PID gain scheduling using fuzzy logic

ISA Transactions 39 (2000) 317±325
www.elsevier.com/locate/isatrans

PID gain scheduling using fuzzy logic
T.P. Blanchett a, G.C. Kember a,*, R. Dubay b
a
Department of Engineering Mathematics, DalTech, Dalhousie University, PO Box 1000, Halifax, NS, Canada B3J 2X4
b
Department of Mechanical Engineering, University of New Brunswick, Federicton, NB, Canada E3B 5A3

Abstract
A simple, yet robust and stable alternative to proportional, integral, derivative (PID) gain scheduling is developed
using fuzzy logic. This fuzzy gain scheduling allows simple online duplication of PID control and the online improvement
of PID control performance. The method is demonstrated with a physical model where PID control performance is
improved to levels comparable to model predictive control. The fuzzy formulation is uniquely characterized by; (i) one
fuzzy input variable involving the PID manipulated variable, (ii) two parameters to be tuned, while previously tuned
PID parameters are retained, and (iii) a gain scheduling di€erential equation which relates the fuzzy and conventional
PID manipulated variables and enables fuzzy gain scheduling. # 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Gain scheduling; Fuzzy control; Model predictive control; PID control

1. Introduction desired and predicted responses. However, chan-
ging to MPC is not justi®ed for the majority of
Most industrial process control continues to rely industrial PID controllers since its control struc-
upon `classical', or `conventional' proportional, tures are very di€erent from PID, are much more
integral, derivative (PID) control. Gain scheduling complicated, and have an increased computational
is the most common PID advancement used in cost.
industry to overcome nonlinear process character- Fuzzy logic approaches have been shown in
istics through the tailoring of controller gains over numerous studies to be a simpler alternative to
local operating bands. This scheduling is compli- improve conventional PID control performance
cated by the need for detailed process knowledge (for example, [1±5] for a recent overview). The pro-
to de®ne operating bands and open loop tests which blem of interest here, is the control of a manipulated
must be performed to locally calibrate the controller variable to a constant set point. Performance
gain within each band. An alternative method is improvements for such a problem are usually
predictive control which uses a `black box' model to demonstrated by reductions in the amplitude of
remove the need for detailed knowledge of process undesirable oscillations in the manipulated vari-
characteristics. For example, in model predictive able around the set point, shorter times to converge
control (MPC), controller moves are determined by to the set point, and the maintenance of control
continuously minimizing the di€erence between the stability seen in conventional PID control. Since
substantial, but similar improvements are found
* Corresponding author. Tel.: +1-902-494-3262; fax: +1- from a wide variety of fuzzy logic schemes, the
902-494-1801. main feature which delineates these approaches is
E-mail address: guy.kember@dal.ca (G.C. Kember). their relative complexity. For those fuzzy logic
0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00024-0

