User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/System Zeros with Feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $$ D $$ as full rank.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,D) $$



\begin{align} \dot x(t) = Ax(t) + Bu(t)\\ y(t) = Cx(t) + Du \end{align} $$

The Data
The matrices needed as inputs are:



\begin{align} A \in \mathbb{R}^{n \times n} \\ B \in \mathbb{R}^{n \times m} \\ N \in \mathbb{R}^{p \times n} \\ \end{align}$$ In this case, $$ m \leq p $$

The LMI: System Zeros with feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B + D $$ are the eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that



\begin{align} P(A-B(D^T D)^{-1} D^T C) + (A-B(D^T D)^{-1} D^T C)^T P < 0 \\ \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$ then gives the zeros of the system.

Related LMIs
LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

Implementation
https://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough