7.8 Further Examples
309
Fig. 7.13 (a) Memristor circuit with a current-controlled voltage-source and (b) resistive circuit
for finding the hybrid representation
and
u(t) =
−q a (t; t 0 )
0
with u x (t) = −q a (t; t 0 ) = −
t
t 0
a(τ )dτ and u y (t) = 0. Note that H 22 = 2R is
nonsingular and [H 22 ] −1 = 1/2R. The representation always exists, even if R < 0.
The state variables in the (v, i)-domain are given by (ϕ M (t), v C (t), i L (t)). We
have x M (t) = ϕ M (t), M x ˙
x(t) = Cv C (t), and M y ˙
y(t) = Li L (t).
From (7.29), we have
K(ϕ M (t), v C (t), i L (t)) =
3
2R
+
ρ
2R 2
ϕ M (t) + f (ϕ M (t))
+Cv C (t) +
Li L (t)
2R
so that on the basis of (7.30) the manifolds are given as
M( ¯
k) = {(ϕ M (t), v C (t), i L (t))
T
∈ R
3
: K(ϕ M (t), v C (t), i L (t)) = ¯
k}
where ¯
k ∈ R. Moreover, from (7.34)