Fuzzy model reference learning control (1)

Fuzzy Model Reference Learning Control∗
Jeffery R. Layne† and Kevin M. Passino‡
Department of Electrical Engineering
The Ohio State University
2015 Neil Avenue
Columbus, OH 43210
Abstract
A “learning system” possesses the capability to improve its performance over time by interaction with its environment.
A learning control system is designed so that its “learning controller” has the ability to improve the performance of the
closed-loop system by generating command inputs to the plant and utilizing feedback information from the plant. In this
brief paper, we introduce a learning controller that is developed by synthesizing several basic ideas from fuzzy set and control
theory, self-organizing control, and conventional adaptive control. We utilize a learning mechanism which observes the plant
outputs and adjusts the membership functions of the rules in a direct fuzzy controller so that the overall system behaves
like a “reference model”. The effectiveness of this “fuzzy model reference learning controller” (FMRLC) is illustrated by
showing that it can achieve high performance learning control for a nonlinear time-varying rocket velocity control problem
and a multi-input multi-output (MIMO) two degree-of-freedom robot manipulator.
I Introduction
Over recent years, fuzzy control has emerged as a practical alternative to classical control schemes when
one is interested in controlling certain time-varying, non-linear, and ill-defined processes. There have in
fact been several successful commercial and industrial applications of fuzzy control [1] -[ 5]. Despite this
success, there exist several significant drawbacks of this approach:
1. The design of fuzzy controllers is usually performed in an ad hoc manner; hence, it is often not clear
exactly how to justify the choices for many parameters in the fuzzy controller (e.g., the membership
functions, defuzzification strategy, and fuzzy inference strategy).
2. The fuzzy controller constructed for the nominal plant may later perform inadequately if significant
and unpredictable plant parameter variations, structural changes, or environmental disturbances
occur.
In this paper a “learning” control algorithm is presented which helps to resolve some of these fuzzy con-troller
design issues. This algorithm employs a reference model (a model of how you would like the plant
to behave) to provide closed-loop performance feedback for synthesizing and tuning a fuzzy controller’s
∗
Bibliographic information for this paper: Layne J.R., Passino K.M., “Fuzzy Model Reference Learning
Control,” Journal of Intelligent andF uzzy Systems, Vol. 4, No. 1, pp. 33–47, 1996.
†
J. Layne gratefully acknowledges the support of the U.S. Air Force Palace Knight Program. J. Layne has recently joined
Wright Laboratories, WL/AAAS-3, Wright Patterson AFB, Ohio 45433-6543
‡
K. Passino was supported in part by an Engineering Foundation Research Initiation Grant and in part by National Science
Foundation Grant IRI-9210332. Please address all correspondence to K. Passino (614-292-5716, email: passino@ee.eng.ohio-state.
edu).
1

knowledge-base. Consequently, this algorithm is referred to as a “fuzzy model reference learning con-troller”
(FMRLC). The FMRLC grew from research on how to improve Procyk and Mamdani’s linguistic
self-organizing controller (SOC) [6] by utilizing certain general ideas in conventional adaptive control [7, 8].
The first advantage that the FMRLC has over the SOC is that it does not rely on the specification of an ex-plicit
inverse model of the process (which can be difficult/impossible to determine for many applications).
In addition, the performance criteria for the linguistic SOC can only characterize what is essentially a
compromise between rise-time and overshoot (and not the relative importance of each) and hence it pro-vides
little flexibility in specifying what performance is to be achieved/maintained (this is the case even
if the “optimized” fuzzy performance evaluator introduced in [9, 10, 11] is used). Via the use of a ref-erence
model, in the FMRLC framework we incorporate a capability for accurately quantifying virtually
any form of desired performance. Next, note that the knowledge-base modification algorithm of Procyk
and Mamdani [6] relies on modification of a fuzzy relation table which describes the relationship between
the fuzzy controller inputs and outputs. Often, this automatically implies that all input and output uni-verses
of discourse must be quantized into discrete levels to implement the fuzzy relation in a computer.
Unfortunately, this will generally result in large memory requirements and computational demands since a
fuzzy relation table often contains many entries for real world applications (some progress has been made
at addressing the computational complexity of knowledge-base modification for the SOC in [9]). In this
article, we use a knowledge-base modification algorithm (similar to the one in [11]) which reduces com-putation
time and memory requirements by utilizing a rule base array table rather than a fuzzy relation
table. The knowledge-base modification approach is flexible enough to be used in both the conventional
SOC approach and the FMRLC (this is shown in [12]). Finally, we note that the linguistic SOC has been
used in robotics applications [9, 10], motor and temperature control [13], blood pressure control [14], and
in satellite control [15, 16, 17]. While in this paper we describe the application of the FMRLC to robotics
and a rocket velocity control problem (where there is a significant underlying process variation resulting
from fuel consumption as the rocket launches), the FMRLC has also recently been used for (i) control of
a cart-pendulum system where certain improvements over SOC were illustrated [12], (ii) anti-skid brake
system control to enhance performance when there are significant variations in the road conditions [18, 19],
(iii) cargo ship steering where in [20, 21] it is shown to have certain advantages over conventional model
reference adaptive control, (iv) vibration damping in a two-link flexible robot where in [22] the authors
develop a fuzzy controller and show how its performance can be enhanced if it is tuned with the FMRLC
(experimental results are also provided for both the direct fuzzy controller and the FMRLC to illustrate its
ability to compensate for the effects of a payload variation), and (v) for aircraft control law reconfiguration
in case of failures [23].
Using conventional adaptive control terminology, the FMRLC and SOC are “direct” adaptive control
schemes since they directly update the parameters of the controller without explicit identification of the
plant parameters. Other relevant literature that focuses on direct adaptive fuzzy control includes the work
in [24] where an adaptive fuzzy system is developed for a continuous casting plant, and the approach in
[25] where a fuzzy system adapts itself to driver characteristics for an automotive speed control device.
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The use of fuzzy systems for estimation/identification [26, 27, 28, 29, 30, 31, 32] is relevant, especially if
“indirect adaptive” [7, 8] fuzzy control techniques (i.e., ones where plant parameters are identified and
used to tune the parameters of the controller) such as those in [33, 34, 35] are used. Also, it is interesting
to note that in [30, 31, 34] there are inherent uses of inverse dynamics of the plant; however, our use of
the fuzzy inverse model is significantly different. Finally, the authors note that since the initial results
in FMRLC have been introduced in [21] some other relevant new adaptive/learning techniques have been
developed [36, 37, 38, 39] where neural approaches are used to tune fuzzy systems and one fuzzy adaptive
control scheme is shown to be stable.
In Section II, the detailed description of the FMRLC algorithm is presented. Then in Section III we
study the performance of the FMRLC for single stage rocket velocity control where there is a significant
variation in the process dynamics due to the change in mass of the rocket as fuel is expended. Moreover,
the FMRLC will be used as a learning controller for a two degree-of-freedom robot manipulator to illustrate
the application of FMRLC for a multi-input, multi-output (MIMO) process. Finally, in the concluding
remarks in Section IV we will discuss the advantages and disadvantages of FMRLC and highlight some
important future research directions.
II Fuzzy Model Reference Learning Control
In this Section, we present a novel learning control technique that was developed by extending some of
the linguistic self-organizing control concepts presented by Procyk and Mamdani in [6] and by utilizing ideas
from conventional “model reference adaptive control” (MRAC) [8, 7]. The learning control technique, which
is shown in Figure 1, utilizes a learning mechanism that: (i) observes data from a fuzzy control system,
(ii) characterizes its current performance, and (iii) automatically synthesizes and/or adjusts the fuzzy
controller so that some pre-specified performance objectives are met. These performance objectives are
characterized via the reference model shown in Figure 1. In an analogous manner to conventional MRAC
[8, 7] where conventional (often linear) controllers are adjusted, the learning mechanism seeks to adjust the
fuzzy controller (a nonlinear controller) so that the closed-loop system (the map from yr(kT) to y(kT)) acts
like a pre-specified reference model (the map from yr(kT) to ym(kT)). We have named the new learning
control technique “fuzzy model reference learning control” (FMRLC) due to its similarities to MRAC,
its unique approach to remembering the adjustments it makes, and according to the prevailing definition
of “learning” and “adaptive” [40] (we avoid the term “self-organizing” due to the differences between
our approach and the work in [6]). For a much more detailed discussion of the key issues encountered
when studying the comparison between adaptive and learning systems see [40, 21]. Next we describe each
component of the FMRLC in more detail.
A. The Fuzzy Controller
The process in Figure 1 is assumed to have r inputs denoted by the r -di mensional vector u(kT) =
[u1(kT) ... ur(kT)]t (T is the sample period) and s outputs denoted by the s -di mensional vector y(kT) =
3

