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

The System:

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

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}}$

where

${\displaystyle {\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\end{aligned}}}$

where ${\displaystyle \delta _{i}(t)}$ lies in either the intervals

${\displaystyle {\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}}$

or the simplex

${\displaystyle {\begin{aligned}\delta (t)\in {\delta$ :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}}

where ${\displaystyle x\in \mathbb {R} ^{m}}$ and ${\displaystyle A\in \mathbb {R} ^{mxm}}$

The Data

The matrix A and ${\displaystyle \Delta _{A(t)}}$ are known

The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty

The system

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}}$

is Quadraticallly Stable over ${\displaystyle \Delta }$ if there exists a P > 0

Theorem
${\displaystyle (A+\Delta ,{\boldsymbol {\Delta }})}$ is quadratically stable over ${\displaystyle {\boldsymbol {\Delta }}:=Co(A_{1},...,A_{k})}$ if and only if there exists a P > 0 such that

${\displaystyle {\begin{aligned}(A+A_{i})^{T}P+P(A+A_{i})<0\quad for\quad all\quad i=1,....,k\end{aligned}}}$

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

${\displaystyle {\begin{aligned}(A+\Delta )^{T}P+P(A+\Delta )<0\quad for\quad all\quad \Delta \in {\boldsymbol {\Delta }}\end{aligned}}}$

Conclusion:

Quadratic Stability Implies Stability of trajectories for any ${\displaystyle \Delta }$ with ${\displaystyle \Delta \in {\boldsymbol {\Delta }}}$ for all ${\displaystyle t\geq 0}$
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 ${\displaystyle \Delta }$ with ${\displaystyle \Delta \in {\boldsymbol {\Delta }}}$
• Also called a parameterized LMI
• Such LMIs are not tractable.
• For polytopic sets, the LMI can be made finite.

Implementation

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

Related LMIs

• Parametric Norm Bounded Uncertain System Quadratic Stability

External Links

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.

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