Synthesis of Stabilization Control on Outputs for a Class of Continuous and Pulse-Modulated Undefined Systems

Abstract

Consider system

\(\left\{ {\begin{array}{*{20}{c}}
{{{\dot x}_1} = {\varphi _1}(.) + {\rho _1}{x_{l + 1}},} \\
{{{\dot x}_m} = {\varphi _m}(.) + {\rho _m}{x_n},} \\
{{{\dot x}_{m + 1}} = {\varphi _{m + 1}}(.) + {\mu _1},} \\
{{{\dot x}_n} = {\varphi _n}(.) + {\mu _1},}
\end{array}} \right.\)

where x₁, …, and x_n is the state of the system, u₁, …, and u_l are controls, n/l is not an integer, and l ≥ 2. It is supposed that only outputs x₁, …, and x_l are measurable, (l > n) and ϕ_i(·) are non-anticipating arbitrary functionals, and 0 < ρ–≤ ρi (t, x₁, …, and x_l) ≤ ρ+. Using the backstepping method, we construct the square Lyapunov function and stabilize the control for the global exponential stability of the closed loop system. The stabilization by means of synchronous modulators with a sufficiently high impulsion frequency is considered as well.