User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/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 $$ m x m $$ 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.

Related LMIs
Nevalinna-Pick Interpolation

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