LMIs in Control/Click here to continue/Integral Quadratic Constraints/Frequency Domain

The System

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We will consider the following feedback interconnection  :

 

where   and   are exogeneous inputs.   and   are two casual operators.

The Problem

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Let   be a measurable Hermitian-valued function,   and   be a bounded casual operator.   such that

 

Then the feedback interconnection of   and   is stable.

The Data

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  is a linear time-invariant system with the state space realization:

 

where   is the state.

Any   can be factorized as   where   and  . Denote the state space realization of   by  .

A state space realization for the system   is  

The LMI

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If there exists a matrix   such that

 

then the feedback interconnection   is stable.

References

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A. Megretski and A. Rantzer, "System analysis via integral quadratic constraints," in IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819-830, June 1997, doi: 10.1109/9.587335

P. Seiler, "Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints," in IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1704-1709, June 2015, doi: 10.1109/TAC.2014.2361004

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