LMIs in Control/Stability Analysis/Stabilizability LMI

Stabilizability LMI

Stabilizability is a essentially a weaker version of the controllability condition. 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. The LMI condition for stabilizability of pair $(A,B)$ is shown below.

The System edit

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

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

The Data edit

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

The LMI: Stabilizability LMI edit

$(A,B)$ 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^{T}X^{-1}x(t)

.

Conclusion: edit

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.