On the L2-stability of a class of nonlinear systems

Sufficient conditions are given for the L₂-stability of a class of feedback systems consisting of a linear operator G and a nonlinear gain function, either odd monotone or restricted by a power-law, in cascade, in a negative feedback loop. The criterion takes the form of a frequency-domain inequality, Re1 + Z(jω)] G(jω) ? δ > 0 ?ω? (?∞, +∞), where Z(jω) is given by, Z(jω) = βY₁(jω) + Y₂(jω)] + (1 ? β)Y₃(jω) ? Y₃(?jω)], with 0 ? β ? 1 and the functions y₁(·), y₂(·) and y₃(·) satisfying the time-domain inequalities, $\text{∝}{}_{\mathrm{?\infty }}{}^{\mathrm{+\infty }}\text{¦y}{}_1\text{(t) + y}{}_2\text{(t)¦ dt ? 1 ? ?, y}{}_1\text{(·) = 0, t < 0}$, y₂(·) = 0, t > 0 and ? > 0, and $\text{∝}{}_0{}^{\mathrm{\infty }}\text{¦y}{}_3\text{(t)¦ dt <}\text{1}\text{2c}{}_2$, c₂ being a constant depending on the order of the power-law restricting the nonlinear function. The criterion is derived using Zames' passive operator theory and is shown to be more general than the existing criteria.