- Email: [email protected]
A Simple Supervisor for the Adaptation of the Similarity Factor in a Fuzzy Controller
Copyright @ IFAC Automatic Systems for Building the Infrastructure in Developing Countries, Ohrid, Republic of Macedonia, 2001
A SIMPLE SUPERVISOR FOR THE ADAPTATION OF THE SIMILARITY FACTOR IN A FUZZY CONTROLLER J. Henriques, M. Jose and A. Dourado CISUC - Centro de Infonmitica e Sistemas, Departarnento de Engenharia Infonnatica Polo 1I da Universidade de Coimbra, 3030 Coimbra Portugal Phone: + 3512397900 00 Fax: + 352 239 701266 e-mail:Uh, dourado} @dei.uc.pt
Abstract: A supervisor for the adaptation of the similarity fuctor of a previously developed fuzzy controller is presented. In the start-up phase, a PI controller is running while the fuzzy selforganizing system fills the rule base. The only needed parameter in the supervisor is a rough estimation of the process gain, computed in a very simple way. Simulation and experimental show that the resulting structure improved capability to control non-linear systems with reduced a-priori knowledge for the regulation and the tracking situations. Copy right @200J IFAC Keywords: Self-organizing fuzzy controller; supervision; on-line learning; experimental research. control system also determines the switching between the PI and the fuzzy controller. The paper is organized as follows. In section 2 a brief review of the fuzzy self-organizing controller is made. Section 3 discusses a strategy to compute the initial value of its Dmax parameter. Section 4 discusses the problem of initial learning and the supervisor. Section 5 presents some results, in simulation and in a real application, showing the improvements that can be obtained with the proposed new features.
I. INTRODUCTION Since the pioneer works of Zadeh (1965, 1973), Mamdani and Assilian (1975), Procyck and Mamdani (1979), Takagi and Sugeno (1985), fuzzy control had a large development in its methodologies and applications, see for example Li et af. (1995, 2001). However, to a large extend, there still remains open the problem of efficient on-line learning and in general the self-organizing capabilities of the fuzzy-systems, Singh (1998). Many authors have proposed solutions to the problem through neuro- fuzzy structures, Yager (1999), Belarbi et al. (2000), Lazzerini et af. (1999), Chen and Peng (1999), Shi and Mizumoto (2000), clustering techniques, Gomez et al. (1999), and other several techniques based on data Wang (1999), Zapata and Galvao (1999), Li and Priemer (1999). However there is a need for simple structures with large generality, avoiding a-priori knowledge about the process. In this sense, in a previous work Dias and Dourado (1999) a fuzzy controller with self-organizing capabilities and with a broad generality for monotone stable systems has been developed. One important characteristic of the controller is that its rule base has a Fixedfuzzy Maximum Number of Rules (FMNR), determined by the operator. Its performance is determined by two parameters: the initial value of Dmax and the value of the observation time-window. When these parameters are not properly fixed, namely at initialisation phase, the performance of the controller maybe not satisfuctory. In this work further developments are presented though a supervisor that is charged to control the adaptation of the Dmax parameter. The fuct that the controller starts with an empty rule base may cause a inadequate performance at start-up, showing oscillations and considerable overshoot. This drawback can be avoided if at start-up, while the initial rule base is constructed, a simple controller, like a PI one, is implemented. The supervisory level of the
2. THE FMNR SOC FUZZY CONTROLLER The controller supposes a stable monotone process of Non-linear Auto-Regressive with eXogenous Input (NARX) type generally defined by (I ): y( k) =
f ( y(k -
1), y(k - 2 ), ... , u(k -
n. ... ,
(1 )
u(k - T-l), ... , e(k))
with f( ·) a linear or non-linear function, time variant or not, u(k -
y(k). y(k-l) .. . .
the outputs of the system and
u(k).
1). .. .. the inputs to the system at the k, k -1, ...
time instants. T represents the discrete pure time delay and e(k) is a stochastic disturbance (assumed white noise and neglected in the following). The specific knowledge about the process is the pure time delay T or its upper bound, the upper bound of the memory (order) of the system and the sign of its monotonicity. The reference should also be known at least T + 1 sampling instants in advance (particularly for tracking problems). Solving (1) for u(k - 1). one obtains (2), u(k -
n
= g ( y(k), y(k - 1), . . . , u(k - T - 1), . . . )
(2)
Applying to (2) a linguistic transformation, and adding T + 1 to every index (time translation) one obtains (3) in the form of a Fuzzy Auto-Regressive with Exogenous Input system (FARX).
75
IF y(k + T + 1) is Al and y(k +
n is A2 and ... and
u(k) is BI and .. . THEN u(k + 1) is C AI, Az, . .. ,
and (3)
D*; =
--J (YI; -
ri + (Y2; - r2) 2 (5)
1
BI , .•• , C are linguistic values at the
w· =
--xD+ I
-
•
proper sampling instants. For the inference of this fuzzy rule, the needed future values of the process output are replaced in (3) by the known future values of the reference; this gives some feed forward effect to the control signal. The rules are constructed accordingly to the experiments. The rule base is a matrix where each row is a rule and each column is a linguistic value until the FMNR is reached. At this point the new rules are written over the older ones, beginning with the rule that occupies the first place in the matrix, so that the FMNR is never exceeded and up-dating the rules to changes in process conditions. The choice of the FMNR must be considered with some care. On one hand the number of rules must be compatible with the time needed to compute the algorithm. On the other hand it should be enough representative of the process behaviour, spanning along the input and output spaces. See Dias and Dourado (1999) for some guidelines. The fuzzification uses (for all the variables) triangular membership functions as in Park et al. (1995). Every crisp Xl value defines the unity vertex of a membership function and all the membership functions have the same slope. All the fuzzy sets overlap on the entire interval [a, b] including all possible values for the
This
ill;
is
Dmax'
called
the
degree of similarity.
Dmaxrepresents the maximum distance considered. If D; is greater than Dmax it is considered that there is no similarity. The consequent fuzzy set for each rule is an interval of values [PI , P2], where Il.... (u(k)) ~ W;, V u(k)
E
(PI> P2]
(6)
The total output is taken as the intersection of all the intervals (one for each rule). From the inference mechanism one obtains a learning interval (7) of possible values for u(k + 1). (7)
flu = [min, max]
The adaptive parameter Dmax
The Dmax parameter is adapted in order to reduce oscillations and steady-state error. Its value is increased by (8) in the presence of the first situation and decreased by (9) in the presence of the second, where UI and U2
are coefficients empirically fixed. Dmax = Dmax x ( 1
+
uI percenlage of oscillation)
Dmax = Dmax x( 1 - u2 percenlage of error)
variable Xl' By this way each membership function is simply defined by its maximum crisp value and the rule base is always complete. For the inference mechanism with dynamics changes, and considering, for example, a I S/ order
(8) (9)
The output behaviour is monitored during a timeobservation window, whose size must be determined by the user. The Defuzzification mechanism
process, each rule has the format: Ify(k) is YI; and y(k - 1) is Y2i then u(k -
n is
If the future values of the
The output of the inference mechanism is an interval [min, max] (7). Let the error in advance be defined as the
reference are
and r2 (respectively in k + T + 1 and
difference between the actual process output (at k
k + T sampling times), the degree of equality between
instant) and the desired reference after the T + 1 instants.
them and each rule antecedent is inspired in Park et al. (1995) by expression but with a new factor to consider changes in process dynamics; by this way it contains information from both the reference values and the real output values, given by (4) and (5),
To compute a value for u(k + 1), a simple but efficient method is proposed, inspired in Park et al. (1995) given by (10) and (11).
rl
U; ".
error_ in_ advance
=
r(k + T + 1) - y(k)
Considering a positive monotonocity: If error_in_advance> 0
(4)
with
then u(k + 1) = El
= YI; - y(k)
E2 = Y2i - y(k - 1)
else u(k + 1) =
u= u;-u(k-n
76
max+u(k-n
2 min+u(k-
2
n
(10)
3. THE INITIAL VALUE OF Dmax AND THE ADAPTATION WINOOW
In the case of a negative monotonocity: If error_ in_ advance> 0 min +u(k-1)
then u(k + 1) = else u(k + 1) =
The value of Dmax represents the maximum distance allowed to compute the similarity degree between a given situation and the antecedents of the rule. If Dmax is
(11)
2 max+u(k-1)
low, the used rules are few, the CJ)i values are low and the resulting control interval presents too large values. If Dmax is high, the reverse happens. On the other hand if the process gain is high, the input must be in a smaller interval. If this gain is low, then the input interval can be larger. From these thoughts one can conclude that for high gain systems Dmax should be higher, while for low gain ones it should be lower. There exists then a relation between the process gain and the initial value of Dmax; this relation is
2
This controller shows good perfonnance but in some situations it may have two main drawbacks. Fig. I shows simulations results for the simple first order i1lustrative process (12). y(k + 1) = 0.8553 y(k)
~
s-"
"
Cl
." ..e:::'" ~
~
(12)
~
2. 5
proposed as (13), where Umax is the maximum admissible value of the control signal.
2.0
~
2"
+ 0.1543 u(k)
y
1.5
(13)
-.- --
1.0
\
0.5
\
The process gain is not known, in general, with accuracy. An estimation of it, even coarse, is however enough for the purpose. To obtain it (14) is used. Empirical evidence showed that it contains sufficient infonnation for the purpose in the used examples. y(k) Gain = u(k _ 1) (14)
--
,
- ---
o
20
40
60
80
100 120 Iterations
140
160
180
The adaptation window has a strong influence in the If it is small, Dmax becomes too oscillating and perfonnance is not good; if it is long, the adaptation of Dmax is not done in proper time and performance is also insufficient. The good value for the adaptation window is related to the maximum number of rules. A good compromise is to consider this window (in sampling intervals) equal to half of the maximum number of rules. By this way it is guaranteed that it is sufficiently representative of the system behaviour and there is no "over adaptation".
