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{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 $$B$$. $$A$$ 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: $$W=\int_0^{\infty} e^{As}BB^Te^{A^Ts} ds$$, with control law $$u(t)=B^TW^{-1}x(t)$$ that will transfer the given initial state $$x(t_0)=x_0$$ to a desired final state $$x(t_f)=x_f$$.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Observability Grammian LMI