User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Polytopic Quadratic 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{align} \dot x(t)&=(A_0+\Delta A(t))x(t)\\ \end{align}$$

where



\begin{align} \Delta A(t) = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \end{align}$$ where $$

\delta_i (t) $$ lies in either the intervals



\begin{align} \delta_i \in [\delta_i^-,\delta_i^+] \\ \end{align} $$ or the simplex

\begin{align} \delta(t) \in { \delta : \Sigma \alpha_i = 1, \alpha \geq 0 } \end{align} $$

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{align} \dot x(t)&=(A_0+\Delta A(t))x(t)\\ \end{align}$$ is Quadraticallly Stable over $$ \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{align} (A+A_i)^TP + P(A+A_i) < 0 \quad

for \quad all \quad i = 1,....,k \end{align} $$ 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{align} (A+\Delta)^TP + P(A+\Delta ) < 0 \quad for \quad all \quad

\Delta \in \boldsymbol{\Delta} \end{align} $$

Conclusion:
Quadratic Stability Implies Stability of trajectories for any $$ \Delta $$ with $$ \Delta \in \boldsymbol{\Delta} $$ for all $$ t \geq 0 $$ 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 $$ \Delta $$ 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 https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

Related LMIs

 * Parametric Norm Bounded Uncertain System Quadratic Stability