Plant
y(kT)
y (kT)
Knowledge-base
modifier
1-z -1
T
1-z-1
T
Knowledge-base
Fuzzy sets Rule-base
Knowledge-base
g
e
g
c
Inference
mechanism
Σ
Σ
gu
Fuzzy controller
e(kT)
g
ye
u(kT)
r (kT)
ye(kT)
+
yc(kT)
+
g
yc
g
p
Inference
mechanism
Reference
model
p(kT)
Storage
Fuzzy inverse model
m
Learning mechanism
Figure 1: Functional Architecture for the FMRLC.
[y1(kT) ... ys(kT)]t. Most often the inputs to the fuzzy controller are generated via some linear function
of the plant output y(kT) and reference input yr(kT). Figure 1 shows a special case of such a linear
map that was found to be useful in many applications. The inputs to the fuzzy controller are the error
e(kT) = [e1(kT) ... es(kT)]t and change in error c(kT) = [c1(kT) ... cs(kT)]t defined as
e(kT) = yr(kT) − y(kT), (1)
c(kT) = e(kT) − e(kT − T)
T
, (2)
respectively, where yr(kT) = [yr1(kT) ... yrs (kT)]t denotes the desired process output.
Often, for greater flexibility in fuzzy controller implementation, the universes of discourse for each
process input are “normalized” to the interval [−1,+1] by means of constant scaling factors. For our fuzzy
controller design, the gains ge, gc, and gu were employed to normalize the universe of discourse for the
error e(kT), change in error c(kT), and controller output u(kT), respectively (e.g., g
e
= [ge1, ..., ges]t so
that geiei(kT) is a scaled input to the fuzzy controller).
For convenience we utilize r multi-input single output (MISO) fuzzy controllers (one for each process
input un) as it is equivalent to using one s input r output MIMO fuzzy controller. The knowledge-base
for the fuzzy controller associated with the nth process input is generated from IF-THEN control rules of
the form:
If ˜e1 is ˜E
j
1 and ... and ˜es is ˜ Ek
s and ˜c1 is ˜ Cl1
and ... and ˜cs is ˜ Cm
s Then ˜un is ˜U
j,...,k,l,...,m
n ,
ba
ba
where ea ˜and ca ˜denote the linguistic variables associated with controller inputs ea and ca, respectively,
un ˜denotes the linguistic variable associated with the controller output un, E˜ and C˜ denote the bth
linguistic values associated with ˜ea and ˜ca, respectively, and ˜U
j,...,k,l,...,m
n denotes the consequent linguistic
4

value associated with ˜un for the rule listed above. The above control rule may be quantified by utilizing
fuzzy set theory to obtain a fuzzy implication of the form:
If Ej
1 and ... and Ek
s and Cl1
and ... and Cm
s Then Uj,...,k,l,...,m
n ,
where Eba
, Cba
n denote the fuzzy sets that quantify the linguistic statements “˜ea is ˜ Eba
, and Uj,...,k,l,...,m
”, “˜cs
is ˜ Cm
s ”, and “˜un is ˜U
j,...,k,l,...,m
n ”, respectively. This fuzzy implication can be represented by a fuzzy relation
Rj,...,k,l,...,m
n = (Ej
1
× ... × Ek
s ) × (Cl1
× ... × Cm
s ) × Uj,...,k,l,...,m
n . (3)
The fuzzy controller decision mechanism for this control rule may be expressed by
Û
j,...,k,l,...,m
n (kT) = ((Ê
1(kT) × Ê
2(kT) × ... × ˆ Es(kT)) ×
( ˆ C1(kT) × ˆ C2(kT) × ... × ˆ Cs(kT))) ◦ Rj,...,k,l,...,m
n (4)
where Ê
j(kT) and ˆ Cj (kT) denote the fuzzified error and change in error, respectively, associated with
the jth element of e(kT) or c(kT), Û
j,...,k,l,...,m
n (kT) denotes the implied fuzzy set, and “◦” denotes Zadeh’s
Composition. See [41] for a more detailed mathematical explanation of Equation 4. Typically in fuzzy
system design, a fuzzy implication exists for every possible combination of fuzzy sets describing the inputs
to the fuzzy system. Therefore, the fuzzy controller is made up of many fuzzy implications whose overall
control action may be computed by the “center of gravity” (COG) method expressed as
un(kT) =

