[<< wikibooks] LMIs in Control/Stability Analysis/Continuous Time/Strong Stabilizability
== The System ==
Consider the continous-time LTI system,

G
:

L

2
e

→

L

2
e

{\displaystyle G:L_{2e}\rightarrow L_{2e}}
with state-space realization (A,B,C,0)

x
˙

(
t
)

=
A
x
(
t
)
+
B
u
(
t
)
,

y
(
t
)

=
C
x
(
t
)
,

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t),\end{aligned}}}
where

A
∈

R

n
×
n

{\displaystyle A\in \mathbb {R} ^{n\times n}}
,

B
∈

R

n
×
m

{\displaystyle B\in \mathbb {R} ^{n\times m}}
,

C
∈

R

p
×
n

{\displaystyle C\in \mathbb {R} ^{p\times n}}
, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable,
and the transfer matrix

G
(
s
)
=
C
(
s
1
−

A

−
1

)
B

{\displaystyle G(s)=C(s1-A^{-1})B}
has no poles on the imaginary axis.

== The Data ==
The matrices

A
,
B
,
C

{\displaystyle A,B,C}
.

== The Optimization Problem ==
The system G is strongly stabilizable if there exist

P
∈

S

n

{\displaystyle P\in \mathbb {S} ^{n}}
,

Z
∈

R

n
×
p

{\displaystyle Z\in \mathbb {R} ^{n\times p}}
, and

γ
∈

R

>
0

{\displaystyle \gamma \in \mathbb {R} _{>0}}
, where

P
>
0

{\displaystyle P>0}
, such that

P
A
+

A

T

+
Z
C
+

C

T

Z

T

<
0

[

−
P
(
A
+
B
F
)
+
(
A
+
B
F

)

T

P
+
Z
C
+

C

T

Z

T

−
Z

−
X
B

∗

−
γ
I

0

∗

∗

−
γ
I

]

<
0

{\displaystyle {\begin{aligned}PA+A^{T}+ZC+C^{T}Z^{T}<0\\{\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&&-Z&&-XB\\*&&-\gamma I&&0\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}}

== Conclusion: ==
where

F
=
−

B

T

X

{\displaystyle F=-B^{T}X}
and

X
∈

S

n

{\displaystyle X\in S_{n}}
,

X
≥
0

{\displaystyle X\geq 0}
is the solution to the Lyapunov equation given by

X
A
+

A

T

X
−
X
B

B

T

X
=
0

{\displaystyle {\begin{aligned}XA+A^{T}X-XBB^{T}X=0\end{aligned}}}
Moreover, a controller that strongly stabilizes G is given by the state-space realization

x
˙

c

=
(
A
+
B
F
+

P

−
1

Z
C
)
x
(
t
)
−

P

−
1

Z
u
(
t
)

y

C

=
−

B

T

X
x
(
t
)

{\displaystyle {\begin{aligned}{\dot {x}}_{c}=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\y_{C}=-B^{T}Xx(t)\end{aligned}}}

== Implementation ==
[1] Example Code

== Related LMIs ==
https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability

http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.