314
7 Pulse Programming of Memristor Circuits
A simple analysis shows that in this case we have
H =
0 −1
1 0
, u(t) =
0
e(t)
hence, as expected, H 22 = 0 is singular.
The state variables in the (v, i)-domain are (ϕ M , v C , i L ). By KVL we obtain
L
di L (t)
dt
= e(t) +
dϕ M (t)
dt
.
Let
K(ϕ M (t), v C (t), i L (t)) = Li L (t) − ϕ M (t).
If e(t) = 0 there are planar invariant manifolds
M(k) = {(ϕ M (t), v C (t), i L (t)) ∈ R
3
: K(ϕ M (t), v C (t), i L (t)) = k}
for any k ∈ R. If e(t) = 0 we have
˙
k(ϕ M (t), i L (t)) = e(t)
and so via the voltage source we are able to programme the different reduced-order
dynamics on manifolds.
Appendix 2: Proof of Theorem 7.3
1. Given any k ∈ R n M , it can be checked that M(k) contains at least point w =
(0, M −1
x (k − F(0)), 0), hence M(k) = ∅. The manifold M(k) is nonplanar
due to the nonlinear function F(·). Finally, function K(w) = k specifies n M
constraints in the (n M + n C + n L )-dimensional state space in the (v, i)-domain.
Hence, the dimension of M(k) is given by the number (n C + n L ) of independent
state variables in the (v, i)-domain.
2. Consider the class of rigid translations x M → x M , ˙
x → ˙
x + Δ ˙
x , ˙
y → ˙
y +
Δ ˙
y . Also consider the equation M x Δ ˙
x − H 12 H
−1
22 M y Δ ˙
y = k 2 − k 1 , that has a
solution Δ ˙
x = M −1
x (k 2 − k 1 ), Δ ˙
y = 0 (and possibly other solutions). We have
K(w + (0, Δ ˙
x , 0)) = K(w) + k 2 − k 1 . This means that, for any w ∈ M(k 1 ),
we have w + (0, Δ ˙
x , 0) ∈ M(k 2 ). Conversely, for any w ∈ M(k 2 ) we have
w − (0, Δ ˙
x , 0) ∈ M(k 1 ).
3. Manifolds are nonintersecting because if there exist w ∈ M(k 1 )
M(k 2 ), with
k 1 = k 2 , then K(w) = k 1 and K(w) = k 2 , i.e., w is mapped by K(·) in two
distinct vectors k 1 and k 2 , while K(·) in (7.29) is a (single-valued) function.
7 Pulse Programming of Memristor Circuits
A simple analysis shows that in this case we have
H =
0 −1
1 0
, u(t) =
0
e(t)
hence, as expected, H 22 = 0 is singular.
The state variables in the (v, i)-domain are (ϕ M , v C , i L ). By KVL we obtain
L
di L (t)
dt
= e(t) +
dϕ M (t)
dt
.
Let
K(ϕ M (t), v C (t), i L (t)) = Li L (t) − ϕ M (t).
If e(t) = 0 there are planar invariant manifolds
M(k) = {(ϕ M (t), v C (t), i L (t)) ∈ R
3
: K(ϕ M (t), v C (t), i L (t)) = k}
for any k ∈ R. If e(t) = 0 we have
˙
k(ϕ M (t), i L (t)) = e(t)
and so via the voltage source we are able to programme the different reduced-order
dynamics on manifolds.
Appendix 2: Proof of Theorem 7.3
1. Given any k ∈ R n M , it can be checked that M(k) contains at least point w =
(0, M −1
x (k − F(0)), 0), hence M(k) = ∅. The manifold M(k) is nonplanar
due to the nonlinear function F(·). Finally, function K(w) = k specifies n M
constraints in the (n M + n C + n L )-dimensional state space in the (v, i)-domain.
Hence, the dimension of M(k) is given by the number (n C + n L ) of independent
state variables in the (v, i)-domain.
2. Consider the class of rigid translations x M → x M , ˙
x → ˙
x + Δ ˙
x , ˙
y → ˙
y +
Δ ˙
y . Also consider the equation M x Δ ˙
x − H 12 H
−1
22 M y Δ ˙
y = k 2 − k 1 , that has a
solution Δ ˙
x = M −1
x (k 2 − k 1 ), Δ ˙
y = 0 (and possibly other solutions). We have
K(w + (0, Δ ˙
x , 0)) = K(w) + k 2 − k 1 . This means that, for any w ∈ M(k 1 ),
we have w + (0, Δ ˙
x , 0) ∈ M(k 2 ). Conversely, for any w ∈ M(k 2 ) we have
w − (0, Δ ˙
x , 0) ∈ M(k 1 ).
3. Manifolds are nonintersecting because if there exist w ∈ M(k 1 )
M(k 2 ), with
k 1 = k 2 , then K(w) = k 1 and K(w) = k 2 , i.e., w is mapped by K(·) in two
distinct vectors k 1 and k 2 , while K(·) in (7.29) is a (single-valued) function.