# LMIs in Control/pages/Discrete Time KYP Lemma with Feedthrough

## The Concept

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

## The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$ , with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$ , where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$  and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$ .

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$
$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

## The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$  and ${\mathcal {D}}_{d}$

## LMI : Discrete-Time KYP Lemma with 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
${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$
2. There exists $Q\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$
3. There exists $P\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$
4. There exists $Q\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$

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
${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$
2. There exists $Q\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$
3. There exists $P\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$
4. There exists $Q\in {\mathcal {S}}^{n},$  where $Q>0$  such that
${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$

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