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. Theorem Suppose is a Banach Algebra and . If , then exists, and furthermore, Proof 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 Conclusion If , then this implies stability, since the higher exponents of in the summation of will converge to , instead of blowing up to infinity. External Links 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. Return to Main Page: