LMIs in Control/pages/Continuous time Quadratic stability

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as . A sufficient condition for this is the existence of a quadratic function , that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call a quadratic Lyapunov function. The System The Data The system coefficient matrix takes the form of where is a known matrix, which represents the nominal system matrix, while is the system matrix perturbation, where are known matrices, which represent the perturbation matrices. which represent the uncertain parameters in the system. is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : that is The LMI: Continuous-Time Quadratic Stability The uncertain system is quadratically stable if and only if there exists , where such that The following statements can be made for particular sets of perturbations. Case 1: Regular Polyhedron Consider the case where the set of perturbation parameters is defined by a regular polyhedron as The uncertain system is quadratically stable if and only if there exists , where such that Case 2: Polytope Consider the case where the set of perturbation parameters is defined by a polytope as The uncertain system is quadratically stable if and only if there exists , where such that Conclusion: If feasible, System is Quadratically stable for any Implementation https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities External Links 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. Return to Main Page: