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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 $x(t_{0})=x_{0}$ 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 grammian". If the grammian 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 Grammian

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

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

where $W$ is the Controllability Grammian.

Conclusion:

The LMI above finds the controllability grammian $W$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 grammian 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 $x(t_{f})=x_{f}$.

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

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
• A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.