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.
Suppose is a Banach Algebra and . If , then exists, and furthermore,
Assuming we have an interconnected system :
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
If , then this implies stability, since the higher exponents of in the summation of will converge to , instead of blowing up to infinity.
A list of references documenting and validating the LMI.
- 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.