Multi-region fuzzy logic control system with auxiliary variables

A fuzzy control system for industrial process control utilizes an auxiliary process variable to determine which of several regions of different gain a non-linear process is operating. Based upon that determination and fuzzy input signals, the fuzzy controller provides a process control signal which corresponds to that region.

BACKGROUND OF THE INVENTION 
The present invention relates to a control system for controlling 
non-linear processes having more than one region of process operation. In 
particular, the present invention relates to a fuzzy logic controller 
which utilizes an auxiliary process variable for determining a region of 
process operation and selecting a fuzzy membership function for 
application to a fuzzy input variable to provide a process control input 
that corresponds to the region of process operation. 
Fuzzy logic involves a series of fuzzy control rules which are expressed by 
a fuzzy implication of the form "if . . . then . . . ." These fuzzy 
implications include fuzzy variables which are often referred to as 
"linguistic variables". Fuzzy reasoning or inferences are accomplished by 
the application of the fuzzy variable to the fuzzy rules. 
Fuzzy logic has been used in process control applications as described for 
example in Tanaka et al., U.S. Pat. No. 5,158,024 incorporated herein by 
reference. In these control applications, the fuzzy logic forms a fuzzy 
controller for controlling process parameters. A typical fuzzy logic 
controller is composed of three basic parts: input signal fuzzification, a 
fuzzy engine that handles rule inference, and defuzzification that 
generates continuous signals for actuators such as control valves. 
There are several advantages to the use of fuzzy control of process 
parameters. One advantage is that human experience can easily be 
integrated into the fuzzy controller because the fuzzy control rules and 
the fuzzy variable are well suited to the human thought process. 
Another advantage of the use of a fuzzy controller is the non-linearity 
resulting from fuzzification, application of the fuzzy rules to the fuzzy 
variables, and defuzzification. This non-linearity inherent in the fuzzy 
control process makes fuzzy controllers well suited to non-linear process 
control. 
However, process controllers that are presently used for process control 
tend to make use of a process error signal and a change in process error 
signal as controller inputs for determining a process control or process 
input signal. These controller inputs do not provide sufficient 
information to the fuzzy controller so that the process control signal can 
account for any process non-linearities. Therefore, these fuzzy logic 
controllers are not capable of compensating for process non-linearities. 
There is a present need for fuzzy controllers capable of providing a 
process control signal for compensating a process having a non-linear 
process variable such as process gain. This fuzzy controller should be 
capable of compensating the process throughout the different regions of 
non-linearity so that the control performance is uniform. 
SUMMARY OF THE INVENTION 
One aspect of the present invention is a fuzzy logic control system suited 
for controlling a non-linear process. The fuzzy control system includes a 
fuzzy controller which utilizes an auxiliary process variable to determine 
where the process is operating and compensate for process non-linearities. 
Another aspect of the present invention is a control system that is suited 
for providing a process control signal for controlling a non-linear 
process having first and second regions of operation. The control system 
includes means for providing an auxiliary process signal having 
predetermined characteristics is each of first and second regions of 
process operation. 
A fuzzy logic controller is provided which includes a first fuzzy 
membership function relating a process error signal to the process control 
signal for process operation in the first region. Also included is a 
second fuzzy membership function relating the process error signal to the 
process control signal for process operation in the second region. A fuzzy 
inference engine is included for applying one of the first and second 
fuzzy membership functions selected based on the auxiliary process signal 
value to a process error signal value for inferring a process control 
signal value. 
In one preferred embodiment, the present invention is a control system for 
controlling a process having non-linear process gain. The process has 
first and second regions of process gain. The control system includes a 
circuit that is connected between a process and a fuzzy logic controller. 
This circuit provides an auxiliary process variable that is indicative for 
first and second regions of process gain. 
The fuzzy logic controller provides a control signal value to the process. 
The fuzzy logic controller includes a first fuzzy membership function that 
is selected to provide proportional and integral control suitable for the 
first region of process gain. A second fuzzy membership function is 
included that is selected to provide proportional and integral control 
suitable for the second region of process gain. Also included is a fuzzy 
engine configured for applying one of the first and second fuzzy 
membership functions (selected based on a process operation region 
determined from an auxiliary process variable value to a process error 
signal value) and a change in process error value. This application of the 
selected fuzzy membership function allows the fuzzy engine to infer the 
control signal value for maintaining proper process operation in the first 
and second regions of process gain. 
An additional aspect of the present invention is a control system for 
controlling a non-linear process as a function of a process error signal. 
The control system includes means for providing an auxiliary variable 
signal which is indicative of which of a plurality of operating regions 
the non-linear process is operating. A fuzzy logic controller is included 
for providing a process control output signal for controlling the 
non-linear process as a function of the process error signal and the 
auxiliary variable signal. 
In one preferred embodiment, the present invention is a control system for 
providing derivative, proportional and integral control of a process 
having a non-linear process gain for each of a plurality of different 
operating regions of the process. The control system includes means for 
providing a process error value and means for providing a change in 
process error value. Included is means for providing an auxiliary value 
indicative of one of a plurality of different operating regions the 
process is operating. Also included is a fuzzy logic controller for 
providing proportional, integral and derivative control of the process as 
a function of the process signal error value, the change in process error 
value and the auxiliary value. 
A further aspect of the present invention is a controller for controlling a 
non-linear process as a function of process error signal and a controller 
parameter value. The controller parameter value is based on which of a 
plurality of operating regions the non-linear process is operating. The 
controller includes a fuzzy engine linked to the non-linear process for 
providing a process control output signal as a function of the process 
error signal, the controller parameter value and one of a plurality of 
sets of fuzzy rules for the different operating regions of the process. 
Yet another aspect of the present invention is a method for controlling a 
non-linear process as a function of a process error signal. The method 
includes determining a region of process operation from a plurality of 
operating regions based on an auxiliary process variable signal. The 
method includes applying one of a plurality of sets of fuzzy rules for the 
different operating regions of the process to the process error signal for 
inferring a process control output signal. The fuzzy rule applied is based 
upon the operating region of the process. 
In one preferred embodiment, the method for controlling the non-linear 
process also includes converting the process error signal to a fuzzy error 
signal value. In this preferred embodiment, applying one of a plurality of 
sets of fuzzy rules includes applying one of a plurality of sets of fuzzy 
rules to the fuzzy process error signal for inferring a process control 
signal value.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Before discussing the fuzzy logic controller of the present invention, it 
is helpful to first review frequently used fuzzy controllers of the prior 
art. FIG. 1 shows a fuzzy controller 10 of the prior art. Fuzzy controller 
10 includes a fuzzification portion 12, a fuzzy engine 14, and a 
defuzzification portion 16. Input signals 18 and 20 provided to the fuzzy 
controller 10 represent continuous signals indicative of process error. 
Output signal 22 provided by the fuzzy controller 10 is a continuous 
signal for providing process control. 
Often, input signals 18 and 20 provided to the fuzzy controller 10 
represent process error (e) and process change in error (.DELTA.e), 
respectively. For the case of a proportional integral (PI) controller, the 
process error signal 18 is frequently the difference between a setpoint 
and a process output signal, and the change in error signal 20 represents 
the change in the process error signal 18 over a selected sampling 
interval. 
The output signal 22 of the fuzzy controller 10 often is provided to the 
process for minimizing or reducing the process error represented by fuzzy 
controller input signals 18 and 20. Output signal 22 is frequently used to 
activate control elements within the process such as control valves and 
actuators. 
The fuzzification portion 12 converts or transforms the continuous input 
signals 18 and 20 into linguistic or fuzzy variables at selected sample 
intervals. These fuzzy variables are derived from human experience such as 
small, large, positive, and negative, just to name a few. The fuzzy 
variables are provided the fuzzy engine 14 for performing a fuzzy 
inference. 
