Memristor Crossbar-based Hardware Implementation of Fuzzy Membership Functions

Abstract

In May 1, 2008, researchers at Hewlett Packard (HP) announced the first physical realization of a fundamental circuit element called memristor that attracted so much interest worldwide. This newly found element can easily be combined with crossbar interconnect technology which this new structure has opened a new field in designing configurable or programmable electronic systems. These systems in return can have applications in signal processing and artificial intelligence. In this paper, based on the simple memristor crossbar structure, we propose new and simple circuits for hardware implementation of fuzzy membership functions. In our proposed circuits, these fuzzy membership functions can have any shapes and resolutions. In addition, these circuits can be used as a basis in the construction of evolutionary systems. Comment: 5 pages, 5 figures, Submitted to ICCAE 2011 conference

Full text

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Figure 1. Structure of the memristor reported by HP researchers and its
equivalent circuit model.
Memristor Crossbar-based Hardware
Implementation of Fuzzy Membership Functions

Farnood Merrikh-Bayat
Electrical Engineering Department
Sharif University of Technology
Tehran, Iran
F_merrikhbayat@ee.sharif.edu
Saeed Bagheri Shouraki
Electrical Engineering Department
Sharif University of Technology
Tehran, Iran
Bagheri-s@sharif.edu


Abstract—In May 1, 2008, researchers at Hewlett Packard
(HP) announced the first physical realization of a fundamental
circuit element called memristor that attracted so much
interest worldwide. This newly found element can easily be
combined with crossbar interconnect technology which this
new structure has opened a new field in designing configurable
or programmable electronic systems. These systems in return
can have applications in signal processing and artificial
intelligence. In this paper, based on the simple memristor
crossbar structure, we propose new and simple circuits for
hardware implementation of fuzzy membership functions. In
our proposed circuits, these fuzzy membership functions can
have any shapes and resolutions. In addition, these circuits can
be used as a basis in the construction of evolutionary systems.
Keywords-Memristor; fuzzy membership function; memristor
crossbar; hardware implementation.
I.
INTRODUCTION
Nowadays, fuzzy inference systems are extensively in
use in so many different applications [1, 2] and for variety of
reasons such as modeling and control [3]. The main
advantage of these systems compared to other conventional
methods is their capacity to tolerate information expressed in
a way that is uncertain and imprecise.
Almost all of the currently working fuzzy systems are
implemented digitally on digital hardware devices such as
Field-Programmable Gate Arrays (FPGAs) [4], Digital
Signal Processors (DSPs) and dedicated Application Specific
Integrated Circuits (ASICs) [5]. However, such computing
paradigms suffer from the constant need of establishing a
trade-off between flexibility and performance. In these kinds
of devices, the arithmetic operations are carried out with
limited computational precision [6–8]. Another drawback of
these digital hardware devices is that they are very poor in
acting as an evolvable hardware. In addition, they are
completely in contrast with the nature of the fuzzy which is
uncertainty. In this paper, we propose a new and simple
hardware based on the memristor crossbar structure for
implementing fuzzy membership functions. Since any
operation is performed in analog in our proposed circuits,
computational precision is theoretically infinite. Here, it is
worth to mention that since the fuzzy membership functions
are implemented through the memristance of the memristors
in the crossbar, they can be simply evolved by changing their
memristance value.
The paper is organized as follows. In Section 2, we
describe the characteristics of memristors and their
application in the field of programmable analog circuits.
Section 3 is devoted to the description of our proposed
memristor crossbar-based hardware implementation of fuzzy
membership functions. Finally, a brief conclusion is
presented in Section 4.
II.
MEMRISTORS AND THEIR CONTRIBUTION IN
CONSTRUCTING PROGRAMMABLE ANALOG CIRCUITS
Memristor is an electrically switchable semiconductor
thin film sandwiched between two metal contacts with a total
length of D and consists of doped and un-doped regions
which its physical structure with its equivalent circuit model
is shown in Fig. 1 [9]. The internal state variable W
determines the length of doped region with low resistance
against un-doped region with high resistivity. This internal
state variable and consequently the total resistivity of the
device can be changed by applying external voltage bias
( )V t . This means that passing current from memristor in one
direction will increase the resistance while changing the
direction of the applied current will decrease its memristance
[10]. On the other hand, it is obvious that in this element,
passing current in one direction for longer period of time will
change the resistance of the memristor more.
As a result, memristor is nothing else than the analog
variable resistor which its resistance can be adjusted by
changing the direction and duration of the applied voltage or
current. Therefore, memristor can be used as a storage device

