LMIs in Control/pages/Polytopic stability

An important result to determine the stability of the system with uncertainties The System: Consider the system with Affine Time-Varying uncertainty (No input) where where lies in either the intervals or the simplex where and The Data The matrix A and are known The Optimization The Definitions: Quadratic Stability for Dynamic Uncertainty The system is Quadraticallly Stable over if there exists a P > 0 Theorem is quadratically stable over if and only if there exists a P > 0 such that The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope. Quadratic Stability MUST be expressed as an LMI The LMI Conclusion: Quadratic Stability Implies Stability of trajectories for any with for all Quadratic Stability is CONSERVATIVE. There are Stable System which are not Quadratically stable. Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI" Meaning it represents an infinite number of LMI constraints. One for each possible value with Also called a parameterized LMI Such LMIs are not tractable. For polytopic sets, the LMI can be made finite. Implementation A link to implementation of the LMI https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m Related LMIs Parametric Norm Bounded Uncertain System Quadratic Stability 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. LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd. Return to Main Page: