# LMIs in Control/pages/Polytopic stability

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 ${\displaystyle t\geq 0}$
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