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LMIs in Control/pages/Polytopic stability

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

The System:

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

where

where lies in either the intervals


or the simplex


where and

The Data

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The matrix A and are known

The Optimization

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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

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Conclusion:

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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

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A link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

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A list of references documenting and validating the LMI.


Return to Main Page:

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