LMIs in Control/pages/Detectability LMI

Detectability LMI

Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $(A,C)$ is shown below.

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t),\\ y(t)& =Cx(t)+Du(t),\\ x(0)& =x_0,\end{array}$

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 $C$. There is no restriction on the stability of $A$.

The LMI: Detectability LMI

$(A,B)$ is detectable if and only if there exists $X>0$ such that

$AX+X{A}^T-B^{T}B<0$.

Conclusion:

If we are able to find $X>0$ such that the above LMI holds it means the matrix pair $(A,C)$ is detectable. In words, a system pair $(A,C)$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $u(t)$ and output $y(t)$.

Implementation

This implementation requires Yalmip and Sedumi.

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

Related LMIs

Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian 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.