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^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$


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


 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 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^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 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^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 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^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix} < 0.$$


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


 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ 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^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ 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^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ 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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough