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

**The System**Edit

Given a square, discrete-time LTI system **G**: *L _{2e} --> L_{2e}* with minimal state-space realization (A

_{d}, B

_{d}, C

_{d}, D

_{d}) where

, , , and with m p. D_{d} is full rank.

The transmission zeros of are the eigenvalues of: .

**The Data**Edit

, , , and with m p. D_{d} is full rank.

**The LMI:**Edit

With the system defined above, it can be seen that **G(z)** is minimum phase if and only if there exists , where P > 0, such that:

.

If the system **G** is square (m = p), then full rank D_{d} implies D_{d}^{-1} exists and the above LMI simplifies to:

.

**Conclusion**Edit

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

**Implementation**Edit

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

**Related LMIs**Edit

## External LinksEdit

- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.