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 $H_{\mathrm{\infty }}$ 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 Template:Indent $H({\lambda }_i)u_i=v_i,$ Template:Spaces $i=1,...,m$ Template:Spaces with $||H|{|}_{\mathrm{\infty }}\le 1$Template:Spaces$(1)$

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 $Im(H(\lambda ))\ge 0$ $(\lambda \in {\pi }^{+})$.

The Data

Given: Template:Indent Initial sequence of data points on real axis ${\lambda }_1,...,{\lambda }_m$ with ${\lambda }_i\in C_{+}\widehat{=}(s|Re(s)>0)$,Template:Indent 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

Template:Spaces$N_{ij}=$ $\frac{u_i^{\ast }{u}_j-v_i^{\ast }{v}_j}{{\lambda }_i^{\ast }+{\lambda }_j}$ Template:Indent

Lyapunov Approach

N can also be found using the Lyapunov equation:

Template:Spaces$A^{\ast }N+NA-(U^{\ast }U-V^{\ast }V)=0$

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 $H({\lambda }_i)u_i=v_i,$Template:Spaces $i=1,...,m$ Template:Spaces with $||H|{|}_{\mathrm{\infty }}\le 1$

Implementation

Implementation requires YALMIP and a linear solver such as sedumi. [1] - MATLAB code for Tangential Nevanlinna-Pick Problem.

Related LMIs

Nevalinna-Pick Interpolation with Scaling

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 $H_{\mathrm{\infty }}$ 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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