[<< wikibooks] LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability

LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability
The System:
Consider the system with Affine Time-Varying
uncertainty (No input)
x
˙
(
t
)
=
(
A
0
+
Δ
A
(
t
)
)
x
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}}
where
Δ
A
(
t
)
=
A
1
δ
1
(
t
)
+
.
.
.
.
+
A
k
δ
k
(
t
)
{\displaystyle {\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\end{aligned}}}
where
δ
i
(
t
)
{\displaystyle \delta _{i}(t)}
lies in either the intervals
δ
i
∈
[
δ
i
−
,
δ
i
+
]
{\displaystyle {\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\end{aligned}}}
or the simplex
δ
(
t
)
∈
δ
:
Σ
α
i
=
1
,
α
≥
0
{\displaystyle {\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}}
The LMI: Polytopic Quadratic Stability
The system is Quadratically Stable over
Δ
{\displaystyle \Delta }
if there exists a P > 0
(
A
+
Δ
A
)
T
+
P
(
A
+
Δ
A
)
<
0
{\displaystyle (A+\Delta A)^{T}+P(A+\Delta A)<0}
for all
Δ
A
∈
Δ
{\displaystyle \Delta A\in \Delta }
Conclusion:
Interpretation of the results of the LMI
Implementation
A link to CodeOcean or other online implementation of the LMI
Related LMIs
Links to other closely-related LMIs
External Links
A list of references documenting and validating the LMI.
LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.Return to Main Page:
Related LMIs: