Nonlinear Systems Stability Analysis

Appendix A3: Stability Analysis of Nonlinear Systems Via Δ-Homogeneous Approximation

It is well known that any n-dimensional function f(x) can be expanded by Δ-homogeneous functions with different order with respect to an arbitrarily dilation ${\text{α}}_r^{}$ as follows:

$\begin{array}{ccc}f\left(x\right)={\displaystyle \sum _{k=-r_n}^{+\infty }{f}^{\left(k\right)}\left(x\right)},& f^{\left(k\right)}\in n_k\text{,}& \forall k\ge -r_n.\end{array}$

(A3.1)

A Δ-homogeneous approximation^* of a nonlinear system is obtained using the first nonzero term of the expansion (A3.1). It is obvious that, if the zero equilibrium state of a Δ-homogeneous approximation of a nonlinear system is globally asymptotically stable, then the equilibrium state of the original nonlinear system is locally asymptotically stable using the same Δ-homogeneous Lyapunov function.

The importance of this method is appreciated ...