358
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
Proof
(i) Immediately follows from Property 9.1.
(ii) We have seen in Property 9.1 that for sufficiently large ρ > 0 the hypercube
K ρ is positively invariant for the dynamics of (9.13). By using a standard result
based on a fixed point theorem (see, e.g., [18, Theorem 8.2, p. 49]) we obtain
that (9.13) has at least an EP within K ρ .
Remark 9.1 On the basis of Property 9.2 we can clarify what we mean by saying
that the nonlinearity ˆ
ϕ(·) is a good approximation of s(·) (cf. Sect. 9.2.1). If we
choose q(0) ∞ ≤ ρ min , by i) of Property 9.2 we have q(t) ∞ ≤ ρ min for any
t ≥ 0, i.e., q(t) evolves within K ρ min for t ≥ 0. So, actually it is enough that
ˆ
ϕ(q i ) is a good approximation of s(·) in I ρ min = {q i ∈ R : |q i | ≤ ρ min }. We have
| ˆ
ϕ(q i ) − s(q i )| < 0.072 in I ρ min up to ρ min = 7.3, | ˆ
ϕ(q i ) − s(q i )| < 0.1 in I ρ min up
to ρ min = 9.7 and | ˆ
ϕ(q i ) − s(q i )| < 0.13 in I ρ min up to ρ min = 12. Note that typical
values of ρ min in the applications are of a few units (cf. examples in Sect. 9.4).
It is also worth remarking that the approximation improves if the memristor ratio
R off /R on increases. In the chapter we considered the HP memristor model where
this ratio equals 160. However, in the literature memristors are reported with values
of this ratio as large as 1000 and well above [1].
Let us now address convergence of M-SCNNs. The main result is as follows.
Theorem 9.1 Suppose that (9.16) is satisfied and the interconnection matrix G is
symmetric. Then, (9.13) is convergent, i.e., any solution of (9.13) converges to an
EP as t → +∞.
Proof We have seen in Property 9.2 that any solution of (9.13) is bounded for
t ≥ 0. For symmetric interconnections and a strictly increasing activation function
it is a standard technique to prove that any solution converges to the set of EPs
via a Lyapunov approach. For the reader convenience we report the details of the
verification. Suppose without loss of generality τ = 1. Let y = ˆ
Φ(q) and consider
as in the case of a symmetric SCNN [8] the candidate Lyapunov function
V (y) =
n
i=1
y i
0
ˆ
ϕ
−1
i (ρ)dρ − y
T
Gy − y
T
Q 0 .
By exploiting the symmetry of G it can be easily checked that
∇V (y) =
∂V (y)
∂y 1
, . . . ,
∂V (y)
∂y n
T
= q − G ˆ
Φ(q) − Q 0
hence the M-SCNN equations can be rewritten as the gradient-type system
dq
dt
= −∇V (y).
(9.19)
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
Proof
(i) Immediately follows from Property 9.1.
(ii) We have seen in Property 9.1 that for sufficiently large ρ > 0 the hypercube
K ρ is positively invariant for the dynamics of (9.13). By using a standard result
based on a fixed point theorem (see, e.g., [18, Theorem 8.2, p. 49]) we obtain
that (9.13) has at least an EP within K ρ .
Remark 9.1 On the basis of Property 9.2 we can clarify what we mean by saying
that the nonlinearity ˆ
ϕ(·) is a good approximation of s(·) (cf. Sect. 9.2.1). If we
choose q(0) ∞ ≤ ρ min , by i) of Property 9.2 we have q(t) ∞ ≤ ρ min for any
t ≥ 0, i.e., q(t) evolves within K ρ min for t ≥ 0. So, actually it is enough that
ˆ
ϕ(q i ) is a good approximation of s(·) in I ρ min = {q i ∈ R : |q i | ≤ ρ min }. We have
| ˆ
ϕ(q i ) − s(q i )| < 0.072 in I ρ min up to ρ min = 7.3, | ˆ
ϕ(q i ) − s(q i )| < 0.1 in I ρ min up
to ρ min = 9.7 and | ˆ
ϕ(q i ) − s(q i )| < 0.13 in I ρ min up to ρ min = 12. Note that typical
values of ρ min in the applications are of a few units (cf. examples in Sect. 9.4).
It is also worth remarking that the approximation improves if the memristor ratio
R off /R on increases. In the chapter we considered the HP memristor model where
this ratio equals 160. However, in the literature memristors are reported with values
of this ratio as large as 1000 and well above [1].
Let us now address convergence of M-SCNNs. The main result is as follows.
Theorem 9.1 Suppose that (9.16) is satisfied and the interconnection matrix G is
symmetric. Then, (9.13) is convergent, i.e., any solution of (9.13) converges to an
EP as t → +∞.
Proof We have seen in Property 9.2 that any solution of (9.13) is bounded for
t ≥ 0. For symmetric interconnections and a strictly increasing activation function
it is a standard technique to prove that any solution converges to the set of EPs
via a Lyapunov approach. For the reader convenience we report the details of the
verification. Suppose without loss of generality τ = 1. Let y = ˆ
Φ(q) and consider
as in the case of a symmetric SCNN [8] the candidate Lyapunov function
V (y) =
n
i=1
y i
0
ˆ
ϕ
−1
i (ρ)dρ − y
T
Gy − y
T
Q 0 .
By exploiting the symmetry of G it can be easily checked that
∇V (y) =
∂V (y)
∂y 1
, . . . ,
∂V (y)
∂y n
T
= q − G ˆ
Φ(q) − Q 0
hence the M-SCNN equations can be rewritten as the gradient-type system
dq
dt
= −∇V (y).
(9.19)