LMIs in Control/pages/LMI for the Controllability Grammian

LMI to Find the Controllability Grammian

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 ${\displaystyle x(t_{0})=x_{0}}$ and transfer it to a desired final state ${\displaystyle x(t_{f})=x_{f}}$. There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability grammian". If the grammian is full rank, the system is controllable and a state transferring control law exists.

The System

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

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

The Data

The matrices necessary for this LMI are ${\displaystyle A}$ and ${\displaystyle B}$. ${\displaystyle A}$ must be stable for the problem to be feasible.

The LMI: LMI to Determine the Controllability Grammian

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

${\displaystyle AW+WA^{T}-BB^{T}<0}$,

where ${\displaystyle W}$ is the Controllability Grammian.

Conclusion:

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

Implementation

This implementation requires Yalmip and Sedumi.