# LMIs in Control/Matrix and LMI Properties and Tools/Discrete Time/Discrete Time System Zeros With Feedthrough

## The System

Given a square, discrete-time LTI system G: L2e --> L2e with minimal state-space realization (Ad, Bd, Cd, Dd) where

${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$, ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$, ${\displaystyle C_{d}\in \mathbb {R} ^{pxn}}$, and ${\displaystyle D_{d}\in \mathbb {R} ^{pxm}}$ with m ${\displaystyle \leq }$ p. Dd is full rank.

The transmission zeros of ${\displaystyle G(z)=C_{d}(z1-A_{d})^{-1}B_{d}+D_{d}}$ are the eigenvalues of: ${\displaystyle A_{d}-B_{d}(D_{d}^{T}D_{d})^{-1}D_{d}^{T}C_{d}}$.

## The Data

${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$, ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$, ${\displaystyle C_{d}\in \mathbb {R} ^{pxn}}$, and ${\displaystyle D_{d}\in \mathbb {R} ^{pxm}}$ with m ${\displaystyle \leq }$ p. Dd is full rank.

## The LMI:

With the system defined above, it can be seen that G(z) is minimum phase if and only if there exists ${\displaystyle P\in \mathbb {S} ^{n}}$, where P > 0, such that:

${\displaystyle {\begin{bmatrix}P&(A_{d}-B_{d}(D_{d}^{T}D_{d})^{-1}D_{d}^{T}C_{d})P\\*&P\end{bmatrix}}>0}$.

If the system G is square (m = p), then full rank Dd implies Dd-1 exists and the above LMI simplifies to:

${\displaystyle {\begin{bmatrix}P&(A_{d}-B_{d}D_{d}^{-1}C_{d})P\\*&P\end{bmatrix}}>0}$.

## Conclusion

With the LMI constructed above, the system zeros for a discrete-time LTI system with feedthrough can be found and verified.

## Implementation

The LMI can be implemented using a platform like YALMIP along with an LMI solver such as MOSEK to compute the result.