It is assumed in the Lemma that the state-space representation (Ad , Bd , Cd , Dd ) is minimal. Then Positive Realness (PR) of the transfer function Cd (SI − Ad )-1 Bd + Dd is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad , Bd , Cd , Dd )=(−α, 0, 0, 1), with α > 0 . The transfer function is H(s) = 0 that is PR
Consider a discrete-time LTI system,
G
:
l
2
e
→
l
2
e
{\displaystyle {\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}}
(
A
d
,
B
d
,
C
d
,
D
d
)
{\displaystyle ({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})}
A
d
∈
R
n
×
n
,
B
d
∈
R
n
×
m
,
C
d
∈
R
p
×
n
,
{\displaystyle {\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},}
D
d
∈
R
p
×
m
{\displaystyle {\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}}
x
(
k
+
1
)
=
A
d
x
(
k
)
+
B
d
u
(
k
)
{\displaystyle x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)}
y
(
k
)
=
C
d
x
(
k
)
+
D
d
u
(
k
)
,
k
=
0
,
1...
{\displaystyle y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...}
The matrices
A
d
,
B
d
,
C
d
{\displaystyle {\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}}
D
d
{\displaystyle {\mathcal {D}}_{d}}
LMI : Discrete-Time KYP Lemma with Feedthrough
edit
The system
G
{\displaystyle {\mathcal {G}}}
1. There exists
P
∈
S
n
,
{\displaystyle P\in {\mathcal {S}}^{n},}
P
>
0
{\displaystyle P>0}
[
A
d
T
P
A
d
−
P
A
d
T
P
B
d
−
C
d
T
(
A
d
T
P
B
d
−
C
d
T
)
T
B
d
T
P
B
d
−
(
D
d
T
+
D
d
)
]
≤
0.
{\displaystyle {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.}
2. There exists
Q
∈
S
n
,
{\displaystyle Q\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
A
d
Q
A
d
T
−
Q
A
d
Q
C
d
T
−
B
d
(
A
d
Q
C
d
T
−
B
d
)
T
C
d
P
C
d
T
−
(
D
d
T
+
D
d
)
]
≤
0.
{\displaystyle {\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.}
3. There exists
P
∈
S
n
,
{\displaystyle P\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
P
P
A
d
P
B
d
(
P
A
d
)
T
P
C
d
T
(
P
B
d
)
T
C
d
D
d
T
+
D
d
]
≥
0.
{\displaystyle {\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.}
4. There exists
Q
∈
S
n
,
{\displaystyle Q\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
Q
A
d
Q
B
d
(
A
d
Q
)
T
Q
Q
C
d
T
(
B
d
)
T
(
Q
C
d
T
)
T
D
d
T
+
D
d
]
≥
0.
{\displaystyle {\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.}
This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0 .
The system
G
{\displaystyle {\mathcal {G}}}
1. There exists
P
∈
S
n
,
{\displaystyle P\in {\mathcal {S}}^{n},}
P
>
0
{\displaystyle P>0}
[
A
d
T
P
A
d
−
P
A
d
T
P
B
d
−
C
d
T
(
A
d
T
P
B
d
−
C
d
T
)
T
B
d
T
P
B
d
−
(
D
d
T
+
D
d
)
]
<
0.
{\displaystyle {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.}
2. There exists
Q
∈
S
n
,
{\displaystyle Q\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
A
d
Q
A
d
T
−
Q
A
d
Q
C
d
T
−
B
d
(
A
d
Q
C
d
T
−
B
d
)
T
C
d
P
C
d
T
−
(
D
d
T
+
D
d
)
]
<
0.
{\displaystyle {\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.}
3. There exists
P
∈
S
n
,
{\displaystyle P\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
P
P
A
d
P
B
d
(
P
A
d
)
T
P
C
d
T
(
P
B
d
)
T
C
d
D
d
T
+
D
d
]
>
0.
{\displaystyle {\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.}
4. There exists
Q
∈
S
n
,
{\displaystyle Q\in {\mathcal {S}}^{n},}
Q
>
0
{\displaystyle Q>0}
[
Q
A
d
Q
B
d
(
A
d
Q
)
T
Q
Q
C
d
T
(
B
d
)
T
(
Q
C
d
T
)
T
D
d
T
+
D
d
]
>
0.
{\displaystyle {\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.}
This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0 . where ε
∈
R
>
0
.
{\displaystyle \in {\mathcal {R}}_{>0}.}
If there exist a positive definite
P
{\displaystyle P}
Q,S and R matrices then the system
G
{\displaystyle {\mathcal {G}}}
Positive Real .
1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