j,...,k,l,...,m
Aˆj,...,k,l,...,m
n (kT) ˆcj,...,k,l,...,m
n (kT)

j,...,k,l,...,m
Aˆj,...,k,l,...,m
n (kT)
, (5)
where Aˆj,...,k,l,...,m
n (kT) and ˆcj,...,k,l,...,m
n (kT) are the area and center of area, respectively, of the membership
function associated with Û
j,...,k,l,...,m
n (kT).
B. The Reference Model
The reference model provides a capability for quantifying the desired performance of the process. In
general, the reference model may be any type of dynamical system (linear or non-linear, time-invariant or
time-varying, discrete or continuous time, etc.). The performance of the overall system is computed with
respect to the reference model by generating an error signal ye(kT) = [ye1 ... yes ]t where
ye(kT) = ym(kT) − y(kT). (6)
Given that the reference model characterizes design criteria such as stability, rise time, overshoot, settling
time, etc. and the input to the reference model is the reference input yr(kT), the desired performance of the
controlled process is met if the learning mechanism forces y
e
(kT) to remain very small for all time. Hence,
the error ye(kT) provides a characterization of the extent to which the desired performance is met at time
t = kT. If the performance is met (ye(kT) ≈ 0) then the learning mechanism will not make significant
modifications to the fuzzy controller. On the other hand if ye(kT) is big, the desired performance is not
achieved and the learning mechanism must adjust the fuzzy controller.
5

C. The LearningMe chanism
As previously mentioned, the learning mechanism performs the function of modifying the knowledge-base of
a direct fuzzy controller so that the closed-loop system behaves like the reference model. These knowledge-base
modifications are made based on observing data from the controlled process, the reference model, and
the fuzzy controller. The learning mechanism consists of two parts: a fuzzy inverse model and a knowledge-base
modifier. The fuzzy inverse model performs the function of mapping necessary changes in the process
output, as expressed by ye(kT), to the relative changes in to process inputs (denoted by p = [p1 ... pr]t)
necessary to achieve these process output changes. The knowledge-base modifier performs the function of
modifying the fuzzy controller’s knowledge-base to affect the needed changes in the process inputs. More
details of this process are discussed next.
The Fuzzy Inverse Model
The fuzzy inverse model was developed by investigating methods to alleviate the problems with using
the inverse process model in the linguistic SOC framework of Procyk and Mamdani [6]. Procyk and
Mamdani’s use of the inverse process model depended on the use of an explicit mathematical model of
the process and ultimately assumptions about the underlying physical process. This dependence on a
mathematical model of the process often causes significant difficulties in applying their approach (e.g.,
they are often forced to assume that the plant will act like a constant gain (matrix) and hope that the
adaptation mechanism can compensate for this inaccuracy).
Using the fact that most often a control engineer will know how to roughly characterize the inverse
model of the plant, we introduce the idea of using a fuzzy system to represent the inverse plant dynamics.
We emphasize that it is not necessary to accurately characterize the inverse dynamics; only an approximate
representation is needed. This “fuzzy inverse model” as it is shown in Figure 1, simply maps y
e
(kT), and
possibly other parameters such as the functions of ye(kT) and the process operating conditions, to the
necessary changes in the process inputs. Hence, the fuzzy inverse model is used to characterize how to
change the plant inputs to force the plant output y(kT) to be as close as possible to y
m
(kT) (i.e., to
make ye(kT) small). Again, we use r MISO fuzzy inverse models. While there exist numerous possible
combinations of inputs to the fuzzy inverse model, in Figure 1 only error ye(kT) and the change in error
yc(kT) are shown for the sake of brevity (e.g. delayed versions and functions of the variables could also be
used). In this paper, we will assume ye(kT) and yc(kT) are always employed as inputs to the fuzzy inverse
model. The reasons for using the change in the desired output change is to provide some “damping” in
the learning mechanism. In other words, since we have information about the rate of change of the desired
output changes, we may quantify that a small value of yei with a small change in yei is more desirable
than a small value of yei with a large change in yei due to the fact that overshoot is likely to occur. (Using
reasoning along similar lines the authors in [22] explain how to view the fuzzy inverse model as a controller
in the adaptation loop and show how this perspective can be used in FMRLC design.)
Note that similar to the fuzzy controller, the fuzzy inverse model shown in Figure 1 contains normalizing
scaling factors, namely gye
, gyc
, and gp, for each universe of discourse. Selection of the normalizing gains
6

can impact the overall performance of the system and a gain selection procedure is given below in Section D.
The knowledge-base for the fuzzy inverse model associated with the nth process input is generated from
e1 and ... and Y k
fuzzy implications of the form: If Y j
c1 and ... and Y m
es and Y l
n , where
cs Then Pj,...,k,l,...,m
Y b
ea and Y b
ca denote the bth fuzzy set associated with the error yea and change in error yca , respectively,
and associated with the ath process output and Pj,...,k,l,...,m
n denotes the consequent fuzzy set for this rule
describing the necessary change in the nth process input. This fuzzy implication can be represented by a
fuzzy relation Sj,...,k,l,...,m
n = (Y j
e1
×...×Y k
es )×(Y l
c1
×...×Y m
cs )×Pj,...,k,l,...,m
n . The fuzzy inverse model decision
n (kT) = ((ˆ Ye1 (kT) × ˆYe2 (kT) × ... ×
mechanism for this fuzzy implication may be expressed by ˆ Pj,...,k,l,...,m
ˆ Yes (kT))×( ˆ Yc1(kT)× ˆYc2 (kT)×...× ˆYcs(kT)))◦Sj,...,k,l,...,m
n where ˆ Yep (kT) and ˆ Ycp (kT) denote the fuzzified
error and change in error, respectively associated with the pth element of ye and yc, ˆ Pj,...,k,l,...,m
n (kT) denotes
the implied fuzzy set for this fuzzy implication describing input changes for the nth process input (actually
the nth direct fuzzy controller). As was the case for the direct fuzzy controller, the overall input changes
for the nth direct fuzzy controller pn(kT) are determined from the COG defuzzification method.
A typical rule base array which may be employed in a fuzzy inverse model for a SISO process is
e and
shown in Table 1 below (this and other fuzzy inverse models are used in the applications) where Y j
Y k
c denote the fuzzy sets associated with ye(kT) and yc(kT), respectively, and the Pj,k
i denote the fuzzy
sets quantifying the desired process input change pi(kT). Note that the body of Table 1 lists the center
values for convex, symmetric, and normal membership functions that are defined on universes of discourse
normalized to [−1, 1].
Table 1: Typical rule base array table for the fuzzy inverse model.
Y k
c
Pj,k
i
−5 −4 −3 −2 −1 +0 +1 +2 +3 +4 +5
−5 −1.0 −1.0 −1.0 −1.0 −1.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0
−4 −1.0 −1.0 −1.0 −1.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 +0.2
−3 −1.0 −1.0 −1.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 +0.2 +0.4
−2 −1.0 −1.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 +0.2 +0.4 +0.6
−1 −1.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8
Y j
e 0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0
+1 −0.8 −0.6 −0.4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0
+2 −0.6 −0.4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0
+3 −0.4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0 +1.0
+4 −0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0 +1.0 +1.0
+5 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0
The fuzzy inverse model rule base array shown in Table 1 was designed to take advantage of the “damp-ing”
feature described above. For example, consider the case where ye(kT) = 0 which is best characterized
by fuzzy set Y j
e where j = 0 since it characterizes the case where ye(kT) is small. The best change in
ye(kT), is yc(kT) = 0 which is characterized in a similar way by Y k
c where k = 0. This zero point (i.e., the
center of Table 1, j = k = 0) represents a point where the fuzzy inverse model indicates that no change in
the input is required to force y(kT) = ym(kT) since this equality is already achieved. If for some time k,
7

we have j = 0 but k = −2 (i.e., currently y(kT) is close to ym(kT) but y(kT) is increasing above ym(kT))
then Table 1 indicates that for Pj,k
i , the center of the fuzzy set is at −0.4 which characterizes the fact
that a negative increment should be added to the process input to ensure that y will not continue to in-crease
(i.e., so that we maintain a small yei ). Similar statements hold for the remaining elements in Table 1.
It is important to note that: (i) development of the fuzzy inverse model does not depend on the exis-tence
and specification of the mathematical model of the plant or its inverse (i.e., the plant inverse need not
exist), (ii) the fuzzy inverse model should not be confused with the mathematical model of the inverse of
the plant that is sometimes used in fixed (i.e., non-adaptive) control where the controller has no ability to
synthesize itself or auto-tune in response to plant parameter changes, and (iii) while the above discussion
provides some general guidelines for the construction of the fuzzy inverse model, and the applications in
Section III show how to construct it for a rocket velocity control problem with time-varying parameters
and a multi-input multi-output robot control problem, if the plant is very complex then it can sometimes
be difficult to specify the fuzzy inverse model. To gain further insight into how to specify the fuzzy inverse
model see [22] where a fuzzy inverse model is developed for a FMRLC which is implemented on a complex
flexible robotic mechanism (the perspective used there is that the fuzzy inverse model acts as a controller
in the adaptation loop). Moreover, see [20, 18] for the details on how to specify the fuzzy inverse model for
the FMRLC for a cargo ship steering application and a anti-skid braking problem. Overall, the FMRLC’s
performance depends on the engineer’s ability to specify a fuzzy inverse model. For the applications listed
above we have found that the fuzzy inverse model is relatively easy to specify and that it does not need
to be extremely accurate since the learning mechanism tends to compensate for the inaccuracies. The
general guidelines given above, coupled with the applications studied in this paper and in [22, 20, 18],
provide significant insights into choosing the fuzzy inverse model so that the FMRLC will be useful for
other applications. Finally, it is interesting to note that other inherent uses of inverse dynamics of the
plant in adaptive fuzzy control schemes can be found in [30, 31, 34].
The Knowledge-Base Modifier
The knowledge-base modifier presented next grew from work that focused on improving the knowledge-base
modification approach for the linguistic SOC [6]. In the linguistic SOC framework, knowledge-base
modification was performed on the overall fuzzy relation (Rn =

j,...,k,l,...,m
n ) used to implement
Rj,...,k,l,...,m
the fuzzy controller. However, this method of knowledge-base modification can be computationally complex
due to the fact that Rn is generally a very large array. Here we use a knowledge-base modification algorithm
(similar to the one in [11]) which increases computational efficiency by modifying the membership functions
of consequent fuzzy sets Uj,...,k,l,...,m
n rather than the fuzzy relation array Rn.
The knowledge-base modifier performs the function of modifying the fuzzy controller so that better
performance is achieved. Given the information about the necessary changes in the input as expressed by
the vector p(kT) from the fuzzy inverse model, the knowledge-base modifier changes the knowledge-base
of the fuzzy controller so that the previously applied control action will be modified by the amount p(kT).
Therefore, consider the previously computed control action, which contributed to the present good/bad
system performance. Note that e(kT −T) and c(kT −T) would have been the process error and change in
error, respectively, at that time. Likewise, u(kT − T) would have been the controller output at that time.
8

