LMIs in Control/Matrix and LMI Properties and Tools/Nevanlinna Pick Interpolation with Scaling
Nevanlinna-Pick Interpolation with Scaling
[edit | edit source]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 System
[edit | edit source]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 Data
[edit | edit source]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 Scaling
[edit | edit source]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 | edit source]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).
Implementation
[edit | edit source]Implementation requires YALMIP and Mosek. [1] - MATLAB code for Nevanlinna-Pick Interpolation.
Related LMIs
[edit | edit source]External Links
[edit | edit source]- 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 .
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[edit | edit source]LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control