IEEE Circuits and Systems Magazine - Q2 2018 - 52
"
x 0 - xu 0
Applying u (t ) = u (t )
Applying u (t ) = u (t )
"
x (t )
x0
0
"
x (t )
xT
Unique Steady State
~
~
xT
x (t )
0
T
Ts
t
Figure 1. development of a unique asymptotic behaviour in
a system endowed with fading memory (see text for details).
III. The Concept of Fading Memory
This section first provides a rigorous definition of the
concept of fading memory, which was first introduced
by S. Boyd and L.O. Chua in 1985 [41], then explains why
ideal and ideal-generic current (voltage)-controlled
memristors may never be subject to this dynamic phenomenon, and, finally, investigates the conditions under which history erase effects may affect the dynamics
of a purely-mathematical non-ideal memristor model,
which was proposed by Pershin et al. in 2009 to capture the learning process in the unicellular organism
amoeba [57].
A. Fading Memory: a Rigorous Definition
Let a nth-order nonlinear system be described by the
following initial value problem (IVP)
.
x = f (x, u), x 0 _ x (0),
(12)
where x ! R n represents a n-dimensional state vector,
u ! C (R) denotes the input6, whereas f : R n # C (R) " R n
stands for the state evolution function. It is further assumed that the IVP expressed by equation (12) defines an
operator N, establishing the state response of the given
system to the application of any input u ! C (R) through
the mathematical expression x = Nu ! R n. The operator
N has fading memory on a set K 1 C (R) if there exists a
decreasing function of time w : R " (0, 1], asymptotically
approaching the null value as t tends to + 3, such that,
for each input u ! K and positive constant f, a further
positive constant d may be found so that, under the application of any other input uu ! K, if the inequality
sup | u (t) - uu (t) | w (- t) 1 d
t#0
(13)
holds, then the norm of the difference between the
states x (t) and xu (t), due to inputs u and uu respectively,
is upper bounded at time t = 0 according to
6
52
C (R) is the space of bounded and continuous functions of R.
IEEE cIrcuIts and systEms magazInE
2
= Nu (0) - Nuu (0)
2
1 f,
(14)
~
where v 2 = v 21 + g + v 2n indicates the 2-norm of the
T
vector v = [v 1, f, v n] T ! R n, while xu 0 = xu (0) . Very importantly, S. Boyd and L.O. Chua [41] proved that a dynamic system with fading memory capability is endowed
with unique steady-state behaviour. Letting the set X include all states, which may be observed in response to
the application of the operator N to all inputs in K, i.e.
X = " x = Nu|u ! K ,,
(15)
the IVP (12) may admit one and only one possible asymptotic solution for any input in K, irrespective of the
initial condition in X. In order to shed further light into
the fading memory capability of a nonlinear system, let
us consider the IVP (12) in the one-dimensional case-
i.e. n = 1-under the application of two input signals
ut (t) ! K and uu (t) ! K, which are different from (identical to) each other for t ! [0, T ) (for t > T ) . If the system, initiated in x 0 at time t = 0, exhibits fading memory, denoting with xt T = x (T ) ! X ^ xu T = xu (T ) ! X h the
value the solution of the IVP attains at time t = T as a
result of the application of the input signal ut (t) ^uu (t) h, as
time proceeds from the time instant t = T , the informat T and xt T
tion embedded in either of the state values x
progressively fades away, leading to the emergence of
a unique asymptotic behaviour, as mathematically expressed by the following limit:
lim
xt (t) - xu (t) = 0.
t"3
(16)
This is graphically illustrated in Fig. 1, where we have
clearly highlighted the time instant TS at which the
modulus of the difference between xt (t) and xu (t) is infinitesimally small. For practical purposes, for t 2 TS the
two solutions to the IVP may be considered identical.
B. Absence of Fading Memory in Ideal and Ideal
Generic Voltage (Current)-Controlled Memristors
In this section we consider the classes of voltage
(current)-controlled ideal or ideal-generic memristors.
As originally proved in [53], the initial condition of ideal
flux (charge)-controlled memristors plays a crucial role
in the asymptotic behaviour of the memory state, i.e.
the flux (charge), which, consequently, does not feature
the fading memory property. To explain this concept
briefly, let us refer to an example. We consider an ideal
flux-controlled memristor with constitutive relation
expressed by q m = qt m ({ m) = a{ m + b{ 3m, where a and
b are two positive real values with units C/(Vs) and
C/(Vs)3, respectively. Note that the memductance-state
map of this device is the function G ({ m) = a + 3b{ 2m .
sEcOnd quartEr 2018
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