# LMIs in Control/pages/Stabilizability LMI

Stabilizability LMI

A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair ${\displaystyle (A,B)}$ is shown below.

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\x(0)&=x_{0},\end{aligned}}}

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$, ${\displaystyle u(t)\in \mathbb {R} ^{m}}$, at any ${\displaystyle t\in \mathbb {R} }$.

## The Data

The matrices necessary for this LMI are ${\displaystyle A}$ and ${\displaystyle B}$. There is no restriction on the stability of A.

## The LMI: Stabilizability LMI

${\displaystyle (A,B)}$ is stabilizable if and only if there exists ${\displaystyle X>0}$ such that

${\displaystyle AX+XA^{T}+BB^{T}<<0}$,

where the stabilizing controller is given by

${\displaystyle u(t)=-{\frac {1}{2}}B^{T}X^{-1}x(t)}$.

## Conclusion:

If we are able to find ${\displaystyle X>0}$ such that the above LMI holds it means the matrix pair ${\displaystyle (A,B)}$ is stabilizable. In words, a system pair ${\displaystyle (A,B)}$ is stabilizable if for any initial state ${\displaystyle x(0)=x_{0}}$ an appropriate input ${\displaystyle u(t)}$ can be found so that the state ${\displaystyle x(t)}$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach ${\displaystyle x(t)=0}$ as ${\displaystyle t\rightarrow \infty }$ whereas controllability requires that the state must reach the origin in a finite time.

## Implementation

This implementation requires Yalmip and Sedumi.