318 T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325

controllers intended to replace existing conven- Note that a fuzzy logic scheme incorporating
tional PID controllers in the industrial setting, the these features is a true gain scheduler Ð a `fuzzy
drive to simplify fuzzy logic controllers is impor- gain scheduler'. Fuzzy gain scheduling is com-
tant to reduce the costs of their implementation pactly and generally formulated in terms of a `gain
[3]. Two features shared by most of these fuzzy scheduling' di€erential equation: the rate of
logic setups are: each error component (taken change of the fuzzy manipulated variable is equa-
from the proportional error and its derivatives) is ted to a function of the rate of change of the con-
de®ned as a separate input, and the fuzzy rule- ventional PID manipulated variable. The form of
bases are redundant, that is, the rulebases show a this function is globally determined by details of
linear dependence upon the error components. the fuzzy formulation and the defuzzi®cation
Such `fuzzy redundancy' together with appro- strategy. The existence of a limiting linear form is
priate input and output bounds has been shown to used to preserve conventional PID control and
lead to stable control in a large class of nonlinear allow the desired online replacement. Then, mod-
control problems [6]. However, a practical obser- i®cation of this linear form, to a nonlinear sig-
vation [6,7] is that fuzzy input variables taken moidal form, yields fuzzy gain scheduling. The use
from linear combinations of the error components of a di€erential equation also makes this approach
(termed here `summed fuzzy input variables') should equally convenient for continuous and discrete
be used to reduce the number of input variables control situations.
where separated inputs would lead to a more The layout of the paper is as follows. The con-
redundant rulebase. Such designs are simpler and trol of a temperature process by conventional PID
thus provide more ecient control than the more is used for illustration (Section 2). The fuzzy gain
redundant fuzzy formulations, yet do not sacri®ce scheduling method and approach to independent
stability [6]. In addition, control robustness with tuning of parameters is developed (Section 3) and
respect to parameter ¯uctuations, seen in most demonstrated with a physical model (Section 4). A
fuzzy designs is related to widespread use of error well-tuned PID controller is substantially improved
components involving the proportional error and to performance levels of the benchmark MPC
its derivatives [6], i.e. there is no integral term of after tuning the fuzzy gain scheduling method with
the error, and such control has been coined `slid- a few tests (Section 5). Excellent control robust-
ing mode control' in [6]. ness and stability to large disturbances and large
Therefore, the aim of this study is to provide a set point modi®cations is also demonstrated.
new fuzzy formulation which provides a signi®cant
simpli®cation over existing fuzzy-PID schemes
intended to improve conventional PID controllers. 2. PID control
The larger simplicity of the method stems from
three features: The control of a temperature process to a set
point temperature is used to illustrate the fuzzy
1. Fuzzy redundancy is eliminated by using gain scheduling developed here. For the control of
only one fuzzy input variable proportional to a temperature process by varying heater power,
the derivative of the conventional PID the heater power is determined in conventional
manipulated variable. PID control by manipulating
2. Online replacement and subsequent improve- … !
ment of PID control is simpli®ed through the 1 t d
À…t† ˆ Kp e…t† ‡ e…u†du ‡ Td e…t† Y …I†
introduction of a di€erential equation relat- Ti 0 dt
ing the fuzzy input and output variables.
3. Online control improvement is achieved by the where the error at time t is e ˆ Ts À T; T is the
independent tuning of only two parameters, process temperature, and Ts is the process set
while the previously tuned conventional PID point temperature. The three PID control para-
parameters Ti and Td are retained. meters are: the proportional gain Kp , the integral

T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 319

time constant Ti , and the derivative time constant with initial condition g…0† ˆ 2…0†. Hence, gain
Td . The heater power, P is equal to À, but P is set scheduling of the input, d2ad(, is modelled in (3)
to 0 or the maximum heater power Pm—x , when À is as a nonlinear dependence of the output dgad(
respectively less than 0, or is greater than Pm—x . upon d2ad(. The positive scaling constants and
The temperature T is conveniently rescaled with are necessary to scale the fuzzy input and output
respect to the set point temperature and the ambient respectively (this is further detailed in Section 3.3),
temperature TI , using 0 ˆ …T À TI †a …Ts À TI †, ”
and the dimensionless heater power, P, equals g
so that TI 4T4Ts corresponds to 04041. The truncated to the range (0,1), i.e. P” ˆ 1 when g b 1,
time is also rescaled, using a timescale ts , as ( ˆ tats . ”
and P ˆ 0 when g ` 0. Conventional PID control
With these de®nitions, the dimensionless error is is generally recovered (`fuzzy logic equivalent')
E ˆ 1 À 0, and if 2 ˆ ÀaPm—x , then the dimen- when g ˆ 2; if f… d2ad( † ˆ d2ad( and ˆ ,
sionless form of the manipulated variable (1) is then integration and application of the initial
… ! condition, g…0† ˆ 2…0†, yields g ˆ 2. Note that,
Ã 1 ( Ã d although fuzzy gain scheduling could also be
2…( † ˆ Kp E…( † ‡ Ã E…u†du ‡ Td E…( † X …P†
Ti 0 d( based upon 2 instead of d2ad(, and this may seem
attractive for 2 perturbed by noise, the tradeo€ is
”
Now, P, the dimensionless heater power, is equal that it introduces an increased sensitivity to para-
”
to 2 truncated to the range [0,1], i.e. P ˆ 1 when metric ¯uctuations. Hence, the approach taken
2 b 1, and P ” ˆ 0 when 2 ` 0. The dimensionless here is to utilize the robustness associated with
PID control parameters are KÃ ˆ Kp …Ts À TI †a
p sliding mode control [6], and to supplement this
Pm—x Y TÃ ˆ Ti ats , and TÃ ˆ Td ats .
i d with explicit signal processing for noise suppres-
sion (Section 5).
A discrete equivalence to PID is also important
3. Fuzzy gain scheduling for discrete control applications, such as pulse
width modulation (Section 4). Assume that the
Fuzzy gain scheduling is in three steps: a fuzzy process is sampled at intervals of ts seconds so
logic system is built that incorporates the features that the dimensionless sampling interval is unity.
listed in the Introduction while preserving con- If d2ad( in (3) is approximated, at ( ˆ n, as
ventional PID control (Section 3.1), gain schedul- 2n À 2nÀ1 [note, any di€erencing scheme produces
ing is then implemented by modifying this system a fuzzy logic equivalent if it is applied to both
(Section 3.2), and two parameters are indepen- sides of (3)], and g at ( ˆ n is gn , then
dently tuned (Section 3.3) to improve PID control
performance. 1
gn ˆ gnÀ1 ‡ f…‰2n À 2nÀ1 Š†X …R†

