User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

The System
Given a state-space representation of a linear system

\begin{align} \ \rho x = Ax + Bu \\ \ y = Cx + Du \\ \end{align}$$

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

The Data
$$ A,B,C,D $$ are system matrices.

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

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

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

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

Letting

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

Implementation
This implementation requires Yalmip and Mosek.
 * https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Hurwitz_Stabilizability.m

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