# LMIs in Control/pages/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{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}} ## 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{aligned}PA+A^{T}P\geq 0\\PB=C^{T}\end{aligned}} 2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that
{\begin{aligned}AQ+QA^{T}\geq 0\\B=QC^{T}\end{aligned}} 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{aligned}PA+A^{T}P<0\\PB=C^{T}\end{aligned}} 2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that
{\begin{aligned}AQ+QA^{T}<0\\B=QC^{T}\end{aligned}} 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