Circuits, Systems, and Signal Processing
https://doi.org/10.1007/s00034-020-01449-z
Positive Realness of Second-order and High-order
Descriptor Systems
Liping Zhang
1
· Guoshan Zhang
1
Received: 11 June 2019 / Revised: 2 May 2020 / Accepted: 5 May 2020
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
This paper is concerned with the positive realness problem for second-order and high-
order descriptor systems. First, without any linearization, necessary and sufficient
conditions are established under which the second-order descriptor systems are strictly
positive real and extended strictly positive real, respectively. Applying the relations
between the positive realness and the optimal control theory, the solutions of the
proposed positive real lemma equations can be represented by the s ymmetric posi-
tive semi-definite solutions for t he second-order generalized Riccati equations. Then,
employing polynomial matrix decomposition techniques, the extended strictly positive
real lemma of high-order descriptor systems is also presented based on the original
coefficient matrices of the system. Furthermore, linear matrix inequality conditions
are given that can effectively test the positive realness and the extended strictly positive
realness of the system. Finally, three numerical examples are provided to verify the
effectiveness of the developed theoretical results.
Keywords Generalized positive real lemma · Descriptor system · Linear quadratic
optimal control · Matrix decomposition
1 Introduction
Positive realness is an important property in linear circuit network and system control
theory, which has found many applications in the realization theory of electric net-
works, absolute stability analysis and inverse optimal control problem [1,6,25,32,33].
It is well known that the descriptor system provides more natural mathematical descrip-
tions than the normal linear system in the modeling of electrical circuit systems,
B
Guoshan Zhang
Liping Zhang
1
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
Circuits, Systems, and Signal Processing
mechanical systems and other areas [13,23]. Over the last decade, the positive real-
ness analysis and positive real control problems for descriptor systems have aroused
the scholars great attention. For instance, [38] first proposed extended strictly posi-
tive real conditions for continuous-time and discrete-time descriptor systems based
on the generalized Riccati equation and inequality. [34] established a new extended
strictly positive real condition for discrete descriptor systems by means of strict LMIs
technique, which is different from the non-strict case in [38]. To improve the previous
results in [35,38] presented another version of positive real lemma for descriptor sys-
tems. But the above articles mainly focus on the impulse-free descriptor system, [7]
removed the extra assumption on the system and put forward the extension version
of Kalman-Yakubovich-Popov(KYP) lemma in the behavioral framework. Applying
the similar idea, [26,27] i nvestigated a more general KYP inequality for differential-
algebraic systems. Regarding the positive real control problem, [19] studied the robust
H
2
control problem for uncertain descriptor systems based on the bounded real lemma.
[8,28] mainly considered how to design feedback control to enable the closed-loop
system to be positive real, strictly positive real and extended strictly positive real,
respectively.
In fact, a large number of physical systems are more suitable to model by second-
order or high-order differential algebraic equations. Second-order descriptor systems
have wide applications in the vibration control of mechanical systems, the control
of electric power systems, the analysis and modelling of flexible beams systems (see
[2,9,29]). The high-order descriptor system consisting of high-order matrix differen-
tial equation with singular coefficient matrix is associated with the matrix polynomial
of degree k > 2, which also arouses broad applications in complex multibody system
with linear constraints [3] and linear multivariable control systems [17 ,31]. Concern-
ing the control of s econd-order systems, most results are focused on controllability
and observability [20,30], pole placement [4,5], eigenstructure assignment [14,36],
stability [16,39] and robust H
∞
control [37]. Regarding the control of the high-order
descriptor system, there are very limited research results; [15] dealt with the robust
pole assignment with the proportional plus derivative state feedback. [21] addressed
the controller design problem based on requirements on tracking performance and
disturbance rejection. But until now, no research results are published that consider
the positive realness for second-order and high-order descriptor systems in the existing
studies. Generally, for analyzing second-order or high-order descriptor systems, the
classical way is to convert the original system into the first-order descriptor system by
linearization approach, while this method needs a quite complicated calculation pro-
cedure, and there is also no standard way to perform the transformation. In particular,
when the leading coefficient matrix is singular, the linearization transformation may
change the infinite zero structures of the original systems [22], so that the solution
space of the original system is different from that of the linearized system. Moreover,
for the large systems, the linearized system suffers from increased dimension, which
greatly reduced the computation efficiency [4]. Therefore, to overcome these draw-
backs, the most effective way is to directly use the original system’s parameters to
study the positive realness of second-order or high-order descriptor systems.
