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{matrix} \dot{x} (t) & = A x (t) + B u (t), \\ x (0) & = x_{0}, \end{matrix}

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

A X + X A^{T} - B B^{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 $t \to \infty$ whereas controllability requires that the state must reach the origin in a finite time.