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: L_2e --> L_2e with minimal state-space realization (A_d, B_d, C_d, D_d) where

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

$A_{d}\in \mathbb {R} ^{nxn}$, $B_{d}\in \mathbb {R} ^{nxm}$, $C_{d}\in \mathbb {R} ^{pxn}$, and $D_{d}\in \mathbb {R} ^{pxm}$ with m $\leq$ p. D_d 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 D_d implies D_d^-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

External Links

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