LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Tangential Nevanlinna Pick
Tangential Nevanlinna-Pick[edit | edit source]
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 System[edit | edit source]
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 Data[edit | edit source]
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- Pick[edit | edit source]
Problem (1) has a solution if and only if the following is true:
Nevanlinna-Pick Approach[edit | edit source]
Lyapunov Approach[edit | edit source]
N can also be found using the Lyapunov equation:
where
The tangential Nevanlinna-Pick problem is then solved by confirming that .
Conclusion:[edit | edit source]
If is positive (semi)-definite, then there exists a norm-bounded analytic function, which satisfies with
Implementation[edit | edit source]
Implementation requires YALMIP and a linear solver such as sedumi. [1] - MATLAB code for Tangential Nevanlinna-Pick Problem.
Related LMIs[edit | edit source]
Nevalinna-Pick Interpolation with Scaling
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 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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LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control