An optimization-based methodology for the multiobjective control of a large class of nonlinear systems is performed.
x
∈
R
n
{\displaystyle x\in \mathbb {R} ^{n}}
u
∈
R
n
u
{\displaystyle u\in \mathbb {R} ^{n_{u}}}
y
∈
R
n
y
{\displaystyle y\in \mathbb {R} ^{n_{y}}}
A
,
B
u
,
C
y
,
D
y
u
{\displaystyle A,B_{u},C_{y},D_{yu}}
A
(
0
)
=
0
{\displaystyle A(0)=0}
C
y
(
0
)
=
0
{\displaystyle C_{y}(0)=0}
B
u
,
D
y
u
{\displaystyle B_{u},D_{yu}}
[
A
(
x
)
B
u
(
x
)
C
y
(
x
)
D
y
u
(
x
)
]
{\displaystyle {\begin{bmatrix}A(x)&B_{u}(x)\\C_{y}(x)&D_{yu}(x)\end{bmatrix}}}
[
A
B
u
C
y
D
y
u
]
{\displaystyle {\begin{bmatrix}A&B_{u}\\C_{y}&D_{yu}\end{bmatrix}}}
[
B
p
D
y
p
]
{\displaystyle {\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}}
△
(
x
)
{\displaystyle \bigtriangleup (x)}
[
C
q
D
q
u
]
{\displaystyle {\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}}
For a given scalar \sigma > 0, we associate with the Linear differential inclusion,
x
˙
=
A
x
+
B
u
u
+
B
p
p
,
{\displaystyle {\dot {x}}=Ax+B_{u}u+B_{p}p,}
q
=
C
q
x
+
D
q
u
u
+
D
q
p
p
,
{\displaystyle q=C_{q}x+D_{qu}u+D_{qp}p,}
y
=
C
y
x
+
D
y
u
u
+
D
y
p
p
,
{\displaystyle y=C_{y}x+D_{yu}u+D_{yp}p,}
p
=△
(
t
)
q
,
‖
△
(
t
)
‖
≤
σ
−
1
,
△
(
t
)
∈
D
(
r
)
,
t
≥
0.
{\displaystyle p=\bigtriangleup (t)q,\|\bigtriangleup (t)\|\leq \sigma ^{-}1,\bigtriangleup (t)\in D(r),t\geq 0.}
The LMI : Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities
edit
For given \sigma > 0, the LDI system is quadratically stable if there exists P, S, and G such that the LMIs
P
>
0
,
S
>
0
,
G
=
−
G
T
,
S
,
G
∈
B
(
r
)
,
{\displaystyle P>0,S>0,G=-G^{T},S,G\in B(r),}
[
A
T
P
+
P
A
+
C
q
T
S
C
q
P
B
p
+
C
q
T
G
+
C
q
T
S
D
q
p
B
p
T
P
−
G
C
q
+
D
q
p
T
S
C
q
D
q
p
T
S
D
q
p
G
−
σ
2
S
+
D
q
p
T
G
−
G
D
q
p
]
<
0
{\displaystyle {\begin{bmatrix}A^{T}P+PA+C_{q}^{T}SC_{q}&PB_{p}+C{q}^{T}G+C_{q}^{T}SD_{q}p\\B_{p}^{T}P-GC_{q}+D_{qp}^{T}SC_{q}&D_{qp}^{T}SD_{qp}G-\sigma ^{2}S+D_{qp}^{T}G-GD_{q}p\end{bmatrix}}<0}
hold. Then, for every
△∈
D
(
r
)
{\displaystyle \bigtriangleup \in D(r)}
‖
△
(
t
)
‖
≤
σ
−
1
,
{\displaystyle \|\bigtriangleup (t)\|\leq \sigma ^{-}1,}
d
e
t
(
I
−
D
p
q
△
)
−
1
≠
0
,
{\displaystyle det(I-D_{pq}\bigtriangleup )^{-1}\neq 0,}
[
y
m
a
x
2
I
C
y
C
y
T
P
]
≥
0.
{\displaystyle {\begin{bmatrix}y_{max}^{2}I&C_{y}\\C_{y}^{T}&P\end{bmatrix}}\geq 0.}
[
A
Q
+
Q
A
T
+
B
p
T
B
p
T
+
B
u
Y
+
T
T
B
u
T
Q
C
q
T
+
Y
T
D
q
u
T
+
B
p
T
D
q
p
T
+
B
p
H
C
q
Q
+
D
q
u
Y
+
D
q
p
T
B
p
T
−
H
B
p
T
D
q
p
T
D
q
p
T
−
σ
2
T
+
D
q
p
H
−
H
D
q
p
T
]
{\displaystyle {\begin{bmatrix}AQ+QA^{T}+B_{p}TB_{p}^{T}+B_{u}Y+T^{T}B_{u}^{T}&QC_{q}^{T}+Y^{T}D_{qu}^{T}+B_{p}TD_{qp}^{T}+B_{p}H\\C_{q}Q+D_{qu}Y+D_{qp}TB_{p}^{T}-HB_{p}^{T}&D_{qp}TD_{qp}^{T}-\sigma ^{2}T+D_{qp}H-HD_{qp}^{T}\end{bmatrix}}}
The above LMIs provide a unified setting, as well as an efficient computational procedure, for answering (possibly conservatively) several control problems pertaining to a quite generic class of nonlinear systems. This method makes an explicit and systematic connection (via LFRs and LMIs) between robust control methods and nonlinear systems.