3.1. Fuzzy logic system

The fuzzy input variable is taken to be equal to ” ” ”
At ( ˆ n, the power P is Pn , and Pn is equal to
the rate of change of the PID manipulated vari- gn truncated to the range (0,1). The fuzzy logic
able 2 in (2). Gain scheduling of d2ad( ensures equivalent now follows the continuous case.
that control is less susceptible to parameter ¯uc- The parameter [(3) and (4)], is necessary to
tuations [6] since control near the set point always scale the input ˆ d2ad( to ˆ O…1† (`O'
corresponds to d2ad( ˆ 0 (sliding mode control means `the order of'). Therefore, the function f,
[6]). Gain scheduling of d2ad( is formulated using that the fuzzy logic system must reproduce to
a di€erential equation where the rate of change of obtain a fuzzy logic equivalent is; f… † ˆ Y 41,
the fuzzy output (manipulated) variable satis®es and it is further assumed that f ˆ 1, 51,
and f ˆ À1Y 4 À 1. Given that f ˆ over 41,
dg 1 d2 it is also clear that the fuzzy logic equivalence
ˆ f Y …Q†
d( d( relating to (3) and (4), requires such


that d2ad( 41. Note that, in practice 2 has increase f, which are respectively denoted as Yd ,
superimposed noisy perturbations and conditions Yn , and Yi and these sets are all unity respectively
Â Ã Â Ã Â Ã
do change between control runs. Hence, a peak
in f P À 3 Y À 1 Y f P À 1 Y 1 , and f P 1 Y 3 , and are
2 2 2 2 2 2
value of d2ad( is normally estimated from pre- 0 otherwise. The consequence of each rule is
vious control runs, and this value is used to deter- represented as a fuzzy set following [7]. For
mine . A fuzzy logic system that reproduces this example, the consequent for Rule 1, labelled as
f… † is developed now. To implement fuzzy gain the set Y1 , is equated to the fuzzy set Yd , where
scheduling it is necessary to at least resolve the the maximum value of Y1 is Xn … †. Following the
scalar inputs into three domains: negative, near same procedure for the three rules gives the
zero and positive (this point is further examined in Â Ã
three consequence sets: (i) Y1 ˆ Xn … †Y f P À 3 Y À 1 ,
Section 3.2 where the generalization to more than Â Ã 2 2
(ii) Y2 ˆ Xz … †Y f P À 1 Y 1 , and (iii) Y3 ˆ Xp … †Y
three domains is also outlined). Therefore, three Â1 3Ã 2 2
rules, relating the scalar input , and the scalar f P 2 Y 2 . It remains to evaluate the scalar output
output f, are introduced: Rule 1; IF is negative f. Adopting an additive centroidal defuzzi®cation
THEN decrease f, Rule 2; IF is zero THEN do strategy [8]
nothing to f, Rule 3; IF is positive THEN
increase f. The three input fuzzy sets are negative, €
3
Aj … †cj
zero, and positive, and the three output fuzzy sets jˆ1
are, decrease f, do nothing to f, and increase f. It is f… † ˆ X …S†
€
3
necessary to convert these three rules into a com- A j … †
jˆ1
putational framework. This requires a means; (i)
to compute the degree of membership of the scalar
input in the input fuzzy sets, or the IF portion of Aj Y j ˆ 1Y 2Y 3, are the areas respectively corre-
each rule, (ii) to evaluate the consequence of sponding to the consequent fuzzy sets Yj Y
membership in each set, or the THEN portion of j ˆ 1Y 2Y 3; A1 ˆ Xn … †, A2 ˆ Xz … †, A3 ˆ Xp … †.
each rule, and (iii) to estimate the scalar output f The values c1 ˆ À1, c2 ˆ 0, c3 ˆ 1 are the respec-
from the three consequences of membership eval- tive centroids of these consequent sets. The sum of
uated in (ii). the areas in the denominator of (5) is always unity
The input fuzzy sets, negative, zero, and posi- since the components x sum to unity. The

