# 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 ${\displaystyle H_{\infty }}$  robust and optimal controller synthesis with structured perturbations.

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

## The System

The scaling factor ${\displaystyle \mathbb {D} }$  is given as a set of ${\displaystyle mxm}$  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 ${\displaystyle Im(H({\lambda }))\geq 0}$  ${\displaystyle ({\lambda }\in {\pi }^{+})}$ . The Nevanlinna LMI matrix ${\displaystyle N}$  is defined as ${\displaystyle N=G_{in}-G_{out}}$ . The matrix ${\displaystyle A}$  is a diagonal matrix of the given sequence of data points ${\displaystyle {\lambda }_{i}\in \mathbb {C} (A=diag({\lambda }_{1},...,{\lambda }_{m})}$

## The Data

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

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

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

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

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

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

## Conclusion:

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

## Implementation

Implementation requires YALMIP and Mosek. [1] - MATLAB code for Nevanlinna-Pick Interpolation.