LMIs in Control/Stability Analysis/KYP Lemma With Feedthrough

Definition

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

LMI 1.1

The system $𝒢$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.

1. There exists $P\in {𝕊}^n$, where $P>0$, such that $\left[\begin{array}{cc}PA+A^{T}P& PB-C^T\\ *& -(D+D^T)\end{array}\right]\le 0$.
2. There exists $Q\in {𝕊}^n$, where $Q>0$, such that $\left[\begin{array}{cc}AQ+Q{A}^T& B-Q{C}^T\\ *& -(D+D^T)\end{array}\right]\le 0$

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 $𝒢$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.

1. There exists $P\in {𝕊}^n$, where $P>0$, such that $\left[\begin{array}{cc}PA+A^{T}P& PB-C^T\\ *& -(D+D^T)\end{array}\right]\le 0$.
2. There exists $Q\in {𝕊}^n$, where $Q>0$, such that $\left[\begin{array}{cc}AQ+Q{A}^T& B-Q{C}^T\\ *& -(D+D^T)\end{array}\right]\le 0$

This is a special case of the KYP Lemma for QSR dissipative systems with $Q=ϵ1$, $S=\frac{1}{2}\cdot 1$, and $R=0$, where $ϵ\in {ℝ}_{>0}$.

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.