10.1 A Class of Extended Memristors
375
10.1 A Class of Extended Memristors
Consider the class of voltage-controlled extended memristor devices described by
i M = G(x, v M )v M
(10.1)
dx
dt
= h(x, v M )
(10.2)
where i M and v M are the current through and the voltage across the memristor,
G(x, v M ) is its memductance, whereas x ∈ R n is a vector of n internal state
variables (possibly) including the memristor flux ϕ M . According to the definition
in Sect. 2.4.4 of Chap. 2, the zero-crossing property of the i M –v M curves, i.e.,
lim
v M →0
i M = lim
v M →0
G(x, v M )v M = 0
(10.3)
requires that, for any x ∈ R n , the memductance G(x, v M ) : R n+1 → R is a bounded
function in the neighbor of (x, 0). Then, (10.1)–(10.2) satisfying (10.3) describe a
generic dynamical system satisfying the distinctive zero-crossing property [7].
In this chapter, the focus is on the class of memristor devices D ext , represented
in Fig. 10.1, given by the parallel connection of an ideal flux-controlled memristor
and a voltage-controlled nonlinear resistor. The ideal memristor is an algebraic
(α, β) = (−1, −1) element characterized by the nonlinear relationship q M =
f (ϕ M ) between the flux ϕ M and charge q M , where f is a locally Lipschitz function. 1
Since v M = v, the CR of the ideal memristor in the (v, i)-domain is
i M = W (ϕ M )v
(10.4)
dϕ M
dt
= v
(10.5)
where W (ϕ M ) = f (ϕ M ) is the memductance. The nonlinear resistor is an algebraic
(α, β) = (0, 0) element defined by (being v R = v)
i R = F R (v)
(10.6)
where F R (·) is locally Lipschitz and F R (0) = 0.
In the (v, i)-domain, D ext is described by the CR
i = W (ϕ M )v + F R (v)
(10.7)
1 This means that, for any ϕ M , there exists a neighborhood U of ϕ M and k f (ϕ M ) > 0 such that
|f (ϕ M,1 ) − f (ϕ M,2 )| ≤ k f (ϕ M )|ϕ M,1 − ϕ M,2 | for any ϕ M,1 , ϕ M,2 ∈ U , where k f (ϕ M ) is the
Lipschitz constant of f at ϕ M . This constant depends on ϕ M and U . If f is locally Lipschitz, then
the derivative f is defined almost everywhere [10].
375
10.1 A Class of Extended Memristors
Consider the class of voltage-controlled extended memristor devices described by
i M = G(x, v M )v M
(10.1)
dx
dt
= h(x, v M )
(10.2)
where i M and v M are the current through and the voltage across the memristor,
G(x, v M ) is its memductance, whereas x ∈ R n is a vector of n internal state
variables (possibly) including the memristor flux ϕ M . According to the definition
in Sect. 2.4.4 of Chap. 2, the zero-crossing property of the i M –v M curves, i.e.,
lim
v M →0
i M = lim
v M →0
G(x, v M )v M = 0
(10.3)
requires that, for any x ∈ R n , the memductance G(x, v M ) : R n+1 → R is a bounded
function in the neighbor of (x, 0). Then, (10.1)–(10.2) satisfying (10.3) describe a
generic dynamical system satisfying the distinctive zero-crossing property [7].
In this chapter, the focus is on the class of memristor devices D ext , represented
in Fig. 10.1, given by the parallel connection of an ideal flux-controlled memristor
and a voltage-controlled nonlinear resistor. The ideal memristor is an algebraic
(α, β) = (−1, −1) element characterized by the nonlinear relationship q M =
f (ϕ M ) between the flux ϕ M and charge q M , where f is a locally Lipschitz function. 1
Since v M = v, the CR of the ideal memristor in the (v, i)-domain is
i M = W (ϕ M )v
(10.4)
dϕ M
dt
= v
(10.5)
where W (ϕ M ) = f (ϕ M ) is the memductance. The nonlinear resistor is an algebraic
(α, β) = (0, 0) element defined by (being v R = v)
i R = F R (v)
(10.6)
where F R (·) is locally Lipschitz and F R (0) = 0.
In the (v, i)-domain, D ext is described by the CR
i = W (ϕ M )v + F R (v)
(10.7)
1 This means that, for any ϕ M , there exists a neighborhood U of ϕ M and k f (ϕ M ) > 0 such that
|f (ϕ M,1 ) − f (ϕ M,2 )| ≤ k f (ϕ M )|ϕ M,1 − ϕ M,2 | for any ϕ M,1 , ϕ M,2 ∈ U , where k f (ϕ M ) is the
Lipschitz constant of f at ϕ M . This constant depends on ϕ M and U . If f is locally Lipschitz, then
the derivative f is defined almost everywhere [10].