Nonlinear Systems Stability Analysis

1	Basic Concepts

Introduction: In this chapter, the stability analysis of a system, the dynamics of which are represented in time domain by nonlinear time-invariant ordinary differential equations, is considered. This chapter consists of the following subsections:

1.1 Mathematical model for nonlinear systems.

1.2 Qualitative behavior of second-order linear time-invariant systems (LTI).

1.1 MATHEMATICAL MODEL FOR NONLINEAR SYSTEMS

A nonlinear system may mathematically be represented in the following form:

$\begin{array}{ll}{\dot{x}}_1\hfill & =f_1(x_1,x_2,\mathrm{\dots },x_n,u_1,u_2,\mathrm{\dots },u_m,t),\hfill \\ {\dot{x}}_2\hfill & =f_2(x_1,x_2,\mathrm{\dots },x_n,u_1,u_2,\mathrm{\dots },u_m,t),\hfill \\ ⋮\hfill & ⋮\hfill \\ {\dot{x}}_n\hfill & =f_n(x_1,x_2,\mathrm{\dots },x_n,u_1,u_2,\mathrm{\dots },u_m,t),\hfill \end{array}$

(1.1)

where ẋ_i, i = 1,2,…,n denotes the derivative of x_i (the ith state variable) with respect to the time variable t and u_j, j = 1, ...