7.0
Dmax behaviour.
6. 0 ~
~
5.0
<;
" ~
] '" v"
4.0 3.0 2.0
'\IV
1.0
0
20
40
u
60
80
100 120 iteratIOns
140
160
180
4. THE START-UP PHASE AND THE SUPERVISOR During the start-up phase there are few rules (initially no rule). The computation ofthe control action results in some cases in an oscillatory behaviour with strong overshoots and undershoots. Only when a significant part of the maximum number of rules is built from experimental data is this problem surpassed. In order to improve the control system behaviour, a simple controller (P, PI, etc.) may be used to start-up the process. After some time, this controller is switched to the fuzzy controller. The switching is a task of the supervisor, according to the following rules (15) found by experimental research. For the regulation problem
Fig. I. The FMNR fuzzy controller behaviour for a step reference.
In the starting phase, it may show osci1lations and overshoot that deteriorates the behavior. The initial value of Dmax is found mainly by trial and error procedures, as well as the size of the observation window. To solve these drawbacks a supervisor is proposed in the following.
77
The fuzzy controller is put to work after the starting phase (during which a PI is working) and a good performance can be observed with a smooth control signal. For the second step in the reference (iteration 100), a significant overshoot is obtained (bad performance) but the fuzzy system changes its own rules to decrease this overshoot (and undershoot) in the following steps. The rule base is being changed with time in such a way that performance is improved.
(constant reference) the switching is done when the output approaches the reference, as (15). If reference is constant And error is less than (15) 10% Then switch to fuzzy controller For the tracking problem (varying references) the switching is made after a certain number of samples fills the rule-base. For example for sinusoidal references (16) is enough.
If reference is not constant And error is less than 10% And the sampling instant is greater than 10 Then switch to fuzzy controller
B. Experimental results with a laboratory process In the laboratory process, depicted in Fig. 3 and Fig. 4, air is forced to circulate by a fan blower through a duct and is heated at its inlet. This is a non-linear process with a pure time delay, which depends on the position of the temperature sensor element and of the airflow rate, depending on the damper position n. The system input u(k), is the voltage on the heating device, which
(16)
The supervisor is charged with the following tasks: computation of the process gain, computation of the initial value of Dmax, controller switching, monitoring the adaptation of Dmax. 5. RESULTS
consists of a mesh of resistor wires, and the output, y(k), is the outlet air temperature.
A. Simulation results
For the process given by (12), with a square wave as reference signal, the results shown in Fig. 2 were obtained. 3. 5
~
3.0
~or
2.5
0
p,.'--------i
,
.
."
"" :2 " "... ~ <>:: " ~
2.0
Y
1.5 1.0 0.5
r-
V0
'''l.l'. ._-----'
_
Fi~ ._ 3. Laboratory
Process.
-- -- --- ---- ------------- ---------------
I
o 50
100
150
200
250
300
350
JI
o
111
Air
Output
c:::)
400
Jt e ralion.~
Y(k)r D ( - 1O.1O( Volts
35
Fig. 4. Schematic diagram of the Laboratory Process.
\
3.0
~
]
u
Sampling time was chosen 400 milliseconds. An experiment with the following parameters was made: maximum number of rules: 100; time-window: 50; initial Dmax = 20. For a square wave as reference, Fig. 5 shows the results. The learning ability can be observed. The performance increases with time and after 200 iterations the behaviour of the control system is good. During the learning phase the controller learns its own rules in order to improve its performance.
f\.
2.5
!'!l
'"
2.0
'-''"
1. 5
l
1.0 0.5
If o
v
50
100
150
200
250
300
350
400
Iterations
Fig. 2. The FMNR controller with the supervisor
78
B
--
':i
-i!.
~
"
0
"&
""'"
t;;
" :2
------- ~fI-----
~
~
'~"
..
3. 5
e
3.0
Cl
~ Cl:
2.
0
2.0
·1
o
100
SO
ISO
200 250 Iteralions
300
350
400
100
SO
0
ISO lterallons
200
250
300
Fig. 6 . Experimental results with PT326; sinusoidal reference
Without the supervisor the controller behaves as shown by Figure 7: the learning phase is very long (two periods of the reference).
~
]
4
~
9
e;:
u
B
c3
8
;;
~
"
0
"""'"
6
:2
o
50
100
150
200 2SO lleralions
300
3SO
5 "
~
'~"
400
4
.
~
<>:
Fig. 5. Experimental results with PT326; square wave reference.
Now for a sinusoidal reference with a period of 38s, the results are presented in Fig. 6. The learning ability can be observed. The perfonnance increases with time and after 200 iterations the behaviour of the control system is good. During the learning phase the controller learns its own rules in order to improve its perfonnance.
o
50
100
150 Iterations
200
250
300
SO
100
ISO Iterollo,..
200
2SO
300
9
8
~ 5.0
~
~
0" ~
" :2 ~ ~
...
] ~
4.0
e;:
3.0 :
(l
5 4
2.0 1.0
... !.
~
Cl:
6
0.0
o
·1.0
Fig. 7 . The FMNR without supervisor; learning phase may be quite 0
50
100
150 Iterations
200
2SO
300
longer, if Dmax is not properly initialised.
79
Li
6. CONCLUSION
c., R. Priemer (1999) - Fuzzy control of unknown multiple-
input-multiple-output plants, Fuzzy Sets and Systems, Vol 104, Iss 2, pp 245-267, Elsevier Science. Li, Hongxing, C. Vincent C. Yen, (1995) - Fuzzy Sets and Fuzzy Decision -Making, CRC Press. Li, Hongxing, C. Philip Chen, H. Huang (2001) - Fuzzy Neural Intelligent Systems, Mathematical Foundations and the Applications in Engineering, CRC Press. Mandani, E., S. Assilian (1975) - An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller, Int. L of ManMachnineStudies, vol. 7, nO I, pp I-B. Park, Young-Moon, U. Moon and L. Kwang (1995) - A SelfOrganizing Fuzzy Logic Controller for Dynamic Systems Using a Fuzzy Auto-Regressive Moving Average (FARMA) Model, lEE Transactions on fuzzy Systems, Vol.3, nOI , pp 75- 82. Procyck T., E. Mamdani (1979) - A Linguistic SelfOrganizing Process Controller, Automatica, Vol. 15, 1530. Shi Y. , M. Mizumoto (2000) - A new approach ofneuro-fuzzy learning algorithm for tuning fuzzy rules, Fuzzy Sets and Systems, Vol 112, Iss I, pp 99-116, Elsevier Science. Singh Y. (1998) - A modified self-organizing controller for real-time process control applications, Fuzzy Sets and Systems, Vo1.96, pp 147-159. Takagi T., M. Sugeno (1985) - Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man and CybernetiCS, Vo1.l5, No. I, pp 116-B2. Wang, L. X. (1999) - Automatic design of fuzzy controllers. Automatica, Vol 35, Iss 8, pp 1471 -1475, PergamonElsevier Science. . Yager R.R. (1999) - Implementing fuzzy logic controllers using a neural network framework, Fuzzy Sets and Systems, Vol lOO, Suppl. S, pp 133-144, Elsevier Science Zadeh, L. A (1965). "Fuzzy sets", Information and Control, Vol. 8, pp 338-358. Zadeh, L. A. (1973) - Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. On Systems. Man and Cybernetics, vol. 3, nO I, pp 28-44. Zapata, G., R. Galvao, T. Yoneyama, (1999) - Extracting fuzzy control rules from experimental human operator data, IEEE Transactions on Systems Man and Cybernetics Part B - Cybernetics, Vol 29, Iss 3, pp 398406.
By experimental evidence, the introduction of a supervisor in the fuzzy controller with a similarity factor and a maximum number of rules allowed increased performance and robustness. The operations implemented in the supervisor are simple and resulting from experimental research. However, more formal developments are needed in order to show generality in the learning capabilities and to support the value of the proposed architecture for general non-linear systems. The monotonicity hypothesis of the process is a limitation, but as most of the processes possess this property, in practice it is not a serious drawback. The objective of this work is to obtain a controller for nonlinear systems with a broad generality, using minimum a-priori knowledge about the process. ACKNOWLEDGEMffiNTS This work has been partially supported by FCTlPraxis XXI through CISUC and the project ALCINE- Online learning in intelligent control and by POSI- Programa Operacional da Sociedade da Informa'Y3o of Portuguese Funda'Y3o para a Ciencia e Tecnologia of Portuguese Government, supported by European Union FEDER.
REFERENCES Belarbi, K., K. Bettou, A. Mezaache (2000) - Fuzzy neural networks for estimation and fuzzy controller design: simulation study for a pulp batch digester, Journal of Process Control, Vol 10, !ss I, pp 35-41, Elsevier Sci. Chen, c., S. Peng (1999) - Intelligent process control using neural fuzzy techniques, Journal of Process Control, Vol 9, !ss 6, pp 493-503 , Elsevier Sci. Dias J., A Dourado (1999) - A Self-Organizing Fuzzy Controller with a Fixed Maximum Number of Rules and an Adaptive Similarity Factor, Fuzzy Sets and Systems, vol. 103, pp 27-48, Elsevier Science. Gomez, A , F. Skarmeta, M. Delgado, M. Vila (1999) - About the use of fuzzy clustering techniques for fuzzy model identification, Fuzzy Sets and Systems, Vol 106, !ss 2, pp 179-188. Lazzerini B., L. Reyneri, M. Chiaberge (1999) - A neurofuzzy approach to hybrid intelligent control, IEEE Transactions on Industry Applications, Vol 35, Iss 2, pp 413-425.
