LMIs in Control/Click here to continue/LMIs in system and stability Theory/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 ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
2. There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$

This is a special case of the KYP Lemma for QSR dissipative systems with $Q=0$, $S={\frac {1}{2}}\cdot 1$, 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 ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
2. There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$

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

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.