# LMIs in Control/Matrix and LMI Properties and Tools/Tangential Nevanlinna Pick

## Tangential Nevanlinna-Pick

The Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly ${\displaystyle H_{\infty }}$  robust and optimal control.

The problem is to try and find a function ${\displaystyle H:C\to C^{pxq}}$  which is analytic in ${\displaystyle C_{+}}$  and satisfies
${\displaystyle H({\lambda }_{i})u_{i}=v_{i},}$      ${\displaystyle i=1,...,m}$        with ${\displaystyle ||H||_{\infty }\leq 1}$                ${\displaystyle (1)}$

## The System

${\displaystyle N_{(ij)}}$  is a set of ${\displaystyle pxq}$  matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if ${\displaystyle Im(H({\lambda }))\geq 0}$  ${\displaystyle ({\lambda }\in {\pi }^{+})}$ .

## The Data

Given:
Initial sequence of data points on real axis ${\displaystyle {\lambda }_{1},...,{\lambda }_{m}}$  with ${\displaystyle {\lambda }_{i}\in C_{+}{\widehat {=}}(s|Re(s)>0)}$ ,
And two sequences of row vectors containing distinct target points ${\displaystyle u_{1},....,u_{m}}$  with ${\displaystyle u_{i}\in C^{q}}$ , and ${\displaystyle v_{1},...,v_{m}}$  with ${\displaystyle v_{i}\in C^{p},i=1,...,m}$ .

## The LMI: Tangential Nevanlinna- Pick

Problem (1) has a solution if and only if the following is true:

##### Nevanlinna-Pick Approach

${\displaystyle N_{ij}=}$  ${\displaystyle {\frac {u_{i}^{\ast }u_{j}-v_{i}^{\ast }v_{j}}{\ {\lambda }_{i}^{\ast }+{\lambda }_{j}}}}$

##### Lyapunov Approach

N can also be found using the Lyapunov equation:

${\displaystyle A^{\ast }N+NA-(U^{\ast }U-V^{\ast }V)=0}$

where ${\displaystyle A=diag({\lambda }_{1},...,{\lambda }_{m}),U=[u_{1}...u_{m}],V=[v1...vm]}$

The tangential Nevanlinna-Pick problem is then solved by confirming that ${\displaystyle N\geq 0}$ .

## Conclusion:

If ${\displaystyle N_{(ij)}}$  is positive (semi)-definite, then there exists a norm-bounded analytic function, ${\displaystyle H}$  which satisfies ${\displaystyle H({\lambda }_{i})u_{i}=v_{i},}$     ${\displaystyle i=1,...,m}$        with ${\displaystyle ||H||_{\infty }\leq 1}$

## Implementation

Implementation requires YALMIP and a linear solver such as sedumi. [1] - MATLAB code for Tangential Nevanlinna-Pick Problem.