tive, are respectively denoted as Xn Y Xz Y Xp . The numerator evaluates to for 41, and is 1 for
typical linear sets are used, i.e. Xn ˆ ÀY and À 1 for 4 À 1. Hence f… † ˆ for
51
À1440 …Xn 1Y 4 À 1, and 0 otherwise), 41, while f… † ˆ 1Y 51, and f… † ˆ À1Y
ˆ
Xz ˆ 1 À Y 41 (0 otherwise), Xp ˆ Y 04
À 4 À 1, that is, f preserves conventional PID.
41 Xp ˆ 151, and 0 otherwise). The degree of
membership of the scalar input in the input 3.2. Gain scheduling
fuzzy sets, is evaluated as the three scalars:
Xn … †Y Xz … †, and Xp … †, and these are stored in
Â Ã Global PID control performance can be
the vector x ˆ Xn … †Y Xz … †Y Xp … † . When 41, improved by scheduling the gain KÃ as a function
p
the control is not truncated, and x ˆ ‰ÀY 1‡ of the derivative of the PID manipulated variable
Y 0ŠY À1440, and x ˆ ‰0Y 1 À Y ŠY 0441. d2ad(. More speci®cally, the sensitivity to small
Similarly, when 51, the control is truncated, and deviations from the set point is increased, and
x ˆ ‰1Y 0Y 0ŠY 4 À 1, and x ˆ ‰0Y 0Y 1ŠY 51. The the reverse is applied to larger deviations, i.e.
locations where the control is truncated are chosen dfad is increased near ˆ 0, and decreased near

without loss of generality as ˆ 1 and ˆ À1, ˆ 1, so that f is sigmoidal. This is achieved
since the inputs are scaled to ensure is order 1. here by applying variable weights to the con-
To evaluate the consequence of membership, it sequent fuzzy sets ([8] does so for an unrelated
is necessary ®rst to de®ne the output fuzzy sets, i.e. problem) so that the defuzzi®cation strategy in (5)
the three fuzzy sets decrease f, do nothing to f, becomes


€
3 right-hand side of (8) is a monotone function of
wj Aj … †cj the derivative of the PID manipulated variable,
jˆ1
f… † ˆ X …T† absolutely bounded by 1a (this is analogous to
€
3
wj Aj … † the statement regarding stability made in [6] noted
jˆ1 in the Introduction). Note, for control problems
where the gain scheduling requires ®ner control of
The weights are positive, and conventional PID the sigmoidal shape of f… †, the number of sets
is recovered in the same fashion as in Section 3.1 may be increased and this simply adds extra
with the additional requirement w1 ˆ w2 ˆ w3 . weights to the defuzzi®cation strategy.
Since the form of (6) is unchanged when the
weights are multiplied by a constant, the weight w2 3.3. Parameter tuning
is set to unity without loss of generality, and
attention is further restricted to symmetric weights There are three parameters in the fuzzy gain
w1 ˆ w3 w. Applying both of these to (6) gives
scheduling in (8): , , and w. The parameter is

