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)

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

{\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\end{aligned}} where $\delta _{i}(t)$ lies in either the intervals

{\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}} or the simplex

{\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}} where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$ The Data

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

The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty

The system

{\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}} is Quadraticallly Stable over $\Delta$ if there exists a P > 0

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

{\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

{\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 $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$ 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 $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ • 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