Stability of Processes. Averaging Method

Abstract

In the preceding sections we studied the stability of the zero equilibrium x = 0 of the system whose CLP is given by (3.1). However, the technique developed there enables us to investigate the stability of an arbitrary solution (a process) of the system

$$ \frac{{dx}}{{dt}} = Ax + bf + q\left( t \right),\sigma = {c^*}x + \psi \left( t \right),f = M\sigma $$

(4.1)

with a given function ψ(t) and a given vector function q(t) (these are external disturbances), and the other notation is the same as in (3.1) where operator M describes a modulation law. Indeed, suppose that x_i (t) and x₂ (t) are solutions of (4.1) with the initial values x₁(0) and x₂(0), respectively, ${\sigma _i}\left( t \right) + \psi \left( t \right)$ and ${f_i}\left( t \right) = M{\sigma _i}\left( t \right)$ ( x = 1,2)Then the deviations ${x_d} = {x_1} - {x_2},{\sigma _d} = {\sigma _1} - {\sigma _2},$ and ${f_d} = {f_1} - {f_2}$ satisfy the equations

$$ \frac{{d{x_d}}}{{dt}} = A{x_d} + b{f_d},\sigma d = {c^*}xd,{f_d} = M{\sigma _1} - M{\sigma _2}. $$

(4.2)