80
A SIMPLE SUPERVISOR FOR THE ADAPTATION OF THE SIMILARITY FACTOR IN A FUZZY CONTROLLER J. Henriques, M. Jose and A. Dourado CISUC - Centro de Infonmitica e Sistemas, Departarnento de Engenharia Infonnatica Polo 1I da Universidade de Coimbra, 3030 Coimbra Portugal Phone: + 3512397900 00 Fax: + 352 239 701266 e-mail:Uh, dourado} @dei.uc.pt
Abstract: A supervisor for the adaptation of the similarity fuctor of a previously developed fuzzy controller is presented. In the start-up phase, a PI controller is running while the fuzzy selforganizing system fills the rule base. The only needed parameter in the supervisor is a rough estimation of the process gain, computed in a very simple way. Simulation and experimental show that the resulting structure improved capability to control non-linear systems with reduced a-priori knowledge for the regulation and the tracking situations. Copy right @200J IFAC Keywords: Self-organizing fuzzy controller; supervision; on-line learning; experimental research. control system also determines the switching between the PI and the fuzzy controller. The paper is organized as follows. In section 2 a brief review of the fuzzy self-organizing controller is made. Section 3 discusses a strategy to compute the initial value of its Dmax parameter. Section 4 discusses the problem of initial learning and the supervisor. Section 5 presents some results, in simulation and in a real application, showing the improvements that can be obtained with the proposed new features.
I. INTRODUCTION Since the pioneer works of Zadeh (1965, 1973), Mamdani and Assilian (1975), Procyck and Mamdani (1979), Takagi and Sugeno (1985), fuzzy control had a large development in its methodologies and applications, see for example Li et af. (1995, 2001). However, to a large extend, there still remains open the problem of efficient on-line learning and in general the self-organizing capabilities of the fuzzy-systems, Singh (1998). Many authors have proposed solutions to the problem through neuro- fuzzy structures, Yager (1999), Belarbi et al. (2000), Lazzerini et af. (1999), Chen and Peng (1999), Shi and Mizumoto (2000), clustering techniques, Gomez et al. (1999), and other several techniques based on data Wang (1999), Zapata and Galvao (1999), Li and Priemer (1999). However there is a need for simple structures with large generality, avoiding a-priori knowledge about the process. In this sense, in a previous work Dias and Dourado (1999) a fuzzy controller with self-organizing capabilities and with a broad generality for monotone stable systems has been developed. One important characteristic of the controller is that its rule base has a Fixedfuzzy Maximum Number of Rules (FMNR), determined by the operator. Its performance is determined by two parameters: the initial value of Dmax and the value of the observation time-window. When these parameters are not properly fixed, namely at initialisation phase, the performance of the controller maybe not satisfuctory. In this work further developments are presented though a supervisor that is charged to control the adaptation of the Dmax parameter. The fuct that the controller starts with an empty rule base may cause a inadequate performance at start-up, showing oscillations and considerable overshoot. This drawback can be avoided if at start-up, while the initial rule base is constructed, a simple controller, like a PI one, is implemented. The supervisory level of the
2. THE FMNR SOC FUZZY CONTROLLER The controller supposes a stable monotone process of Non-linear Auto-Regressive with eXogenous Input (NARX) type generally defined by (I ): y( k) =
f ( y(k -
1), y(k - 2 ), ... , u(k -
n. ... ,
(1 )
u(k - T-l), ... , e(k))
with f( ·) a linear or non-linear function, time variant or not, u(k -
y(k). y(k-l) .. . .
the outputs of the system and
u(k).
1). .. .. the inputs to the system at the k, k -1, ...
time instants. T represents the discrete pure time delay and e(k) is a stochastic disturbance (assumed white noise and neglected in the following). The specific knowledge about the process is the pure time delay T or its upper bound, the upper bound of the memory (order) of the system and the sign of its monotonicity. The reference should also be known at least T + 1 sampling instants in advance (particularly for tracking problems). Solving (1) for u(k - 1). one obtains (2), u(k -
n
= g ( y(k), y(k - 1), . . . , u(k - T - 1), . . . )
(2)
Applying to (2) a linguistic transformation, and adding T + 1 to every index (time translation) one obtains (3) in the form of a Fuzzy Auto-Regressive with Exogenous Input system (FARX).
75
IF y(k + T + 1) is Al and y(k +
n is A2 and ... and
u(k) is BI and .. . THEN u(k + 1) is C AI, Az, . .. ,
and (3)
D*; =
--J (YI; -
ri + (Y2; - r2) 2 (5)
1
BI , .•• , C are linguistic values at the
w· =
--xD+ I
-
•
proper sampling instants. For the inference of this fuzzy rule, the needed future values of the process output are replaced in (3) by the known future values of the reference; this gives some feed forward effect to the control signal. The rules are constructed accordingly to the experiments. The rule base is a matrix where each row is a rule and each column is a linguistic value until the FMNR is reached. At this point the new rules are written over the older ones, beginning with the rule that occupies the first place in the matrix, so that the FMNR is never exceeded and up-dating the rules to changes in process conditions. The choice of the FMNR must be considered with some care. On one hand the number of rules must be compatible with the time needed to compute the algorithm. On the other hand it should be enough representative of the process behaviour, spanning along the input and output spaces. See Dias and Dourado (1999) for some guidelines. The fuzzification uses (for all the variables) triangular membership functions as in Park et al. (1995). Every crisp Xl value defines the unity vertex of a membership function and all the membership functions have the same slope. All the fuzzy sets overlap on the entire interval [a, b] including all possible values for the
This
ill;
is
Dmax'
called
the
degree of similarity.
Dmaxrepresents the maximum distance considered. If D; is greater than Dmax it is considered that there is no similarity. The consequent fuzzy set for each rule is an interval of values [PI , P2], where Il.... (u(k)) ~ W;, V u(k)
E
(PI> P2]
(6)
The total output is taken as the intersection of all the intervals (one for each rule). From the inference mechanism one obtains a learning interval (7) of possible values for u(k + 1). (7)
flu = [min, max]
The adaptive parameter Dmax
The Dmax parameter is adapted in order to reduce oscillations and steady-state error. Its value is increased by (8) in the presence of the first situation and decreased by (9) in the presence of the second, where UI and U2
are coefficients empirically fixed. Dmax = Dmax x ( 1
+
uI percenlage of oscillation)
Dmax = Dmax x( 1 - u2 percenlage of error)
variable Xl' By this way each membership function is simply defined by its maximum crisp value and the rule base is always complete. For the inference mechanism with dynamics changes, and considering, for example, a I S/ order
(8) (9)
The output behaviour is monitored during a timeobservation window, whose size must be determined by the user. The Defuzzification mechanism
process, each rule has the format: Ify(k) is YI; and y(k - 1) is Y2i then u(k -
n is
If the future values of the
The output of the inference mechanism is an interval [min, max] (7). Let the error in advance be defined as the
reference are
and r2 (respectively in k + T + 1 and
difference between the actual process output (at k
k + T sampling times), the degree of equality between
instant) and the desired reference after the T + 1 instants.
them and each rule antecedent is inspired in Park et al. (1995) by expression but with a new factor to consider changes in process dynamics; by this way it contains information from both the reference values and the real output values, given by (4) and (5),
To compute a value for u(k + 1), a simple but efficient method is proposed, inspired in Park et al. (1995) given by (10) and (11).
rl
U; ".
error_ in_ advance
=
r(k + T + 1) - y(k)
Considering a positive monotonocity: If error_in_advance> 0
(4)
with
then u(k + 1) = El
= YI; - y(k)
E2 = Y2i - y(k - 1)
else u(k + 1) =
u= u;-u(k-n
76
max+u(k-n
2 min+u(k-
2
n
(10)
3. THE INITIAL VALUE OF Dmax AND THE ADAPTATION WINOOW
In the case of a negative monotonocity: If error_ in_ advance> 0 min +u(k-1)
then u(k + 1) = else u(k + 1) =
The value of Dmax represents the maximum distance allowed to compute the similarity degree between a given situation and the antecedents of the rule. If Dmax is
(11)
2 max+u(k-1)
low, the used rules are few, the CJ)i values are low and the resulting control interval presents too large values. If Dmax is high, the reverse happens. On the other hand if the process gain is high, the input must be in a smaller interval. If this gain is low, then the input interval can be larger. From these thoughts one can conclude that for high gain systems Dmax should be higher, while for low gain ones it should be lower. There exists then a relation between the process gain and the initial value of Dmax; this relation is
2
This controller shows good perfonnance but in some situations it may have two main drawbacks. Fig. I shows simulations results for the simple first order i1lustrative process (12). y(k + 1) = 0.8553 y(k)
~
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~
(12)
~
2. 5
proposed as (13), where Umax is the maximum admissible value of the control signal.
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+ 0.1543 u(k)
y
1.5
(13)
-.- --
1.0
\
0.5
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The process gain is not known, in general, with accuracy. An estimation of it, even coarse, is however enough for the purpose. To obtain it (14) is used. Empirical evidence showed that it contains sufficient infonnation for the purpose in the used examples. y(k) Gain = u(k _ 1) (14)
--
,
- ---
o
20
40
60
80
100 120 Iterations
140
160
180
The adaptation window has a strong influence in the If it is small, Dmax becomes too oscillating and perfonnance is not good; if it is long, the adaptation of Dmax is not done in proper time and performance is also insufficient. The good value for the adaptation window is related to the maximum number of rules. A good compromise is to consider this window (in sampling intervals) equal to half of the maximum number of rules. By this way it is guaranteed that it is sufficiently representative of the system behaviour and there is no "over adaptation".
7.0
Dmax behaviour.
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~
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<;
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4.0 3.0 2.0
'\IV
1.0
0
20
40
u
60
80
100 120 iteratIOns
140
160
180
4. THE START-UP PHASE AND THE SUPERVISOR During the start-up phase there are few rules (initially no rule). The computation ofthe control action results in some cases in an oscillatory behaviour with strong overshoots and undershoots. Only when a significant part of the maximum number of rules is built from experimental data is this problem surpassed. In order to improve the control system behaviour, a simple controller (P, PI, etc.) may be used to start-up the process. After some time, this controller is switched to the fuzzy controller. The switching is a task of the supervisor, according to the following rules (15) found by experimental research. For the regulation problem
Fig. I. The FMNR fuzzy controller behaviour for a step reference.
In the starting phase, it may show osci1lations and overshoot that deteriorates the behavior. The initial value of Dmax is found mainly by trial and error procedures, as well as the size of the observation window. To solve these drawbacks a supervisor is proposed in the following.
77
The fuzzy controller is put to work after the starting phase (during which a PI is working) and a good performance can be observed with a smooth control signal. For the second step in the reference (iteration 100), a significant overshoot is obtained (bad performance) but the fuzzy system changes its own rules to decrease this overshoot (and undershoot) in the following steps. The rule base is being changed with time in such a way that performance is improved.
(constant reference) the switching is done when the output approaches the reference, as (15). If reference is constant And error is less than (15) 10% Then switch to fuzzy controller For the tracking problem (varying references) the switching is made after a certain number of samples fills the rule-base. For example for sinusoidal references (16) is enough.
If reference is not constant And error is less than 10% And the sampling instant is greater than 10 Then switch to fuzzy controller
B. Experimental results with a laboratory process In the laboratory process, depicted in Fig. 3 and Fig. 4, air is forced to circulate by a fan blower through a duct and is heated at its inlet. This is a non-linear process with a pure time delay, which depends on the position of the temperature sensor element and of the airflow rate, depending on the damper position n. The system input u(k), is the voltage on the heating device, which
(16)
The supervisor is charged with the following tasks: computation of the process gain, computation of the initial value of Dmax, controller switching, monitoring the adaptation of Dmax. 5. RESULTS
consists of a mesh of resistor wires, and the output, y(k), is the outlet air temperature.
A. Simulation results
For the process given by (12), with a square wave as reference signal, the results shown in Fig. 2 were obtained. 3. 5
~
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~or
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0
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,
.
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Fi~ ._ 3. Laboratory
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I
o 50
100
150
200
250
300
350
JI
o
111
Air
Output
c:::)
400
Jt e ralion.~
Y(k)r D ( - 1O.1O( Volts
35
Fig. 4. Schematic diagram of the Laboratory Process.
\
3.0
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u
Sampling time was chosen 400 milliseconds. An experiment with the following parameters was made: maximum number of rules: 100; time-window: 50; initial Dmax = 20. For a square wave as reference, Fig. 5 shows the results. The learning ability can be observed. The performance increases with time and after 200 iterations the behaviour of the control system is good. During the learning phase the controller learns its own rules in order to improve its performance.
f\.
2.5
!'!l
'"
2.0
'-''"
1. 5
l
1.0 0.5
If o
v
50
100
150
200
250
300
350
400
Iterations
Fig. 2. The FMNR controller with the supervisor
78
B
--
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------- ~fI-----
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e
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o
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SO
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200 250 Iteralions
300
350
400
100
SO
0
ISO lterallons
200
250
300
Fig. 6 . Experimental results with PT326; sinusoidal reference
Without the supervisor the controller behaves as shown by Figure 7: the learning phase is very long (two periods of the reference).
~
]
4
~
9
e;:
u
B
c3
8
;;
~
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0
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6
:2
o
50
100
150
200 2SO lleralions
300
3SO
5 "
~
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400
4
.
~
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Fig. 5. Experimental results with PT326; square wave reference.
Now for a sinusoidal reference with a period of 38s, the results are presented in Fig. 6. The learning ability can be observed. The perfonnance increases with time and after 200 iterations the behaviour of the control system is good. During the learning phase the controller learns its own rules in order to improve its perfonnance.
o
50
100
150 Iterations
200
250
300
SO
100
ISO Iterollo,..
200
2SO
300
9
8
~ 5.0
~
~
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e;:
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6
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o
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Fig. 7 . The FMNR without supervisor; learning phase may be quite 0
50
100
150 Iterations
200
2SO
300
longer, if Dmax is not properly initialised.
79
Li
6. CONCLUSION
c., R. Priemer (1999) - Fuzzy control of unknown multiple-
input-multiple-output plants, Fuzzy Sets and Systems, Vol 104, Iss 2, pp 245-267, Elsevier Science. Li, Hongxing, C. Vincent C. Yen, (1995) - Fuzzy Sets and Fuzzy Decision -Making, CRC Press. Li, Hongxing, C. Philip Chen, H. Huang (2001) - Fuzzy Neural Intelligent Systems, Mathematical Foundations and the Applications in Engineering, CRC Press. Mandani, E., S. Assilian (1975) - An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller, Int. L of ManMachnineStudies, vol. 7, nO I, pp I-B. Park, Young-Moon, U. Moon and L. Kwang (1995) - A SelfOrganizing Fuzzy Logic Controller for Dynamic Systems Using a Fuzzy Auto-Regressive Moving Average (FARMA) Model, lEE Transactions on fuzzy Systems, Vol.3, nOI , pp 75- 82. Procyck T., E. Mamdani (1979) - A Linguistic SelfOrganizing Process Controller, Automatica, Vol. 15, 1530. Shi Y. , M. Mizumoto (2000) - A new approach ofneuro-fuzzy learning algorithm for tuning fuzzy rules, Fuzzy Sets and Systems, Vol 112, Iss I, pp 99-116, Elsevier Science. Singh Y. (1998) - A modified self-organizing controller for real-time process control applications, Fuzzy Sets and Systems, Vo1.96, pp 147-159. Takagi T., M. Sugeno (1985) - Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man and CybernetiCS, Vo1.l5, No. I, pp 116-B2. Wang, L. X. (1999) - Automatic design of fuzzy controllers. Automatica, Vol 35, Iss 8, pp 1471 -1475, PergamonElsevier Science. . Yager R.R. (1999) - Implementing fuzzy logic controllers using a neural network framework, Fuzzy Sets and Systems, Vol lOO, Suppl. S, pp 133-144, Elsevier Science Zadeh, L. A (1965). "Fuzzy sets", Information and Control, Vol. 8, pp 338-358. Zadeh, L. A. (1973) - Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. On Systems. Man and Cybernetics, vol. 3, nO I, pp 28-44. Zapata, G., R. Galvao, T. Yoneyama, (1999) - Extracting fuzzy control rules from experimental human operator data, IEEE Transactions on Systems Man and Cybernetics Part B - Cybernetics, Vol 29, Iss 3, pp 398406.
By experimental evidence, the introduction of a supervisor in the fuzzy controller with a similarity factor and a maximum number of rules allowed increased performance and robustness. The operations implemented in the supervisor are simple and resulting from experimental research. However, more formal developments are needed in order to show generality in the learning capabilities and to support the value of the proposed architecture for general non-linear systems. The monotonicity hypothesis of the process is a limitation, but as most of the processes possess this property, in practice it is not a serious drawback. The objective of this work is to obtain a controller for nonlinear systems with a broad generality, using minimum a-priori knowledge about the process. ACKNOWLEDGEMffiNTS This work has been partially supported by FCTlPraxis XXI through CISUC and the project ALCINE- Online learning in intelligent control and by POSI- Programa Operacional da Sociedade da Informa'Y3o of Portuguese Funda'Y3o para a Ciencia e Tecnologia of Portuguese Government, supported by European Union FEDER.
REFERENCES Belarbi, K., K. Bettou, A. Mezaache (2000) - Fuzzy neural networks for estimation and fuzzy controller design: simulation study for a pulp batch digester, Journal of Process Control, Vol 10, !ss I, pp 35-41, Elsevier Sci. Chen, c., S. Peng (1999) - Intelligent process control using neural fuzzy techniques, Journal of Process Control, Vol 9, !ss 6, pp 493-503 , Elsevier Sci. Dias J., A Dourado (1999) - A Self-Organizing Fuzzy Controller with a Fixed Maximum Number of Rules and an Adaptive Similarity Factor, Fuzzy Sets and Systems, vol. 103, pp 27-48, Elsevier Science. Gomez, A , F. Skarmeta, M. Delgado, M. Vila (1999) - About the use of fuzzy clustering techniques for fuzzy model identification, Fuzzy Sets and Systems, Vol 106, !ss 2, pp 179-188. Lazzerini B., L. Reyneri, M. Chiaberge (1999) - A neurofuzzy approach to hybrid intelligent control, IEEE Transactions on Industry Applications, Vol 35, Iss 2, pp 413-425.
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