In this paper, our main purpose is to determine the conditions under which second-
order descriptor systems are strictly positive real without linearization transformations.
Circuits, Systems, and Signal Processing
We first derive a second-order generalized positive real lemma, i.e., a necessary and suf-
ficient condition that the second-order descriptor system is strictly positive real, which
is to fill up the theory gap of positive realness for second-order descriptor systems. The
sufficiency of the proposed positive real lemma is proved by decomposing the polyno-
mial matrix of the system into the product of two block polynomial matrices; a series
of matrix constraint equations are derived by directly using the original coefficient
matrices of the system. Next, the necessity of the positive real lemma is more diffi-
cult and challenging, which is addressed from the optimal control point of view. The
solutions for the positive real equations are constructed based on non-positive definite
solutions of the generalized second-order Riccati equation and computed by solving a
corresponding LMI. Furthermore, we investigate the extended strictly positive realness
for the high-order descriptor system and establish the extended strictly positive real
lemma based on the original coefficient matrices of the high-order descriptor system.
Finally, we provide three examples to illustrate the validity of the theoretical results.
The paper is organized as follows. Section 2 briefly reviews some definitions on
positive realness for first-order descriptor system. In Sect. 3, a novel generalized pos-
itive real lemma is established for the controllable second-order descriptor system.
In Sect. 4, the extended strictly positive realness of the high-order descriptor sys-
tem is studied without any linearization. Section 5 provides three typical examples.
Conclusions are given in Sect. 6.
Notation 1 Let R
n×m
be the set of n × m real matrices. A
T
and A
stand for the
transpose and the conjugate transpose of a matrix A, respectively. j =
√
−1 means
the imaginary unit, and ¯s is the complex conjugate of s ∈ C, where C is the set of
complex numbers. A > 0(≥ 0 ) denotes a positive definite (semi-definite) symmetric
matrix. The symmetric terms in a symmetric matrix are denoted by ∗.
2 Problem Statements
Firstly, we briefly review the positive real theory for the first-order descriptor system,
which is represented by
⎧
⎨
⎩
E ˙x(t) = Ax (t) + Bu(t)
y(t) = Cx(t) + Du(t)
Ex(0) = Ex
0
,
(1)
where E, A ∈ R
n×n
, B ∈ R
n×m
, C ∈ R
m×n
and D ∈ R
m×m
are coefficient matrices,
respectively. The matrix E may be singular, i.e., rankE ≤ n. The transfer function
from the input u(t) to the output y(t) of the system (1)is
G(s) = C(sE − A)
−1
B + D. (2)
Definition 1 [38]
1. System (1) is said to be positive real (PR) if the transfer function G(s) in (2)is
analytic in Re[s] > 0 and satisfies G(s) + G
(¯s) ≥ 0.
Circuits, Systems, and Signal Processing
2. System (1) is said to be strictly positive real (SPR) if the transfer function G(s) in
(2) is analytic in Re[s]≥0 and satisfies G( jω) +G
(−jω) > 0, for ω ∈[0, ∞).
3. System (1) is said to be extended strictly positive real (ESPR) if the system (1)is
SPR and G( j∞) + G
(−j∞)>0.
