User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/KYP Lemma Without Feedthrough

The Concept
It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)-1B + D  is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System
Consider a contiuous-time LTI system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization (A, B, C, 0), where $$\mathcal{A} \in \mathcal{R}^{n\times n}, \mathcal{B} \in \mathcal{R}^{n\times m}, $$ and $$\mathcal{C} \in \mathcal{R}^{m\times n}, $$.



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

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

LMI : KYP Lemma without Feedthrough
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.


 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that


 * $$\begin{align}

PA + A^{T}P \ge 0 \\ PB = C^{T} \end{align}$$


 * 2. There exists $$ Q\in \mathcal{S}^{n}, $$  where $$ Q > 0 $$ such that


 * $$\begin{align}

AQ + QA^{T} \ge 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

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


 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that


 * $$\begin{align}

PA + A^{T}P < 0 \\ PB = C^{T} \end{align}$$


 * 2. There exists $$ Q\in \mathcal{S}^{n}, $$  where $$ Q > 0 $$ such that


 * $$\begin{align}

AQ + QA^{T} < 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $$ \in \mathcal{R}_{>0}.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is Positive Real.

Implementation
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs
KYP Lemma State Space Stability Discrete Time KYP Lemma with Feedthrough