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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 $mxm$ 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 $Im(H({\lambda }))\geq 0$ $({\lambda }\in {\pi }^{+})$. 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. [1] - MATLAB code for Nevanlinna-Pick Interpolation.

Related LMIs

Nevalinna-Pick Interpolation

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 $H_{\infty }$.

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