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

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

\begin{align} \dot x(k)&=A_dx(k)+B_du(k),\\ y(k)&=C_dx(k), \end{align}$$

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

The Data
The matrices $$ A_d,B_d,C_d $$.

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}

{\displaystyle {\begin{aligned}{\begin{bmatrix}A^T_dPA_d-P-A_d^TZC_d-C^T_dZ^TA_d && C_d^TZ^T\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11} &&(A_d +B_dF)^TZ&& XB_d &&C_d^TZ^T\\ \end{align}$$
 * &&-P\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\
 * && -\gamma I && 0 && Z^T\\
 * && * && -\gamma I && 0\\
 * &&*&&*&&-P\end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$N_{11}=(A_d+B_dF)^TP(A_d+B_dF)-P+(A_d+B_dF)^TZC_d^TZ^T(A_d+B_dF), F=-B_d^TX,X=Y$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the discrete-time Lyapunov equation given by

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



\begin{align} \dot x_{c,k+1} = (A_d+B_dF+P^{-1}ZC_d)x_k-P^{-1}Zu_k\\ y_{c,k}=-B_d^TXx_k \end{align}$$

Implementation

 * -example code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIsinControl/StabilityAnalysis/ContinuousTime/StrongStabilizability - Continuous Time Strong Stabilizability