LMIs in Control/pages/Continuous time Quadratic stability

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 SystemEdit

 

The DataEdit

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 StabilityEdit

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 PolyhedronEdit

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: PolytopeEdit

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:Edit

If feasible, System is Quadratically stable for any  

ImplementationEdit

https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities

External LinksEdit


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