The fuzzy engine 14 includes a set of fuzzy rules that are a series of 
statements of the form "if . . . then . . . " which involve fuzzy 
variables. The fuzzy variables provided by fuzzification portion 12 are 
applied to the fuzzy rules by fuzzy engine 14 to infer a control action. 
This control action which is in the form of a fuzzy variable is provided 
to the defuzzification portion 16. 
The defuzzification portion 16 converts the fuzzy variable representing an 
inferred control action to a continuous signal represented by output 
signal 22. 
FIG. 2a and 2b are two examples illustrating the process input and process 
output relationship for two non-linear processes. In each of these figures 
the horizontal axis represents the process input represented by the 
variable U and the vertical axis represents the process output represented 
by the variable Y. A plot 24 shown in FIG. 2a has a sinusoidal shape 
having two low process gain regions indicated by numerals I and III and 
one high process gain region indicated by numeral II. A plot 26 shown in 
FIG. 2b is another example of non-linear process gain. Plot 26 has a 
region of positive process gain as indicated by numeral I and a region of 
negative process gain as indicated by numeral II. These examples in FIGS. 
2a and 2b are just two examples for illustrative purposes that show 
process non-linearities which should be compensated for by a process 
controller. 
As discussed previously, PI-type fuzzy controller 10 as shown in FIG. 1 
(which makes use of a process error signal 18 and a change in process 
error signal 20) does not differentiate between high gain regions 
represented by numeral II (FIG. 2) and low gain regions represented by 
numerals I and III and therefore cannot provide satisfactory performance 
in all regions. Furthermore, the fuzzy controller 10 shown in FIG. 1 
cannot differentiate between positive gain regions represented by numeral 
I in FIG. 2b and negative process gain regions represented by numeral II 
and therefore is not designed to give satisfactory performance in both of 
these regions. 
One method for controlling non-linear processes having a plurality of 
regions of operation has been to use a plurality of conventional linear 
PID controllers. A scheduler is provided to monitor the process conditions 
and select one of the plurality of conventional linear PID controllers 
based on process conditions. This method of process control is known and 
is frequently referred to as gain scheduling. 
FIG. 3 shows the process control system 30 of the present invention which 
is capable of determining in which region of a non-linear process the 
process is operating and provides a process control signal which 
corresponds to this region. In this manner, the fuzzy control system 30 is 
capable of assuring stability in the high process gain region indicated by 
numeral II without sacrificing performance in the low process gain regions 
indicated by numerals I and III of FIG. 2a. In addition, as another 
example, the fuzzy control system 30 is capable of distinguishing between 
the positive process gain region indicated by numeral I in FIG. 2b from 
the negative process gain region indicated by numeral II and thereby 
providing negative feedback in one region and positive feedback in the 
other, respectively. 
The fuzzy control system 30 of the present invention includes a non-linear 
process 32 and a fuzzy logic controller 34 for providing a control input 
to the process so that a process output is uniform across each of the 
regions of process operation. 
The fuzzy control system 30 receives a continuous input signal or setpoint 
36 and provides a continuous process output signal 38 designated, y, that 
has a fuzzy variable value designated y.sub.K. An error signal 40 
designated, e, that has a fuzzy variable value designated e.sub.K is 
provided by a combining means 42 which combines the process output signal 
38 with the process input signal 36. The fuzzy logic controller 34 also 
receives a change in process error signal 44 designated .DELTA.e that has 
a fuzzy variable value designated .DELTA.e.sub.K from signal combining 
means 42. The change in process error signal 44 represents the change in 
the process error signal 40 over a selected sampling interval. 
An important aspect of the fuzzy control system 30 of the present invention 
is the use of an auxiliary variable signal 46 designated AV that has a 
fuzzy variable value designated AV.sub.K which is provided by the process 
32 to the fuzzy logic controller 34 for determining where the non-linear 
process is operating so that the fuzzy logic controller 34 can provide 
satisfactory performance in all regions of process operation. Fuzzy 
control systems that utilize fuzzy logic controllers such as that shown in 
FIG. 1 which do not make use of an auxiliary variable signal 46 cannot 
differentiate between operation at different points in the process 
non-linearity and therefore cannot properly compensate for this process 
non-linearity. Therefore, the use of an auxiliary variable signal 46 by 
the fuzzy logic controller 34 improves the process performance in all 
regions of operation. 
The fuzzy logic controller 34 includes a fuzzification portion, a fuzzy 
engine, and a defuzzification portion similar to the fuzzy logic 
controller 10 shown in FIG. 1. However, instead of converting only the 
process error signal (e) and the process change in error signal 
(.DELTA.e), the fuzzification portion of fuzzy logic controller 34 also 
converts the auxiliary process variable signal 46 (AV) into a fuzzy 
variable. These fuzzy variables are applied to fuzzy rules by the fuzzy 
engine to provide a fuzzy inference. This fuzzy inference, which is in the 
form of a fuzzy variable, is converted to a continuous signal by the 
defuzzification portion of the fuzzy logic controller 34. This defuzzified 
output signal from fuzzy logic controller 34 represents a process input 
signal 48 that is designated .DELTA.u and that has a fuzzy variable value 
designated .DELTA.u.sub.K. 
The process input or control signal 48 (.DELTA.u.sub.K) provided by the 
fuzzy logic controller 34 is a change in process input signal. Therefore, 
the instantaneous process input 50 designated u.sub.K is determined by a 
summation of all previous control actions. This summation is represented 
by delay portion 52 which provides the previous instantaneous process 
input signal u.sub.K which is summed with the change in process input 
signal 48 (.DELTA.u.sub.K) by summing means 54. 
The auxiliary variable (AV.sub.K) should be selected so that the operation 
regions of the non-linear variable can be defined. The instantaneous 
process input 50 (u.sub.K) or the process output signal 38 (y.sub.K) may 
be used as the auxiliary variable (AV.sub.K) depending on how the 
operation regions are defined. For example, the process output signal 
(y.sub.K) may be used as the auxiliary variable (AV.sub.K) for the case 
where the gain is sinusoidal as shown by plot 24 in FIG. 2a. However, in 
FIG. 2b the process input signal (u.sub.K) should be used as the auxiliary 
variable (AV.sub.K) for the case when the process gain changes signs as 
shown by plot 26 in FIG. 2b. 
The fuzzy membership functions or rules which are stored in the fuzzy logic 
controller 34 are defined based on prior knowledge or predetermined 
characteristics of the process 32. The membership function is defined so 
that the process 32 is compensated for undesirable system behavior. As an 
example, a membership function for a process having a sinusoidal 
non-linear gain as shown in FIG. 2a is used to illustrate the definition 
of the membership function. For the case of the sinusoidal non-linear 
system the auxiliary variable (AV.sub.K) should have three regions 
corresponding to the three distinct process gain regions of FIG. 2a. The 
membership function for the auxiliary variable (AV.sub.K) is shown in FIG. 
4b. The fuzzy variable associated with each of the three regions 
designated by numerals I, II, and III are "low", "medium", and "high", 
respectively. The fuzzy membership functions for change in process error 
(.DELTA.e.sub.K), process error (e.sub.K) and change in process input 
(.DELTA.u.sub.K) are shown in FIGS. 4c, 4b, and 4d, respectively. In FIGS. 
4b, 4c, and 4d, NL represents the fuzzy variable negative large, NM 
represents the fuzzy variable negative medium, NS represents the fuzzy 
variable negative small, ZO represents the fuzzy variable zero, PS 
represents the fuzzy variable positive small, PM represents the fuzzy 
variable positive medium, and PL represents the fuzzy variable positive 
large. Each of the fuzzy variables, control error (e), change of control 
error (.DELTA.e) and change in process input (.DELTA.u) are normalized in 
a conventional manner as shown in equations 1, 2, and 3, respectively. 
EQU e*=e/S.sub.e (1) 
EQU .DELTA.e*=.DELTA.e/S.sub..DELTA..sbsb.e (2) 
EQU .DELTA.u*=.DELTA.u/S.sub..DELTA..sbsb.u (3) 
Where e*, .DELTA.e*, and .DELTA.u* are the scaled membership functions 
shown in FIGS. 4b, 4c, and 4d, respectively and S.sub.e, S.sub..DELTA.e, 
and S.sub..DELTA.u, are scaling factors for the fuzzy variables e, 
.DELTA.e, and .DELTA.u, respectively. 
After defining the fuzzy variables for the controlled process, the fuzzy 
inference rules must then be defined. An example of a general fuzzy 
inference rule is described as follows: 
EQU If AV is A.sub.i and e is B.sub.i and .DELTA.e is C.sub.i, then make 
.DELTA.u D.sub.i, (4) 
where A.sub.i, B.sub.i, C.sub.i, and D.sub.i are adjectives representing 
variable values for AV, e, .DELTA.e, and .DELTA.u, respectively. These 
adjectives or variable values could be descriptors such as negative small, 
positive large, and zero. Fuzzy rules are derived from experience of human 
operators controlling the process for which the fuzzy controller will be 
used. 
The fuzzy inference rules define the response by the fuzzy logic controller 
34 to a particular process condition. For example, when the process is 
operating in the low gain region, aggressive control action is required. 
On the other hand, when the process is in a high gain region, mild to low 
control action should be used to ensure stability of the system. Within 
each region I, II, and III of the process shown in FIG. 2a for which there 
is a corresponding region for the auxiliary variable (AV), rules can be 
defined for a conventional fuzzy-type controller. In this manner, three 
proportional-integral (PI) type fuzzy controllers are designed in a 
conventional manner to provide proper compensation when the process is 
operated in the selected region. An example of fuzzy inference rules for 
the sinusoidal non-linear process of FIG. 2a is shown in FIG. 5. 
As shown in FIG. 5, these fuzzy inference rules are in three separate sets 
with the first set corresponding to the auxiliary variable (AV.sub.K) in a 
low condition as defined in FIG. 4a, the second set corresponding to the 
auxiliary variable (AV.sub.K) in a medium condition as defined in FIG. 4a 
and the third set corresponding to the auxiliary variable (AV.sub.K) in a 
high condition as defined in FIG. 4a. In this manner, the auxiliary 
variable AV.sub.K (which has three regions having variable values low, 
medium, and high that correspond to the three regions shown in FIG. 2a 
labeled numeral I, II, and III, respectively) can be used to select the 
corresponding set of fuzzy rules designed for control of that particular 
region of process operation. 
From FIG. 5, when the auxiliary variable is either low or high and when the 
error and change in error variables have values that are either negative 
large or negative small, the fuzzy controller output signal has a value 
(u.sub.K) that is equal to positive large. The fuzzy logic output 
variables are either positive large or negative large when the auxiliary 
value is either high or low which corresponds to low gain regions of the 
non-linear process. The process control is more aggressive in these low 
gain regions to provide better response times. In contrast, the fuzzy 
membership rules which correspond to the auxiliary variable having a 
medium value which corresponds to the high gain region the control 
variable is never large in order to maintain stability in this operating 
region. 
An important aspect of the present invention is that the fuzzy rules for 
each region of the non-linear process can be consolidated where possible. 
For example, the process control variable has a zero value for the same 
value of variables e.sub.K and .DELTA.e.sub.K, no matter what the 
auxiliary value AV.sub.K is. By combining these rules in FIG. 5, ten rules 
can be eliminated without changing functionality. Moreover, if the fuzzy 
control rules where identical or both conditions where the auxiliary 
variable value low or high, then 25 of the fuzzy control rules can be 
eliminated. In this manner, rule reduction can be done by combining 
identical rules for different values of the auxiliary variable. 
For a three-region fuzzy controller using five variables, there are 75 
possible rules. If the controller requires the use of seven adjectives, 
then 147 rules are required. Therefore, the total number of rules can be 
reduced by reducing the number of variables. In addition, as discussed 
previously, the number of rules can be reduced by combining or 
consolidating identical rules used in different regions of process 
operation. This reduction of rules reduces both the computing time 
required by the fuzzy logic controller 34 and the memory required to store 
each of these rules within the fuzzy logic controller. 
