User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time strong stabilizability

The System
Consider the continous-time LTI system, $$G:L_{2e}\rightarrow L_{2e}$$ with state-space realization (A,B,C,0)

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

where $$A\in \R^{n\times n}$$, $$B\in \R^{n\times m}$$, $$C\in \R^{p\times n}$$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $$G(s)=C(s1-A^{-1})B$$ has no poles on the imaginary axis.

The Data
The matrices $$ A,B,C $$.

The Optimization Problem
The system G is strongly stabilizable if there exist $$P \in \mathbb S^n$$, $$Z\in \R^{n\times p}$$, and $$\gamma\in \R_{>0}$$, where $$P > 0$$, such that



\begin{align} PA+A^T+ZC+C^TZ^T<0\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^TP+ZC+C^TZ^T&&-Z&&-XB\\ \end{align}$$
 * &&-\gamma I && 0\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$F=-B^TX$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the Lyapunov equation given by

\begin{align} XA+A^TX-XBB^TX=0 \end{align}$$ Moreover, a controller that strongly stabilizes G is given by the state-space realization



\begin{align} \dot x_c = (A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\ y_C=-B^TXx(t) \end{align}$$

Implementation

 * Example Code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability