LMIs in Control/Stability Analysis/KYP Lemma With Feedthrough

Definition
Consider a square, continuous-time LTI system, $$\mathcal{G}:\mathcal{L}_{2e}\longrightarrow \mathcal{L}_{2e}$$, with minimal state-space realization ($$A$$, $$B$$, $$C$$, $$D$$), where $$A\in\mathbb{R}^{n \times n}$$, $$B\in\mathbb{R}^{n \times m}$$, $$C\in\mathbb{R}^{m \times n}$$, and $$D\in\mathbb{R}^{m \times m}$$.

LMI 1.1
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.


 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=0$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$.

LMIs 1.2
The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.


 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=\epsilon1$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$, where $$\epsilon\in\mathbb{R}_{>0}$$.