[<< 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 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
Related LMIs: