Stabilizability LMI

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. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair ${\displaystyle (A,B)}$ is shown below.

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 ${\displaystyle B}$. There is no restriction on the stability of A.

The LMI: Stabilizability LMI

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

${\displaystyle AX+XA^{T}+BB^{T}<<0}$,

where the stabilizing controller is given by

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

Conclusion:

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

Hurwitz Stability 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.

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