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. [1] - MATLAB code for Tangential Nevanlinna-Pick Problem.

External Links

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Generalized Interpolation in $H_{\infty }$ by Donald Sarason.
Tangential Nevanlinna-Pick Interpolation Problem With Boundary Nodes in the Nevanlinna Class And The Related Moment Problem by Yong Jian Hu and Xiu Ping Zhang.

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