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(a)

(b)

(c)

(d)

(e)
Figure 2. First example of memristor crossbar-based hardware implementation of membership functions; (a) Continuous form of the membership
functions; (b) Discrete form of the membership functions; (c) Memristor crossbar-based hardware implementation of membership functions; (d)
Equivalent circuit; (e) A negative narrow pulse with amplitude -1 which is applied to the columns of the crossbar.
in which analog values can be stored as impedance instead of
voltage.
These analog memories can be simply combined by the
crossbar structure. A crossbar array basically consists of two
sets of conductive parallel wires intersecting each other
perpendicularly. The intersections (or crosspoints) are
separated by a thin film material which its properties such as
its resistance can be changed by controlling the voltage
applied to it such as memristor. Now, it is well-known that
these memristor crossbar structures can offer so many new
applications in the field of programmable electronic systems
[11, 12].
III. MEMRISTOR CROSSBAR-BASED HARDWARE
IMPLEMENTATION OF FUZZY MEMBERSHIP FUNCTIONS
In this section, we will show that how any membership
function with any shape can be implemented with a simple
memristor crossbar. Assume that A is a fuzzy set defined on
the input domain x and
A
µ ⋅ denotes the membership
function of it. In fact, by hardware implementation of
membership function we mean that we are going to construct
a system that if the input sample
system, the output becomes
illustrate the idea of using memristor crossbar for
( )
0x is presented to this
)
. Hereafter, we will
0
(
Ax
µ
implementing membership functions by some examples.
First of all, consider a simplest case in which two fuzzy
sets A and B with membership functions
respectively are defined on input domain x as shown in Fig.
2(a). In almost all of the current applications, input signal is
sampled and quantized with finite resolution. Without loss of
generality, in this example we assume that the input is
quantized with a quantizer which its quantization step is
equal to 1. In this case, the input variable x can only have
discrete integer values and therefore, fuzzy sets A and B and
their corresponding membership functions
will become discrete as well as shown in Fig. 2(b) for
convenience. Figure 2(c) shows the memristor crossbar-
based circuit that we propose as a hardware implementation
of fuzzy sets of Fig. 2(a). In this figure, the crossbar has two
rows where each of them corresponds to one of the fuzzy
sets A and B. Fuzzy membership function
implemented in lower row and fuzzy membership function
( )
B
µ ⋅ is implemented in upper row of this crossbar. In
addition, each column of this crossbar represents a discrete
value of the input variable x. Memristors at those crosspoints
which are specified by black dots should be programmed to
have memristance values equal to those which are shown in
Fig. 2(c) (written near to them). In fact, memristance of the
( )
A
µ ⋅ and ( )
B
µ ⋅
( )
A
µ ⋅ and
( )
B
µ ⋅
( )
A
µ ⋅
is

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(a)

(b)

(c)
Figure 3. Another example demonstrating how any number of membership functions with any shapes can be implemented by memristor crossbars.
memristor located at the crossing point of the lower
horizontal wire and the vertical wire corresponds to x0 should
be equal to
0
()
A
Rx
µ
function of the fuzzy set A at point
resistor of the opamps of Fig. 2(c). Any other memristors in
this crossbar which is not programmed (
have the memristance of
off
R
possible memristance value of the memristor.
The circuit of Fig. 2(c) works as follows. Suppose that
the input sample
0
3x = is observed. In this case, a negative
narrow pulse with amplitude -1 such as the one shown in
Fig. 2(e) is applied to the 10th column of the crossbar (which
corresponds to x = 3) while other columns are connected to
high impedance. Note that the width of the pulse should be
chosen in a way that its application does not change the
memristance of the memristors. In this case, the memristor
crossbar of Fig. 2(c) will be equal to the circuit shown in Fig.
2(d) which in this configuration, outputs of opamps A and B
are 0.25 and 0.25 respectively. It is evident that these outputs
are equal to
0
( 3)
µ=
and
µ
circuit of Fig. 2(c), the input is the quantized sample of the
input variable x and the outputs are the membership
functions of fuzzy sets A and B, i.e.
Now, consider a situation in which the input observed
sample is
0
8x = . In this case, the output of the lower opamp
will be
off
R R
Consequently, resistance R should be chosen in a way that
R R
be as much small as possible. It is worth to mention
that in this example, samples of the input variable are
considered to be a singleton instead of the fuzzy number (the
reason of applying negative narrow pulse to one column
while connecting others to high impedance).
Figure 3 shows another example of hardware
where
0
()
Ax
µ
is the membership
0x and R is the feedback
0
()
Ax
µ
R
is zero) will
is the highest where
off
Ax
0
( 3)
Bx
=
. Therefore, in the
( )
Ax
µ
and ( )
Bx
µ
.
whereas ideally it should be zero.
off
implementation of fuzzy membership functions. Figure 3(a)
shows the continuous form of the fuzzy sets and their
corresponding membership functions while the discrete
versions of them are depicted in Fig. 3(b). Similar to the
previous example, these fuzzy sets simply can be
implemented by our proposed structure like the one
demonstrated in Fig. 3(c). By applying a negative pulse of
Fig. 2(e) to any column of this circuit, values of the
membership functions at that point will appear at the outputs
of the opamps.
Through these two examples, it should become clear that
any number of fuzzy sets with any shapes can be simply
implemented by our memristor crossbar-based circuits. In
addition, it is evident that by increasing the number of
columns of the crossbar, it is possible to increase the
resolution of the input signal. For instance, assume that we
use finer quantizer for the quantization of the input signal,
i.e. x in our first example. In this case, if the quantization
step of the utilized quantizer becomes
fuzzy sets in Fig. 2 can be implemented such as the one
shown in Fig. 4.
In all of the above examples, the input signal is assumed
to be a singleton. Now, consider the case in which the input
signal is a fuzzy number. In this case, the output of the
structure should be a fuzzy number as well. As an example,
consider the typical fuzzy set and the input fuzzy number
which are shown in Fig. 5(a) and Fig. 5(b) respectively. In its
simplest form, at the output we should have two fuzzy
numbers which are the componentwise multiplication of the
input fuzzy number with the membership functions of two
fuzzy sets. This process can simply be implemented by the
memristor crossbar-based circuit shown in Fig. 5(c).
Memristance of the memristors at those crosspoints which
are specified by black dots in Fig. 5(c) should be adjusted to
the values written near to them in this figure. Any other
memristors in the crossbar should have their highest
0.5q
∆ =
, then the two

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(a)

(b)

Figure 4. Hardware implementation of membership functions of Fig. 2(a) with higher resolution.
memristance value, i.e.
implement the fuzzy set of Fig. 5(a) in the antidiagonal of the
crossbar through the memristance of the memristor. Now, if
the samples of the fuzzy number of Fig. 5(b) are interpreted
as a voltage and are applied to the columns of the crossbar,
voltages at the outputs of the opamps will be the output
fuzzy numbers. By applying standard opamp circuit analysis
techniques, it is easy to see that the outputs of the opamps
are the samples of the componentwise multiplication of the
fuzzy sets (Fig. 5(a)) and fuzzy input number (Fig. 5(b)).
Here it is worth to mention that memristor crossbar
structures like the one proposed in this paper have this
potential that they can be used as a platform for
implementing evolvable hardware. This is because of the fact
that in these kinds of systems, variable parameters are mostly
implemented at the crosspoints through the memristance of
the memristors which can in return be simply modified by
applying the suitable voltage or current. Note that
modification of the memristance of the memristors can be
done even during the execution time of the system.
off
R
. In fact by this way, we
IV. CONCLUSION
In this paper we illustrated how any number of
membership functions with any shapes and resolution can
simply be implemented through the memristor crossbar-
based structure. In our proposed method, membership
functions are programmed into the memristance of the
memristors at the crosspoints. One of the most particular
advantages of our proposed circuits is that they have a
capability of being used in evolutionary systems.
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Figure 5. Memristor crossbar-based hardware implementation of membership functions in the case that both of the input and output of the system are
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