The controller output which would have been desired is expressed by
¯u(kT − T) = u(kT − T) + p(kT). (7)
Next we will show that by modifying the fuzzy controller’s knowledge-base we may force the fuzzy controller
to produce this desired output given similar controller inputs.
Assume that only symmetric membership functions are defined for the fuzzy controller’s output so that
cj,...,k,l,...,m
n denotes the center value of the membership function associated with the fuzzy set Uj,...,k,l,...,m
n .
Knowledge-base modification is performed by shifting centers of the membership functions of the fuzzy
sets Uj,...,k,l,...,m
n which are associated with the fuzzy implications that contributed to the previous control
action u(kT − T). This modification involves shifting these membership functions by an amount specified
by p(kT) = [p1(kT) ... pr(kT)]t so that
cj,...,k,l,...,m
n (kT) = cj,...,k,l,...,m
n (kT − T) + pn(kT). (8)
The degree of contribution for a particular fuzzy implication in the fuzzy controller whose fuzzy relation
is denoted Rj,...,k,l,...,m
n is determined by its “activation level”, defined
δj,...,k,l,...,m
n (t) =min{μEj
1
(e1(t)), ..., μEk
s
(es(t)),
μCl1
s (cs(t))}, (9)
(c1(t)), ..., μCm
where μA denotes the membership function of the fuzzy set A and t denotes the current time. Only
those fuzzy implications Rj,...,k,l,...,m
n (kT − T) whose activation level δj,...,k,l,...,m
n (kT − T) 0 are modified.
All others remain unchanged (this allows for local learning and hence memory for our learning controller
[40, 21]).
Consider the effect that the above knowledge-base modification has on the COG defuzzification (for
the direct fuzzy controller) expressed in Equation 5. Notice that since the area of the implied fuzzy sets is
proportional to the “activation level” of the fuzzy relation (i.e., Aj,...,k,l,...,m
n (kT −T) ∝ δj,...,k,l,...,m
n (kT −T))
only those fuzzy relations whose activation levels are greater than zero affect the center of gravity, or
control action. Furthermore, notice that since symmetric membership functions are utilized, a shift in the
membership function associated with fuzzy set Uj,...,k,l,...,m
n (kT) will also shift, by the same amount, the
centers of the membership functions associated with the previous implied fuzzy sets ˆU
j,...,k,l,...,m
n (kT − T).
Therefore, given the previous controller inputs e(kT − T) and c(kT − T) and the new fuzzy relation
Rj,...,k,l,...,m
n (kT) obtained after shifting the consequent fuzzy set, the new center value of the membership
function associated with the implied fuzzy set ˆU
j,...,k,l,...,m
n (kT − T) is expressed as
¯ˆcj,...,k,l,...,m
n (kT − T) = ˆcj,...,k,l,...,m
n (kT − T) + pn(kT). (10)
Substituting this new center value into Equation 5 we obtain
¯un(kT − T) =

n (kT − T) ¯ˆcj,...,k,l,...,m
j,...,k,l,...,mAj,...,k,l,...,m
n (kT − T)

j,...,k,l,...,mAj,...,k,l,...,m
n (kT − T)
(11)
where ¯un(kT − T) is the new control action that is obtained. Simplifying, it is easy to see that
¯un(kT − T) = un(kT − T) + pn(kT), (12)
9

which is the desired effect. Notice that this approach also achieves “generalization” as it is called in learning
theory [40] since it will at the same time learn how to deal with values that are near to those considered
(i.e., near to e(kT), c(kT), and y(kT)). Along these lines it is interesting to note that our knowledge-base
modification procedure implements a form of local adaptation and hence utilizes memory. Different
parts of the rule-base are “filled in” based on different operating conditions of the system, and when one
area of the rule-base is updated, the other rules are not affected. Hence, the controller adapts to new sit-uations
and also remembers how it has adapted to past situations (this is why the term “learning” is used).
As an example of the knowledge-base modification procedure, Table 2 shows a knowledge-base array
table where its entries represent the center values of symmetric membership functions associated with the
implied fuzzy sets Uj,k defined on a normalized universe of discourse (normalized to [−1, 1]). Given that
the fuzzy controller employs a knowledge-base array table similar to Table 2, the process of knowledge-base
modification reduces to a simple two step algorithm which is expressed below.
1. Determine which fuzzy implications in the knowledge-base array table contributed to the previously
applied input. In other words, determine the fuzzy implications whose premise element has an
activation level above zero (i.e., the rule is “on” -its implied fuzzy set is not null).
2. Modify the entries in the knowledge-base array for those fuzzy implications.
Table 2: Typical knowledge-base array table.
Ck
Uj,k −5 −4 −3 −2 −1 +0 +1 +2 +3 +4 +5
−5 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0
−4 +1.0 +1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2
−3 +1.0 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4
−2 +1.0 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6
−1 +1.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8
Ej 0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0
+1 +0.8 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0 −1.0
+2 +0.6 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0 −1.0 −1.0
+3 +0.4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0 −1.0 −1.0 −1.0
+4 +0.2 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0 −1.0 −1.0 −1.0 −1.0
+5 +0.0 −0.2 −0.4 −0.6 −0.8 −1.0 −1.0 −1.0 −1.0 −1.0 −1.0
For example, assume that the previous process error e(kT−T) took on a value such that the membership
functions associated with sets E+3 and E+4 shown in Table 2 evaluated to be greater than zero. Similarly,
assume that the previous change in the process error c(kT − T) was best characterized by the fuzzy sets
C−4 and C−5. The fuzzy implications which contributed (i.e., had δ 0) to the previously applied process
input are illustrated by the boxed entries in Table 2 (i.e., they are all the implications which contain as
left-hand-side elements the boxed entries – 4 rules in this case). Suppose that pi(kT) = guip
i(kT) so that
p
i(kT) is the normalized desired change in the process input. For our example, assume that p
i(kT) = 0.1,
then after knowledge-base modification for the boxed values 0.4, 0.2, 0.0, and 0.2 in Table 2 we get 0.5,
0.3, 0.1 and 0.3, respectively (all other entries in the Table remain unchanged at this time).
10

D. FMRLC Design Procedure and Implementation Issues
The design procedure for the FMRLC involves: (i) the specification of a direct fuzzy controller with
consequent membership functions that can be tuned (this fuzzy controller can be chosen via conventional
(heuristic) fuzzy control design techniques for the nominal plant), (ii) specifying the reference model which
characterizes the desired system performance, (iii) specifying the fuzzy inverse model which characterizes
how the inputs to the plant should be changed so that the desired performance is achieved, and (iv)
selection of the normalizing gains for the fuzzy controller and the fuzzy inverse model. As selection of
the normalizing gains for both the fuzzy controller and the fuzzy inverse model can impact the overall
performance, next we will provide a procedure to choose these parameters. It is important to note that
although it is often not highlighted, most learning/adaptive control approaches require some type of initial
choice of controller structure and parameters (e.g., the choice of an adaptation gain or initial controller
parameters in a conventional adaptive controller). The gain selection procedure to be presented next
provides a systematic methodology to select such initial parameters for the FMRLC.
Due to physical constraints for a given system, the range of values for the process inputs and outputs
is generally known from a qualitative analysis of the process. As a result, we can determine the range of
values or the universe of discourse for e(kT), u(kT), ye(kT), and p(kT). Consequently, ge, gu, gye
, and
gp are chosen so that the appropriate universes of discourse are mapped to [−1,+1]. To determine gc we
disconnect the adaptive mechanism and pick it using standard fuzzy control system design techniques or
by iteratively applying inputs to yr, observing c(kT), and finding a scaling factors to map the universes of
discourse to [−1, 1].
The vector gyc
is left as a vector of tuning parameters for the FMRLC. Recall that the scaling factors
g
yc
associated with the change in the desired output changes has the effect of providing “damping” to the
controller modifications. Moreover, the “damping” effect is increased as the elements of the scaling factor
gare increased. A suitable selection of gyc
yc
may be obtained by monitoring the response of the overall
process with respect to the reference model response. If undesirable oscillations exist between a given
process and the associated reference model output response, it is likely that the element of g
yc
associated
with this ouput is too small and should be increased. Likewise, if a given element of gyc
is too large, the
process will be unable to keep up with the reference model due to the resulting damping.
Below a simple procedure is presented for selecting the gains:
1. Select the controller gains g
e
, g
u
, and g
ye
so that each universe of discourse is mapped to the interval
[−1, 1].
2. Choose the controller gains gpi to be the same as for the fuzzy controller output gain gui . This will
allow the pi(kT) to take on values as large as the largest possible inputs ui(kT).
3. Using standard fuzzy control design techniques (i.e., ones that use human expertise) or simple ex-periments
choose gc to map the universes of discourse of c(kT) to [−1, 1].
4. Assign the numerical value 0 to the scaling factors associated with the changes in the desired output
changes (i.e., all elements of gyc
are set equal to 0).
5. Apply a step input to the process which is of a magnitude that may be typical for the process during
normal operation. Observe the process response and the reference model response.
11

6. Three cases:
(a) If there exist unacceptable oscillations in a given process output response about the reference
model response, then increase the associated element of gyc
. Go to step 5.
(b) If a given process output response is unable to “keep up” with the reference model response,
then decrease the associated element of gyc
. Go to step 5.
(c) If the process response is acceptable with respect to the reference model response, then the
controller design is completed.
For the applications presented in this paper, the above gain selection procedure has proven very successful.
However, given that the procedure is a result of practical experience with the FMRLC rather than strict
mathematical analysis, it is likely that it will not work for all processes. For some applications (none of
the ones we’ve studied in [21, 12, 19, 20, 18, 22]), the procedure may result in an unstable process. In such
situations, it may be necessary to modify other controller parameters such as the controller sampling period
T or the number of fuzzy controller rules. Clearly, the stability analysis of the FMRLC is an important
research direction.
Note that when implementing (and simulating) the FMRLC one must be concerned with the “curse of
dimensionality”. Particularly, assume that: (i) the fuzzy controller has s inputs (where s = αs with α = 2
for the case shown in Figure 1), r outputs, and N membership functions on each of its input universes
of discourse; (ii) the fuzzy inverse model has s inputs (where s = βs with β = 2 for the case shown in
Figure 1), r outputs, and M membership functions on each of its input universes of discourse; and (iii)
that both the fuzzy controller and the fuzzy inverse model use the maximum number of rules possible (for
completeness). In this case there are r(Ns + Ms ) rules in the FMRLC. As is the case with standard
fuzzy control, increasing the number of inputs causes an exponential increase in the number of rules. It
is important to note that if one assumes that the membership functions are uniformly distributed across
the input universes of discourse so that at most two overlap for any point (this is in fact what we do in
all the applications in this paper), then at most r(2s + 2s ) rules will be on at one time and hence the
code implementing the FMRLC is much less complex than one might think at first glance. It is in fact
this characteristic that we exploit when we implemented the FMRLC for the flexible robotic system in [22]
where the FMRLC had 1150 rules and operated with a sampling interval of 15 milliseconds on a 386-based
computer.
III Applications
A. Rocket Velocity Control
In this section we illustrate the performance of a FMRLC which is employed to control the velocity of
a single stage rocket. A mathematical model for this process is presented by Barr´ere et al. in [42] and
Mandell et al. in [43] and is expressed by the following differential equation:
d v(t)
dt
= c(t)

m
M −m t

− go

R
R + y(t)

− 0.5 v2(t)

ρa A Cd
M −m t

, (13)
where v(t) is the rocket velocity at time t, y(t) is the altitude of the rocket (above sea level), c(t) is the
velocity of the exhaust gases, and for our simulation: (i) M = 15000.0 kg -in itial mass of the rocket and
12

fuel, (ii) m = 100.0 kg
s -e xhaust gases mass flow rate (approximately constant for some solid propellant
rockets), (iii) A = 1.0 meter2 -maxim um cross sectional area of the rocket, (iv) go = 9.8 meters
s2 -th e
acceleration due to gravity at sea level, (v) R = 6.37×106 meters -r adius of the earth, (vi) ρa = 1.21 kg
m3
-d ensity of air, and (vii) Cd = 0.3 -dr ag coefficient for the rocket.
The mathematical model in Equation 13 was developed based on the simple dynamics of a point mass.
However, in general, rockets dynamics are studied in the realm of exterior ballistics. This type of analysis
often tends to be very complex and falls outside the scope of this paper. However, even in this restricted
context the modeled dynamics have characteristics which make for difficult control. For example, due to
the loss of fuel resulting from combustion and exhaust the rocket has a time-varying mass. Furthermore,
it can be determined by inspection of Equation 13 that the system is a non-linear process. Indeed, the
primary purpose for considering this control application is to investigate the capability of the FMRLC
algorithm for controlling non-linear time-varying processes.
As stated before, the control objective is to control the velocity of the rocket. To accomplish this task
the rocket is assumed to have one input, namely the velocity of the exhaust gases c(t). In general, the
exhaust gas velocity is proportional to the cross-sectional area of the nozzle. Consequently, the exhaust
gases may be controlled by changing this cross sectional area. However, for this controller implementation,
we assume that the dynamics of the actuators which change the nozzle area and the dynamics of the
exhaust gases are fast relative the rocket velocity dynamics and therefore may be eliminated from the
model.
1. FMRLC Design
The inputs to the fuzzy controller are the velocity error and change in error and the controller output is
the velocity of the exhaust gases. In this fuzzy controller design, 11 fuzzy sets are defined for each controller
input (using the structure of Figure 1) such that the membership functions are triangular shaped and evenly
distributed on the appropriate universe of discourse (of course the outer-most membership functions are
trapezoidal). The normalizing controller gains for the error, change error, and the controller output are
chosen to be ge = 1
4000, gc = 1
2000, and gu = 10000, respectively. The fuzzy sets for the fuzzy controller
output are also assumed to be triangular shaped with a width of 0.4 on the normalized universe of discourse.
The knowledge-base array was initially chosen with all zero entries. The fuzzy controller sampling period
was chosen to be T = 100 milliseconds.
The reference model for this process was chosen to represent somewhat realistic performance specifica-tions
and is expressed by the following differential equation
d ym(t)
dt
= −0.2 ym(t) + 0.2 yr(t), (14)
where ym(t) specifies the desired system performance for the rocket velocity v(t) and the input to the
reference model yr(t) is equal to the desired rocket velocity.
The inputs to the fuzzy inverse model include the error and change in error between the reference model
and the rocket velocity expressed as ye(kT) = ym(kT)−v(kT), and yc(kT) = ye(kT )−ye(kT−T )
T , respectively.
For these inputs, 11 fuzzy sets are defined with triangular shaped membership functions which are evenly
distributed on the appropriate universes of discourse. The normalizing controller gains associated with
ye(kT), yc(kT), and p(kT) are chosen to be gye = 1
4000, gyc = 1
2000, and gp = 10000, respectively. For
13

the rocket process, for an increase in the exhaust gas velocity we would generally expect an increase in
the process output. Consequently, the knowledge-base array shown in Table 1 was employed for the fuzzy
inverse model.
2. Simulation Results
The simulation results for the FMRLC of the rocket are shown below in Figure 2. Note that the system
1500
1000
500
0
Ref. input
Ref. Model Response
Actual Response
0 20 40 60 80 100 120
Time (sec)
Rocket Velocity (m/s)
Rocket Response - FMRLC
10000
8000
6000
4000
2000
0
0 20 40 60 80 100 120
Time (sec)
Exhaust Gas Velocity (m/s)
Velocity of the Exhaust Gas
40
30
20
10
0
-10
0 20 40 60 80 100 120
Time (sec)
Ref Model Error (m/s)
Error Between the Reference Model Output and the Rocket Velocity
8 x104
6
4
2
0
0 20 40 60 80 100 120
Time (sec)
Rocket Altitude (m)
Altitude of the Rocket
Figure 2: Simulation results for FMRLC control of the rocket system.
exhibits “good” tracking with the reference model even after the mass of the rocket has been reduced
significantly from the initial mass due to fuel loss (note that lower amounts of exhaust gas velocity, the
control input, are needed as more fuel is used). This application clearly illustrates the effectiveness of the
FMRLC algorithm for controlling a nonlinear time varying process.
14

B. Two-Degree of Freedom Robot Manipulator
Figure 3 illustrates the physical model of a two degree of freedom manipulator. It consists of two links
where link #1 is mounted on a rigid base by means frictionless hinge and link #2 is mounted at the end
of link #1 by means of a frictionless ball bearing. This control problem is provided to illustrate the
θ1
θ2
l
l
l
l
c
c
2
1
2
1
I 2
I 1
Link #1
Link #2
Joint #1
Joint #2
Figure 3: Graphical representation of a 2-link robot.
application of the FMRLC to a nonlinear MIMO system. The inputs to the process are the torques τ1
and τ2 which are applied to the links at joints #1 and #2. The outputs are the joint positions θ1 and θ2.
The model for the robotic system was developed using the well-known Lagrangian equations in classical
dynamics and is expressed by the following matrix differential equation [44, 8]:

H11 H12
H21 H22

¨θ1
¨θ2

+

−hθ˙2 −hθ˙1 − hθ˙2
hθ˙1 0

θ˙1
θ˙2

+

g1
g2

=

τ1
τ2

(15)
where
H11 = m1l2
c1 + I1 + m2[l2
1 + l2
c2 + 2l1lc2 cos(θ2)] + I2 (16)
H22 = m2l2
c2 + I2 (17)
H12 = H21 = m2l1lc2 cos(θ2) +m2l2
c2 + I2 (18)
h = m2l1lc2 sin(θ2) (19)
g1 = m1lc1g cos(θ1) + m2g[lc2 cos(θ1 + θ2) + l1 cos(θ1)] (20)
g2 = m2lc2g cos(θ1 + θ2), (21)
15

and where θ = [θ1 θ2]T are the two joint angles, τ = [τ1 τ2]T are the input joint torques. For purpose of
our simulation the robot parameters are given by: (i) m1 = 1.0 kg -m ass of link #1, (ii) m2 = 1.0 kg
-m ass of link #2, (iii) l1 = 1.0 meters -le ngth of link #1, (iv) l2 = 1.0 meters -le ngth of link #2,
(v) lc1 = 0.5 meters -d istance from joint #1 to the center of gravity of link #1, (vi) lc2 = 0.5 meters -
distance from joint #2 to the center of gravity of link #2, (vi) I1 = 0.2 kg − m2 -le ngthwise centroidal
inertia of link #1, and (vii) I2 = 0.2 kg −m2 -le ngthwise centroidal inertia of link #2.
1. FMRLC Design
For this application, the process contains two inputs, namely τ1 and τ2. Consequently, two MISO fuzzy
controllers are needed for this process (one for each process input). The inputs to the fuzzy controller are
the robot joint position error e = [e1 e2]T and change in error c = [c1 c2]T . The fuzzy controllers have
outputs τ1 for the first controller and τ2 for the second controller. For both fuzzy controller designs, 11
fuzzy sets are defined for each controller input such that the membership functions are triangular shaped
(with base widths of 0.4) and evenly distributed on appropriate universes of discourse (the outer-most
membership functions are trapezoidal). Also, the normalizing controller gains for the error, change error,
and the controller output are chosen to be g= [ 1
1
]T , g= [ 1
1
]T, and g= [100 25]T , respectively.
e 2π
2π c 20
20 u The knowledge-base array for both fuzzy controllers was initially chosen with all zero entries. The fuzzy
controller sampling period was chosen to be T = 5 milliseconds.
The reference model for this FMRLC design is given by the following differential equation

y˙m1 (t)
y˙m2 (t)

