Stabilization by output of continuous and pulse-modulated uncertain systems

Abstract

The system ẋ_i = ϕ_i(⋅) + x_i+2, \(i \in \overline {1,n - 2} \), ẋ_n−1 = ϕ_n−1(⋅) + u₁, ẋ_n = ϕ_n(⋅) + u₂,where ϕ_i(·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u₁ and u₂ are the controls is considered. It is assumed that only the outputs x₁ and x₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.