LMIs in Control/Matrix and LMI Properties and Tools/Nevanlinna Pick Interpolation with Scaling

Nevanlinna-Pick Interpolation with ScalingEdit

The Nevanlinna-Pick problem arises in multi-input, multi-output (MIMO) control theory, particularly   robust and optimal controller synthesis with structured perturbations.

The problem is to try and find   such that   is analytic in       and   define the scaling, and finally,
                

The SystemEdit

The scaling factor   is given as a set of   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    . The Nevanlinna LMI matrix   is defined as  . The matrix   is a diagonal matrix of the given sequence of data points  

The DataEdit

Given:
Initial sequence of data points in the complex plane   with  .
Two sequences of row vectors containing distinct target points   with  , and   with  .

The LMI: Nevanlinna- Pick Interpolation with ScalingEdit

First, implement a change of variables for   and  .

From this substitution it can be concluded that   is the smallest positive   such that there exists a   such that the following is true:

       ,

       ,

       

Conclusion:Edit

If the LMI constraints are met, then there exists a   norm-bounded optimal gain   which satisfies the scaled Nevanlinna-Pick interpolation objective defined above in Problem (1).

ImplementationEdit

Implementation requires YALMIP and Mosek. [1] - MATLAB code for Nevanlinna-Pick Interpolation.

Related LMIsEdit

Nevalinna-Pick Interpolation

External LinksEdit


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