LMIs in Control/Stability Analysis/Stabilizability LMI

Stabilizability LMI

Stabilizability is a essentially a weaker version of the controllability condition. A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. The LMI condition for stabilizability of pair $(A,B)$ is shown below.

The System

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

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

The LMI: Stabilizability LMI

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

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

where the stabilizing controller is given by

$u(t)=-\frac{1}{2}{B}^T{X}^{-1}x(t)$.

Conclusion:

If we are able to find $X>0$ such that the above LMI holds it means the matrix pair $(A,B)$ is stabilizable. In words, a system pair $(A,B)$ is stabilizable if for any initial state $x(0)=x_0$ an appropriate input $u(t)$ can be found so that the state $x(t)$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $x(t)=0$ as $t\to \mathrm{\infty }$ whereas controllability requires that the state must reach the origin in a finite time.

Implementation

This implementation requires Yalmip and Sedumi.

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

Related LMIs

WIP: Will be linked once they have been created.

Controllability LMI

Hurwitz Stability LMI

Schur Stability LMI

Observability LMI

Detectability 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.