LMIs in Control/pages/LMI for the Controllability Grammian

LMI to Find the Controllability Gramian

Being able to adjust a system in a desired manor using feedback and sensors is a very important part of control engineering. However, not all systems are able to be adjusted. This ability to be adjusted refers to the idea of a "controllable" system and motivates the necessity of determining the "controllability" of the system. Controllability refers to the ability to accurately and precisely manipulate the state of a system using inputs. Essentially if a system is controllable then it implies that there is a control law that will transfer a given initial state $x(t_{0})=x_{0}$ and transfer it to a desired final state $x(t_{f})=x_{f}$ . There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability Gramian". If the Gramian is full rank, the system is controllable and a state transferring control law exists.

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 $B$ . $A$ must be stable for the problem to be feasible.

The LMI: LMI to Determine the Controllability Gramian

$(A,B)$ is controllable if and only if $W>0$ is the unique solution to

AW+WA^{T}+BB^{T}<0

,

where $W$ is the Controllability Gramian.

Conclusion:

The LMI above finds the controllability Gramian $W$ of the system $(A,B)$ . If the problem is feasible and a unique $W$ can be found, then we also will be able to say the system is controllable. The controllability Gramian of the system $(A,B)$ can also be computed as: $W=\int _{0}^{\infty }e^{As}BB^{T}e^{A^{T}s}ds$ , with control law $u(t)=B^{T}W^{-1}x(t)$ that will transfer the given initial state $x(t_{0})=x_{0}$ to a desired final state $x(t_{f})=x_{f}$ .