieh, J. S. and Chen, C. S. (2003). Genetic algorithm applied in online autotuning PID parameters of a liquid-level control system, Transactions of the institute of Measurement and Control, 25 (5), pp.433450. Sharma, S. K., McLoone, S. F. and Irwin, G. W. (2005). Genetic algorithms for local controller network construction, IEE Proc.-Control Theory Appl., 152 (5), pp.587-597. Hsu, C.F., Lin, C.M. and Chen, T.Y. (2005). Neural network identification based adaptive control of wing rock motions, IEE Proc.-Control Theory Appl., 152 (1), pp.65-71.
Fig.10. The response of the pressure system. When the parameters of the pressure model of the gas collector pipes of the coke oven 1# and the coke oven 2# are increased by 25%, the response of the system of the two control methods is shown in Fig.11. After the parameters of the pressure model of the gas collector pipes of the coke oven1# and the coke oven2# are decreased by 25%, the response of the system of the two control methods is shown in Fig.12. It is clearly seen that the MRAC is robust.
Fig.11. The response of the pressure system.
Appendix A. ACKNOWLEDGMENT This work was supported by the project of science and technology development plan of Beijing Municipal Commission of Education under Grant KM201011417015, Beijing, P. R. China.
Fig.12. The response of the system in parameters of the gas collector pipes are decreased by 25%. 659