User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Static Output Feedback Stabilizability

In view of applications, static feedback of the full state is not feasible in general: only a few of the state variables (or a linear combination of them, $$y= Cx(t)$$, called the output) can be actually measured and re-injected into the system. '''So, we are led to the notion of static output feedback '''

The System
Consider the continuous-time LTI system, with generalized state-space realization  $$(A,B,C,0)$$



\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)\\

\end{align}$$

The Data

 * $$A\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n}$$


 * $$x\in\mathbb{R}^n, y\in\mathbb{R}^p, u\in\mathbb{R}^m$$

The Optimization Problem
This system is static output feedback stabilizable (SOFS) if there exists a matrix F such that the closed-loop system $$ \dot x = (A-BKC)x $$ (obtained by replacing $$u = -Ky$$ which means applying static output feedback) is asymptotically stable at the origin

The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability
The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:


 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - PBB^TP & PB+C^TK^T \\ KC+B^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$


 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - QC^TCQ & BK+QC^T \\ CQ^T+K^TB^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$


 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - BB^T & B+QC^TK^T \\ B^T+KCQ^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$


 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - C^TC & PBK+C^T \\ K^TB^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

Conclusion
On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix $$P$$ (or $$Q$$) and $$K$$

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Discrete time Static Output Feedback Stabilizability Static Feedback Stabilizability