# LMIs in Control/pages/Polytopic stability

An important result to determine the stability of the system with uncertainties

**The System:**[edit | edit source]

Consider the system with Affine Time-Varying uncertainty (No input)

where

where
lies in either the intervals

or the simplex

where and

**The Data**[edit | edit source]

The matrix A and are known

**The Optimization**[edit | edit source]

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**[edit | edit source]

**Conclusion:**[edit | edit source]

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**[edit | edit source]

A link to implementation of the LMI

https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

**Related LMIs**[edit | edit source]

**External Links**[edit | edit source]

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.