User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Stabilizability

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 $$(A,B)$$ is shown below.

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

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

The LMI: Stabilizability LMI
$$(A,B)$$ is stabilizable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T+BB^T<<0$$,

where the stabilizing controller is given by
 * $$u(t)=-\frac{1}{2} B^TX^{-1}x(t)$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,B)$$ is stabilizable. In words, a system pair $$(A,B)$$ is stabilizable if for any initial state $$x(0)=x_0$$ an appropriate input $$u(t)$$ can be found so that the state $$x(t)$$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $$x(t)=0$$ as $$t\rightarrow \infty$$ whereas controllability requires that the state must reach the origin in a finite time.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Stabilizability_LMI.m

Related LMIs
Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI