LMIs in Control/pages/Small Gain Theorem

LMIs in Control/Matrix and LMI Properties and Tools/Small Gain Theorem

The Small Gain Theorem provides a sufficient condition for the stability of a feedback connection.


TheoremEdit

Suppose   is a Banach Algebra and  . If  , then   exists, and furthermore,

                     

ProofEdit

Assuming we have an interconnected system  :

  and,  


The above equations can be represented in matrix form as

 


Making   the subject, we then have:

 


If   is well-behaved, then the interconnection is stable. For   to be well-behaved,   must be finite.

Hence, we have  

  and   for the higher exponents of   to converge to  


ConclusionEdit

If  , then this implies stability, since the higher exponents of   in the summation of   will converge to  , instead of blowing up to infinity.


External LinksEdit

A list of references documenting and validating the LMI.


Return to Main Page:Edit