# 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 $H_{\infty }$  robust and optimal control.

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

## The System

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

## The Data

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

## The LMI: Tangential Nevanlinna- Pick

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

##### Nevanlinna-Pick Approach

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

##### Lyapunov Approach

N can also be found using the Lyapunov equation:

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

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

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

## Conclusion:

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

## Implementation

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