Consider the following second-order descriptor system given by
⎧
⎨
⎩
A
2
¨x(t) + A
1
˙x(t) + A
0
x(t) = Bu(t)
y(t) = C
1
x(t) + C
2
˙x(t) + Du(t)
A
1
x(0) = A
1
x
0
, A
2
˙x(0) = A
2
x
1
,
(3)
where A
2
, A
1
, A
0
∈ R
n×n
, B ∈ R
n×m
, C
1
, C
2
∈ R
m×n
, D ∈ R
m×m
are coefficient
matrices, respectively. x(t) is the state vector, u(t ) is control input, y(t) is output
vector. A
1
x
0
and A
2
x
1
are admissible initial conditions. The nonzero matrix A
2
may be
singular, i.e., rankA
2
≤ n. The system is called to be regular if det( A
2
s
2
+A
1
s+A
0
) ≡
0fors ∈ C, the regularity is equivalent to the unique solvability of system (3). The
system is called to be stable if the roots of det(A
2
s
2
+ A
1
s + A
0
) = 0 lie in the
left half of the complex plane. System (3)issaidtobeR
2
-controllable if and only if
rank
A
2
s
2
+ A
1
s + A
0
B
= n, ∀s ∈ C (see [20]). In this paper, we assume that the
second-order descriptor system (3) is regular and R
2
-controllable.
The main purpose of this paper is to determine the conditions to guarantee the
positive realness of the second-order descriptor system.
3 Positive Real Lemma of Second-Order Descriptor Systems
In this section, we derive a necessary and sufficient condition for the second-order
descriptor system (3) to be strictly positive real.
Taking Laplace transformation for system (3) with zero initial conditions, we get
A(s)X (s) = BU(s)
Y (s) = C(s)X(s) + DU (s),
where the polynomial matrices A(s) = A
2
s
2
+ A
1
s + A
0
and C(s) = C
1
+C
2
s,the
corresponding transfer function is
G
2
(s) = C(s) A(s)
−1
B + D
= (C
1
+ C
2
s)( A
2
s
2
+ A
1
s + A
0
)
−1
B + D ∈ R
m×m
(s). (4)
Before analyzing main results, we firstly consider the linear quadratic optimal control
problem of system (3); it means that we need to find a feedback control u(t) to minimize
the cost function (or storage function [6])
V =
∞
0
ω(x(s), ˙x(s), u(s))ds, (5)
Circuits, Systems, and Signal Processing
with supply rate [6]
ω(x, ˙x, u) = (A
2
˙x)
T
Q
1
(A
2
˙x) + (A
2
˙x + A
1
x)Q
2
(A
2
˙x + A
1
x)
+ u
T
Ru + 2(C
1
x + C
2
˙x)
T
u, (6)
where Q
1
= Q
T
1
≥ 0, Q
2
= Q
T
2
≥ 0 and R = D + D
T
> 0.
When Q
1
= Q
2
= 0, the supply rate ω(x, ˙x, u) = 2y
T
u = u
T
Ru + 2(C
1
x +
C
2
˙x)
T
u, the quadratic performance function is defined as:
J
2
=
∞
0
u
T
(t)Ru(t) + 2(C
1
x(t) + C
2
˙x(t))
T
u(t)dt . (7)
The solutions of the optimal control problem will be helpful to construct the solutions
for matrix constraint equations in second-order generalized positive real lemma. The
following theorem characterizes the optimal control u
∗
(t) which can minimize the
performance index J
2
and stabilize system (3).
Theorem 1 Consider the controllable second-order descriptor system (3) with the
performance function ( 7). If there exist symmetric matrices X
1
, X
2
∈ R
n×n
satisfying
the following second-order generalized Riccati equation:
A
T
0
X
2
A
1
+ A
T
1
X
2
A
0
A
T
0
X
1
A
2
+ A
T
0
X
2
A
2
A
T
2
X
1
A
0
+ A
T
2
X
2
A
0
A
T
1
X
1
A
2
+ A
T
2
X
1
A
1
+
C
T
1
C
T
2
+
A
T
1
0
A
T
2
A
T
2
X
2
X
1
B
R
−1
C
1
C
2
+ B
T
X
2
X
1
A
1
A
2
0 A
2
= 0,
(8)
then, the optimal performance index is
J
∗
2
=˙x
T
0
A
T
2
X
1
A
2
˙x
0
+ ( A
2
˙x
0
+ A
1
x
0
)
T
X
2
(A
2
˙x
0
+ A
1
x
0
),
and the associated optimal control is given by
u
∗
(t) =−R
−1
C
1
C
2
+ B
T
X
2
X
1
A
1
A
2
0 A
2
x(t)
˙x(t)
. (9)
Furthermore, system (3) is stable under the optimal control (9).
Proof To solve the optimal control problem, we define the following Hamillton func-
tion for the second-order system (3)as
H
2
(x, ˙x,λ
1
,λ
2
, u) = 2(C
1
x + C
2
˙x)
T
u + u
T
Ru
+ λ
T
1
(t)(−A
1
˙x − A
0
x + Bu) + λ
T
2
(t)(−A
0
x + Bu), (10)
Circuits, Systems, and Signal Processing
where λ
1
(t), λ
2
(t) ∈ R
n
are the associated Lagrange multipliers. By the Bellman’s
principle of optimality, and Theorem 3.1 in [18], we can get the Hamilton-Jacobi-
Bellman (HJB) equation for the second-order system (3) as follows
−
∂ J
∗
2
∂t
= min
u
H
2
(x, ˙x, u(t), λ
1
(t), λ
2
(t)).
Based on the minimum value principle, the necessary conditions for J
2
to be minimal
are that
∂ H
2
∂u
= Ru +
C
1
C
2
x
˙x
+ B
T
λ
2
λ
1
= 0
A
1
A
2
0 A
2
˙x
¨x
=
∂ H
2
(x , ˙x,λ
1
,λ
2
,u)
∂λ
2
∂ H
2
(x , ˙x,λ
1
,λ
2
,u)
∂λ
1
=
−A
0
0
−A
0
−A
1
x
˙x
+
B
B
u
A
1
A
2
0 A
2
T
˙
λ
2
(t)
˙
λ
1
(t)
=−
∂ H
2
(x , ˙x,λ
1
,λ
2
,u)
∂x
∂ H
2
(x , ˙x,λ
1
,λ
2
,u)
∂ ˙x
=−2
C
T
1
C
T
2
u +
A
T
0
A
T
0
0 A
T
1
λ
2
(t)
λ
1
(t)
.
(11)
Define two auxiliary variables: Z
1
= A
2
˙x and Z
2
= A
2
˙x + A
1
x. Consider a contin-
uously differentiable function V
2
:
V
2
(Z
1
, Z
2
) = Z
T
1
X
1
Z
1
+ Z
T
2
X
2
Z
2
= (A
2
˙x)
T
X
1
(A
2
˙x) + (A
2
˙x + A
1
x)
T
X
2
(A
2
˙x + A
1
x) ≥ 0(12)
such that
0 =
∂ V
2
(Z
1
, Z
2
)
∂t
+ H
2
(x, ˙x, u
∗
(t), λ
1
(t), λ
2
(t)),
where
λ
1
(t) =
∂ V
2
(Z
1
, Z
2
)
∂ Z
1
, and λ
2
(t) =
∂ V
2
(Z
1
, Z
2
)
∂ Z
2
.
By virtue of the third equation in Eq.(11), we have
A
1
A
2
0 A
2
T
X
2
X
1
A
1
A
2
0 A
2
˙x
¨x
=
A
1
A
2
0 A
2
T
X
2
X
1
−A
0
0
−A
0
−A
1
x
˙x
+
B
B
u
=−
C
T
1
C
T
2
u +
A
0
0
A
0
A
1
T
X
2
X
1
A
1
A
2
0 A
2
x
˙x
,
Circuits, Systems, and Signal Processing
then we can get
A
T
0
X
2
A
1
+ A
T
1
X
2
A
0
A
T
0
X
1
A
2
+ A
T
0
X
2
A
2
A
T
2
X
1
A
0
+ A
T
2
X
2
A
0
A
T
1
X
1
A
2
+ A
T
2
X
1
A
1
x
˙x
+
A
1
A
2
0 A
2
T
X
2
X
1
B +
C
T
1
C
T
2
R
−1
A
1
A
2
0 A
2
T
X
2
X
1
B +
C
T
1
C
T
2
T
x
˙x
= 0,
that is
H
2
(x, ˙x, u
∗
(t)) = 0
and the optimal control is
u
∗
=−R
−1
B
T
X
2
X
1
A
1
A
2
0 A
2
+
C
1
C
2
x
˙x
,
where X
1
and X
2
are defined as the solutions of the Riccati Eq. (8), and the optimal
performance index is
J
∗
2
= V
2
(A
2
˙x
0
+ A
1
x
0
) =˙x
T
0
A
T
2
X
1
A
2
˙x
0
+ ( A
2
˙x
0
+ A
1
x
0
)
T
X
2
(A
2
˙x
0
+ A
1
x
0
).
Next, using the Lyapunov method, we consider the stability of the s econd-order system
(3) under the optimal control (9). Differentiating (12) with respect to time t leads to
˙
V
2
=−
x
T
˙x
T
A
T
0
X
2
A
1
+ A
T
1
X
2
A
0
A
T
0
X
1
A
2
+ A
T
0
X
2
A
2
A
T
2
X
1
A
0
+ A
T
2
X
2
A
0
A
T
1
X
1
A
2
+ A
T
2
X
1
A
1
x
˙x
+ 2
x
T
˙x
T
A
1
A
2
0 A
2
T
X
2
X
1
Bu,
=−
x
T
˙x
T
C
T
1
C
T
2
+
A
1
A
2
0 A
2
T
X
2
X
1
B
R
−1
×
C
1
C
2
+ B
T
X
2
X
1
A
1
A
2
0 A
2
x
˙x
≤ 0.
Therefore, the system is stable. The proof is completed.
Remark 1 According to Schur complementary lemma, the existence of the solution
for the second-order generalized Riccati equation (8) is equivalent to the feasibility of
the following LMI
⎡
⎢
⎢
⎣
A
T
0
X
2
A
1
+ A
T
1
X
2
A
0
A
T
0
(X
1
+ X
2
)A
2
C
T
1
+ A
T
1
X
2
B
A
T
2
(X
1
+ X
2
)A
0
A
T
1
X
1
A
2
+ A
T
2
X
1
A
1
C
T
2
+ A
T
2
(X
1
+ X
2
)B
B
T
X
2
A
1
+ C
1
B
T
(X
1
+ X
2
)A
2
+ C
2
−R
⎤
⎥
⎥
⎦
< 0,
Circuits, Systems, and Signal Processing
with the assumption of R
2
-controllability, the solutions of the above LMI provide
the optimal feedback control corresponding to the optimal control problem with cost
function (7).
Remark 2 Different from normal linear systems, the regularity is the special feature
of descriptor systems, which guarantees the existence and uniqueness of solutions to
second-order descriptor systems. The R
2
-controllability assumption on system (3)
implies the existence of some control u
∗
(t) such that J
∗
2
= V
2
(A
2
˙x
0
+ A
1
x
0
)>−∞
is lower bounded, i.e., there are stabilizing solutions X
1
, X
2
satisfying Riccati Eq. (8).
Motivated by the analysis though in [1], which provides the relation between the
optimal control and the positive realness for normal linear systems. In the current
study, employing a similar derivation, we can obtain similar results suitable for system
(3) in following lemmas.
Lemma 1 Assume that D + D
T
= R is positive definite. Then, there exists a lower
bound for the performance function J
2
defined by (7) if and only if the second-order
system (3)isSPR.
