LMIs in Control/Matrix and LMI Properties and Tools/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 $$p(x) = R^2 - x^Tx $$. [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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