LMIs in Control/Stability Analysis/Continuous Time/Stability of Structured, Norm-Bounded Uncertainty
Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and
Θ
{\displaystyle \Theta }
The System
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x
˙
(
t
)
=
A
x
(
t
)
+
M
p
(
t
)
,
p
(
t
)
=
Δ
(
t
)
q
(
t
)
,
q
(
t
)
=
N
x
(
t
)
+
Q
p
(
t
)
,
Δ
∈
Δ
,
|
|
Δ
|
|
≤
1
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}}}
The Data
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The matrices
A
,
M
,
N
,
Q
{\displaystyle A,M,N,Q}
The LMI:
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Find
P
>
0
:
[
A
P
+
P
A
T
P
N
T
N
P
0
]
+
[
M
Θ
M
T
M
Θ
Q
T
Q
Θ
M
T
Q
Θ
Q
T
−
Θ
]
<
0
{\displaystyle {\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+{\begin{bmatrix}M\Theta M^{T}&M\Theta Q^{T}\\Q\Theta M^{T}&Q\Theta Q^{T}-\Theta \end{bmatrix}}<0\\\end{aligned}}}
Conclusion:
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The system above is quadratically stable if and only if there exists some
Θ
∈
P
Θ
and
P
>
0
such that the LMI is feasible.
{\displaystyle {\begin{aligned}{\text{The system above is quadratically stable if and only if there exists some }}\Theta \in P\Theta {\text{ and }}P>0{\text{ such that the LMI is feasible.}}\end{aligned}}}
Implementation
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Related LMIs
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External Links
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