LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Lyapunov Stability of a System With Polynomial Dynamics

[UNDER CONSTRUCTION] - CME

Find the local Lyapunov function for a system with known polynomial dynamics.

Required Data

The system dynamics to be tested in the form ${\dot {x}}=f(x)$ where $f(x)$ is a polynomial function of the states $x$.

A function or functions representing the local domain over which stability is to be determined, $X:=\{x:p_{i}(x)\geq 0,i=1,...,k\}$. Commonly this is a ball of specified radius such that $p(x)=R^{2}-x^{T}x$. [Is this assuming P-Compactness of the set?]

Find the Lyapunov Function

Suppose there exists Lyapunov function $V(x)$, a scalar $\epsilon >0$, and $s_{0},s_{i},t_{0},t_{i}\in \Sigma _{s}$ (where $\Sigma _{s}$ is the set of Sum-of-Squares polynomials, all of which are positive) such that:

$V(x)=s_{0}(x)+\sum \limits _{i}s_{i}(x)p_{i}(x)+\epsilon x^{T}x$

$-{\dot {V}}(x)=-\nabla V(x)^{T}f(x)=t_{0}(x)+\sum \limits _{i}t_{i}(x)p_{i}(x)+\epsilon x^{T}x$

The Lyaponov function, if it exists, will be positive and always decreasing over the desired set $X$

Example

Code example in SOStools probably worthwhile!

Reference

SOS tools manual is a good one

Tie in the reference from SOS tools? The Khalil?

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