# LMIs in Control/pages/Continuoustimequadraticstability

LMIs in Control/pages/Continuoustimequadraticstability

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**[edit]

**The Data**[edit]

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[edit]

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[edit]

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[edit]

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

**Implementation**[edit]

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

## External Links[edit]

- 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.