[<< wikibooks] LMIs in Control/Stability of Lure's Systems
== The System ==

x
˙

(
t
)

=
A
x
(
t
)
+

B

p

p
(
t
)
+

B

w

w
(
t
)
,

z
(
t
)

=

C

z

x
(
t
)

p

i

(
t
)

=

ϕ

i

(

q

i

(
t
)
)
,
i
=
1
,
…
,

n

p

q

=

C

q

x
,

0

≤
σ

ϕ

i

(
σ
)
≤

σ

2

∀
σ
∈

R

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}}

== The Data ==
The matrices

A
,

B

p

,

B

w

,

C

q

,

C

z

{\displaystyle A,B_{p},B_{w},C_{q},C_{z}}
.

== The LMI: The Lure's System's Stability ==
The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

Find

P
>
0
,
Λ
=
d
i
a
g
(

λ

1

,
…
,

λ

n

p

)
⪰
0
,
T
=
d
i
a
g
(

τ

1

,
…
,

τ

n

p

)
⪰
0
:

[

A

⊤

P
+
P
A

P

B

p

+

A

⊤

C

q

⊤

Λ
+

C

q

⊤

T

B

p

⊤

P
+
Λ

C

q

A
+
T

C

q

Λ

C

q

B

p

+

B

p

⊤

C

q

⊤

Λ
−
2
T

]

≺
0

{\displaystyle {\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0:\\&{\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\prec 0\\\end{aligned}}}

== Implementation ==
https://codeocean.com/capsule/0232754/tree

== Conclusion ==
If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

== Remark ==
The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when

n

p

=
1

{\displaystyle n_{p}=1}
.

== External Links ==
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.

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