Page 158 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 149

5.1 KALMAN–YAKUBOVICH–POPOV LEMMA

Let us discuss the relationship between Assumption (G1) and the famous
Kalman–Yakubovich–Popov lemma. Let A ∈ R n×n , B ∈ R n×m , and C ∈ R m×n .
We say that the representation (A,B,C) is minimal if (A,B) is controllable
2
and (A,C) is observable, that is, the matrices (B AB A B ... A n−1 B) and
2
) have full rank. Let us now consider the real rational
(C CA CA ... CA n−1 T
matrix-valued transfer function H : C → C m×m given by
H(s) = C(sI n − A) −1 B. (5.16)

Definition 4. We say that H is positive real if H is analytic in

+
C ={s ∈ C : Re[s] > 0}
T
+
and for all s ∈ C , the matrix H(s) + H (¯s) is positive semidefinite (¯s denotes
the conjugate of s).
The following result is called the Kalman–Yakubovich–Popov lemma [58],
[77], [85] (see also e.g. [27], [78]).
Lemma 1. Let (A,B,C) be a minimal realization, and let H be defined
in (5.16). The transfer function matrix H is positive real if and only if there exist
a symmetric and positive definite matrix P ∈ R n×n and a matrix L ∈ R n×m such
that
T
T
PA + A P =−LL ,
(5.17)
T
PB = C .
So, if the realization (A,B,C) is minimal and the transfer function H is
positive real, then there exist a symmetric and positive definite matrix P ∈ R n×n
T
T
T
and a matrix L ∈ R n×m such that PA + A P =−LL and PB = C .Let R
T
be the symmetric square root of P , that is, R = R , R is positive definite, and
2
2
T
R = P . Then we see that B R = C and thus
T
R −2 C = B.
It results that assumption (G1) holds.
5.1.1 A Nonregular Circuit
Let us consider the following dynamics that corresponds to the circuit depicted
in Fig. 5.2:

1 t

L 3 x (t)+ x 2 (s)ds +R 3 x 2 (t)+R 1 (x 2 (t)−x 3 (t))+y L,1 (t)+y L,2 (t) = 0
2
C 4 0