for 41 used to scale d2ad( to order 1, so that control

far from the set point is d2ad( ˆ O…1†, and

w near to the set point is d2ad( ( 1. More pre-
f… † ˆ Y …U†
…w À 1† ‡ 1 cisely, the necessity of preserving conventional
PID control is used to ®x ; if is such that

where f ˆ 1 for b 1, and f ˆ À1 for ` À1. The d2ad( 41, where d2ad( is taken from the
end point values of f…Æ1† ˆ Æ1, and f…0† ˆ 0 are existing PID manipulated variable, then the fuzzy
independent of w, in contrast to the slopes
logic equivalent follows from ˆ , and w ˆ 1.
dfad ˆ0 ˆ w, and dfad ˆÆ1 ˆ 1awY f… † is sig- Therefore, it is only necessary to tune two para-
moidal when w b 1 and this is the desired gain meters, and w to globally improve the existing
scheduling described above. The derivative of f… † PID control performance. A key observation
is also continuous at ˆ 0 so that special treat- leading to independent tuning of and w is that
ment of control near, and across ˆ 0 [6] is avoi- for improvement of well-tuned PID, ˆ O…†.
ded and this justi®es the restriction to symmetric Then dgad( ˆ O…wd2ad( † near the set point and
weights. Furthermore, the parameter 3 de®nes the control sensitivity near there is O…w†. Therefore,
extent to which inputs near 0 in¯uence the output control sensitivity near the set point is increased by
relative to those further away from 0 and thus it is setting equal to and independently tuning w b
necessary to de®ne at least three sets (as in Section 1 to reduce maximum set point overshoot. Next,
3.1) since inputs can at least be, near zero, large is independently modi®ed to b to reduce
and positive, or large and negative. control sensitivity far from the set point and fur-
Substituting f… † (3) gives explicitly for
ther reduce maximum set point overshoot. Whilst,
d2ad( 41 is varied, the sensitivity near the set point is
maintained at the previously tuned w, by modifying
dg 1 w d2ad( w such that wa is unchanged. A physical model
ˆ Y …V† (Section 4) is now used to demonstrate (Section 5)
d( …w À 1†d2ad( ‡ 1
improvement of well-tuned PID control.

where dgad( ˆ 1a for d2ad( b 1, and dgad( ˆ
À1a for d2ad( ` À1. To recover conventional 4. Physical model
PID control; 3 ˆ 1, ˆ , and is chosen such

that d2ad( 41, whereupon (8) reduces to The control of a temperature process, depicted
dgad( ˆ d2ad(, and then integration and appli- in the schematic in Fig. 1, is conducted on a solid
cation of g…0† ˆ 2…0† yields the desired g ˆ 2. cylindrical block of aluminum, 5 cm diameter and
Control based upon (8) is stable, since the 12.5 cm in length. The block is externally heated


Fig. 1. Experimental setup for control of temperature process by pulse width modulation.

by a 300 watt electrical heater band wrapped power setting, a continuous variable, it is easily
around the block circumference. A type E, modi®ed to the discrete pulse width modulation.
ungrounded thermocouple, measures the object's To avoid confusion, the nomenclature in Sections
temperature at its center, and these analog mea- 2 and 3.1 is adopted. During the nth duty cycle,
surements are converted to digital readings using a the on time of the heater, or pulse width Pn s, is
12 bit analog-to-digital converter. Process control determined by the control algorithm, while the
is over contiguous duty cycles of constant dura- heater power setting is held ®xed between duty
tion. The heater is on for a portion of a duty cycle, cycles. The maximum pulse width, Pm—x s, is equal
starting at the beginning, and then o€ for the to the duty cycle duration. The average error
remainder; the heater on time during a duty cycle within a duty cycle is en ˆ Ts À Tn where Tn is the
is termed the pulse width. A pulse is implemented average temperature over a duty cycle. The time
using a 16 bit digital timing board, and an opti- scale, for the dimensionless form, is taken to be
cally isolated solid state SSR-20 electronic relay. the duty cycle duration (the sampling interval of
Two logic states, on and o€, corresponding to the the average temperature), and the nth duty cycle is
heater being on or o€, are generated by the digital then over n À 14(4n. The average dimensionless
counter and are inputted to the relay. The process error within the nth duty cycle, is En ˆ 1 À 0n ,
control algorithm determines the duration of the ”
where 0n ˆ …Tn À TI †a…Ts À TI †. Finally, Pn Y 2n ,
on logic state for each duty cycle, or modulates the and gn , follow the description in Section 3.1.
pulse width between duty cycles Ð hence pulse The general approach followed here to ®lter
width modulation. The duty cycle duration is noisy ¯uctuations from the error components (i.e.
empirically set at 4.25 s, and at steady state this the error variable and its derivatives), does not
corresponds to a maximum error, over a duty rely upon features of the control setup or choice of
cycle, of less than 1% (the thermocouple accuracy sampling period (prone to aliasing errors). Rather,
is about 1%). the error variable is ®rst sampled at a high enough
rate to establish all features relevant for the con-
trol application. Then, each independent error
5. Results component is separately processed for noise sup-
pression. For the experiments conducted here, the
Although the process control described in Sec- average temperature Tn during a duty cycle is the
tion 2 is based on the determination of heater average of 10, equally spaced temperature mea-


