# LMIs in Control/Stability Analysis/Strong Stabilizability
## Continuous-Time Strong Stabilizability
### Definition

Consider a continuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\longrightarrow {\mathcal {L}}_{2e}$, with state-space realization ($A$, $B$, $C$, $0$), where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, and $C\in \mathbb {R} ^{p\times n}$, and it is assumed that ($A$, $B$) is stabilizable, ($A$, $C$) is detectable, and the transfer matrix $G(s)=C(s1-A)^{-1}B$ has no poles on the imaginary axis.
The system ${\mathcal {G}}$ is strongly stabilizable if there exist $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that

### LMIs

$PA+A^{T}P+ZC+C^{T}Z^{T}<0$
${\begin{bmatrix}P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&-Z&-XB\\*&-\gamma 1&0\\*&*&-\gamma 1\end{bmatrix}}<0$
where $F=-B^{T}X$ and $X\in \mathbb {S} _{n}$, $X\geq 0$ is the solution to the Lyapunov equation given by
$XA+A^{T}X-XBB^{T}X=0$
Moreover, a controller that strongly stabilizes ${\mathcal {G}}$ is given by the state-space realization
${\dot {x_{c}}}=(A+BF+P^{-1}ZC)x-P^{-1}Zu$
$y_{c}=-B^{T}Xx$

## Discrete-Time Strong Stabilizability
### Definition

Consider a discrete-time LTI system, ${\mathcal {G}}:\ell _{2e}\longrightarrow \ell _{2e}$, with state-space realization ($A_{d}$, $B_{d}$, $C_{d}$, $0$), where $A_{d}\in \mathbb {R} ^{n\times n}$, $B_{d}\in \mathbb {R} ^{n\times m}$, and $C_{d}\in \mathbb {R} ^{p\times n}$, and it is assumed that ($A_{d}$, $B_{d}$) is stabilizable, ($A_{d}$, $C_{d}$) is detectable, and the transfer matrix $G(z)=C_{d}(s1-A_{d})^{-1}B_{d}$ has no poles on the unit circle.
The system ${\mathcal {G}}$ is strongly stabilizable is there exist $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that

### LMIs

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-A_{d}^{T}ZC_{d}-C_{d}^{T}Z^{T}A_{d}&C_{d}^{T}Z^{T}\\*&-P\end{bmatrix}}<0$,
${\begin{bmatrix}N_{11}&(A_{d}+B_{d}F)^{T}Z&XB_{d}&C_{d}^{T}Z^{T}\\*&-\gamma 1&0&Z^{T}\\*&*&-\gamma 1&0\\*&*&*&-P\end{bmatrix}}<0$,
where
$N_{11}=(A_{d}+B_{d}F)^{T}P(A_{d}+B_{d}F)-P+(A_{d}+B_{d}F)^{T}ZC_{d}+C_{d}^{T}Z^{T}(A_{d}+B_{d}F)$,
$F=B_{d}^{T}X$,
$X=Y$,
$Y\in \mathbb {S} _{n}$,
$Y\geq 0$
is the solution to the discrete-time Lyapunov equation given by
$A_{d}YA_{d}^{T}-Y-B_{d}B_{d}^{T}=0$.
Moreover, a discrete-time controller that strongly stabilizes ${\mathcal {G}}$ is given by the state-space realization
$x_{c,k+1}=(A_{d}+B_{d}F+P^{-1}ZC_{d})x_{k}+P^{-1}Zu_{k}$,
$y_{c,k}=-B_{d}^{T}Xx_{k}$.

### Proof

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-A_{d}^{T}ZC_{d}-C_{d}^{T}Z^{T}A_{d}&C_{d}^{T}Z^{T}\\*&-P\end{bmatrix}}<0$
this ensures that the feedback controller defined by $x_{c,k+1}=(A_{d}+B_{d}F+P^{-1}ZC_{d})x_{k}+P^{-1}Zu_{k}$ and $y_{c,k}=-B_{d}^{T}Xx_{k}$,
renders the closed-loop system asymptotically stable and
${\begin{bmatrix}N_{11}&(A_{d}+B_{d}F)^{T}Z&XB_{d}&C_{d}^{T}Z^{T}\\*&-\gamma 1&0&Z^{T}\\*&*&-\gamma 1&0\\*&*&*&-P\end{bmatrix}}<0$
ensures that the feedback controller defined by $x_{c,k+1}=(A_{d}+B_{d}F+P^{-1}ZC_{d})x_{k}+P^{-1}Zu_{k}$ and $y_{c,k}=-B_{d}^{T}Xx_{k}$ has a finite ${\mathcal {H}}_{\infty }$ norm, and thus is asymptotically stable.