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

$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{p x n}$$, and $$D_d \in \mathbb{R}^{p x m}$$ with m $$\leq$$ p. Dd is full rank.

The transmission zeros of $$G(z) = C_d(z1 - A_d)^{-1} B_d + D_d $$ are the eigenvalues of: $$A_d - B_d (D_d^T D_d)^{-1} D_d^T C_d$$.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{p x n}$$, and $$D_d \in \mathbb{R}^{p x m}$$ with m $$\leq$$ 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 $$P \in \mathbb{S}^{n}$$, where P > 0, such that:

$$\begin{bmatrix} P & (A_d - B_d(D_d^T D_d)^{-1} D_d^T C_d)P \\ * & P \end{bmatrix} > 0$$.

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

$$\begin{bmatrix} P & (A_d - B_d D_d^{-1} C_d)P \\ * & P \end{bmatrix} > 0$$.

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.

Related LMIs

 * System Zeros without feedthrough
 * System zeros with feedthrough