After both the fuzzy variables are selected and the fuzzy rules are 
defined, the multi-region fuzzy controller 34 must be tuned. Tuning is a 
process which includes: (i) tuning of the scaling factors, (ii) tuning of 
the fuzzy membership functions, (iii) tuning of the fuzzy rules, and (iv) 
tuning of the auxiliary variable (AV) membership functions for smooth 
regional transitions. The scaling factors should be tuned first because 
they are global tuning parameters that affect the overall control 
performance. A membership function, which has an effect on one subset of 
rules, can be tuned second. Individual fuzzy rules should be tuned last 
because they affect only specific singular outcomes. 
Within each region, the fuzzy controller 34 can be tuned using conventional 
techniques similar to that used for a fuzzy PI controller. For a PI 
controller that can be represented by the following equation: 
##EQU1## 
K.sub.p and T.sub.i are related to the scaling factors by the following 
equations: 
EQU K.sub.p =S.sub..DELTA..sbsb.u /S.sub..DELTA..sbsb.e (6) 
EQU T.sub.i =(S.sub.e /S.sub..DELTA..sbsb.e).DELTA.t (7) 
Where K.sub.p is the proportional gain, T.sub.i is the integral time 
constant, and .DELTA.t is the controller sampling time constant. 
Therefore, the proportional action is increased by either increasing 
S.sub..DELTA.u or decreasing S.sub..DELTA.e. The integral control action 
is increased by either decreasing S.sub.e or increasing S.sub..DELTA.e 
because the small integral time constant represents strong integral 
control action. Although a change in the controller sampling time 
(.DELTA.e) also affects the integral time, controller sampling time is 
normally not used as a tuning factor. 
After the fuzzy controller 34 is tuned in each region, the fuzzy controller 
tuning should be coordinated over all regions. If the scaling factor 
S.sub..DELTA.u is used to tune the low gain region, the high gain region 
can only be tuned by adjusting the position of the related membership 
functions. To achieve smooth transition between regions, the membership 
functions for AV can be tuned by experience and trial and error. In 
summary, the following procedures used to tune the multi-region fuzzy 
controller 34: (1) tune the scaling factors (S.sub.e, S.sub..DELTA.e, and 
S.sub..DELTA.u) for the low gain region; (2) tune the position of the 
related membership function for .DELTA.u in the high gain region; (3) tune 
the fuzzy controller 34 over all regions to achieve smooth transitions of 
control; and (4) fine tune the membership functions and the rules to 
achieve desired control performance. 
The following is an example of the fuzzy control system 30 of the present 
invention that is used to control a continuously stirred tank reactor 
(CSTR) for pH titration as shown in FIG. 6. The CSTR 60 includes a tank 62 
having a stirring means 64 and two input streams, an acid stream 66 having 
a flow rate designated F.sub.1 and a concentration represented by the 
variable C.sub.1, and a base stream 68 having a flow rate designated 
F.sub.2 and a concentration represented by the variable C.sub.2. In this 
example, the flow rate F.sub.1 of the acid stream 66 is used to control 
the pH value of a solution 70 within tank 62. The concentration C.sub.2 of 
the base stream 68 is used as a disturbance which is to be compensated for 
by the fuzzy control system 30 of the present invention. The pH of the 
solution 70 is measured as the solution streams from the tank 62 and 70. 
Valves 72A, 72B and 72C are provided for controlling the flow of acid 
stream 66, base stream 68 and solution stream 74. 
Assuming an ideal CSTR 60 that is perfectly controlled, the relationships 
between the volume of solution 70 in tank 62 and the flow rates F.sub.1, 
F.sub.2 of the acid and the base and the concentrations C.sub.1 and 
C.sub.2 of the acid and base are known, as is the relationship of these 
flow rates to the pH of the tank, shown generally in the following 
equations: 
##EQU2## 
EQU [H.sup.+ ].sup.3 +(K.sub.4 +.xi.)[H.sup.+].sup.2 +(K.sub.4 
(.xi.-.zeta.)-K.sub.4)[H.sup.+ ]-K.sub.W K.sub.4 =0 (10) 
EQU pH=-log.sub.10 [H.sup.+ ] (11) 
EQU .zeta..ident.[HAC]+[AC.sup.- ] (12) 
EQU .xi..ident.[Na.sup.+ ] (13) 
Where the physical meaning of each variable and associated initial values 
are shown in Table 1. 
TABLE 1 
______________________________________ 
Variable Meaning Initial Setting 
______________________________________ 
V Volume of Tank 1000 Liters 
F.sub.1 Flow Rate of Acid 
81 Liters/min 
F.sub.2 Flow Rate of Base 
512 Liters/min 
C.sub.1 Concentration of Acid in F.sub.1 
0.32 moles/liter 
C.sub.2 Concentration of Acid in F.sub.2 
.05005 moles/liter 
K.sub.a Acid Equilibrium Constant 
1.8 .times. 10.sup.-5 
K.sub.w Water Equilibrium Constant 
1.0 .times. 10.sup.-14 
______________________________________ 
The steady state non-linear relationship between the flow rate of acid 
(F.sub.1) and the pH of the solution 70 is shown by plot 76 in FIG. 7. The 
plot 76 in FIG. 7 has three regions of non-linear gain indicated by 
numerals I, II, and III which correspond to low pH, medium pH, and high 
pH, respectively. The process gain is very small in the regions of low and 
high pH identified by numerals I and III, while the process gain is 
extremely high in the region of medium pH identified by numeral II. In the 
high gain region indicated by numeral II, a small change in the acid flow 
rate (F.sub.1) results in a large change in pH value, while in the low 
gain regions identified by numerals I and III, considerable change must be 
made in the acid flow rate (F.sub.1) to make an appreciable change in the 
pH value. The CSTR system 60 presents a difficult task for a conventional 
PI-type controller. The process input signal provided by the controller 
must be relatively small to maintain good stability in the high process 
gain region without sacrificing too much dynamic response in the low 
process gain region. 
The fuzzy logic control system 30 of the present invention provides good 
control of pH values over each of the three regions identified by numerals 
I, II, and III in FIG. 7. Using the pH value as an auxiliary variable 
(AV), a set of fuzzy rules can be defined for each of the three regions of 
process gain defined in FIG. 7. These fuzzy rules are defined using human 
knowledge of the process depicted in the steady state relationship of pH 
and acid flow rate (F.sub.1) shown in FIG. 7. 
FIGS. 8a, 8b, and 8c represent the relationship between the change in 
process input on control signal (.DELTA.u), process error (e) and change 
in process error (.DELTA.e) for each region of process gain pH high, pH 
medium, and pH low, respectively. The process control action versus 
control error and change in control error shown in FIGS. 8a, 8b, and 8c 
are frequently referred to as "control surfaces" or "control response 
surfaces" indicated by reference numerals 78a, 78b and 78c, respectively. 
It can be seen from these control surfaces 78a, 78b and 78c that for 
identical process control error and change in process control error, the 
control action provided by the fuzzy logic controller 34 is different 
depending on whether the processes operating in a high gain region or a 
low gain region. For process operation in either of the low gain regions 
shown in FIG. 8a and FIG. 8c, the control action is more aggressive as 
evidenced by the magnitude of the change in process input signal 
(.DELTA.u). In contrast, process operation in the high gain region shown 
in FIG. 8b, the control action is much less aggressive as evidenced by the 
relatively small change in process input signal (.DELTA.u). This variation 
in control response in regions of differing gain cannot be accomplished by 
the conventional fuzzy controller 10 shown in FIG. 1. In addition, the 
control surfaces shown in FIGS. 8a, 8b, and 8c are fairly non-linear, 
which is a feature which cannot be achieved by conventional PID-type 
controllers. The transient regions between each of the three gain regions, 
low, medium, and high are an interpolation of the three control surfaces 
implied defuzzification. 
FIG. 9a shows a step response of the CSTR 60 controlled by the three-region 
control system 30 of the present invention. The step response represents 
the relationship between pH of the solution 70 and time for step changes 
in the setpoint signal 36. The setpoint is changed from 5.5 to 7.0, to 
10.0, to 11.5, and then from 11.5 to 10.0, to 7.0, and to 5.5 so that the 
control performance is evaluated over each of the three process gain 
regions. For each of the setpoint changes the dynamic response or time 
required to compensate for these changes can be seen. The setpoint value 
is shown by plot 90 and the measured pH value of solution 70 is shown by 
plot 92. The system response time varies depending on the gain region and 
direction of change. For example, the step change from a low gain region 
to a high gain region produces a large overshoot. However, a step change 
from a high gain region to a low gain region produces little overshoot. 
FIG. 9b is the step response of the pH controller for the continuously 
stirred tank reactor using a conventional PI-type fuzzy controller for 
comparison. FIG. 9b shows a plot 94 of the pH of solution 70 and the 
identical step response 90 from FIG. 9a with relation to time. To assure 
stability in all three regions, the controller 34 must have a gain that is 
carefully adjusted so that the continuously stirred tank reactor is stable 
in the high process gain region indicated by numeral II in FIG. 7. A 
comparison of the response of the three-region fuzzy logic controller of 
the present invention (shown in FIG. 9a) with the response of the 
conventional one-region fuzzy logic controller (shown in FIG. 9b) 
illustrates that the three-region fuzzy logic controller has less 
overshoot in moving from the low gain region to the high gain region. In 
addition, the overall settling time for the three-region controller is 
less than that for the one-region controller. 
FIG. 10 shows the three-region fuzzy controller performance in the presence 
of disturbances in base concentration. A plot 96 shows the base 
concentration (C.sub.2) with respect to time and plot 98 is the pH of 
solution 70 with respect to time. Both positive and negative base 
concentration disturbances are applied while the pH setpoint is fixed at 
7. As can be seen from plot 98, the three region fuzzy controller performs 
well in rejecting disturbance changes in either positive or negative 
directions. 
In conclusion, the multi-region fuzzy logic control system 30 of the 
present invention is capable of compensating for non-linear process gain 
and yields better control performance than a single-region fuzzy 
controller. The control system 30 uses an additional process variable as 
the auxiliary variable to detect where the process is operating. The 
multi-region fuzzy control system 30 is well suited for controlling 
processes having dramatic non-linear gain changes. 
Although the fuzzy logic controller 34 is described as using the fuzzy 
engine to distinguish between operation in different portions of the 
process, this function can be performed as well outside the fuzzy engine. 
For example, a separate circuit can be used to monitor the auxiliary 
variable and select the appropriate set of fuzzy rules based on the region 
the process is operating. In addition, the rule acquisition for the fuzzy 
controller 34 was previously described as being acquired from a human 
operator. However, a fuzzy/neuro architecture can be used instead for 
acquisition of fuzzy rules. 
In operation, the fuzzy logic controller 34 of the multi-region fuzzy 
control system 30 of the present invention determines a region of process 
operation based on an auxiliary variable value AV.sub.K of process 32. A 
fuzzy membership function associated with the region of process operation 
is selected from at least two fuzzy membership functions each having an 
associated region of process operation. A process control signal value is 
inferred by the fuzzy logic controller 34 based on the selected fuzzy 
membership function and the process error signal values e.sub.K and 
.DELTA.e.sub.K. 
The above process is performed at selected time intervals so that changes 
in regions of process operation are detected and the appropriate process 
control signal is provided to the process 32 to ensure proper process 
operation across all regions of process operation. 
Although the present invention has been described with reference to 
preferred embodiments, workers skilled in the art will recognize that 
changes may be made in form and detail without departing from the spirit 
and scope of the invention.