=

−0.75 0.0
0.0 −1.5

ym1 (t)
ym2 (t)

+

+0.75 0.0
0.0 +1.5

yr1 (t)
yr2 (t)

, (22)
where ym1 and ym2 specify the system performance for θ1 and θ2, respectively. For FMRLC implementation,
the inputs to the reference model yr1 and yr2 are equal to the desired position of joints #1 and #2,
respectively.
For this FMRLC design, two fuzzy inverse models are needed, one for each fuzzy controller. In general,
both process inputs will affect both process outputs. However, for this fuzzy inverse model design we will
assume that the cross-coupling between the inputs is negligible (i.e., τ1 affects only θ1 and τ2 affects only
θ2). As a result, the input to a given fuzzy inverse model includes the error and change in error between
the associated reference model output and robot position. Therefore, for the ith fuzzy inverse model, these
inputs may expressed as yei (kT) = ymi (kT) − θi(kT) and yci (kT) = yei (kT )−yei (kT−T )
T respectively. For
these inputs, 11 fuzzy sets are defined with triangular shaped membership functions which are evenly
distributed on the appropriate universe of discourse. The normalizing fuzzy system gains associated with
y(kT), y(kT), and p(kT) are chosen to be g= [ 1
1
]T , g= [1 1
]T, and g= [100 25]T , respectively.
ecye
2π
2π yc
2 p For the robot process for an increase in the torque τ1 we would generally expect an increase in the process
output θ1. Likewise, for an increase in the torque τ2 we would generally expect an increase in the process
output θ2. Consequently, the knowledge-base array shown in Table 1 was employed for both fuzzy inverse
models.
16

2. Simulation Results
The simulation results for the FMRLC of the two degree-of-freedom robot manipulator are shown
below in Figure 4 for joint #1 and Figure 5 for joint #2. Once again the FMRLC provided good
250
200
150
100
50
0
Ref. Input
Ref. Model Response
Actual Response
0 10 20 30 40 50 60 70 80 90 100
Time (sec)
Joint #1 Position (deg)
Robot Response - FMRLC
Figure 4: Simulation results for joint #1 of FMRLC controlled robot system.
150
100
50
0
-50
-100
-150
Ref. Input
Ref. Model Response
Actual Response
0 10 20 30 40 50 60 70 80 90 100
Time (sec)
Joint #2 Position (deg)
Robot Response - FMRLC
Figure 5: Simulation results for joint #2 of FMRLC controlled robot system.
system tracking with respect to the reference model. As a result, the system exhibits good steady state
and transient response. In fact, the response for joint #1 in Figure 4 was so close to the response of the
reference model that the two almost perfectly overlap.
IV Concluding Remarks
The principal objectives of this paper were to: (i) introduce the FMRLC, (ii) provide a design method-ology
for the FMRLC, (iii) design a FMRLC for a nonlinear time-varying rocket velocity control problem,
17

and (iv) develop a MIMO FMRLC for a nonlinear two degree-of-freedom robot manipulator. The key
advantages that the FMRLC seems to offer as a learning controller may be summarized as follows:
• A detailed analytical model of the process is not needed to develop the FMRLC.
• The FMRLC provides an automatic method to synthesize a portion of the knowledge-base (specif-ically,
the right-hand-sides of the rules) for the direct fuzzy controller while at the same time it
ensures that the system will behave in a desirable fashion (in particular, there is no need to “learn
from drastic failures” as is often the case for other learning control techniques -e. g., as is often done
for the inverted pendulum).
• The learning/adaptation mechanism in the FMRLC dynamically and continually updates the rule-base
in the direct fuzzy controller in response to process parameter variations and/or disturbances
(e.g., see the rocket velocity control application). In this way if unpredictable changes occur within
the plant, the FMRLC can make on-line adjustments to a direct fuzzy controller to maintain adequate
performance levels.
Basically, by combining learning/adaptive control concepts with fuzzy system theory, we have developed
a control scheme which often has a fast rate of convergence and often provides an appropriate nonlinear
mapping between controller inputs and outputs (i.e., it automatically performs “function approximation”
[40] to achieve learning control).
Despite these apparant advantages of the FMRLC algorithm, several drawbacks do exist: (1) the design
procedure (e.g., selection of the normalizing gains) tends to be somewhat ad hoc, (2) there have been no
investigations for the FMRLC (or any other fuzzy adaptive technique) to theoretically show that the fuzzy
controller can in fact be tuned so that the performance specified in the reference model can be achieved
(this problem is very well studied in conventional adaptive control where linear controllers are tuned so
that performance specified in linear reference models is achieved), (3) conditions for stability and conver-gence
of the FMRLC algorithm are yet to be found, (4) persistent excitation [7, 8] issues for the FMRLC
need to be mathematically investigated (since the reference input affects the ability of the fuzzy controller
parameters to converge to values that result in the reference model behavior being achieved), and (5)
although it provides certain improvements over SOC (as shown in [12]), the FMRLC algorithm is still
computationally intensive. These disadvantages provide several future research directions. For example,
future research involving the FMRLC algorithm should include a mathematical analysis of the controller to
better quantify the effect of controller design parameters. Perhaps the most important research direction is
to perform stability and convergence analysis in the spirit of the extensive and significant contributions in
stability analysis of conventional adaptive control [7, 8] and the important recent (actually yet to appear)
results in [38]. Such stability analysis can be quite involved as the learning mechanism for the FMRLC ad-justs
a nonlinear (fuzzy) controller as opposed to the conventional adaptive control case where often linear
controllers are adjusted (there is, however, a growing body of literature on adapting nonlinear controllers).
Finally, research must be directed towards developing faster algorithms for FMRLC computations to en-sure
that the FMRLC can be employed in real world applications (along the same lines as was done in [22]).
Acknowledgement: The authors would like to thank the reviewers for their helpful comments.
18

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