LMIs in Control/Stability Analysis/Strong Stabilizability

Definition
Consider a continuous-time LTI system, $$\mathcal{G}:\mathcal{L}_{2e}\longrightarrow \mathcal{L}_{2e}$$, with state-space realization ($$A$$, $$B$$, $$C$$, $$0$$), where $$A\in\mathbb{R}^{n \times n}$$, $$B\in\mathbb{R}^{n \times m}$$, and $$C\in\mathbb{R}^{p \times n}$$, 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 system $$\mathcal{G}$$ is strongly stabilizable if there exist $$P\in\mathbb{S}^{n}$$, $$Z\in\mathbb{R}^{n \times p}$$, and $$\gamma\in\mathbb{R}_{>0}$$, where $$P>0$$, such that

LMIs
$$PA+A^{T}P+ZC+C^{T} Z^{T}<0$$

$$\begin{bmatrix} P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T} & -Z & -XB \\ * & -\gamma1 & 0 \\
 * & * & -\gamma1 \end{bmatrix}<0$$

where $$F=-B^{T}X$$ and $$X\in\mathbb{S}_{n}$$, $$X\geq0$$ is the solution to the Lyapunov equation given by

$$XA+A^{T}X-XBB^{T}X=0$$

Moreover, a controller that strongly stabilizes $$\mathcal{G}$$ is given by the state-space realization

$$\dot{x_{c}}=(A+BF+P^{ -1}ZC)x-P^{ -1} Zu$$

$$y_{c}=-B^{T}Xx$$

Definition
Consider a discrete-time LTI system, $$\mathcal{G}:\ell_{2e}\longrightarrow \ell_{2e}$$, with state-space realization ($$A_{d}$$, $$B_{d}$$, $$C_{d}$$, $$0$$), where $$A_{d}\in\mathbb{R}^{n \times n}$$, $$B_{d}\in\mathbb{R}^{n \times m}$$, and $$C_{d}\in\mathbb{R}^{p \times n}$$,  and it is assumed that ($$A_{d}$$, $$B_{d}$$) is stabilizable, ($$A_{d}$$, $$C_{d}$$) is detectable, and the transfer matrix $$G(z)=C_{d}(s1-A_{d})^{ -1}B_{d}$$ has no poles on the unit circle.

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

LMIs
$$\begin{bmatrix} A^{T}_{d}PA_{d}-P-A^{T}_{d}ZC_{d}-C^{T}_{d}Z^{T}A_{d} & C^{T}_{d}Z^{T} \\
 * & -P \end{bmatrix}<0$$,

$$\begin{bmatrix} N_{11} & (A_{d}+B_{d}F)^{T}Z & XB_{d} & C^{T}_{d}Z^{T} \\
 * & -\gamma1 & 0 & Z^{T} \\ * & * & -\gamma1 & 0 \\ * & * & * & -P \end{bmatrix}<0$$,

where

$$N_{11}=(A_{d}+B_{d}F)^{T}P(A_{d}+B_{d}F)-P+(A_{d}+B_{d}F)^{T}ZC_{d}+C^{T}_{d}Z^{T}(A_{d}+B_{d}F)$$,

$$F=B^{T}_{d}X$$,

$$X=Y$$,

$$Y\in\mathbb{S}_{n}$$,

$$Y\geq0$$

is the solution to the discrete-time Lyapunov equation given by

$$A_{d}YA^{T}_{d}-Y-B_{d}B^{T}_{d}=0$$.

Moreover, a discrete-time controller that strongly stabilizes $$\mathcal{G}$$ is given by the state-space realization

$$x_{c,k+1}=(A_{d}+B_{d}F+P^{ -1}ZC_{d})x_{k}+P^{ -1}Zu_{k}$$,

$$y_{c,k}=-B^{T}_{d}Xx_{k}$$.

Proof
$$\begin{bmatrix} A^{T}_{d}PA_{d}-P-A^{T}_{d}ZC_{d}-C^{T}_{d}Z^{T}A_{d} & C^{T}_{d}Z^{T} \\
 * & -P \end{bmatrix}<0$$

this ensures that the feedback controller defined by $$x_{c,k+1}=(A_{d}+B_{d}F+P^{ -1}ZC_{d})x_{k}+P^{ -1}Zu_{k}$$ and $$y_{c,k}=-B^{T}_{d}Xx_{k}$$,

renders the closed-loop system asymptotically stable and

$$\begin{bmatrix} N_{11} & (A_{d}+B_{d}F)^{T}Z & XB_{d} & C^{T}_{d}Z^{T} \\
 * & -\gamma1 & 0 & Z^{T} \\ * & * & -\gamma1 & 0 \\ * & * & * & -P \end{bmatrix}<0$$

ensures that the feedback controller defined by $$x_{c,k+1}=(A_{d}+B_{d}F+P^{ -1}ZC_{d})x_{k}+P^{ -1}Zu_{k}$$ and $$y_{c,k}=-B^{T}_{d}Xx_{k}$$ has a finite $$\mathcal{H}_\infty$$ norm, and thus is asymptotically stable.