# LMIs in Control/Matrix and LMI Properties and Tools/Nevanlinna Pick Interpolation with Scaling

## Nevanlinna-Pick Interpolation with Scaling

The Nevanlinna-Pick problem arises in multi-input, multi-output (MIMO) control theory, particularly $H_{\infty }$  robust and optimal controller synthesis with structured perturbations.

The problem is to try and find ${\gamma }_{opt}=inf(||DHD^{-1}||_{\infty })$  such that $H$  is analytic in $C_{+},$    $D=D^{\ast }>0,$  and $D\in \mathbb {D}$  define the scaling, and finally,
$H({\lambda }_{i})u_{i}=v_{i}$     $i=1,...,m$          $(1)$

## The System

The scaling factor $\mathbb {D}$  is given as a set of $mxm$  block-diagonal matrices with specified block structure. 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 Nevanlinna LMI matrix $N$  is defined as $N=G_{in}-G_{out}$ . The matrix $A$  is a diagonal matrix of the given sequence of data points ${\lambda }_{i}\in \mathbb {C} (A=diag({\lambda }_{1},...,{\lambda }_{m})$

## The Data

Given:
Initial sequence of data points in the complex plane ${\lambda }_{1},...,{\lambda }_{m}$  with ${\lambda }_{i}\in C_{+}{\widehat {=}}(s|Re(s)>0)$ .
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: Nevanlinna- Pick Interpolation with Scaling

First, implement a change of variables for $P=D^{\ast }D$  and $N=G_{in}-G_{out}$ .

From this substitution it can be concluded that ${\gamma }_{opt}$  is the smallest positive ${\gamma }$  such that there exists a $P>0,P\in \mathbb {D}$  such that the following is true:

$A^{\ast }G_{in}+G_{in}A-U^{\ast }PU=0$ ,

$A^{\ast }G_{out}+G_{out}A-V^{\ast }PV=0$ ,

${\gamma }^{2}G_{in}-G_{out}\geq 0$

## Conclusion:

If the LMI constraints are met, then there exists a $H_{\infty }$  norm-bounded optimal gain ${\gamma }$  which satisfies the scaled Nevanlinna-Pick interpolation objective defined above in Problem (1).

## Implementation

Implementation requires YALMIP and Mosek.  - MATLAB code for Nevanlinna-Pick Interpolation.