User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/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 $$ H: C \to C^{p x q} $$ 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 $$ p x q $$ 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 $$ N \ge 0 $$.

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.

Related LMIs
Nevalinna-Pick Interpolation with Scaling

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