Lemma 2 Consider the controllable second-order system (3). If the transfer function
G
2
(s) defined by (4) with D + D
T
= R > 0 is SPR, then there exist non-positive
definite matrices X
1
, X
2
∈ R
n×n
satisfying (8).
Theorem 2 Consider the R
2
-controllable second-order descriptor system (3), and the
transfer function G
2
(s) be given by (4). Then G
2
(s) is SPR if and only if there exist
matrices P
1
= P
T
1
, P
2
= P
T
2
∈ R
n×n
,L
1
, L
2
∈ R
m×n
, W ∈ R
m×m
such that
⎧
⎨
⎩
A
T
0
P
2
A
1
+ A
T
1
P
2
A
0
A
T
0
P
1
A
2
+ A
T
0
P
2
A
2
A
T
2
P
2
A
0
+ A
T
2
P
1
A
0
A
T
2
P
1
A
1
+ A
T
1
P
1
A
2
=
L
T
1
L
T
2
L
1
L
2
> 0
A
T
1
0
A
T
2
A
T
2
P
2
P
1
B −
C
T
1
C
T
2
=−
L
T
1
L
T
2
W
D + D
T
= W
T
W ≥ 0
(13)
and rank
A( jω) B
L
1
+ jL
2
ω −W
= n + m, for all ω ∈[0, ∞).
Proof Sufficiency:
Assume that the matrix constraint equations (13) hold true, we need to prove that
the transfer function G
2
(s) is SPR.
Define a continuously differentiable function V
l
:
V
l
(A
2
˙x + A
1
x) = ( A
2
˙x)
T
P
1
(A
2
˙x) + (A
2
˙x + A
1
x)
T
P
2
(A
2
˙x + A
1
x). (14)
Differentiating (14) with respect to time t along the uncontrolled (B = 0) system’s
trajectories yields
˙
V
l
=−
x
T
˙x
T
A
T
0
P
2
A
1
+ A
T
1
P
2
A
0
A
T
0
P
1
A
2
+ A
T
0
P
2
A
2
A
T
2
P
1
A
0
+ A
T
2
P
2
A
0
A
T
1
P
1
A
2
+ A
T
2
P
1
A
1
x
˙x
.
Circuits, Systems, and Signal Processing
According to the first equation in (13), we can get
˙
V
l
=−
x
T
˙x
T
L
T
1
L
T
2
L
1
L
2
x
˙x
< 0,
which implies that system (3) is asymptotically stable and G
2
(s) is analytic in Re[s]≥
0.
Next, we need to prove that G
2
( jω) + G
2
(−jω) > 0forω ∈[0, ∞).
Let F(s) = A(s)
−1
B, then B
T
= F
(¯s) A
(¯s). Decompose the polynomial matrix
A(s) as
A(s) =
A
0
A
1
+ A
2
s
I
n
sI
n
, A
(¯s) =
I
n
¯sI
n
A
T
0
A
T
1
+ A
T
2
¯s
,
therefore
G
2
(s) + G
2
(¯s) = D + D
T
+
C
1
C
2
1
s
A(s)
−1
B + B
T
(A
−1
(¯s))
1 ¯s
C
T
1
C
T
2
= W
T
W + F
(¯s)
1 ¯s
A
T
0
A
T
1
+ A
T
2
¯s
P
2
P
1
A
1
A
2
0 A
2
1
s
F(s)
+ F
(¯s)
1 ¯s
A
T
1
0
A
T
2
A
T
2
P
2
P
1
A
0
A
1
+ A
2
s
1
s
F(s)
+ W
T
L
1
L
2
1
s
F(s) + F
(¯s)
1 ¯s
L
T
1
L
T
2
W
1
= W
T
W + W
T
L
1
L
2
1
s
F(s) + F
(¯s)
1 ¯s
L
T
1
L
T
2
W
1
+ F
(¯s)
1 ¯s
A
T
0
