LMIs in Control/Matrix and LMI Properties and Tools/Tangential Nevanlinna Pick

Tangential Nevanlinna-PickEdit

The Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly   robust and optimal control.

The problem is to try and find a function   which is analytic in   and satisfies
              with                  

The SystemEdit

  is a set of   matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if    .

The DataEdit

Given:
Initial sequence of data points on real axis   with  ,
And two sequences of row vectors containing distinct target points   with  , and   with  .

The LMI: Tangential Nevanlinna- PickEdit

Problem (1) has a solution if and only if the following is true:

Nevanlinna-Pick ApproachEdit

        

Lyapunov ApproachEdit

N can also be found using the Lyapunov equation:

      

where  

The tangential Nevanlinna-Pick problem is then solved by confirming that  .

Conclusion:Edit

If   is positive (semi)-definite, then there exists a norm-bounded analytic function,   which satisfies              with  

ImplementationEdit

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

Related LMIsEdit

Nevalinna-Pick Interpolation with Scaling

External LinksEdit


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