Page 126 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 117

4.13 A GENERAL FRAMEWORK IN ELECTRONICS

The practice (see [6] and [20]) shows that a large class of circuits can be studied
via the following general mathematical formalism.
m
Let ∈ (R ;R ∪{+∞}),let A ∈ R n×n , B ∈ R n×m , C ∈ R m×n , and D ∈
p
R n×p , and let u ∈ R . We consider the following problem:
m
n
NRM(A,B,C,D,u, ):Find (x,y L ) ∈ R × R such that
Ax − By L + Du = 0, (4.87)
y = Cx, (4.88)

and
y L ∈ ∂ (y). (4.89)

The matrices A, B,C, and D in (4.87) are structural matrices used to state
the Kirchhoff voltage and current laws in matrix form. The matrix A depends
on electrical parameters like resistances, capacitances, and inductances. Usually,
u is a control vector that drives the system, x denotes a current vector, and y L is
a voltage vector corresponding to electrical devices like diodes whose (possibly
set-valued) ampere–volt characteristics can be described as in (4.89).
It is worth noting that (4.87)–(4.89) may represent the equations of a static
circuit but also the generalized equation that is to be satisfied by the equilib-
rium points of a dynamical circuit, or more generally of a class of differential
inclusions (see [20] for applications concerning the absolute stability problem).
Let us now make the following two assumptions.

n
Assumption (H1): There exists ¯x 0 ∈ R such that is finite and continuous at
¯ y 0 = C ¯x 0 .
Assumption (H2): There exists an invertible matrix P ∈ R n×n such that
T
PB = C .
We set
n
(∀x ∈ R ) : (x) = (Cx). (4.90)
Then
n
D( ) ={x ∈ R : Cx ∈ D( )}. (4.91)
n
Assumption (H1) entails that D( )
= ∅, and it is clear that : R → R ∪
{+∞} is a proper convex lower semicontinuous function. The following result
shows that our general framework can be reduced to a variational inequality of
the second kind.