P
2
A
1
+ A
T
1
P
2
A
0
A
T
0
P
1
A
2
+ A
T
0
P
2
A
2
A
T
2
P
2
A
0
+ A
T
2
P
1
A
0
A
T
2
P
1
A
1
+ A
T
1
P
1
A
2
1
s
F(s)
+ F
(¯s)
1 ¯s
0 A
T
1
P
2
A
1
A
T
1
P
2
A
1
A
T
2
P
2
A
1
+ A
T
1
P
2
A
2
1
s
F(s)
+ F
(¯s)
1 ¯s
0 A
T
1
P
2
A
2
s
A
T
2
P
2
A
1
¯s 2Re[s]A
T
2
(P
1
+ P
2
)A
2
1
s
F(s)
= (W +
L
1
L
2
1
s
F(s))
(W +
L
1
L
2
1
s
F(s))
+ 2Re[s]F
(¯s) A
T
1
P
2
A
1
F(s) + 2Re[s](sF(s))
A
T
2
(P
1
+ P
2
)A
2
(sF(s))
+ 2Re[s]F
(¯s)(Re[s]( A
T
2
P
2
A
1
+ A
T
1
P
2
A
2
))F(s)
− 2Re[s]F
(¯s)(I
m
[s](A
T
2
P
2
A
1
− A
T
1
P
2
A
2
))F(s). (15)
Circuits, Systems, and Signal Processing
Let s = jω, then the last three parts on the right-hand side of Eq.(15) are equal to
zero; we can have G
2
( jω) + G
2
(−jω) ≥ 0. In addition, it follows from
rank
A( jω) B
L
1
+ jL
2
ω −W
= rank
A( jω) B
0 −(L
1
+ jL
2
ω)(A( jω))
−1
B − W
= n + m
that −(L
1
+ jL
2
ω)(A( jω))
−1
B − W is full rank, then G( j ω) + G
(−jω) > 0for
all ω ∈[0, ∞), i.e., G
2
(s) is SPR.
Necessity: When D + D
T
= R is full rank, since G
2
(s) is SPR, according to
Theorem 1 and Lemma 2, there exist non-positive definite matrices X
1
and X
2
∈ R
n×n
satisfying the following equation:
A
T
0
X
2
A
1
+ A
T
1
X
2
A
0
A
T
0
X
1
A
2
+ A
T
0
X
2
A
2
A
T
2
X
1
A
0
+ A
T
2
X
2
A
0
A
T
1
X
1
A
2
+ A
T
2
X
1
A
1
+
C
T
1
C
T
2
+
A
T
1
0
A
T
2
A
T
2
X
2
X
1
B
R
−1
C
T
1
C
T
2
+
A
T
1
0
A
T
2
A
T
2
X
2
X
1
B
T
= 0.
We choose
P
1
=−X
1
, P
2
=−X
2
,
L
1
L
2
= R
−
1
2
C
1
C
2
+ B
T
X
2
X
1
A
1
A
2
0 A
2
,
W = R
1
2
.
(16)
Due to the non-positive definite property of matrices X
1
, X
2
, we can get P
1
, P
2
≥ 0.
Therefore, we have constructed a set of matrices P
1
, P
2
, L
1
, L
2
and W that satisfy the
positive real equations (13).
When D + D
T
= R is singular, that is, rankR = r < m, we perform the
following procedure. Since R is a symmetric matrix, there exists an orthogonal matrix
U =[U
1
, U
2
]∈R
m×m
such that
U
T
RU =
U
T
1
U
T
2
R
U
1
U
2
=
R
1
0
00
,
BU =
B
1
B
2
,
C
T
1
C
T
2
U =
C
T
11
C
T
12
C
T
21
C
T
22
,
where U
1
∈ R
m×r
, U
2
∈ R
m×(m−r)
and R
1
∈ R
r×r
is a positive definite matrix.
Because G
2
(s) is SPR, from Theorem 1 and Lemma 2, there exist matrices X
1
, X
2
