Page 154 - Complementarity and Variational Inequalities in Electronics

Chapter 5

The Nonregular Dynamical

System

In this chapter, we introduce a general formalism whose study has been initiated
in [21]. We also refer the reader to [5], [25], and [42] for some related recent
works.
m
Let A ∈ R n×n , B ∈ R n×m , C ∈ R m×n , and D ∈ R n×p , and let : R → R.
m
We assume that ∈ 0 (R ;R ∪{+∞}). Our aim is to introduce a system de-
scribed by a transfer function

H(s) = C(sI − A) −1 B

and a feedback branch containing a sector static nonlinearity as depicted in
Fig. 5.1.
The feedback nonlinearity that describes the graph (y,y L ) is here defined by
the model:

y L ∈ ∂ (y).
Moreover, the system is driven by inputs Du for some given function

p
u :[0,+∞[ → R ;t → u(t).
The state-space equations of such a system are given by
dx
(t) = Ax(t) − By L (t) + Du(t), (5.1)
dt
y(t) = Cx(t), (5.2)

FIGURE 5.1 Illustration of the circuit with a feedback branch.
Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00005-2
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