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 $(A,B)$ is shown below.

The System

{\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

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^{T}X^{-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.