# LMIs in Control/Stability Analysis/Continuous Time/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

## The System

Given a state-space representation of a linear system

{\displaystyle {\begin{aligned}\ \rho x=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}}

Where ${\displaystyle \rho }$ represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). ${\displaystyle x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}}$ are the state, output and input vectors respectively.

## The Data

${\displaystyle A,B,C,D}$ are system matrices.

## Definition

The system , or the matrix pair ${\displaystyle (A,B)}$ is Hurwitz Stabilizable if there exists a real matrix ${\displaystyle K}$ such that ${\displaystyle (A+BK)}$ is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:

{\displaystyle {\begin{aligned}\ rank{\begin{bmatrix}sI-A&B\end{bmatrix}}&=n,\forall s\in \mathbb {C} ,Re(s)\geq 0\\\end{aligned}}}

(1)

The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

## LMI Condition

The system, or matrix pair ${\displaystyle (A,B)}$ is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix ${\displaystyle P}$ and ${\displaystyle W}$ such that:

{\displaystyle {\begin{aligned}AP+PA^{T}+BW+W^{T}B^{T}<0\\\end{aligned}}}

(2)

Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix ${\displaystyle K}$ and a matrix ${\displaystyle P>0}$ satisfying:

{\displaystyle {\begin{aligned}(A+BK)P+P(A+BK)^{T}<0\\\end{aligned}}}

(3)

Letting

{\displaystyle {\begin{aligned}W=KP\\\end{aligned}}}

(4)

Putting (4) in (3) gives us (2).

## Implementation

This implementation requires Yalmip and Mosek.

## Conclusion

Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.