surements. A least squares regression is also used and the fuzzy logic equivalent corresponds to ˆ
to reduce noise in the error and the numerical ˆ 38 and w ˆ 1. Fuzzy gain scheduling is used
approximation of its derivatives. Speci®cally, the to improve upon the existing PID control perfor-
error value En (essentially Tn ) and its ®rst deriva- mance by tuning the parameters and w away
tive are calculated from a line, and the second from ˆ and w ˆ 1. Firstly, maximum set point
derivative from a quadratic polynomial, all least overshoot is reduced by increasing control sensi-
squares regressed on measurements taken from the tivity near the set point. This is achieved by inde-
most recent 16 duty cycles. The choice of 16 duty pendently tuning 3 (Section 3.3) with four
cycles (about 1 min) is arbitrary, but is chosen to experiments w ˆ 2Y 4Y 6Y 8, where ˆ ˆ 38. The
be much smaller than the process time constant value w ˆ 6 is chosen (see Fig. 2; w ˆ 2Y 4Y 8 are
(about 500 duty cycles or 30 min). The least not shown) since it gives about a ®vefold reduc-
squares regression is eciently implemented as a tion in maximum overshoot and w ˆ 8 provides
convolution using the Savitzky±Golay formula- marginal additional improvement. The maximum
tion [9]. The set point temperature is chosen as overshoot is somewhat reduced again, by decreas-
Ts=100 C and the ambient temperature is ing the control response, dgad(, away from the set
approximately 25 C. The dimensionless PID con- point and this corresponds to b . The pre-
trol parameters KpÃ ˆ 1Y TiÃ ˆ 37X5, and TdÃ ˆ 9X4, viously tuned control sensitivity near the set point
and the temperature response 0 is shown in Fig. 2 is maintained at w ˆ 6 by varying w such that wa
[well-tuned PID control ( ˆ ˆ 38, w ˆ 1) in the is constant, while is increased by 20, 30, and
®gure] as a function of the dimensionless time (. 40%. The overshoot for each of these values is
This control (tuned for minimum overshoot) shows about 3, 2, and 10%, respectively. Hence, as is
approximately one quarter amplitude damping increased the maximum overshoot is ®rst reduced
with settling time approximately the process time by the initial reduction in control sensitivity far
constant and is typical of well-tuned PID. This from the set point, but then increases as the control
temperature response can be greatly improved by sensitivity becomes too reduced. A 30% increase over
fuzzy gain scheduling. is chosen to scale
ˆ 38 is chosen and the ®nal results are shown in
d2ad( to order 1. From the existing PID Fig. 2 for ˆ 50 which also corresponds to w ˆ 7X8
manipulated variable (not shown), if % 38, then
(a 30% increase over 3 ˆ 6). The maximum over-
d2ad( 41 for the duration of the PID control, shoot has now been reduced about sixfold to 2.5%
maximum overshoot. Five indices are also used to
assess the overall control performance: the max-
imum overshoot and undershoot of the tempera-
ture expressed as a percentage of the set point, the
rise time, which is the time needed to rise to within
90% of the set point, the settling time, or time the
process requires to fall within Æ2.5% of the set
point, and the steady state error. These ®ve indices
are presented for the conventional PID control
…w ˆ 1Y ˆ ˆ 38† fuzzy gain scheduled control
…w ˆ 6Y ˆ 38Y —nd Y w ˆ 7X8Y ˆ 50†, and MPC
control in Table 1. From Table 1 and Fig. 2, it is
clear that fuzzy gain scheduling ( ˆ 38Y w ˆ 7X8,
and ˆ 50) provides much better control perfor-
mance than the well-tuned PID control …w ˆ 1Y
ˆ ˆ 38†. In particular, the settling time is
Fig. 2. Temperature response for conventional PID control,
reduced to about one half the process time con-
fuzzy gain scheduled PID, and model predictive control stant, and the percentage maximum overshoot is
(MPC). reduced from 15 to 2.5%.


The benchmark experiment is based on MPC. In in 50 cm3 of water at 10 C for 5 s), and large
the MPC approach used here (details are in [10]) a (40 C) changes in the set point (Fig. 4), is clearly
discrete step response of the physical model is demonstrated for the tuned fuzzy gain scheduling
obtained by an open loop test. The method utilizes … ˆ 38Y w ˆ 7X8Y ˆ 50† where the manipulated
two horizons: a `control' horizon equal to the
number of predicted control moves, and a `predic-
tion' horizon equal to the number of sampling
intervals to reach 95% of the open loop steady
state. Predictions of the physical model output are
made within the prediction horizon, and these are
compared to the desired set point pro®le. Least
squares minimization of the di€erence between the
predictions and the set point pro®le, over the pre-
diction horizon, is used to determine the manipu-
lated variable within the control horizon. A
control horizon of length 2 and a prediction horizon
of length 139 was used here for controlling the
temperature. Although MPC control is funda-
mentally di€erent from conventional PID, it pro-
duces control actions similar to PID control, but
shows a very reduced overshoot and settling time to
the set point due to its predictive capability. Thus, Fig. 3. Temperature response of fuzzy gain scheduled PID to a
MPC is practically useful to provide a range of disturbance (applied at ( % 280), and the same for model pre-
comparison to fuzzy gain scheduling. A surprising dictive control (MPC) (applied at ( % 350). The fuzzy gain
scheduled response is almost Identical to that seen for MPC.
result, evident in Fig. 2, is that the fuzzy gain sche-
duling fares very well in comparison to the more
sophisticated benchmark MPC control. Better per-
formance indices were also obtained for gain
scheduling, w ˆ 7X8Y ˆ 50, over MPC control
when only the rise time and settling time are con-
sidered, and marginally worse results for percen-
tage maximum overshoot and undershoot.
Control robustness to a short, cooling disturbance
(Fig. 3) (the cylindrical block was suddenly placed

Table 1
Control performance indices

Performance indices w ˆ 1, w ˆ 6, w ˆ 7X8, MPC
ˆ 38, ˆ 38 ˆ 38
ˆ 38 ˆ 38 ˆ 38

Percentage 15 3.2 2.5 0
maximum overshoot
Percentage 4 1 1 0
maximum undershoot
Rise time (min) 7.7 7.3 6.9 14
Fig. 4. Temperature response (a) and manipulated variable (b)
Setting time (min) 30 15 14 19
for fuzzy gain scheduled PID. Control is depicted for set points
Percentage Æ0.5 Æ0.5 Æ0.5 Æ0.5
40% larger …Ts ˆ 140 g† and smaller …Ts ˆ 60 g† than that
steady state error
used for the parameter tuning …Ts ˆ 100 g†.


variable is shown together with the temperature operating grants held by G.C.K. and R.D., and an
response. NSERC postgraduate scholarship held by T.P.B.
The authors would like to thank Dr. Gordon
Fenton, Dr. Adam Bell and thoughtful reviewers
6. Summary and conclusion for comments.

A fuzzy gain scheduling scheme that allows for
the online replacement and subsequent improve- References
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PID gain scheduling using fuzzy logic

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