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)

$\begin{array}{cc}\dot{x}(t)& =(A_0+\mathrm{\Delta }A(t))x(t)\\ \end{array}$

where

$\begin{array}{c}\mathrm{\Delta }A(t)=A_1{\delta }_1(t)+....+A_k{\delta }_k(t)\\ \end{array}$

where ${\delta }_i(t)$ lies in either the intervals

$\begin{array}{c}{\delta }_i\in [{\delta }_i^{-},{\delta }_i^{+}]\\ \end{array}$

or the simplex

$\begin{array}{c}\delta (t)\in \delta :\mathrm{\Sigma }{\alpha }_i=1,\alpha \ge 0\end{array}$

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

The Data

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

The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty

The system

$\begin{array}{cc}\dot{x}(t)& =(A_0+\mathrm{\Delta }A(t))x(t)\\ \end{array}$

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

Theorem
$(A+\mathrm{\Delta },\mathrm{\Delta })$ is quadratically stable over $\mathrm{\Delta }:=Co(A_1,...,A_k)$ if and only if there exists a P > 0 such that

$\begin{array}{c}(A+A_i{)}^{T}P+P(A+A_i)<0foralli=1,....,k\end{array}$

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

$\begin{array}{c}(A+\mathrm{\Delta }{)}^{T}P+P(A+\mathrm{\Delta })<0forall\mathrm{\Delta }\in \mathrm{\Delta }\end{array}$

Conclusion:

Quadratic Stability Implies Stability of trajectories for any $\mathrm{\Delta }$ with $\mathrm{\Delta }\in \mathrm{\Delta }$ for all $t\ge 0$
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 $\mathrm{\Delta }$ with $\mathrm{\Delta }\in \mathrm{\Delta }$
• 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.